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

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

[Out]

-a^2/(2*x^2) + (b^2*x^2)/2 + 2*a*b*Log[x]

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Rubi [A]  time = 0.0135157, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {266, 43} \[ -\frac{a^2}{2 x^2}+2 a b \log (x)+\frac{b^2 x^2}{2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^2/x^3,x]

[Out]

-a^2/(2*x^2) + (b^2*x^2)/2 + 2*a*b*Log[x]

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{\left (a+b x^2\right )^2}{x^3} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{(a+b x)^2}{x^2} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (b^2+\frac{a^2}{x^2}+\frac{2 a b}{x}\right ) \, dx,x,x^2\right )\\ &=-\frac{a^2}{2 x^2}+\frac{b^2 x^2}{2}+2 a b \log (x)\\ \end{align*}

Mathematica [A]  time = 0.0007728, size = 27, normalized size = 1. \[ -\frac{a^2}{2 x^2}+2 a b \log (x)+\frac{b^2 x^2}{2} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^2/x^3,x]

[Out]

-a^2/(2*x^2) + (b^2*x^2)/2 + 2*a*b*Log[x]

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Maple [A]  time = 0.005, size = 24, normalized size = 0.9 \begin{align*} -{\frac{{a}^{2}}{2\,{x}^{2}}}+{\frac{{b}^{2}{x}^{2}}{2}}+2\,ab\ln \left ( x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^2/x^3,x)

[Out]

-1/2/x^2*a^2+1/2*b^2*x^2+2*a*b*ln(x)

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Maxima [A]  time = 1.33831, size = 32, normalized size = 1.19 \begin{align*} \frac{1}{2} \, b^{2} x^{2} + a b \log \left (x^{2}\right ) - \frac{a^{2}}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*b^2*x^2 + a*b*log(x^2) - 1/2*a^2/x^2

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Fricas [A]  time = 1.39969, size = 59, normalized size = 2.19 \begin{align*} \frac{b^{2} x^{4} + 4 \, a b x^{2} \log \left (x\right ) - a^{2}}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(b^2*x^4 + 4*a*b*x^2*log(x) - a^2)/x^2

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Sympy [A]  time = 0.270194, size = 24, normalized size = 0.89 \begin{align*} - \frac{a^{2}}{2 x^{2}} + 2 a b \log{\left (x \right )} + \frac{b^{2} x^{2}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**2/x**3,x)

[Out]

-a**2/(2*x**2) + 2*a*b*log(x) + b**2*x**2/2

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Giac [A]  time = 1.55442, size = 43, normalized size = 1.59 \begin{align*} \frac{1}{2} \, b^{2} x^{2} + a b \log \left (x^{2}\right ) - \frac{2 \, a b x^{2} + a^{2}}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*b^2*x^2 + a*b*log(x^2) - 1/2*(2*a*b*x^2 + a^2)/x^2