3.28 \(\int x^3 (A+B x) (b x+c x^2)^3 \, dx\)

Optimal. Leaf size=75 \[ \frac{1}{8} b^2 x^8 (3 A c+b B)+\frac{1}{7} A b^3 x^7+\frac{1}{10} c^2 x^{10} (A c+3 b B)+\frac{1}{3} b c x^9 (A c+b B)+\frac{1}{11} B c^3 x^{11} \]

[Out]

(A*b^3*x^7)/7 + (b^2*(b*B + 3*A*c)*x^8)/8 + (b*c*(b*B + A*c)*x^9)/3 + (c^2*(3*b*B + A*c)*x^10)/10 + (B*c^3*x^1
1)/11

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Rubi [A]  time = 0.081268, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {765} \[ \frac{1}{8} b^2 x^8 (3 A c+b B)+\frac{1}{7} A b^3 x^7+\frac{1}{10} c^2 x^{10} (A c+3 b B)+\frac{1}{3} b c x^9 (A c+b B)+\frac{1}{11} B c^3 x^{11} \]

Antiderivative was successfully verified.

[In]

Int[x^3*(A + B*x)*(b*x + c*x^2)^3,x]

[Out]

(A*b^3*x^7)/7 + (b^2*(b*B + 3*A*c)*x^8)/8 + (b*c*(b*B + A*c)*x^9)/3 + (c^2*(3*b*B + A*c)*x^10)/10 + (B*c^3*x^1
1)/11

Rule 765

Int[((e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[Expand
Integrand[(e*x)^m*(f + g*x)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, e, f, g, m}, x] && IntegerQ[p] && (
GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

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

Mathematica [A]  time = 0.0132916, size = 75, normalized size = 1. \[ \frac{1}{8} b^2 x^8 (3 A c+b B)+\frac{1}{7} A b^3 x^7+\frac{1}{10} c^2 x^{10} (A c+3 b B)+\frac{1}{3} b c x^9 (A c+b B)+\frac{1}{11} B c^3 x^{11} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(A*b^3*x^7)/7 + (b^2*(b*B + 3*A*c)*x^8)/8 + (b*c*(b*B + A*c)*x^9)/3 + (c^2*(3*b*B + A*c)*x^10)/10 + (B*c^3*x^1
1)/11

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Maple [A]  time = 0.002, size = 76, normalized size = 1. \begin{align*}{\frac{B{c}^{3}{x}^{11}}{11}}+{\frac{ \left ( A{c}^{3}+3\,Bb{c}^{2} \right ){x}^{10}}{10}}+{\frac{ \left ( 3\,Ab{c}^{2}+3\,B{b}^{2}c \right ){x}^{9}}{9}}+{\frac{ \left ( 3\,A{b}^{2}c+{b}^{3}B \right ){x}^{8}}{8}}+{\frac{A{b}^{3}{x}^{7}}{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/11*B*c^3*x^11+1/10*(A*c^3+3*B*b*c^2)*x^10+1/9*(3*A*b*c^2+3*B*b^2*c)*x^9+1/8*(3*A*b^2*c+B*b^3)*x^8+1/7*A*b^3*
x^7

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Maxima [A]  time = 1.10543, size = 99, normalized size = 1.32 \begin{align*} \frac{1}{11} \, B c^{3} x^{11} + \frac{1}{7} \, A b^{3} x^{7} + \frac{1}{10} \,{\left (3 \, B b c^{2} + A c^{3}\right )} x^{10} + \frac{1}{3} \,{\left (B b^{2} c + A b c^{2}\right )} x^{9} + \frac{1}{8} \,{\left (B b^{3} + 3 \, A b^{2} c\right )} x^{8} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/11*B*c^3*x^11 + 1/7*A*b^3*x^7 + 1/10*(3*B*b*c^2 + A*c^3)*x^10 + 1/3*(B*b^2*c + A*b*c^2)*x^9 + 1/8*(B*b^3 + 3
*A*b^2*c)*x^8

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Fricas [A]  time = 1.5707, size = 190, normalized size = 2.53 \begin{align*} \frac{1}{11} x^{11} c^{3} B + \frac{3}{10} x^{10} c^{2} b B + \frac{1}{10} x^{10} c^{3} A + \frac{1}{3} x^{9} c b^{2} B + \frac{1}{3} x^{9} c^{2} b A + \frac{1}{8} x^{8} b^{3} B + \frac{3}{8} x^{8} c b^{2} A + \frac{1}{7} x^{7} b^{3} A \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/11*x^11*c^3*B + 3/10*x^10*c^2*b*B + 1/10*x^10*c^3*A + 1/3*x^9*c*b^2*B + 1/3*x^9*c^2*b*A + 1/8*x^8*b^3*B + 3/
8*x^8*c*b^2*A + 1/7*x^7*b^3*A

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Sympy [A]  time = 0.101604, size = 80, normalized size = 1.07 \begin{align*} \frac{A b^{3} x^{7}}{7} + \frac{B c^{3} x^{11}}{11} + x^{10} \left (\frac{A c^{3}}{10} + \frac{3 B b c^{2}}{10}\right ) + x^{9} \left (\frac{A b c^{2}}{3} + \frac{B b^{2} c}{3}\right ) + x^{8} \left (\frac{3 A b^{2} c}{8} + \frac{B b^{3}}{8}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

A*b**3*x**7/7 + B*c**3*x**11/11 + x**10*(A*c**3/10 + 3*B*b*c**2/10) + x**9*(A*b*c**2/3 + B*b**2*c/3) + x**8*(3
*A*b**2*c/8 + B*b**3/8)

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Giac [A]  time = 1.12685, size = 104, normalized size = 1.39 \begin{align*} \frac{1}{11} \, B c^{3} x^{11} + \frac{3}{10} \, B b c^{2} x^{10} + \frac{1}{10} \, A c^{3} x^{10} + \frac{1}{3} \, B b^{2} c x^{9} + \frac{1}{3} \, A b c^{2} x^{9} + \frac{1}{8} \, B b^{3} x^{8} + \frac{3}{8} \, A b^{2} c x^{8} + \frac{1}{7} \, A b^{3} x^{7} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/11*B*c^3*x^11 + 3/10*B*b*c^2*x^10 + 1/10*A*c^3*x^10 + 1/3*B*b^2*c*x^9 + 1/3*A*b*c^2*x^9 + 1/8*B*b^3*x^8 + 3/
8*A*b^2*c*x^8 + 1/7*A*b^3*x^7