Optimal. Leaf size=88 \[ \frac{3 b x}{\sqrt [4]{a+b x^2}}-\frac{\left (a+b x^2\right )^{3/4}}{x}-\frac{3 \sqrt{a} \sqrt{b} \sqrt [4]{\frac{b x^2}{a}+1} E\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{\sqrt [4]{a+b x^2}} \]
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Rubi [A] time = 0.0238425, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {277, 229, 227, 196} \[ \frac{3 b x}{\sqrt [4]{a+b x^2}}-\frac{\left (a+b x^2\right )^{3/4}}{x}-\frac{3 \sqrt{a} \sqrt{b} \sqrt [4]{\frac{b x^2}{a}+1} E\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{\sqrt [4]{a+b x^2}} \]
Antiderivative was successfully verified.
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Rule 277
Rule 229
Rule 227
Rule 196
Rubi steps
\begin{align*} \int \frac{\left (a+b x^2\right )^{3/4}}{x^2} \, dx &=-\frac{\left (a+b x^2\right )^{3/4}}{x}+\frac{1}{2} (3 b) \int \frac{1}{\sqrt [4]{a+b x^2}} \, dx\\ &=-\frac{\left (a+b x^2\right )^{3/4}}{x}+\frac{\left (3 b \sqrt [4]{1+\frac{b x^2}{a}}\right ) \int \frac{1}{\sqrt [4]{1+\frac{b x^2}{a}}} \, dx}{2 \sqrt [4]{a+b x^2}}\\ &=\frac{3 b x}{\sqrt [4]{a+b x^2}}-\frac{\left (a+b x^2\right )^{3/4}}{x}-\frac{\left (3 b \sqrt [4]{1+\frac{b x^2}{a}}\right ) \int \frac{1}{\left (1+\frac{b x^2}{a}\right )^{5/4}} \, dx}{2 \sqrt [4]{a+b x^2}}\\ &=\frac{3 b x}{\sqrt [4]{a+b x^2}}-\frac{\left (a+b x^2\right )^{3/4}}{x}-\frac{3 \sqrt{a} \sqrt{b} \sqrt [4]{1+\frac{b x^2}{a}} E\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{\sqrt [4]{a+b x^2}}\\ \end{align*}
Mathematica [C] time = 0.0082458, size = 49, normalized size = 0.56 \[ -\frac{\left (a+b x^2\right )^{3/4} \, _2F_1\left (-\frac{3}{4},-\frac{1}{2};\frac{1}{2};-\frac{b x^2}{a}\right )}{x \left (\frac{b x^2}{a}+1\right )^{3/4}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.026, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{2}} \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{{\left (b x^{2} + a\right )}^{\frac{3}{4}}}{x^{2}}\,{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{3}{4}}}{x^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 1.02385, size = 29, normalized size = 0.33 \begin{align*} - \frac{a^{\frac{3}{4}}{{}_{2}F_{1}\left (\begin{matrix} - \frac{3}{4}, - \frac{1}{2} \\ \frac{1}{2} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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