Optimal. Leaf size=85 \[ -\frac {\left (a-b x^4\right )^{3/4}}{2 a x^2}-\frac {\sqrt {b} \sqrt [4]{1-\frac {b x^4}{a}} E\left (\left .\frac {1}{2} \sin ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{2 \sqrt {a} \sqrt [4]{a-b x^4}} \]
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Rubi [A] time = 0.05, antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {275, 325, 229, 228} \[ -\frac {\left (a-b x^4\right )^{3/4}}{2 a x^2}-\frac {\sqrt {b} \sqrt [4]{1-\frac {b x^4}{a}} E\left (\left .\frac {1}{2} \sin ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{2 \sqrt {a} \sqrt [4]{a-b x^4}} \]
Antiderivative was successfully verified.
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Rule 228
Rule 229
Rule 275
Rule 325
Rubi steps
\begin {align*} \int \frac {1}{x^3 \sqrt [4]{a-b x^4}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{x^2 \sqrt [4]{a-b x^2}} \, dx,x,x^2\right )\\ &=-\frac {\left (a-b x^4\right )^{3/4}}{2 a x^2}-\frac {b \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{a-b x^2}} \, dx,x,x^2\right )}{4 a}\\ &=-\frac {\left (a-b x^4\right )^{3/4}}{2 a x^2}-\frac {\left (b \sqrt [4]{1-\frac {b x^4}{a}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{1-\frac {b x^2}{a}}} \, dx,x,x^2\right )}{4 a \sqrt [4]{a-b x^4}}\\ &=-\frac {\left (a-b x^4\right )^{3/4}}{2 a x^2}-\frac {\sqrt {b} \sqrt [4]{1-\frac {b x^4}{a}} E\left (\left .\frac {1}{2} \sin ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{2 \sqrt {a} \sqrt [4]{a-b x^4}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 52, normalized size = 0.61 \[ -\frac {\sqrt [4]{1-\frac {b x^4}{a}} \, _2F_1\left (-\frac {1}{2},\frac {1}{4};\frac {1}{2};\frac {b x^4}{a}\right )}{2 x^2 \sqrt [4]{a-b x^4}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.59, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (-b x^{4} + a\right )}^{\frac {3}{4}}}{b x^{7} - a x^{3}}, x\right ) \]
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 \[ \int \frac {1}{{\left (-b x^{4} + a\right )}^{\frac {1}{4}} x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.15, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (-b \,x^{4}+a \right )^{\frac {1}{4}} x^{3}}\, dx \]
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 \[ \int \frac {1}{{\left (-b x^{4} + a\right )}^{\frac {1}{4}} x^{3}}\,{d x} \]
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 \[ \int \frac {1}{x^3\,{\left (a-b\,x^4\right )}^{1/4}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 1.41, size = 32, normalized size = 0.38 \[ - \frac {{{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {1}{4} \\ \frac {1}{2} \end {matrix}\middle | {\frac {b x^{4} e^{2 i \pi }}{a}} \right )}}{2 \sqrt [4]{a} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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