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

Optimal. Leaf size=55 \[ \frac{1}{4} A b^2 x^4+\frac{1}{6} c x^6 (A c+2 b B)+\frac{1}{5} b x^5 (2 A c+b B)+\frac{1}{7} B c^2 x^7 \]

[Out]

(A*b^2*x^4)/4 + (b*(b*B + 2*A*c)*x^5)/5 + (c*(2*b*B + A*c)*x^6)/6 + (B*c^2*x^7)/7

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Rubi [A]  time = 0.0485926, antiderivative size = 55, 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} \[ \frac{1}{4} A b^2 x^4+\frac{1}{6} c x^6 (A c+2 b B)+\frac{1}{5} b x^5 (2 A c+b B)+\frac{1}{7} B c^2 x^7 \]

Antiderivative was successfully verified.

[In]

Int[x*(A + B*x)*(b*x + c*x^2)^2,x]

[Out]

(A*b^2*x^4)/4 + (b*(b*B + 2*A*c)*x^5)/5 + (c*(2*b*B + A*c)*x^6)/6 + (B*c^2*x^7)/7

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

Mathematica [A]  time = 0.0086164, size = 55, normalized size = 1. \[ \frac{1}{4} A b^2 x^4+\frac{1}{6} c x^6 (A c+2 b B)+\frac{1}{5} b x^5 (2 A c+b B)+\frac{1}{7} B c^2 x^7 \]

Antiderivative was successfully verified.

[In]

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

[Out]

(A*b^2*x^4)/4 + (b*(b*B + 2*A*c)*x^5)/5 + (c*(2*b*B + A*c)*x^6)/6 + (B*c^2*x^7)/7

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Maple [A]  time = 0., size = 52, normalized size = 1. \begin{align*}{\frac{B{c}^{2}{x}^{7}}{7}}+{\frac{ \left ( A{c}^{2}+2\,Bbc \right ){x}^{6}}{6}}+{\frac{ \left ( 2\,Abc+{b}^{2}B \right ){x}^{5}}{5}}+{\frac{A{b}^{2}{x}^{4}}{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*B*c^2*x^7+1/6*(A*c^2+2*B*b*c)*x^6+1/5*(2*A*b*c+B*b^2)*x^5+1/4*A*b^2*x^4

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Maxima [A]  time = 1.14083, size = 69, normalized size = 1.25 \begin{align*} \frac{1}{7} \, B c^{2} x^{7} + \frac{1}{4} \, A b^{2} x^{4} + \frac{1}{6} \,{\left (2 \, B b c + A c^{2}\right )} x^{6} + \frac{1}{5} \,{\left (B b^{2} + 2 \, A b c\right )} 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)^2,x, algorithm="maxima")

[Out]

1/7*B*c^2*x^7 + 1/4*A*b^2*x^4 + 1/6*(2*B*b*c + A*c^2)*x^6 + 1/5*(B*b^2 + 2*A*b*c)*x^5

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Fricas [A]  time = 1.6214, size = 128, normalized size = 2.33 \begin{align*} \frac{1}{7} x^{7} c^{2} B + \frac{1}{3} x^{6} c b B + \frac{1}{6} x^{6} c^{2} A + \frac{1}{5} x^{5} b^{2} B + \frac{2}{5} x^{5} c b A + \frac{1}{4} x^{4} b^{2} A \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*x^7*c^2*B + 1/3*x^6*c*b*B + 1/6*x^6*c^2*A + 1/5*x^5*b^2*B + 2/5*x^5*c*b*A + 1/4*x^4*b^2*A

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Sympy [A]  time = 0.069056, size = 54, normalized size = 0.98 \begin{align*} \frac{A b^{2} x^{4}}{4} + \frac{B c^{2} x^{7}}{7} + x^{6} \left (\frac{A c^{2}}{6} + \frac{B b c}{3}\right ) + x^{5} \left (\frac{2 A b c}{5} + \frac{B b^{2}}{5}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

A*b**2*x**4/4 + B*c**2*x**7/7 + x**6*(A*c**2/6 + B*b*c/3) + x**5*(2*A*b*c/5 + B*b**2/5)

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Giac [A]  time = 1.12792, size = 72, normalized size = 1.31 \begin{align*} \frac{1}{7} \, B c^{2} x^{7} + \frac{1}{3} \, B b c x^{6} + \frac{1}{6} \, A c^{2} x^{6} + \frac{1}{5} \, B b^{2} x^{5} + \frac{2}{5} \, A b c x^{5} + \frac{1}{4} \, A b^{2} x^{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*B*c^2*x^7 + 1/3*B*b*c*x^6 + 1/6*A*c^2*x^6 + 1/5*B*b^2*x^5 + 2/5*A*b*c*x^5 + 1/4*A*b^2*x^4