Optimal. Leaf size=153 \[ \frac{15 b^{7/4} \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac{1}{2}\right )}{28 a^{13/4} \sqrt{a+b x^4}}+\frac{15 b \sqrt{a+b x^4}}{14 a^3 x^3}-\frac{9 \sqrt{a+b x^4}}{14 a^2 x^7}+\frac{1}{2 a x^7 \sqrt{a+b x^4}} \]
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Rubi [A] time = 0.0503625, antiderivative size = 153, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, 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 290
Rule 325
Rule 220
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*}
Mathematica [C] time = 0.0087661, 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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Maple [C] time = 0.02, size = 135, normalized size = 0.9 \begin{align*} -{\frac{1}{7\,{a}^{2}{x}^{7}}\sqrt{b{x}^{4}+a}}+{\frac{4\,b}{7\,{a}^{3}{x}^{3}}\sqrt{b{x}^{4}+a}}+{\frac{{b}^{2}x}{2\,{a}^{3}}{\frac{1}{\sqrt{ \left ({x}^{4}+{\frac{a}{b}} \right ) b}}}}+{\frac{15\,{b}^{2}}{14\,{a}^{3}}\sqrt{1-{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}{\it EllipticF} \left ( x\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}},i \right ){\frac{1}{\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{b{x}^{4}+a}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b x^{4} + a\right )}^{\frac{3}{2}} x^{8}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{b x^{4} + a}}{b^{2} x^{16} + 2 \, a b x^{12} + a^{2} x^{8}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 2.26804, size = 44, normalized size = 0.29 \begin{align*} \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 )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b x^{4} + a\right )}^{\frac{3}{2}} x^{8}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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