Optimal. Leaf size=102 \[ \frac {5 b^{7/4} \sqrt {1-\frac {b x^4}{a}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{21 a^{7/4} \sqrt {a-b x^4}}-\frac {5 b \sqrt {a-b x^4}}{21 a^2 x^3}-\frac {\sqrt {a-b x^4}}{7 a x^7} \]
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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 = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {325, 224, 221} \[ \frac {5 b^{7/4} \sqrt {1-\frac {b x^4}{a}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{21 a^{7/4} \sqrt {a-b x^4}}-\frac {5 b \sqrt {a-b x^4}}{21 a^2 x^3}-\frac {\sqrt {a-b x^4}}{7 a x^7} \]
Antiderivative was successfully verified.
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Rule 221
Rule 224
Rule 325
Rubi steps
\begin {align*} \int \frac {1}{x^8 \sqrt {a-b x^4}} \, dx &=-\frac {\sqrt {a-b x^4}}{7 a x^7}+\frac {(5 b) \int \frac {1}{x^4 \sqrt {a-b x^4}} \, dx}{7 a}\\ &=-\frac {\sqrt {a-b x^4}}{7 a x^7}-\frac {5 b \sqrt {a-b x^4}}{21 a^2 x^3}+\frac {\left (5 b^2\right ) \int \frac {1}{\sqrt {a-b x^4}} \, dx}{21 a^2}\\ &=-\frac {\sqrt {a-b x^4}}{7 a x^7}-\frac {5 b \sqrt {a-b x^4}}{21 a^2 x^3}+\frac {\left (5 b^2 \sqrt {1-\frac {b x^4}{a}}\right ) \int \frac {1}{\sqrt {1-\frac {b x^4}{a}}} \, dx}{21 a^2 \sqrt {a-b x^4}}\\ &=-\frac {\sqrt {a-b x^4}}{7 a x^7}-\frac {5 b \sqrt {a-b x^4}}{21 a^2 x^3}+\frac {5 b^{7/4} \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 )}{21 a^{7/4} \sqrt {a-b x^4}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 52, normalized size = 0.51 \[ -\frac {\sqrt {1-\frac {b x^4}{a}} \, _2F_1\left (-\frac {7}{4},\frac {1}{2};-\frac {3}{4};\frac {b x^4}{a}\right )}{7 x^7 \sqrt {a-b x^4}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.94, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-b x^{4} + a}}{b x^{12} - a x^{8}}, x\right ) \]
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 \[ \int \frac {1}{\sqrt {-b x^{4} + a} x^{8}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 109, normalized size = 1.07 \[ \frac {5 \sqrt {-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}+1}\, \sqrt {\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}+1}\, b^{2} \EllipticF \left (\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, x , i\right )}{21 \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}\, a^{2}}-\frac {5 \sqrt {-b \,x^{4}+a}\, b}{21 a^{2} x^{3}}-\frac {\sqrt {-b \,x^{4}+a}}{7 a \,x^{7}} \]
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 \[ \int \frac {1}{\sqrt {-b x^{4} + a} x^{8}}\,{d x} \]
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 \[ \int \frac {1}{x^8\,\sqrt {a-b\,x^4}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.40, size = 46, normalized size = 0.45 \[ \frac {\Gamma \left (- \frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {7}{4}, \frac {1}{2} \\ - \frac {3}{4} \end {matrix}\middle | {\frac {b x^{4} e^{2 i \pi }}{a}} \right )}}{4 \sqrt {a} x^{7} \Gamma \left (- \frac {3}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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