Optimal. Leaf size=114 \[ \frac{5 b^2 \sqrt{b x^2+c x^4}}{16 c^3}-\frac{5 b^3 \tanh ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{b x^2+c x^4}}\right )}{16 c^{7/2}}-\frac{5 b x^2 \sqrt{b x^2+c x^4}}{24 c^2}+\frac{x^4 \sqrt{b x^2+c x^4}}{6 c} \]
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Rubi [A] time = 0.12934, antiderivative size = 114, normalized size of antiderivative = 1., number of steps used = 6, 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} \[ \frac{5 b^2 \sqrt{b x^2+c x^4}}{16 c^3}-\frac{5 b^3 \tanh ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{b x^2+c x^4}}\right )}{16 c^{7/2}}-\frac{5 b x^2 \sqrt{b x^2+c x^4}}{24 c^2}+\frac{x^4 \sqrt{b x^2+c x^4}}{6 c} \]
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^7}{\sqrt{b x^2+c x^4}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x^3}{\sqrt{b x+c x^2}} \, dx,x,x^2\right )\\ &=\frac{x^4 \sqrt{b x^2+c x^4}}{6 c}-\frac{(5 b) \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{b x+c x^2}} \, dx,x,x^2\right )}{12 c}\\ &=-\frac{5 b x^2 \sqrt{b x^2+c x^4}}{24 c^2}+\frac{x^4 \sqrt{b x^2+c x^4}}{6 c}+\frac{\left (5 b^2\right ) \operatorname{Subst}\left (\int \frac{x}{\sqrt{b x+c x^2}} \, dx,x,x^2\right )}{16 c^2}\\ &=\frac{5 b^2 \sqrt{b x^2+c x^4}}{16 c^3}-\frac{5 b x^2 \sqrt{b x^2+c x^4}}{24 c^2}+\frac{x^4 \sqrt{b x^2+c x^4}}{6 c}-\frac{\left (5 b^3\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{b x+c x^2}} \, dx,x,x^2\right )}{32 c^3}\\ &=\frac{5 b^2 \sqrt{b x^2+c x^4}}{16 c^3}-\frac{5 b x^2 \sqrt{b x^2+c x^4}}{24 c^2}+\frac{x^4 \sqrt{b x^2+c x^4}}{6 c}-\frac{\left (5 b^3\right ) \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^3}\\ &=\frac{5 b^2 \sqrt{b x^2+c x^4}}{16 c^3}-\frac{5 b x^2 \sqrt{b x^2+c x^4}}{24 c^2}+\frac{x^4 \sqrt{b x^2+c x^4}}{6 c}-\frac{5 b^3 \tanh ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{b x^2+c x^4}}\right )}{16 c^{7/2}}\\ \end{align*}
Mathematica [A] time = 0.0528215, size = 100, normalized size = 0.88 \[ \frac{x \left (\sqrt{c} x \left (5 b^2 c x^2+15 b^3-2 b c^2 x^4+8 c^3 x^6\right )-15 b^3 \sqrt{b+c x^2} \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b+c x^2}}\right )\right )}{48 c^{7/2} \sqrt{x^2 \left (b+c x^2\right )}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.051, size = 105, normalized size = 0.9 \begin{align*}{\frac{x}{48}\sqrt{c{x}^{2}+b} \left ( 8\,{x}^{5}\sqrt{c{x}^{2}+b}{c}^{7/2}-10\,{c}^{5/2}\sqrt{c{x}^{2}+b}{x}^{3}b+15\,{c}^{3/2}\sqrt{c{x}^{2}+b}x{b}^{2}-15\,\ln \left ( x\sqrt{c}+\sqrt{c{x}^{2}+b} \right ){b}^{3}c \right ){\frac{1}{\sqrt{c{x}^{4}+b{x}^{2}}}}{c}^{-{\frac{9}{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 = 1.34681, size = 378, normalized size = 3.32 \begin{align*} \left [\frac{15 \, 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} - 10 \, b c^{2} x^{2} + 15 \, b^{2} c\right )} \sqrt{c x^{4} + b x^{2}}}{96 \, c^{4}}, \frac{15 \, 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} - 10 \, b c^{2} x^{2} + 15 \, b^{2} c\right )} \sqrt{c x^{4} + b x^{2}}}{48 \, c^{4}}\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}}{\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^{7}}{\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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