Optimal. Leaf size=326 \[ \frac{7 c^{9/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right ),\frac{1}{2}\right )}{15 b^{11/4} \sqrt{b x^2+c x^4}}+\frac{14 c^{5/2} x^{3/2} \left (b+c x^2\right )}{15 b^3 \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{b x^2+c x^4}}-\frac{14 c^2 \sqrt{b x^2+c x^4}}{15 b^3 x^{3/2}}-\frac{14 c^{9/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{15 b^{11/4} \sqrt{b x^2+c x^4}}+\frac{14 c \sqrt{b x^2+c x^4}}{45 b^2 x^{7/2}}-\frac{2 \sqrt{b x^2+c x^4}}{9 b x^{11/2}} \]
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Rubi [A] time = 0.358624, antiderivative size = 326, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {2025, 2032, 329, 305, 220, 1196} \[ \frac{14 c^{5/2} x^{3/2} \left (b+c x^2\right )}{15 b^3 \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{b x^2+c x^4}}-\frac{14 c^2 \sqrt{b x^2+c x^4}}{15 b^3 x^{3/2}}+\frac{7 c^{9/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{15 b^{11/4} \sqrt{b x^2+c x^4}}-\frac{14 c^{9/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{15 b^{11/4} \sqrt{b x^2+c x^4}}+\frac{14 c \sqrt{b x^2+c x^4}}{45 b^2 x^{7/2}}-\frac{2 \sqrt{b x^2+c x^4}}{9 b x^{11/2}} \]
Antiderivative was successfully verified.
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Rule 2025
Rule 2032
Rule 329
Rule 305
Rule 220
Rule 1196
Rubi steps
\begin{align*} \int \frac{1}{x^{9/2} \sqrt{b x^2+c x^4}} \, dx &=-\frac{2 \sqrt{b x^2+c x^4}}{9 b x^{11/2}}-\frac{(7 c) \int \frac{1}{x^{5/2} \sqrt{b x^2+c x^4}} \, dx}{9 b}\\ &=-\frac{2 \sqrt{b x^2+c x^4}}{9 b x^{11/2}}+\frac{14 c \sqrt{b x^2+c x^4}}{45 b^2 x^{7/2}}+\frac{\left (7 c^2\right ) \int \frac{1}{\sqrt{x} \sqrt{b x^2+c x^4}} \, dx}{15 b^2}\\ &=-\frac{2 \sqrt{b x^2+c x^4}}{9 b x^{11/2}}+\frac{14 c \sqrt{b x^2+c x^4}}{45 b^2 x^{7/2}}-\frac{14 c^2 \sqrt{b x^2+c x^4}}{15 b^3 x^{3/2}}+\frac{\left (7 c^3\right ) \int \frac{x^{3/2}}{\sqrt{b x^2+c x^4}} \, dx}{15 b^3}\\ &=-\frac{2 \sqrt{b x^2+c x^4}}{9 b x^{11/2}}+\frac{14 c \sqrt{b x^2+c x^4}}{45 b^2 x^{7/2}}-\frac{14 c^2 \sqrt{b x^2+c x^4}}{15 b^3 x^{3/2}}+\frac{\left (7 c^3 x \sqrt{b+c x^2}\right ) \int \frac{\sqrt{x}}{\sqrt{b+c x^2}} \, dx}{15 b^3 \sqrt{b x^2+c x^4}}\\ &=-\frac{2 \sqrt{b x^2+c x^4}}{9 b x^{11/2}}+\frac{14 c \sqrt{b x^2+c x^4}}{45 b^2 x^{7/2}}-\frac{14 c^2 \sqrt{b x^2+c x^4}}{15 b^3 x^{3/2}}+\frac{\left (14 c^3 x \sqrt{b+c x^2}\right ) \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{b+c x^4}} \, dx,x,\sqrt{x}\right )}{15 b^3 \sqrt{b x^2+c x^4}}\\ &=-\frac{2 \sqrt{b x^2+c x^4}}{9 b x^{11/2}}+\frac{14 c \sqrt{b x^2+c x^4}}{45 b^2 x^{7/2}}-\frac{14 c^2 \sqrt{b x^2+c x^4}}{15 b^3 x^{3/2}}+\frac{\left (14 c^{5/2} x \sqrt{b+c x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{b+c x^4}} \, dx,x,\sqrt{x}\right )}{15 b^{5/2} \sqrt{b x^2+c x^4}}-\frac{\left (14 c^{5/2} x \sqrt{b+c x^2}\right ) \operatorname{Subst}\left (\int \frac{1-\frac{\sqrt{c} x^2}{\sqrt{b}}}{\sqrt{b+c x^4}} \, dx,x,\sqrt{x}\right )}{15 b^{5/2} \sqrt{b x^2+c x^4}}\\ &=\frac{14 c^{5/2} x^{3/2} \left (b+c x^2\right )}{15 b^3 \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{b x^2+c x^4}}-\frac{2 \sqrt{b x^2+c x^4}}{9 b x^{11/2}}+\frac{14 c \sqrt{b x^2+c x^4}}{45 b^2 x^{7/2}}-\frac{14 c^2 \sqrt{b x^2+c x^4}}{15 b^3 x^{3/2}}-\frac{14 c^{9/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{15 b^{11/4} \sqrt{b x^2+c x^4}}+\frac{7 c^{9/4} 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 )}{15 b^{11/4} \sqrt{b x^2+c x^4}}\\ \end{align*}
Mathematica [C] time = 0.0132543, size = 57, normalized size = 0.17 \[ -\frac{2 \sqrt{\frac{c x^2}{b}+1} \, _2F_1\left (-\frac{9}{4},\frac{1}{2};-\frac{5}{4};-\frac{c x^2}{b}\right )}{9 x^{7/2} \sqrt{x^2 \left (b+c x^2\right )}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.188, size = 230, normalized size = 0.7 \begin{align*}{\frac{1}{45\,{b}^{3}} \left ( 42\,\sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{-{\frac{cx}{\sqrt{-bc}}}}{\it EllipticE} \left ( \sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}},1/2\,\sqrt{2} \right ){x}^{4}b{c}^{2}-21\,\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 ){x}^{4}b{c}^{2}-42\,{c}^{3}{x}^{6}-28\,b{c}^{2}{x}^{4}+4\,{b}^{2}c{x}^{2}-10\,{b}^{3} \right ){x}^{-{\frac{7}{2}}}{\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{1}{\sqrt{c x^{4} + b x^{2}} x^{\frac{9}{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}} \sqrt{x}}{c x^{9} + b x^{7}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x^{\frac{9}{2}} \sqrt{x^{2} \left (b + c x^{2}\right )}}\, dx \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{1}{\sqrt{c x^{4} + b x^{2}} x^{\frac{9}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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