Optimal. Leaf size=79 \[ \frac {b^{3/4} \sqrt {1-\frac {b x^4}{a}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{3 a^{3/4} \sqrt {a-b x^4}}-\frac {\sqrt {a-b x^4}}{3 a x^3} \]
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Rubi [A] time = 0.02, antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {325, 224, 221} \[ \frac {b^{3/4} \sqrt {1-\frac {b x^4}{a}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{3 a^{3/4} \sqrt {a-b x^4}}-\frac {\sqrt {a-b x^4}}{3 a x^3} \]
Antiderivative was successfully verified.
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Rule 221
Rule 224
Rule 325
Rubi steps
\begin {align*} \int \frac {1}{x^4 \sqrt {a-b x^4}} \, dx &=-\frac {\sqrt {a-b x^4}}{3 a x^3}+\frac {b \int \frac {1}{\sqrt {a-b x^4}} \, dx}{3 a}\\ &=-\frac {\sqrt {a-b x^4}}{3 a x^3}+\frac {\left (b \sqrt {1-\frac {b x^4}{a}}\right ) \int \frac {1}{\sqrt {1-\frac {b x^4}{a}}} \, dx}{3 a \sqrt {a-b x^4}}\\ &=-\frac {\sqrt {a-b x^4}}{3 a x^3}+\frac {b^{3/4} \sqrt {1-\frac {b x^4}{a}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{3 a^{3/4} \sqrt {a-b x^4}}\\ \end {align*}
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Mathematica [C] time = 0.03, size = 52, normalized size = 0.66 \[ -\frac {\sqrt {1-\frac {b x^4}{a}} \, _2F_1\left (-\frac {3}{4},\frac {1}{2};\frac {1}{4};\frac {b x^4}{a}\right )}{3 x^3 \sqrt {a-b x^4}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.66, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-b x^{4} + a}}{b x^{8} - a x^{4}}, x\right ) \]
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 \[ \int \frac {1}{\sqrt {-b x^{4} + a} x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 88, normalized size = 1.11 \[ \frac {\sqrt {-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}+1}\, \sqrt {\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}+1}\, b \EllipticF \left (\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, x , i\right )}{3 \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}\, a}-\frac {\sqrt {-b \,x^{4}+a}}{3 a \,x^{3}} \]
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 \[ \int \frac {1}{\sqrt {-b x^{4} + a} x^{4}}\,{d x} \]
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 \[ \int \frac {1}{x^4\,\sqrt {a-b\,x^4}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.98, size = 42, normalized size = 0.53 \[ \frac {\Gamma \left (- \frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, \frac {1}{2} \\ \frac {1}{4} \end {matrix}\middle | {\frac {b x^{4} e^{2 i \pi }}{a}} \right )}}{4 \sqrt {a} x^{3} \Gamma \left (\frac {1}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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