Optimal. Leaf size=80 \[ \frac{3 b^2 \tanh ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{b x^2+c x^4}}\right )}{8 \sqrt{c}}+\frac{3}{8} b \sqrt{b x^2+c x^4}+\frac{\left (b x^2+c x^4\right )^{3/2}}{4 x^2} \]
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Rubi [A] time = 0.0999827, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {2018, 664, 620, 206} \[ \frac{3 b^2 \tanh ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{b x^2+c x^4}}\right )}{8 \sqrt{c}}+\frac{3}{8} b \sqrt{b x^2+c x^4}+\frac{\left (b x^2+c x^4\right )^{3/2}}{4 x^2} \]
Antiderivative was successfully verified.
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Rule 2018
Rule 664
Rule 620
Rule 206
Rubi steps
\begin{align*} \int \frac{\left (b x^2+c x^4\right )^{3/2}}{x^3} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{\left (b x+c x^2\right )^{3/2}}{x^2} \, dx,x,x^2\right )\\ &=\frac{\left (b x^2+c x^4\right )^{3/2}}{4 x^2}+\frac{1}{8} (3 b) \operatorname{Subst}\left (\int \frac{\sqrt{b x+c x^2}}{x} \, dx,x,x^2\right )\\ &=\frac{3}{8} b \sqrt{b x^2+c x^4}+\frac{\left (b x^2+c x^4\right )^{3/2}}{4 x^2}+\frac{1}{16} \left (3 b^2\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{b x+c x^2}} \, dx,x,x^2\right )\\ &=\frac{3}{8} b \sqrt{b x^2+c x^4}+\frac{\left (b x^2+c x^4\right )^{3/2}}{4 x^2}+\frac{1}{8} \left (3 b^2\right ) \operatorname{Subst}\left (\int \frac{1}{1-c x^2} \, dx,x,\frac{x^2}{\sqrt{b x^2+c x^4}}\right )\\ &=\frac{3}{8} b \sqrt{b x^2+c x^4}+\frac{\left (b x^2+c x^4\right )^{3/2}}{4 x^2}+\frac{3 b^2 \tanh ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{b x^2+c x^4}}\right )}{8 \sqrt{c}}\\ \end{align*}
Mathematica [A] time = 0.111658, size = 71, normalized size = 0.89 \[ \frac{1}{8} \sqrt{x^2 \left (b+c x^2\right )} \left (\frac{3 b^{3/2} \sinh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{\sqrt{c} x \sqrt{\frac{c x^2}{b}+1}}+5 b+2 c x^2\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.048, size = 84, normalized size = 1.1 \begin{align*}{\frac{1}{8\,{x}^{3}} \left ( c{x}^{4}+b{x}^{2} \right ) ^{{\frac{3}{2}}} \left ( 2\,x \left ( c{x}^{2}+b \right ) ^{3/2}\sqrt{c}+3\,\sqrt{c}\sqrt{c{x}^{2}+b}xb+3\,\ln \left ( x\sqrt{c}+\sqrt{c{x}^{2}+b} \right ){b}^{2} \right ) \left ( c{x}^{2}+b \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{c}}}} \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 = 1.37256, size = 321, normalized size = 4.01 \begin{align*} \left [\frac{3 \, b^{2} \sqrt{c} \log \left (-2 \, c x^{2} - b - 2 \, \sqrt{c x^{4} + b x^{2}} \sqrt{c}\right ) + 2 \, \sqrt{c x^{4} + b x^{2}}{\left (2 \, c^{2} x^{2} + 5 \, b c\right )}}{16 \, c}, -\frac{3 \, b^{2} \sqrt{-c} \arctan \left (\frac{\sqrt{c x^{4} + b x^{2}} \sqrt{-c}}{c x^{2} + b}\right ) - \sqrt{c x^{4} + b x^{2}}{\left (2 \, c^{2} x^{2} + 5 \, b c\right )}}{8 \, c}\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^{3}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.29204, size = 92, normalized size = 1.15 \begin{align*} -\frac{3 \, b^{2} \log \left ({\left | -\sqrt{c} x + \sqrt{c x^{2} + b} \right |}\right ) \mathrm{sgn}\left (x\right )}{8 \, \sqrt{c}} + \frac{3 \, b^{2} \log \left ({\left | b \right |}\right ) \mathrm{sgn}\left (x\right )}{16 \, \sqrt{c}} + \frac{1}{8} \,{\left (2 \, c x^{2} \mathrm{sgn}\left (x\right ) + 5 \, b \mathrm{sgn}\left (x\right )\right )} \sqrt{c x^{2} + b} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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