Optimal. Leaf size=40 \[ 3 a^2 b \log (x)-\frac{a^3}{4 x^4}+\frac{3}{4} a b^2 x^4+\frac{b^3 x^8}{8} \]
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Rubi [A] time = 0.0214073, antiderivative size = 40, 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} \[ 3 a^2 b \log (x)-\frac{a^3}{4 x^4}+\frac{3}{4} a b^2 x^4+\frac{b^3 x^8}{8} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{\left (a+b x^4\right )^3}{x^5} \, dx &=\frac{1}{4} \operatorname{Subst}\left (\int \frac{(a+b x)^3}{x^2} \, dx,x,x^4\right )\\ &=\frac{1}{4} \operatorname{Subst}\left (\int \left (3 a b^2+\frac{a^3}{x^2}+\frac{3 a^2 b}{x}+b^3 x\right ) \, dx,x,x^4\right )\\ &=-\frac{a^3}{4 x^4}+\frac{3}{4} a b^2 x^4+\frac{b^3 x^8}{8}+3 a^2 b \log (x)\\ \end{align*}
Mathematica [A] time = 0.0043256, size = 40, normalized size = 1. \[ 3 a^2 b \log (x)-\frac{a^3}{4 x^4}+\frac{3}{4} a b^2 x^4+\frac{b^3 x^8}{8} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 35, normalized size = 0.9 \begin{align*} -{\frac{{a}^{3}}{4\,{x}^{4}}}+{\frac{3\,a{b}^{2}{x}^{4}}{4}}+{\frac{{b}^{3}{x}^{8}}{8}}+3\,{a}^{2}b\ln \left ( x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.954185, size = 49, normalized size = 1.22 \begin{align*} \frac{1}{8} \, b^{3} x^{8} + \frac{3}{4} \, a b^{2} x^{4} + \frac{3}{4} \, a^{2} b \log \left (x^{4}\right ) - \frac{a^{3}}{4 \, x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.4121, size = 86, normalized size = 2.15 \begin{align*} \frac{b^{3} x^{12} + 6 \, a b^{2} x^{8} + 24 \, a^{2} b x^{4} \log \left (x\right ) - 2 \, a^{3}}{8 \, x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.347013, size = 37, normalized size = 0.92 \begin{align*} - \frac{a^{3}}{4 x^{4}} + 3 a^{2} b \log{\left (x \right )} + \frac{3 a b^{2} x^{4}}{4} + \frac{b^{3} x^{8}}{8} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14398, size = 62, normalized size = 1.55 \begin{align*} \frac{1}{8} \, b^{3} x^{8} + \frac{3}{4} \, a b^{2} x^{4} + \frac{3}{4} \, a^{2} b \log \left (x^{4}\right ) - \frac{3 \, a^{2} b x^{4} + a^{3}}{4 \, x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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