3.17.75 \(\int (A+B x) (d+e x)^5 (a^2+2 a b x+b^2 x^2)^2 \, dx\) [1675]

Optimal. Leaf size=206 \[ -\frac {(b d-a e)^4 (B d-A e) (d+e x)^6}{6 e^6}+\frac {(b d-a e)^3 (5 b B d-4 A b e-a B e) (d+e x)^7}{7 e^6}-\frac {b (b d-a e)^2 (5 b B d-3 A b e-2 a B e) (d+e x)^8}{4 e^6}+\frac {2 b^2 (b d-a e) (5 b B d-2 A b e-3 a B e) (d+e x)^9}{9 e^6}-\frac {b^3 (5 b B d-A b e-4 a B e) (d+e x)^{10}}{10 e^6}+\frac {b^4 B (d+e x)^{11}}{11 e^6} \]

[Out]

-1/6*(-a*e+b*d)^4*(-A*e+B*d)*(e*x+d)^6/e^6+1/7*(-a*e+b*d)^3*(-4*A*b*e-B*a*e+5*B*b*d)*(e*x+d)^7/e^6-1/4*b*(-a*e
+b*d)^2*(-3*A*b*e-2*B*a*e+5*B*b*d)*(e*x+d)^8/e^6+2/9*b^2*(-a*e+b*d)*(-2*A*b*e-3*B*a*e+5*B*b*d)*(e*x+d)^9/e^6-1
/10*b^3*(-A*b*e-4*B*a*e+5*B*b*d)*(e*x+d)^10/e^6+1/11*b^4*B*(e*x+d)^11/e^6

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Rubi [A]
time = 0.36, antiderivative size = 206, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {27, 78} \begin {gather*} -\frac {b^3 (d+e x)^{10} (-4 a B e-A b e+5 b B d)}{10 e^6}+\frac {2 b^2 (d+e x)^9 (b d-a e) (-3 a B e-2 A b e+5 b B d)}{9 e^6}-\frac {b (d+e x)^8 (b d-a e)^2 (-2 a B e-3 A b e+5 b B d)}{4 e^6}+\frac {(d+e x)^7 (b d-a e)^3 (-a B e-4 A b e+5 b B d)}{7 e^6}-\frac {(d+e x)^6 (b d-a e)^4 (B d-A e)}{6 e^6}+\frac {b^4 B (d+e x)^{11}}{11 e^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(A + B*x)*(d + e*x)^5*(a^2 + 2*a*b*x + b^2*x^2)^2,x]

[Out]

-1/6*((b*d - a*e)^4*(B*d - A*e)*(d + e*x)^6)/e^6 + ((b*d - a*e)^3*(5*b*B*d - 4*A*b*e - a*B*e)*(d + e*x)^7)/(7*
e^6) - (b*(b*d - a*e)^2*(5*b*B*d - 3*A*b*e - 2*a*B*e)*(d + e*x)^8)/(4*e^6) + (2*b^2*(b*d - a*e)*(5*b*B*d - 2*A
*b*e - 3*a*B*e)*(d + e*x)^9)/(9*e^6) - (b^3*(5*b*B*d - A*b*e - 4*a*B*e)*(d + e*x)^10)/(10*e^6) + (b^4*B*(d + e
*x)^11)/(11*e^6)

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 78

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

\begin {align*} \int (A+B x) (d+e x)^5 \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx &=\int (a+b x)^4 (A+B x) (d+e x)^5 \, dx\\ &=\int \left (\frac {(-b d+a e)^4 (-B d+A e) (d+e x)^5}{e^5}+\frac {(-b d+a e)^3 (-5 b B d+4 A b e+a B e) (d+e x)^6}{e^5}+\frac {2 b (b d-a e)^2 (-5 b B d+3 A b e+2 a B e) (d+e x)^7}{e^5}-\frac {2 b^2 (b d-a e) (-5 b B d+2 A b e+3 a B e) (d+e x)^8}{e^5}+\frac {b^3 (-5 b B d+A b e+4 a B e) (d+e x)^9}{e^5}+\frac {b^4 B (d+e x)^{10}}{e^5}\right ) \, dx\\ &=-\frac {(b d-a e)^4 (B d-A e) (d+e x)^6}{6 e^6}+\frac {(b d-a e)^3 (5 b B d-4 A b e-a B e) (d+e x)^7}{7 e^6}-\frac {b (b d-a e)^2 (5 b B d-3 A b e-2 a B e) (d+e x)^8}{4 e^6}+\frac {2 b^2 (b d-a e) (5 b B d-2 A b e-3 a B e) (d+e x)^9}{9 e^6}-\frac {b^3 (5 b B d-A b e-4 a B e) (d+e x)^{10}}{10 e^6}+\frac {b^4 B (d+e x)^{11}}{11 e^6}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(615\) vs. \(2(206)=412\).
time = 0.15, size = 615, normalized size = 2.99 \begin {gather*} a^4 A d^5 x+\frac {1}{2} a^3 d^4 (4 A b d+a B d+5 a A e) x^2+\frac {1}{3} a^2 d^3 \left (a B d (4 b d+5 a e)+2 A \left (3 b^2 d^2+10 a b d e+5 a^2 e^2\right )\right ) x^3+\frac {1}{2} a d^2 \left (a B d \left (3 b^2 d^2+10 a b d e+5 a^2 e^2\right )+A \left (2 b^3 d^3+15 a b^2 d^2 e+20 a^2 b d e^2+5 a^3 e^3\right )\right ) x^4+\frac {1}{5} d \left (2 a B d \left (2 b^3 d^3+15 a b^2 d^2 e+20 a^2 b d e^2+5 a^3 e^3\right )+A \left (b^4 d^4+20 a b^3 d^3 e+60 a^2 b^2 d^2 e^2+40 a^3 b d e^3+5 a^4 e^4\right )\right ) x^5+\frac {1}{6} \left (60 a^2 b^2 d^2 e^2 (B d+A e)+20 a^3 b d e^3 (2 B d+A e)+a^4 e^4 (5 B d+A e)+20 a b^3 d^3 e (B d+2 A e)+b^4 d^4 (B d+5 A e)\right ) x^6+\frac {1}{7} e \left (a^4 B e^4+40 a b^3 d^2 e (B d+A e)+30 a^2 b^2 d e^2 (2 B d+A e)+4 a^3 b e^3 (5 B d+A e)+5 b^4 d^3 (B d+2 A e)\right ) x^7+\frac {1}{4} b e^2 \left (2 a^3 B e^3+5 b^3 d^2 (B d+A e)+10 a b^2 d e (2 B d+A e)+3 a^2 b e^2 (5 B d+A e)\right ) x^8+\frac {1}{9} b^2 e^3 \left (6 a^2 B e^2+5 b^2 d (2 B d+A e)+4 a b e (5 B d+A e)\right ) x^9+\frac {1}{10} b^3 e^4 (5 b B d+A b e+4 a B e) x^{10}+\frac {1}{11} b^4 B e^5 x^{11} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x)*(d + e*x)^5*(a^2 + 2*a*b*x + b^2*x^2)^2,x]

[Out]

a^4*A*d^5*x + (a^3*d^4*(4*A*b*d + a*B*d + 5*a*A*e)*x^2)/2 + (a^2*d^3*(a*B*d*(4*b*d + 5*a*e) + 2*A*(3*b^2*d^2 +
10*a*b*d*e + 5*a^2*e^2))*x^3)/3 + (a*d^2*(a*B*d*(3*b^2*d^2 + 10*a*b*d*e + 5*a^2*e^2) + A*(2*b^3*d^3 + 15*a*b^
2*d^2*e + 20*a^2*b*d*e^2 + 5*a^3*e^3))*x^4)/2 + (d*(2*a*B*d*(2*b^3*d^3 + 15*a*b^2*d^2*e + 20*a^2*b*d*e^2 + 5*a
^3*e^3) + A*(b^4*d^4 + 20*a*b^3*d^3*e + 60*a^2*b^2*d^2*e^2 + 40*a^3*b*d*e^3 + 5*a^4*e^4))*x^5)/5 + ((60*a^2*b^
2*d^2*e^2*(B*d + A*e) + 20*a^3*b*d*e^3*(2*B*d + A*e) + a^4*e^4*(5*B*d + A*e) + 20*a*b^3*d^3*e*(B*d + 2*A*e) +
b^4*d^4*(B*d + 5*A*e))*x^6)/6 + (e*(a^4*B*e^4 + 40*a*b^3*d^2*e*(B*d + A*e) + 30*a^2*b^2*d*e^2*(2*B*d + A*e) +
4*a^3*b*e^3*(5*B*d + A*e) + 5*b^4*d^3*(B*d + 2*A*e))*x^7)/7 + (b*e^2*(2*a^3*B*e^3 + 5*b^3*d^2*(B*d + A*e) + 10
*a*b^2*d*e*(2*B*d + A*e) + 3*a^2*b*e^2*(5*B*d + A*e))*x^8)/4 + (b^2*e^3*(6*a^2*B*e^2 + 5*b^2*d*(2*B*d + A*e) +
4*a*b*e*(5*B*d + A*e))*x^9)/9 + (b^3*e^4*(5*b*B*d + A*b*e + 4*a*B*e)*x^10)/10 + (b^4*B*e^5*x^11)/11

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(691\) vs. \(2(194)=388\).
time = 0.94, size = 692, normalized size = 3.36 Too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(e*x+d)^5*(b^2*x^2+2*a*b*x+a^2)^2,x,method=_RETURNVERBOSE)

[Out]

1/11*B*b^4*e^5*x^11+1/10*((A*e^5+5*B*d*e^4)*b^4+4*B*a*b^3*e^5)*x^10+1/9*((5*A*d*e^4+10*B*d^2*e^3)*b^4+4*(A*e^5
+5*B*d*e^4)*a*b^3+6*B*a^2*b^2*e^5)*x^9+1/8*((10*A*d^2*e^3+10*B*d^3*e^2)*b^4+4*(5*A*d*e^4+10*B*d^2*e^3)*a*b^3+6
*(A*e^5+5*B*d*e^4)*a^2*b^2+4*B*a^3*b*e^5)*x^8+1/7*((10*A*d^3*e^2+5*B*d^4*e)*b^4+4*(10*A*d^2*e^3+10*B*d^3*e^2)*
a*b^3+6*(5*A*d*e^4+10*B*d^2*e^3)*a^2*b^2+4*(A*e^5+5*B*d*e^4)*a^3*b+B*e^5*a^4)*x^7+1/6*((5*A*d^4*e+B*d^5)*b^4+4
*(10*A*d^3*e^2+5*B*d^4*e)*a*b^3+6*(10*A*d^2*e^3+10*B*d^3*e^2)*a^2*b^2+4*(5*A*d*e^4+10*B*d^2*e^3)*a^3*b+(A*e^5+
5*B*d*e^4)*a^4)*x^6+1/5*(A*b^4*d^5+4*(5*A*d^4*e+B*d^5)*a*b^3+6*(10*A*d^3*e^2+5*B*d^4*e)*a^2*b^2+4*(10*A*d^2*e^
3+10*B*d^3*e^2)*a^3*b+(5*A*d*e^4+10*B*d^2*e^3)*a^4)*x^5+1/4*(4*A*a*b^3*d^5+6*(5*A*d^4*e+B*d^5)*a^2*b^2+4*(10*A
*d^3*e^2+5*B*d^4*e)*a^3*b+(10*A*d^2*e^3+10*B*d^3*e^2)*a^4)*x^4+1/3*(6*A*a^2*b^2*d^5+4*(5*A*d^4*e+B*d^5)*a^3*b+
(10*A*d^3*e^2+5*B*d^4*e)*a^4)*x^3+1/2*(4*A*a^3*b*d^5+(5*A*d^4*e+B*d^5)*a^4)*x^2+a^4*A*d^5*x

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 699 vs. \(2 (207) = 414\).
time = 0.28, size = 699, normalized size = 3.39 \begin {gather*} \frac {1}{11} \, B b^{4} x^{11} e^{5} + A a^{4} d^{5} x + \frac {1}{10} \, {\left (5 \, B b^{4} d e^{4} + 4 \, B a b^{3} e^{5} + A b^{4} e^{5}\right )} x^{10} + \frac {1}{9} \, {\left (10 \, B b^{4} d^{2} e^{3} + 6 \, B a^{2} b^{2} e^{5} + 4 \, A a b^{3} e^{5} + 5 \, {\left (4 \, B a b^{3} e^{4} + A b^{4} e^{4}\right )} d\right )} x^{9} + \frac {1}{4} \, {\left (5 \, B b^{4} d^{3} e^{2} + 2 \, B a^{3} b e^{5} + 3 \, A a^{2} b^{2} e^{5} + 5 \, {\left (4 \, B a b^{3} e^{3} + A b^{4} e^{3}\right )} d^{2} + 5 \, {\left (3 \, B a^{2} b^{2} e^{4} + 2 \, A a b^{3} e^{4}\right )} d\right )} x^{8} + \frac {1}{7} \, {\left (5 \, B b^{4} d^{4} e + B a^{4} e^{5} + 4 \, A a^{3} b e^{5} + 10 \, {\left (4 \, B a b^{3} e^{2} + A b^{4} e^{2}\right )} d^{3} + 20 \, {\left (3 \, B a^{2} b^{2} e^{3} + 2 \, A a b^{3} e^{3}\right )} d^{2} + 10 \, {\left (2 \, B a^{3} b e^{4} + 3 \, A a^{2} b^{2} e^{4}\right )} d\right )} x^{7} + \frac {1}{6} \, {\left (B b^{4} d^{5} + A a^{4} e^{5} + 5 \, {\left (4 \, B a b^{3} e + A b^{4} e\right )} d^{4} + 20 \, {\left (3 \, B a^{2} b^{2} e^{2} + 2 \, A a b^{3} e^{2}\right )} d^{3} + 20 \, {\left (2 \, B a^{3} b e^{3} + 3 \, A a^{2} b^{2} e^{3}\right )} d^{2} + 5 \, {\left (B a^{4} e^{4} + 4 \, A a^{3} b e^{4}\right )} d\right )} x^{6} + \frac {1}{5} \, {\left (5 \, A a^{4} d e^{4} + {\left (4 \, B a b^{3} + A b^{4}\right )} d^{5} + 10 \, {\left (3 \, B a^{2} b^{2} e + 2 \, A a b^{3} e\right )} d^{4} + 20 \, {\left (2 \, B a^{3} b e^{2} + 3 \, A a^{2} b^{2} e^{2}\right )} d^{3} + 10 \, {\left (B a^{4} e^{3} + 4 \, A a^{3} b e^{3}\right )} d^{2}\right )} x^{5} + \frac {1}{2} \, {\left (5 \, A a^{4} d^{2} e^{3} + {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{5} + 5 \, {\left (2 \, B a^{3} b e + 3 \, A a^{2} b^{2} e\right )} d^{4} + 5 \, {\left (B a^{4} e^{2} + 4 \, A a^{3} b e^{2}\right )} d^{3}\right )} x^{4} + \frac {1}{3} \, {\left (10 \, A a^{4} d^{3} e^{2} + 2 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d^{5} + 5 \, {\left (B a^{4} e + 4 \, A a^{3} b e\right )} d^{4}\right )} x^{3} + \frac {1}{2} \, {\left (5 \, A a^{4} d^{4} e + {\left (B a^{4} + 4 \, A a^{3} b\right )} d^{5}\right )} x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^5*(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="maxima")

[Out]

1/11*B*b^4*x^11*e^5 + A*a^4*d^5*x + 1/10*(5*B*b^4*d*e^4 + 4*B*a*b^3*e^5 + A*b^4*e^5)*x^10 + 1/9*(10*B*b^4*d^2*
e^3 + 6*B*a^2*b^2*e^5 + 4*A*a*b^3*e^5 + 5*(4*B*a*b^3*e^4 + A*b^4*e^4)*d)*x^9 + 1/4*(5*B*b^4*d^3*e^2 + 2*B*a^3*
b*e^5 + 3*A*a^2*b^2*e^5 + 5*(4*B*a*b^3*e^3 + A*b^4*e^3)*d^2 + 5*(3*B*a^2*b^2*e^4 + 2*A*a*b^3*e^4)*d)*x^8 + 1/7
*(5*B*b^4*d^4*e + B*a^4*e^5 + 4*A*a^3*b*e^5 + 10*(4*B*a*b^3*e^2 + A*b^4*e^2)*d^3 + 20*(3*B*a^2*b^2*e^3 + 2*A*a
*b^3*e^3)*d^2 + 10*(2*B*a^3*b*e^4 + 3*A*a^2*b^2*e^4)*d)*x^7 + 1/6*(B*b^4*d^5 + A*a^4*e^5 + 5*(4*B*a*b^3*e + A*
b^4*e)*d^4 + 20*(3*B*a^2*b^2*e^2 + 2*A*a*b^3*e^2)*d^3 + 20*(2*B*a^3*b*e^3 + 3*A*a^2*b^2*e^3)*d^2 + 5*(B*a^4*e^
4 + 4*A*a^3*b*e^4)*d)*x^6 + 1/5*(5*A*a^4*d*e^4 + (4*B*a*b^3 + A*b^4)*d^5 + 10*(3*B*a^2*b^2*e + 2*A*a*b^3*e)*d^
4 + 20*(2*B*a^3*b*e^2 + 3*A*a^2*b^2*e^2)*d^3 + 10*(B*a^4*e^3 + 4*A*a^3*b*e^3)*d^2)*x^5 + 1/2*(5*A*a^4*d^2*e^3
+ (3*B*a^2*b^2 + 2*A*a*b^3)*d^5 + 5*(2*B*a^3*b*e + 3*A*a^2*b^2*e)*d^4 + 5*(B*a^4*e^2 + 4*A*a^3*b*e^2)*d^3)*x^4
+ 1/3*(10*A*a^4*d^3*e^2 + 2*(2*B*a^3*b + 3*A*a^2*b^2)*d^5 + 5*(B*a^4*e + 4*A*a^3*b*e)*d^4)*x^3 + 1/2*(5*A*a^4
*d^4*e + (B*a^4 + 4*A*a^3*b)*d^5)*x^2

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 689 vs. \(2 (207) = 414\).
time = 1.95, size = 689, normalized size = 3.34 \begin {gather*} \frac {1}{6} \, B b^{4} d^{5} x^{6} + A a^{4} d^{5} x + \frac {1}{5} \, {\left (4 \, B a b^{3} + A b^{4}\right )} d^{5} x^{5} + \frac {1}{2} \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{5} x^{4} + \frac {2}{3} \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d^{5} x^{3} + \frac {1}{2} \, {\left (B a^{4} + 4 \, A a^{3} b\right )} d^{5} x^{2} + \frac {1}{13860} \, {\left (1260 \, B b^{4} x^{11} + 2310 \, A a^{4} x^{6} + 1386 \, {\left (4 \, B a b^{3} + A b^{4}\right )} x^{10} + 3080 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} x^{9} + 3465 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} x^{8} + 1980 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} x^{7}\right )} e^{5} + \frac {1}{252} \, {\left (126 \, B b^{4} d x^{10} + 252 \, A a^{4} d x^{5} + 140 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d x^{9} + 315 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d x^{8} + 360 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d x^{7} + 210 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} d x^{6}\right )} e^{4} + \frac {1}{252} \, {\left (280 \, B b^{4} d^{2} x^{9} + 630 \, A a^{4} d^{2} x^{4} + 315 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d^{2} x^{8} + 720 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{2} x^{7} + 840 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d^{2} x^{6} + 504 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} d^{2} x^{5}\right )} e^{3} + \frac {1}{84} \, {\left (105 \, B b^{4} d^{3} x^{8} + 280 \, A a^{4} d^{3} x^{3} + 120 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d^{3} x^{7} + 280 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{3} x^{6} + 336 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d^{3} x^{5} + 210 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} d^{3} x^{4}\right )} e^{2} + \frac {1}{42} \, {\left (30 \, B b^{4} d^{4} x^{7} + 105 \, A a^{4} d^{4} x^{2} + 35 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d^{4} x^{6} + 84 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{4} x^{5} + 105 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d^{4} x^{4} + 70 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} d^{4} x^{3}\right )} e \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^5*(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="fricas")

[Out]

1/6*B*b^4*d^5*x^6 + A*a^4*d^5*x + 1/5*(4*B*a*b^3 + A*b^4)*d^5*x^5 + 1/2*(3*B*a^2*b^2 + 2*A*a*b^3)*d^5*x^4 + 2/
3*(2*B*a^3*b + 3*A*a^2*b^2)*d^5*x^3 + 1/2*(B*a^4 + 4*A*a^3*b)*d^5*x^2 + 1/13860*(1260*B*b^4*x^11 + 2310*A*a^4*
x^6 + 1386*(4*B*a*b^3 + A*b^4)*x^10 + 3080*(3*B*a^2*b^2 + 2*A*a*b^3)*x^9 + 3465*(2*B*a^3*b + 3*A*a^2*b^2)*x^8
+ 1980*(B*a^4 + 4*A*a^3*b)*x^7)*e^5 + 1/252*(126*B*b^4*d*x^10 + 252*A*a^4*d*x^5 + 140*(4*B*a*b^3 + A*b^4)*d*x^
9 + 315*(3*B*a^2*b^2 + 2*A*a*b^3)*d*x^8 + 360*(2*B*a^3*b + 3*A*a^2*b^2)*d*x^7 + 210*(B*a^4 + 4*A*a^3*b)*d*x^6)
*e^4 + 1/252*(280*B*b^4*d^2*x^9 + 630*A*a^4*d^2*x^4 + 315*(4*B*a*b^3 + A*b^4)*d^2*x^8 + 720*(3*B*a^2*b^2 + 2*A
*a*b^3)*d^2*x^7 + 840*(2*B*a^3*b + 3*A*a^2*b^2)*d^2*x^6 + 504*(B*a^4 + 4*A*a^3*b)*d^2*x^5)*e^3 + 1/84*(105*B*b
^4*d^3*x^8 + 280*A*a^4*d^3*x^3 + 120*(4*B*a*b^3 + A*b^4)*d^3*x^7 + 280*(3*B*a^2*b^2 + 2*A*a*b^3)*d^3*x^6 + 336
*(2*B*a^3*b + 3*A*a^2*b^2)*d^3*x^5 + 210*(B*a^4 + 4*A*a^3*b)*d^3*x^4)*e^2 + 1/42*(30*B*b^4*d^4*x^7 + 105*A*a^4
*d^4*x^2 + 35*(4*B*a*b^3 + A*b^4)*d^4*x^6 + 84*(3*B*a^2*b^2 + 2*A*a*b^3)*d^4*x^5 + 105*(2*B*a^3*b + 3*A*a^2*b^
2)*d^4*x^4 + 70*(B*a^4 + 4*A*a^3*b)*d^4*x^3)*e

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 884 vs. \(2 (202) = 404\).
time = 0.06, size = 884, normalized size = 4.29 \begin {gather*} A a^{4} d^{5} x + \frac {B b^{4} e^{5} x^{11}}{11} + x^{10} \left (\frac {A b^{4} e^{5}}{10} + \frac {2 B a b^{3} e^{5}}{5} + \frac {B b^{4} d e^{4}}{2}\right ) + x^{9} \cdot \left (\frac {4 A a b^{3} e^{5}}{9} + \frac {5 A b^{4} d e^{4}}{9} + \frac {2 B a^{2} b^{2} e^{5}}{3} + \frac {20 B a b^{3} d e^{4}}{9} + \frac {10 B b^{4} d^{2} e^{3}}{9}\right ) + x^{8} \cdot \left (\frac {3 A a^{2} b^{2} e^{5}}{4} + \frac {5 A a b^{3} d e^{4}}{2} + \frac {5 A b^{4} d^{2} e^{3}}{4} + \frac {B a^{3} b e^{5}}{2} + \frac {15 B a^{2} b^{2} d e^{4}}{4} + 5 B a b^{3} d^{2} e^{3} + \frac {5 B b^{4} d^{3} e^{2}}{4}\right ) + x^{7} \cdot \left (\frac {4 A a^{3} b e^{5}}{7} + \frac {30 A a^{2} b^{2} d e^{4}}{7} + \frac {40 A a b^{3} d^{2} e^{3}}{7} + \frac {10 A b^{4} d^{3} e^{2}}{7} + \frac {B a^{4} e^{5}}{7} + \frac {20 B a^{3} b d e^{4}}{7} + \frac {60 B a^{2} b^{2} d^{2} e^{3}}{7} + \frac {40 B a b^{3} d^{3} e^{2}}{7} + \frac {5 B b^{4} d^{4} e}{7}\right ) + x^{6} \left (\frac {A a^{4} e^{5}}{6} + \frac {10 A a^{3} b d e^{4}}{3} + 10 A a^{2} b^{2} d^{2} e^{3} + \frac {20 A a b^{3} d^{3} e^{2}}{3} + \frac {5 A b^{4} d^{4} e}{6} + \frac {5 B a^{4} d e^{4}}{6} + \frac {20 B a^{3} b d^{2} e^{3}}{3} + 10 B a^{2} b^{2} d^{3} e^{2} + \frac {10 B a b^{3} d^{4} e}{3} + \frac {B b^{4} d^{5}}{6}\right ) + x^{5} \left (A a^{4} d e^{4} + 8 A a^{3} b d^{2} e^{3} + 12 A a^{2} b^{2} d^{3} e^{2} + 4 A a b^{3} d^{4} e + \frac {A b^{4} d^{5}}{5} + 2 B a^{4} d^{2} e^{3} + 8 B a^{3} b d^{3} e^{2} + 6 B a^{2} b^{2} d^{4} e + \frac {4 B a b^{3} d^{5}}{5}\right ) + x^{4} \cdot \left (\frac {5 A a^{4} d^{2} e^{3}}{2} + 10 A a^{3} b d^{3} e^{2} + \frac {15 A a^{2} b^{2} d^{4} e}{2} + A a b^{3} d^{5} + \frac {5 B a^{4} d^{3} e^{2}}{2} + 5 B a^{3} b d^{4} e + \frac {3 B a^{2} b^{2} d^{5}}{2}\right ) + x^{3} \cdot \left (\frac {10 A a^{4} d^{3} e^{2}}{3} + \frac {20 A a^{3} b d^{4} e}{3} + 2 A a^{2} b^{2} d^{5} + \frac {5 B a^{4} d^{4} e}{3} + \frac {4 B a^{3} b d^{5}}{3}\right ) + x^{2} \cdot \left (\frac {5 A a^{4} d^{4} e}{2} + 2 A a^{3} b d^{5} + \frac {B a^{4} d^{5}}{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)**5*(b**2*x**2+2*a*b*x+a**2)**2,x)

[Out]

A*a**4*d**5*x + B*b**4*e**5*x**11/11 + x**10*(A*b**4*e**5/10 + 2*B*a*b**3*e**5/5 + B*b**4*d*e**4/2) + x**9*(4*
A*a*b**3*e**5/9 + 5*A*b**4*d*e**4/9 + 2*B*a**2*b**2*e**5/3 + 20*B*a*b**3*d*e**4/9 + 10*B*b**4*d**2*e**3/9) + x
**8*(3*A*a**2*b**2*e**5/4 + 5*A*a*b**3*d*e**4/2 + 5*A*b**4*d**2*e**3/4 + B*a**3*b*e**5/2 + 15*B*a**2*b**2*d*e*
*4/4 + 5*B*a*b**3*d**2*e**3 + 5*B*b**4*d**3*e**2/4) + x**7*(4*A*a**3*b*e**5/7 + 30*A*a**2*b**2*d*e**4/7 + 40*A
*a*b**3*d**2*e**3/7 + 10*A*b**4*d**3*e**2/7 + B*a**4*e**5/7 + 20*B*a**3*b*d*e**4/7 + 60*B*a**2*b**2*d**2*e**3/
7 + 40*B*a*b**3*d**3*e**2/7 + 5*B*b**4*d**4*e/7) + x**6*(A*a**4*e**5/6 + 10*A*a**3*b*d*e**4/3 + 10*A*a**2*b**2
*d**2*e**3 + 20*A*a*b**3*d**3*e**2/3 + 5*A*b**4*d**4*e/6 + 5*B*a**4*d*e**4/6 + 20*B*a**3*b*d**2*e**3/3 + 10*B*
a**2*b**2*d**3*e**2 + 10*B*a*b**3*d**4*e/3 + B*b**4*d**5/6) + x**5*(A*a**4*d*e**4 + 8*A*a**3*b*d**2*e**3 + 12*
A*a**2*b**2*d**3*e**2 + 4*A*a*b**3*d**4*e + A*b**4*d**5/5 + 2*B*a**4*d**2*e**3 + 8*B*a**3*b*d**3*e**2 + 6*B*a*
*2*b**2*d**4*e + 4*B*a*b**3*d**5/5) + x**4*(5*A*a**4*d**2*e**3/2 + 10*A*a**3*b*d**3*e**2 + 15*A*a**2*b**2*d**4
*e/2 + A*a*b**3*d**5 + 5*B*a**4*d**3*e**2/2 + 5*B*a**3*b*d**4*e + 3*B*a**2*b**2*d**5/2) + x**3*(10*A*a**4*d**3
*e**2/3 + 20*A*a**3*b*d**4*e/3 + 2*A*a**2*b**2*d**5 + 5*B*a**4*d**4*e/3 + 4*B*a**3*b*d**5/3) + x**2*(5*A*a**4*
d**4*e/2 + 2*A*a**3*b*d**5 + B*a**4*d**5/2)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 826 vs. \(2 (207) = 414\).
time = 1.02, size = 826, normalized size = 4.01 \begin {gather*} \frac {1}{11} \, B b^{4} x^{11} e^{5} + \frac {1}{2} \, B b^{4} d x^{10} e^{4} + \frac {10}{9} \, B b^{4} d^{2} x^{9} e^{3} + \frac {5}{4} \, B b^{4} d^{3} x^{8} e^{2} + \frac {5}{7} \, B b^{4} d^{4} x^{7} e + \frac {1}{6} \, B b^{4} d^{5} x^{6} + \frac {2}{5} \, B a b^{3} x^{10} e^{5} + \frac {1}{10} \, A b^{4} x^{10} e^{5} + \frac {20}{9} \, B a b^{3} d x^{9} e^{4} + \frac {5}{9} \, A b^{4} d x^{9} e^{4} + 5 \, B a b^{3} d^{2} x^{8} e^{3} + \frac {5}{4} \, A b^{4} d^{2} x^{8} e^{3} + \frac {40}{7} \, B a b^{3} d^{3} x^{7} e^{2} + \frac {10}{7} \, A b^{4} d^{3} x^{7} e^{2} + \frac {10}{3} \, B a b^{3} d^{4} x^{6} e + \frac {5}{6} \, A b^{4} d^{4} x^{6} e + \frac {4}{5} \, B a b^{3} d^{5} x^{5} + \frac {1}{5} \, A b^{4} d^{5} x^{5} + \frac {2}{3} \, B a^{2} b^{2} x^{9} e^{5} + \frac {4}{9} \, A a b^{3} x^{9} e^{5} + \frac {15}{4} \, B a^{2} b^{2} d x^{8} e^{4} + \frac {5}{2} \, A a b^{3} d x^{8} e^{4} + \frac {60}{7} \, B a^{2} b^{2} d^{2} x^{7} e^{3} + \frac {40}{7} \, A a b^{3} d^{2} x^{7} e^{3} + 10 \, B a^{2} b^{2} d^{3} x^{6} e^{2} + \frac {20}{3} \, A a b^{3} d^{3} x^{6} e^{2} + 6 \, B a^{2} b^{2} d^{4} x^{5} e + 4 \, A a b^{3} d^{4} x^{5} e + \frac {3}{2} \, B a^{2} b^{2} d^{5} x^{4} + A a b^{3} d^{5} x^{4} + \frac {1}{2} \, B a^{3} b x^{8} e^{5} + \frac {3}{4} \, A a^{2} b^{2} x^{8} e^{5} + \frac {20}{7} \, B a^{3} b d x^{7} e^{4} + \frac {30}{7} \, A a^{2} b^{2} d x^{7} e^{4} + \frac {20}{3} \, B a^{3} b d^{2} x^{6} e^{3} + 10 \, A a^{2} b^{2} d^{2} x^{6} e^{3} + 8 \, B a^{3} b d^{3} x^{5} e^{2} + 12 \, A a^{2} b^{2} d^{3} x^{5} e^{2} + 5 \, B a^{3} b d^{4} x^{4} e + \frac {15}{2} \, A a^{2} b^{2} d^{4} x^{4} e + \frac {4}{3} \, B a^{3} b d^{5} x^{3} + 2 \, A a^{2} b^{2} d^{5} x^{3} + \frac {1}{7} \, B a^{4} x^{7} e^{5} + \frac {4}{7} \, A a^{3} b x^{7} e^{5} + \frac {5}{6} \, B a^{4} d x^{6} e^{4} + \frac {10}{3} \, A a^{3} b d x^{6} e^{4} + 2 \, B a^{4} d^{2} x^{5} e^{3} + 8 \, A a^{3} b d^{2} x^{5} e^{3} + \frac {5}{2} \, B a^{4} d^{3} x^{4} e^{2} + 10 \, A a^{3} b d^{3} x^{4} e^{2} + \frac {5}{3} \, B a^{4} d^{4} x^{3} e + \frac {20}{3} \, A a^{3} b d^{4} x^{3} e + \frac {1}{2} \, B a^{4} d^{5} x^{2} + 2 \, A a^{3} b d^{5} x^{2} + \frac {1}{6} \, A a^{4} x^{6} e^{5} + A a^{4} d x^{5} e^{4} + \frac {5}{2} \, A a^{4} d^{2} x^{4} e^{3} + \frac {10}{3} \, A a^{4} d^{3} x^{3} e^{2} + \frac {5}{2} \, A a^{4} d^{4} x^{2} e + A a^{4} d^{5} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^5*(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="giac")

[Out]

1/11*B*b^4*x^11*e^5 + 1/2*B*b^4*d*x^10*e^4 + 10/9*B*b^4*d^2*x^9*e^3 + 5/4*B*b^4*d^3*x^8*e^2 + 5/7*B*b^4*d^4*x^
7*e + 1/6*B*b^4*d^5*x^6 + 2/5*B*a*b^3*x^10*e^5 + 1/10*A*b^4*x^10*e^5 + 20/9*B*a*b^3*d*x^9*e^4 + 5/9*A*b^4*d*x^
9*e^4 + 5*B*a*b^3*d^2*x^8*e^3 + 5/4*A*b^4*d^2*x^8*e^3 + 40/7*B*a*b^3*d^3*x^7*e^2 + 10/7*A*b^4*d^3*x^7*e^2 + 10
/3*B*a*b^3*d^4*x^6*e + 5/6*A*b^4*d^4*x^6*e + 4/5*B*a*b^3*d^5*x^5 + 1/5*A*b^4*d^5*x^5 + 2/3*B*a^2*b^2*x^9*e^5 +
4/9*A*a*b^3*x^9*e^5 + 15/4*B*a^2*b^2*d*x^8*e^4 + 5/2*A*a*b^3*d*x^8*e^4 + 60/7*B*a^2*b^2*d^2*x^7*e^3 + 40/7*A*
a*b^3*d^2*x^7*e^3 + 10*B*a^2*b^2*d^3*x^6*e^2 + 20/3*A*a*b^3*d^3*x^6*e^2 + 6*B*a^2*b^2*d^4*x^5*e + 4*A*a*b^3*d^
4*x^5*e + 3/2*B*a^2*b^2*d^5*x^4 + A*a*b^3*d^5*x^4 + 1/2*B*a^3*b*x^8*e^5 + 3/4*A*a^2*b^2*x^8*e^5 + 20/7*B*a^3*b
*d*x^7*e^4 + 30/7*A*a^2*b^2*d*x^7*e^4 + 20/3*B*a^3*b*d^2*x^6*e^3 + 10*A*a^2*b^2*d^2*x^6*e^3 + 8*B*a^3*b*d^3*x^
5*e^2 + 12*A*a^2*b^2*d^3*x^5*e^2 + 5*B*a^3*b*d^4*x^4*e + 15/2*A*a^2*b^2*d^4*x^4*e + 4/3*B*a^3*b*d^5*x^3 + 2*A*
a^2*b^2*d^5*x^3 + 1/7*B*a^4*x^7*e^5 + 4/7*A*a^3*b*x^7*e^5 + 5/6*B*a^4*d*x^6*e^4 + 10/3*A*a^3*b*d*x^6*e^4 + 2*B
*a^4*d^2*x^5*e^3 + 8*A*a^3*b*d^2*x^5*e^3 + 5/2*B*a^4*d^3*x^4*e^2 + 10*A*a^3*b*d^3*x^4*e^2 + 5/3*B*a^4*d^4*x^3*
e + 20/3*A*a^3*b*d^4*x^3*e + 1/2*B*a^4*d^5*x^2 + 2*A*a^3*b*d^5*x^2 + 1/6*A*a^4*x^6*e^5 + A*a^4*d*x^5*e^4 + 5/2
*A*a^4*d^2*x^4*e^3 + 10/3*A*a^4*d^3*x^3*e^2 + 5/2*A*a^4*d^4*x^2*e + A*a^4*d^5*x

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Mupad [B]
time = 0.25, size = 711, normalized size = 3.45 \begin {gather*} x^5\,\left (2\,B\,a^4\,d^2\,e^3+A\,a^4\,d\,e^4+8\,B\,a^3\,b\,d^3\,e^2+8\,A\,a^3\,b\,d^2\,e^3+6\,B\,a^2\,b^2\,d^4\,e+12\,A\,a^2\,b^2\,d^3\,e^2+\frac {4\,B\,a\,b^3\,d^5}{5}+4\,A\,a\,b^3\,d^4\,e+\frac {A\,b^4\,d^5}{5}\right )+x^7\,\left (\frac {B\,a^4\,e^5}{7}+\frac {20\,B\,a^3\,b\,d\,e^4}{7}+\frac {4\,A\,a^3\,b\,e^5}{7}+\frac {60\,B\,a^2\,b^2\,d^2\,e^3}{7}+\frac {30\,A\,a^2\,b^2\,d\,e^4}{7}+\frac {40\,B\,a\,b^3\,d^3\,e^2}{7}+\frac {40\,A\,a\,b^3\,d^2\,e^3}{7}+\frac {5\,B\,b^4\,d^4\,e}{7}+\frac {10\,A\,b^4\,d^3\,e^2}{7}\right )+x^4\,\left (\frac {5\,B\,a^4\,d^3\,e^2}{2}+\frac {5\,A\,a^4\,d^2\,e^3}{2}+5\,B\,a^3\,b\,d^4\,e+10\,A\,a^3\,b\,d^3\,e^2+\frac {3\,B\,a^2\,b^2\,d^5}{2}+\frac {15\,A\,a^2\,b^2\,d^4\,e}{2}+A\,a\,b^3\,d^5\right )+x^8\,\left (\frac {B\,a^3\,b\,e^5}{2}+\frac {15\,B\,a^2\,b^2\,d\,e^4}{4}+\frac {3\,A\,a^2\,b^2\,e^5}{4}+5\,B\,a\,b^3\,d^2\,e^3+\frac {5\,A\,a\,b^3\,d\,e^4}{2}+\frac {5\,B\,b^4\,d^3\,e^2}{4}+\frac {5\,A\,b^4\,d^2\,e^3}{4}\right )+x^3\,\left (\frac {5\,B\,a^4\,d^4\,e}{3}+\frac {10\,A\,a^4\,d^3\,e^2}{3}+\frac {4\,B\,a^3\,b\,d^5}{3}+\frac {20\,A\,a^3\,b\,d^4\,e}{3}+2\,A\,a^2\,b^2\,d^5\right )+x^9\,\left (\frac {2\,B\,a^2\,b^2\,e^5}{3}+\frac {20\,B\,a\,b^3\,d\,e^4}{9}+\frac {4\,A\,a\,b^3\,e^5}{9}+\frac {10\,B\,b^4\,d^2\,e^3}{9}+\frac {5\,A\,b^4\,d\,e^4}{9}\right )+x^6\,\left (\frac {5\,B\,a^4\,d\,e^4}{6}+\frac {A\,a^4\,e^5}{6}+\frac {20\,B\,a^3\,b\,d^2\,e^3}{3}+\frac {10\,A\,a^3\,b\,d\,e^4}{3}+10\,B\,a^2\,b^2\,d^3\,e^2+10\,A\,a^2\,b^2\,d^2\,e^3+\frac {10\,B\,a\,b^3\,d^4\,e}{3}+\frac {20\,A\,a\,b^3\,d^3\,e^2}{3}+\frac {B\,b^4\,d^5}{6}+\frac {5\,A\,b^4\,d^4\,e}{6}\right )+\frac {a^3\,d^4\,x^2\,\left (5\,A\,a\,e+4\,A\,b\,d+B\,a\,d\right )}{2}+\frac {b^3\,e^4\,x^{10}\,\left (A\,b\,e+4\,B\,a\,e+5\,B\,b\,d\right )}{10}+A\,a^4\,d^5\,x+\frac {B\,b^4\,e^5\,x^{11}}{11} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*x)*(d + e*x)^5*(a^2 + b^2*x^2 + 2*a*b*x)^2,x)

[Out]

x^5*((A*b^4*d^5)/5 + (4*B*a*b^3*d^5)/5 + A*a^4*d*e^4 + 2*B*a^4*d^2*e^3 + 8*A*a^3*b*d^2*e^3 + 6*B*a^2*b^2*d^4*e
+ 8*B*a^3*b*d^3*e^2 + 12*A*a^2*b^2*d^3*e^2 + 4*A*a*b^3*d^4*e) + x^7*((B*a^4*e^5)/7 + (4*A*a^3*b*e^5)/7 + (5*B
*b^4*d^4*e)/7 + (10*A*b^4*d^3*e^2)/7 + (40*A*a*b^3*d^2*e^3)/7 + (30*A*a^2*b^2*d*e^4)/7 + (40*B*a*b^3*d^3*e^2)/
7 + (60*B*a^2*b^2*d^2*e^3)/7 + (20*B*a^3*b*d*e^4)/7) + x^4*(A*a*b^3*d^5 + (3*B*a^2*b^2*d^5)/2 + (5*A*a^4*d^2*e
^3)/2 + (5*B*a^4*d^3*e^2)/2 + (15*A*a^2*b^2*d^4*e)/2 + 10*A*a^3*b*d^3*e^2 + 5*B*a^3*b*d^4*e) + x^8*((B*a^3*b*e
^5)/2 + (3*A*a^2*b^2*e^5)/4 + (5*A*b^4*d^2*e^3)/4 + (5*B*b^4*d^3*e^2)/4 + 5*B*a*b^3*d^2*e^3 + (15*B*a^2*b^2*d*
e^4)/4 + (5*A*a*b^3*d*e^4)/2) + x^3*((4*B*a^3*b*d^5)/3 + (5*B*a^4*d^4*e)/3 + 2*A*a^2*b^2*d^5 + (10*A*a^4*d^3*e
^2)/3 + (20*A*a^3*b*d^4*e)/3) + x^9*((4*A*a*b^3*e^5)/9 + (5*A*b^4*d*e^4)/9 + (2*B*a^2*b^2*e^5)/3 + (10*B*b^4*d
^2*e^3)/9 + (20*B*a*b^3*d*e^4)/9) + x^6*((A*a^4*e^5)/6 + (B*b^4*d^5)/6 + (5*A*b^4*d^4*e)/6 + (5*B*a^4*d*e^4)/6
+ (20*A*a*b^3*d^3*e^2)/3 + (20*B*a^3*b*d^2*e^3)/3 + 10*A*a^2*b^2*d^2*e^3 + 10*B*a^2*b^2*d^3*e^2 + (10*A*a^3*b
*d*e^4)/3 + (10*B*a*b^3*d^4*e)/3) + (a^3*d^4*x^2*(5*A*a*e + 4*A*b*d + B*a*d))/2 + (b^3*e^4*x^10*(A*b*e + 4*B*a
*e + 5*B*b*d))/10 + A*a^4*d^5*x + (B*b^4*e^5*x^11)/11

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