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

Optimal. Leaf size=24 \[ -\frac{a^2}{4 x^4}-\frac{a b}{x^2}+b^2 \log (x) \]

[Out]

-a^2/(4*x^4) - (a*b)/x^2 + b^2*Log[x]

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Rubi [A]  time = 0.01285, antiderivative size = 24, 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} \[ -\frac{a^2}{4 x^4}-\frac{a b}{x^2}+b^2 \log (x) \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^2/x^5,x]

[Out]

-a^2/(4*x^4) - (a*b)/x^2 + b^2*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^5} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{(a+b x)^2}{x^3} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{a^2}{x^3}+\frac{2 a b}{x^2}+\frac{b^2}{x}\right ) \, dx,x,x^2\right )\\ &=-\frac{a^2}{4 x^4}-\frac{a b}{x^2}+b^2 \log (x)\\ \end{align*}

Mathematica [A]  time = 0.0007085, size = 24, normalized size = 1. \[ -\frac{a^2}{4 x^4}-\frac{a b}{x^2}+b^2 \log (x) \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^2/x^5,x]

[Out]

-a^2/(4*x^4) - (a*b)/x^2 + b^2*Log[x]

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Maple [A]  time = 0.005, size = 23, normalized size = 1. \begin{align*} -{\frac{{a}^{2}}{4\,{x}^{4}}}-{\frac{ab}{{x}^{2}}}+{b}^{2}\ln \left ( x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^2/x^5,x)

[Out]

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

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Maxima [A]  time = 2.40295, size = 35, normalized size = 1.46 \begin{align*} \frac{1}{2} \, b^{2} \log \left (x^{2}\right ) - \frac{4 \, a b x^{2} + a^{2}}{4 \, x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^2/x^5,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.44523, size = 62, normalized size = 2.58 \begin{align*} \frac{4 \, b^{2} x^{4} \log \left (x\right ) - 4 \, a b x^{2} - a^{2}}{4 \, x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^2/x^5,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.31295, size = 22, normalized size = 0.92 \begin{align*} b^{2} \log{\left (x \right )} - \frac{a^{2} + 4 a b x^{2}}{4 x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**2/x**5,x)

[Out]

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

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Giac [A]  time = 2.70646, size = 46, normalized size = 1.92 \begin{align*} \frac{1}{2} \, b^{2} \log \left (x^{2}\right ) - \frac{3 \, b^{2} x^{4} + 4 \, a b x^{2} + a^{2}}{4 \, x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^2/x^5,x, algorithm="giac")

[Out]

1/2*b^2*log(x^2) - 1/4*(3*b^2*x^4 + 4*a*b*x^2 + a^2)/x^4