3.631 \(\int \frac{(a+b x^4)^2}{x^5} \, dx\)

Optimal. Leaf size=27 \[ -\frac{a^2}{4 x^4}+2 a b \log (x)+\frac{b^2 x^4}{4} \]

[Out]

-a^2/(4*x^4) + (b^2*x^4)/4 + 2*a*b*Log[x]

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Rubi [A]  time = 0.0132604, 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} \[ -\frac{a^2}{4 x^4}+2 a b \log (x)+\frac{b^2 x^4}{4} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^4)^2/x^5,x]

[Out]

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

Mathematica [A]  time = 0.0006649, size = 27, normalized size = 1. \[ -\frac{a^2}{4 x^4}+2 a b \log (x)+\frac{b^2 x^4}{4} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^4)^2/x^5,x]

[Out]

-a^2/(4*x^4) + (b^2*x^4)/4 + 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}}{4\,{x}^{4}}}+{\frac{{b}^{2}{x}^{4}}{4}}+2\,ab\ln \left ( x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^4+a)^2/x^5,x)

[Out]

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

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Maxima [A]  time = 0.979432, size = 34, normalized size = 1.26 \begin{align*} \frac{1}{4} \, b^{2} x^{4} + \frac{1}{2} \, a b \log \left (x^{4}\right ) - \frac{a^{2}}{4 \, x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^4+a)^2/x^5,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.36255, size = 59, normalized size = 2.19 \begin{align*} \frac{b^{2} x^{8} + 8 \, a b x^{4} \log \left (x\right ) - a^{2}}{4 \, x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^4+a)^2/x^5,x, algorithm="fricas")

[Out]

1/4*(b^2*x^8 + 8*a*b*x^4*log(x) - a^2)/x^4

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Sympy [A]  time = 0.39181, size = 24, normalized size = 0.89 \begin{align*} - \frac{a^{2}}{4 x^{4}} + 2 a b \log{\left (x \right )} + \frac{b^{2} x^{4}}{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**4+a)**2/x**5,x)

[Out]

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

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Giac [A]  time = 1.08996, size = 45, normalized size = 1.67 \begin{align*} \frac{1}{4} \, b^{2} x^{4} + \frac{1}{2} \, a b \log \left (x^{4}\right ) - \frac{2 \, a b x^{4} + a^{2}}{4 \, x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^4+a)^2/x^5,x, algorithm="giac")

[Out]

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