Optimal. Leaf size=350 \[ \frac{28 b^{17/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right ),\frac{1}{2}\right )}{1105 c^{11/4} \sqrt{b x^2+c x^4}}+\frac{56 b^4 x^{3/2} \left (b+c x^2\right )}{1105 c^{5/2} \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{b x^2+c x^4}}-\frac{56 b^3 \sqrt{x} \sqrt{b x^2+c x^4}}{3315 c^2}-\frac{56 b^{17/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{1105 c^{11/4} \sqrt{b x^2+c x^4}}+\frac{8 b^2 x^{5/2} \sqrt{b x^2+c x^4}}{663 c}+\frac{12}{221} b x^{9/2} \sqrt{b x^2+c x^4}+\frac{2}{17} x^{5/2} \left (b x^2+c x^4\right )^{3/2} \]
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Rubi [A] time = 0.433556, antiderivative size = 350, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {2021, 2024, 2032, 329, 305, 220, 1196} \[ \frac{56 b^4 x^{3/2} \left (b+c x^2\right )}{1105 c^{5/2} \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{b x^2+c x^4}}-\frac{56 b^3 \sqrt{x} \sqrt{b x^2+c x^4}}{3315 c^2}+\frac{28 b^{17/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{1105 c^{11/4} \sqrt{b x^2+c x^4}}-\frac{56 b^{17/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{1105 c^{11/4} \sqrt{b x^2+c x^4}}+\frac{8 b^2 x^{5/2} \sqrt{b x^2+c x^4}}{663 c}+\frac{12}{221} b x^{9/2} \sqrt{b x^2+c x^4}+\frac{2}{17} x^{5/2} \left (b x^2+c x^4\right )^{3/2} \]
Antiderivative was successfully verified.
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Rule 2021
Rule 2024
Rule 2032
Rule 329
Rule 305
Rule 220
Rule 1196
Rubi steps
\begin{align*} \int x^{3/2} \left (b x^2+c x^4\right )^{3/2} \, dx &=\frac{2}{17} x^{5/2} \left (b x^2+c x^4\right )^{3/2}+\frac{1}{17} (6 b) \int x^{7/2} \sqrt{b x^2+c x^4} \, dx\\ &=\frac{12}{221} b x^{9/2} \sqrt{b x^2+c x^4}+\frac{2}{17} x^{5/2} \left (b x^2+c x^4\right )^{3/2}+\frac{1}{221} \left (12 b^2\right ) \int \frac{x^{11/2}}{\sqrt{b x^2+c x^4}} \, dx\\ &=\frac{8 b^2 x^{5/2} \sqrt{b x^2+c x^4}}{663 c}+\frac{12}{221} b x^{9/2} \sqrt{b x^2+c x^4}+\frac{2}{17} x^{5/2} \left (b x^2+c x^4\right )^{3/2}-\frac{\left (28 b^3\right ) \int \frac{x^{7/2}}{\sqrt{b x^2+c x^4}} \, dx}{663 c}\\ &=-\frac{56 b^3 \sqrt{x} \sqrt{b x^2+c x^4}}{3315 c^2}+\frac{8 b^2 x^{5/2} \sqrt{b x^2+c x^4}}{663 c}+\frac{12}{221} b x^{9/2} \sqrt{b x^2+c x^4}+\frac{2}{17} x^{5/2} \left (b x^2+c x^4\right )^{3/2}+\frac{\left (28 b^4\right ) \int \frac{x^{3/2}}{\sqrt{b x^2+c x^4}} \, dx}{1105 c^2}\\ &=-\frac{56 b^3 \sqrt{x} \sqrt{b x^2+c x^4}}{3315 c^2}+\frac{8 b^2 x^{5/2} \sqrt{b x^2+c x^4}}{663 c}+\frac{12}{221} b x^{9/2} \sqrt{b x^2+c x^4}+\frac{2}{17} x^{5/2} \left (b x^2+c x^4\right )^{3/2}+\frac{\left (28 b^4 x \sqrt{b+c x^2}\right ) \int \frac{\sqrt{x}}{\sqrt{b+c x^2}} \, dx}{1105 c^2 \sqrt{b x^2+c x^4}}\\ &=-\frac{56 b^3 \sqrt{x} \sqrt{b x^2+c x^4}}{3315 c^2}+\frac{8 b^2 x^{5/2} \sqrt{b x^2+c x^4}}{663 c}+\frac{12}{221} b x^{9/2} \sqrt{b x^2+c x^4}+\frac{2}{17} x^{5/2} \left (b x^2+c x^4\right )^{3/2}+\frac{\left (56 b^4 x \sqrt{b+c x^2}\right ) \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{b+c x^4}} \, dx,x,\sqrt{x}\right )}{1105 c^2 \sqrt{b x^2+c x^4}}\\ &=-\frac{56 b^3 \sqrt{x} \sqrt{b x^2+c x^4}}{3315 c^2}+\frac{8 b^2 x^{5/2} \sqrt{b x^2+c x^4}}{663 c}+\frac{12}{221} b x^{9/2} \sqrt{b x^2+c x^4}+\frac{2}{17} x^{5/2} \left (b x^2+c x^4\right )^{3/2}+\frac{\left (56 b^{9/2} x \sqrt{b+c x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{b+c x^4}} \, dx,x,\sqrt{x}\right )}{1105 c^{5/2} \sqrt{b x^2+c x^4}}-\frac{\left (56 b^{9/2} 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 )}{1105 c^{5/2} \sqrt{b x^2+c x^4}}\\ &=\frac{56 b^4 x^{3/2} \left (b+c x^2\right )}{1105 c^{5/2} \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{b x^2+c x^4}}-\frac{56 b^3 \sqrt{x} \sqrt{b x^2+c x^4}}{3315 c^2}+\frac{8 b^2 x^{5/2} \sqrt{b x^2+c x^4}}{663 c}+\frac{12}{221} b x^{9/2} \sqrt{b x^2+c x^4}+\frac{2}{17} x^{5/2} \left (b x^2+c x^4\right )^{3/2}-\frac{56 b^{17/4} 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 )}{1105 c^{11/4} \sqrt{b x^2+c x^4}}+\frac{28 b^{17/4} 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 )}{1105 c^{11/4} \sqrt{b x^2+c x^4}}\\ \end{align*}
Mathematica [C] time = 0.0618634, size = 101, normalized size = 0.29 \[ \frac{2 \sqrt{x} \sqrt{x^2 \left (b+c x^2\right )} \left (7 b^3 \, _2F_1\left (-\frac{3}{2},\frac{3}{4};\frac{7}{4};-\frac{c x^2}{b}\right )-\left (7 b-13 c x^2\right ) \left (b+c x^2\right )^2 \sqrt{\frac{c x^2}{b}+1}\right )}{221 c^2 \sqrt{\frac{c x^2}{b}+1}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.198, size = 248, normalized size = 0.7 \begin{align*}{\frac{2}{3315\, \left ( c{x}^{2}+b \right ) ^{2}{c}^{3}} \left ( c{x}^{4}+b{x}^{2} \right ) ^{{\frac{3}{2}}} \left ( 195\,{x}^{10}{c}^{5}+480\,{x}^{8}b{c}^{4}+305\,{x}^{6}{b}^{2}{c}^{3}+84\,{b}^{5}\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 ) -42\,{b}^{5}\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 ) -8\,{x}^{4}{b}^{3}{c}^{2}-28\,{x}^{2}{b}^{4}c \right ){x}^{-{\frac{7}{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{\left (c x^{4} + b x^{2}\right )}^{\frac{3}{2}} x^{\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 ({\left (c x^{5} + b x^{3}\right )} \sqrt{c x^{4} + b x^{2}} \sqrt{x}, 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{\left (c x^{4} + b x^{2}\right )}^{\frac{3}{2}} x^{\frac{3}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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