Optimal. Leaf size=109 \[ \frac{15 b^2 \tanh ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{b x^2+c x^4}}\right )}{8 c^{7/2}}+\frac{5 x^2 \sqrt{b x^2+c x^4}}{4 c^2}-\frac{15 b \sqrt{b x^2+c x^4}}{8 c^3}-\frac{x^6}{c \sqrt{b x^2+c x^4}} \]
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Rubi [A] time = 0.128751, antiderivative size = 109, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.316, Rules used = {2018, 668, 670, 640, 620, 206} \[ \frac{15 b^2 \tanh ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{b x^2+c x^4}}\right )}{8 c^{7/2}}+\frac{5 x^2 \sqrt{b x^2+c x^4}}{4 c^2}-\frac{15 b \sqrt{b x^2+c x^4}}{8 c^3}-\frac{x^6}{c \sqrt{b x^2+c x^4}} \]
Antiderivative was successfully verified.
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Rule 2018
Rule 668
Rule 670
Rule 640
Rule 620
Rule 206
Rubi steps
\begin{align*} \int \frac{x^9}{\left (b x^2+c x^4\right )^{3/2}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x^4}{\left (b x+c x^2\right )^{3/2}} \, dx,x,x^2\right )\\ &=-\frac{x^6}{c \sqrt{b x^2+c x^4}}+\frac{5 \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{b x+c x^2}} \, dx,x,x^2\right )}{2 c}\\ &=-\frac{x^6}{c \sqrt{b x^2+c x^4}}+\frac{5 x^2 \sqrt{b x^2+c x^4}}{4 c^2}-\frac{(15 b) \operatorname{Subst}\left (\int \frac{x}{\sqrt{b x+c x^2}} \, dx,x,x^2\right )}{8 c^2}\\ &=-\frac{x^6}{c \sqrt{b x^2+c x^4}}-\frac{15 b \sqrt{b x^2+c x^4}}{8 c^3}+\frac{5 x^2 \sqrt{b x^2+c x^4}}{4 c^2}+\frac{\left (15 b^2\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{b x+c x^2}} \, dx,x,x^2\right )}{16 c^3}\\ &=-\frac{x^6}{c \sqrt{b x^2+c x^4}}-\frac{15 b \sqrt{b x^2+c x^4}}{8 c^3}+\frac{5 x^2 \sqrt{b x^2+c x^4}}{4 c^2}+\frac{\left (15 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^3}\\ &=-\frac{x^6}{c \sqrt{b x^2+c x^4}}-\frac{15 b \sqrt{b x^2+c x^4}}{8 c^3}+\frac{5 x^2 \sqrt{b x^2+c x^4}}{4 c^2}+\frac{15 b^2 \tanh ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{b x^2+c x^4}}\right )}{8 c^{7/2}}\\ \end{align*}
Mathematica [A] time = 0.0512658, size = 88, normalized size = 0.81 \[ \frac{x \left (\sqrt{c} x \left (-15 b^2-5 b c x^2+2 c^2 x^4\right )+15 b^{5/2} \sqrt{\frac{c x^2}{b}+1} \sinh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )\right )}{8 c^{7/2} \sqrt{x^2 \left (b+c x^2\right )}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.051, size = 87, normalized size = 0.8 \begin{align*}{\frac{{x}^{3} \left ( c{x}^{2}+b \right ) }{8} \left ( 2\,{x}^{5}{c}^{7/2}-5\,{c}^{5/2}{x}^{3}b-15\,{c}^{3/2}x{b}^{2}+15\,\sqrt{c{x}^{2}+b}\ln \left ( x\sqrt{c}+\sqrt{c{x}^{2}+b} \right ){b}^{2}c \right ) \left ( c{x}^{4}+b{x}^{2} \right ) ^{-{\frac{3}{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 = 1.60523, size = 451, normalized size = 4.14 \begin{align*} \left [\frac{15 \,{\left (b^{2} c x^{2} + b^{3}\right )} \sqrt{c} \log \left (-2 \, c x^{2} - b - 2 \, \sqrt{c x^{4} + b x^{2}} \sqrt{c}\right ) + 2 \,{\left (2 \, c^{3} x^{4} - 5 \, b c^{2} x^{2} - 15 \, b^{2} c\right )} \sqrt{c x^{4} + b x^{2}}}{16 \,{\left (c^{5} x^{2} + b c^{4}\right )}}, -\frac{15 \,{\left (b^{2} c x^{2} + b^{3}\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{c x^{4} + b x^{2}} \sqrt{-c}}{c x^{2} + b}\right ) -{\left (2 \, c^{3} x^{4} - 5 \, b c^{2} x^{2} - 15 \, b^{2} c\right )} \sqrt{c x^{4} + b x^{2}}}{8 \,{\left (c^{5} x^{2} + b c^{4}\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^{9}}{\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^{9}}{{\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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