3.9 \(\int \frac{(A+B x) (b x+c x^2)}{x^4} \, dx\)

Optimal. Leaf size=27 \[ -\frac{A c+b B}{x}-\frac{A b}{2 x^2}+B c \log (x) \]

[Out]

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

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Rubi [A]  time = 0.0165446, antiderivative size = 27, 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} \[ -\frac{A c+b B}{x}-\frac{A b}{2 x^2}+B c \log (x) \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(b*x + c*x^2))/x^4,x]

[Out]

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

Mathematica [A]  time = 0.00983, size = 28, normalized size = 1.04 \[ \frac{-A c-b B}{x}-\frac{A b}{2 x^2}+B c \log (x) \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(b*x + c*x^2))/x^4,x]

[Out]

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

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Maple [A]  time = 0.006, size = 28, normalized size = 1. \begin{align*} Bc\ln \left ( x \right ) -{\frac{Ab}{2\,{x}^{2}}}-{\frac{Ac}{x}}-{\frac{bB}{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(c*x^2+b*x)/x^4,x)

[Out]

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

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Maxima [A]  time = 0.991116, size = 34, normalized size = 1.26 \begin{align*} B c \log \left (x\right ) - \frac{A b + 2 \,{\left (B b + A 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)/x^4,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.7421, size = 70, normalized size = 2.59 \begin{align*} \frac{2 \, B c x^{2} \log \left (x\right ) - A b - 2 \,{\left (B b + A 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)/x^4,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.399622, size = 26, normalized size = 0.96 \begin{align*} B c \log{\left (x \right )} - \frac{A b + x \left (2 A c + 2 B b\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)/x**4,x)

[Out]

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

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Giac [A]  time = 1.1383, size = 35, normalized size = 1.3 \begin{align*} B c \log \left ({\left | x \right |}\right ) - \frac{A b + 2 \,{\left (B b + A 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)/x^4,x, algorithm="giac")

[Out]

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