Optimal. Leaf size=86 \[ \frac{3 b^2 \tanh ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{b x^2+c x^4}}\right )}{8 c^{5/2}}-\frac{3 b \sqrt{b x^2+c x^4}}{8 c^2}+\frac{x^2 \sqrt{b x^2+c x^4}}{4 c} \]
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Rubi [A] time = 0.103347, antiderivative size = 86, 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, 670, 640, 620, 206} \[ \frac{3 b^2 \tanh ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{b x^2+c x^4}}\right )}{8 c^{5/2}}-\frac{3 b \sqrt{b x^2+c x^4}}{8 c^2}+\frac{x^2 \sqrt{b x^2+c x^4}}{4 c} \]
Antiderivative was successfully verified.
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Rule 2018
Rule 670
Rule 640
Rule 620
Rule 206
Rubi steps
\begin{align*} \int \frac{x^5}{\sqrt{b x^2+c x^4}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{b x+c x^2}} \, dx,x,x^2\right )\\ &=\frac{x^2 \sqrt{b x^2+c x^4}}{4 c}-\frac{(3 b) \operatorname{Subst}\left (\int \frac{x}{\sqrt{b x+c x^2}} \, dx,x,x^2\right )}{8 c}\\ &=-\frac{3 b \sqrt{b x^2+c x^4}}{8 c^2}+\frac{x^2 \sqrt{b x^2+c x^4}}{4 c}+\frac{\left (3 b^2\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{b x+c x^2}} \, dx,x,x^2\right )}{16 c^2}\\ &=-\frac{3 b \sqrt{b x^2+c x^4}}{8 c^2}+\frac{x^2 \sqrt{b x^2+c x^4}}{4 c}+\frac{\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 )}{8 c^2}\\ &=-\frac{3 b \sqrt{b x^2+c x^4}}{8 c^2}+\frac{x^2 \sqrt{b x^2+c x^4}}{4 c}+\frac{3 b^2 \tanh ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{b x^2+c x^4}}\right )}{8 c^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.0369363, size = 89, normalized size = 1.03 \[ \frac{x \left (\sqrt{c} x \left (-3 b^2-b c x^2+2 c^2 x^4\right )+3 b^2 \sqrt{b+c x^2} \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b+c x^2}}\right )\right )}{8 c^{5/2} \sqrt{x^2 \left (b+c x^2\right )}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 85, normalized size = 1. \begin{align*}{\frac{x}{8}\sqrt{c{x}^{2}+b} \left ( 2\,{x}^{3}\sqrt{c{x}^{2}+b}{c}^{5/2}-3\,\sqrt{c{x}^{2}+b}{c}^{3/2}xb+3\,\ln \left ( x\sqrt{c}+\sqrt{c{x}^{2}+b} \right ){b}^{2}c \right ){\frac{1}{\sqrt{c{x}^{4}+b{x}^{2}}}}{c}^{-{\frac{7}{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 = 1.38589, size = 327, normalized size = 3.8 \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} - 3 \, b c\right )}}{16 \, c^{3}}, -\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} - 3 \, b c\right )}}{8 \, c^{3}}\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{x^{5}}{\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{x^{5}}{\sqrt{c x^{4} + b x^{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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