3.10.25 \(\int (d+e x)^m (f+g x)^2 (a+b x+c x^2)^2 \, dx\) [925]
Optimal. Leaf size=525 \[ \frac {\left (c d^2-b d e+a e^2\right )^2 (e f-d g)^2 (d+e x)^{1+m}}{e^7 (1+m)}-\frac {2 \left (c d^2-b d e+a e^2\right ) (e f-d g) (c d (2 e f-3 d g)-e (b e f-2 b d g+a e g)) (d+e x)^{2+m}}{e^7 (2+m)}+\frac {\left (c^2 d^2 \left (6 e^2 f^2-20 d e f g+15 d^2 g^2\right )+e^2 \left (a^2 e^2 g^2+2 a b e g (2 e f-3 d g)+b^2 \left (e^2 f^2-6 d e f g+6 d^2 g^2\right )\right )+2 c e \left (a e \left (e^2 f^2-6 d e f g+6 d^2 g^2\right )-b d \left (3 e^2 f^2-12 d e f g+10 d^2 g^2\right )\right )\right ) (d+e x)^{3+m}}{e^7 (3+m)}+\frac {2 \left (b e^2 g (b e f-2 b d g+a e g)-2 c^2 d \left (e^2 f^2-5 d e f g+5 d^2 g^2\right )+c e \left (2 a e g (e f-2 d g)+b \left (e^2 f^2-8 d e f g+10 d^2 g^2\right )\right )\right ) (d+e x)^{4+m}}{e^7 (4+m)}+\frac {\left (b^2 e^2 g^2+2 c e g (2 b e f-5 b d g+a e g)+c^2 \left (e^2 f^2-10 d e f g+15 d^2 g^2\right )\right ) (d+e x)^{5+m}}{e^7 (5+m)}+\frac {2 c g (c e f-3 c d g+b e g) (d+e x)^{6+m}}{e^7 (6+m)}+\frac {c^2 g^2 (d+e x)^{7+m}}{e^7 (7+m)} \]
[Out]
(a*e^2-b*d*e+c*d^2)^2*(-d*g+e*f)^2*(e*x+d)^(1+m)/e^7/(1+m)-2*(a*e^2-b*d*e+c*d^2)*(-d*g+e*f)*(c*d*(-3*d*g+2*e*f
)-e*(a*e*g-2*b*d*g+b*e*f))*(e*x+d)^(2+m)/e^7/(2+m)+(c^2*d^2*(15*d^2*g^2-20*d*e*f*g+6*e^2*f^2)+e^2*(a^2*e^2*g^2
+2*a*b*e*g*(-3*d*g+2*e*f)+b^2*(6*d^2*g^2-6*d*e*f*g+e^2*f^2))+2*c*e*(a*e*(6*d^2*g^2-6*d*e*f*g+e^2*f^2)-b*d*(10*
d^2*g^2-12*d*e*f*g+3*e^2*f^2)))*(e*x+d)^(3+m)/e^7/(3+m)+2*(b*e^2*g*(a*e*g-2*b*d*g+b*e*f)-2*c^2*d*(5*d^2*g^2-5*
d*e*f*g+e^2*f^2)+c*e*(2*a*e*g*(-2*d*g+e*f)+b*(10*d^2*g^2-8*d*e*f*g+e^2*f^2)))*(e*x+d)^(4+m)/e^7/(4+m)+(b^2*e^2
*g^2+2*c*e*g*(a*e*g-5*b*d*g+2*b*e*f)+c^2*(15*d^2*g^2-10*d*e*f*g+e^2*f^2))*(e*x+d)^(5+m)/e^7/(5+m)+2*c*g*(b*e*g
-3*c*d*g+c*e*f)*(e*x+d)^(6+m)/e^7/(6+m)+c^2*g^2*(e*x+d)^(7+m)/e^7/(7+m)
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Rubi [A]
time = 0.38, antiderivative size = 525, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.037, Rules used = {961}
\begin {gather*} \frac {(d+e x)^{m+3} \left (e^2 \left (a^2 e^2 g^2+2 a b e g (2 e f-3 d g)+b^2 \left (6 d^2 g^2-6 d e f g+e^2 f^2\right )\right )+2 c e \left (a e \left (6 d^2 g^2-6 d e f g+e^2 f^2\right )-b d \left (10 d^2 g^2-12 d e f g+3 e^2 f^2\right )\right )+c^2 d^2 \left (15 d^2 g^2-20 d e f g+6 e^2 f^2\right )\right )}{e^7 (m+3)}+\frac {(d+e x)^{m+5} \left (2 c e g (a e g-5 b d g+2 b e f)+b^2 e^2 g^2+c^2 \left (15 d^2 g^2-10 d e f g+e^2 f^2\right )\right )}{e^7 (m+5)}+\frac {2 (d+e x)^{m+4} \left (c e \left (2 a e g (e f-2 d g)+b \left (10 d^2 g^2-8 d e f g+e^2 f^2\right )\right )+b e^2 g (a e g-2 b d g+b e f)-2 c^2 d \left (5 d^2 g^2-5 d e f g+e^2 f^2\right )\right )}{e^7 (m+4)}+\frac {(e f-d g)^2 (d+e x)^{m+1} \left (a e^2-b d e+c d^2\right )^2}{e^7 (m+1)}-\frac {2 (e f-d g) (d+e x)^{m+2} \left (a e^2-b d e+c d^2\right ) (c d (2 e f-3 d g)-e (a e g-2 b d g+b e f))}{e^7 (m+2)}+\frac {2 c g (d+e x)^{m+6} (b e g-3 c d g+c e f)}{e^7 (m+6)}+\frac {c^2 g^2 (d+e x)^{m+7}}{e^7 (m+7)} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[(d + e*x)^m*(f + g*x)^2*(a + b*x + c*x^2)^2,x]
[Out]
((c*d^2 - b*d*e + a*e^2)^2*(e*f - d*g)^2*(d + e*x)^(1 + m))/(e^7*(1 + m)) - (2*(c*d^2 - b*d*e + a*e^2)*(e*f -
d*g)*(c*d*(2*e*f - 3*d*g) - e*(b*e*f - 2*b*d*g + a*e*g))*(d + e*x)^(2 + m))/(e^7*(2 + m)) + ((c^2*d^2*(6*e^2*f
^2 - 20*d*e*f*g + 15*d^2*g^2) + e^2*(a^2*e^2*g^2 + 2*a*b*e*g*(2*e*f - 3*d*g) + b^2*(e^2*f^2 - 6*d*e*f*g + 6*d^
2*g^2)) + 2*c*e*(a*e*(e^2*f^2 - 6*d*e*f*g + 6*d^2*g^2) - b*d*(3*e^2*f^2 - 12*d*e*f*g + 10*d^2*g^2)))*(d + e*x)
^(3 + m))/(e^7*(3 + m)) + (2*(b*e^2*g*(b*e*f - 2*b*d*g + a*e*g) - 2*c^2*d*(e^2*f^2 - 5*d*e*f*g + 5*d^2*g^2) +
c*e*(2*a*e*g*(e*f - 2*d*g) + b*(e^2*f^2 - 8*d*e*f*g + 10*d^2*g^2)))*(d + e*x)^(4 + m))/(e^7*(4 + m)) + ((b^2*e
^2*g^2 + 2*c*e*g*(2*b*e*f - 5*b*d*g + a*e*g) + c^2*(e^2*f^2 - 10*d*e*f*g + 15*d^2*g^2))*(d + e*x)^(5 + m))/(e^
7*(5 + m)) + (2*c*g*(c*e*f - 3*c*d*g + b*e*g)*(d + e*x)^(6 + m))/(e^7*(6 + m)) + (c^2*g^2*(d + e*x)^(7 + m))/(
e^7*(7 + m))
Rule 961
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :
> Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] &
& NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IGtQ[p, 0] && (IGtQ[m, 0] || (E
qQ[m, -2] && EqQ[p, 1] && EqQ[2*c*d - b*e, 0]))
Rubi steps
\begin {align*} \int (d+e x)^m (f+g x)^2 \left (a+b x+c x^2\right )^2 \, dx &=\int \left (\frac {\left (c d^2-b d e+a e^2\right )^2 (e f-d g)^2 (d+e x)^m}{e^6}+\frac {2 \left (c d^2-b d e+a e^2\right ) (e f-d g) (-c d (2 e f-3 d g)+e (b e f-2 b d g+a e g)) (d+e x)^{1+m}}{e^6}+\frac {\left (c^2 d^2 \left (6 e^2 f^2-20 d e f g+15 d^2 g^2\right )+e^2 \left (a^2 e^2 g^2+2 a b e g (2 e f-3 d g)+b^2 \left (e^2 f^2-6 d e f g+6 d^2 g^2\right )\right )+2 c e \left (a e \left (e^2 f^2-6 d e f g+6 d^2 g^2\right )-b d \left (3 e^2 f^2-12 d e f g+10 d^2 g^2\right )\right )\right ) (d+e x)^{2+m}}{e^6}+\frac {2 \left (b e^2 g (b e f-2 b d g+a e g)-2 c^2 d \left (e^2 f^2-5 d e f g+5 d^2 g^2\right )+c e \left (2 a e g (e f-2 d g)+b \left (e^2 f^2-8 d e f g+10 d^2 g^2\right )\right )\right ) (d+e x)^{3+m}}{e^6}+\frac {\left (b^2 e^2 g^2+2 c e g (2 b e f-5 b d g+a e g)+c^2 \left (e^2 f^2-10 d e f g+15 d^2 g^2\right )\right ) (d+e x)^{4+m}}{e^6}+\frac {2 c g (c e f-3 c d g+b e g) (d+e x)^{5+m}}{e^6}+\frac {c^2 g^2 (d+e x)^{6+m}}{e^6}\right ) \, dx\\ &=\frac {\left (c d^2-b d e+a e^2\right )^2 (e f-d g)^2 (d+e x)^{1+m}}{e^7 (1+m)}-\frac {2 \left (c d^2-b d e+a e^2\right ) (e f-d g) (c d (2 e f-3 d g)-e (b e f-2 b d g+a e g)) (d+e x)^{2+m}}{e^7 (2+m)}+\frac {\left (c^2 d^2 \left (6 e^2 f^2-20 d e f g+15 d^2 g^2\right )+e^2 \left (a^2 e^2 g^2+2 a b e g (2 e f-3 d g)+b^2 \left (e^2 f^2-6 d e f g+6 d^2 g^2\right )\right )+2 c e \left (a e \left (e^2 f^2-6 d e f g+6 d^2 g^2\right )-b d \left (3 e^2 f^2-12 d e f g+10 d^2 g^2\right )\right )\right ) (d+e x)^{3+m}}{e^7 (3+m)}+\frac {2 \left (b e^2 g (b e f-2 b d g+a e g)-2 c^2 d \left (e^2 f^2-5 d e f g+5 d^2 g^2\right )+c e \left (2 a e g (e f-2 d g)+b \left (e^2 f^2-8 d e f g+10 d^2 g^2\right )\right )\right ) (d+e x)^{4+m}}{e^7 (4+m)}+\frac {\left (b^2 e^2 g^2+2 c e g (2 b e f-5 b d g+a e g)+c^2 \left (e^2 f^2-10 d e f g+15 d^2 g^2\right )\right ) (d+e x)^{5+m}}{e^7 (5+m)}+\frac {2 c g (c e f-3 c d g+b e g) (d+e x)^{6+m}}{e^7 (6+m)}+\frac {c^2 g^2 (d+e x)^{7+m}}{e^7 (7+m)}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(1263\) vs. \(2(525)=1050\).
time = 1.86, size = 1263, normalized size = 2.41 \begin {gather*} \frac {(d+e x)^{1+m} \left (c^2 \left (720 d^6 g^2-240 d^5 e g (f (7+m)+3 g (1+m) x)+24 d^4 e^2 \left (f^2 \left (42+13 m+m^2\right )+10 f g \left (7+8 m+m^2\right ) x+15 g^2 \left (2+3 m+m^2\right ) x^2\right )-24 d^3 e^3 (1+m) x \left (f^2 \left (42+13 m+m^2\right )+5 f g \left (14+9 m+m^2\right ) x+5 g^2 \left (6+5 m+m^2\right ) x^2\right )+2 d^2 e^4 \left (2+3 m+m^2\right ) x^2 \left (6 f^2 \left (42+13 m+m^2\right )+20 f g \left (21+10 m+m^2\right ) x+15 g^2 \left (12+7 m+m^2\right ) x^2\right )-2 d e^5 \left (6+11 m+6 m^2+m^3\right ) x^3 \left (2 f^2 \left (42+13 m+m^2\right )+5 f g \left (28+11 m+m^2\right ) x+3 g^2 \left (20+9 m+m^2\right ) x^2\right )+e^6 \left (24+50 m+35 m^2+10 m^3+m^4\right ) x^4 \left (f^2 \left (42+13 m+m^2\right )+2 f g \left (35+12 m+m^2\right ) x+g^2 \left (30+11 m+m^2\right ) x^2\right )\right )+e^2 \left (42+13 m+m^2\right ) \left (a^2 e^2 \left (20+9 m+m^2\right ) \left (2 d^2 g^2-2 d e g (f (3+m)+g (1+m) x)+e^2 \left (f^2 \left (6+5 m+m^2\right )+2 f g \left (3+4 m+m^2\right ) x+g^2 \left (2+3 m+m^2\right ) x^2\right )\right )+2 a b e (5+m) \left (-6 d^3 g^2+2 d^2 e g (2 f (4+m)+3 g (1+m) x)-d e^2 \left (f^2 \left (12+7 m+m^2\right )+4 f g \left (4+5 m+m^2\right ) x+3 g^2 \left (2+3 m+m^2\right ) x^2\right )+e^3 (1+m) x \left (f^2 \left (12+7 m+m^2\right )+2 f g \left (8+6 m+m^2\right ) x+g^2 \left (6+5 m+m^2\right ) x^2\right )\right )+b^2 \left (24 d^4 g^2-12 d^3 e g (f (5+m)+2 g (1+m) x)+2 d^2 e^2 \left (f^2 \left (20+9 m+m^2\right )+6 f g \left (5+6 m+m^2\right ) x+6 g^2 \left (2+3 m+m^2\right ) x^2\right )-2 d e^3 (1+m) x \left (f^2 \left (20+9 m+m^2\right )+3 f g \left (10+7 m+m^2\right ) x+2 g^2 \left (6+5 m+m^2\right ) x^2\right )+e^4 \left (2+3 m+m^2\right ) x^2 \left (f^2 \left (20+9 m+m^2\right )+2 f g \left (15+8 m+m^2\right ) x+g^2 \left (12+7 m+m^2\right ) x^2\right )\right )\right )+2 c e (7+m) \left (a e (6+m) \left (24 d^4 g^2-12 d^3 e g (f (5+m)+2 g (1+m) x)+2 d^2 e^2 \left (f^2 \left (20+9 m+m^2\right )+6 f g \left (5+6 m+m^2\right ) x+6 g^2 \left (2+3 m+m^2\right ) x^2\right )-2 d e^3 (1+m) x \left (f^2 \left (20+9 m+m^2\right )+3 f g \left (10+7 m+m^2\right ) x+2 g^2 \left (6+5 m+m^2\right ) x^2\right )+e^4 \left (2+3 m+m^2\right ) x^2 \left (f^2 \left (20+9 m+m^2\right )+2 f g \left (15+8 m+m^2\right ) x+g^2 \left (12+7 m+m^2\right ) x^2\right )\right )+b \left (-120 d^5 g^2+24 d^4 e g (2 f (6+m)+5 g (1+m) x)-6 d^3 e^2 \left (f^2 \left (30+11 m+m^2\right )+8 f g \left (6+7 m+m^2\right ) x+10 g^2 \left (2+3 m+m^2\right ) x^2\right )+2 d^2 e^3 (1+m) x \left (3 f^2 \left (30+11 m+m^2\right )+12 f g \left (12+8 m+m^2\right ) x+10 g^2 \left (6+5 m+m^2\right ) x^2\right )-d e^4 \left (2+3 m+m^2\right ) x^2 \left (3 f^2 \left (30+11 m+m^2\right )+8 f g \left (18+9 m+m^2\right ) x+5 g^2 \left (12+7 m+m^2\right ) x^2\right )+e^5 \left (6+11 m+6 m^2+m^3\right ) x^3 \left (f^2 \left (30+11 m+m^2\right )+2 f g \left (24+10 m+m^2\right ) x+g^2 \left (20+9 m+m^2\right ) x^2\right )\right )\right )\right )}{e^7 (1+m) (2+m) (3+m) (4+m) (5+m) (6+m) (7+m)} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(d + e*x)^m*(f + g*x)^2*(a + b*x + c*x^2)^2,x]
[Out]
((d + e*x)^(1 + m)*(c^2*(720*d^6*g^2 - 240*d^5*e*g*(f*(7 + m) + 3*g*(1 + m)*x) + 24*d^4*e^2*(f^2*(42 + 13*m +
m^2) + 10*f*g*(7 + 8*m + m^2)*x + 15*g^2*(2 + 3*m + m^2)*x^2) - 24*d^3*e^3*(1 + m)*x*(f^2*(42 + 13*m + m^2) +
5*f*g*(14 + 9*m + m^2)*x + 5*g^2*(6 + 5*m + m^2)*x^2) + 2*d^2*e^4*(2 + 3*m + m^2)*x^2*(6*f^2*(42 + 13*m + m^2)
+ 20*f*g*(21 + 10*m + m^2)*x + 15*g^2*(12 + 7*m + m^2)*x^2) - 2*d*e^5*(6 + 11*m + 6*m^2 + m^3)*x^3*(2*f^2*(42
+ 13*m + m^2) + 5*f*g*(28 + 11*m + m^2)*x + 3*g^2*(20 + 9*m + m^2)*x^2) + e^6*(24 + 50*m + 35*m^2 + 10*m^3 +
m^4)*x^4*(f^2*(42 + 13*m + m^2) + 2*f*g*(35 + 12*m + m^2)*x + g^2*(30 + 11*m + m^2)*x^2)) + e^2*(42 + 13*m + m
^2)*(a^2*e^2*(20 + 9*m + m^2)*(2*d^2*g^2 - 2*d*e*g*(f*(3 + m) + g*(1 + m)*x) + e^2*(f^2*(6 + 5*m + m^2) + 2*f*
g*(3 + 4*m + m^2)*x + g^2*(2 + 3*m + m^2)*x^2)) + 2*a*b*e*(5 + m)*(-6*d^3*g^2 + 2*d^2*e*g*(2*f*(4 + m) + 3*g*(
1 + m)*x) - d*e^2*(f^2*(12 + 7*m + m^2) + 4*f*g*(4 + 5*m + m^2)*x + 3*g^2*(2 + 3*m + m^2)*x^2) + e^3*(1 + m)*x
*(f^2*(12 + 7*m + m^2) + 2*f*g*(8 + 6*m + m^2)*x + g^2*(6 + 5*m + m^2)*x^2)) + b^2*(24*d^4*g^2 - 12*d^3*e*g*(f
*(5 + m) + 2*g*(1 + m)*x) + 2*d^2*e^2*(f^2*(20 + 9*m + m^2) + 6*f*g*(5 + 6*m + m^2)*x + 6*g^2*(2 + 3*m + m^2)*
x^2) - 2*d*e^3*(1 + m)*x*(f^2*(20 + 9*m + m^2) + 3*f*g*(10 + 7*m + m^2)*x + 2*g^2*(6 + 5*m + m^2)*x^2) + e^4*(
2 + 3*m + m^2)*x^2*(f^2*(20 + 9*m + m^2) + 2*f*g*(15 + 8*m + m^2)*x + g^2*(12 + 7*m + m^2)*x^2))) + 2*c*e*(7 +
m)*(a*e*(6 + m)*(24*d^4*g^2 - 12*d^3*e*g*(f*(5 + m) + 2*g*(1 + m)*x) + 2*d^2*e^2*(f^2*(20 + 9*m + m^2) + 6*f*
g*(5 + 6*m + m^2)*x + 6*g^2*(2 + 3*m + m^2)*x^2) - 2*d*e^3*(1 + m)*x*(f^2*(20 + 9*m + m^2) + 3*f*g*(10 + 7*m +
m^2)*x + 2*g^2*(6 + 5*m + m^2)*x^2) + e^4*(2 + 3*m + m^2)*x^2*(f^2*(20 + 9*m + m^2) + 2*f*g*(15 + 8*m + m^2)*
x + g^2*(12 + 7*m + m^2)*x^2)) + b*(-120*d^5*g^2 + 24*d^4*e*g*(2*f*(6 + m) + 5*g*(1 + m)*x) - 6*d^3*e^2*(f^2*(
30 + 11*m + m^2) + 8*f*g*(6 + 7*m + m^2)*x + 10*g^2*(2 + 3*m + m^2)*x^2) + 2*d^2*e^3*(1 + m)*x*(3*f^2*(30 + 11
*m + m^2) + 12*f*g*(12 + 8*m + m^2)*x + 10*g^2*(6 + 5*m + m^2)*x^2) - d*e^4*(2 + 3*m + m^2)*x^2*(3*f^2*(30 + 1
1*m + m^2) + 8*f*g*(18 + 9*m + m^2)*x + 5*g^2*(12 + 7*m + m^2)*x^2) + e^5*(6 + 11*m + 6*m^2 + m^3)*x^3*(f^2*(3
0 + 11*m + m^2) + 2*f*g*(24 + 10*m + m^2)*x + g^2*(20 + 9*m + m^2)*x^2)))))/(e^7*(1 + m)*(2 + m)*(3 + m)*(4 +
m)*(5 + m)*(6 + m)*(7 + m))
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(4652\) vs.
\(2(525)=1050\).
time = 0.16, size = 4653, normalized size = 8.86
| | |
| method |
result |
size |
| | |
| norman |
\(\text {Expression too large to display}\) |
\(4653\) |
| gosper |
\(\text {Expression too large to display}\) |
\(5890\) |
| risch |
\(\text {Expression too large to display}\) |
\(7342\) |
| | |
|
|
|
|
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((e*x+d)^m*(g*x+f)^2*(c*x^2+b*x+a)^2,x,method=_RETURNVERBOSE)
[Out]
d*(a^2*e^6*f^2*m^6-2*a^2*d*e^5*f*g*m^5+27*a^2*e^6*f^2*m^5-2*a*b*d*e^5*f^2*m^5+2*a^2*d^2*e^4*g^2*m^4-50*a^2*d*e
^5*f*g*m^4+295*a^2*e^6*f^2*m^4+8*a*b*d^2*e^4*f*g*m^4-50*a*b*d*e^5*f^2*m^4+4*a*c*d^2*e^4*f^2*m^4+2*b^2*d^2*e^4*
f^2*m^4+44*a^2*d^2*e^4*g^2*m^3-490*a^2*d*e^5*f*g*m^3+1665*a^2*e^6*f^2*m^3-12*a*b*d^3*e^3*g^2*m^3+176*a*b*d^2*e
^4*f*g*m^3-490*a*b*d*e^5*f^2*m^3-24*a*c*d^3*e^3*f*g*m^3+88*a*c*d^2*e^4*f^2*m^3-12*b^2*d^3*e^3*f*g*m^3+44*b^2*d
^2*e^4*f^2*m^3-12*b*c*d^3*e^3*f^2*m^3+358*a^2*d^2*e^4*g^2*m^2-2350*a^2*d*e^5*f*g*m^2+5104*a^2*e^6*f^2*m^2-216*
a*b*d^3*e^3*g^2*m^2+1432*a*b*d^2*e^4*f*g*m^2-2350*a*b*d*e^5*f^2*m^2+48*a*c*d^4*e^2*g^2*m^2-432*a*c*d^3*e^3*f*g
*m^2+716*a*c*d^2*e^4*f^2*m^2+24*b^2*d^4*e^2*g^2*m^2-216*b^2*d^3*e^3*f*g*m^2+358*b^2*d^2*e^4*f^2*m^2+96*b*c*d^4
*e^2*f*g*m^2-216*b*c*d^3*e^3*f^2*m^2+24*c^2*d^4*e^2*f^2*m^2+1276*a^2*d^2*e^4*g^2*m-5508*a^2*d*e^5*f*g*m+8028*a
^2*e^6*f^2*m-1284*a*b*d^3*e^3*g^2*m+5104*a*b*d^2*e^4*f*g*m-5508*a*b*d*e^5*f^2*m+624*a*c*d^4*e^2*g^2*m-2568*a*c
*d^3*e^3*f*g*m+2552*a*c*d^2*e^4*f^2*m+312*b^2*d^4*e^2*g^2*m-1284*b^2*d^3*e^3*f*g*m+1276*b^2*d^2*e^4*f^2*m-240*
b*c*d^5*e*g^2*m+1248*b*c*d^4*e^2*f*g*m-1284*b*c*d^3*e^3*f^2*m-240*c^2*d^5*e*f*g*m+312*c^2*d^4*e^2*f^2*m+1680*a
^2*d^2*e^4*g^2-5040*a^2*d*e^5*f*g+5040*a^2*e^6*f^2-2520*a*b*d^3*e^3*g^2+6720*a*b*d^2*e^4*f*g-5040*a*b*d*e^5*f^
2+2016*a*c*d^4*e^2*g^2-5040*a*c*d^3*e^3*f*g+3360*a*c*d^2*e^4*f^2+1008*b^2*d^4*e^2*g^2-2520*b^2*d^3*e^3*f*g+168
0*b^2*d^2*e^4*f^2-1680*b*c*d^5*e*g^2+4032*b*c*d^4*e^2*f*g-2520*b*c*d^3*e^3*f^2+720*c^2*d^6*g^2-1680*c^2*d^5*e*
f*g+1008*c^2*d^4*e^2*f^2)/e^7/(m^7+28*m^6+322*m^5+1960*m^4+6769*m^3+13132*m^2+13068*m+5040)*exp(m*ln(e*x+d))+g
^2*c^2/(7+m)*x^7*exp(m*ln(e*x+d))+(2*a*c*e^2*g^2*m^2+b^2*e^2*g^2*m^2+2*b*c*d*e*g^2*m^2+4*b*c*e^2*f*g*m^2+2*c^2
*d*e*f*g*m^2+c^2*e^2*f^2*m^2+26*a*c*e^2*g^2*m+13*b^2*e^2*g^2*m+14*b*c*d*e*g^2*m+52*b*c*e^2*f*g*m-6*c^2*d^2*g^2
*m+14*c^2*d*e*f*g*m+13*c^2*e^2*f^2*m+84*a*c*e^2*g^2+42*b^2*e^2*g^2+168*b*c*e^2*f*g+42*c^2*e^2*f^2)/e^2/(m^3+18
*m^2+107*m+210)*x^5*exp(m*ln(e*x+d))+(2*a*b*e^3*g^2*m^3+2*a*c*d*e^2*g^2*m^3+4*a*c*e^3*f*g*m^3+b^2*d*e^2*g^2*m^
3+2*b^2*e^3*f*g*m^3+4*b*c*d*e^2*f*g*m^3+2*b*c*e^3*f^2*m^3+c^2*d*e^2*f^2*m^3+36*a*b*e^3*g^2*m^2+26*a*c*d*e^2*g^
2*m^2+72*a*c*e^3*f*g*m^2+13*b^2*d*e^2*g^2*m^2+36*b^2*e^3*f*g*m^2-10*b*c*d^2*e*g^2*m^2+52*b*c*d*e^2*f*g*m^2+36*
b*c*e^3*f^2*m^2-10*c^2*d^2*e*f*g*m^2+13*c^2*d*e^2*f^2*m^2+214*a*b*e^3*g^2*m+84*a*c*d*e^2*g^2*m+428*a*c*e^3*f*g
*m+42*b^2*d*e^2*g^2*m+214*b^2*e^3*f*g*m-70*b*c*d^2*e*g^2*m+168*b*c*d*e^2*f*g*m+214*b*c*e^3*f^2*m+30*c^2*d^3*g^
2*m-70*c^2*d^2*e*f*g*m+42*c^2*d*e^2*f^2*m+420*a*b*e^3*g^2+840*a*c*e^3*f*g+420*b^2*e^3*f*g+420*b*c*e^3*f^2)/e^3
/(m^4+22*m^3+179*m^2+638*m+840)*x^4*exp(m*ln(e*x+d))+(a^2*e^4*g^2*m^4+2*a*b*d*e^3*g^2*m^4+4*a*b*e^4*f*g*m^4+4*
a*c*d*e^3*f*g*m^4+2*a*c*e^4*f^2*m^4+2*b^2*d*e^3*f*g*m^4+b^2*e^4*f^2*m^4+2*b*c*d*e^3*f^2*m^4+22*a^2*e^4*g^2*m^3
+36*a*b*d*e^3*g^2*m^3+88*a*b*e^4*f*g*m^3-8*a*c*d^2*e^2*g^2*m^3+72*a*c*d*e^3*f*g*m^3+44*a*c*e^4*f^2*m^3-4*b^2*d
^2*e^2*g^2*m^3+36*b^2*d*e^3*f*g*m^3+22*b^2*e^4*f^2*m^3-16*b*c*d^2*e^2*f*g*m^3+36*b*c*d*e^3*f^2*m^3-4*c^2*d^2*e
^2*f^2*m^3+179*a^2*e^4*g^2*m^2+214*a*b*d*e^3*g^2*m^2+716*a*b*e^4*f*g*m^2-104*a*c*d^2*e^2*g^2*m^2+428*a*c*d*e^3
*f*g*m^2+358*a*c*e^4*f^2*m^2-52*b^2*d^2*e^2*g^2*m^2+214*b^2*d*e^3*f*g*m^2+179*b^2*e^4*f^2*m^2+40*b*c*d^3*e*g^2
*m^2-208*b*c*d^2*e^2*f*g*m^2+214*b*c*d*e^3*f^2*m^2+40*c^2*d^3*e*f*g*m^2-52*c^2*d^2*e^2*f^2*m^2+638*a^2*e^4*g^2
*m+420*a*b*d*e^3*g^2*m+2552*a*b*e^4*f*g*m-336*a*c*d^2*e^2*g^2*m+840*a*c*d*e^3*f*g*m+1276*a*c*e^4*f^2*m-168*b^2
*d^2*e^2*g^2*m+420*b^2*d*e^3*f*g*m+638*b^2*e^4*f^2*m+280*b*c*d^3*e*g^2*m-672*b*c*d^2*e^2*f*g*m+420*b*c*d*e^3*f
^2*m-120*c^2*d^4*g^2*m+280*c^2*d^3*e*f*g*m-168*c^2*d^2*e^2*f^2*m+840*a^2*e^4*g^2+3360*a*b*e^4*f*g+1680*a*c*e^4
*f^2+840*b^2*e^4*f^2)/e^4/(m^5+25*m^4+245*m^3+1175*m^2+2754*m+2520)*x^3*exp(m*ln(e*x+d))+(a^2*d*e^4*g^2*m^5+2*
a^2*e^5*f*g*m^5+4*a*b*d*e^4*f*g*m^5+2*a*b*e^5*f^2*m^5+2*a*c*d*e^4*f^2*m^5+b^2*d*e^4*f^2*m^5+22*a^2*d*e^4*g^2*m
^4+50*a^2*e^5*f*g*m^4-6*a*b*d^2*e^3*g^2*m^4+88*a*b*d*e^4*f*g*m^4+50*a*b*e^5*f^2*m^4-12*a*c*d^2*e^3*f*g*m^4+44*
a*c*d*e^4*f^2*m^4-6*b^2*d^2*e^3*f*g*m^4+22*b^2*d*e^4*f^2*m^4-6*b*c*d^2*e^3*f^2*m^4+179*a^2*d*e^4*g^2*m^3+490*a
^2*e^5*f*g*m^3-108*a*b*d^2*e^3*g^2*m^3+716*a*b*d*e^4*f*g*m^3+490*a*b*e^5*f^2*m^3+24*a*c*d^3*e^2*g^2*m^3-216*a*
c*d^2*e^3*f*g*m^3+358*a*c*d*e^4*f^2*m^3+12*b^2*d^3*e^2*g^2*m^3-108*b^2*d^2*e^3*f*g*m^3+179*b^2*d*e^4*f^2*m^3+4
8*b*c*d^3*e^2*f*g*m^3-108*b*c*d^2*e^3*f^2*m^3+12*c^2*d^3*e^2*f^2*m^3+638*a^2*d*e^4*g^2*m^2+2350*a^2*e^5*f*g*m^
2-642*a*b*d^2*e^3*g^2*m^2+2552*a*b*d*e^4*f*g*m^2+2350*a*b*e^5*f^2*m^2+312*a*c*d^3*e^2*g^2*m^2-1284*a*c*d^2*e^3
*f*g*m^2+1276*a*c*d*e^4*f^2*m^2+156*b^2*d^3*e^2*g^2*m^2-642*b^2*d^2*e^3*f*g*m^2+638*b^2*d*e^4*f^2*m^2-120*b*c*
d^4*e*g^2*m^2+624*b*c*d^3*e^2*f*g*m^2-642*b*c*d^2*e^3*f^2*m^2-120*c^2*d^4*e*f*g*m^2+156*c^2*d^3*e^2*f^2*m^2+84
0*a^2*d*e^4*g^2*m+5508*a^2*e^5*f*g*m-1260*a*b*d^2*e^3*g^2*m+3360*a*b*d*e^4*f*g*m+5508*a*b*e^5*f^2*m+1008*a*c*d
^3*e^2*g^2*m-2520*a*c*d^2*e^3*f*g*m+1680*a*c*d*...
________________________________________________________________________________________
Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 2031 vs.
\(2 (550) = 1100\).
time = 0.34, size = 2031, normalized size = 3.87 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x+d)^m*(g*x+f)^2*(c*x^2+b*x+a)^2,x, algorithm="maxima")
[Out]
(x*e + d)^(m + 1)*a^2*f^2*e^(-1)/(m + 1) + 2*((m + 1)*x^2*e^2 + d*m*x*e - d^2)*a*b*f^2*e^(m*log(x*e + d) - 2)/
(m^2 + 3*m + 2) + 2*((m + 1)*x^2*e^2 + d*m*x*e - d^2)*a^2*f*g*e^(m*log(x*e + d) - 2)/(m^2 + 3*m + 2) + ((m^2 +
3*m + 2)*x^3*e^3 + (m^2 + m)*d*x^2*e^2 - 2*d^2*m*x*e + 2*d^3)*b^2*f^2*e^(m*log(x*e + d) - 3)/(m^3 + 6*m^2 + 1
1*m + 6) + 2*((m^2 + 3*m + 2)*x^3*e^3 + (m^2 + m)*d*x^2*e^2 - 2*d^2*m*x*e + 2*d^3)*a*c*f^2*e^(m*log(x*e + d) -
3)/(m^3 + 6*m^2 + 11*m + 6) + 4*((m^2 + 3*m + 2)*x^3*e^3 + (m^2 + m)*d*x^2*e^2 - 2*d^2*m*x*e + 2*d^3)*a*b*f*g
*e^(m*log(x*e + d) - 3)/(m^3 + 6*m^2 + 11*m + 6) + ((m^2 + 3*m + 2)*x^3*e^3 + (m^2 + m)*d*x^2*e^2 - 2*d^2*m*x*
e + 2*d^3)*a^2*g^2*e^(m*log(x*e + d) - 3)/(m^3 + 6*m^2 + 11*m + 6) + 2*((m^3 + 6*m^2 + 11*m + 6)*x^4*e^4 + (m^
3 + 3*m^2 + 2*m)*d*x^3*e^3 - 3*(m^2 + m)*d^2*x^2*e^2 + 6*d^3*m*x*e - 6*d^4)*b*c*f^2*e^(m*log(x*e + d) - 4)/(m^
4 + 10*m^3 + 35*m^2 + 50*m + 24) + 2*((m^3 + 6*m^2 + 11*m + 6)*x^4*e^4 + (m^3 + 3*m^2 + 2*m)*d*x^3*e^3 - 3*(m^
2 + m)*d^2*x^2*e^2 + 6*d^3*m*x*e - 6*d^4)*b^2*f*g*e^(m*log(x*e + d) - 4)/(m^4 + 10*m^3 + 35*m^2 + 50*m + 24) +
4*((m^3 + 6*m^2 + 11*m + 6)*x^4*e^4 + (m^3 + 3*m^2 + 2*m)*d*x^3*e^3 - 3*(m^2 + m)*d^2*x^2*e^2 + 6*d^3*m*x*e -
6*d^4)*a*c*f*g*e^(m*log(x*e + d) - 4)/(m^4 + 10*m^3 + 35*m^2 + 50*m + 24) + 2*((m^3 + 6*m^2 + 11*m + 6)*x^4*e
^4 + (m^3 + 3*m^2 + 2*m)*d*x^3*e^3 - 3*(m^2 + m)*d^2*x^2*e^2 + 6*d^3*m*x*e - 6*d^4)*a*b*g^2*e^(m*log(x*e + d)
- 4)/(m^4 + 10*m^3 + 35*m^2 + 50*m + 24) + ((m^4 + 10*m^3 + 35*m^2 + 50*m + 24)*x^5*e^5 + (m^4 + 6*m^3 + 11*m^
2 + 6*m)*d*x^4*e^4 - 4*(m^3 + 3*m^2 + 2*m)*d^2*x^3*e^3 + 12*(m^2 + m)*d^3*x^2*e^2 - 24*d^4*m*x*e + 24*d^5)*c^2
*f^2*e^(m*log(x*e + d) - 5)/(m^5 + 15*m^4 + 85*m^3 + 225*m^2 + 274*m + 120) + 4*((m^4 + 10*m^3 + 35*m^2 + 50*m
+ 24)*x^5*e^5 + (m^4 + 6*m^3 + 11*m^2 + 6*m)*d*x^4*e^4 - 4*(m^3 + 3*m^2 + 2*m)*d^2*x^3*e^3 + 12*(m^2 + m)*d^3
*x^2*e^2 - 24*d^4*m*x*e + 24*d^5)*b*c*f*g*e^(m*log(x*e + d) - 5)/(m^5 + 15*m^4 + 85*m^3 + 225*m^2 + 274*m + 12
0) + ((m^4 + 10*m^3 + 35*m^2 + 50*m + 24)*x^5*e^5 + (m^4 + 6*m^3 + 11*m^2 + 6*m)*d*x^4*e^4 - 4*(m^3 + 3*m^2 +
2*m)*d^2*x^3*e^3 + 12*(m^2 + m)*d^3*x^2*e^2 - 24*d^4*m*x*e + 24*d^5)*b^2*g^2*e^(m*log(x*e + d) - 5)/(m^5 + 15*
m^4 + 85*m^3 + 225*m^2 + 274*m + 120) + 2*((m^4 + 10*m^3 + 35*m^2 + 50*m + 24)*x^5*e^5 + (m^4 + 6*m^3 + 11*m^2
+ 6*m)*d*x^4*e^4 - 4*(m^3 + 3*m^2 + 2*m)*d^2*x^3*e^3 + 12*(m^2 + m)*d^3*x^2*e^2 - 24*d^4*m*x*e + 24*d^5)*a*c*
g^2*e^(m*log(x*e + d) - 5)/(m^5 + 15*m^4 + 85*m^3 + 225*m^2 + 274*m + 120) + 2*((m^5 + 15*m^4 + 85*m^3 + 225*m
^2 + 274*m + 120)*x^6*e^6 + (m^5 + 10*m^4 + 35*m^3 + 50*m^2 + 24*m)*d*x^5*e^5 - 5*(m^4 + 6*m^3 + 11*m^2 + 6*m)
*d^2*x^4*e^4 + 20*(m^3 + 3*m^2 + 2*m)*d^3*x^3*e^3 - 60*(m^2 + m)*d^4*x^2*e^2 + 120*d^5*m*x*e - 120*d^6)*c^2*f*
g*e^(m*log(x*e + d) - 6)/(m^6 + 21*m^5 + 175*m^4 + 735*m^3 + 1624*m^2 + 1764*m + 720) + 2*((m^5 + 15*m^4 + 85*
m^3 + 225*m^2 + 274*m + 120)*x^6*e^6 + (m^5 + 10*m^4 + 35*m^3 + 50*m^2 + 24*m)*d*x^5*e^5 - 5*(m^4 + 6*m^3 + 11
*m^2 + 6*m)*d^2*x^4*e^4 + 20*(m^3 + 3*m^2 + 2*m)*d^3*x^3*e^3 - 60*(m^2 + m)*d^4*x^2*e^2 + 120*d^5*m*x*e - 120*
d^6)*b*c*g^2*e^(m*log(x*e + d) - 6)/(m^6 + 21*m^5 + 175*m^4 + 735*m^3 + 1624*m^2 + 1764*m + 720) + ((m^6 + 21*
m^5 + 175*m^4 + 735*m^3 + 1624*m^2 + 1764*m + 720)*x^7*e^7 + (m^6 + 15*m^5 + 85*m^4 + 225*m^3 + 274*m^2 + 120*
m)*d*x^6*e^6 - 6*(m^5 + 10*m^4 + 35*m^3 + 50*m^2 + 24*m)*d^2*x^5*e^5 + 30*(m^4 + 6*m^3 + 11*m^2 + 6*m)*d^3*x^4
*e^4 - 120*(m^3 + 3*m^2 + 2*m)*d^4*x^3*e^3 + 360*(m^2 + m)*d^5*x^2*e^2 - 720*d^6*m*x*e + 720*d^7)*c^2*g^2*e^(m
*log(x*e + d) - 7)/(m^7 + 28*m^6 + 322*m^5 + 1960*m^4 + 6769*m^3 + 13132*m^2 + 13068*m + 5040)
________________________________________________________________________________________
Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 4356 vs.
\(2 (550) = 1100\).
time = 2.88, size = 4356, normalized size = 8.30 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x+d)^m*(g*x+f)^2*(c*x^2+b*x+a)^2,x, algorithm="fricas")
[Out]
(720*c^2*d^7*g^2 + ((c^2*g^2*m^6 + 21*c^2*g^2*m^5 + 175*c^2*g^2*m^4 + 735*c^2*g^2*m^3 + 1624*c^2*g^2*m^2 + 176
4*c^2*g^2*m + 720*c^2*g^2)*x^7 + 2*((c^2*f*g + b*c*g^2)*m^6 + 22*(c^2*f*g + b*c*g^2)*m^5 + 190*(c^2*f*g + b*c*
g^2)*m^4 + 840*c^2*f*g + 840*b*c*g^2 + 820*(c^2*f*g + b*c*g^2)*m^3 + 1849*(c^2*f*g + b*c*g^2)*m^2 + 2038*(c^2*
f*g + b*c*g^2)*m)*x^6 + ((c^2*f^2 + 4*b*c*f*g + (b^2 + 2*a*c)*g^2)*m^6 + 23*(c^2*f^2 + 4*b*c*f*g + (b^2 + 2*a*
c)*g^2)*m^5 + 207*(c^2*f^2 + 4*b*c*f*g + (b^2 + 2*a*c)*g^2)*m^4 + 1008*c^2*f^2 + 4032*b*c*f*g + 925*(c^2*f^2 +
4*b*c*f*g + (b^2 + 2*a*c)*g^2)*m^3 + 1008*(b^2 + 2*a*c)*g^2 + 2144*(c^2*f^2 + 4*b*c*f*g + (b^2 + 2*a*c)*g^2)*
m^2 + 2412*(c^2*f^2 + 4*b*c*f*g + (b^2 + 2*a*c)*g^2)*m)*x^5 + 2*((b*c*f^2 + a*b*g^2 + (b^2 + 2*a*c)*f*g)*m^6 +
24*(b*c*f^2 + a*b*g^2 + (b^2 + 2*a*c)*f*g)*m^5 + 226*(b*c*f^2 + a*b*g^2 + (b^2 + 2*a*c)*f*g)*m^4 + 1260*b*c*f
^2 + 1260*a*b*g^2 + 1056*(b*c*f^2 + a*b*g^2 + (b^2 + 2*a*c)*f*g)*m^3 + 1260*(b^2 + 2*a*c)*f*g + 2545*(b*c*f^2
+ a*b*g^2 + (b^2 + 2*a*c)*f*g)*m^2 + 2952*(b*c*f^2 + a*b*g^2 + (b^2 + 2*a*c)*f*g)*m)*x^4 + ((4*a*b*f*g + a^2*g
^2 + (b^2 + 2*a*c)*f^2)*m^6 + 25*(4*a*b*f*g + a^2*g^2 + (b^2 + 2*a*c)*f^2)*m^5 + 247*(4*a*b*f*g + a^2*g^2 + (b
^2 + 2*a*c)*f^2)*m^4 + 6720*a*b*f*g + 1680*a^2*g^2 + 1219*(4*a*b*f*g + a^2*g^2 + (b^2 + 2*a*c)*f^2)*m^3 + 1680
*(b^2 + 2*a*c)*f^2 + 3112*(4*a*b*f*g + a^2*g^2 + (b^2 + 2*a*c)*f^2)*m^2 + 3796*(4*a*b*f*g + a^2*g^2 + (b^2 + 2
*a*c)*f^2)*m)*x^3 + 2*((a*b*f^2 + a^2*f*g)*m^6 + 26*(a*b*f^2 + a^2*f*g)*m^5 + 270*(a*b*f^2 + a^2*f*g)*m^4 + 25
20*a*b*f^2 + 2520*a^2*f*g + 1420*(a*b*f^2 + a^2*f*g)*m^3 + 3929*(a*b*f^2 + a^2*f*g)*m^2 + 5274*(a*b*f^2 + a^2*
f*g)*m)*x^2 + (a^2*f^2*m^6 + 27*a^2*f^2*m^5 + 295*a^2*f^2*m^4 + 1665*a^2*f^2*m^3 + 5104*a^2*f^2*m^2 + 8028*a^2
*f^2*m + 5040*a^2*f^2)*x)*e^7 + (a^2*d*f^2*m^6 + 27*a^2*d*f^2*m^5 + 295*a^2*d*f^2*m^4 + 1665*a^2*d*f^2*m^3 + 5
104*a^2*d*f^2*m^2 + (c^2*d*g^2*m^6 + 15*c^2*d*g^2*m^5 + 85*c^2*d*g^2*m^4 + 225*c^2*d*g^2*m^3 + 274*c^2*d*g^2*m
^2 + 120*c^2*d*g^2*m)*x^6 + 8028*a^2*d*f^2*m + 2*((c^2*d*f*g + b*c*d*g^2)*m^6 + 17*(c^2*d*f*g + b*c*d*g^2)*m^5
+ 105*(c^2*d*f*g + b*c*d*g^2)*m^4 + 295*(c^2*d*f*g + b*c*d*g^2)*m^3 + 374*(c^2*d*f*g + b*c*d*g^2)*m^2 + 168*(
c^2*d*f*g + b*c*d*g^2)*m)*x^5 + 5040*a^2*d*f^2 + ((c^2*d*f^2 + 4*b*c*d*f*g + (b^2 + 2*a*c)*d*g^2)*m^6 + 19*(c^
2*d*f^2 + 4*b*c*d*f*g + (b^2 + 2*a*c)*d*g^2)*m^5 + 131*(c^2*d*f^2 + 4*b*c*d*f*g + (b^2 + 2*a*c)*d*g^2)*m^4 + 4
01*(c^2*d*f^2 + 4*b*c*d*f*g + (b^2 + 2*a*c)*d*g^2)*m^3 + 540*(c^2*d*f^2 + 4*b*c*d*f*g + (b^2 + 2*a*c)*d*g^2)*m
^2 + 252*(c^2*d*f^2 + 4*b*c*d*f*g + (b^2 + 2*a*c)*d*g^2)*m)*x^4 + 2*((b*c*d*f^2 + a*b*d*g^2 + (b^2 + 2*a*c)*d*
f*g)*m^6 + 21*(b*c*d*f^2 + a*b*d*g^2 + (b^2 + 2*a*c)*d*f*g)*m^5 + 163*(b*c*d*f^2 + a*b*d*g^2 + (b^2 + 2*a*c)*d
*f*g)*m^4 + 567*(b*c*d*f^2 + a*b*d*g^2 + (b^2 + 2*a*c)*d*f*g)*m^3 + 844*(b*c*d*f^2 + a*b*d*g^2 + (b^2 + 2*a*c)
*d*f*g)*m^2 + 420*(b*c*d*f^2 + a*b*d*g^2 + (b^2 + 2*a*c)*d*f*g)*m)*x^3 + ((4*a*b*d*f*g + a^2*d*g^2 + (b^2 + 2*
a*c)*d*f^2)*m^6 + 23*(4*a*b*d*f*g + a^2*d*g^2 + (b^2 + 2*a*c)*d*f^2)*m^5 + 201*(4*a*b*d*f*g + a^2*d*g^2 + (b^2
+ 2*a*c)*d*f^2)*m^4 + 817*(4*a*b*d*f*g + a^2*d*g^2 + (b^2 + 2*a*c)*d*f^2)*m^3 + 1478*(4*a*b*d*f*g + a^2*d*g^2
+ (b^2 + 2*a*c)*d*f^2)*m^2 + 840*(4*a*b*d*f*g + a^2*d*g^2 + (b^2 + 2*a*c)*d*f^2)*m)*x^2 + 2*((a*b*d*f^2 + a^2
*d*f*g)*m^6 + 25*(a*b*d*f^2 + a^2*d*f*g)*m^5 + 245*(a*b*d*f^2 + a^2*d*f*g)*m^4 + 1175*(a*b*d*f^2 + a^2*d*f*g)*
m^3 + 2754*(a*b*d*f^2 + a^2*d*f*g)*m^2 + 2520*(a*b*d*f^2 + a^2*d*f*g)*m)*x)*e^6 - 2*(2520*a*b*d^2*f^2 + 2520*a
^2*d^2*f*g + (a*b*d^2*f^2 + a^2*d^2*f*g)*m^5 + 3*(c^2*d^2*g^2*m^5 + 10*c^2*d^2*g^2*m^4 + 35*c^2*d^2*g^2*m^3 +
50*c^2*d^2*g^2*m^2 + 24*c^2*d^2*g^2*m)*x^5 + 25*(a*b*d^2*f^2 + a^2*d^2*f*g)*m^4 + 5*((c^2*d^2*f*g + b*c*d^2*g^
2)*m^5 + 13*(c^2*d^2*f*g + b*c*d^2*g^2)*m^4 + 53*(c^2*d^2*f*g + b*c*d^2*g^2)*m^3 + 83*(c^2*d^2*f*g + b*c*d^2*g
^2)*m^2 + 42*(c^2*d^2*f*g + b*c*d^2*g^2)*m)*x^4 + 245*(a*b*d^2*f^2 + a^2*d^2*f*g)*m^3 + 2*((c^2*d^2*f^2 + 4*b*
c*d^2*f*g + (b^2 + 2*a*c)*d^2*g^2)*m^5 + 16*(c^2*d^2*f^2 + 4*b*c*d^2*f*g + (b^2 + 2*a*c)*d^2*g^2)*m^4 + 83*(c^
2*d^2*f^2 + 4*b*c*d^2*f*g + (b^2 + 2*a*c)*d^2*g^2)*m^3 + 152*(c^2*d^2*f^2 + 4*b*c*d^2*f*g + (b^2 + 2*a*c)*d^2*
g^2)*m^2 + 84*(c^2*d^2*f^2 + 4*b*c*d^2*f*g + (b^2 + 2*a*c)*d^2*g^2)*m)*x^3 + 1175*(a*b*d^2*f^2 + a^2*d^2*f*g)*
m^2 + 3*((b*c*d^2*f^2 + a*b*d^2*g^2 + (b^2 + 2*a*c)*d^2*f*g)*m^5 + 19*(b*c*d^2*f^2 + a*b*d^2*g^2 + (b^2 + 2*a*
c)*d^2*f*g)*m^4 + 125*(b*c*d^2*f^2 + a*b*d^2*g^2 + (b^2 + 2*a*c)*d^2*f*g)*m^3 + 317*(b*c*d^2*f^2 + a*b*d^2*g^2
+ (b^2 + 2*a*c)*d^2*f*g)*m^2 + 210*(b*c*d^2*f^2 + a*b*d^2*g^2 + (b^2 + 2*a*c)*d^2*f*g)*m)*x^2 + 2754*(a*b*d^2
*f^2 + a^2*d^2*f*g)*m + ((4*a*b*d^2*f*g + a^2*d^2*g^2 + (b^2 + 2*a*c)*d^2*f^2)*m^5 + 22*(4*a*b*d^2*f*g + a^2*d
^2*g^2 + (b^2 + 2*a*c)*d^2*f^2)*m^4 + 179*(4*a*b*d^2*f*g + a^2*d^2*g^2 + (b^2 + 2*a*c)*d^2*f^2)*m^3 + 638*(4*a
*b*d^2*f*g + a^2*d^2*g^2 + (b^2 + 2*a*c)*d^2*f^2)*m^2 + 840*(4*a*b*d^2*f*g + a^2*d^2*g^2 + (b^2 + 2*a*c)*d^2*f
^2)*m)*x)*e^5 + 2*(3360*a*b*d^3*f*g + 840*a^2*d...
________________________________________________________________________________________
Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 74400 vs.
\(2 (537) = 1074\).
time = 13.84, size = 74400, normalized size = 141.71 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x+d)**m*(g*x+f)**2*(c*x**2+b*x+a)**2,x)
[Out]
Piecewise((d**m*(a**2*f**2*x + a**2*f*g*x**2 + a**2*g**2*x**3/3 + a*b*f**2*x**2 + 4*a*b*f*g*x**3/3 + a*b*g**2*
x**4/2 + 2*a*c*f**2*x**3/3 + a*c*f*g*x**4 + 2*a*c*g**2*x**5/5 + b**2*f**2*x**3/3 + b**2*f*g*x**4/2 + b**2*g**2
*x**5/5 + b*c*f**2*x**4/2 + 4*b*c*f*g*x**5/5 + b*c*g**2*x**6/3 + c**2*f**2*x**5/5 + c**2*f*g*x**6/3 + c**2*g**
2*x**7/7), Eq(e, 0)), (-a**2*d**2*e**4*g**2/(60*d**6*e**7 + 360*d**5*e**8*x + 900*d**4*e**9*x**2 + 1200*d**3*e
**10*x**3 + 900*d**2*e**11*x**4 + 360*d*e**12*x**5 + 60*e**13*x**6) - 4*a**2*d*e**5*f*g/(60*d**6*e**7 + 360*d*
*5*e**8*x + 900*d**4*e**9*x**2 + 1200*d**3*e**10*x**3 + 900*d**2*e**11*x**4 + 360*d*e**12*x**5 + 60*e**13*x**6
) - 6*a**2*d*e**5*g**2*x/(60*d**6*e**7 + 360*d**5*e**8*x + 900*d**4*e**9*x**2 + 1200*d**3*e**10*x**3 + 900*d**
2*e**11*x**4 + 360*d*e**12*x**5 + 60*e**13*x**6) - 10*a**2*e**6*f**2/(60*d**6*e**7 + 360*d**5*e**8*x + 900*d**
4*e**9*x**2 + 1200*d**3*e**10*x**3 + 900*d**2*e**11*x**4 + 360*d*e**12*x**5 + 60*e**13*x**6) - 24*a**2*e**6*f*
g*x/(60*d**6*e**7 + 360*d**5*e**8*x + 900*d**4*e**9*x**2 + 1200*d**3*e**10*x**3 + 900*d**2*e**11*x**4 + 360*d*
e**12*x**5 + 60*e**13*x**6) - 15*a**2*e**6*g**2*x**2/(60*d**6*e**7 + 360*d**5*e**8*x + 900*d**4*e**9*x**2 + 12
00*d**3*e**10*x**3 + 900*d**2*e**11*x**4 + 360*d*e**12*x**5 + 60*e**13*x**6) - 2*a*b*d**3*e**3*g**2/(60*d**6*e
**7 + 360*d**5*e**8*x + 900*d**4*e**9*x**2 + 1200*d**3*e**10*x**3 + 900*d**2*e**11*x**4 + 360*d*e**12*x**5 + 6
0*e**13*x**6) - 4*a*b*d**2*e**4*f*g/(60*d**6*e**7 + 360*d**5*e**8*x + 900*d**4*e**9*x**2 + 1200*d**3*e**10*x**
3 + 900*d**2*e**11*x**4 + 360*d*e**12*x**5 + 60*e**13*x**6) - 12*a*b*d**2*e**4*g**2*x/(60*d**6*e**7 + 360*d**5
*e**8*x + 900*d**4*e**9*x**2 + 1200*d**3*e**10*x**3 + 900*d**2*e**11*x**4 + 360*d*e**12*x**5 + 60*e**13*x**6)
- 4*a*b*d*e**5*f**2/(60*d**6*e**7 + 360*d**5*e**8*x + 900*d**4*e**9*x**2 + 1200*d**3*e**10*x**3 + 900*d**2*e**
11*x**4 + 360*d*e**12*x**5 + 60*e**13*x**6) - 24*a*b*d*e**5*f*g*x/(60*d**6*e**7 + 360*d**5*e**8*x + 900*d**4*e
**9*x**2 + 1200*d**3*e**10*x**3 + 900*d**2*e**11*x**4 + 360*d*e**12*x**5 + 60*e**13*x**6) - 30*a*b*d*e**5*g**2
*x**2/(60*d**6*e**7 + 360*d**5*e**8*x + 900*d**4*e**9*x**2 + 1200*d**3*e**10*x**3 + 900*d**2*e**11*x**4 + 360*
d*e**12*x**5 + 60*e**13*x**6) - 24*a*b*e**6*f**2*x/(60*d**6*e**7 + 360*d**5*e**8*x + 900*d**4*e**9*x**2 + 1200
*d**3*e**10*x**3 + 900*d**2*e**11*x**4 + 360*d*e**12*x**5 + 60*e**13*x**6) - 60*a*b*e**6*f*g*x**2/(60*d**6*e**
7 + 360*d**5*e**8*x + 900*d**4*e**9*x**2 + 1200*d**3*e**10*x**3 + 900*d**2*e**11*x**4 + 360*d*e**12*x**5 + 60*
e**13*x**6) - 40*a*b*e**6*g**2*x**3/(60*d**6*e**7 + 360*d**5*e**8*x + 900*d**4*e**9*x**2 + 1200*d**3*e**10*x**
3 + 900*d**2*e**11*x**4 + 360*d*e**12*x**5 + 60*e**13*x**6) - 4*a*c*d**4*e**2*g**2/(60*d**6*e**7 + 360*d**5*e*
*8*x + 900*d**4*e**9*x**2 + 1200*d**3*e**10*x**3 + 900*d**2*e**11*x**4 + 360*d*e**12*x**5 + 60*e**13*x**6) - 4
*a*c*d**3*e**3*f*g/(60*d**6*e**7 + 360*d**5*e**8*x + 900*d**4*e**9*x**2 + 1200*d**3*e**10*x**3 + 900*d**2*e**1
1*x**4 + 360*d*e**12*x**5 + 60*e**13*x**6) - 24*a*c*d**3*e**3*g**2*x/(60*d**6*e**7 + 360*d**5*e**8*x + 900*d**
4*e**9*x**2 + 1200*d**3*e**10*x**3 + 900*d**2*e**11*x**4 + 360*d*e**12*x**5 + 60*e**13*x**6) - 2*a*c*d**2*e**4
*f**2/(60*d**6*e**7 + 360*d**5*e**8*x + 900*d**4*e**9*x**2 + 1200*d**3*e**10*x**3 + 900*d**2*e**11*x**4 + 360*
d*e**12*x**5 + 60*e**13*x**6) - 24*a*c*d**2*e**4*f*g*x/(60*d**6*e**7 + 360*d**5*e**8*x + 900*d**4*e**9*x**2 +
1200*d**3*e**10*x**3 + 900*d**2*e**11*x**4 + 360*d*e**12*x**5 + 60*e**13*x**6) - 60*a*c*d**2*e**4*g**2*x**2/(6
0*d**6*e**7 + 360*d**5*e**8*x + 900*d**4*e**9*x**2 + 1200*d**3*e**10*x**3 + 900*d**2*e**11*x**4 + 360*d*e**12*
x**5 + 60*e**13*x**6) - 12*a*c*d*e**5*f**2*x/(60*d**6*e**7 + 360*d**5*e**8*x + 900*d**4*e**9*x**2 + 1200*d**3*
e**10*x**3 + 900*d**2*e**11*x**4 + 360*d*e**12*x**5 + 60*e**13*x**6) - 60*a*c*d*e**5*f*g*x**2/(60*d**6*e**7 +
360*d**5*e**8*x + 900*d**4*e**9*x**2 + 1200*d**3*e**10*x**3 + 900*d**2*e**11*x**4 + 360*d*e**12*x**5 + 60*e**1
3*x**6) - 80*a*c*d*e**5*g**2*x**3/(60*d**6*e**7 + 360*d**5*e**8*x + 900*d**4*e**9*x**2 + 1200*d**3*e**10*x**3
+ 900*d**2*e**11*x**4 + 360*d*e**12*x**5 + 60*e**13*x**6) - 30*a*c*e**6*f**2*x**2/(60*d**6*e**7 + 360*d**5*e**
8*x + 900*d**4*e**9*x**2 + 1200*d**3*e**10*x**3 + 900*d**2*e**11*x**4 + 360*d*e**12*x**5 + 60*e**13*x**6) - 80
*a*c*e**6*f*g*x**3/(60*d**6*e**7 + 360*d**5*e**8*x + 900*d**4*e**9*x**2 + 1200*d**3*e**10*x**3 + 900*d**2*e**1
1*x**4 + 360*d*e**12*x**5 + 60*e**13*x**6) - 60*a*c*e**6*g**2*x**4/(60*d**6*e**7 + 360*d**5*e**8*x + 900*d**4*
e**9*x**2 + 1200*d**3*e**10*x**3 + 900*d**2*e**11*x**4 + 360*d*e**12*x**5 + 60*e**13*x**6) - 2*b**2*d**4*e**2*
g**2/(60*d**6*e**7 + 360*d**5*e**8*x + 900*d**4*e**9*x**2 + 1200*d**3*e**10*x**3 + 900*d**2*e**11*x**4 + 360*d
*e**12*x**5 + 60*e**13*x**6) - 2*b**2*d**3*e**3*f*g/(60*d**6*e**7 + 360*d**5*e**8*x + 900*d**4*e**9*x**2 + 120
0*d**3*e**10*x**3 + 900*d**2*e**11*x**4 + 360*d*e**12*x**5 + 60*e**13*x**6) - 12*b**2*d**3*e**3*g**2*x/(60*d**
6*e**7 + 360*d**5*e**8*x + 900*d**4*e**9*x**2 +...
________________________________________________________________________________________
Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 10489 vs.
\(2 (550) = 1100\).
time = 1.92, size = 10489, normalized size = 19.98 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x+d)^m*(g*x+f)^2*(c*x^2+b*x+a)^2,x, algorithm="giac")
[Out]
((x*e + d)^m*c^2*g^2*m^6*x^7*e^7 + (x*e + d)^m*c^2*d*g^2*m^6*x^6*e^6 + 2*(x*e + d)^m*c^2*f*g*m^6*x^6*e^7 + 2*(
x*e + d)^m*b*c*g^2*m^6*x^6*e^7 + 21*(x*e + d)^m*c^2*g^2*m^5*x^7*e^7 + 2*(x*e + d)^m*c^2*d*f*g*m^6*x^5*e^6 + 2*
(x*e + d)^m*b*c*d*g^2*m^6*x^5*e^6 + 15*(x*e + d)^m*c^2*d*g^2*m^5*x^6*e^6 - 6*(x*e + d)^m*c^2*d^2*g^2*m^5*x^5*e
^5 + (x*e + d)^m*c^2*f^2*m^6*x^5*e^7 + 4*(x*e + d)^m*b*c*f*g*m^6*x^5*e^7 + (x*e + d)^m*b^2*g^2*m^6*x^5*e^7 + 2
*(x*e + d)^m*a*c*g^2*m^6*x^5*e^7 + 44*(x*e + d)^m*c^2*f*g*m^5*x^6*e^7 + 44*(x*e + d)^m*b*c*g^2*m^5*x^6*e^7 + 1
75*(x*e + d)^m*c^2*g^2*m^4*x^7*e^7 + (x*e + d)^m*c^2*d*f^2*m^6*x^4*e^6 + 4*(x*e + d)^m*b*c*d*f*g*m^6*x^4*e^6 +
(x*e + d)^m*b^2*d*g^2*m^6*x^4*e^6 + 2*(x*e + d)^m*a*c*d*g^2*m^6*x^4*e^6 + 34*(x*e + d)^m*c^2*d*f*g*m^5*x^5*e^
6 + 34*(x*e + d)^m*b*c*d*g^2*m^5*x^5*e^6 + 85*(x*e + d)^m*c^2*d*g^2*m^4*x^6*e^6 - 10*(x*e + d)^m*c^2*d^2*f*g*m
^5*x^4*e^5 - 10*(x*e + d)^m*b*c*d^2*g^2*m^5*x^4*e^5 - 60*(x*e + d)^m*c^2*d^2*g^2*m^4*x^5*e^5 + 30*(x*e + d)^m*
c^2*d^3*g^2*m^4*x^4*e^4 + 2*(x*e + d)^m*b*c*f^2*m^6*x^4*e^7 + 2*(x*e + d)^m*b^2*f*g*m^6*x^4*e^7 + 4*(x*e + d)^
m*a*c*f*g*m^6*x^4*e^7 + 2*(x*e + d)^m*a*b*g^2*m^6*x^4*e^7 + 23*(x*e + d)^m*c^2*f^2*m^5*x^5*e^7 + 92*(x*e + d)^
m*b*c*f*g*m^5*x^5*e^7 + 23*(x*e + d)^m*b^2*g^2*m^5*x^5*e^7 + 46*(x*e + d)^m*a*c*g^2*m^5*x^5*e^7 + 380*(x*e + d
)^m*c^2*f*g*m^4*x^6*e^7 + 380*(x*e + d)^m*b*c*g^2*m^4*x^6*e^7 + 735*(x*e + d)^m*c^2*g^2*m^3*x^7*e^7 + 2*(x*e +
d)^m*b*c*d*f^2*m^6*x^3*e^6 + 2*(x*e + d)^m*b^2*d*f*g*m^6*x^3*e^6 + 4*(x*e + d)^m*a*c*d*f*g*m^6*x^3*e^6 + 2*(x
*e + d)^m*a*b*d*g^2*m^6*x^3*e^6 + 19*(x*e + d)^m*c^2*d*f^2*m^5*x^4*e^6 + 76*(x*e + d)^m*b*c*d*f*g*m^5*x^4*e^6
+ 19*(x*e + d)^m*b^2*d*g^2*m^5*x^4*e^6 + 38*(x*e + d)^m*a*c*d*g^2*m^5*x^4*e^6 + 210*(x*e + d)^m*c^2*d*f*g*m^4*
x^5*e^6 + 210*(x*e + d)^m*b*c*d*g^2*m^4*x^5*e^6 + 225*(x*e + d)^m*c^2*d*g^2*m^3*x^6*e^6 - 4*(x*e + d)^m*c^2*d^
2*f^2*m^5*x^3*e^5 - 16*(x*e + d)^m*b*c*d^2*f*g*m^5*x^3*e^5 - 4*(x*e + d)^m*b^2*d^2*g^2*m^5*x^3*e^5 - 8*(x*e +
d)^m*a*c*d^2*g^2*m^5*x^3*e^5 - 130*(x*e + d)^m*c^2*d^2*f*g*m^4*x^4*e^5 - 130*(x*e + d)^m*b*c*d^2*g^2*m^4*x^4*e
^5 - 210*(x*e + d)^m*c^2*d^2*g^2*m^3*x^5*e^5 + 40*(x*e + d)^m*c^2*d^3*f*g*m^4*x^3*e^4 + 40*(x*e + d)^m*b*c*d^3
*g^2*m^4*x^3*e^4 + 180*(x*e + d)^m*c^2*d^3*g^2*m^3*x^4*e^4 - 120*(x*e + d)^m*c^2*d^4*g^2*m^3*x^3*e^3 + (x*e +
d)^m*b^2*f^2*m^6*x^3*e^7 + 2*(x*e + d)^m*a*c*f^2*m^6*x^3*e^7 + 4*(x*e + d)^m*a*b*f*g*m^6*x^3*e^7 + (x*e + d)^m
*a^2*g^2*m^6*x^3*e^7 + 48*(x*e + d)^m*b*c*f^2*m^5*x^4*e^7 + 48*(x*e + d)^m*b^2*f*g*m^5*x^4*e^7 + 96*(x*e + d)^
m*a*c*f*g*m^5*x^4*e^7 + 48*(x*e + d)^m*a*b*g^2*m^5*x^4*e^7 + 207*(x*e + d)^m*c^2*f^2*m^4*x^5*e^7 + 828*(x*e +
d)^m*b*c*f*g*m^4*x^5*e^7 + 207*(x*e + d)^m*b^2*g^2*m^4*x^5*e^7 + 414*(x*e + d)^m*a*c*g^2*m^4*x^5*e^7 + 1640*(x
*e + d)^m*c^2*f*g*m^3*x^6*e^7 + 1640*(x*e + d)^m*b*c*g^2*m^3*x^6*e^7 + 1624*(x*e + d)^m*c^2*g^2*m^2*x^7*e^7 +
(x*e + d)^m*b^2*d*f^2*m^6*x^2*e^6 + 2*(x*e + d)^m*a*c*d*f^2*m^6*x^2*e^6 + 4*(x*e + d)^m*a*b*d*f*g*m^6*x^2*e^6
+ (x*e + d)^m*a^2*d*g^2*m^6*x^2*e^6 + 42*(x*e + d)^m*b*c*d*f^2*m^5*x^3*e^6 + 42*(x*e + d)^m*b^2*d*f*g*m^5*x^3*
e^6 + 84*(x*e + d)^m*a*c*d*f*g*m^5*x^3*e^6 + 42*(x*e + d)^m*a*b*d*g^2*m^5*x^3*e^6 + 131*(x*e + d)^m*c^2*d*f^2*
m^4*x^4*e^6 + 524*(x*e + d)^m*b*c*d*f*g*m^4*x^4*e^6 + 131*(x*e + d)^m*b^2*d*g^2*m^4*x^4*e^6 + 262*(x*e + d)^m*
a*c*d*g^2*m^4*x^4*e^6 + 590*(x*e + d)^m*c^2*d*f*g*m^3*x^5*e^6 + 590*(x*e + d)^m*b*c*d*g^2*m^3*x^5*e^6 + 274*(x
*e + d)^m*c^2*d*g^2*m^2*x^6*e^6 - 6*(x*e + d)^m*b*c*d^2*f^2*m^5*x^2*e^5 - 6*(x*e + d)^m*b^2*d^2*f*g*m^5*x^2*e^
5 - 12*(x*e + d)^m*a*c*d^2*f*g*m^5*x^2*e^5 - 6*(x*e + d)^m*a*b*d^2*g^2*m^5*x^2*e^5 - 64*(x*e + d)^m*c^2*d^2*f^
2*m^4*x^3*e^5 - 256*(x*e + d)^m*b*c*d^2*f*g*m^4*x^3*e^5 - 64*(x*e + d)^m*b^2*d^2*g^2*m^4*x^3*e^5 - 128*(x*e +
d)^m*a*c*d^2*g^2*m^4*x^3*e^5 - 530*(x*e + d)^m*c^2*d^2*f*g*m^3*x^4*e^5 - 530*(x*e + d)^m*b*c*d^2*g^2*m^3*x^4*e
^5 - 300*(x*e + d)^m*c^2*d^2*g^2*m^2*x^5*e^5 + 12*(x*e + d)^m*c^2*d^3*f^2*m^4*x^2*e^4 + 48*(x*e + d)^m*b*c*d^3
*f*g*m^4*x^2*e^4 + 12*(x*e + d)^m*b^2*d^3*g^2*m^4*x^2*e^4 + 24*(x*e + d)^m*a*c*d^3*g^2*m^4*x^2*e^4 + 400*(x*e
+ d)^m*c^2*d^3*f*g*m^3*x^3*e^4 + 400*(x*e + d)^m*b*c*d^3*g^2*m^3*x^3*e^4 + 330*(x*e + d)^m*c^2*d^3*g^2*m^2*x^4
*e^4 - 120*(x*e + d)^m*c^2*d^4*f*g*m^3*x^2*e^3 - 120*(x*e + d)^m*b*c*d^4*g^2*m^3*x^2*e^3 - 360*(x*e + d)^m*c^2
*d^4*g^2*m^2*x^3*e^3 + 360*(x*e + d)^m*c^2*d^5*g^2*m^2*x^2*e^2 + 2*(x*e + d)^m*a*b*f^2*m^6*x^2*e^7 + 2*(x*e +
d)^m*a^2*f*g*m^6*x^2*e^7 + 25*(x*e + d)^m*b^2*f^2*m^5*x^3*e^7 + 50*(x*e + d)^m*a*c*f^2*m^5*x^3*e^7 + 100*(x*e
+ d)^m*a*b*f*g*m^5*x^3*e^7 + 25*(x*e + d)^m*a^2*g^2*m^5*x^3*e^7 + 452*(x*e + d)^m*b*c*f^2*m^4*x^4*e^7 + 452*(x
*e + d)^m*b^2*f*g*m^4*x^4*e^7 + 904*(x*e + d)^m*a*c*f*g*m^4*x^4*e^7 + 452*(x*e + d)^m*a*b*g^2*m^4*x^4*e^7 + 92
5*(x*e + d)^m*c^2*f^2*m^3*x^5*e^7 + 3700*(x*e + d)^m*b*c*f*g*m^3*x^5*e^7 + 925*(x*e + d)^m*b^2*g^2*m^3*x^5*e^7
+ 1850*(x*e + d)^m*a*c*g^2*m^3*x^5*e^7 + 3698*(x*e + d)^m*c^2*f*g*m^2*x^6*e^7 + 3698*(x*e + d)^m*b*c*g^2*m^2*
x^6*e^7 + 1764*(x*e + d)^m*c^2*g^2*m*x^7*e^7 + ...
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Mupad [B]
time = 5.38, size = 2500, normalized size = 4.76 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((f + g*x)^2*(d + e*x)^m*(a + b*x + c*x^2)^2,x)
[Out]
((d + e*x)^m*(720*c^2*d^7*g^2 + 5040*a^2*d*e^6*f^2 + 1680*a^2*d^3*e^4*g^2 + 1680*b^2*d^3*e^4*f^2 + 1008*b^2*d^
5*e^2*g^2 + 1008*c^2*d^5*e^2*f^2 - 1680*b*c*d^6*e*g^2 - 1680*c^2*d^6*e*f*g + 358*a^2*d^3*e^4*g^2*m^2 + 358*b^2
*d^3*e^4*f^2*m^2 + 44*a^2*d^3*e^4*g^2*m^3 + 44*b^2*d^3*e^4*f^2*m^3 + 2*a^2*d^3*e^4*g^2*m^4 + 2*b^2*d^3*e^4*f^2
*m^4 + 24*b^2*d^5*e^2*g^2*m^2 + 24*c^2*d^5*e^2*f^2*m^2 - 5040*a*b*d^2*e^5*f^2 - 2520*a*b*d^4*e^3*g^2 + 3360*a*
c*d^3*e^4*f^2 + 2016*a*c*d^5*e^2*g^2 - 2520*b*c*d^4*e^3*f^2 - 5040*a^2*d^2*e^5*f*g - 2520*b^2*d^4*e^3*f*g + 80
28*a^2*d*e^6*f^2*m + 5104*a^2*d*e^6*f^2*m^2 + 1665*a^2*d*e^6*f^2*m^3 + 295*a^2*d*e^6*f^2*m^4 + 27*a^2*d*e^6*f^
2*m^5 + a^2*d*e^6*f^2*m^6 + 1276*a^2*d^3*e^4*g^2*m + 1276*b^2*d^3*e^4*f^2*m + 312*b^2*d^5*e^2*g^2*m + 312*c^2*
d^5*e^2*f^2*m - 2350*a*b*d^2*e^5*f^2*m^2 - 490*a*b*d^2*e^5*f^2*m^3 - 50*a*b*d^2*e^5*f^2*m^4 - 2*a*b*d^2*e^5*f^
2*m^5 - 216*a*b*d^4*e^3*g^2*m^2 + 716*a*c*d^3*e^4*f^2*m^2 - 12*a*b*d^4*e^3*g^2*m^3 + 88*a*c*d^3*e^4*f^2*m^3 +
4*a*c*d^3*e^4*f^2*m^4 + 48*a*c*d^5*e^2*g^2*m^2 - 216*b*c*d^4*e^3*f^2*m^2 - 12*b*c*d^4*e^3*f^2*m^3 - 2350*a^2*d
^2*e^5*f*g*m^2 - 490*a^2*d^2*e^5*f*g*m^3 - 50*a^2*d^2*e^5*f*g*m^4 - 2*a^2*d^2*e^5*f*g*m^5 - 216*b^2*d^4*e^3*f*
g*m^2 - 12*b^2*d^4*e^3*f*g*m^3 + 6720*a*b*d^3*e^4*f*g - 5040*a*c*d^4*e^3*f*g + 4032*b*c*d^5*e^2*f*g - 240*b*c*
d^6*e*g^2*m - 240*c^2*d^6*e*f*g*m - 5508*a*b*d^2*e^5*f^2*m - 1284*a*b*d^4*e^3*g^2*m + 2552*a*c*d^3*e^4*f^2*m +
624*a*c*d^5*e^2*g^2*m - 1284*b*c*d^4*e^3*f^2*m - 5508*a^2*d^2*e^5*f*g*m - 1284*b^2*d^4*e^3*f*g*m + 1432*a*b*d
^3*e^4*f*g*m^2 + 176*a*b*d^3*e^4*f*g*m^3 + 8*a*b*d^3*e^4*f*g*m^4 - 432*a*c*d^4*e^3*f*g*m^2 - 24*a*c*d^4*e^3*f*
g*m^3 + 96*b*c*d^5*e^2*f*g*m^2 + 5104*a*b*d^3*e^4*f*g*m - 2568*a*c*d^4*e^3*f*g*m + 1248*b*c*d^5*e^2*f*g*m))/(e
^7*(13068*m + 13132*m^2 + 6769*m^3 + 1960*m^4 + 322*m^5 + 28*m^6 + m^7 + 5040)) + (x*(d + e*x)^m*(5040*a^2*e^7
*f^2 + 8028*a^2*e^7*f^2*m + 5104*a^2*e^7*f^2*m^2 + 1665*a^2*e^7*f^2*m^3 + 295*a^2*e^7*f^2*m^4 + 27*a^2*e^7*f^2
*m^5 + a^2*e^7*f^2*m^6 - 1276*a^2*d^2*e^5*g^2*m^2 - 1276*b^2*d^2*e^5*f^2*m^2 - 358*a^2*d^2*e^5*g^2*m^3 - 358*b
^2*d^2*e^5*f^2*m^3 - 44*a^2*d^2*e^5*g^2*m^4 - 44*b^2*d^2*e^5*f^2*m^4 - 2*a^2*d^2*e^5*g^2*m^5 - 2*b^2*d^2*e^5*f
^2*m^5 - 312*b^2*d^4*e^3*g^2*m^2 - 312*c^2*d^4*e^3*f^2*m^2 - 24*b^2*d^4*e^3*g^2*m^3 - 24*c^2*d^4*e^3*f^2*m^3 -
720*c^2*d^6*e*g^2*m - 1680*a^2*d^2*e^5*g^2*m - 1680*b^2*d^2*e^5*f^2*m - 1008*b^2*d^4*e^3*g^2*m - 1008*c^2*d^4
*e^3*f^2*m + 1284*a*b*d^3*e^4*g^2*m^2 - 2552*a*c*d^2*e^5*f^2*m^2 + 216*a*b*d^3*e^4*g^2*m^3 - 716*a*c*d^2*e^5*f
^2*m^3 + 12*a*b*d^3*e^4*g^2*m^4 - 88*a*c*d^2*e^5*f^2*m^4 - 4*a*c*d^2*e^5*f^2*m^5 - 624*a*c*d^4*e^3*g^2*m^2 + 1
284*b*c*d^3*e^4*f^2*m^2 - 48*a*c*d^4*e^3*g^2*m^3 + 216*b*c*d^3*e^4*f^2*m^3 + 12*b*c*d^3*e^4*f^2*m^4 + 240*b*c*
d^5*e^2*g^2*m^2 + 1284*b^2*d^3*e^4*f*g*m^2 + 216*b^2*d^3*e^4*f*g*m^3 + 12*b^2*d^3*e^4*f*g*m^4 + 240*c^2*d^5*e^
2*f*g*m^2 + 5040*a*b*d*e^6*f^2*m + 5040*a^2*d*e^6*f*g*m + 5508*a*b*d*e^6*f^2*m^2 + 2350*a*b*d*e^6*f^2*m^3 + 49
0*a*b*d*e^6*f^2*m^4 + 50*a*b*d*e^6*f^2*m^5 + 2*a*b*d*e^6*f^2*m^6 + 2520*a*b*d^3*e^4*g^2*m - 3360*a*c*d^2*e^5*f
^2*m - 2016*a*c*d^4*e^3*g^2*m + 2520*b*c*d^3*e^4*f^2*m + 1680*b*c*d^5*e^2*g^2*m + 5508*a^2*d*e^6*f*g*m^2 + 235
0*a^2*d*e^6*f*g*m^3 + 490*a^2*d*e^6*f*g*m^4 + 50*a^2*d*e^6*f*g*m^5 + 2*a^2*d*e^6*f*g*m^6 + 2520*b^2*d^3*e^4*f*
g*m + 1680*c^2*d^5*e^2*f*g*m - 5104*a*b*d^2*e^5*f*g*m^2 - 1432*a*b*d^2*e^5*f*g*m^3 - 176*a*b*d^2*e^5*f*g*m^4 -
8*a*b*d^2*e^5*f*g*m^5 + 2568*a*c*d^3*e^4*f*g*m^2 + 432*a*c*d^3*e^4*f*g*m^3 + 24*a*c*d^3*e^4*f*g*m^4 - 1248*b*
c*d^4*e^3*f*g*m^2 - 96*b*c*d^4*e^3*f*g*m^3 - 6720*a*b*d^2*e^5*f*g*m + 5040*a*c*d^3*e^4*f*g*m - 4032*b*c*d^4*e^
3*f*g*m))/(e^7*(13068*m + 13132*m^2 + 6769*m^3 + 1960*m^4 + 322*m^5 + 28*m^6 + m^7 + 5040)) + (x^3*(d + e*x)^m
*(3*m + m^2 + 2)*(840*a^2*e^4*g^2 + 840*b^2*e^4*f^2 + 638*a^2*e^4*g^2*m + 638*b^2*e^4*f^2*m - 120*c^2*d^4*g^2*
m + 179*a^2*e^4*g^2*m^2 + 179*b^2*e^4*f^2*m^2 + 22*a^2*e^4*g^2*m^3 + 22*b^2*e^4*f^2*m^3 + a^2*e^4*g^2*m^4 + b^
2*e^4*f^2*m^4 + 1680*a*c*e^4*f^2 + 1276*a*c*e^4*f^2*m - 52*b^2*d^2*e^2*g^2*m^2 - 52*c^2*d^2*e^2*f^2*m^2 - 4*b^
2*d^2*e^2*g^2*m^3 - 4*c^2*d^2*e^2*f^2*m^3 + 358*a*c*e^4*f^2*m^2 + 44*a*c*e^4*f^2*m^3 + 2*a*c*e^4*f^2*m^4 + 336
0*a*b*e^4*f*g - 168*b^2*d^2*e^2*g^2*m - 168*c^2*d^2*e^2*f^2*m + 2552*a*b*e^4*f*g*m - 104*a*c*d^2*e^2*g^2*m^2 -
8*a*c*d^2*e^2*g^2*m^3 + 420*a*b*d*e^3*g^2*m + 420*b*c*d*e^3*f^2*m + 280*b*c*d^3*e*g^2*m + 716*a*b*e^4*f*g*m^2
+ 88*a*b*e^4*f*g*m^3 + 4*a*b*e^4*f*g*m^4 + 420*b^2*d*e^3*f*g*m + 280*c^2*d^3*e*f*g*m + 214*a*b*d*e^3*g^2*m^2
+ 36*a*b*d*e^3*g^2*m^3 + 2*a*b*d*e^3*g^2*m^4 - 336*a*c*d^2*e^2*g^2*m + 214*b*c*d*e^3*f^2*m^2 + 36*b*c*d*e^3*f^
2*m^3 + 2*b*c*d*e^3*f^2*m^4 + 40*b*c*d^3*e*g^2*m^2 + 214*b^2*d*e^3*f*g*m^2 + 36*b^2*d*e^3*f*g*m^3 + 2*b^2*d*e^
3*f*g*m^4 + 40*c^2*d^3*e*f*g*m^2 - 208*b*c*d^2*e^2*f*g*m^2 - 16*b*c*d^2*e^2*f*g*m^3 + 840*a*c*d*e^3*f*g*m + 42
8*a*c*d*e^3*f*g*m^2 + 72*a*c*d*e^3*f*g*m^3 + 4*a*c*d*e^3*f*g*m^4 - 672*b*c*d^2*e^2*f*g*m))/(e^4*(13068*m + 131
32*m^2 + 6769*m^3 + 1960*m^4 + 322*m^5 + 28*m^6...
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