3.23 \(\int \frac{(A+B x) (b x+c x^2)^2}{x^5} \, dx\)

Optimal. Leaf size=44 \[ -\frac{A b^2}{2 x^2}-\frac{b (2 A c+b B)}{x}+c \log (x) (A c+2 b B)+B c^2 x \]

[Out]

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

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Rubi [A]  time = 0.0299489, antiderivative size = 44, 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} \[ -\frac{A b^2}{2 x^2}-\frac{b (2 A c+b B)}{x}+c \log (x) (A c+2 b B)+B c^2 x \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(b*x + c*x^2)^2)/x^5,x]

[Out]

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

Mathematica [A]  time = 0.0252772, size = 44, normalized size = 1. \[ -\frac{A b^2}{2 x^2}-\frac{b (2 A c+b B)}{x}+c \log (x) (A c+2 b B)+B c^2 x \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(b*x + c*x^2)^2)/x^5,x]

[Out]

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

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Maple [A]  time = 0.007, size = 48, normalized size = 1.1 \begin{align*} B{c}^{2}x+A\ln \left ( x \right ){c}^{2}+2\,B\ln \left ( x \right ) bc-2\,{\frac{Abc}{x}}-{\frac{{b}^{2}B}{x}}-{\frac{A{b}^{2}}{2\,{x}^{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^5,x)

[Out]

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

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Maxima [A]  time = 1.16344, size = 62, normalized size = 1.41 \begin{align*} B c^{2} x +{\left (2 \, B b c + A c^{2}\right )} \log \left (x\right ) - \frac{A b^{2} + 2 \,{\left (B b^{2} + 2 \, A b c\right )} x}{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^5,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.76216, size = 119, normalized size = 2.7 \begin{align*} \frac{2 \, B c^{2} x^{3} + 2 \,{\left (2 \, B b c + A c^{2}\right )} x^{2} \log \left (x\right ) - A b^{2} - 2 \,{\left (B b^{2} + 2 \, A b c\right )} x}{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^5,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.565436, size = 44, normalized size = 1. \begin{align*} B c^{2} x + c \left (A c + 2 B b\right ) \log{\left (x \right )} - \frac{A b^{2} + x \left (4 A b c + 2 B b^{2}\right )}{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**5,x)

[Out]

B*c**2*x + c*(A*c + 2*B*b)*log(x) - (A*b**2 + x*(4*A*b*c + 2*B*b**2))/(2*x**2)

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Giac [A]  time = 1.15981, size = 63, normalized size = 1.43 \begin{align*} B c^{2} x +{\left (2 \, B b c + A c^{2}\right )} \log \left ({\left | x \right |}\right ) - \frac{A b^{2} + 2 \,{\left (B b^{2} + 2 \, A b c\right )} x}{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^5,x, algorithm="giac")

[Out]

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