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

Optimal. Leaf size=55 \[ \frac{1}{5} A b^2 x^5+\frac{1}{7} c x^7 (A c+2 b B)+\frac{1}{6} b x^6 (2 A c+b B)+\frac{1}{8} B c^2 x^8 \]

[Out]

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

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Rubi [A]  time = 0.0558134, antiderivative size = 55, 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} \[ \frac{1}{5} A b^2 x^5+\frac{1}{7} c x^7 (A c+2 b B)+\frac{1}{6} b x^6 (2 A c+b B)+\frac{1}{8} B c^2 x^8 \]

Antiderivative was successfully verified.

[In]

Int[x^2*(A + B*x)*(b*x + c*x^2)^2,x]

[Out]

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

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

Mathematica [A]  time = 0.0088347, size = 55, normalized size = 1. \[ \frac{1}{5} A b^2 x^5+\frac{1}{7} c x^7 (A c+2 b B)+\frac{1}{6} b x^6 (2 A c+b B)+\frac{1}{8} B c^2 x^8 \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.001, size = 52, normalized size = 1. \begin{align*}{\frac{B{c}^{2}{x}^{8}}{8}}+{\frac{ \left ( A{c}^{2}+2\,Bbc \right ){x}^{7}}{7}}+{\frac{ \left ( 2\,Abc+{b}^{2}B \right ){x}^{6}}{6}}+{\frac{A{b}^{2}{x}^{5}}{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.12066, size = 69, normalized size = 1.25 \begin{align*} \frac{1}{8} \, B c^{2} x^{8} + \frac{1}{5} \, A b^{2} x^{5} + \frac{1}{7} \,{\left (2 \, B b c + A c^{2}\right )} x^{7} + \frac{1}{6} \,{\left (B b^{2} + 2 \, A b c\right )} x^{6} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.56158, size = 128, normalized size = 2.33 \begin{align*} \frac{1}{8} x^{8} c^{2} B + \frac{2}{7} x^{7} c b B + \frac{1}{7} x^{7} c^{2} A + \frac{1}{6} x^{6} b^{2} B + \frac{1}{3} x^{6} c b A + \frac{1}{5} x^{5} b^{2} A \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.06951, size = 54, normalized size = 0.98 \begin{align*} \frac{A b^{2} x^{5}}{5} + \frac{B c^{2} x^{8}}{8} + x^{7} \left (\frac{A c^{2}}{7} + \frac{2 B b c}{7}\right ) + x^{6} \left (\frac{A b c}{3} + \frac{B b^{2}}{6}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.1172, size = 72, normalized size = 1.31 \begin{align*} \frac{1}{8} \, B c^{2} x^{8} + \frac{2}{7} \, B b c x^{7} + \frac{1}{7} \, A c^{2} x^{7} + \frac{1}{6} \, B b^{2} x^{6} + \frac{1}{3} \, A b c x^{6} + \frac{1}{5} \, A b^{2} x^{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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