Optimal. Leaf size=134 \[ \frac{\left (-8 a c+3 b^2-2 b c x^2\right ) \sqrt{a+b x^2+c x^4}}{2 c^2 \left (b^2-4 a c\right )}+\frac{x^4 \left (2 a+b x^2\right )}{\left (b^2-4 a c\right ) \sqrt{a+b x^2+c x^4}}-\frac{3 b \tanh ^{-1}\left (\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right )}{4 c^{5/2}} \]
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Rubi [A] time = 0.111292, antiderivative size = 134, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {1114, 738, 779, 621, 206} \[ \frac{\left (-8 a c+3 b^2-2 b c x^2\right ) \sqrt{a+b x^2+c x^4}}{2 c^2 \left (b^2-4 a c\right )}+\frac{x^4 \left (2 a+b x^2\right )}{\left (b^2-4 a c\right ) \sqrt{a+b x^2+c x^4}}-\frac{3 b \tanh ^{-1}\left (\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right )}{4 c^{5/2}} \]
Antiderivative was successfully verified.
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Rule 1114
Rule 738
Rule 779
Rule 621
Rule 206
Rubi steps
\begin{align*} \int \frac{x^7}{\left (a+b x^2+c x^4\right )^{3/2}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x^3}{\left (a+b x+c x^2\right )^{3/2}} \, dx,x,x^2\right )\\ &=\frac{x^4 \left (2 a+b x^2\right )}{\left (b^2-4 a c\right ) \sqrt{a+b x^2+c x^4}}-\frac{\operatorname{Subst}\left (\int \frac{x (4 a+2 b x)}{\sqrt{a+b x+c x^2}} \, dx,x,x^2\right )}{b^2-4 a c}\\ &=\frac{x^4 \left (2 a+b x^2\right )}{\left (b^2-4 a c\right ) \sqrt{a+b x^2+c x^4}}+\frac{\left (3 b^2-8 a c-2 b c x^2\right ) \sqrt{a+b x^2+c x^4}}{2 c^2 \left (b^2-4 a c\right )}-\frac{(3 b) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x+c x^2}} \, dx,x,x^2\right )}{4 c^2}\\ &=\frac{x^4 \left (2 a+b x^2\right )}{\left (b^2-4 a c\right ) \sqrt{a+b x^2+c x^4}}+\frac{\left (3 b^2-8 a c-2 b c x^2\right ) \sqrt{a+b x^2+c x^4}}{2 c^2 \left (b^2-4 a c\right )}-\frac{(3 b) \operatorname{Subst}\left (\int \frac{1}{4 c-x^2} \, dx,x,\frac{b+2 c x^2}{\sqrt{a+b x^2+c x^4}}\right )}{2 c^2}\\ &=\frac{x^4 \left (2 a+b x^2\right )}{\left (b^2-4 a c\right ) \sqrt{a+b x^2+c x^4}}+\frac{\left (3 b^2-8 a c-2 b c x^2\right ) \sqrt{a+b x^2+c x^4}}{2 c^2 \left (b^2-4 a c\right )}-\frac{3 b \tanh ^{-1}\left (\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right )}{4 c^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.119625, size = 137, normalized size = 1.02 \[ \frac{\frac{2 \sqrt{c} \left (8 a^2 c+a \left (-3 b^2+10 b c x^2+4 c^2 x^4\right )-b^2 x^2 \left (3 b+c x^2\right )\right )}{\sqrt{a+b x^2+c x^4}}+3 b \left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right )}{4 c^{5/2} \left (4 a c-b^2\right )} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.173, size = 264, normalized size = 2. \begin{align*}{\frac{{x}^{4}}{2\,c}{\frac{1}{\sqrt{c{x}^{4}+b{x}^{2}+a}}}}+{\frac{3\,b{x}^{2}}{4\,{c}^{2}}{\frac{1}{\sqrt{c{x}^{4}+b{x}^{2}+a}}}}-{\frac{3\,{b}^{2}}{8\,{c}^{3}}{\frac{1}{\sqrt{c{x}^{4}+b{x}^{2}+a}}}}-{\frac{3\,{b}^{3}{x}^{2}}{4\,{c}^{2} \left ( 4\,ac-{b}^{2} \right ) }{\frac{1}{\sqrt{c{x}^{4}+b{x}^{2}+a}}}}-{\frac{3\,{b}^{4}}{8\,{c}^{3} \left ( 4\,ac-{b}^{2} \right ) }{\frac{1}{\sqrt{c{x}^{4}+b{x}^{2}+a}}}}-{\frac{3\,b}{4}\ln \left ({ \left ({\frac{b}{2}}+c{x}^{2} \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{4}+b{x}^{2}+a} \right ){c}^{-{\frac{5}{2}}}}+{\frac{a}{{c}^{2}}{\frac{1}{\sqrt{c{x}^{4}+b{x}^{2}+a}}}}+2\,{\frac{ab{x}^{2}}{c \left ( 4\,ac-{b}^{2} \right ) \sqrt{c{x}^{4}+b{x}^{2}+a}}}+{\frac{{b}^{2}a}{{c}^{2} \left ( 4\,ac-{b}^{2} \right ) }{\frac{1}{\sqrt{c{x}^{4}+b{x}^{2}+a}}}} \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 = 2.01745, size = 980, normalized size = 7.31 \begin{align*} \left [\frac{3 \,{\left ({\left (b^{3} c - 4 \, a b c^{2}\right )} x^{4} + a b^{3} - 4 \, a^{2} b c +{\left (b^{4} - 4 \, a b^{2} c\right )} x^{2}\right )} \sqrt{c} \log \left (-8 \, c^{2} x^{4} - 8 \, b c x^{2} - b^{2} + 4 \, \sqrt{c x^{4} + b x^{2} + a}{\left (2 \, c x^{2} + b\right )} \sqrt{c} - 4 \, a c\right ) + 4 \,{\left ({\left (b^{2} c^{2} - 4 \, a c^{3}\right )} x^{4} + 3 \, a b^{2} c - 8 \, a^{2} c^{2} +{\left (3 \, b^{3} c - 10 \, a b c^{2}\right )} x^{2}\right )} \sqrt{c x^{4} + b x^{2} + a}}{8 \,{\left (a b^{2} c^{3} - 4 \, a^{2} c^{4} +{\left (b^{2} c^{4} - 4 \, a c^{5}\right )} x^{4} +{\left (b^{3} c^{3} - 4 \, a b c^{4}\right )} x^{2}\right )}}, \frac{3 \,{\left ({\left (b^{3} c - 4 \, a b c^{2}\right )} x^{4} + a b^{3} - 4 \, a^{2} b c +{\left (b^{4} - 4 \, a b^{2} c\right )} x^{2}\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{c x^{4} + b x^{2} + a}{\left (2 \, c x^{2} + b\right )} \sqrt{-c}}{2 \,{\left (c^{2} x^{4} + b c x^{2} + a c\right )}}\right ) + 2 \,{\left ({\left (b^{2} c^{2} - 4 \, a c^{3}\right )} x^{4} + 3 \, a b^{2} c - 8 \, a^{2} c^{2} +{\left (3 \, b^{3} c - 10 \, a b c^{2}\right )} x^{2}\right )} \sqrt{c x^{4} + b x^{2} + a}}{4 \,{\left (a b^{2} c^{3} - 4 \, a^{2} c^{4} +{\left (b^{2} c^{4} - 4 \, a c^{5}\right )} x^{4} +{\left (b^{3} c^{3} - 4 \, a b c^{4}\right )} x^{2}\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 (a + b x^{2} + c x^{4}\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.37653, size = 359, normalized size = 2.68 \begin{align*} \frac{{\left (\frac{{\left (b^{4} c - 8 \, a b^{2} c^{2} + 16 \, a^{2} c^{3}\right )} x^{2}}{b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}} + \frac{3 \, b^{5} - 22 \, a b^{3} c + 40 \, a^{2} b c^{2}}{b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}}\right )} x^{2} + \frac{3 \, a b^{4} - 20 \, a^{2} b^{2} c + 32 \, a^{3} c^{2}}{b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}}}{2 \, \sqrt{c x^{4} + b x^{2} + a}} + \frac{3 \,{\left (b^{5} - 8 \, a b^{3} c + 16 \, a^{2} b c^{2}\right )} \log \left ({\left | -2 \,{\left (\sqrt{c} x^{2} - \sqrt{c x^{4} + b x^{2} + a}\right )} \sqrt{c} - b \right |}\right )}{4 \,{\left (b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} \sqrt{c}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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