Optimal. Leaf size=179 \[ -\frac{15 b^{11/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right ),\frac{1}{2}\right )}{77 c^{13/4} \sqrt{b x^2+c x^4}}+\frac{30 b^2 \sqrt{b x^2+c x^4}}{77 c^3 \sqrt{x}}-\frac{18 b x^{3/2} \sqrt{b x^2+c x^4}}{77 c^2}+\frac{2 x^{7/2} \sqrt{b x^2+c x^4}}{11 c} \]
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Rubi [A] time = 0.244054, antiderivative size = 179, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {2024, 2032, 329, 220} \[ \frac{30 b^2 \sqrt{b x^2+c x^4}}{77 c^3 \sqrt{x}}-\frac{15 b^{11/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{77 c^{13/4} \sqrt{b x^2+c x^4}}-\frac{18 b x^{3/2} \sqrt{b x^2+c x^4}}{77 c^2}+\frac{2 x^{7/2} \sqrt{b x^2+c x^4}}{11 c} \]
Antiderivative was successfully verified.
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Rule 2024
Rule 2032
Rule 329
Rule 220
Rubi steps
\begin{align*} \int \frac{x^{13/2}}{\sqrt{b x^2+c x^4}} \, dx &=\frac{2 x^{7/2} \sqrt{b x^2+c x^4}}{11 c}-\frac{(9 b) \int \frac{x^{9/2}}{\sqrt{b x^2+c x^4}} \, dx}{11 c}\\ &=-\frac{18 b x^{3/2} \sqrt{b x^2+c x^4}}{77 c^2}+\frac{2 x^{7/2} \sqrt{b x^2+c x^4}}{11 c}+\frac{\left (45 b^2\right ) \int \frac{x^{5/2}}{\sqrt{b x^2+c x^4}} \, dx}{77 c^2}\\ &=\frac{30 b^2 \sqrt{b x^2+c x^4}}{77 c^3 \sqrt{x}}-\frac{18 b x^{3/2} \sqrt{b x^2+c x^4}}{77 c^2}+\frac{2 x^{7/2} \sqrt{b x^2+c x^4}}{11 c}-\frac{\left (15 b^3\right ) \int \frac{\sqrt{x}}{\sqrt{b x^2+c x^4}} \, dx}{77 c^3}\\ &=\frac{30 b^2 \sqrt{b x^2+c x^4}}{77 c^3 \sqrt{x}}-\frac{18 b x^{3/2} \sqrt{b x^2+c x^4}}{77 c^2}+\frac{2 x^{7/2} \sqrt{b x^2+c x^4}}{11 c}-\frac{\left (15 b^3 x \sqrt{b+c x^2}\right ) \int \frac{1}{\sqrt{x} \sqrt{b+c x^2}} \, dx}{77 c^3 \sqrt{b x^2+c x^4}}\\ &=\frac{30 b^2 \sqrt{b x^2+c x^4}}{77 c^3 \sqrt{x}}-\frac{18 b x^{3/2} \sqrt{b x^2+c x^4}}{77 c^2}+\frac{2 x^{7/2} \sqrt{b x^2+c x^4}}{11 c}-\frac{\left (30 b^3 x \sqrt{b+c x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{b+c x^4}} \, dx,x,\sqrt{x}\right )}{77 c^3 \sqrt{b x^2+c x^4}}\\ &=\frac{30 b^2 \sqrt{b x^2+c x^4}}{77 c^3 \sqrt{x}}-\frac{18 b x^{3/2} \sqrt{b x^2+c x^4}}{77 c^2}+\frac{2 x^{7/2} \sqrt{b x^2+c x^4}}{11 c}-\frac{15 b^{11/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{77 c^{13/4} \sqrt{b x^2+c x^4}}\\ \end{align*}
Mathematica [C] time = 0.0352537, size = 97, normalized size = 0.54 \[ \frac{2 x^{3/2} \left (-15 b^3 \sqrt{\frac{c x^2}{b}+1} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{5}{4};-\frac{c x^2}{b}\right )+6 b^2 c x^2+15 b^3-2 b c^2 x^4+7 c^3 x^6\right )}{77 c^3 \sqrt{x^2 \left (b+c x^2\right )}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.19, size = 148, normalized size = 0.8 \begin{align*} -{\frac{1}{77\,{c}^{4}}\sqrt{x} \left ( -14\,{x}^{7}{c}^{4}+15\,{b}^{3}\sqrt{-bc}\sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{-{\frac{cx}{\sqrt{-bc}}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}},1/2\,\sqrt{2} \right ) +4\,{x}^{5}b{c}^{3}-12\,{b}^{2}{c}^{2}{x}^{3}-30\,x{b}^{3}c \right ){\frac{1}{\sqrt{c{x}^{4}+b{x}^{2}}}}} \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^{\frac{13}{2}}}{\sqrt{c x^{4} + b x^{2}}}\,{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{c x^{4} + b x^{2}} x^{\frac{9}{2}}}{c x^{2} + b}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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^{\frac{13}{2}}}{\sqrt{c x^{4} + b x^{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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