Optimal. Leaf size=153 \[ \frac {15 b^{7/4} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{28 a^{13/4} \sqrt {a+b x^4}}+\frac {15 b \sqrt {a+b x^4}}{14 a^3 x^3}-\frac {9 \sqrt {a+b x^4}}{14 a^2 x^7}+\frac {1}{2 a x^7 \sqrt {a+b x^4}} \]
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Rubi [A] time = 0.05, antiderivative size = 153, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {290, 325, 220} \[ \frac {15 b^{7/4} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{28 a^{13/4} \sqrt {a+b x^4}}+\frac {15 b \sqrt {a+b x^4}}{14 a^3 x^3}-\frac {9 \sqrt {a+b x^4}}{14 a^2 x^7}+\frac {1}{2 a x^7 \sqrt {a+b x^4}} \]
Antiderivative was successfully verified.
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Rule 220
Rule 290
Rule 325
Rubi steps
\begin {align*} \int \frac {1}{x^8 \left (a+b x^4\right )^{3/2}} \, dx &=\frac {1}{2 a x^7 \sqrt {a+b x^4}}+\frac {9 \int \frac {1}{x^8 \sqrt {a+b x^4}} \, dx}{2 a}\\ &=\frac {1}{2 a x^7 \sqrt {a+b x^4}}-\frac {9 \sqrt {a+b x^4}}{14 a^2 x^7}-\frac {(45 b) \int \frac {1}{x^4 \sqrt {a+b x^4}} \, dx}{14 a^2}\\ &=\frac {1}{2 a x^7 \sqrt {a+b x^4}}-\frac {9 \sqrt {a+b x^4}}{14 a^2 x^7}+\frac {15 b \sqrt {a+b x^4}}{14 a^3 x^3}+\frac {\left (15 b^2\right ) \int \frac {1}{\sqrt {a+b x^4}} \, dx}{14 a^3}\\ &=\frac {1}{2 a x^7 \sqrt {a+b x^4}}-\frac {9 \sqrt {a+b x^4}}{14 a^2 x^7}+\frac {15 b \sqrt {a+b x^4}}{14 a^3 x^3}+\frac {15 b^{7/4} \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 )}{28 a^{13/4} \sqrt {a+b x^4}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 54, normalized size = 0.35 \[ -\frac {\sqrt {\frac {b x^4}{a}+1} \, _2F_1\left (-\frac {7}{4},\frac {3}{2};-\frac {3}{4};-\frac {b x^4}{a}\right )}{7 a x^7 \sqrt {a+b x^4}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.90, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {b x^{4} + a}}{b^{2} x^{16} + 2 \, a b x^{12} + a^{2} x^{8}}, x\right ) \]
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 \[ \int \frac {1}{{\left (b x^{4} + a\right )}^{\frac {3}{2}} x^{8}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.02, size = 135, normalized size = 0.88 \[ \frac {b^{2} x}{2 \sqrt {\left (x^{4}+\frac {a}{b}\right ) b}\, a^{3}}+\frac {15 \sqrt {-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}+1}\, \sqrt {\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}+1}\, b^{2} \EllipticF \left (\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, x , i\right )}{14 \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}\, a^{3}}+\frac {4 \sqrt {b \,x^{4}+a}\, b}{7 a^{3} x^{3}}-\frac {\sqrt {b \,x^{4}+a}}{7 a^{2} x^{7}} \]
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 \[ \int \frac {1}{{\left (b x^{4} + a\right )}^{\frac {3}{2}} x^{8}}\,{d x} \]
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 \[ \int \frac {1}{x^8\,{\left (b\,x^4+a\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 1.56, size = 44, normalized size = 0.29 \[ \frac {\Gamma \left (- \frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {7}{4}, \frac {3}{2} \\ - \frac {3}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 a^{\frac {3}{2}} x^{7} \Gamma \left (- \frac {3}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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