3.10.83 \(\int (A+B x) (d+e x)^4 (b x+c x^2)^3 \, dx\)

Optimal. Leaf size=412 \[ \frac {1}{4} A b^3 d^4 x^4+\frac {1}{10} c e^2 x^{10} \left (A c e (3 b e+4 c d)+3 B \left (b^2 e^2+4 b c d e+2 c^2 d^2\right )\right )+\frac {1}{6} b d^2 x^6 \left (2 b^2 e (3 A e+2 B d)+3 b c d (4 A e+B d)+3 A c^2 d^2\right )+\frac {1}{5} b^2 d^3 x^5 (4 A b e+3 A c d+b B d)+\frac {1}{7} d x^7 \left (2 b^3 e^2 (2 A e+3 B d)+6 b^2 c d e (3 A e+2 B d)+3 b c^2 d^2 (4 A e+B d)+A c^3 d^3\right )+\frac {1}{9} e x^9 \left (3 A c e \left (b^2 e^2+4 b c d e+2 c^2 d^2\right )+B \left (b^3 e^3+12 b^2 c d e^2+18 b c^2 d^2 e+4 c^3 d^3\right )\right )+\frac {1}{8} x^8 \left (A e \left (b^3 e^3+12 b^2 c d e^2+18 b c^2 d^2 e+4 c^3 d^3\right )+B d \left (4 b^3 e^3+18 b^2 c d e^2+12 b c^2 d^2 e+c^3 d^3\right )\right )+\frac {1}{11} c^2 e^3 x^{11} (A c e+3 b B e+4 B c d)+\frac {1}{12} B c^3 e^4 x^{12} \]

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Rubi [A]  time = 0.51, antiderivative size = 412, 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 {1}{10} c e^2 x^{10} \left (A c e (3 b e+4 c d)+3 B \left (b^2 e^2+4 b c d e+2 c^2 d^2\right )\right )+\frac {1}{9} e x^9 \left (3 A c e \left (b^2 e^2+4 b c d e+2 c^2 d^2\right )+B \left (12 b^2 c d e^2+b^3 e^3+18 b c^2 d^2 e+4 c^3 d^3\right )\right )+\frac {1}{8} x^8 \left (A e \left (12 b^2 c d e^2+b^3 e^3+18 b c^2 d^2 e+4 c^3 d^3\right )+B d \left (18 b^2 c d e^2+4 b^3 e^3+12 b c^2 d^2 e+c^3 d^3\right )\right )+\frac {1}{7} d x^7 \left (6 b^2 c d e (3 A e+2 B d)+2 b^3 e^2 (2 A e+3 B d)+3 b c^2 d^2 (4 A e+B d)+A c^3 d^3\right )+\frac {1}{6} b d^2 x^6 \left (2 b^2 e (3 A e+2 B d)+3 b c d (4 A e+B d)+3 A c^2 d^2\right )+\frac {1}{5} b^2 d^3 x^5 (4 A b e+3 A c d+b B d)+\frac {1}{4} A b^3 d^4 x^4+\frac {1}{11} c^2 e^3 x^{11} (A c e+3 b B e+4 B c d)+\frac {1}{12} B c^3 e^4 x^{12} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(A + B*x)*(d + e*x)^4*(b*x + c*x^2)^3,x]

[Out]

(A*b^3*d^4*x^4)/4 + (b^2*d^3*(b*B*d + 3*A*c*d + 4*A*b*e)*x^5)/5 + (b*d^2*(3*A*c^2*d^2 + 2*b^2*e*(2*B*d + 3*A*e
) + 3*b*c*d*(B*d + 4*A*e))*x^6)/6 + (d*(A*c^3*d^3 + 2*b^3*e^2*(3*B*d + 2*A*e) + 6*b^2*c*d*e*(2*B*d + 3*A*e) +
3*b*c^2*d^2*(B*d + 4*A*e))*x^7)/7 + ((A*e*(4*c^3*d^3 + 18*b*c^2*d^2*e + 12*b^2*c*d*e^2 + b^3*e^3) + B*d*(c^3*d
^3 + 12*b*c^2*d^2*e + 18*b^2*c*d*e^2 + 4*b^3*e^3))*x^8)/8 + (e*(3*A*c*e*(2*c^2*d^2 + 4*b*c*d*e + b^2*e^2) + B*
(4*c^3*d^3 + 18*b*c^2*d^2*e + 12*b^2*c*d*e^2 + b^3*e^3))*x^9)/9 + (c*e^2*(A*c*e*(4*c*d + 3*b*e) + 3*B*(2*c^2*d
^2 + 4*b*c*d*e + b^2*e^2))*x^10)/10 + (c^2*e^3*(4*B*c*d + 3*b*B*e + A*c*e)*x^11)/11 + (B*c^3*e^4*x^12)/12

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)^4 \left (b x+c x^2\right )^3 \, dx &=\int \left (A b^3 d^4 x^3+b^2 d^3 (b B d+3 A c d+4 A b e) x^4+b d^2 \left (3 A c^2 d^2+2 b^2 e (2 B d+3 A e)+3 b c d (B d+4 A e)\right ) x^5+d \left (A c^3 d^3+2 b^3 e^2 (3 B d+2 A e)+6 b^2 c d e (2 B d+3 A e)+3 b c^2 d^2 (B d+4 A e)\right ) x^6+\left (A e \left (4 c^3 d^3+18 b c^2 d^2 e+12 b^2 c d e^2+b^3 e^3\right )+B d \left (c^3 d^3+12 b c^2 d^2 e+18 b^2 c d e^2+4 b^3 e^3\right )\right ) x^7+e \left (3 A c e \left (2 c^2 d^2+4 b c d e+b^2 e^2\right )+B \left (4 c^3 d^3+18 b c^2 d^2 e+12 b^2 c d e^2+b^3 e^3\right )\right ) x^8+c e^2 \left (A c e (4 c d+3 b e)+3 B \left (2 c^2 d^2+4 b c d e+b^2 e^2\right )\right ) x^9+c^2 e^3 (4 B c d+3 b B e+A c e) x^{10}+B c^3 e^4 x^{11}\right ) \, dx\\ &=\frac {1}{4} A b^3 d^4 x^4+\frac {1}{5} b^2 d^3 (b B d+3 A c d+4 A b e) x^5+\frac {1}{6} b d^2 \left (3 A c^2 d^2+2 b^2 e (2 B d+3 A e)+3 b c d (B d+4 A e)\right ) x^6+\frac {1}{7} d \left (A c^3 d^3+2 b^3 e^2 (3 B d+2 A e)+6 b^2 c d e (2 B d+3 A e)+3 b c^2 d^2 (B d+4 A e)\right ) x^7+\frac {1}{8} \left (A e \left (4 c^3 d^3+18 b c^2 d^2 e+12 b^2 c d e^2+b^3 e^3\right )+B d \left (c^3 d^3+12 b c^2 d^2 e+18 b^2 c d e^2+4 b^3 e^3\right )\right ) x^8+\frac {1}{9} e \left (3 A c e \left (2 c^2 d^2+4 b c d e+b^2 e^2\right )+B \left (4 c^3 d^3+18 b c^2 d^2 e+12 b^2 c d e^2+b^3 e^3\right )\right ) x^9+\frac {1}{10} c e^2 \left (A c e (4 c d+3 b e)+3 B \left (2 c^2 d^2+4 b c d e+b^2 e^2\right )\right ) x^{10}+\frac {1}{11} c^2 e^3 (4 B c d+3 b B e+A c e) x^{11}+\frac {1}{12} B c^3 e^4 x^{12}\\ \end {align*}

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Mathematica [A]  time = 0.16, size = 412, normalized size = 1.00 \begin {gather*} \frac {1}{4} A b^3 d^4 x^4+\frac {1}{10} c e^2 x^{10} \left (A c e (3 b e+4 c d)+3 B \left (b^2 e^2+4 b c d e+2 c^2 d^2\right )\right )+\frac {1}{6} b d^2 x^6 \left (2 b^2 e (3 A e+2 B d)+3 b c d (4 A e+B d)+3 A c^2 d^2\right )+\frac {1}{5} b^2 d^3 x^5 (4 A b e+3 A c d+b B d)+\frac {1}{7} d x^7 \left (2 b^3 e^2 (2 A e+3 B d)+6 b^2 c d e (3 A e+2 B d)+3 b c^2 d^2 (4 A e+B d)+A c^3 d^3\right )+\frac {1}{9} e x^9 \left (3 A c e \left (b^2 e^2+4 b c d e+2 c^2 d^2\right )+B \left (b^3 e^3+12 b^2 c d e^2+18 b c^2 d^2 e+4 c^3 d^3\right )\right )+\frac {1}{8} x^8 \left (A e \left (b^3 e^3+12 b^2 c d e^2+18 b c^2 d^2 e+4 c^3 d^3\right )+B d \left (4 b^3 e^3+18 b^2 c d e^2+12 b c^2 d^2 e+c^3 d^3\right )\right )+\frac {1}{11} c^2 e^3 x^{11} (A c e+3 b B e+4 B c d)+\frac {1}{12} B c^3 e^4 x^{12} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x)*(d + e*x)^4*(b*x + c*x^2)^3,x]

[Out]

(A*b^3*d^4*x^4)/4 + (b^2*d^3*(b*B*d + 3*A*c*d + 4*A*b*e)*x^5)/5 + (b*d^2*(3*A*c^2*d^2 + 2*b^2*e*(2*B*d + 3*A*e
) + 3*b*c*d*(B*d + 4*A*e))*x^6)/6 + (d*(A*c^3*d^3 + 2*b^3*e^2*(3*B*d + 2*A*e) + 6*b^2*c*d*e*(2*B*d + 3*A*e) +
3*b*c^2*d^2*(B*d + 4*A*e))*x^7)/7 + ((A*e*(4*c^3*d^3 + 18*b*c^2*d^2*e + 12*b^2*c*d*e^2 + b^3*e^3) + B*d*(c^3*d
^3 + 12*b*c^2*d^2*e + 18*b^2*c*d*e^2 + 4*b^3*e^3))*x^8)/8 + (e*(3*A*c*e*(2*c^2*d^2 + 4*b*c*d*e + b^2*e^2) + B*
(4*c^3*d^3 + 18*b*c^2*d^2*e + 12*b^2*c*d*e^2 + b^3*e^3))*x^9)/9 + (c*e^2*(A*c*e*(4*c*d + 3*b*e) + 3*B*(2*c^2*d
^2 + 4*b*c*d*e + b^2*e^2))*x^10)/10 + (c^2*e^3*(4*B*c*d + 3*b*B*e + A*c*e)*x^11)/11 + (B*c^3*e^4*x^12)/12

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int (A+B x) (d+e x)^4 \left (b x+c x^2\right )^3 \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(A + B*x)*(d + e*x)^4*(b*x + c*x^2)^3,x]

[Out]

IntegrateAlgebraic[(A + B*x)*(d + e*x)^4*(b*x + c*x^2)^3, x]

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fricas [A]  time = 0.37, size = 540, normalized size = 1.31 \begin {gather*} \frac {1}{12} x^{12} e^{4} c^{3} B + \frac {4}{11} x^{11} e^{3} d c^{3} B + \frac {3}{11} x^{11} e^{4} c^{2} b B + \frac {1}{11} x^{11} e^{4} c^{3} A + \frac {3}{5} x^{10} e^{2} d^{2} c^{3} B + \frac {6}{5} x^{10} e^{3} d c^{2} b B + \frac {3}{10} x^{10} e^{4} c b^{2} B + \frac {2}{5} x^{10} e^{3} d c^{3} A + \frac {3}{10} x^{10} e^{4} c^{2} b A + \frac {4}{9} x^{9} e d^{3} c^{3} B + 2 x^{9} e^{2} d^{2} c^{2} b B + \frac {4}{3} x^{9} e^{3} d c b^{2} B + \frac {1}{9} x^{9} e^{4} b^{3} B + \frac {2}{3} x^{9} e^{2} d^{2} c^{3} A + \frac {4}{3} x^{9} e^{3} d c^{2} b A + \frac {1}{3} x^{9} e^{4} c b^{2} A + \frac {1}{8} x^{8} d^{4} c^{3} B + \frac {3}{2} x^{8} e d^{3} c^{2} b B + \frac {9}{4} x^{8} e^{2} d^{2} c b^{2} B + \frac {1}{2} x^{8} e^{3} d b^{3} B + \frac {1}{2} x^{8} e d^{3} c^{3} A + \frac {9}{4} x^{8} e^{2} d^{2} c^{2} b A + \frac {3}{2} x^{8} e^{3} d c b^{2} A + \frac {1}{8} x^{8} e^{4} b^{3} A + \frac {3}{7} x^{7} d^{4} c^{2} b B + \frac {12}{7} x^{7} e d^{3} c b^{2} B + \frac {6}{7} x^{7} e^{2} d^{2} b^{3} B + \frac {1}{7} x^{7} d^{4} c^{3} A + \frac {12}{7} x^{7} e d^{3} c^{2} b A + \frac {18}{7} x^{7} e^{2} d^{2} c b^{2} A + \frac {4}{7} x^{7} e^{3} d b^{3} A + \frac {1}{2} x^{6} d^{4} c b^{2} B + \frac {2}{3} x^{6} e d^{3} b^{3} B + \frac {1}{2} x^{6} d^{4} c^{2} b A + 2 x^{6} e d^{3} c b^{2} A + x^{6} e^{2} d^{2} b^{3} A + \frac {1}{5} x^{5} d^{4} b^{3} B + \frac {3}{5} x^{5} d^{4} c b^{2} A + \frac {4}{5} x^{5} e d^{3} b^{3} A + \frac {1}{4} x^{4} d^{4} b^{3} A \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^4*(c*x^2+b*x)^3,x, algorithm="fricas")

[Out]

1/12*x^12*e^4*c^3*B + 4/11*x^11*e^3*d*c^3*B + 3/11*x^11*e^4*c^2*b*B + 1/11*x^11*e^4*c^3*A + 3/5*x^10*e^2*d^2*c
^3*B + 6/5*x^10*e^3*d*c^2*b*B + 3/10*x^10*e^4*c*b^2*B + 2/5*x^10*e^3*d*c^3*A + 3/10*x^10*e^4*c^2*b*A + 4/9*x^9
*e*d^3*c^3*B + 2*x^9*e^2*d^2*c^2*b*B + 4/3*x^9*e^3*d*c*b^2*B + 1/9*x^9*e^4*b^3*B + 2/3*x^9*e^2*d^2*c^3*A + 4/3
*x^9*e^3*d*c^2*b*A + 1/3*x^9*e^4*c*b^2*A + 1/8*x^8*d^4*c^3*B + 3/2*x^8*e*d^3*c^2*b*B + 9/4*x^8*e^2*d^2*c*b^2*B
+ 1/2*x^8*e^3*d*b^3*B + 1/2*x^8*e*d^3*c^3*A + 9/4*x^8*e^2*d^2*c^2*b*A + 3/2*x^8*e^3*d*c*b^2*A + 1/8*x^8*e^4*b
^3*A + 3/7*x^7*d^4*c^2*b*B + 12/7*x^7*e*d^3*c*b^2*B + 6/7*x^7*e^2*d^2*b^3*B + 1/7*x^7*d^4*c^3*A + 12/7*x^7*e*d
^3*c^2*b*A + 18/7*x^7*e^2*d^2*c*b^2*A + 4/7*x^7*e^3*d*b^3*A + 1/2*x^6*d^4*c*b^2*B + 2/3*x^6*e*d^3*b^3*B + 1/2*
x^6*d^4*c^2*b*A + 2*x^6*e*d^3*c*b^2*A + x^6*e^2*d^2*b^3*A + 1/5*x^5*d^4*b^3*B + 3/5*x^5*d^4*c*b^2*A + 4/5*x^5*
e*d^3*b^3*A + 1/4*x^4*d^4*b^3*A

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giac [A]  time = 0.16, size = 524, normalized size = 1.27 \begin {gather*} \frac {1}{12} \, B c^{3} x^{12} e^{4} + \frac {4}{11} \, B c^{3} d x^{11} e^{3} + \frac {3}{5} \, B c^{3} d^{2} x^{10} e^{2} + \frac {4}{9} \, B c^{3} d^{3} x^{9} e + \frac {1}{8} \, B c^{3} d^{4} x^{8} + \frac {3}{11} \, B b c^{2} x^{11} e^{4} + \frac {1}{11} \, A c^{3} x^{11} e^{4} + \frac {6}{5} \, B b c^{2} d x^{10} e^{3} + \frac {2}{5} \, A c^{3} d x^{10} e^{3} + 2 \, B b c^{2} d^{2} x^{9} e^{2} + \frac {2}{3} \, A c^{3} d^{2} x^{9} e^{2} + \frac {3}{2} \, B b c^{2} d^{3} x^{8} e + \frac {1}{2} \, A c^{3} d^{3} x^{8} e + \frac {3}{7} \, B b c^{2} d^{4} x^{7} + \frac {1}{7} \, A c^{3} d^{4} x^{7} + \frac {3}{10} \, B b^{2} c x^{10} e^{4} + \frac {3}{10} \, A b c^{2} x^{10} e^{4} + \frac {4}{3} \, B b^{2} c d x^{9} e^{3} + \frac {4}{3} \, A b c^{2} d x^{9} e^{3} + \frac {9}{4} \, B b^{2} c d^{2} x^{8} e^{2} + \frac {9}{4} \, A b c^{2} d^{2} x^{8} e^{2} + \frac {12}{7} \, B b^{2} c d^{3} x^{7} e + \frac {12}{7} \, A b c^{2} d^{3} x^{7} e + \frac {1}{2} \, B b^{2} c d^{4} x^{6} + \frac {1}{2} \, A b c^{2} d^{4} x^{6} + \frac {1}{9} \, B b^{3} x^{9} e^{4} + \frac {1}{3} \, A b^{2} c x^{9} e^{4} + \frac {1}{2} \, B b^{3} d x^{8} e^{3} + \frac {3}{2} \, A b^{2} c d x^{8} e^{3} + \frac {6}{7} \, B b^{3} d^{2} x^{7} e^{2} + \frac {18}{7} \, A b^{2} c d^{2} x^{7} e^{2} + \frac {2}{3} \, B b^{3} d^{3} x^{6} e + 2 \, A b^{2} c d^{3} x^{6} e + \frac {1}{5} \, B b^{3} d^{4} x^{5} + \frac {3}{5} \, A b^{2} c d^{4} x^{5} + \frac {1}{8} \, A b^{3} x^{8} e^{4} + \frac {4}{7} \, A b^{3} d x^{7} e^{3} + A b^{3} d^{2} x^{6} e^{2} + \frac {4}{5} \, A b^{3} d^{3} x^{5} e + \frac {1}{4} \, A b^{3} d^{4} x^{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^4*(c*x^2+b*x)^3,x, algorithm="giac")

[Out]

1/12*B*c^3*x^12*e^4 + 4/11*B*c^3*d*x^11*e^3 + 3/5*B*c^3*d^2*x^10*e^2 + 4/9*B*c^3*d^3*x^9*e + 1/8*B*c^3*d^4*x^8
+ 3/11*B*b*c^2*x^11*e^4 + 1/11*A*c^3*x^11*e^4 + 6/5*B*b*c^2*d*x^10*e^3 + 2/5*A*c^3*d*x^10*e^3 + 2*B*b*c^2*d^2
*x^9*e^2 + 2/3*A*c^3*d^2*x^9*e^2 + 3/2*B*b*c^2*d^3*x^8*e + 1/2*A*c^3*d^3*x^8*e + 3/7*B*b*c^2*d^4*x^7 + 1/7*A*c
^3*d^4*x^7 + 3/10*B*b^2*c*x^10*e^4 + 3/10*A*b*c^2*x^10*e^4 + 4/3*B*b^2*c*d*x^9*e^3 + 4/3*A*b*c^2*d*x^9*e^3 + 9
/4*B*b^2*c*d^2*x^8*e^2 + 9/4*A*b*c^2*d^2*x^8*e^2 + 12/7*B*b^2*c*d^3*x^7*e + 12/7*A*b*c^2*d^3*x^7*e + 1/2*B*b^2
*c*d^4*x^6 + 1/2*A*b*c^2*d^4*x^6 + 1/9*B*b^3*x^9*e^4 + 1/3*A*b^2*c*x^9*e^4 + 1/2*B*b^3*d*x^8*e^3 + 3/2*A*b^2*c
*d*x^8*e^3 + 6/7*B*b^3*d^2*x^7*e^2 + 18/7*A*b^2*c*d^2*x^7*e^2 + 2/3*B*b^3*d^3*x^6*e + 2*A*b^2*c*d^3*x^6*e + 1/
5*B*b^3*d^4*x^5 + 3/5*A*b^2*c*d^4*x^5 + 1/8*A*b^3*x^8*e^4 + 4/7*A*b^3*d*x^7*e^3 + A*b^3*d^2*x^6*e^2 + 4/5*A*b^
3*d^3*x^5*e + 1/4*A*b^3*d^4*x^4

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maple [A]  time = 0.04, size = 444, normalized size = 1.08 \begin {gather*} \frac {B \,c^{3} e^{4} x^{12}}{12}+\frac {A \,b^{3} d^{4} x^{4}}{4}+\frac {\left (3 B b \,c^{2} e^{4}+\left (A \,e^{4}+4 B d \,e^{3}\right ) c^{3}\right ) x^{11}}{11}+\frac {\left (3 B \,b^{2} c \,e^{4}+3 \left (A \,e^{4}+4 B d \,e^{3}\right ) b \,c^{2}+\left (4 A d \,e^{3}+6 B \,d^{2} e^{2}\right ) c^{3}\right ) x^{10}}{10}+\frac {\left (B \,b^{3} e^{4}+3 \left (A \,e^{4}+4 B d \,e^{3}\right ) b^{2} c +3 \left (4 A d \,e^{3}+6 B \,d^{2} e^{2}\right ) b \,c^{2}+\left (6 A \,d^{2} e^{2}+4 B \,d^{3} e \right ) c^{3}\right ) x^{9}}{9}+\frac {\left (\left (A \,e^{4}+4 B d \,e^{3}\right ) b^{3}+3 \left (4 A d \,e^{3}+6 B \,d^{2} e^{2}\right ) b^{2} c +3 \left (6 A \,d^{2} e^{2}+4 B \,d^{3} e \right ) b \,c^{2}+\left (4 A \,d^{3} e +B \,d^{4}\right ) c^{3}\right ) x^{8}}{8}+\frac {\left (A \,c^{3} d^{4}+\left (4 A d \,e^{3}+6 B \,d^{2} e^{2}\right ) b^{3}+3 \left (6 A \,d^{2} e^{2}+4 B \,d^{3} e \right ) b^{2} c +3 \left (4 A \,d^{3} e +B \,d^{4}\right ) b \,c^{2}\right ) x^{7}}{7}+\frac {\left (3 A b \,c^{2} d^{4}+\left (6 A \,d^{2} e^{2}+4 B \,d^{3} e \right ) b^{3}+3 \left (4 A \,d^{3} e +B \,d^{4}\right ) b^{2} c \right ) x^{6}}{6}+\frac {\left (3 A \,b^{2} c \,d^{4}+\left (4 A \,d^{3} e +B \,d^{4}\right ) b^{3}\right ) x^{5}}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(e*x+d)^4*(c*x^2+b*x)^3,x)

[Out]

1/12*B*c^3*e^4*x^12+1/11*((A*e^4+4*B*d*e^3)*c^3+3*B*e^4*b*c^2)*x^11+1/10*((4*A*d*e^3+6*B*d^2*e^2)*c^3+3*(A*e^4
+4*B*d*e^3)*b*c^2+3*B*e^4*b^2*c)*x^10+1/9*((6*A*d^2*e^2+4*B*d^3*e)*c^3+3*(4*A*d*e^3+6*B*d^2*e^2)*b*c^2+3*(A*e^
4+4*B*d*e^3)*b^2*c+B*e^4*b^3)*x^9+1/8*((4*A*d^3*e+B*d^4)*c^3+3*(6*A*d^2*e^2+4*B*d^3*e)*b*c^2+3*(4*A*d*e^3+6*B*
d^2*e^2)*b^2*c+(A*e^4+4*B*d*e^3)*b^3)*x^8+1/7*(A*d^4*c^3+3*(4*A*d^3*e+B*d^4)*b*c^2+3*(6*A*d^2*e^2+4*B*d^3*e)*b
^2*c+(4*A*d*e^3+6*B*d^2*e^2)*b^3)*x^7+1/6*(3*A*d^4*b*c^2+3*(4*A*d^3*e+B*d^4)*b^2*c+(6*A*d^2*e^2+4*B*d^3*e)*b^3
)*x^6+1/5*(3*A*d^4*b^2*c+(4*A*d^3*e+B*d^4)*b^3)*x^5+1/4*A*b^3*d^4*x^4

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maxima [A]  time = 0.51, size = 428, normalized size = 1.04 \begin {gather*} \frac {1}{12} \, B c^{3} e^{4} x^{12} + \frac {1}{4} \, A b^{3} d^{4} x^{4} + \frac {1}{11} \, {\left (4 \, B c^{3} d e^{3} + {\left (3 \, B b c^{2} + A c^{3}\right )} e^{4}\right )} x^{11} + \frac {1}{10} \, {\left (6 \, B c^{3} d^{2} e^{2} + 4 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d e^{3} + 3 \, {\left (B b^{2} c + A b c^{2}\right )} e^{4}\right )} x^{10} + \frac {1}{9} \, {\left (4 \, B c^{3} d^{3} e + 6 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{2} e^{2} + 12 \, {\left (B b^{2} c + A b c^{2}\right )} d e^{3} + {\left (B b^{3} + 3 \, A b^{2} c\right )} e^{4}\right )} x^{9} + \frac {1}{8} \, {\left (B c^{3} d^{4} + A b^{3} e^{4} + 4 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{3} e + 18 \, {\left (B b^{2} c + A b c^{2}\right )} d^{2} e^{2} + 4 \, {\left (B b^{3} + 3 \, A b^{2} c\right )} d e^{3}\right )} x^{8} + \frac {1}{7} \, {\left (4 \, A b^{3} d e^{3} + {\left (3 \, B b c^{2} + A c^{3}\right )} d^{4} + 12 \, {\left (B b^{2} c + A b c^{2}\right )} d^{3} e + 6 \, {\left (B b^{3} + 3 \, A b^{2} c\right )} d^{2} e^{2}\right )} x^{7} + \frac {1}{6} \, {\left (6 \, A b^{3} d^{2} e^{2} + 3 \, {\left (B b^{2} c + A b c^{2}\right )} d^{4} + 4 \, {\left (B b^{3} + 3 \, A b^{2} c\right )} d^{3} e\right )} x^{6} + \frac {1}{5} \, {\left (4 \, A b^{3} d^{3} e + {\left (B b^{3} + 3 \, A b^{2} c\right )} d^{4}\right )} x^{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^4*(c*x^2+b*x)^3,x, algorithm="maxima")

[Out]

1/12*B*c^3*e^4*x^12 + 1/4*A*b^3*d^4*x^4 + 1/11*(4*B*c^3*d*e^3 + (3*B*b*c^2 + A*c^3)*e^4)*x^11 + 1/10*(6*B*c^3*
d^2*e^2 + 4*(3*B*b*c^2 + A*c^3)*d*e^3 + 3*(B*b^2*c + A*b*c^2)*e^4)*x^10 + 1/9*(4*B*c^3*d^3*e + 6*(3*B*b*c^2 +
A*c^3)*d^2*e^2 + 12*(B*b^2*c + A*b*c^2)*d*e^3 + (B*b^3 + 3*A*b^2*c)*e^4)*x^9 + 1/8*(B*c^3*d^4 + A*b^3*e^4 + 4*
(3*B*b*c^2 + A*c^3)*d^3*e + 18*(B*b^2*c + A*b*c^2)*d^2*e^2 + 4*(B*b^3 + 3*A*b^2*c)*d*e^3)*x^8 + 1/7*(4*A*b^3*d
*e^3 + (3*B*b*c^2 + A*c^3)*d^4 + 12*(B*b^2*c + A*b*c^2)*d^3*e + 6*(B*b^3 + 3*A*b^2*c)*d^2*e^2)*x^7 + 1/6*(6*A*
b^3*d^2*e^2 + 3*(B*b^2*c + A*b*c^2)*d^4 + 4*(B*b^3 + 3*A*b^2*c)*d^3*e)*x^6 + 1/5*(4*A*b^3*d^3*e + (B*b^3 + 3*A
*b^2*c)*d^4)*x^5

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mupad [B]  time = 0.16, size = 445, normalized size = 1.08 \begin {gather*} x^6\,\left (\frac {2\,B\,b^3\,d^3\,e}{3}+A\,b^3\,d^2\,e^2+\frac {B\,b^2\,c\,d^4}{2}+2\,A\,b^2\,c\,d^3\,e+\frac {A\,b\,c^2\,d^4}{2}\right )+x^{10}\,\left (\frac {3\,B\,b^2\,c\,e^4}{10}+\frac {6\,B\,b\,c^2\,d\,e^3}{5}+\frac {3\,A\,b\,c^2\,e^4}{10}+\frac {3\,B\,c^3\,d^2\,e^2}{5}+\frac {2\,A\,c^3\,d\,e^3}{5}\right )+x^8\,\left (\frac {B\,b^3\,d\,e^3}{2}+\frac {A\,b^3\,e^4}{8}+\frac {9\,B\,b^2\,c\,d^2\,e^2}{4}+\frac {3\,A\,b^2\,c\,d\,e^3}{2}+\frac {3\,B\,b\,c^2\,d^3\,e}{2}+\frac {9\,A\,b\,c^2\,d^2\,e^2}{4}+\frac {B\,c^3\,d^4}{8}+\frac {A\,c^3\,d^3\,e}{2}\right )+x^7\,\left (\frac {6\,B\,b^3\,d^2\,e^2}{7}+\frac {4\,A\,b^3\,d\,e^3}{7}+\frac {12\,B\,b^2\,c\,d^3\,e}{7}+\frac {18\,A\,b^2\,c\,d^2\,e^2}{7}+\frac {3\,B\,b\,c^2\,d^4}{7}+\frac {12\,A\,b\,c^2\,d^3\,e}{7}+\frac {A\,c^3\,d^4}{7}\right )+x^9\,\left (\frac {B\,b^3\,e^4}{9}+\frac {4\,B\,b^2\,c\,d\,e^3}{3}+\frac {A\,b^2\,c\,e^4}{3}+2\,B\,b\,c^2\,d^2\,e^2+\frac {4\,A\,b\,c^2\,d\,e^3}{3}+\frac {4\,B\,c^3\,d^3\,e}{9}+\frac {2\,A\,c^3\,d^2\,e^2}{3}\right )+\frac {b^2\,d^3\,x^5\,\left (4\,A\,b\,e+3\,A\,c\,d+B\,b\,d\right )}{5}+\frac {c^2\,e^3\,x^{11}\,\left (A\,c\,e+3\,B\,b\,e+4\,B\,c\,d\right )}{11}+\frac {A\,b^3\,d^4\,x^4}{4}+\frac {B\,c^3\,e^4\,x^{12}}{12} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x + c*x^2)^3*(A + B*x)*(d + e*x)^4,x)

[Out]

x^6*((A*b*c^2*d^4)/2 + (B*b^2*c*d^4)/2 + (2*B*b^3*d^3*e)/3 + A*b^3*d^2*e^2 + 2*A*b^2*c*d^3*e) + x^10*((3*A*b*c
^2*e^4)/10 + (3*B*b^2*c*e^4)/10 + (2*A*c^3*d*e^3)/5 + (3*B*c^3*d^2*e^2)/5 + (6*B*b*c^2*d*e^3)/5) + x^8*((A*b^3
*e^4)/8 + (B*c^3*d^4)/8 + (A*c^3*d^3*e)/2 + (B*b^3*d*e^3)/2 + (9*A*b*c^2*d^2*e^2)/4 + (9*B*b^2*c*d^2*e^2)/4 +
(3*A*b^2*c*d*e^3)/2 + (3*B*b*c^2*d^3*e)/2) + x^7*((A*c^3*d^4)/7 + (3*B*b*c^2*d^4)/7 + (4*A*b^3*d*e^3)/7 + (6*B
*b^3*d^2*e^2)/7 + (18*A*b^2*c*d^2*e^2)/7 + (12*A*b*c^2*d^3*e)/7 + (12*B*b^2*c*d^3*e)/7) + x^9*((B*b^3*e^4)/9 +
(A*b^2*c*e^4)/3 + (4*B*c^3*d^3*e)/9 + (2*A*c^3*d^2*e^2)/3 + 2*B*b*c^2*d^2*e^2 + (4*A*b*c^2*d*e^3)/3 + (4*B*b^
2*c*d*e^3)/3) + (b^2*d^3*x^5*(4*A*b*e + 3*A*c*d + B*b*d))/5 + (c^2*e^3*x^11*(A*c*e + 3*B*b*e + 4*B*c*d))/11 +
(A*b^3*d^4*x^4)/4 + (B*c^3*e^4*x^12)/12

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sympy [A]  time = 0.15, size = 564, normalized size = 1.37 \begin {gather*} \frac {A b^{3} d^{4} x^{4}}{4} + \frac {B c^{3} e^{4} x^{12}}{12} + x^{11} \left (\frac {A c^{3} e^{4}}{11} + \frac {3 B b c^{2} e^{4}}{11} + \frac {4 B c^{3} d e^{3}}{11}\right ) + x^{10} \left (\frac {3 A b c^{2} e^{4}}{10} + \frac {2 A c^{3} d e^{3}}{5} + \frac {3 B b^{2} c e^{4}}{10} + \frac {6 B b c^{2} d e^{3}}{5} + \frac {3 B c^{3} d^{2} e^{2}}{5}\right ) + x^{9} \left (\frac {A b^{2} c e^{4}}{3} + \frac {4 A b c^{2} d e^{3}}{3} + \frac {2 A c^{3} d^{2} e^{2}}{3} + \frac {B b^{3} e^{4}}{9} + \frac {4 B b^{2} c d e^{3}}{3} + 2 B b c^{2} d^{2} e^{2} + \frac {4 B c^{3} d^{3} e}{9}\right ) + x^{8} \left (\frac {A b^{3} e^{4}}{8} + \frac {3 A b^{2} c d e^{3}}{2} + \frac {9 A b c^{2} d^{2} e^{2}}{4} + \frac {A c^{3} d^{3} e}{2} + \frac {B b^{3} d e^{3}}{2} + \frac {9 B b^{2} c d^{2} e^{2}}{4} + \frac {3 B b c^{2} d^{3} e}{2} + \frac {B c^{3} d^{4}}{8}\right ) + x^{7} \left (\frac {4 A b^{3} d e^{3}}{7} + \frac {18 A b^{2} c d^{2} e^{2}}{7} + \frac {12 A b c^{2} d^{3} e}{7} + \frac {A c^{3} d^{4}}{7} + \frac {6 B b^{3} d^{2} e^{2}}{7} + \frac {12 B b^{2} c d^{3} e}{7} + \frac {3 B b c^{2} d^{4}}{7}\right ) + x^{6} \left (A b^{3} d^{2} e^{2} + 2 A b^{2} c d^{3} e + \frac {A b c^{2} d^{4}}{2} + \frac {2 B b^{3} d^{3} e}{3} + \frac {B b^{2} c d^{4}}{2}\right ) + x^{5} \left (\frac {4 A b^{3} d^{3} e}{5} + \frac {3 A b^{2} c d^{4}}{5} + \frac {B b^{3} d^{4}}{5}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)**4*(c*x**2+b*x)**3,x)

[Out]

A*b**3*d**4*x**4/4 + B*c**3*e**4*x**12/12 + x**11*(A*c**3*e**4/11 + 3*B*b*c**2*e**4/11 + 4*B*c**3*d*e**3/11) +
x**10*(3*A*b*c**2*e**4/10 + 2*A*c**3*d*e**3/5 + 3*B*b**2*c*e**4/10 + 6*B*b*c**2*d*e**3/5 + 3*B*c**3*d**2*e**2
/5) + x**9*(A*b**2*c*e**4/3 + 4*A*b*c**2*d*e**3/3 + 2*A*c**3*d**2*e**2/3 + B*b**3*e**4/9 + 4*B*b**2*c*d*e**3/3
+ 2*B*b*c**2*d**2*e**2 + 4*B*c**3*d**3*e/9) + x**8*(A*b**3*e**4/8 + 3*A*b**2*c*d*e**3/2 + 9*A*b*c**2*d**2*e**
2/4 + A*c**3*d**3*e/2 + B*b**3*d*e**3/2 + 9*B*b**2*c*d**2*e**2/4 + 3*B*b*c**2*d**3*e/2 + B*c**3*d**4/8) + x**7
*(4*A*b**3*d*e**3/7 + 18*A*b**2*c*d**2*e**2/7 + 12*A*b*c**2*d**3*e/7 + A*c**3*d**4/7 + 6*B*b**3*d**2*e**2/7 +
12*B*b**2*c*d**3*e/7 + 3*B*b*c**2*d**4/7) + x**6*(A*b**3*d**2*e**2 + 2*A*b**2*c*d**3*e + A*b*c**2*d**4/2 + 2*B
*b**3*d**3*e/3 + B*b**2*c*d**4/2) + x**5*(4*A*b**3*d**3*e/5 + 3*A*b**2*c*d**4/5 + B*b**3*d**4/5)

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