Optimal. Leaf size=106 \[ -\frac {\sqrt [4]{a-b x^2}}{3 a x^3}-\frac {5 b \sqrt [4]{a-b x^2}}{6 a^2 x}+\frac {5 b^{3/2} \left (1-\frac {b x^2}{a}\right )^{3/4} F\left (\left .\frac {1}{2} \sin ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{6 a^{3/2} \left (a-b x^2\right )^{3/4}} \]
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Rubi [A]
time = 0.03, antiderivative size = 106, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {331, 239, 238}
\begin {gather*} \frac {5 b^{3/2} \left (1-\frac {b x^2}{a}\right )^{3/4} F\left (\left .\frac {1}{2} \text {ArcSin}\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} \end {gather*}
Antiderivative was successfully verified.
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Rule 238
Rule 239
Rule 331
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} \sin ^{-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] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 10.02, size = 52, normalized size = 0.49 \begin {gather*} -\frac {\left (1-\frac {b x^2}{a}\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}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.00, size = 0, normalized size = 0.00 \[\int \frac {1}{x^{4} \left (-b \,x^{2}+a \right )^{\frac {3}{4}}}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 0.56, size = 34, normalized size = 0.32 \begin {gather*} - \frac {{{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{2}, \frac {3}{4} \\ - \frac {1}{2} \end {matrix}\middle | {\frac {b x^{2} e^{2 i \pi }}{a}} \right )}}{3 a^{\frac {3}{4}} x^{3}} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {1}{x^4\,{\left (a-b\,x^2\right )}^{3/4}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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