3.15.56 \(\int (A+B x) (d+e x)^4 (a^2+2 a b x+b^2 x^2)^2 \, dx\)

Optimal. Leaf size=204 \[ \frac {e^3 (a+b x)^9 (-5 a B e+A b e+4 b B d)}{9 b^6}+\frac {e^2 (a+b x)^8 (b d-a e) (-5 a B e+2 A b e+3 b B d)}{4 b^6}+\frac {2 e (a+b x)^7 (b d-a e)^2 (-5 a B e+3 A b e+2 b B d)}{7 b^6}+\frac {(a+b x)^6 (b d-a e)^3 (-5 a B e+4 A b e+b B d)}{6 b^6}+\frac {(a+b x)^5 (A b-a B) (b d-a e)^4}{5 b^6}+\frac {B e^4 (a+b x)^{10}}{10 b^6} \]

________________________________________________________________________________________

Rubi [A]  time = 0.46, antiderivative size = 204, 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, 77} \begin {gather*} \frac {e^3 (a+b x)^9 (-5 a B e+A b e+4 b B d)}{9 b^6}+\frac {e^2 (a+b x)^8 (b d-a e) (-5 a B e+2 A b e+3 b B d)}{4 b^6}+\frac {2 e (a+b x)^7 (b d-a e)^2 (-5 a B e+3 A b e+2 b B d)}{7 b^6}+\frac {(a+b x)^6 (b d-a e)^3 (-5 a B e+4 A b e+b B d)}{6 b^6}+\frac {(a+b x)^5 (A b-a B) (b d-a e)^4}{5 b^6}+\frac {B e^4 (a+b x)^{10}}{10 b^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(A + B*x)*(d + e*x)^4*(a^2 + 2*a*b*x + b^2*x^2)^2,x]

[Out]

((A*b - a*B)*(b*d - a*e)^4*(a + b*x)^5)/(5*b^6) + ((b*d - a*e)^3*(b*B*d + 4*A*b*e - 5*a*B*e)*(a + b*x)^6)/(6*b
^6) + (2*e*(b*d - a*e)^2*(2*b*B*d + 3*A*b*e - 5*a*B*e)*(a + b*x)^7)/(7*b^6) + (e^2*(b*d - a*e)*(3*b*B*d + 2*A*
b*e - 5*a*B*e)*(a + b*x)^8)/(4*b^6) + (e^3*(4*b*B*d + A*b*e - 5*a*B*e)*(a + b*x)^9)/(9*b^6) + (B*e^4*(a + b*x)
^10)/(10*b^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 77

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

________________________________________________________________________________________

Mathematica [B]  time = 0.16, size = 512, normalized size = 2.51 \begin {gather*} a^4 A d^4 x+\frac {1}{2} a^3 d^3 x^2 (4 A (a e+b d)+a B d)+\frac {2}{3} a^2 d^2 x^3 \left (A \left (3 a^2 e^2+8 a b d e+3 b^2 d^2\right )+2 a B d (a e+b d)\right )+\frac {1}{4} b^2 e^2 x^8 \left (3 a^2 B e^2+2 a b e (A e+4 B d)+b^2 d (2 A e+3 B d)\right )+\frac {2}{7} b e x^7 \left (2 a^3 B e^3+3 a^2 b e^2 (A e+4 B d)+4 a b^2 d e (2 A e+3 B d)+b^3 d^2 (3 A e+2 B d)\right )+\frac {1}{2} a d x^4 \left (a B d \left (3 a^2 e^2+8 a b d e+3 b^2 d^2\right )+2 A \left (a^3 e^3+6 a^2 b d e^2+6 a b^2 d^2 e+b^3 d^3\right )\right )+\frac {1}{6} x^6 \left (a^4 B e^4+4 a^3 b e^3 (A e+4 B d)+12 a^2 b^2 d e^2 (2 A e+3 B d)+8 a b^3 d^2 e (3 A e+2 B d)+b^4 d^3 (4 A e+B d)\right )+\frac {1}{5} x^5 \left (4 a B d \left (a^3 e^3+6 a^2 b d e^2+6 a b^2 d^2 e+b^3 d^3\right )+A \left (a^4 e^4+16 a^3 b d e^3+36 a^2 b^2 d^2 e^2+16 a b^3 d^3 e+b^4 d^4\right )\right )+\frac {1}{9} b^3 e^3 x^9 (4 a B e+A b e+4 b B d)+\frac {1}{10} b^4 B e^4 x^{10} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x)*(d + e*x)^4*(a^2 + 2*a*b*x + b^2*x^2)^2,x]

[Out]

a^4*A*d^4*x + (a^3*d^3*(a*B*d + 4*A*(b*d + a*e))*x^2)/2 + (2*a^2*d^2*(2*a*B*d*(b*d + a*e) + A*(3*b^2*d^2 + 8*a
*b*d*e + 3*a^2*e^2))*x^3)/3 + (a*d*(a*B*d*(3*b^2*d^2 + 8*a*b*d*e + 3*a^2*e^2) + 2*A*(b^3*d^3 + 6*a*b^2*d^2*e +
6*a^2*b*d*e^2 + a^3*e^3))*x^4)/2 + ((4*a*B*d*(b^3*d^3 + 6*a*b^2*d^2*e + 6*a^2*b*d*e^2 + a^3*e^3) + A*(b^4*d^4
+ 16*a*b^3*d^3*e + 36*a^2*b^2*d^2*e^2 + 16*a^3*b*d*e^3 + a^4*e^4))*x^5)/5 + ((a^4*B*e^4 + 4*a^3*b*e^3*(4*B*d
+ A*e) + 12*a^2*b^2*d*e^2*(3*B*d + 2*A*e) + 8*a*b^3*d^2*e*(2*B*d + 3*A*e) + b^4*d^3*(B*d + 4*A*e))*x^6)/6 + (2
*b*e*(2*a^3*B*e^3 + 3*a^2*b*e^2*(4*B*d + A*e) + 4*a*b^2*d*e*(3*B*d + 2*A*e) + b^3*d^2*(2*B*d + 3*A*e))*x^7)/7
+ (b^2*e^2*(3*a^2*B*e^2 + 2*a*b*e*(4*B*d + A*e) + b^2*d*(3*B*d + 2*A*e))*x^8)/4 + (b^3*e^3*(4*b*B*d + A*b*e +
4*a*B*e)*x^9)/9 + (b^4*B*e^4*x^10)/10

________________________________________________________________________________________

IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int (A+B x) (d+e x)^4 \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(A + B*x)*(d + e*x)^4*(a^2 + 2*a*b*x + b^2*x^2)^2,x]

[Out]

IntegrateAlgebraic[(A + B*x)*(d + e*x)^4*(a^2 + 2*a*b*x + b^2*x^2)^2, x]

________________________________________________________________________________________

fricas [B]  time = 0.37, size = 696, normalized size = 3.41 \begin {gather*} \frac {1}{10} x^{10} e^{4} b^{4} B + \frac {4}{9} x^{9} e^{3} d b^{4} B + \frac {4}{9} x^{9} e^{4} b^{3} a B + \frac {1}{9} x^{9} e^{4} b^{4} A + \frac {3}{4} x^{8} e^{2} d^{2} b^{4} B + 2 x^{8} e^{3} d b^{3} a B + \frac {3}{4} x^{8} e^{4} b^{2} a^{2} B + \frac {1}{2} x^{8} e^{3} d b^{4} A + \frac {1}{2} x^{8} e^{4} b^{3} a A + \frac {4}{7} x^{7} e d^{3} b^{4} B + \frac {24}{7} x^{7} e^{2} d^{2} b^{3} a B + \frac {24}{7} x^{7} e^{3} d b^{2} a^{2} B + \frac {4}{7} x^{7} e^{4} b a^{3} B + \frac {6}{7} x^{7} e^{2} d^{2} b^{4} A + \frac {16}{7} x^{7} e^{3} d b^{3} a A + \frac {6}{7} x^{7} e^{4} b^{2} a^{2} A + \frac {1}{6} x^{6} d^{4} b^{4} B + \frac {8}{3} x^{6} e d^{3} b^{3} a B + 6 x^{6} e^{2} d^{2} b^{2} a^{2} B + \frac {8}{3} x^{6} e^{3} d b a^{3} B + \frac {1}{6} x^{6} e^{4} a^{4} B + \frac {2}{3} x^{6} e d^{3} b^{4} A + 4 x^{6} e^{2} d^{2} b^{3} a A + 4 x^{6} e^{3} d b^{2} a^{2} A + \frac {2}{3} x^{6} e^{4} b a^{3} A + \frac {4}{5} x^{5} d^{4} b^{3} a B + \frac {24}{5} x^{5} e d^{3} b^{2} a^{2} B + \frac {24}{5} x^{5} e^{2} d^{2} b a^{3} B + \frac {4}{5} x^{5} e^{3} d a^{4} B + \frac {1}{5} x^{5} d^{4} b^{4} A + \frac {16}{5} x^{5} e d^{3} b^{3} a A + \frac {36}{5} x^{5} e^{2} d^{2} b^{2} a^{2} A + \frac {16}{5} x^{5} e^{3} d b a^{3} A + \frac {1}{5} x^{5} e^{4} a^{4} A + \frac {3}{2} x^{4} d^{4} b^{2} a^{2} B + 4 x^{4} e d^{3} b a^{3} B + \frac {3}{2} x^{4} e^{2} d^{2} a^{4} B + x^{4} d^{4} b^{3} a A + 6 x^{4} e d^{3} b^{2} a^{2} A + 6 x^{4} e^{2} d^{2} b a^{3} A + x^{4} e^{3} d a^{4} A + \frac {4}{3} x^{3} d^{4} b a^{3} B + \frac {4}{3} x^{3} e d^{3} a^{4} B + 2 x^{3} d^{4} b^{2} a^{2} A + \frac {16}{3} x^{3} e d^{3} b a^{3} A + 2 x^{3} e^{2} d^{2} a^{4} A + \frac {1}{2} x^{2} d^{4} a^{4} B + 2 x^{2} d^{4} b a^{3} A + 2 x^{2} e d^{3} a^{4} A + x d^{4} a^{4} A \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^4*(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="fricas")

[Out]

1/10*x^10*e^4*b^4*B + 4/9*x^9*e^3*d*b^4*B + 4/9*x^9*e^4*b^3*a*B + 1/9*x^9*e^4*b^4*A + 3/4*x^8*e^2*d^2*b^4*B +
2*x^8*e^3*d*b^3*a*B + 3/4*x^8*e^4*b^2*a^2*B + 1/2*x^8*e^3*d*b^4*A + 1/2*x^8*e^4*b^3*a*A + 4/7*x^7*e*d^3*b^4*B
+ 24/7*x^7*e^2*d^2*b^3*a*B + 24/7*x^7*e^3*d*b^2*a^2*B + 4/7*x^7*e^4*b*a^3*B + 6/7*x^7*e^2*d^2*b^4*A + 16/7*x^7
*e^3*d*b^3*a*A + 6/7*x^7*e^4*b^2*a^2*A + 1/6*x^6*d^4*b^4*B + 8/3*x^6*e*d^3*b^3*a*B + 6*x^6*e^2*d^2*b^2*a^2*B +
8/3*x^6*e^3*d*b*a^3*B + 1/6*x^6*e^4*a^4*B + 2/3*x^6*e*d^3*b^4*A + 4*x^6*e^2*d^2*b^3*a*A + 4*x^6*e^3*d*b^2*a^2
*A + 2/3*x^6*e^4*b*a^3*A + 4/5*x^5*d^4*b^3*a*B + 24/5*x^5*e*d^3*b^2*a^2*B + 24/5*x^5*e^2*d^2*b*a^3*B + 4/5*x^5
*e^3*d*a^4*B + 1/5*x^5*d^4*b^4*A + 16/5*x^5*e*d^3*b^3*a*A + 36/5*x^5*e^2*d^2*b^2*a^2*A + 16/5*x^5*e^3*d*b*a^3*
A + 1/5*x^5*e^4*a^4*A + 3/2*x^4*d^4*b^2*a^2*B + 4*x^4*e*d^3*b*a^3*B + 3/2*x^4*e^2*d^2*a^4*B + x^4*d^4*b^3*a*A
+ 6*x^4*e*d^3*b^2*a^2*A + 6*x^4*e^2*d^2*b*a^3*A + x^4*e^3*d*a^4*A + 4/3*x^3*d^4*b*a^3*B + 4/3*x^3*e*d^3*a^4*B
+ 2*x^3*d^4*b^2*a^2*A + 16/3*x^3*e*d^3*b*a^3*A + 2*x^3*e^2*d^2*a^4*A + 1/2*x^2*d^4*a^4*B + 2*x^2*d^4*b*a^3*A +
2*x^2*e*d^3*a^4*A + x*d^4*a^4*A

________________________________________________________________________________________

giac [B]  time = 0.16, size = 676, normalized size = 3.31 \begin {gather*} \frac {1}{10} \, B b^{4} x^{10} e^{4} + \frac {4}{9} \, B b^{4} d x^{9} e^{3} + \frac {3}{4} \, B b^{4} d^{2} x^{8} e^{2} + \frac {4}{7} \, B b^{4} d^{3} x^{7} e + \frac {1}{6} \, B b^{4} d^{4} x^{6} + \frac {4}{9} \, B a b^{3} x^{9} e^{4} + \frac {1}{9} \, A b^{4} x^{9} e^{4} + 2 \, B a b^{3} d x^{8} e^{3} + \frac {1}{2} \, A b^{4} d x^{8} e^{3} + \frac {24}{7} \, B a b^{3} d^{2} x^{7} e^{2} + \frac {6}{7} \, A b^{4} d^{2} x^{7} e^{2} + \frac {8}{3} \, B a b^{3} d^{3} x^{6} e + \frac {2}{3} \, A b^{4} d^{3} x^{6} e + \frac {4}{5} \, B a b^{3} d^{4} x^{5} + \frac {1}{5} \, A b^{4} d^{4} x^{5} + \frac {3}{4} \, B a^{2} b^{2} x^{8} e^{4} + \frac {1}{2} \, A a b^{3} x^{8} e^{4} + \frac {24}{7} \, B a^{2} b^{2} d x^{7} e^{3} + \frac {16}{7} \, A a b^{3} d x^{7} e^{3} + 6 \, B a^{2} b^{2} d^{2} x^{6} e^{2} + 4 \, A a b^{3} d^{2} x^{6} e^{2} + \frac {24}{5} \, B a^{2} b^{2} d^{3} x^{5} e + \frac {16}{5} \, A a b^{3} d^{3} x^{5} e + \frac {3}{2} \, B a^{2} b^{2} d^{4} x^{4} + A a b^{3} d^{4} x^{4} + \frac {4}{7} \, B a^{3} b x^{7} e^{4} + \frac {6}{7} \, A a^{2} b^{2} x^{7} e^{4} + \frac {8}{3} \, B a^{3} b d x^{6} e^{3} + 4 \, A a^{2} b^{2} d x^{6} e^{3} + \frac {24}{5} \, B a^{3} b d^{2} x^{5} e^{2} + \frac {36}{5} \, A a^{2} b^{2} d^{2} x^{5} e^{2} + 4 \, B a^{3} b d^{3} x^{4} e + 6 \, A a^{2} b^{2} d^{3} x^{4} e + \frac {4}{3} \, B a^{3} b d^{4} x^{3} + 2 \, A a^{2} b^{2} d^{4} x^{3} + \frac {1}{6} \, B a^{4} x^{6} e^{4} + \frac {2}{3} \, A a^{3} b x^{6} e^{4} + \frac {4}{5} \, B a^{4} d x^{5} e^{3} + \frac {16}{5} \, A a^{3} b d x^{5} e^{3} + \frac {3}{2} \, B a^{4} d^{2} x^{4} e^{2} + 6 \, A a^{3} b d^{2} x^{4} e^{2} + \frac {4}{3} \, B a^{4} d^{3} x^{3} e + \frac {16}{3} \, A a^{3} b d^{3} x^{3} e + \frac {1}{2} \, B a^{4} d^{4} x^{2} + 2 \, A a^{3} b d^{4} x^{2} + \frac {1}{5} \, A a^{4} x^{5} e^{4} + A a^{4} d x^{4} e^{3} + 2 \, A a^{4} d^{2} x^{3} e^{2} + 2 \, A a^{4} d^{3} x^{2} e + A a^{4} d^{4} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^4*(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="giac")

[Out]

1/10*B*b^4*x^10*e^4 + 4/9*B*b^4*d*x^9*e^3 + 3/4*B*b^4*d^2*x^8*e^2 + 4/7*B*b^4*d^3*x^7*e + 1/6*B*b^4*d^4*x^6 +
4/9*B*a*b^3*x^9*e^4 + 1/9*A*b^4*x^9*e^4 + 2*B*a*b^3*d*x^8*e^3 + 1/2*A*b^4*d*x^8*e^3 + 24/7*B*a*b^3*d^2*x^7*e^2
+ 6/7*A*b^4*d^2*x^7*e^2 + 8/3*B*a*b^3*d^3*x^6*e + 2/3*A*b^4*d^3*x^6*e + 4/5*B*a*b^3*d^4*x^5 + 1/5*A*b^4*d^4*x
^5 + 3/4*B*a^2*b^2*x^8*e^4 + 1/2*A*a*b^3*x^8*e^4 + 24/7*B*a^2*b^2*d*x^7*e^3 + 16/7*A*a*b^3*d*x^7*e^3 + 6*B*a^2
*b^2*d^2*x^6*e^2 + 4*A*a*b^3*d^2*x^6*e^2 + 24/5*B*a^2*b^2*d^3*x^5*e + 16/5*A*a*b^3*d^3*x^5*e + 3/2*B*a^2*b^2*d
^4*x^4 + A*a*b^3*d^4*x^4 + 4/7*B*a^3*b*x^7*e^4 + 6/7*A*a^2*b^2*x^7*e^4 + 8/3*B*a^3*b*d*x^6*e^3 + 4*A*a^2*b^2*d
*x^6*e^3 + 24/5*B*a^3*b*d^2*x^5*e^2 + 36/5*A*a^2*b^2*d^2*x^5*e^2 + 4*B*a^3*b*d^3*x^4*e + 6*A*a^2*b^2*d^3*x^4*e
+ 4/3*B*a^3*b*d^4*x^3 + 2*A*a^2*b^2*d^4*x^3 + 1/6*B*a^4*x^6*e^4 + 2/3*A*a^3*b*x^6*e^4 + 4/5*B*a^4*d*x^5*e^3 +
16/5*A*a^3*b*d*x^5*e^3 + 3/2*B*a^4*d^2*x^4*e^2 + 6*A*a^3*b*d^2*x^4*e^2 + 4/3*B*a^4*d^3*x^3*e + 16/3*A*a^3*b*d
^3*x^3*e + 1/2*B*a^4*d^4*x^2 + 2*A*a^3*b*d^4*x^2 + 1/5*A*a^4*x^5*e^4 + A*a^4*d*x^4*e^3 + 2*A*a^4*d^2*x^3*e^2 +
2*A*a^4*d^3*x^2*e + A*a^4*d^4*x

________________________________________________________________________________________

maple [B]  time = 0.05, size = 563, normalized size = 2.76 \begin {gather*} \frac {B \,b^{4} e^{4} x^{10}}{10}+A \,a^{4} d^{4} x +\frac {\left (4 B a \,b^{3} e^{4}+\left (A \,e^{4}+4 B d \,e^{3}\right ) b^{4}\right ) x^{9}}{9}+\frac {\left (6 B \,a^{2} b^{2} e^{4}+4 \left (A \,e^{4}+4 B d \,e^{3}\right ) a \,b^{3}+\left (4 A d \,e^{3}+6 B \,d^{2} e^{2}\right ) b^{4}\right ) x^{8}}{8}+\frac {\left (4 B \,a^{3} b \,e^{4}+6 \left (A \,e^{4}+4 B d \,e^{3}\right ) a^{2} b^{2}+4 \left (4 A d \,e^{3}+6 B \,d^{2} e^{2}\right ) a \,b^{3}+\left (6 A \,d^{2} e^{2}+4 B \,d^{3} e \right ) b^{4}\right ) x^{7}}{7}+\frac {\left (B \,a^{4} e^{4}+4 \left (A \,e^{4}+4 B d \,e^{3}\right ) a^{3} b +6 \left (4 A d \,e^{3}+6 B \,d^{2} e^{2}\right ) a^{2} b^{2}+4 \left (6 A \,d^{2} e^{2}+4 B \,d^{3} e \right ) a \,b^{3}+\left (4 A \,d^{3} e +B \,d^{4}\right ) b^{4}\right ) x^{6}}{6}+\frac {\left (A \,b^{4} d^{4}+\left (A \,e^{4}+4 B d \,e^{3}\right ) a^{4}+4 \left (4 A d \,e^{3}+6 B \,d^{2} e^{2}\right ) a^{3} b +6 \left (6 A \,d^{2} e^{2}+4 B \,d^{3} e \right ) a^{2} b^{2}+4 \left (4 A \,d^{3} e +B \,d^{4}\right ) a \,b^{3}\right ) x^{5}}{5}+\frac {\left (4 A a \,b^{3} d^{4}+\left (4 A d \,e^{3}+6 B \,d^{2} e^{2}\right ) a^{4}+4 \left (6 A \,d^{2} e^{2}+4 B \,d^{3} e \right ) a^{3} b +6 \left (4 A \,d^{3} e +B \,d^{4}\right ) a^{2} b^{2}\right ) x^{4}}{4}+\frac {\left (6 A \,a^{2} b^{2} d^{4}+\left (6 A \,d^{2} e^{2}+4 B \,d^{3} e \right ) a^{4}+4 \left (4 A \,d^{3} e +B \,d^{4}\right ) a^{3} b \right ) x^{3}}{3}+\frac {\left (4 A \,a^{3} b \,d^{4}+\left (4 A \,d^{3} e +B \,d^{4}\right ) a^{4}\right ) x^{2}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(e*x+d)^4*(b^2*x^2+2*a*b*x+a^2)^2,x)

[Out]

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

________________________________________________________________________________________

maxima [B]  time = 0.56, size = 562, normalized size = 2.75 \begin {gather*} \frac {1}{10} \, B b^{4} e^{4} x^{10} + A a^{4} d^{4} x + \frac {1}{9} \, {\left (4 \, B b^{4} d e^{3} + {\left (4 \, B a b^{3} + A b^{4}\right )} e^{4}\right )} x^{9} + \frac {1}{4} \, {\left (3 \, B b^{4} d^{2} e^{2} + 2 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d e^{3} + {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} e^{4}\right )} x^{8} + \frac {2}{7} \, {\left (2 \, B b^{4} d^{3} e + 3 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d^{2} e^{2} + 4 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d e^{3} + {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} e^{4}\right )} x^{7} + \frac {1}{6} \, {\left (B b^{4} d^{4} + 4 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d^{3} e + 12 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{2} e^{2} + 8 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d e^{3} + {\left (B a^{4} + 4 \, A a^{3} b\right )} e^{4}\right )} x^{6} + \frac {1}{5} \, {\left (A a^{4} e^{4} + {\left (4 \, B a b^{3} + A b^{4}\right )} d^{4} + 8 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{3} e + 12 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d^{2} e^{2} + 4 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} d e^{3}\right )} x^{5} + \frac {1}{2} \, {\left (2 \, A a^{4} d e^{3} + {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{4} + 4 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d^{3} e + 3 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} d^{2} e^{2}\right )} x^{4} + \frac {2}{3} \, {\left (3 \, A a^{4} d^{2} e^{2} + {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d^{4} + 2 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} d^{3} e\right )} x^{3} + \frac {1}{2} \, {\left (4 \, A a^{4} d^{3} e + {\left (B a^{4} + 4 \, A a^{3} b\right )} d^{4}\right )} x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^4*(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="maxima")

[Out]

1/10*B*b^4*e^4*x^10 + A*a^4*d^4*x + 1/9*(4*B*b^4*d*e^3 + (4*B*a*b^3 + A*b^4)*e^4)*x^9 + 1/4*(3*B*b^4*d^2*e^2 +
2*(4*B*a*b^3 + A*b^4)*d*e^3 + (3*B*a^2*b^2 + 2*A*a*b^3)*e^4)*x^8 + 2/7*(2*B*b^4*d^3*e + 3*(4*B*a*b^3 + A*b^4)
*d^2*e^2 + 4*(3*B*a^2*b^2 + 2*A*a*b^3)*d*e^3 + (2*B*a^3*b + 3*A*a^2*b^2)*e^4)*x^7 + 1/6*(B*b^4*d^4 + 4*(4*B*a*
b^3 + A*b^4)*d^3*e + 12*(3*B*a^2*b^2 + 2*A*a*b^3)*d^2*e^2 + 8*(2*B*a^3*b + 3*A*a^2*b^2)*d*e^3 + (B*a^4 + 4*A*a
^3*b)*e^4)*x^6 + 1/5*(A*a^4*e^4 + (4*B*a*b^3 + A*b^4)*d^4 + 8*(3*B*a^2*b^2 + 2*A*a*b^3)*d^3*e + 12*(2*B*a^3*b
+ 3*A*a^2*b^2)*d^2*e^2 + 4*(B*a^4 + 4*A*a^3*b)*d*e^3)*x^5 + 1/2*(2*A*a^4*d*e^3 + (3*B*a^2*b^2 + 2*A*a*b^3)*d^4
+ 4*(2*B*a^3*b + 3*A*a^2*b^2)*d^3*e + 3*(B*a^4 + 4*A*a^3*b)*d^2*e^2)*x^4 + 2/3*(3*A*a^4*d^2*e^2 + (2*B*a^3*b
+ 3*A*a^2*b^2)*d^4 + 2*(B*a^4 + 4*A*a^3*b)*d^3*e)*x^3 + 1/2*(4*A*a^4*d^3*e + (B*a^4 + 4*A*a^3*b)*d^4)*x^2

________________________________________________________________________________________

mupad [B]  time = 2.17, size = 576, normalized size = 2.82 \begin {gather*} x^5\,\left (\frac {4\,B\,a^4\,d\,e^3}{5}+\frac {A\,a^4\,e^4}{5}+\frac {24\,B\,a^3\,b\,d^2\,e^2}{5}+\frac {16\,A\,a^3\,b\,d\,e^3}{5}+\frac {24\,B\,a^2\,b^2\,d^3\,e}{5}+\frac {36\,A\,a^2\,b^2\,d^2\,e^2}{5}+\frac {4\,B\,a\,b^3\,d^4}{5}+\frac {16\,A\,a\,b^3\,d^3\,e}{5}+\frac {A\,b^4\,d^4}{5}\right )+x^6\,\left (\frac {B\,a^4\,e^4}{6}+\frac {8\,B\,a^3\,b\,d\,e^3}{3}+\frac {2\,A\,a^3\,b\,e^4}{3}+6\,B\,a^2\,b^2\,d^2\,e^2+4\,A\,a^2\,b^2\,d\,e^3+\frac {8\,B\,a\,b^3\,d^3\,e}{3}+4\,A\,a\,b^3\,d^2\,e^2+\frac {B\,b^4\,d^4}{6}+\frac {2\,A\,b^4\,d^3\,e}{3}\right )+x^3\,\left (\frac {4\,B\,a^4\,d^3\,e}{3}+2\,A\,a^4\,d^2\,e^2+\frac {4\,B\,a^3\,b\,d^4}{3}+\frac {16\,A\,a^3\,b\,d^3\,e}{3}+2\,A\,a^2\,b^2\,d^4\right )+x^8\,\left (\frac {3\,B\,a^2\,b^2\,e^4}{4}+2\,B\,a\,b^3\,d\,e^3+\frac {A\,a\,b^3\,e^4}{2}+\frac {3\,B\,b^4\,d^2\,e^2}{4}+\frac {A\,b^4\,d\,e^3}{2}\right )+x^4\,\left (\frac {3\,B\,a^4\,d^2\,e^2}{2}+A\,a^4\,d\,e^3+4\,B\,a^3\,b\,d^3\,e+6\,A\,a^3\,b\,d^2\,e^2+\frac {3\,B\,a^2\,b^2\,d^4}{2}+6\,A\,a^2\,b^2\,d^3\,e+A\,a\,b^3\,d^4\right )+x^7\,\left (\frac {4\,B\,a^3\,b\,e^4}{7}+\frac {24\,B\,a^2\,b^2\,d\,e^3}{7}+\frac {6\,A\,a^2\,b^2\,e^4}{7}+\frac {24\,B\,a\,b^3\,d^2\,e^2}{7}+\frac {16\,A\,a\,b^3\,d\,e^3}{7}+\frac {4\,B\,b^4\,d^3\,e}{7}+\frac {6\,A\,b^4\,d^2\,e^2}{7}\right )+\frac {a^3\,d^3\,x^2\,\left (4\,A\,a\,e+4\,A\,b\,d+B\,a\,d\right )}{2}+\frac {b^3\,e^3\,x^9\,\left (A\,b\,e+4\,B\,a\,e+4\,B\,b\,d\right )}{9}+A\,a^4\,d^4\,x+\frac {B\,b^4\,e^4\,x^{10}}{10} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*x)*(d + e*x)^4*(a^2 + b^2*x^2 + 2*a*b*x)^2,x)

[Out]

x^5*((A*a^4*e^4)/5 + (A*b^4*d^4)/5 + (4*B*a*b^3*d^4)/5 + (4*B*a^4*d*e^3)/5 + (24*B*a^2*b^2*d^3*e)/5 + (24*B*a^
3*b*d^2*e^2)/5 + (36*A*a^2*b^2*d^2*e^2)/5 + (16*A*a*b^3*d^3*e)/5 + (16*A*a^3*b*d*e^3)/5) + x^6*((B*a^4*e^4)/6
+ (B*b^4*d^4)/6 + (2*A*a^3*b*e^4)/3 + (2*A*b^4*d^3*e)/3 + 4*A*a*b^3*d^2*e^2 + 4*A*a^2*b^2*d*e^3 + 6*B*a^2*b^2*
d^2*e^2 + (8*B*a*b^3*d^3*e)/3 + (8*B*a^3*b*d*e^3)/3) + x^3*((4*B*a^3*b*d^4)/3 + (4*B*a^4*d^3*e)/3 + 2*A*a^2*b^
2*d^4 + 2*A*a^4*d^2*e^2 + (16*A*a^3*b*d^3*e)/3) + x^8*((A*a*b^3*e^4)/2 + (A*b^4*d*e^3)/2 + (3*B*a^2*b^2*e^4)/4
+ (3*B*b^4*d^2*e^2)/4 + 2*B*a*b^3*d*e^3) + x^4*(A*a*b^3*d^4 + A*a^4*d*e^3 + (3*B*a^2*b^2*d^4)/2 + (3*B*a^4*d^
2*e^2)/2 + 6*A*a^2*b^2*d^3*e + 6*A*a^3*b*d^2*e^2 + 4*B*a^3*b*d^3*e) + x^7*((4*B*a^3*b*e^4)/7 + (4*B*b^4*d^3*e)
/7 + (6*A*a^2*b^2*e^4)/7 + (6*A*b^4*d^2*e^2)/7 + (24*B*a*b^3*d^2*e^2)/7 + (24*B*a^2*b^2*d*e^3)/7 + (16*A*a*b^3
*d*e^3)/7) + (a^3*d^3*x^2*(4*A*a*e + 4*A*b*d + B*a*d))/2 + (b^3*e^3*x^9*(A*b*e + 4*B*a*e + 4*B*b*d))/9 + A*a^4
*d^4*x + (B*b^4*e^4*x^10)/10

________________________________________________________________________________________

sympy [B]  time = 0.16, size = 717, normalized size = 3.51 \begin {gather*} A a^{4} d^{4} x + \frac {B b^{4} e^{4} x^{10}}{10} + x^{9} \left (\frac {A b^{4} e^{4}}{9} + \frac {4 B a b^{3} e^{4}}{9} + \frac {4 B b^{4} d e^{3}}{9}\right ) + x^{8} \left (\frac {A a b^{3} e^{4}}{2} + \frac {A b^{4} d e^{3}}{2} + \frac {3 B a^{2} b^{2} e^{4}}{4} + 2 B a b^{3} d e^{3} + \frac {3 B b^{4} d^{2} e^{2}}{4}\right ) + x^{7} \left (\frac {6 A a^{2} b^{2} e^{4}}{7} + \frac {16 A a b^{3} d e^{3}}{7} + \frac {6 A b^{4} d^{2} e^{2}}{7} + \frac {4 B a^{3} b e^{4}}{7} + \frac {24 B a^{2} b^{2} d e^{3}}{7} + \frac {24 B a b^{3} d^{2} e^{2}}{7} + \frac {4 B b^{4} d^{3} e}{7}\right ) + x^{6} \left (\frac {2 A a^{3} b e^{4}}{3} + 4 A a^{2} b^{2} d e^{3} + 4 A a b^{3} d^{2} e^{2} + \frac {2 A b^{4} d^{3} e}{3} + \frac {B a^{4} e^{4}}{6} + \frac {8 B a^{3} b d e^{3}}{3} + 6 B a^{2} b^{2} d^{2} e^{2} + \frac {8 B a b^{3} d^{3} e}{3} + \frac {B b^{4} d^{4}}{6}\right ) + x^{5} \left (\frac {A a^{4} e^{4}}{5} + \frac {16 A a^{3} b d e^{3}}{5} + \frac {36 A a^{2} b^{2} d^{2} e^{2}}{5} + \frac {16 A a b^{3} d^{3} e}{5} + \frac {A b^{4} d^{4}}{5} + \frac {4 B a^{4} d e^{3}}{5} + \frac {24 B a^{3} b d^{2} e^{2}}{5} + \frac {24 B a^{2} b^{2} d^{3} e}{5} + \frac {4 B a b^{3} d^{4}}{5}\right ) + x^{4} \left (A a^{4} d e^{3} + 6 A a^{3} b d^{2} e^{2} + 6 A a^{2} b^{2} d^{3} e + A a b^{3} d^{4} + \frac {3 B a^{4} d^{2} e^{2}}{2} + 4 B a^{3} b d^{3} e + \frac {3 B a^{2} b^{2} d^{4}}{2}\right ) + x^{3} \left (2 A a^{4} d^{2} e^{2} + \frac {16 A a^{3} b d^{3} e}{3} + 2 A a^{2} b^{2} d^{4} + \frac {4 B a^{4} d^{3} e}{3} + \frac {4 B a^{3} b d^{4}}{3}\right ) + x^{2} \left (2 A a^{4} d^{3} e + 2 A a^{3} b d^{4} + \frac {B a^{4} d^{4}}{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)**4*(b**2*x**2+2*a*b*x+a**2)**2,x)

[Out]

A*a**4*d**4*x + B*b**4*e**4*x**10/10 + x**9*(A*b**4*e**4/9 + 4*B*a*b**3*e**4/9 + 4*B*b**4*d*e**3/9) + x**8*(A*
a*b**3*e**4/2 + A*b**4*d*e**3/2 + 3*B*a**2*b**2*e**4/4 + 2*B*a*b**3*d*e**3 + 3*B*b**4*d**2*e**2/4) + x**7*(6*A
*a**2*b**2*e**4/7 + 16*A*a*b**3*d*e**3/7 + 6*A*b**4*d**2*e**2/7 + 4*B*a**3*b*e**4/7 + 24*B*a**2*b**2*d*e**3/7
+ 24*B*a*b**3*d**2*e**2/7 + 4*B*b**4*d**3*e/7) + x**6*(2*A*a**3*b*e**4/3 + 4*A*a**2*b**2*d*e**3 + 4*A*a*b**3*d
**2*e**2 + 2*A*b**4*d**3*e/3 + B*a**4*e**4/6 + 8*B*a**3*b*d*e**3/3 + 6*B*a**2*b**2*d**2*e**2 + 8*B*a*b**3*d**3
*e/3 + B*b**4*d**4/6) + x**5*(A*a**4*e**4/5 + 16*A*a**3*b*d*e**3/5 + 36*A*a**2*b**2*d**2*e**2/5 + 16*A*a*b**3*
d**3*e/5 + A*b**4*d**4/5 + 4*B*a**4*d*e**3/5 + 24*B*a**3*b*d**2*e**2/5 + 24*B*a**2*b**2*d**3*e/5 + 4*B*a*b**3*
d**4/5) + x**4*(A*a**4*d*e**3 + 6*A*a**3*b*d**2*e**2 + 6*A*a**2*b**2*d**3*e + A*a*b**3*d**4 + 3*B*a**4*d**2*e*
*2/2 + 4*B*a**3*b*d**3*e + 3*B*a**2*b**2*d**4/2) + x**3*(2*A*a**4*d**2*e**2 + 16*A*a**3*b*d**3*e/3 + 2*A*a**2*
b**2*d**4 + 4*B*a**4*d**3*e/3 + 4*B*a**3*b*d**4/3) + x**2*(2*A*a**4*d**3*e + 2*A*a**3*b*d**4 + B*a**4*d**4/2)

________________________________________________________________________________________