Optimal. Leaf size=88 \[ -\frac{b^3 \tanh ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{b x^2+c x^4}}\right )}{16 c^{3/2}}+\frac{b \left (b+2 c x^2\right ) \sqrt{b x^2+c x^4}}{16 c}+\frac{1}{6} \left (b x^2+c x^4\right )^{3/2} \]
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Rubi [A] time = 0.0991041, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263, Rules used = {2018, 664, 612, 620, 206} \[ -\frac{b^3 \tanh ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{b x^2+c x^4}}\right )}{16 c^{3/2}}+\frac{b \left (b+2 c x^2\right ) \sqrt{b x^2+c x^4}}{16 c}+\frac{1}{6} \left (b x^2+c x^4\right )^{3/2} \]
Antiderivative was successfully verified.
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Rule 2018
Rule 664
Rule 612
Rule 620
Rule 206
Rubi steps
\begin{align*} \int \frac{\left (b x^2+c x^4\right )^{3/2}}{x} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{\left (b x+c x^2\right )^{3/2}}{x} \, dx,x,x^2\right )\\ &=\frac{1}{6} \left (b x^2+c x^4\right )^{3/2}+\frac{1}{4} b \operatorname{Subst}\left (\int \sqrt{b x+c x^2} \, dx,x,x^2\right )\\ &=\frac{b \left (b+2 c x^2\right ) \sqrt{b x^2+c x^4}}{16 c}+\frac{1}{6} \left (b x^2+c x^4\right )^{3/2}-\frac{b^3 \operatorname{Subst}\left (\int \frac{1}{\sqrt{b x+c x^2}} \, dx,x,x^2\right )}{32 c}\\ &=\frac{b \left (b+2 c x^2\right ) \sqrt{b x^2+c x^4}}{16 c}+\frac{1}{6} \left (b x^2+c x^4\right )^{3/2}-\frac{b^3 \operatorname{Subst}\left (\int \frac{1}{1-c x^2} \, dx,x,\frac{x^2}{\sqrt{b x^2+c x^4}}\right )}{16 c}\\ &=\frac{b \left (b+2 c x^2\right ) \sqrt{b x^2+c x^4}}{16 c}+\frac{1}{6} \left (b x^2+c x^4\right )^{3/2}-\frac{b^3 \tanh ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{b x^2+c x^4}}\right )}{16 c^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.079886, size = 104, normalized size = 1.18 \[ \frac{\sqrt{x^2 \left (b+c x^2\right )} \left (\sqrt{c} x \sqrt{\frac{c x^2}{b}+1} \left (3 b^2+14 b c x^2+8 c^2 x^4\right )-3 b^{5/2} \sinh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )\right )}{48 c^{3/2} x \sqrt{\frac{c x^2}{b}+1}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.049, size = 102, normalized size = 1.2 \begin{align*}{\frac{1}{48\,{x}^{3}} \left ( c{x}^{4}+b{x}^{2} \right ) ^{{\frac{3}{2}}} \left ( 8\,x \left ( c{x}^{2}+b \right ) ^{5/2}\sqrt{c}-2\, \left ( c{x}^{2}+b \right ) ^{3/2}\sqrt{c}xb-3\,\sqrt{c{x}^{2}+b}\sqrt{c}x{b}^{2}-3\,\ln \left ( x\sqrt{c}+\sqrt{c{x}^{2}+b} \right ){b}^{3} \right ) \left ( c{x}^{2}+b \right ) ^{-{\frac{3}{2}}}{c}^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.36909, size = 373, normalized size = 4.24 \begin{align*} \left [\frac{3 \, b^{3} \sqrt{c} \log \left (-2 \, c x^{2} - b + 2 \, \sqrt{c x^{4} + b x^{2}} \sqrt{c}\right ) + 2 \,{\left (8 \, c^{3} x^{4} + 14 \, b c^{2} x^{2} + 3 \, b^{2} c\right )} \sqrt{c x^{4} + b x^{2}}}{96 \, c^{2}}, \frac{3 \, b^{3} \sqrt{-c} \arctan \left (\frac{\sqrt{c x^{4} + b x^{2}} \sqrt{-c}}{c x^{2} + b}\right ) +{\left (8 \, c^{3} x^{4} + 14 \, b c^{2} x^{2} + 3 \, b^{2} c\right )} \sqrt{c x^{4} + b x^{2}}}{48 \, c^{2}}\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{\left (x^{2} \left (b + c x^{2}\right )\right )^{\frac{3}{2}}}{x}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.2987, size = 113, normalized size = 1.28 \begin{align*} \frac{b^{3} \log \left ({\left | -\sqrt{c} x + \sqrt{c x^{2} + b} \right |}\right ) \mathrm{sgn}\left (x\right )}{16 \, c^{\frac{3}{2}}} - \frac{b^{3} \log \left ({\left | b \right |}\right ) \mathrm{sgn}\left (x\right )}{32 \, c^{\frac{3}{2}}} + \frac{1}{48} \,{\left (2 \,{\left (4 \, c x^{2} \mathrm{sgn}\left (x\right ) + 7 \, b \mathrm{sgn}\left (x\right )\right )} x^{2} + \frac{3 \, b^{2} \mathrm{sgn}\left (x\right )}{c}\right )} \sqrt{c x^{2} + b} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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