Optimal. Leaf size=203 \[ \frac{4 c^{15/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right ),\frac{1}{2}\right )}{231 b^{9/4} \sqrt{b x^2+c x^4}}+\frac{8 c^3 \sqrt{b x^2+c x^4}}{231 b^2 x^{5/2}}-\frac{8 c^2 \sqrt{b x^2+c x^4}}{385 b x^{9/2}}-\frac{4 c \sqrt{b x^2+c x^4}}{55 x^{13/2}}-\frac{2 \left (b x^2+c x^4\right )^{3/2}}{15 x^{21/2}} \]
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Rubi [A] time = 0.295741, antiderivative size = 203, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238, Rules used = {2020, 2025, 2032, 329, 220} \[ \frac{8 c^3 \sqrt{b x^2+c x^4}}{231 b^2 x^{5/2}}+\frac{4 c^{15/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{231 b^{9/4} \sqrt{b x^2+c x^4}}-\frac{8 c^2 \sqrt{b x^2+c x^4}}{385 b x^{9/2}}-\frac{4 c \sqrt{b x^2+c x^4}}{55 x^{13/2}}-\frac{2 \left (b x^2+c x^4\right )^{3/2}}{15 x^{21/2}} \]
Antiderivative was successfully verified.
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Rule 2020
Rule 2025
Rule 2032
Rule 329
Rule 220
Rubi steps
\begin{align*} \int \frac{\left (b x^2+c x^4\right )^{3/2}}{x^{23/2}} \, dx &=-\frac{2 \left (b x^2+c x^4\right )^{3/2}}{15 x^{21/2}}+\frac{1}{5} (2 c) \int \frac{\sqrt{b x^2+c x^4}}{x^{15/2}} \, dx\\ &=-\frac{4 c \sqrt{b x^2+c x^4}}{55 x^{13/2}}-\frac{2 \left (b x^2+c x^4\right )^{3/2}}{15 x^{21/2}}+\frac{1}{55} \left (4 c^2\right ) \int \frac{1}{x^{7/2} \sqrt{b x^2+c x^4}} \, dx\\ &=-\frac{4 c \sqrt{b x^2+c x^4}}{55 x^{13/2}}-\frac{8 c^2 \sqrt{b x^2+c x^4}}{385 b x^{9/2}}-\frac{2 \left (b x^2+c x^4\right )^{3/2}}{15 x^{21/2}}-\frac{\left (4 c^3\right ) \int \frac{1}{x^{3/2} \sqrt{b x^2+c x^4}} \, dx}{77 b}\\ &=-\frac{4 c \sqrt{b x^2+c x^4}}{55 x^{13/2}}-\frac{8 c^2 \sqrt{b x^2+c x^4}}{385 b x^{9/2}}+\frac{8 c^3 \sqrt{b x^2+c x^4}}{231 b^2 x^{5/2}}-\frac{2 \left (b x^2+c x^4\right )^{3/2}}{15 x^{21/2}}+\frac{\left (4 c^4\right ) \int \frac{\sqrt{x}}{\sqrt{b x^2+c x^4}} \, dx}{231 b^2}\\ &=-\frac{4 c \sqrt{b x^2+c x^4}}{55 x^{13/2}}-\frac{8 c^2 \sqrt{b x^2+c x^4}}{385 b x^{9/2}}+\frac{8 c^3 \sqrt{b x^2+c x^4}}{231 b^2 x^{5/2}}-\frac{2 \left (b x^2+c x^4\right )^{3/2}}{15 x^{21/2}}+\frac{\left (4 c^4 x \sqrt{b+c x^2}\right ) \int \frac{1}{\sqrt{x} \sqrt{b+c x^2}} \, dx}{231 b^2 \sqrt{b x^2+c x^4}}\\ &=-\frac{4 c \sqrt{b x^2+c x^4}}{55 x^{13/2}}-\frac{8 c^2 \sqrt{b x^2+c x^4}}{385 b x^{9/2}}+\frac{8 c^3 \sqrt{b x^2+c x^4}}{231 b^2 x^{5/2}}-\frac{2 \left (b x^2+c x^4\right )^{3/2}}{15 x^{21/2}}+\frac{\left (8 c^4 x \sqrt{b+c x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{b+c x^4}} \, dx,x,\sqrt{x}\right )}{231 b^2 \sqrt{b x^2+c x^4}}\\ &=-\frac{4 c \sqrt{b x^2+c x^4}}{55 x^{13/2}}-\frac{8 c^2 \sqrt{b x^2+c x^4}}{385 b x^{9/2}}+\frac{8 c^3 \sqrt{b x^2+c x^4}}{231 b^2 x^{5/2}}-\frac{2 \left (b x^2+c x^4\right )^{3/2}}{15 x^{21/2}}+\frac{4 c^{15/4} 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 )}{231 b^{9/4} \sqrt{b x^2+c x^4}}\\ \end{align*}
Mathematica [C] time = 0.0185506, size = 58, normalized size = 0.29 \[ -\frac{2 b \sqrt{x^2 \left (b+c x^2\right )} \, _2F_1\left (-\frac{15}{4},-\frac{3}{2};-\frac{11}{4};-\frac{c x^2}{b}\right )}{15 x^{17/2} \sqrt{\frac{c x^2}{b}+1}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.206, size = 167, normalized size = 0.8 \begin{align*}{\frac{2}{1155\, \left ( c{x}^{2}+b \right ) ^{2}{b}^{2}} \left ( c{x}^{4}+b{x}^{2} \right ) ^{{\frac{3}{2}}} \left ( 10\,\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}^{7}{c}^{3}+20\,{x}^{8}{c}^{4}+8\,{x}^{6}b{c}^{3}-131\,{x}^{4}{b}^{2}{c}^{2}-196\,{x}^{2}{b}^{3}c-77\,{b}^{4} \right ){x}^{-{\frac{21}{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{23}{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{19}{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{23}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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