Optimal. Leaf size=124 \[ -\frac{b^2 x}{2 a^2 \sqrt [4]{a+b x^2}}+\frac{b^{3/2} \sqrt [4]{\frac{b x^2}{a}+1} E\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{2 a^{3/2} \sqrt [4]{a+b x^2}}+\frac{b \left (a+b x^2\right )^{3/4}}{2 a^2 x}-\frac{\left (a+b x^2\right )^{3/4}}{3 a x^3} \]
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Rubi [A] time = 0.0361695, antiderivative size = 124, 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 = {325, 229, 227, 196} \[ -\frac{b^2 x}{2 a^2 \sqrt [4]{a+b x^2}}+\frac{b^{3/2} \sqrt [4]{\frac{b x^2}{a}+1} E\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{2 a^{3/2} \sqrt [4]{a+b x^2}}+\frac{b \left (a+b x^2\right )^{3/4}}{2 a^2 x}-\frac{\left (a+b x^2\right )^{3/4}}{3 a x^3} \]
Antiderivative was successfully verified.
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Rule 325
Rule 229
Rule 227
Rule 196
Rubi steps
\begin{align*} \int \frac{1}{x^4 \sqrt [4]{a+b x^2}} \, dx &=-\frac{\left (a+b x^2\right )^{3/4}}{3 a x^3}-\frac{b \int \frac{1}{x^2 \sqrt [4]{a+b x^2}} \, dx}{2 a}\\ &=-\frac{\left (a+b x^2\right )^{3/4}}{3 a x^3}+\frac{b \left (a+b x^2\right )^{3/4}}{2 a^2 x}-\frac{b^2 \int \frac{1}{\sqrt [4]{a+b x^2}} \, dx}{4 a^2}\\ &=-\frac{\left (a+b x^2\right )^{3/4}}{3 a x^3}+\frac{b \left (a+b x^2\right )^{3/4}}{2 a^2 x}-\frac{\left (b^2 \sqrt [4]{1+\frac{b x^2}{a}}\right ) \int \frac{1}{\sqrt [4]{1+\frac{b x^2}{a}}} \, dx}{4 a^2 \sqrt [4]{a+b x^2}}\\ &=-\frac{b^2 x}{2 a^2 \sqrt [4]{a+b x^2}}-\frac{\left (a+b x^2\right )^{3/4}}{3 a x^3}+\frac{b \left (a+b x^2\right )^{3/4}}{2 a^2 x}+\frac{\left (b^2 \sqrt [4]{1+\frac{b x^2}{a}}\right ) \int \frac{1}{\left (1+\frac{b x^2}{a}\right )^{5/4}} \, dx}{4 a^2 \sqrt [4]{a+b x^2}}\\ &=-\frac{b^2 x}{2 a^2 \sqrt [4]{a+b x^2}}-\frac{\left (a+b x^2\right )^{3/4}}{3 a x^3}+\frac{b \left (a+b x^2\right )^{3/4}}{2 a^2 x}+\frac{b^{3/2} \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 )}{2 a^{3/2} \sqrt [4]{a+b x^2}}\\ \end{align*}
Mathematica [C] time = 0.0085505, size = 51, normalized size = 0.41 \[ -\frac{\sqrt [4]{\frac{b x^2}{a}+1} \, _2F_1\left (-\frac{3}{2},\frac{1}{4};-\frac{1}{2};-\frac{b x^2}{a}\right )}{3 x^3 \sqrt [4]{a+b x^2}} \]
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}}{\frac{1}{\sqrt [4]{b{x}^{2}+a}}}}\, 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{1}{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{3}{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 = 0.921953, size = 32, normalized size = 0.26 \begin{align*} - \frac{{{}_{2}F_{1}\left (\begin{matrix} - \frac{3}{2}, \frac{1}{4} \\ - \frac{1}{2} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{3 \sqrt [4]{a} 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{1}{4}} x^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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