Optimal. Leaf size=158 \[ \frac {3 b^{5/4} \sqrt {1-\frac {b x^4}{a}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{5 a^{5/4} \sqrt {a-b x^4}}-\frac {3 b^{5/4} \sqrt {1-\frac {b x^4}{a}} E\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{5 a^{5/4} \sqrt {a-b x^4}}-\frac {3 b \sqrt {a-b x^4}}{5 a^2 x}-\frac {\sqrt {a-b x^4}}{5 a x^5} \]
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Rubi [A] time = 0.09, antiderivative size = 158, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {325, 307, 224, 221, 1200, 1199, 424} \[ \frac {3 b^{5/4} \sqrt {1-\frac {b x^4}{a}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{5 a^{5/4} \sqrt {a-b x^4}}-\frac {3 b^{5/4} \sqrt {1-\frac {b x^4}{a}} E\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{5 a^{5/4} \sqrt {a-b x^4}}-\frac {3 b \sqrt {a-b x^4}}{5 a^2 x}-\frac {\sqrt {a-b x^4}}{5 a x^5} \]
Antiderivative was successfully verified.
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Rule 221
Rule 224
Rule 307
Rule 325
Rule 424
Rule 1199
Rule 1200
Rubi steps
\begin {align*} \int \frac {1}{x^6 \sqrt {a-b x^4}} \, dx &=-\frac {\sqrt {a-b x^4}}{5 a x^5}+\frac {(3 b) \int \frac {1}{x^2 \sqrt {a-b x^4}} \, dx}{5 a}\\ &=-\frac {\sqrt {a-b x^4}}{5 a x^5}-\frac {3 b \sqrt {a-b x^4}}{5 a^2 x}-\frac {\left (3 b^2\right ) \int \frac {x^2}{\sqrt {a-b x^4}} \, dx}{5 a^2}\\ &=-\frac {\sqrt {a-b x^4}}{5 a x^5}-\frac {3 b \sqrt {a-b x^4}}{5 a^2 x}+\frac {\left (3 b^{3/2}\right ) \int \frac {1}{\sqrt {a-b x^4}} \, dx}{5 a^{3/2}}-\frac {\left (3 b^{3/2}\right ) \int \frac {1+\frac {\sqrt {b} x^2}{\sqrt {a}}}{\sqrt {a-b x^4}} \, dx}{5 a^{3/2}}\\ &=-\frac {\sqrt {a-b x^4}}{5 a x^5}-\frac {3 b \sqrt {a-b x^4}}{5 a^2 x}+\frac {\left (3 b^{3/2} \sqrt {1-\frac {b x^4}{a}}\right ) \int \frac {1}{\sqrt {1-\frac {b x^4}{a}}} \, dx}{5 a^{3/2} \sqrt {a-b x^4}}-\frac {\left (3 b^{3/2} \sqrt {1-\frac {b x^4}{a}}\right ) \int \frac {1+\frac {\sqrt {b} x^2}{\sqrt {a}}}{\sqrt {1-\frac {b x^4}{a}}} \, dx}{5 a^{3/2} \sqrt {a-b x^4}}\\ &=-\frac {\sqrt {a-b x^4}}{5 a x^5}-\frac {3 b \sqrt {a-b x^4}}{5 a^2 x}+\frac {3 b^{5/4} \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 )}{5 a^{5/4} \sqrt {a-b x^4}}-\frac {\left (3 b^{3/2} \sqrt {1-\frac {b x^4}{a}}\right ) \int \frac {\sqrt {1+\frac {\sqrt {b} x^2}{\sqrt {a}}}}{\sqrt {1-\frac {\sqrt {b} x^2}{\sqrt {a}}}} \, dx}{5 a^{3/2} \sqrt {a-b x^4}}\\ &=-\frac {\sqrt {a-b x^4}}{5 a x^5}-\frac {3 b \sqrt {a-b x^4}}{5 a^2 x}-\frac {3 b^{5/4} \sqrt {1-\frac {b x^4}{a}} E\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{5 a^{5/4} \sqrt {a-b x^4}}+\frac {3 b^{5/4} \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 )}{5 a^{5/4} \sqrt {a-b x^4}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 52, normalized size = 0.33 \[ -\frac {\sqrt {1-\frac {b x^4}{a}} \, _2F_1\left (-\frac {5}{4},\frac {1}{2};-\frac {1}{4};\frac {b x^4}{a}\right )}{5 x^5 \sqrt {a-b x^4}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.86, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-b x^{4} + a}}{b x^{10} - a x^{6}}, x\right ) \]
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 \[ \int \frac {1}{\sqrt {-b x^{4} + a} x^{6}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 126, normalized size = 0.80 \[ \frac {3 \sqrt {-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}+1}\, \sqrt {\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}+1}\, \left (-\EllipticE \left (\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, x , i\right )+\EllipticF \left (\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, x , i\right )\right ) b^{\frac {3}{2}}}{5 \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}\, a^{\frac {3}{2}}}-\frac {3 \sqrt {-b \,x^{4}+a}\, b}{5 a^{2} x}-\frac {\sqrt {-b \,x^{4}+a}}{5 a \,x^{5}} \]
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 \[ \int \frac {1}{\sqrt {-b x^{4} + a} x^{6}}\,{d x} \]
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 \[ \int \frac {1}{x^6\,\sqrt {a-b\,x^4}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.48, size = 39, normalized size = 0.25 \[ - \frac {i \Gamma \left (- \frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {7}{4} \\ \frac {11}{4} \end {matrix}\middle | {\frac {a}{b x^{4}}} \right )}}{4 \sqrt {b} x^{7} \Gamma \left (- \frac {3}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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