3.15.57 \(\int (A+B x) (d+e x)^3 (a^2+2 a b x+b^2 x^2)^2 \, dx\)

Optimal. Leaf size=159 \[ \frac {e^2 (a+b x)^8 (-4 a B e+A b e+3 b B d)}{8 b^5}+\frac {3 e (a+b x)^7 (b d-a e) (-2 a B e+A b e+b B d)}{7 b^5}+\frac {(a+b x)^6 (b d-a e)^2 (-4 a B e+3 A b e+b B d)}{6 b^5}+\frac {(a+b x)^5 (A b-a B) (b d-a e)^3}{5 b^5}+\frac {B e^3 (a+b x)^9}{9 b^5} \]

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Rubi [A]  time = 0.31, antiderivative size = 159, 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^2 (a+b x)^8 (-4 a B e+A b e+3 b B d)}{8 b^5}+\frac {3 e (a+b x)^7 (b d-a e) (-2 a B e+A b e+b B d)}{7 b^5}+\frac {(a+b x)^6 (b d-a e)^2 (-4 a B e+3 A b e+b B d)}{6 b^5}+\frac {(a+b x)^5 (A b-a B) (b d-a e)^3}{5 b^5}+\frac {B e^3 (a+b x)^9}{9 b^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(A + B*x)*(d + e*x)^3*(a^2 + 2*a*b*x + b^2*x^2)^2,x]

[Out]

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

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

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Mathematica [B]  time = 0.14, size = 402, normalized size = 2.53 \begin {gather*} a^4 A d^3 x+\frac {1}{2} a^3 d^2 x^2 (3 a A e+a B d+4 A b d)+\frac {1}{3} a^2 d x^3 \left (3 A \left (a^2 e^2+4 a b d e+2 b^2 d^2\right )+a B d (3 a e+4 b d)\right )+\frac {1}{7} b^2 e x^7 \left (6 a^2 B e^2+4 a b e (A e+3 B d)+3 b^2 d (A e+B d)\right )+\frac {1}{6} b x^6 \left (4 a^3 B e^3+6 a^2 b e^2 (A e+3 B d)+12 a b^2 d e (A e+B d)+b^3 d^2 (3 A e+B d)\right )+\frac {1}{5} x^5 \left (A b \left (4 a^3 e^3+18 a^2 b d e^2+12 a b^2 d^2 e+b^3 d^3\right )+a B \left (a^3 e^3+12 a^2 b d e^2+18 a b^2 d^2 e+4 b^3 d^3\right )\right )+\frac {1}{4} a x^4 \left (3 a B d \left (a^2 e^2+4 a b d e+2 b^2 d^2\right )+A \left (a^3 e^3+12 a^2 b d e^2+18 a b^2 d^2 e+4 b^3 d^3\right )\right )+\frac {1}{8} b^3 e^2 x^8 (4 a B e+A b e+3 b B d)+\frac {1}{9} b^4 B e^3 x^9 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x)*(d + e*x)^3*(a^2 + 2*a*b*x + b^2*x^2)^2,x]

[Out]

a^4*A*d^3*x + (a^3*d^2*(4*A*b*d + a*B*d + 3*a*A*e)*x^2)/2 + (a^2*d*(a*B*d*(4*b*d + 3*a*e) + 3*A*(2*b^2*d^2 + 4
*a*b*d*e + a^2*e^2))*x^3)/3 + (a*(3*a*B*d*(2*b^2*d^2 + 4*a*b*d*e + a^2*e^2) + A*(4*b^3*d^3 + 18*a*b^2*d^2*e +
12*a^2*b*d*e^2 + a^3*e^3))*x^4)/4 + ((a*B*(4*b^3*d^3 + 18*a*b^2*d^2*e + 12*a^2*b*d*e^2 + a^3*e^3) + A*b*(b^3*d
^3 + 12*a*b^2*d^2*e + 18*a^2*b*d*e^2 + 4*a^3*e^3))*x^5)/5 + (b*(4*a^3*B*e^3 + 12*a*b^2*d*e*(B*d + A*e) + 6*a^2
*b*e^2*(3*B*d + A*e) + b^3*d^2*(B*d + 3*A*e))*x^6)/6 + (b^2*e*(6*a^2*B*e^2 + 3*b^2*d*(B*d + A*e) + 4*a*b*e*(3*
B*d + A*e))*x^7)/7 + (b^3*e^2*(3*b*B*d + A*b*e + 4*a*B*e)*x^8)/8 + (b^4*B*e^3*x^9)/9

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int (A+B x) (d+e x)^3 \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)^3*(a^2 + 2*a*b*x + b^2*x^2)^2,x]

[Out]

IntegrateAlgebraic[(A + B*x)*(d + e*x)^3*(a^2 + 2*a*b*x + b^2*x^2)^2, x]

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fricas [B]  time = 0.37, size = 534, normalized size = 3.36 \begin {gather*} \frac {1}{9} x^{9} e^{3} b^{4} B + \frac {3}{8} x^{8} e^{2} d b^{4} B + \frac {1}{2} x^{8} e^{3} b^{3} a B + \frac {1}{8} x^{8} e^{3} b^{4} A + \frac {3}{7} x^{7} e d^{2} b^{4} B + \frac {12}{7} x^{7} e^{2} d b^{3} a B + \frac {6}{7} x^{7} e^{3} b^{2} a^{2} B + \frac {3}{7} x^{7} e^{2} d b^{4} A + \frac {4}{7} x^{7} e^{3} b^{3} a A + \frac {1}{6} x^{6} d^{3} b^{4} B + 2 x^{6} e d^{2} b^{3} a B + 3 x^{6} e^{2} d b^{2} a^{2} B + \frac {2}{3} x^{6} e^{3} b a^{3} B + \frac {1}{2} x^{6} e d^{2} b^{4} A + 2 x^{6} e^{2} d b^{3} a A + x^{6} e^{3} b^{2} a^{2} A + \frac {4}{5} x^{5} d^{3} b^{3} a B + \frac {18}{5} x^{5} e d^{2} b^{2} a^{2} B + \frac {12}{5} x^{5} e^{2} d b a^{3} B + \frac {1}{5} x^{5} e^{3} a^{4} B + \frac {1}{5} x^{5} d^{3} b^{4} A + \frac {12}{5} x^{5} e d^{2} b^{3} a A + \frac {18}{5} x^{5} e^{2} d b^{2} a^{2} A + \frac {4}{5} x^{5} e^{3} b a^{3} A + \frac {3}{2} x^{4} d^{3} b^{2} a^{2} B + 3 x^{4} e d^{2} b a^{3} B + \frac {3}{4} x^{4} e^{2} d a^{4} B + x^{4} d^{3} b^{3} a A + \frac {9}{2} x^{4} e d^{2} b^{2} a^{2} A + 3 x^{4} e^{2} d b a^{3} A + \frac {1}{4} x^{4} e^{3} a^{4} A + \frac {4}{3} x^{3} d^{3} b a^{3} B + x^{3} e d^{2} a^{4} B + 2 x^{3} d^{3} b^{2} a^{2} A + 4 x^{3} e d^{2} b a^{3} A + x^{3} e^{2} d a^{4} A + \frac {1}{2} x^{2} d^{3} a^{4} B + 2 x^{2} d^{3} b a^{3} A + \frac {3}{2} x^{2} e d^{2} a^{4} A + x d^{3} a^{4} A \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^3*(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="fricas")

[Out]

1/9*x^9*e^3*b^4*B + 3/8*x^8*e^2*d*b^4*B + 1/2*x^8*e^3*b^3*a*B + 1/8*x^8*e^3*b^4*A + 3/7*x^7*e*d^2*b^4*B + 12/7
*x^7*e^2*d*b^3*a*B + 6/7*x^7*e^3*b^2*a^2*B + 3/7*x^7*e^2*d*b^4*A + 4/7*x^7*e^3*b^3*a*A + 1/6*x^6*d^3*b^4*B + 2
*x^6*e*d^2*b^3*a*B + 3*x^6*e^2*d*b^2*a^2*B + 2/3*x^6*e^3*b*a^3*B + 1/2*x^6*e*d^2*b^4*A + 2*x^6*e^2*d*b^3*a*A +
x^6*e^3*b^2*a^2*A + 4/5*x^5*d^3*b^3*a*B + 18/5*x^5*e*d^2*b^2*a^2*B + 12/5*x^5*e^2*d*b*a^3*B + 1/5*x^5*e^3*a^4
*B + 1/5*x^5*d^3*b^4*A + 12/5*x^5*e*d^2*b^3*a*A + 18/5*x^5*e^2*d*b^2*a^2*A + 4/5*x^5*e^3*b*a^3*A + 3/2*x^4*d^3
*b^2*a^2*B + 3*x^4*e*d^2*b*a^3*B + 3/4*x^4*e^2*d*a^4*B + x^4*d^3*b^3*a*A + 9/2*x^4*e*d^2*b^2*a^2*A + 3*x^4*e^2
*d*b*a^3*A + 1/4*x^4*e^3*a^4*A + 4/3*x^3*d^3*b*a^3*B + x^3*e*d^2*a^4*B + 2*x^3*d^3*b^2*a^2*A + 4*x^3*e*d^2*b*a
^3*A + x^3*e^2*d*a^4*A + 1/2*x^2*d^3*a^4*B + 2*x^2*d^3*b*a^3*A + 3/2*x^2*e*d^2*a^4*A + x*d^3*a^4*A

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giac [B]  time = 0.17, size = 524, normalized size = 3.30 \begin {gather*} \frac {1}{9} \, B b^{4} x^{9} e^{3} + \frac {3}{8} \, B b^{4} d x^{8} e^{2} + \frac {3}{7} \, B b^{4} d^{2} x^{7} e + \frac {1}{6} \, B b^{4} d^{3} x^{6} + \frac {1}{2} \, B a b^{3} x^{8} e^{3} + \frac {1}{8} \, A b^{4} x^{8} e^{3} + \frac {12}{7} \, B a b^{3} d x^{7} e^{2} + \frac {3}{7} \, A b^{4} d x^{7} e^{2} + 2 \, B a b^{3} d^{2} x^{6} e + \frac {1}{2} \, A b^{4} d^{2} x^{6} e + \frac {4}{5} \, B a b^{3} d^{3} x^{5} + \frac {1}{5} \, A b^{4} d^{3} x^{5} + \frac {6}{7} \, B a^{2} b^{2} x^{7} e^{3} + \frac {4}{7} \, A a b^{3} x^{7} e^{3} + 3 \, B a^{2} b^{2} d x^{6} e^{2} + 2 \, A a b^{3} d x^{6} e^{2} + \frac {18}{5} \, B a^{2} b^{2} d^{2} x^{5} e + \frac {12}{5} \, A a b^{3} d^{2} x^{5} e + \frac {3}{2} \, B a^{2} b^{2} d^{3} x^{4} + A a b^{3} d^{3} x^{4} + \frac {2}{3} \, B a^{3} b x^{6} e^{3} + A a^{2} b^{2} x^{6} e^{3} + \frac {12}{5} \, B a^{3} b d x^{5} e^{2} + \frac {18}{5} \, A a^{2} b^{2} d x^{5} e^{2} + 3 \, B a^{3} b d^{2} x^{4} e + \frac {9}{2} \, A a^{2} b^{2} d^{2} x^{4} e + \frac {4}{3} \, B a^{3} b d^{3} x^{3} + 2 \, A a^{2} b^{2} d^{3} x^{3} + \frac {1}{5} \, B a^{4} x^{5} e^{3} + \frac {4}{5} \, A a^{3} b x^{5} e^{3} + \frac {3}{4} \, B a^{4} d x^{4} e^{2} + 3 \, A a^{3} b d x^{4} e^{2} + B a^{4} d^{2} x^{3} e + 4 \, A a^{3} b d^{2} x^{3} e + \frac {1}{2} \, B a^{4} d^{3} x^{2} + 2 \, A a^{3} b d^{3} x^{2} + \frac {1}{4} \, A a^{4} x^{4} e^{3} + A a^{4} d x^{3} e^{2} + \frac {3}{2} \, A a^{4} d^{2} x^{2} e + A a^{4} d^{3} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^3*(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="giac")

[Out]

1/9*B*b^4*x^9*e^3 + 3/8*B*b^4*d*x^8*e^2 + 3/7*B*b^4*d^2*x^7*e + 1/6*B*b^4*d^3*x^6 + 1/2*B*a*b^3*x^8*e^3 + 1/8*
A*b^4*x^8*e^3 + 12/7*B*a*b^3*d*x^7*e^2 + 3/7*A*b^4*d*x^7*e^2 + 2*B*a*b^3*d^2*x^6*e + 1/2*A*b^4*d^2*x^6*e + 4/5
*B*a*b^3*d^3*x^5 + 1/5*A*b^4*d^3*x^5 + 6/7*B*a^2*b^2*x^7*e^3 + 4/7*A*a*b^3*x^7*e^3 + 3*B*a^2*b^2*d*x^6*e^2 + 2
*A*a*b^3*d*x^6*e^2 + 18/5*B*a^2*b^2*d^2*x^5*e + 12/5*A*a*b^3*d^2*x^5*e + 3/2*B*a^2*b^2*d^3*x^4 + A*a*b^3*d^3*x
^4 + 2/3*B*a^3*b*x^6*e^3 + A*a^2*b^2*x^6*e^3 + 12/5*B*a^3*b*d*x^5*e^2 + 18/5*A*a^2*b^2*d*x^5*e^2 + 3*B*a^3*b*d
^2*x^4*e + 9/2*A*a^2*b^2*d^2*x^4*e + 4/3*B*a^3*b*d^3*x^3 + 2*A*a^2*b^2*d^3*x^3 + 1/5*B*a^4*x^5*e^3 + 4/5*A*a^3
*b*x^5*e^3 + 3/4*B*a^4*d*x^4*e^2 + 3*A*a^3*b*d*x^4*e^2 + B*a^4*d^2*x^3*e + 4*A*a^3*b*d^2*x^3*e + 1/2*B*a^4*d^3
*x^2 + 2*A*a^3*b*d^3*x^2 + 1/4*A*a^4*x^4*e^3 + A*a^4*d*x^3*e^2 + 3/2*A*a^4*d^2*x^2*e + A*a^4*d^3*x

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maple [B]  time = 0.04, size = 434, normalized size = 2.73 \begin {gather*} \frac {B \,b^{4} e^{3} x^{9}}{9}+A \,a^{4} d^{3} x +\frac {\left (4 B a \,b^{3} e^{3}+\left (A \,e^{3}+3 B d \,e^{2}\right ) b^{4}\right ) x^{8}}{8}+\frac {\left (6 B \,a^{2} b^{2} e^{3}+4 \left (A \,e^{3}+3 B d \,e^{2}\right ) a \,b^{3}+\left (3 A d \,e^{2}+3 B \,d^{2} e \right ) b^{4}\right ) x^{7}}{7}+\frac {\left (4 B \,a^{3} b \,e^{3}+6 \left (A \,e^{3}+3 B d \,e^{2}\right ) a^{2} b^{2}+4 \left (3 A d \,e^{2}+3 B \,d^{2} e \right ) a \,b^{3}+\left (3 A \,d^{2} e +B \,d^{3}\right ) b^{4}\right ) x^{6}}{6}+\frac {\left (A \,b^{4} d^{3}+B \,a^{4} e^{3}+4 \left (A \,e^{3}+3 B d \,e^{2}\right ) a^{3} b +6 \left (3 A d \,e^{2}+3 B \,d^{2} e \right ) a^{2} b^{2}+4 \left (3 A \,d^{2} e +B \,d^{3}\right ) a \,b^{3}\right ) x^{5}}{5}+\frac {\left (4 A a \,b^{3} d^{3}+\left (A \,e^{3}+3 B d \,e^{2}\right ) a^{4}+4 \left (3 A d \,e^{2}+3 B \,d^{2} e \right ) a^{3} b +6 \left (3 A \,d^{2} e +B \,d^{3}\right ) a^{2} b^{2}\right ) x^{4}}{4}+\frac {\left (6 A \,a^{2} b^{2} d^{3}+\left (3 A d \,e^{2}+3 B \,d^{2} e \right ) a^{4}+4 \left (3 A \,d^{2} e +B \,d^{3}\right ) a^{3} b \right ) x^{3}}{3}+\frac {\left (4 A \,a^{3} b \,d^{3}+\left (3 A \,d^{2} e +B \,d^{3}\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)^3*(b^2*x^2+2*a*b*x+a^2)^2,x)

[Out]

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

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maxima [B]  time = 0.50, size = 444, normalized size = 2.79 \begin {gather*} \frac {1}{9} \, B b^{4} e^{3} x^{9} + A a^{4} d^{3} x + \frac {1}{8} \, {\left (3 \, B b^{4} d e^{2} + {\left (4 \, B a b^{3} + A b^{4}\right )} e^{3}\right )} x^{8} + \frac {1}{7} \, {\left (3 \, B b^{4} d^{2} e + 3 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d e^{2} + 2 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} e^{3}\right )} x^{7} + \frac {1}{6} \, {\left (B b^{4} d^{3} + 3 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d^{2} e + 6 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d e^{2} + 2 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} e^{3}\right )} x^{6} + \frac {1}{5} \, {\left ({\left (4 \, B a b^{3} + A b^{4}\right )} d^{3} + 6 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{2} e + 6 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d e^{2} + {\left (B a^{4} + 4 \, A a^{3} b\right )} e^{3}\right )} x^{5} + \frac {1}{4} \, {\left (A a^{4} e^{3} + 2 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{3} + 6 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d^{2} e + 3 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} d e^{2}\right )} x^{4} + \frac {1}{3} \, {\left (3 \, A a^{4} d e^{2} + 2 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d^{3} + 3 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} d^{2} e\right )} x^{3} + \frac {1}{2} \, {\left (3 \, A a^{4} d^{2} e + {\left (B a^{4} + 4 \, A a^{3} b\right )} d^{3}\right )} x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^3*(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.15, size = 439, normalized size = 2.76 \begin {gather*} x^3\,\left (B\,a^4\,d^2\,e+A\,a^4\,d\,e^2+\frac {4\,B\,a^3\,b\,d^3}{3}+4\,A\,a^3\,b\,d^2\,e+2\,A\,a^2\,b^2\,d^3\right )+x^7\,\left (\frac {6\,B\,a^2\,b^2\,e^3}{7}+\frac {12\,B\,a\,b^3\,d\,e^2}{7}+\frac {4\,A\,a\,b^3\,e^3}{7}+\frac {3\,B\,b^4\,d^2\,e}{7}+\frac {3\,A\,b^4\,d\,e^2}{7}\right )+x^5\,\left (\frac {B\,a^4\,e^3}{5}+\frac {12\,B\,a^3\,b\,d\,e^2}{5}+\frac {4\,A\,a^3\,b\,e^3}{5}+\frac {18\,B\,a^2\,b^2\,d^2\,e}{5}+\frac {18\,A\,a^2\,b^2\,d\,e^2}{5}+\frac {4\,B\,a\,b^3\,d^3}{5}+\frac {12\,A\,a\,b^3\,d^2\,e}{5}+\frac {A\,b^4\,d^3}{5}\right )+x^4\,\left (\frac {3\,B\,a^4\,d\,e^2}{4}+\frac {A\,a^4\,e^3}{4}+3\,B\,a^3\,b\,d^2\,e+3\,A\,a^3\,b\,d\,e^2+\frac {3\,B\,a^2\,b^2\,d^3}{2}+\frac {9\,A\,a^2\,b^2\,d^2\,e}{2}+A\,a\,b^3\,d^3\right )+x^6\,\left (\frac {2\,B\,a^3\,b\,e^3}{3}+3\,B\,a^2\,b^2\,d\,e^2+A\,a^2\,b^2\,e^3+2\,B\,a\,b^3\,d^2\,e+2\,A\,a\,b^3\,d\,e^2+\frac {B\,b^4\,d^3}{6}+\frac {A\,b^4\,d^2\,e}{2}\right )+\frac {a^3\,d^2\,x^2\,\left (3\,A\,a\,e+4\,A\,b\,d+B\,a\,d\right )}{2}+\frac {b^3\,e^2\,x^8\,\left (A\,b\,e+4\,B\,a\,e+3\,B\,b\,d\right )}{8}+A\,a^4\,d^3\,x+\frac {B\,b^4\,e^3\,x^9}{9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*x)*(d + e*x)^3*(a^2 + b^2*x^2 + 2*a*b*x)^2,x)

[Out]

x^3*((4*B*a^3*b*d^3)/3 + A*a^4*d*e^2 + B*a^4*d^2*e + 2*A*a^2*b^2*d^3 + 4*A*a^3*b*d^2*e) + x^7*((4*A*a*b^3*e^3)
/7 + (3*A*b^4*d*e^2)/7 + (3*B*b^4*d^2*e)/7 + (6*B*a^2*b^2*e^3)/7 + (12*B*a*b^3*d*e^2)/7) + x^5*((A*b^4*d^3)/5
+ (B*a^4*e^3)/5 + (4*A*a^3*b*e^3)/5 + (4*B*a*b^3*d^3)/5 + (18*A*a^2*b^2*d*e^2)/5 + (18*B*a^2*b^2*d^2*e)/5 + (1
2*A*a*b^3*d^2*e)/5 + (12*B*a^3*b*d*e^2)/5) + x^4*((A*a^4*e^3)/4 + A*a*b^3*d^3 + (3*B*a^4*d*e^2)/4 + (3*B*a^2*b
^2*d^3)/2 + (9*A*a^2*b^2*d^2*e)/2 + 3*A*a^3*b*d*e^2 + 3*B*a^3*b*d^2*e) + x^6*((B*b^4*d^3)/6 + (2*B*a^3*b*e^3)/
3 + (A*b^4*d^2*e)/2 + A*a^2*b^2*e^3 + 3*B*a^2*b^2*d*e^2 + 2*A*a*b^3*d*e^2 + 2*B*a*b^3*d^2*e) + (a^3*d^2*x^2*(3
*A*a*e + 4*A*b*d + B*a*d))/2 + (b^3*e^2*x^8*(A*b*e + 4*B*a*e + 3*B*b*d))/8 + A*a^4*d^3*x + (B*b^4*e^3*x^9)/9

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sympy [B]  time = 0.14, size = 546, normalized size = 3.43 \begin {gather*} A a^{4} d^{3} x + \frac {B b^{4} e^{3} x^{9}}{9} + x^{8} \left (\frac {A b^{4} e^{3}}{8} + \frac {B a b^{3} e^{3}}{2} + \frac {3 B b^{4} d e^{2}}{8}\right ) + x^{7} \left (\frac {4 A a b^{3} e^{3}}{7} + \frac {3 A b^{4} d e^{2}}{7} + \frac {6 B a^{2} b^{2} e^{3}}{7} + \frac {12 B a b^{3} d e^{2}}{7} + \frac {3 B b^{4} d^{2} e}{7}\right ) + x^{6} \left (A a^{2} b^{2} e^{3} + 2 A a b^{3} d e^{2} + \frac {A b^{4} d^{2} e}{2} + \frac {2 B a^{3} b e^{3}}{3} + 3 B a^{2} b^{2} d e^{2} + 2 B a b^{3} d^{2} e + \frac {B b^{4} d^{3}}{6}\right ) + x^{5} \left (\frac {4 A a^{3} b e^{3}}{5} + \frac {18 A a^{2} b^{2} d e^{2}}{5} + \frac {12 A a b^{3} d^{2} e}{5} + \frac {A b^{4} d^{3}}{5} + \frac {B a^{4} e^{3}}{5} + \frac {12 B a^{3} b d e^{2}}{5} + \frac {18 B a^{2} b^{2} d^{2} e}{5} + \frac {4 B a b^{3} d^{3}}{5}\right ) + x^{4} \left (\frac {A a^{4} e^{3}}{4} + 3 A a^{3} b d e^{2} + \frac {9 A a^{2} b^{2} d^{2} e}{2} + A a b^{3} d^{3} + \frac {3 B a^{4} d e^{2}}{4} + 3 B a^{3} b d^{2} e + \frac {3 B a^{2} b^{2} d^{3}}{2}\right ) + x^{3} \left (A a^{4} d e^{2} + 4 A a^{3} b d^{2} e + 2 A a^{2} b^{2} d^{3} + B a^{4} d^{2} e + \frac {4 B a^{3} b d^{3}}{3}\right ) + x^{2} \left (\frac {3 A a^{4} d^{2} e}{2} + 2 A a^{3} b d^{3} + \frac {B a^{4} d^{3}}{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)**3*(b**2*x**2+2*a*b*x+a**2)**2,x)

[Out]

A*a**4*d**3*x + B*b**4*e**3*x**9/9 + x**8*(A*b**4*e**3/8 + B*a*b**3*e**3/2 + 3*B*b**4*d*e**2/8) + x**7*(4*A*a*
b**3*e**3/7 + 3*A*b**4*d*e**2/7 + 6*B*a**2*b**2*e**3/7 + 12*B*a*b**3*d*e**2/7 + 3*B*b**4*d**2*e/7) + x**6*(A*a
**2*b**2*e**3 + 2*A*a*b**3*d*e**2 + A*b**4*d**2*e/2 + 2*B*a**3*b*e**3/3 + 3*B*a**2*b**2*d*e**2 + 2*B*a*b**3*d*
*2*e + B*b**4*d**3/6) + x**5*(4*A*a**3*b*e**3/5 + 18*A*a**2*b**2*d*e**2/5 + 12*A*a*b**3*d**2*e/5 + A*b**4*d**3
/5 + B*a**4*e**3/5 + 12*B*a**3*b*d*e**2/5 + 18*B*a**2*b**2*d**2*e/5 + 4*B*a*b**3*d**3/5) + x**4*(A*a**4*e**3/4
+ 3*A*a**3*b*d*e**2 + 9*A*a**2*b**2*d**2*e/2 + A*a*b**3*d**3 + 3*B*a**4*d*e**2/4 + 3*B*a**3*b*d**2*e + 3*B*a*
*2*b**2*d**3/2) + x**3*(A*a**4*d*e**2 + 4*A*a**3*b*d**2*e + 2*A*a**2*b**2*d**3 + B*a**4*d**2*e + 4*B*a**3*b*d*
*3/3) + x**2*(3*A*a**4*d**2*e/2 + 2*A*a**3*b*d**3 + B*a**4*d**3/2)

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