3.10.69 \(\int (A+B x) (d+e x)^m (b x+c x^2)^2 \, dx\)
Optimal. Leaf size=282 \[ \frac {(d+e x)^{m+3} \left (A e \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )-B d \left (3 b^2 e^2-12 b c d e+10 c^2 d^2\right )\right )}{e^6 (m+3)}-\frac {(d+e x)^{m+4} \left (2 A c e (2 c d-b e)-B \left (b^2 e^2-8 b c d e+10 c^2 d^2\right )\right )}{e^6 (m+4)}-\frac {d^2 (B d-A e) (c d-b e)^2 (d+e x)^{m+1}}{e^6 (m+1)}+\frac {d (c d-b e) (d+e x)^{m+2} (B d (5 c d-3 b e)-2 A e (2 c d-b e))}{e^6 (m+2)}-\frac {c (d+e x)^{m+5} (-A c e-2 b B e+5 B c d)}{e^6 (m+5)}+\frac {B c^2 (d+e x)^{m+6}}{e^6 (m+6)} \]
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Rubi [A] time = 0.23, antiderivative size = 282, normalized size of antiderivative = 1.00,
number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used =
{771} \begin {gather*} \frac {(d+e x)^{m+3} \left (A e \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )-B d \left (3 b^2 e^2-12 b c d e+10 c^2 d^2\right )\right )}{e^6 (m+3)}-\frac {(d+e x)^{m+4} \left (2 A c e (2 c d-b e)-B \left (b^2 e^2-8 b c d e+10 c^2 d^2\right )\right )}{e^6 (m+4)}-\frac {d^2 (B d-A e) (c d-b e)^2 (d+e x)^{m+1}}{e^6 (m+1)}+\frac {d (c d-b e) (d+e x)^{m+2} (B d (5 c d-3 b e)-2 A e (2 c d-b e))}{e^6 (m+2)}-\frac {c (d+e x)^{m+5} (-A c e-2 b B e+5 B c d)}{e^6 (m+5)}+\frac {B c^2 (d+e x)^{m+6}}{e^6 (m+6)} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[(A + B*x)*(d + e*x)^m*(b*x + c*x^2)^2,x]
[Out]
-((d^2*(B*d - A*e)*(c*d - b*e)^2*(d + e*x)^(1 + m))/(e^6*(1 + m))) + (d*(c*d - b*e)*(B*d*(5*c*d - 3*b*e) - 2*A
*e*(2*c*d - b*e))*(d + e*x)^(2 + m))/(e^6*(2 + m)) + ((A*e*(6*c^2*d^2 - 6*b*c*d*e + b^2*e^2) - B*d*(10*c^2*d^2
- 12*b*c*d*e + 3*b^2*e^2))*(d + e*x)^(3 + m))/(e^6*(3 + m)) - ((2*A*c*e*(2*c*d - b*e) - B*(10*c^2*d^2 - 8*b*c
*d*e + b^2*e^2))*(d + e*x)^(4 + m))/(e^6*(4 + m)) - (c*(5*B*c*d - 2*b*B*e - A*c*e)*(d + e*x)^(5 + m))/(e^6*(5
+ m)) + (B*c^2*(d + e*x)^(6 + m))/(e^6*(6 + m))
Rule 771
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> In
t[ExpandIntegrand[(d + e*x)^m*(f + g*x)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && N
eQ[b^2 - 4*a*c, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))
Rubi steps
\begin {align*} \int (A+B x) (d+e x)^m \left (b x+c x^2\right )^2 \, dx &=\int \left (-\frac {d^2 (B d-A e) (c d-b e)^2 (d+e x)^m}{e^5}+\frac {d (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e)) (d+e x)^{1+m}}{e^5}+\frac {\left (A e \left (6 c^2 d^2-6 b c d e+b^2 e^2\right )-B d \left (10 c^2 d^2-12 b c d e+3 b^2 e^2\right )\right ) (d+e x)^{2+m}}{e^5}+\frac {\left (-2 A c e (2 c d-b e)+B \left (10 c^2 d^2-8 b c d e+b^2 e^2\right )\right ) (d+e x)^{3+m}}{e^5}+\frac {c (-5 B c d+2 b B e+A c e) (d+e x)^{4+m}}{e^5}+\frac {B c^2 (d+e x)^{5+m}}{e^5}\right ) \, dx\\ &=-\frac {d^2 (B d-A e) (c d-b e)^2 (d+e x)^{1+m}}{e^6 (1+m)}+\frac {d (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e)) (d+e x)^{2+m}}{e^6 (2+m)}+\frac {\left (A e \left (6 c^2 d^2-6 b c d e+b^2 e^2\right )-B d \left (10 c^2 d^2-12 b c d e+3 b^2 e^2\right )\right ) (d+e x)^{3+m}}{e^6 (3+m)}-\frac {\left (2 A c e (2 c d-b e)-B \left (10 c^2 d^2-8 b c d e+b^2 e^2\right )\right ) (d+e x)^{4+m}}{e^6 (4+m)}-\frac {c (5 B c d-2 b B e-A c e) (d+e x)^{5+m}}{e^6 (5+m)}+\frac {B c^2 (d+e x)^{6+m}}{e^6 (6+m)}\\ \end {align*}
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Mathematica [A] time = 0.44, size = 309, normalized size = 1.10 \begin {gather*} \frac {(d+e x)^{m+1} \left (A e \left (\frac {(d+e x)^2 \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )}{m+3}+\frac {d^2 (c d-b e)^2}{m+1}-\frac {2 c (d+e x)^3 (2 c d-b e)}{m+4}-\frac {2 d (d+e x) (c d-b e) (2 c d-b e)}{m+2}+\frac {c^2 (d+e x)^4}{m+5}\right )+B \left (\frac {(d+e x)^3 \left (b^2 e^2-8 b c d e+10 c^2 d^2\right )}{m+4}-\frac {d (d+e x)^2 \left (3 b^2 e^2-12 b c d e+10 c^2 d^2\right )}{m+3}-\frac {d^3 (c d-b e)^2}{m+1}+\frac {d^2 (d+e x) (5 c d-3 b e) (c d-b e)}{m+2}-\frac {c (d+e x)^4 (5 c d-2 b e)}{m+5}+\frac {c^2 (d+e x)^5}{m+6}\right )\right )}{e^6} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(A + B*x)*(d + e*x)^m*(b*x + c*x^2)^2,x]
[Out]
((d + e*x)^(1 + m)*(A*e*((d^2*(c*d - b*e)^2)/(1 + m) - (2*d*(c*d - b*e)*(2*c*d - b*e)*(d + e*x))/(2 + m) + ((6
*c^2*d^2 - 6*b*c*d*e + b^2*e^2)*(d + e*x)^2)/(3 + m) - (2*c*(2*c*d - b*e)*(d + e*x)^3)/(4 + m) + (c^2*(d + e*x
)^4)/(5 + m)) + B*(-((d^3*(c*d - b*e)^2)/(1 + m)) + (d^2*(5*c*d - 3*b*e)*(c*d - b*e)*(d + e*x))/(2 + m) - (d*(
10*c^2*d^2 - 12*b*c*d*e + 3*b^2*e^2)*(d + e*x)^2)/(3 + m) + ((10*c^2*d^2 - 8*b*c*d*e + b^2*e^2)*(d + e*x)^3)/(
4 + m) - (c*(5*c*d - 2*b*e)*(d + e*x)^4)/(5 + m) + (c^2*(d + e*x)^5)/(6 + m))))/e^6
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IntegrateAlgebraic [F] time = 0.14, size = 0, normalized size = 0.00 \begin {gather*} \int (A+B x) (d+e x)^m \left (b x+c x^2\right )^2 \, dx \end {gather*}
Verification is not applicable to the result.
[In]
IntegrateAlgebraic[(A + B*x)*(d + e*x)^m*(b*x + c*x^2)^2,x]
[Out]
Defer[IntegrateAlgebraic][(A + B*x)*(d + e*x)^m*(b*x + c*x^2)^2, x]
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fricas [B] time = 0.46, size = 1417, normalized size = 5.02
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((B*x+A)*(e*x+d)^m*(c*x^2+b*x)^2,x, algorithm="fricas")
[Out]
(2*A*b^2*d^3*e^3*m^3 - 120*B*c^2*d^6 + 240*A*b^2*d^3*e^3 + 144*(2*B*b*c + A*c^2)*d^5*e - 180*(B*b^2 + 2*A*b*c)
*d^4*e^2 + (B*c^2*e^6*m^5 + 15*B*c^2*e^6*m^4 + 85*B*c^2*e^6*m^3 + 225*B*c^2*e^6*m^2 + 274*B*c^2*e^6*m + 120*B*
c^2*e^6)*x^6 + (144*(2*B*b*c + A*c^2)*e^6 + (B*c^2*d*e^5 + (2*B*b*c + A*c^2)*e^6)*m^5 + 2*(5*B*c^2*d*e^5 + 8*(
2*B*b*c + A*c^2)*e^6)*m^4 + 5*(7*B*c^2*d*e^5 + 19*(2*B*b*c + A*c^2)*e^6)*m^3 + 10*(5*B*c^2*d*e^5 + 26*(2*B*b*c
+ A*c^2)*e^6)*m^2 + 12*(2*B*c^2*d*e^5 + 27*(2*B*b*c + A*c^2)*e^6)*m)*x^5 + (180*(B*b^2 + 2*A*b*c)*e^6 + ((2*B
*b*c + A*c^2)*d*e^5 + (B*b^2 + 2*A*b*c)*e^6)*m^5 - (5*B*c^2*d^2*e^4 - 12*(2*B*b*c + A*c^2)*d*e^5 - 17*(B*b^2 +
2*A*b*c)*e^6)*m^4 - (30*B*c^2*d^2*e^4 - 47*(2*B*b*c + A*c^2)*d*e^5 - 107*(B*b^2 + 2*A*b*c)*e^6)*m^3 - (55*B*c
^2*d^2*e^4 - 72*(2*B*b*c + A*c^2)*d*e^5 - 307*(B*b^2 + 2*A*b*c)*e^6)*m^2 - 6*(5*B*c^2*d^2*e^4 - 6*(2*B*b*c + A
*c^2)*d*e^5 - 66*(B*b^2 + 2*A*b*c)*e^6)*m)*x^4 + (240*A*b^2*e^6 + (A*b^2*e^6 + (B*b^2 + 2*A*b*c)*d*e^5)*m^5 +
2*(9*A*b^2*e^6 - 2*(2*B*b*c + A*c^2)*d^2*e^4 + 7*(B*b^2 + 2*A*b*c)*d*e^5)*m^4 + (20*B*c^2*d^3*e^3 + 121*A*b^2*
e^6 - 36*(2*B*b*c + A*c^2)*d^2*e^4 + 65*(B*b^2 + 2*A*b*c)*d*e^5)*m^3 + 4*(15*B*c^2*d^3*e^3 + 93*A*b^2*e^6 - 20
*(2*B*b*c + A*c^2)*d^2*e^4 + 28*(B*b^2 + 2*A*b*c)*d*e^5)*m^2 + 4*(10*B*c^2*d^3*e^3 + 127*A*b^2*e^6 - 12*(2*B*b
*c + A*c^2)*d^2*e^4 + 15*(B*b^2 + 2*A*b*c)*d*e^5)*m)*x^3 + 6*(5*A*b^2*d^3*e^3 - (B*b^2 + 2*A*b*c)*d^4*e^2)*m^2
+ (A*b^2*d*e^5*m^5 + (16*A*b^2*d*e^5 - 3*(B*b^2 + 2*A*b*c)*d^2*e^4)*m^4 + (89*A*b^2*d*e^5 + 12*(2*B*b*c + A*c
^2)*d^3*e^3 - 36*(B*b^2 + 2*A*b*c)*d^2*e^4)*m^3 - (60*B*c^2*d^4*e^2 - 194*A*b^2*d*e^5 - 84*(2*B*b*c + A*c^2)*d
^3*e^3 + 123*(B*b^2 + 2*A*b*c)*d^2*e^4)*m^2 - 6*(10*B*c^2*d^4*e^2 - 20*A*b^2*d*e^5 - 12*(2*B*b*c + A*c^2)*d^3*
e^3 + 15*(B*b^2 + 2*A*b*c)*d^2*e^4)*m)*x^2 + 2*(74*A*b^2*d^3*e^3 + 12*(2*B*b*c + A*c^2)*d^5*e - 33*(B*b^2 + 2*
A*b*c)*d^4*e^2)*m - 2*(A*b^2*d^2*e^4*m^4 + 3*(5*A*b^2*d^2*e^4 - (B*b^2 + 2*A*b*c)*d^3*e^3)*m^3 + (74*A*b^2*d^2
*e^4 + 12*(2*B*b*c + A*c^2)*d^4*e^2 - 33*(B*b^2 + 2*A*b*c)*d^3*e^3)*m^2 - 6*(10*B*c^2*d^5*e - 20*A*b^2*d^2*e^4
- 12*(2*B*b*c + A*c^2)*d^4*e^2 + 15*(B*b^2 + 2*A*b*c)*d^3*e^3)*m)*x)*(e*x + d)^m/(e^6*m^6 + 21*e^6*m^5 + 175*
e^6*m^4 + 735*e^6*m^3 + 1624*e^6*m^2 + 1764*e^6*m + 720*e^6)
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giac [B] time = 0.27, size = 2827, normalized size = 10.02
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((B*x+A)*(e*x+d)^m*(c*x^2+b*x)^2,x, algorithm="giac")
[Out]
((x*e + d)^m*B*c^2*m^5*x^6*e^6 + (x*e + d)^m*B*c^2*d*m^5*x^5*e^5 + 2*(x*e + d)^m*B*b*c*m^5*x^5*e^6 + (x*e + d)
^m*A*c^2*m^5*x^5*e^6 + 15*(x*e + d)^m*B*c^2*m^4*x^6*e^6 + 2*(x*e + d)^m*B*b*c*d*m^5*x^4*e^5 + (x*e + d)^m*A*c^
2*d*m^5*x^4*e^5 + 10*(x*e + d)^m*B*c^2*d*m^4*x^5*e^5 - 5*(x*e + d)^m*B*c^2*d^2*m^4*x^4*e^4 + (x*e + d)^m*B*b^2
*m^5*x^4*e^6 + 2*(x*e + d)^m*A*b*c*m^5*x^4*e^6 + 32*(x*e + d)^m*B*b*c*m^4*x^5*e^6 + 16*(x*e + d)^m*A*c^2*m^4*x
^5*e^6 + 85*(x*e + d)^m*B*c^2*m^3*x^6*e^6 + (x*e + d)^m*B*b^2*d*m^5*x^3*e^5 + 2*(x*e + d)^m*A*b*c*d*m^5*x^3*e^
5 + 24*(x*e + d)^m*B*b*c*d*m^4*x^4*e^5 + 12*(x*e + d)^m*A*c^2*d*m^4*x^4*e^5 + 35*(x*e + d)^m*B*c^2*d*m^3*x^5*e
^5 - 8*(x*e + d)^m*B*b*c*d^2*m^4*x^3*e^4 - 4*(x*e + d)^m*A*c^2*d^2*m^4*x^3*e^4 - 30*(x*e + d)^m*B*c^2*d^2*m^3*
x^4*e^4 + 20*(x*e + d)^m*B*c^2*d^3*m^3*x^3*e^3 + (x*e + d)^m*A*b^2*m^5*x^3*e^6 + 17*(x*e + d)^m*B*b^2*m^4*x^4*
e^6 + 34*(x*e + d)^m*A*b*c*m^4*x^4*e^6 + 190*(x*e + d)^m*B*b*c*m^3*x^5*e^6 + 95*(x*e + d)^m*A*c^2*m^3*x^5*e^6
+ 225*(x*e + d)^m*B*c^2*m^2*x^6*e^6 + (x*e + d)^m*A*b^2*d*m^5*x^2*e^5 + 14*(x*e + d)^m*B*b^2*d*m^4*x^3*e^5 + 2
8*(x*e + d)^m*A*b*c*d*m^4*x^3*e^5 + 94*(x*e + d)^m*B*b*c*d*m^3*x^4*e^5 + 47*(x*e + d)^m*A*c^2*d*m^3*x^4*e^5 +
50*(x*e + d)^m*B*c^2*d*m^2*x^5*e^5 - 3*(x*e + d)^m*B*b^2*d^2*m^4*x^2*e^4 - 6*(x*e + d)^m*A*b*c*d^2*m^4*x^2*e^4
- 72*(x*e + d)^m*B*b*c*d^2*m^3*x^3*e^4 - 36*(x*e + d)^m*A*c^2*d^2*m^3*x^3*e^4 - 55*(x*e + d)^m*B*c^2*d^2*m^2*
x^4*e^4 + 24*(x*e + d)^m*B*b*c*d^3*m^3*x^2*e^3 + 12*(x*e + d)^m*A*c^2*d^3*m^3*x^2*e^3 + 60*(x*e + d)^m*B*c^2*d
^3*m^2*x^3*e^3 - 60*(x*e + d)^m*B*c^2*d^4*m^2*x^2*e^2 + 18*(x*e + d)^m*A*b^2*m^4*x^3*e^6 + 107*(x*e + d)^m*B*b
^2*m^3*x^4*e^6 + 214*(x*e + d)^m*A*b*c*m^3*x^4*e^6 + 520*(x*e + d)^m*B*b*c*m^2*x^5*e^6 + 260*(x*e + d)^m*A*c^2
*m^2*x^5*e^6 + 274*(x*e + d)^m*B*c^2*m*x^6*e^6 + 16*(x*e + d)^m*A*b^2*d*m^4*x^2*e^5 + 65*(x*e + d)^m*B*b^2*d*m
^3*x^3*e^5 + 130*(x*e + d)^m*A*b*c*d*m^3*x^3*e^5 + 144*(x*e + d)^m*B*b*c*d*m^2*x^4*e^5 + 72*(x*e + d)^m*A*c^2*
d*m^2*x^4*e^5 + 24*(x*e + d)^m*B*c^2*d*m*x^5*e^5 - 2*(x*e + d)^m*A*b^2*d^2*m^4*x*e^4 - 36*(x*e + d)^m*B*b^2*d^
2*m^3*x^2*e^4 - 72*(x*e + d)^m*A*b*c*d^2*m^3*x^2*e^4 - 160*(x*e + d)^m*B*b*c*d^2*m^2*x^3*e^4 - 80*(x*e + d)^m*
A*c^2*d^2*m^2*x^3*e^4 - 30*(x*e + d)^m*B*c^2*d^2*m*x^4*e^4 + 6*(x*e + d)^m*B*b^2*d^3*m^3*x*e^3 + 12*(x*e + d)^
m*A*b*c*d^3*m^3*x*e^3 + 168*(x*e + d)^m*B*b*c*d^3*m^2*x^2*e^3 + 84*(x*e + d)^m*A*c^2*d^3*m^2*x^2*e^3 + 40*(x*e
+ d)^m*B*c^2*d^3*m*x^3*e^3 - 48*(x*e + d)^m*B*b*c*d^4*m^2*x*e^2 - 24*(x*e + d)^m*A*c^2*d^4*m^2*x*e^2 - 60*(x*
e + d)^m*B*c^2*d^4*m*x^2*e^2 + 120*(x*e + d)^m*B*c^2*d^5*m*x*e + 121*(x*e + d)^m*A*b^2*m^3*x^3*e^6 + 307*(x*e
+ d)^m*B*b^2*m^2*x^4*e^6 + 614*(x*e + d)^m*A*b*c*m^2*x^4*e^6 + 648*(x*e + d)^m*B*b*c*m*x^5*e^6 + 324*(x*e + d)
^m*A*c^2*m*x^5*e^6 + 120*(x*e + d)^m*B*c^2*x^6*e^6 + 89*(x*e + d)^m*A*b^2*d*m^3*x^2*e^5 + 112*(x*e + d)^m*B*b^
2*d*m^2*x^3*e^5 + 224*(x*e + d)^m*A*b*c*d*m^2*x^3*e^5 + 72*(x*e + d)^m*B*b*c*d*m*x^4*e^5 + 36*(x*e + d)^m*A*c^
2*d*m*x^4*e^5 - 30*(x*e + d)^m*A*b^2*d^2*m^3*x*e^4 - 123*(x*e + d)^m*B*b^2*d^2*m^2*x^2*e^4 - 246*(x*e + d)^m*A
*b*c*d^2*m^2*x^2*e^4 - 96*(x*e + d)^m*B*b*c*d^2*m*x^3*e^4 - 48*(x*e + d)^m*A*c^2*d^2*m*x^3*e^4 + 2*(x*e + d)^m
*A*b^2*d^3*m^3*e^3 + 66*(x*e + d)^m*B*b^2*d^3*m^2*x*e^3 + 132*(x*e + d)^m*A*b*c*d^3*m^2*x*e^3 + 144*(x*e + d)^
m*B*b*c*d^3*m*x^2*e^3 + 72*(x*e + d)^m*A*c^2*d^3*m*x^2*e^3 - 6*(x*e + d)^m*B*b^2*d^4*m^2*e^2 - 12*(x*e + d)^m*
A*b*c*d^4*m^2*e^2 - 288*(x*e + d)^m*B*b*c*d^4*m*x*e^2 - 144*(x*e + d)^m*A*c^2*d^4*m*x*e^2 + 48*(x*e + d)^m*B*b
*c*d^5*m*e + 24*(x*e + d)^m*A*c^2*d^5*m*e - 120*(x*e + d)^m*B*c^2*d^6 + 372*(x*e + d)^m*A*b^2*m^2*x^3*e^6 + 39
6*(x*e + d)^m*B*b^2*m*x^4*e^6 + 792*(x*e + d)^m*A*b*c*m*x^4*e^6 + 288*(x*e + d)^m*B*b*c*x^5*e^6 + 144*(x*e + d
)^m*A*c^2*x^5*e^6 + 194*(x*e + d)^m*A*b^2*d*m^2*x^2*e^5 + 60*(x*e + d)^m*B*b^2*d*m*x^3*e^5 + 120*(x*e + d)^m*A
*b*c*d*m*x^3*e^5 - 148*(x*e + d)^m*A*b^2*d^2*m^2*x*e^4 - 90*(x*e + d)^m*B*b^2*d^2*m*x^2*e^4 - 180*(x*e + d)^m*
A*b*c*d^2*m*x^2*e^4 + 30*(x*e + d)^m*A*b^2*d^3*m^2*e^3 + 180*(x*e + d)^m*B*b^2*d^3*m*x*e^3 + 360*(x*e + d)^m*A
*b*c*d^3*m*x*e^3 - 66*(x*e + d)^m*B*b^2*d^4*m*e^2 - 132*(x*e + d)^m*A*b*c*d^4*m*e^2 + 288*(x*e + d)^m*B*b*c*d^
5*e + 144*(x*e + d)^m*A*c^2*d^5*e + 508*(x*e + d)^m*A*b^2*m*x^3*e^6 + 180*(x*e + d)^m*B*b^2*x^4*e^6 + 360*(x*e
+ d)^m*A*b*c*x^4*e^6 + 120*(x*e + d)^m*A*b^2*d*m*x^2*e^5 - 240*(x*e + d)^m*A*b^2*d^2*m*x*e^4 + 148*(x*e + d)^
m*A*b^2*d^3*m*e^3 - 180*(x*e + d)^m*B*b^2*d^4*e^2 - 360*(x*e + d)^m*A*b*c*d^4*e^2 + 240*(x*e + d)^m*A*b^2*x^3*
e^6 + 240*(x*e + d)^m*A*b^2*d^3*e^3)/(m^6*e^6 + 21*m^5*e^6 + 175*m^4*e^6 + 735*m^3*e^6 + 1624*m^2*e^6 + 1764*m
*e^6 + 720*e^6)
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maple [B] time = 0.06, size = 1616, normalized size = 5.73
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((B*x+A)*(e*x+d)^m*(c*x^2+b*x)^2,x)
[Out]
(e*x+d)^(m+1)*(B*c^2*e^5*m^5*x^5+A*c^2*e^5*m^5*x^4+2*B*b*c*e^5*m^5*x^4+15*B*c^2*e^5*m^4*x^5+2*A*b*c*e^5*m^5*x^
3+16*A*c^2*e^5*m^4*x^4+B*b^2*e^5*m^5*x^3+32*B*b*c*e^5*m^4*x^4-5*B*c^2*d*e^4*m^4*x^4+85*B*c^2*e^5*m^3*x^5+A*b^2
*e^5*m^5*x^2+34*A*b*c*e^5*m^4*x^3-4*A*c^2*d*e^4*m^4*x^3+95*A*c^2*e^5*m^3*x^4+17*B*b^2*e^5*m^4*x^3-8*B*b*c*d*e^
4*m^4*x^3+190*B*b*c*e^5*m^3*x^4-50*B*c^2*d*e^4*m^3*x^4+225*B*c^2*e^5*m^2*x^5+18*A*b^2*e^5*m^4*x^2-6*A*b*c*d*e^
4*m^4*x^2+214*A*b*c*e^5*m^3*x^3-48*A*c^2*d*e^4*m^3*x^3+260*A*c^2*e^5*m^2*x^4-3*B*b^2*d*e^4*m^4*x^2+107*B*b^2*e
^5*m^3*x^3-96*B*b*c*d*e^4*m^3*x^3+520*B*b*c*e^5*m^2*x^4+20*B*c^2*d^2*e^3*m^3*x^3-175*B*c^2*d*e^4*m^2*x^4+274*B
*c^2*e^5*m*x^5-2*A*b^2*d*e^4*m^4*x+121*A*b^2*e^5*m^3*x^2-84*A*b*c*d*e^4*m^3*x^2+614*A*b*c*e^5*m^2*x^3+12*A*c^2
*d^2*e^3*m^3*x^2-188*A*c^2*d*e^4*m^2*x^3+324*A*c^2*e^5*m*x^4-42*B*b^2*d*e^4*m^3*x^2+307*B*b^2*e^5*m^2*x^3+24*B
*b*c*d^2*e^3*m^3*x^2-376*B*b*c*d*e^4*m^2*x^3+648*B*b*c*e^5*m*x^4+120*B*c^2*d^2*e^3*m^2*x^3-250*B*c^2*d*e^4*m*x
^4+120*B*c^2*e^5*x^5-32*A*b^2*d*e^4*m^3*x+372*A*b^2*e^5*m^2*x^2+12*A*b*c*d^2*e^3*m^3*x-390*A*b*c*d*e^4*m^2*x^2
+792*A*b*c*e^5*m*x^3+108*A*c^2*d^2*e^3*m^2*x^2-288*A*c^2*d*e^4*m*x^3+144*A*c^2*e^5*x^4+6*B*b^2*d^2*e^3*m^3*x-1
95*B*b^2*d*e^4*m^2*x^2+396*B*b^2*e^5*m*x^3+216*B*b*c*d^2*e^3*m^2*x^2-576*B*b*c*d*e^4*m*x^3+288*B*b*c*e^5*x^4-6
0*B*c^2*d^3*e^2*m^2*x^2+220*B*c^2*d^2*e^3*m*x^3-120*B*c^2*d*e^4*x^4+2*A*b^2*d^2*e^3*m^3-178*A*b^2*d*e^4*m^2*x+
508*A*b^2*e^5*m*x^2+144*A*b*c*d^2*e^3*m^2*x-672*A*b*c*d*e^4*m*x^2+360*A*b*c*e^5*x^3-24*A*c^2*d^3*e^2*m^2*x+240
*A*c^2*d^2*e^3*m*x^2-144*A*c^2*d*e^4*x^3+72*B*b^2*d^2*e^3*m^2*x-336*B*b^2*d*e^4*m*x^2+180*B*b^2*e^5*x^3-48*B*b
*c*d^3*e^2*m^2*x+480*B*b*c*d^2*e^3*m*x^2-288*B*b*c*d*e^4*x^3-180*B*c^2*d^3*e^2*m*x^2+120*B*c^2*d^2*e^3*x^3+30*
A*b^2*d^2*e^3*m^2-388*A*b^2*d*e^4*m*x+240*A*b^2*e^5*x^2-12*A*b*c*d^3*e^2*m^2+492*A*b*c*d^2*e^3*m*x-360*A*b*c*d
*e^4*x^2-168*A*c^2*d^3*e^2*m*x+144*A*c^2*d^2*e^3*x^2-6*B*b^2*d^3*e^2*m^2+246*B*b^2*d^2*e^3*m*x-180*B*b^2*d*e^4
*x^2-336*B*b*c*d^3*e^2*m*x+288*B*b*c*d^2*e^3*x^2+120*B*c^2*d^4*e*m*x-120*B*c^2*d^3*e^2*x^2+148*A*b^2*d^2*e^3*m
-240*A*b^2*d*e^4*x-132*A*b*c*d^3*e^2*m+360*A*b*c*d^2*e^3*x+24*A*c^2*d^4*e*m-144*A*c^2*d^3*e^2*x-66*B*b^2*d^3*e
^2*m+180*B*b^2*d^2*e^3*x+48*B*b*c*d^4*e*m-288*B*b*c*d^3*e^2*x+120*B*c^2*d^4*e*x+240*A*b^2*d^2*e^3-360*A*b*c*d^
3*e^2+144*A*c^2*d^4*e-180*B*b^2*d^3*e^2+288*B*b*c*d^4*e-120*B*c^2*d^5)/e^6/(m^6+21*m^5+175*m^4+735*m^3+1624*m^
2+1764*m+720)
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maxima [B] time = 0.76, size = 755, normalized size = 2.68 \begin {gather*} \frac {{\left ({\left (m^{2} + 3 \, m + 2\right )} e^{3} x^{3} + {\left (m^{2} + m\right )} d e^{2} x^{2} - 2 \, d^{2} e m x + 2 \, d^{3}\right )} {\left (e x + d\right )}^{m} A b^{2}}{{\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} e^{3}} + \frac {{\left ({\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} e^{4} x^{4} + {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} d e^{3} x^{3} - 3 \, {\left (m^{2} + m\right )} d^{2} e^{2} x^{2} + 6 \, d^{3} e m x - 6 \, d^{4}\right )} {\left (e x + d\right )}^{m} B b^{2}}{{\left (m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24\right )} e^{4}} + \frac {2 \, {\left ({\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} e^{4} x^{4} + {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} d e^{3} x^{3} - 3 \, {\left (m^{2} + m\right )} d^{2} e^{2} x^{2} + 6 \, d^{3} e m x - 6 \, d^{4}\right )} {\left (e x + d\right )}^{m} A b c}{{\left (m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24\right )} e^{4}} + \frac {2 \, {\left ({\left (m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24\right )} e^{5} x^{5} + {\left (m^{4} + 6 \, m^{3} + 11 \, m^{2} + 6 \, m\right )} d e^{4} x^{4} - 4 \, {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} d^{2} e^{3} x^{3} + 12 \, {\left (m^{2} + m\right )} d^{3} e^{2} x^{2} - 24 \, d^{4} e m x + 24 \, d^{5}\right )} {\left (e x + d\right )}^{m} B b c}{{\left (m^{5} + 15 \, m^{4} + 85 \, m^{3} + 225 \, m^{2} + 274 \, m + 120\right )} e^{5}} + \frac {{\left ({\left (m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24\right )} e^{5} x^{5} + {\left (m^{4} + 6 \, m^{3} + 11 \, m^{2} + 6 \, m\right )} d e^{4} x^{4} - 4 \, {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} d^{2} e^{3} x^{3} + 12 \, {\left (m^{2} + m\right )} d^{3} e^{2} x^{2} - 24 \, d^{4} e m x + 24 \, d^{5}\right )} {\left (e x + d\right )}^{m} A c^{2}}{{\left (m^{5} + 15 \, m^{4} + 85 \, m^{3} + 225 \, m^{2} + 274 \, m + 120\right )} e^{5}} + \frac {{\left ({\left (m^{5} + 15 \, m^{4} + 85 \, m^{3} + 225 \, m^{2} + 274 \, m + 120\right )} e^{6} x^{6} + {\left (m^{5} + 10 \, m^{4} + 35 \, m^{3} + 50 \, m^{2} + 24 \, m\right )} d e^{5} x^{5} - 5 \, {\left (m^{4} + 6 \, m^{3} + 11 \, m^{2} + 6 \, m\right )} d^{2} e^{4} x^{4} + 20 \, {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} d^{3} e^{3} x^{3} - 60 \, {\left (m^{2} + m\right )} d^{4} e^{2} x^{2} + 120 \, d^{5} e m x - 120 \, d^{6}\right )} {\left (e x + d\right )}^{m} B c^{2}}{{\left (m^{6} + 21 \, m^{5} + 175 \, m^{4} + 735 \, m^{3} + 1624 \, m^{2} + 1764 \, m + 720\right )} e^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((B*x+A)*(e*x+d)^m*(c*x^2+b*x)^2,x, algorithm="maxima")
[Out]
((m^2 + 3*m + 2)*e^3*x^3 + (m^2 + m)*d*e^2*x^2 - 2*d^2*e*m*x + 2*d^3)*(e*x + d)^m*A*b^2/((m^3 + 6*m^2 + 11*m +
6)*e^3) + ((m^3 + 6*m^2 + 11*m + 6)*e^4*x^4 + (m^3 + 3*m^2 + 2*m)*d*e^3*x^3 - 3*(m^2 + m)*d^2*e^2*x^2 + 6*d^3
*e*m*x - 6*d^4)*(e*x + d)^m*B*b^2/((m^4 + 10*m^3 + 35*m^2 + 50*m + 24)*e^4) + 2*((m^3 + 6*m^2 + 11*m + 6)*e^4*
x^4 + (m^3 + 3*m^2 + 2*m)*d*e^3*x^3 - 3*(m^2 + m)*d^2*e^2*x^2 + 6*d^3*e*m*x - 6*d^4)*(e*x + d)^m*A*b*c/((m^4 +
10*m^3 + 35*m^2 + 50*m + 24)*e^4) + 2*((m^4 + 10*m^3 + 35*m^2 + 50*m + 24)*e^5*x^5 + (m^4 + 6*m^3 + 11*m^2 +
6*m)*d*e^4*x^4 - 4*(m^3 + 3*m^2 + 2*m)*d^2*e^3*x^3 + 12*(m^2 + m)*d^3*e^2*x^2 - 24*d^4*e*m*x + 24*d^5)*(e*x +
d)^m*B*b*c/((m^5 + 15*m^4 + 85*m^3 + 225*m^2 + 274*m + 120)*e^5) + ((m^4 + 10*m^3 + 35*m^2 + 50*m + 24)*e^5*x^
5 + (m^4 + 6*m^3 + 11*m^2 + 6*m)*d*e^4*x^4 - 4*(m^3 + 3*m^2 + 2*m)*d^2*e^3*x^3 + 12*(m^2 + m)*d^3*e^2*x^2 - 24
*d^4*e*m*x + 24*d^5)*(e*x + d)^m*A*c^2/((m^5 + 15*m^4 + 85*m^3 + 225*m^2 + 274*m + 120)*e^5) + ((m^5 + 15*m^4
+ 85*m^3 + 225*m^2 + 274*m + 120)*e^6*x^6 + (m^5 + 10*m^4 + 35*m^3 + 50*m^2 + 24*m)*d*e^5*x^5 - 5*(m^4 + 6*m^3
+ 11*m^2 + 6*m)*d^2*e^4*x^4 + 20*(m^3 + 3*m^2 + 2*m)*d^3*e^3*x^3 - 60*(m^2 + m)*d^4*e^2*x^2 + 120*d^5*e*m*x -
120*d^6)*(e*x + d)^m*B*c^2/((m^6 + 21*m^5 + 175*m^4 + 735*m^3 + 1624*m^2 + 1764*m + 720)*e^6)
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mupad [B] time = 2.10, size = 1176, normalized size = 4.17 \begin {gather*} \frac {{\left (d+e\,x\right )}^m\,\left (-6\,B\,b^2\,d^4\,e^2\,m^2-66\,B\,b^2\,d^4\,e^2\,m-180\,B\,b^2\,d^4\,e^2+2\,A\,b^2\,d^3\,e^3\,m^3+30\,A\,b^2\,d^3\,e^3\,m^2+148\,A\,b^2\,d^3\,e^3\,m+240\,A\,b^2\,d^3\,e^3+48\,B\,b\,c\,d^5\,e\,m+288\,B\,b\,c\,d^5\,e-12\,A\,b\,c\,d^4\,e^2\,m^2-132\,A\,b\,c\,d^4\,e^2\,m-360\,A\,b\,c\,d^4\,e^2-120\,B\,c^2\,d^6+24\,A\,c^2\,d^5\,e\,m+144\,A\,c^2\,d^5\,e\right )}{e^6\,\left (m^6+21\,m^5+175\,m^4+735\,m^3+1624\,m^2+1764\,m+720\right )}+\frac {x^3\,{\left (d+e\,x\right )}^m\,\left (m^2+3\,m+2\right )\,\left (B\,b^2\,d\,e^2\,m^3+11\,B\,b^2\,d\,e^2\,m^2+30\,B\,b^2\,d\,e^2\,m+A\,b^2\,e^3\,m^3+15\,A\,b^2\,e^3\,m^2+74\,A\,b^2\,e^3\,m+120\,A\,b^2\,e^3-8\,B\,b\,c\,d^2\,e\,m^2-48\,B\,b\,c\,d^2\,e\,m+2\,A\,b\,c\,d\,e^2\,m^3+22\,A\,b\,c\,d\,e^2\,m^2+60\,A\,b\,c\,d\,e^2\,m+20\,B\,c^2\,d^3\,m-4\,A\,c^2\,d^2\,e\,m^2-24\,A\,c^2\,d^2\,e\,m\right )}{e^3\,\left (m^6+21\,m^5+175\,m^4+735\,m^3+1624\,m^2+1764\,m+720\right )}+\frac {x^4\,{\left (d+e\,x\right )}^m\,\left (m^3+6\,m^2+11\,m+6\right )\,\left (B\,b^2\,e^2\,m^2+11\,B\,b^2\,e^2\,m+30\,B\,b^2\,e^2+2\,B\,b\,c\,d\,e\,m^2+12\,B\,b\,c\,d\,e\,m+2\,A\,b\,c\,e^2\,m^2+22\,A\,b\,c\,e^2\,m+60\,A\,b\,c\,e^2-5\,B\,c^2\,d^2\,m+A\,c^2\,d\,e\,m^2+6\,A\,c^2\,d\,e\,m\right )}{e^2\,\left (m^6+21\,m^5+175\,m^4+735\,m^3+1624\,m^2+1764\,m+720\right )}+\frac {B\,c^2\,x^6\,{\left (d+e\,x\right )}^m\,\left (m^5+15\,m^4+85\,m^3+225\,m^2+274\,m+120\right )}{m^6+21\,m^5+175\,m^4+735\,m^3+1624\,m^2+1764\,m+720}+\frac {c\,x^5\,{\left (d+e\,x\right )}^m\,\left (m^4+10\,m^3+35\,m^2+50\,m+24\right )\,\left (6\,A\,c\,e+12\,B\,b\,e+A\,c\,e\,m+2\,B\,b\,e\,m+B\,c\,d\,m\right )}{e\,\left (m^6+21\,m^5+175\,m^4+735\,m^3+1624\,m^2+1764\,m+720\right )}-\frac {2\,d^2\,m\,x\,{\left (d+e\,x\right )}^m\,\left (-3\,B\,b^2\,d\,e^2\,m^2-33\,B\,b^2\,d\,e^2\,m-90\,B\,b^2\,d\,e^2+A\,b^2\,e^3\,m^3+15\,A\,b^2\,e^3\,m^2+74\,A\,b^2\,e^3\,m+120\,A\,b^2\,e^3+24\,B\,b\,c\,d^2\,e\,m+144\,B\,b\,c\,d^2\,e-6\,A\,b\,c\,d\,e^2\,m^2-66\,A\,b\,c\,d\,e^2\,m-180\,A\,b\,c\,d\,e^2-60\,B\,c^2\,d^3+12\,A\,c^2\,d^2\,e\,m+72\,A\,c^2\,d^2\,e\right )}{e^5\,\left (m^6+21\,m^5+175\,m^4+735\,m^3+1624\,m^2+1764\,m+720\right )}+\frac {d\,m\,x^2\,\left (m+1\right )\,{\left (d+e\,x\right )}^m\,\left (-3\,B\,b^2\,d\,e^2\,m^2-33\,B\,b^2\,d\,e^2\,m-90\,B\,b^2\,d\,e^2+A\,b^2\,e^3\,m^3+15\,A\,b^2\,e^3\,m^2+74\,A\,b^2\,e^3\,m+120\,A\,b^2\,e^3+24\,B\,b\,c\,d^2\,e\,m+144\,B\,b\,c\,d^2\,e-6\,A\,b\,c\,d\,e^2\,m^2-66\,A\,b\,c\,d\,e^2\,m-180\,A\,b\,c\,d\,e^2-60\,B\,c^2\,d^3+12\,A\,c^2\,d^2\,e\,m+72\,A\,c^2\,d^2\,e\right )}{e^4\,\left (m^6+21\,m^5+175\,m^4+735\,m^3+1624\,m^2+1764\,m+720\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((b*x + c*x^2)^2*(A + B*x)*(d + e*x)^m,x)
[Out]
((d + e*x)^m*(144*A*c^2*d^5*e - 120*B*c^2*d^6 + 240*A*b^2*d^3*e^3 - 180*B*b^2*d^4*e^2 + 148*A*b^2*d^3*e^3*m -
66*B*b^2*d^4*e^2*m + 288*B*b*c*d^5*e + 30*A*b^2*d^3*e^3*m^2 + 2*A*b^2*d^3*e^3*m^3 - 6*B*b^2*d^4*e^2*m^2 - 360*
A*b*c*d^4*e^2 + 24*A*c^2*d^5*e*m - 132*A*b*c*d^4*e^2*m - 12*A*b*c*d^4*e^2*m^2 + 48*B*b*c*d^5*e*m))/(e^6*(1764*
m + 1624*m^2 + 735*m^3 + 175*m^4 + 21*m^5 + m^6 + 720)) + (x^3*(d + e*x)^m*(3*m + m^2 + 2)*(120*A*b^2*e^3 + 74
*A*b^2*e^3*m + 20*B*c^2*d^3*m + 15*A*b^2*e^3*m^2 + A*b^2*e^3*m^3 - 4*A*c^2*d^2*e*m^2 + 11*B*b^2*d*e^2*m^2 + B*
b^2*d*e^2*m^3 - 24*A*c^2*d^2*e*m + 30*B*b^2*d*e^2*m + 22*A*b*c*d*e^2*m^2 + 2*A*b*c*d*e^2*m^3 - 8*B*b*c*d^2*e*m
^2 + 60*A*b*c*d*e^2*m - 48*B*b*c*d^2*e*m))/(e^3*(1764*m + 1624*m^2 + 735*m^3 + 175*m^4 + 21*m^5 + m^6 + 720))
+ (x^4*(d + e*x)^m*(11*m + 6*m^2 + m^3 + 6)*(30*B*b^2*e^2 + 60*A*b*c*e^2 + 11*B*b^2*e^2*m - 5*B*c^2*d^2*m + B*
b^2*e^2*m^2 + 22*A*b*c*e^2*m + 6*A*c^2*d*e*m + 2*A*b*c*e^2*m^2 + A*c^2*d*e*m^2 + 12*B*b*c*d*e*m + 2*B*b*c*d*e*
m^2))/(e^2*(1764*m + 1624*m^2 + 735*m^3 + 175*m^4 + 21*m^5 + m^6 + 720)) + (B*c^2*x^6*(d + e*x)^m*(274*m + 225
*m^2 + 85*m^3 + 15*m^4 + m^5 + 120))/(1764*m + 1624*m^2 + 735*m^3 + 175*m^4 + 21*m^5 + m^6 + 720) + (c*x^5*(d
+ e*x)^m*(50*m + 35*m^2 + 10*m^3 + m^4 + 24)*(6*A*c*e + 12*B*b*e + A*c*e*m + 2*B*b*e*m + B*c*d*m))/(e*(1764*m
+ 1624*m^2 + 735*m^3 + 175*m^4 + 21*m^5 + m^6 + 720)) - (2*d^2*m*x*(d + e*x)^m*(120*A*b^2*e^3 - 60*B*c^2*d^3 +
72*A*c^2*d^2*e - 90*B*b^2*d*e^2 + 74*A*b^2*e^3*m + 15*A*b^2*e^3*m^2 + A*b^2*e^3*m^3 - 3*B*b^2*d*e^2*m^2 - 180
*A*b*c*d*e^2 + 144*B*b*c*d^2*e + 12*A*c^2*d^2*e*m - 33*B*b^2*d*e^2*m - 6*A*b*c*d*e^2*m^2 - 66*A*b*c*d*e^2*m +
24*B*b*c*d^2*e*m))/(e^5*(1764*m + 1624*m^2 + 735*m^3 + 175*m^4 + 21*m^5 + m^6 + 720)) + (d*m*x^2*(m + 1)*(d +
e*x)^m*(120*A*b^2*e^3 - 60*B*c^2*d^3 + 72*A*c^2*d^2*e - 90*B*b^2*d*e^2 + 74*A*b^2*e^3*m + 15*A*b^2*e^3*m^2 + A
*b^2*e^3*m^3 - 3*B*b^2*d*e^2*m^2 - 180*A*b*c*d*e^2 + 144*B*b*c*d^2*e + 12*A*c^2*d^2*e*m - 33*B*b^2*d*e^2*m - 6
*A*b*c*d*e^2*m^2 - 66*A*b*c*d*e^2*m + 24*B*b*c*d^2*e*m))/(e^4*(1764*m + 1624*m^2 + 735*m^3 + 175*m^4 + 21*m^5
+ m^6 + 720))
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((B*x+A)*(e*x+d)**m*(c*x**2+b*x)**2,x)
[Out]
Timed out
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