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

Optimal. Leaf size=75 \[ \frac{1}{6} b^2 x^6 (3 A c+b B)+\frac{1}{5} A b^3 x^5+\frac{1}{8} c^2 x^8 (A c+3 b B)+\frac{3}{7} b c x^7 (A c+b B)+\frac{1}{9} B c^3 x^9 \]

[Out]

(A*b^3*x^5)/5 + (b^2*(b*B + 3*A*c)*x^6)/6 + (3*b*c*(b*B + A*c)*x^7)/7 + (c^2*(3*b*B + A*c)*x^8)/8 + (B*c^3*x^9
)/9

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Rubi [A]  time = 0.0573346, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056, Rules used = {765} \[ \frac{1}{6} b^2 x^6 (3 A c+b B)+\frac{1}{5} A b^3 x^5+\frac{1}{8} c^2 x^8 (A c+3 b B)+\frac{3}{7} b c x^7 (A c+b B)+\frac{1}{9} B c^3 x^9 \]

Antiderivative was successfully verified.

[In]

Int[x*(A + B*x)*(b*x + c*x^2)^3,x]

[Out]

(A*b^3*x^5)/5 + (b^2*(b*B + 3*A*c)*x^6)/6 + (3*b*c*(b*B + A*c)*x^7)/7 + (c^2*(3*b*B + A*c)*x^8)/8 + (B*c^3*x^9
)/9

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

Mathematica [A]  time = 0.0120437, size = 75, normalized size = 1. \[ \frac{1}{6} b^2 x^6 (3 A c+b B)+\frac{1}{5} A b^3 x^5+\frac{1}{8} c^2 x^8 (A c+3 b B)+\frac{3}{7} b c x^7 (A c+b B)+\frac{1}{9} B c^3 x^9 \]

Antiderivative was successfully verified.

[In]

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

[Out]

(A*b^3*x^5)/5 + (b^2*(b*B + 3*A*c)*x^6)/6 + (3*b*c*(b*B + A*c)*x^7)/7 + (c^2*(3*b*B + A*c)*x^8)/8 + (B*c^3*x^9
)/9

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Maple [A]  time = 0.001, size = 76, normalized size = 1. \begin{align*}{\frac{B{c}^{3}{x}^{9}}{9}}+{\frac{ \left ( A{c}^{3}+3\,Bb{c}^{2} \right ){x}^{8}}{8}}+{\frac{ \left ( 3\,Ab{c}^{2}+3\,B{b}^{2}c \right ){x}^{7}}{7}}+{\frac{ \left ( 3\,A{b}^{2}c+{b}^{3}B \right ){x}^{6}}{6}}+{\frac{A{b}^{3}{x}^{5}}{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*B*c^3*x^9+1/8*(A*c^3+3*B*b*c^2)*x^8+1/7*(3*A*b*c^2+3*B*b^2*c)*x^7+1/6*(3*A*b^2*c+B*b^3)*x^6+1/5*A*b^3*x^5

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Maxima [A]  time = 1.1048, size = 99, normalized size = 1.32 \begin{align*} \frac{1}{9} \, B c^{3} x^{9} + \frac{1}{5} \, A b^{3} x^{5} + \frac{1}{8} \,{\left (3 \, B b c^{2} + A c^{3}\right )} x^{8} + \frac{3}{7} \,{\left (B b^{2} c + A b c^{2}\right )} x^{7} + \frac{1}{6} \,{\left (B b^{3} + 3 \, A b^{2} c\right )} x^{6} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*B*c^3*x^9 + 1/5*A*b^3*x^5 + 1/8*(3*B*b*c^2 + A*c^3)*x^8 + 3/7*(B*b^2*c + A*b*c^2)*x^7 + 1/6*(B*b^3 + 3*A*b
^2*c)*x^6

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Fricas [A]  time = 1.49758, size = 182, normalized size = 2.43 \begin{align*} \frac{1}{9} x^{9} c^{3} B + \frac{3}{8} x^{8} c^{2} b B + \frac{1}{8} x^{8} c^{3} A + \frac{3}{7} x^{7} c b^{2} B + \frac{3}{7} x^{7} c^{2} b A + \frac{1}{6} x^{6} b^{3} B + \frac{1}{2} x^{6} c b^{2} A + \frac{1}{5} x^{5} b^{3} A \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*x^9*c^3*B + 3/8*x^8*c^2*b*B + 1/8*x^8*c^3*A + 3/7*x^7*c*b^2*B + 3/7*x^7*c^2*b*A + 1/6*x^6*b^3*B + 1/2*x^6*
c*b^2*A + 1/5*x^5*b^3*A

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Sympy [A]  time = 0.140414, size = 82, normalized size = 1.09 \begin{align*} \frac{A b^{3} x^{5}}{5} + \frac{B c^{3} x^{9}}{9} + x^{8} \left (\frac{A c^{3}}{8} + \frac{3 B b c^{2}}{8}\right ) + x^{7} \left (\frac{3 A b c^{2}}{7} + \frac{3 B b^{2} c}{7}\right ) + x^{6} \left (\frac{A b^{2} c}{2} + \frac{B b^{3}}{6}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

A*b**3*x**5/5 + B*c**3*x**9/9 + x**8*(A*c**3/8 + 3*B*b*c**2/8) + x**7*(3*A*b*c**2/7 + 3*B*b**2*c/7) + x**6*(A*
b**2*c/2 + B*b**3/6)

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Giac [A]  time = 1.15231, size = 104, normalized size = 1.39 \begin{align*} \frac{1}{9} \, B c^{3} x^{9} + \frac{3}{8} \, B b c^{2} x^{8} + \frac{1}{8} \, A c^{3} x^{8} + \frac{3}{7} \, B b^{2} c x^{7} + \frac{3}{7} \, A b c^{2} x^{7} + \frac{1}{6} \, B b^{3} x^{6} + \frac{1}{2} \, A b^{2} c x^{6} + \frac{1}{5} \, A b^{3} x^{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*B*c^3*x^9 + 3/8*B*b*c^2*x^8 + 1/8*A*c^3*x^8 + 3/7*B*b^2*c*x^7 + 3/7*A*b*c^2*x^7 + 1/6*B*b^3*x^6 + 1/2*A*b^
2*c*x^6 + 1/5*A*b^3*x^5