Optimal. Leaf size=102 \[ \frac{5 b^{3/2} \left (\frac{b x^2}{a}+1\right )^{3/4} \text{EllipticF}\left (\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ),2\right )}{6 a^{3/2} \left (a+b x^2\right )^{3/4}}+\frac{5 b \sqrt [4]{a+b x^2}}{6 a^2 x}-\frac{\sqrt [4]{a+b x^2}}{3 a x^3} \]
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Rubi [A] time = 0.0314378, antiderivative size = 102, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {325, 233, 231} \[ \frac{5 b^{3/2} \left (\frac{b x^2}{a}+1\right )^{3/4} F\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{6 a^{3/2} \left (a+b x^2\right )^{3/4}}+\frac{5 b \sqrt [4]{a+b x^2}}{6 a^2 x}-\frac{\sqrt [4]{a+b x^2}}{3 a x^3} \]
Antiderivative was successfully verified.
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Rule 325
Rule 233
Rule 231
Rubi steps
\begin{align*} \int \frac{1}{x^4 \left (a+b x^2\right )^{3/4}} \, dx &=-\frac{\sqrt [4]{a+b x^2}}{3 a x^3}-\frac{(5 b) \int \frac{1}{x^2 \left (a+b x^2\right )^{3/4}} \, dx}{6 a}\\ &=-\frac{\sqrt [4]{a+b x^2}}{3 a x^3}+\frac{5 b \sqrt [4]{a+b x^2}}{6 a^2 x}+\frac{\left (5 b^2\right ) \int \frac{1}{\left (a+b x^2\right )^{3/4}} \, dx}{12 a^2}\\ &=-\frac{\sqrt [4]{a+b x^2}}{3 a x^3}+\frac{5 b \sqrt [4]{a+b x^2}}{6 a^2 x}+\frac{\left (5 b^2 \left (1+\frac{b x^2}{a}\right )^{3/4}\right ) \int \frac{1}{\left (1+\frac{b x^2}{a}\right )^{3/4}} \, dx}{12 a^2 \left (a+b x^2\right )^{3/4}}\\ &=-\frac{\sqrt [4]{a+b x^2}}{3 a x^3}+\frac{5 b \sqrt [4]{a+b x^2}}{6 a^2 x}+\frac{5 b^{3/2} \left (1+\frac{b x^2}{a}\right )^{3/4} F\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{6 a^{3/2} \left (a+b x^2\right )^{3/4}}\\ \end{align*}
Mathematica [C] time = 0.0091158, size = 51, normalized size = 0.5 \[ -\frac{\left (\frac{b x^2}{a}+1\right )^{3/4} \, _2F_1\left (-\frac{3}{2},\frac{3}{4};-\frac{1}{2};-\frac{b x^2}{a}\right )}{3 x^3 \left (a+b x^2\right )^{3/4}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.026, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{4}} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{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{1}{{\left (b x^{2} + a\right )}^{\frac{3}{4}} x^{4}}\,{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^{2} + a\right )}^{\frac{1}{4}}}{b x^{6} + a x^{4}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 1.09522, size = 32, normalized size = 0.31 \begin{align*} - \frac{{{}_{2}F_{1}\left (\begin{matrix} - \frac{3}{2}, \frac{3}{4} \\ - \frac{1}{2} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{3 a^{\frac{3}{4}} x^{3}} \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{1}{{\left (b x^{2} + a\right )}^{\frac{3}{4}} x^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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