3.19 \(\int \frac{(A+B x) (b x+c x^2)^2}{x} \, dx\)

Optimal. Leaf size=55 \[ \frac{1}{2} A b^2 x^2+\frac{1}{4} c x^4 (A c+2 b B)+\frac{1}{3} b x^3 (2 A c+b B)+\frac{1}{5} B c^2 x^5 \]

[Out]

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

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Rubi [A]  time = 0.0358139, 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} \[ \frac{1}{2} A b^2 x^2+\frac{1}{4} c x^4 (A c+2 b B)+\frac{1}{3} b x^3 (2 A c+b B)+\frac{1}{5} B c^2 x^5 \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(b*x + c*x^2)^2)/x,x]

[Out]

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

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

Mathematica [A]  time = 0.0102534, size = 49, normalized size = 0.89 \[ \frac{1}{60} x^2 \left (30 A b^2+15 c x^2 (A c+2 b B)+20 b x (2 A c+b B)+12 B c^2 x^3\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x^2*(30*A*b^2 + 20*b*(b*B + 2*A*c)*x + 15*c*(2*b*B + A*c)*x^2 + 12*B*c^2*x^3))/60

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Maple [A]  time = 0.001, size = 52, normalized size = 1. \begin{align*}{\frac{B{c}^{2}{x}^{5}}{5}}+{\frac{ \left ( A{c}^{2}+2\,Bbc \right ){x}^{4}}{4}}+{\frac{ \left ( 2\,Abc+{b}^{2}B \right ){x}^{3}}{3}}+{\frac{A{b}^{2}{x}^{2}}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*B*c^2*x^5+1/4*(A*c^2+2*B*b*c)*x^4+1/3*(2*A*b*c+B*b^2)*x^3+1/2*A*b^2*x^2

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Maxima [A]  time = 1.13337, size = 69, normalized size = 1.25 \begin{align*} \frac{1}{5} \, B c^{2} x^{5} + \frac{1}{2} \, A b^{2} x^{2} + \frac{1}{4} \,{\left (2 \, B b c + A c^{2}\right )} x^{4} + \frac{1}{3} \,{\left (B b^{2} + 2 \, A b c\right )} x^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*B*c^2*x^5 + 1/2*A*b^2*x^2 + 1/4*(2*B*b*c + A*c^2)*x^4 + 1/3*(B*b^2 + 2*A*b*c)*x^3

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Fricas [A]  time = 1.73533, size = 117, normalized size = 2.13 \begin{align*} \frac{1}{5} \, B c^{2} x^{5} + \frac{1}{2} \, A b^{2} x^{2} + \frac{1}{4} \,{\left (2 \, B b c + A c^{2}\right )} x^{4} + \frac{1}{3} \,{\left (B b^{2} + 2 \, A b c\right )} x^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*B*c^2*x^5 + 1/2*A*b^2*x^2 + 1/4*(2*B*b*c + A*c^2)*x^4 + 1/3*(B*b^2 + 2*A*b*c)*x^3

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Sympy [A]  time = 0.071358, size = 54, normalized size = 0.98 \begin{align*} \frac{A b^{2} x^{2}}{2} + \frac{B c^{2} x^{5}}{5} + x^{4} \left (\frac{A c^{2}}{4} + \frac{B b c}{2}\right ) + x^{3} \left (\frac{2 A b c}{3} + \frac{B b^{2}}{3}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.1302, size = 72, normalized size = 1.31 \begin{align*} \frac{1}{5} \, B c^{2} x^{5} + \frac{1}{2} \, B b c x^{4} + \frac{1}{4} \, A c^{2} x^{4} + \frac{1}{3} \, B b^{2} x^{3} + \frac{2}{3} \, A b c x^{3} + \frac{1}{2} \, A b^{2} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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