Optimal. Leaf size=143 \[ \frac{4 c^{7/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right ),\frac{1}{2}\right )}{7 \sqrt [4]{b} \sqrt{b x^2+c x^4}}-\frac{4 c \sqrt{b x^2+c x^4}}{7 x^{5/2}}-\frac{2 \left (b x^2+c x^4\right )^{3/2}}{7 x^{13/2}} \]
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Rubi [A] time = 0.18338, antiderivative size = 143, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {2020, 2032, 329, 220} \[ \frac{4 c^{7/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{7 \sqrt [4]{b} \sqrt{b x^2+c x^4}}-\frac{4 c \sqrt{b x^2+c x^4}}{7 x^{5/2}}-\frac{2 \left (b x^2+c x^4\right )^{3/2}}{7 x^{13/2}} \]
Antiderivative was successfully verified.
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Rule 2020
Rule 2032
Rule 329
Rule 220
Rubi steps
\begin{align*} \int \frac{\left (b x^2+c x^4\right )^{3/2}}{x^{15/2}} \, dx &=-\frac{2 \left (b x^2+c x^4\right )^{3/2}}{7 x^{13/2}}+\frac{1}{7} (6 c) \int \frac{\sqrt{b x^2+c x^4}}{x^{7/2}} \, dx\\ &=-\frac{4 c \sqrt{b x^2+c x^4}}{7 x^{5/2}}-\frac{2 \left (b x^2+c x^4\right )^{3/2}}{7 x^{13/2}}+\frac{1}{7} \left (4 c^2\right ) \int \frac{\sqrt{x}}{\sqrt{b x^2+c x^4}} \, dx\\ &=-\frac{4 c \sqrt{b x^2+c x^4}}{7 x^{5/2}}-\frac{2 \left (b x^2+c x^4\right )^{3/2}}{7 x^{13/2}}+\frac{\left (4 c^2 x \sqrt{b+c x^2}\right ) \int \frac{1}{\sqrt{x} \sqrt{b+c x^2}} \, dx}{7 \sqrt{b x^2+c x^4}}\\ &=-\frac{4 c \sqrt{b x^2+c x^4}}{7 x^{5/2}}-\frac{2 \left (b x^2+c x^4\right )^{3/2}}{7 x^{13/2}}+\frac{\left (8 c^2 x \sqrt{b+c x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{b+c x^4}} \, dx,x,\sqrt{x}\right )}{7 \sqrt{b x^2+c x^4}}\\ &=-\frac{4 c \sqrt{b x^2+c x^4}}{7 x^{5/2}}-\frac{2 \left (b x^2+c x^4\right )^{3/2}}{7 x^{13/2}}+\frac{4 c^{7/4} 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 )}{7 \sqrt [4]{b} \sqrt{b x^2+c x^4}}\\ \end{align*}
Mathematica [C] time = 0.0166989, size = 58, normalized size = 0.41 \[ -\frac{2 b \sqrt{x^2 \left (b+c x^2\right )} \, _2F_1\left (-\frac{7}{4},-\frac{3}{2};-\frac{3}{4};-\frac{c x^2}{b}\right )}{7 x^{9/2} \sqrt{\frac{c x^2}{b}+1}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.185, size = 140, normalized size = 1. \begin{align*}{\frac{2}{7\, \left ( c{x}^{2}+b \right ) ^{2}} \left ( c{x}^{4}+b{x}^{2} \right ) ^{{\frac{3}{2}}} \left ( 2\,\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 ) \sqrt{-bc}{x}^{3}c-3\,{c}^{2}{x}^{4}-4\,bc{x}^{2}-{b}^{2} \right ){x}^{-{\frac{13}{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{{\left (c x^{4} + b x^{2}\right )}^{\frac{3}{2}}}{x^{\frac{15}{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}}{\left (c x^{2} + b\right )}}{x^{\frac{11}{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{{\left (c x^{4} + b x^{2}\right )}^{\frac{3}{2}}}{x^{\frac{15}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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