Optimal. Leaf size=146 \[ -\frac{5 b^{3/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right ),\frac{1}{2}\right )}{6 c^{9/4} \sqrt{b x^2+c x^4}}+\frac{5 \sqrt{b x^2+c x^4}}{3 c^2 \sqrt{x}}-\frac{x^{7/2}}{c \sqrt{b x^2+c x^4}} \]
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Rubi [A] time = 0.185086, antiderivative size = 146, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238, Rules used = {2022, 2024, 2032, 329, 220} \[ -\frac{5 b^{3/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{6 c^{9/4} \sqrt{b x^2+c x^4}}+\frac{5 \sqrt{b x^2+c x^4}}{3 c^2 \sqrt{x}}-\frac{x^{7/2}}{c \sqrt{b x^2+c x^4}} \]
Antiderivative was successfully verified.
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Rule 2022
Rule 2024
Rule 2032
Rule 329
Rule 220
Rubi steps
\begin{align*} \int \frac{x^{13/2}}{\left (b x^2+c x^4\right )^{3/2}} \, dx &=-\frac{x^{7/2}}{c \sqrt{b x^2+c x^4}}+\frac{5 \int \frac{x^{5/2}}{\sqrt{b x^2+c x^4}} \, dx}{2 c}\\ &=-\frac{x^{7/2}}{c \sqrt{b x^2+c x^4}}+\frac{5 \sqrt{b x^2+c x^4}}{3 c^2 \sqrt{x}}-\frac{(5 b) \int \frac{\sqrt{x}}{\sqrt{b x^2+c x^4}} \, dx}{6 c^2}\\ &=-\frac{x^{7/2}}{c \sqrt{b x^2+c x^4}}+\frac{5 \sqrt{b x^2+c x^4}}{3 c^2 \sqrt{x}}-\frac{\left (5 b x \sqrt{b+c x^2}\right ) \int \frac{1}{\sqrt{x} \sqrt{b+c x^2}} \, dx}{6 c^2 \sqrt{b x^2+c x^4}}\\ &=-\frac{x^{7/2}}{c \sqrt{b x^2+c x^4}}+\frac{5 \sqrt{b x^2+c x^4}}{3 c^2 \sqrt{x}}-\frac{\left (5 b x \sqrt{b+c x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{b+c x^4}} \, dx,x,\sqrt{x}\right )}{3 c^2 \sqrt{b x^2+c x^4}}\\ &=-\frac{x^{7/2}}{c \sqrt{b x^2+c x^4}}+\frac{5 \sqrt{b x^2+c x^4}}{3 c^2 \sqrt{x}}-\frac{5 b^{3/4} 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 )}{6 c^{9/4} \sqrt{b x^2+c x^4}}\\ \end{align*}
Mathematica [C] time = 0.026614, size = 73, normalized size = 0.5 \[ \frac{x^{3/2} \left (-5 b \sqrt{\frac{c x^2}{b}+1} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{5}{4};-\frac{c x^2}{b}\right )+5 b+2 c x^2\right )}{3 c^2 \sqrt{x^2 \left (b+c x^2\right )}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.186, size = 131, normalized size = 0.9 \begin{align*} -{\frac{c{x}^{2}+b}{6\,{c}^{3}}{x}^{{\frac{5}{2}}} \left ( 5\,b\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\,{c}^{2}{x}^{3}-10\,bcx \right ) \left ( c{x}^{4}+b{x}^{2} \right ) ^{-{\frac{3}{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}}}{{\left (c x^{4} + b x^{2}\right )}^{\frac{3}{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{5}{2}}}{c^{2} x^{4} + 2 \, b c x^{2} + b^{2}}, 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}}}{{\left (c x^{4} + b x^{2}\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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