Optimal. Leaf size=282 \[ -\frac {21 b^{5/4} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{20 a^{11/4} \sqrt {a+b x^4}}+\frac {21 b^{5/4} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{10 a^{11/4} \sqrt {a+b x^4}}-\frac {21 b^{3/2} x \sqrt {a+b x^4}}{10 a^3 \left (\sqrt {a}+\sqrt {b} x^2\right )}+\frac {21 b \sqrt {a+b x^4}}{10 a^3 x}-\frac {7 \sqrt {a+b x^4}}{10 a^2 x^5}+\frac {1}{2 a x^5 \sqrt {a+b x^4}} \]
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Rubi [A] time = 0.12, antiderivative size = 282, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {290, 325, 305, 220, 1196} \[ -\frac {21 b^{3/2} x \sqrt {a+b x^4}}{10 a^3 \left (\sqrt {a}+\sqrt {b} x^2\right )}-\frac {21 b^{5/4} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{20 a^{11/4} \sqrt {a+b x^4}}+\frac {21 b^{5/4} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{10 a^{11/4} \sqrt {a+b x^4}}+\frac {21 b \sqrt {a+b x^4}}{10 a^3 x}-\frac {7 \sqrt {a+b x^4}}{10 a^2 x^5}+\frac {1}{2 a x^5 \sqrt {a+b x^4}} \]
Antiderivative was successfully verified.
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Rule 220
Rule 290
Rule 305
Rule 325
Rule 1196
Rubi steps
\begin {align*} \int \frac {1}{x^6 \left (a+b x^4\right )^{3/2}} \, dx &=\frac {1}{2 a x^5 \sqrt {a+b x^4}}+\frac {7 \int \frac {1}{x^6 \sqrt {a+b x^4}} \, dx}{2 a}\\ &=\frac {1}{2 a x^5 \sqrt {a+b x^4}}-\frac {7 \sqrt {a+b x^4}}{10 a^2 x^5}-\frac {(21 b) \int \frac {1}{x^2 \sqrt {a+b x^4}} \, dx}{10 a^2}\\ &=\frac {1}{2 a x^5 \sqrt {a+b x^4}}-\frac {7 \sqrt {a+b x^4}}{10 a^2 x^5}+\frac {21 b \sqrt {a+b x^4}}{10 a^3 x}-\frac {\left (21 b^2\right ) \int \frac {x^2}{\sqrt {a+b x^4}} \, dx}{10 a^3}\\ &=\frac {1}{2 a x^5 \sqrt {a+b x^4}}-\frac {7 \sqrt {a+b x^4}}{10 a^2 x^5}+\frac {21 b \sqrt {a+b x^4}}{10 a^3 x}-\frac {\left (21 b^{3/2}\right ) \int \frac {1}{\sqrt {a+b x^4}} \, dx}{10 a^{5/2}}+\frac {\left (21 b^{3/2}\right ) \int \frac {1-\frac {\sqrt {b} x^2}{\sqrt {a}}}{\sqrt {a+b x^4}} \, dx}{10 a^{5/2}}\\ &=\frac {1}{2 a x^5 \sqrt {a+b x^4}}-\frac {7 \sqrt {a+b x^4}}{10 a^2 x^5}+\frac {21 b \sqrt {a+b x^4}}{10 a^3 x}-\frac {21 b^{3/2} x \sqrt {a+b x^4}}{10 a^3 \left (\sqrt {a}+\sqrt {b} x^2\right )}+\frac {21 b^{5/4} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{10 a^{11/4} \sqrt {a+b x^4}}-\frac {21 b^{5/4} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{20 a^{11/4} \sqrt {a+b x^4}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 54, normalized size = 0.19 \[ -\frac {\sqrt {\frac {b x^4}{a}+1} \, _2F_1\left (-\frac {5}{4},\frac {3}{2};-\frac {1}{4};-\frac {b x^4}{a}\right )}{5 a x^5 \sqrt {a+b x^4}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.87, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {b x^{4} + a}}{b^{2} x^{14} + 2 \, a b x^{10} + a^{2} x^{6}}, x\right ) \]
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 \[ \int \frac {1}{{\left (b x^{4} + a\right )}^{\frac {3}{2}} x^{6}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.02, size = 157, normalized size = 0.56 \[ \frac {b^{2} x^{3}}{2 \sqrt {\left (x^{4}+\frac {a}{b}\right ) b}\, a^{3}}-\frac {21 i \sqrt {-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}+1}\, \sqrt {\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}+1}\, \left (-\EllipticE \left (\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, x , i\right )+\EllipticF \left (\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, x , i\right )\right ) b^{\frac {3}{2}}}{10 \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}\, a^{\frac {5}{2}}}+\frac {8 \sqrt {b \,x^{4}+a}\, b}{5 a^{3} x}-\frac {\sqrt {b \,x^{4}+a}}{5 a^{2} x^{5}} \]
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 \[ \int \frac {1}{{\left (b x^{4} + a\right )}^{\frac {3}{2}} x^{6}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {1}{x^6\,{\left (b\,x^4+a\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 1.80, size = 44, normalized size = 0.16 \[ \frac {\Gamma \left (- \frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {5}{4}, \frac {3}{2} \\ - \frac {1}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 a^{\frac {3}{2}} x^{5} \Gamma \left (- \frac {1}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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