Optimal. Leaf size=102 \[ \frac {5 b^{3/2} \left (\frac {b x^2}{a}+1\right )^{3/4} \operatorname {EllipticF}\left (\frac {1}{2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),2\right )}{6 a^{3/2} \left (a+b x^2\right )^{3/4}}+\frac {5 b \sqrt [4]{a+b x^2}}{6 a^2 x}-\frac {\sqrt [4]{a+b x^2}}{3 a x^3} \]
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Rubi [A] time = 0.03, antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, 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 231
Rule 233
Rule 325
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*}
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Mathematica [C] time = 0.01, size = 51, normalized size = 0.50 \[ -\frac {\left (\frac {b x^2}{a}+1\right )^{3/4} \, _2F_1\left (-\frac {3}{2},\frac {3}{4};-\frac {1}{2};-\frac {b x^2}{a}\right )}{3 x^3 \left (a+b x^2\right )^{3/4}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.94, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b x^{2} + a\right )}^{\frac {1}{4}}}{b x^{6} + a x^{4}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b x^{2} + a\right )}^{\frac {3}{4}} x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.29, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (b \,x^{2}+a \right )^{\frac {3}{4}} x^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b x^{2} + a\right )}^{\frac {3}{4}} x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{x^4\,{\left (b\,x^2+a\right )}^{3/4}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 1.09, size = 32, normalized size = 0.31 \[ - \frac {{{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{2}, \frac {3}{4} \\ - \frac {1}{2} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{3 a^{\frac {3}{4}} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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