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

Optimal. Leaf size=26 \[ -\frac{a A}{2 x^2}-\frac{a B}{x}+A c \log (x)+B c x \]

[Out]

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

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Rubi [A]  time = 0.0142789, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062, Rules used = {766} \[ -\frac{a A}{2 x^2}-\frac{a B}{x}+A c \log (x)+B c x \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(a + c*x^2))/x^3,x]

[Out]

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

Rule 766

Int[((e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(e*x
)^m*(f + g*x)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, e, f, g, m}, x] && IGtQ[p, 0]

Rubi steps

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

Mathematica [A]  time = 0.0040867, size = 26, normalized size = 1. \[ -\frac{a A}{2 x^2}-\frac{a B}{x}+A c \log (x)+B c x \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.006, size = 25, normalized size = 1. \begin{align*} -{\frac{aA}{2\,{x}^{2}}}-{\frac{aB}{x}}+Bcx+Ac\ln \left ( x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.02767, size = 32, normalized size = 1.23 \begin{align*} B c x + A c \log \left (x\right ) - \frac{2 \, B a x + A a}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.51101, size = 76, normalized size = 2.92 \begin{align*} \frac{2 \, B c x^{3} + 2 \, A c x^{2} \log \left (x\right ) - 2 \, B a x - A a}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.370478, size = 26, normalized size = 1. \begin{align*} A c \log{\left (x \right )} + B c x - \frac{A a + 2 B a x}{2 x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x**2+a)/x**3,x)

[Out]

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

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Giac [A]  time = 1.13927, size = 34, normalized size = 1.31 \begin{align*} B c x + A c \log \left ({\left | x \right |}\right ) - \frac{2 \, B a x + A a}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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