3.21 \(\int \frac{(A+B x) (b x+c x^2)^2}{x^3} \, dx\)

Optimal. Leaf size=46 \[ A b^2 \log (x)+\frac{1}{2} c x^2 (A c+2 b B)+b x (2 A c+b B)+\frac{1}{3} B c^2 x^3 \]

[Out]

b*(b*B + 2*A*c)*x + (c*(2*b*B + A*c)*x^2)/2 + (B*c^2*x^3)/3 + A*b^2*Log[x]

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Rubi [A]  time = 0.0271827, antiderivative size = 46, 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} \[ A b^2 \log (x)+\frac{1}{2} c x^2 (A c+2 b B)+b x (2 A c+b B)+\frac{1}{3} B c^2 x^3 \]

Antiderivative was successfully verified.

[In]

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

[Out]

b*(b*B + 2*A*c)*x + (c*(2*b*B + A*c)*x^2)/2 + (B*c^2*x^3)/3 + A*b^2*Log[x]

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

Mathematica [A]  time = 0.0142341, size = 43, normalized size = 0.93 \[ A b^2 \log (x)+b c x (2 A+B x)+\frac{1}{6} c^2 x^2 (3 A+2 B x)+b^2 B x \]

Antiderivative was successfully verified.

[In]

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

[Out]

b^2*B*x + b*c*x*(2*A + B*x) + (c^2*x^2*(3*A + 2*B*x))/6 + A*b^2*Log[x]

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Maple [A]  time = 0.002, size = 46, normalized size = 1. \begin{align*}{\frac{B{c}^{2}{x}^{3}}{3}}+{\frac{A{c}^{2}{x}^{2}}{2}}+B{x}^{2}bc+2\,Abcx+{b}^{2}Bx+A{b}^{2}\ln \left ( x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.14898, size = 62, normalized size = 1.35 \begin{align*} \frac{1}{3} \, B c^{2} x^{3} + A b^{2} \log \left (x\right ) + \frac{1}{2} \,{\left (2 \, B b c + A c^{2}\right )} x^{2} +{\left (B b^{2} + 2 \, A b c\right )} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.61684, size = 108, normalized size = 2.35 \begin{align*} \frac{1}{3} \, B c^{2} x^{3} + A b^{2} \log \left (x\right ) + \frac{1}{2} \,{\left (2 \, B b c + A c^{2}\right )} x^{2} +{\left (B b^{2} + 2 \, A b c\right )} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.297472, size = 46, normalized size = 1. \begin{align*} A b^{2} \log{\left (x \right )} + \frac{B c^{2} x^{3}}{3} + x^{2} \left (\frac{A c^{2}}{2} + B b c\right ) + x \left (2 A b c + B b^{2}\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**3,x)

[Out]

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

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Giac [A]  time = 1.11434, size = 62, normalized size = 1.35 \begin{align*} \frac{1}{3} \, B c^{2} x^{3} + B b c x^{2} + \frac{1}{2} \, A c^{2} x^{2} + B b^{2} x + 2 \, A b c x + A b^{2} \log \left ({\left | x \right |}\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^3,x, algorithm="giac")

[Out]

1/3*B*c^2*x^3 + B*b*c*x^2 + 1/2*A*c^2*x^2 + B*b^2*x + 2*A*b*c*x + A*b^2*log(abs(x))