Optimal. Leaf size=154 \[ \frac{\left (16 a c+15 b^2\right ) \sqrt{-a+b x^2+c x^4}}{48 a^3 x^2}-\frac{b \left (12 a c+5 b^2\right ) \tan ^{-1}\left (\frac{2 a-b x^2}{2 \sqrt{a} \sqrt{-a+b x^2+c x^4}}\right )}{32 a^{7/2}}+\frac{5 b \sqrt{-a+b x^2+c x^4}}{24 a^2 x^4}+\frac{\sqrt{-a+b x^2+c x^4}}{6 a x^6} \]
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Rubi [A] time = 0.171621, antiderivative size = 154, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {1114, 744, 834, 806, 724, 204} \[ \frac{\left (16 a c+15 b^2\right ) \sqrt{-a+b x^2+c x^4}}{48 a^3 x^2}-\frac{b \left (12 a c+5 b^2\right ) \tan ^{-1}\left (\frac{2 a-b x^2}{2 \sqrt{a} \sqrt{-a+b x^2+c x^4}}\right )}{32 a^{7/2}}+\frac{5 b \sqrt{-a+b x^2+c x^4}}{24 a^2 x^4}+\frac{\sqrt{-a+b x^2+c x^4}}{6 a x^6} \]
Antiderivative was successfully verified.
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Rule 1114
Rule 744
Rule 834
Rule 806
Rule 724
Rule 204
Rubi steps
\begin{align*} \int \frac{1}{x^7 \sqrt{-a+b x^2+c x^4}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{x^4 \sqrt{-a+b x+c x^2}} \, dx,x,x^2\right )\\ &=\frac{\sqrt{-a+b x^2+c x^4}}{6 a x^6}+\frac{\operatorname{Subst}\left (\int \frac{\frac{5 b}{2}+2 c x}{x^3 \sqrt{-a+b x+c x^2}} \, dx,x,x^2\right )}{6 a}\\ &=\frac{\sqrt{-a+b x^2+c x^4}}{6 a x^6}+\frac{5 b \sqrt{-a+b x^2+c x^4}}{24 a^2 x^4}+\frac{\operatorname{Subst}\left (\int \frac{\frac{1}{4} \left (15 b^2+16 a c\right )+\frac{5 b c x}{2}}{x^2 \sqrt{-a+b x+c x^2}} \, dx,x,x^2\right )}{12 a^2}\\ &=\frac{\sqrt{-a+b x^2+c x^4}}{6 a x^6}+\frac{5 b \sqrt{-a+b x^2+c x^4}}{24 a^2 x^4}+\frac{\left (15 b^2+16 a c\right ) \sqrt{-a+b x^2+c x^4}}{48 a^3 x^2}+\frac{\left (b \left (5 b^2+12 a c\right )\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{-a+b x+c x^2}} \, dx,x,x^2\right )}{32 a^3}\\ &=\frac{\sqrt{-a+b x^2+c x^4}}{6 a x^6}+\frac{5 b \sqrt{-a+b x^2+c x^4}}{24 a^2 x^4}+\frac{\left (15 b^2+16 a c\right ) \sqrt{-a+b x^2+c x^4}}{48 a^3 x^2}-\frac{\left (b \left (5 b^2+12 a c\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-4 a-x^2} \, dx,x,\frac{-2 a+b x^2}{\sqrt{-a+b x^2+c x^4}}\right )}{16 a^3}\\ &=\frac{\sqrt{-a+b x^2+c x^4}}{6 a x^6}+\frac{5 b \sqrt{-a+b x^2+c x^4}}{24 a^2 x^4}+\frac{\left (15 b^2+16 a c\right ) \sqrt{-a+b x^2+c x^4}}{48 a^3 x^2}-\frac{b \left (5 b^2+12 a c\right ) \tan ^{-1}\left (\frac{2 a-b x^2}{2 \sqrt{a} \sqrt{-a+b x^2+c x^4}}\right )}{32 a^{7/2}}\\ \end{align*}
Mathematica [A] time = 0.0767074, size = 116, normalized size = 0.75 \[ \frac{\sqrt{-a+b x^2+c x^4} \left (8 a^2+2 a \left (5 b x^2+8 c x^4\right )+15 b^2 x^4\right )}{48 a^3 x^6}+\frac{b \left (12 a c+5 b^2\right ) \tan ^{-1}\left (\frac{b x^2-2 a}{2 \sqrt{a} \sqrt{-a+b x^2+c x^4}}\right )}{32 a^{7/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.17, size = 202, normalized size = 1.3 \begin{align*}{\frac{1}{6\,a{x}^{6}}\sqrt{c{x}^{4}+b{x}^{2}-a}}+{\frac{5\,b}{24\,{a}^{2}{x}^{4}}\sqrt{c{x}^{4}+b{x}^{2}-a}}+{\frac{5\,{b}^{2}}{16\,{x}^{2}{a}^{3}}\sqrt{c{x}^{4}+b{x}^{2}-a}}-{\frac{5\,{b}^{3}}{32\,{a}^{3}}\ln \left ({\frac{1}{{x}^{2}} \left ( -2\,a+b{x}^{2}+2\,\sqrt{-a}\sqrt{c{x}^{4}+b{x}^{2}-a} \right ) } \right ){\frac{1}{\sqrt{-a}}}}-{\frac{3\,bc}{8\,{a}^{2}}\ln \left ({\frac{1}{{x}^{2}} \left ( -2\,a+b{x}^{2}+2\,\sqrt{-a}\sqrt{c{x}^{4}+b{x}^{2}-a} \right ) } \right ){\frac{1}{\sqrt{-a}}}}+{\frac{c}{3\,{a}^{2}{x}^{2}}\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 = 1.87892, size = 609, normalized size = 3.95 \begin{align*} \left [-\frac{3 \,{\left (5 \, b^{3} + 12 \, a b c\right )} \sqrt{-a} x^{6} \log \left (\frac{{\left (b^{2} - 4 \, a c\right )} x^{4} - 8 \, a b x^{2} - 4 \, \sqrt{c x^{4} + b x^{2} - a}{\left (b x^{2} - 2 \, a\right )} \sqrt{-a} + 8 \, a^{2}}{x^{4}}\right ) - 4 \,{\left (10 \, a^{2} b x^{2} +{\left (15 \, a b^{2} + 16 \, a^{2} c\right )} x^{4} + 8 \, a^{3}\right )} \sqrt{c x^{4} + b x^{2} - a}}{192 \, a^{4} x^{6}}, \frac{3 \,{\left (5 \, b^{3} + 12 \, a b c\right )} \sqrt{a} x^{6} \arctan \left (\frac{\sqrt{c x^{4} + b x^{2} - a}{\left (b x^{2} - 2 \, a\right )} \sqrt{a}}{2 \,{\left (a c x^{4} + a b x^{2} - a^{2}\right )}}\right ) + 2 \,{\left (10 \, a^{2} b x^{2} +{\left (15 \, a b^{2} + 16 \, a^{2} c\right )} x^{4} + 8 \, a^{3}\right )} \sqrt{c x^{4} + b x^{2} - a}}{96 \, a^{4} x^{6}}\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{1}{x^{7} \sqrt{- a + b x^{2} + c x^{4}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.79854, size = 158, normalized size = 1.03 \begin{align*} \frac{1}{48} \, \sqrt{c + \frac{b}{x^{2}} - \frac{a}{x^{4}}}{\left (\frac{2 \,{\left (\frac{5 \, b}{a^{2}} + \frac{4}{a x^{2}}\right )}}{x^{2}} + \frac{15 \, a b^{2} + 16 \, a^{2} c}{a^{4}}\right )} + \frac{{\left (5 \, a b^{3} + 12 \, a^{2} b c\right )} \log \left ({\left | -2 \, \sqrt{-a}{\left (\sqrt{c + \frac{b}{x^{2}} - \frac{a}{x^{4}}} - \frac{\sqrt{-a}}{x^{2}}\right )} + b \right |}\right )}{32 \, \sqrt{-a} a^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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