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

Optimal. Leaf size=26 \[ -\frac{a A}{x}+a B \log (x)+A c x+\frac{1}{2} B c x^2 \]

[Out]

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

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Rubi [A]  time = 0.0143074, 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} \[ -\frac{a A}{x}+a B \log (x)+A c x+\frac{1}{2} B c x^2 \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(a + c*x^2))/x^2,x]

[Out]

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

Mathematica [A]  time = 0.00318, size = 26, normalized size = 1. \[ -\frac{a A}{x}+a B \log (x)+A c x+\frac{1}{2} B c x^2 \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.007, size = 25, normalized size = 1. \begin{align*} -{\frac{aA}{x}}+Acx+{\frac{Bc{x}^{2}}{2}}+aB\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^2,x)

[Out]

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

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Maxima [A]  time = 1.03454, size = 32, normalized size = 1.23 \begin{align*} \frac{1}{2} \, B c x^{2} + A c x + B a \log \left (x\right ) - \frac{A a}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.55518, size = 73, normalized size = 2.81 \begin{align*} \frac{B c x^{3} + 2 \, A c x^{2} + 2 \, B a x \log \left (x\right ) - 2 \, A a}{2 \, x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.287171, size = 24, normalized size = 0.92 \begin{align*} - \frac{A a}{x} + A c x + B a \log{\left (x \right )} + \frac{B c x^{2}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.10117, size = 34, normalized size = 1.31 \begin{align*} \frac{1}{2} \, B c x^{2} + A c x + B a \log \left ({\left | x \right |}\right ) - \frac{A a}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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