Optimal. Leaf size=98 \[ \frac{5 \sqrt{a} b^{3/2} \left (\frac{b x^4}{a}+1\right )^{3/4} \text{EllipticF}\left (\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right ),2\right )}{12 \left (a+b x^4\right )^{3/4}}-\frac{5 b \sqrt [4]{a+b x^4}}{12 x^2}-\frac{\left (a+b x^4\right )^{5/4}}{6 x^6} \]
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Rubi [A] time = 0.0611447, antiderivative size = 98, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {275, 277, 233, 231} \[ \frac{5 \sqrt{a} b^{3/2} \left (\frac{b x^4}{a}+1\right )^{3/4} F\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{12 \left (a+b x^4\right )^{3/4}}-\frac{5 b \sqrt [4]{a+b x^4}}{12 x^2}-\frac{\left (a+b x^4\right )^{5/4}}{6 x^6} \]
Antiderivative was successfully verified.
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Rule 275
Rule 277
Rule 233
Rule 231
Rubi steps
\begin{align*} \int \frac{\left (a+b x^4\right )^{5/4}}{x^7} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{\left (a+b x^2\right )^{5/4}}{x^4} \, dx,x,x^2\right )\\ &=-\frac{\left (a+b x^4\right )^{5/4}}{6 x^6}+\frac{1}{12} (5 b) \operatorname{Subst}\left (\int \frac{\sqrt [4]{a+b x^2}}{x^2} \, dx,x,x^2\right )\\ &=-\frac{5 b \sqrt [4]{a+b x^4}}{12 x^2}-\frac{\left (a+b x^4\right )^{5/4}}{6 x^6}+\frac{1}{24} \left (5 b^2\right ) \operatorname{Subst}\left (\int \frac{1}{\left (a+b x^2\right )^{3/4}} \, dx,x,x^2\right )\\ &=-\frac{5 b \sqrt [4]{a+b x^4}}{12 x^2}-\frac{\left (a+b x^4\right )^{5/4}}{6 x^6}+\frac{\left (5 b^2 \left (1+\frac{b x^4}{a}\right )^{3/4}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (1+\frac{b x^2}{a}\right )^{3/4}} \, dx,x,x^2\right )}{24 \left (a+b x^4\right )^{3/4}}\\ &=-\frac{5 b \sqrt [4]{a+b x^4}}{12 x^2}-\frac{\left (a+b x^4\right )^{5/4}}{6 x^6}+\frac{5 \sqrt{a} b^{3/2} \left (1+\frac{b x^4}{a}\right )^{3/4} F\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{12 \left (a+b x^4\right )^{3/4}}\\ \end{align*}
Mathematica [C] time = 0.0121595, size = 52, normalized size = 0.53 \[ -\frac{a \sqrt [4]{a+b x^4} \, _2F_1\left (-\frac{3}{2},-\frac{5}{4};-\frac{1}{2};-\frac{b x^4}{a}\right )}{6 x^6 \sqrt [4]{\frac{b x^4}{a}+1}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.03, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{7}} \left ( b{x}^{4}+a \right ) ^{{\frac{5}{4}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x^{4} + a\right )}^{\frac{5}{4}}}{x^{7}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (b x^{4} + a\right )}^{\frac{5}{4}}}{x^{7}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 2.5999, size = 34, normalized size = 0.35 \begin{align*} - \frac{a^{\frac{5}{4}}{{}_{2}F_{1}\left (\begin{matrix} - \frac{3}{2}, - \frac{5}{4} \\ - \frac{1}{2} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{6 x^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x^{4} + a\right )}^{\frac{5}{4}}}{x^{7}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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