Optimal. Leaf size=81 \[ \frac{3 \sqrt{b x^2+c x^4}}{2 c^2}-\frac{3 b \tanh ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{b x^2+c x^4}}\right )}{2 c^{5/2}}-\frac{x^4}{c \sqrt{b x^2+c x^4}} \]
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Rubi [A] time = 0.108931, antiderivative size = 81, 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, 668, 640, 620, 206} \[ \frac{3 \sqrt{b x^2+c x^4}}{2 c^2}-\frac{3 b \tanh ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{b x^2+c x^4}}\right )}{2 c^{5/2}}-\frac{x^4}{c \sqrt{b x^2+c x^4}} \]
Antiderivative was successfully verified.
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Rule 2018
Rule 668
Rule 640
Rule 620
Rule 206
Rubi steps
\begin{align*} \int \frac{x^7}{\left (b x^2+c x^4\right )^{3/2}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x^3}{\left (b x+c x^2\right )^{3/2}} \, dx,x,x^2\right )\\ &=-\frac{x^4}{c \sqrt{b x^2+c x^4}}+\frac{3 \operatorname{Subst}\left (\int \frac{x}{\sqrt{b x+c x^2}} \, dx,x,x^2\right )}{2 c}\\ &=-\frac{x^4}{c \sqrt{b x^2+c x^4}}+\frac{3 \sqrt{b x^2+c x^4}}{2 c^2}-\frac{(3 b) \operatorname{Subst}\left (\int \frac{1}{\sqrt{b x+c x^2}} \, dx,x,x^2\right )}{4 c^2}\\ &=-\frac{x^4}{c \sqrt{b x^2+c x^4}}+\frac{3 \sqrt{b x^2+c x^4}}{2 c^2}-\frac{(3 b) \operatorname{Subst}\left (\int \frac{1}{1-c x^2} \, dx,x,\frac{x^2}{\sqrt{b x^2+c x^4}}\right )}{2 c^2}\\ &=-\frac{x^4}{c \sqrt{b x^2+c x^4}}+\frac{3 \sqrt{b x^2+c x^4}}{2 c^2}-\frac{3 b \tanh ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{b x^2+c x^4}}\right )}{2 c^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.0418383, size = 76, normalized size = 0.94 \[ \frac{x \left (\sqrt{c} x \left (3 b+c x^2\right )-3 b^{3/2} \sqrt{\frac{c x^2}{b}+1} \sinh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )\right )}{2 c^{5/2} \sqrt{x^2 \left (b+c x^2\right )}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.048, size = 73, normalized size = 0.9 \begin{align*}{\frac{{x}^{3} \left ( c{x}^{2}+b \right ) }{2} \left ({x}^{3}{c}^{{\frac{5}{2}}}+3\,{c}^{3/2}xb-3\,\ln \left ( x\sqrt{c}+\sqrt{c{x}^{2}+b} \right ) \sqrt{c{x}^{2}+b}bc \right ) \left ( c{x}^{4}+b{x}^{2} \right ) ^{-{\frac{3}{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 = 1.29421, size = 389, normalized size = 4.8 \begin{align*} \left [\frac{3 \,{\left (b c x^{2} + b^{2}\right )} \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 (c^{2} x^{2} + 3 \, b c\right )}}{4 \,{\left (c^{4} x^{2} + b c^{3}\right )}}, \frac{3 \,{\left (b c x^{2} + b^{2}\right )} \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 (c^{2} x^{2} + 3 \, b c\right )}}{2 \,{\left (c^{4} x^{2} + b c^{3}\right )}}\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^{7}}{\left (x^{2} \left (b + c x^{2}\right )\right )^{\frac{3}{2}}}\, 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^{7}}{{\left (c x^{4} + b x^{2}\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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