Optimal. Leaf size=261 \[ \frac{7 a^{9/4} \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac{1}{2}\right )}{30 b^{11/4} \sqrt{a+b x^4}}+\frac{7 a^2 x \sqrt{a+b x^4}}{15 b^{5/2} \left (\sqrt{a}+\sqrt{b} x^2\right )}-\frac{7 a^{9/4} \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{15 b^{11/4} \sqrt{a+b x^4}}-\frac{7 a x^3 \sqrt{a+b x^4}}{45 b^2}+\frac{x^7 \sqrt{a+b x^4}}{9 b} \]
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Rubi [A] time = 0.0918937, antiderivative size = 261, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {321, 305, 220, 1196} \[ \frac{7 a^2 x \sqrt{a+b x^4}}{15 b^{5/2} \left (\sqrt{a}+\sqrt{b} x^2\right )}+\frac{7 a^{9/4} \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{30 b^{11/4} \sqrt{a+b x^4}}-\frac{7 a^{9/4} \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{15 b^{11/4} \sqrt{a+b x^4}}-\frac{7 a x^3 \sqrt{a+b x^4}}{45 b^2}+\frac{x^7 \sqrt{a+b x^4}}{9 b} \]
Antiderivative was successfully verified.
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Rule 321
Rule 305
Rule 220
Rule 1196
Rubi steps
\begin{align*} \int \frac{x^{10}}{\sqrt{a+b x^4}} \, dx &=\frac{x^7 \sqrt{a+b x^4}}{9 b}-\frac{(7 a) \int \frac{x^6}{\sqrt{a+b x^4}} \, dx}{9 b}\\ &=-\frac{7 a x^3 \sqrt{a+b x^4}}{45 b^2}+\frac{x^7 \sqrt{a+b x^4}}{9 b}+\frac{\left (7 a^2\right ) \int \frac{x^2}{\sqrt{a+b x^4}} \, dx}{15 b^2}\\ &=-\frac{7 a x^3 \sqrt{a+b x^4}}{45 b^2}+\frac{x^7 \sqrt{a+b x^4}}{9 b}+\frac{\left (7 a^{5/2}\right ) \int \frac{1}{\sqrt{a+b x^4}} \, dx}{15 b^{5/2}}-\frac{\left (7 a^{5/2}\right ) \int \frac{1-\frac{\sqrt{b} x^2}{\sqrt{a}}}{\sqrt{a+b x^4}} \, dx}{15 b^{5/2}}\\ &=-\frac{7 a x^3 \sqrt{a+b x^4}}{45 b^2}+\frac{x^7 \sqrt{a+b x^4}}{9 b}+\frac{7 a^2 x \sqrt{a+b x^4}}{15 b^{5/2} \left (\sqrt{a}+\sqrt{b} x^2\right )}-\frac{7 a^{9/4} \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 )}{15 b^{11/4} \sqrt{a+b x^4}}+\frac{7 a^{9/4} \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 )}{30 b^{11/4} \sqrt{a+b x^4}}\\ \end{align*}
Mathematica [C] time = 0.0252055, size = 80, normalized size = 0.31 \[ \frac{x^3 \left (7 a^2 \sqrt{\frac{b x^4}{a}+1} \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{7}{4};-\frac{b x^4}{a}\right )-7 a^2-2 a b x^4+5 b^2 x^8\right )}{45 b^2 \sqrt{a+b x^4}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.007, size = 133, normalized size = 0.5 \begin{align*}{\frac{{x}^{7}}{9\,b}\sqrt{b{x}^{4}+a}}-{\frac{7\,a{x}^{3}}{45\,{b}^{2}}\sqrt{b{x}^{4}+a}}+{{\frac{7\,i}{15}}{a}^{{\frac{5}{2}}}\sqrt{1-{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}} \left ({\it EllipticF} \left ( x\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}},i \right ) -{\it EllipticE} \left ( x\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}},i \right ) \right ){b}^{-{\frac{5}{2}}}{\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{x^{10}}{\sqrt{b x^{4} + a}}\,{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{x^{10}}{\sqrt{b x^{4} + a}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 1.66232, size = 37, normalized size = 0.14 \begin{align*} \frac{x^{11} \Gamma \left (\frac{11}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{11}{4} \\ \frac{15}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 \sqrt{a} \Gamma \left (\frac{15}{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{x^{10}}{\sqrt{b x^{4} + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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