Optimal. Leaf size=177 \[ -\frac{\sqrt{a+b x^2+c x^4} \left (-16 a A c-18 a b B+15 A b^2\right )}{48 a^3 x^2}+\frac{\left (8 a^2 B c-12 a A b c-6 a b^2 B+5 A b^3\right ) \tanh ^{-1}\left (\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right )}{32 a^{7/2}}+\frac{(5 A b-6 a B) \sqrt{a+b x^2+c x^4}}{24 a^2 x^4}-\frac{A \sqrt{a+b x^2+c x^4}}{6 a x^6} \]
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Rubi [A] time = 0.238614, antiderivative size = 177, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185, Rules used = {1251, 834, 806, 724, 206} \[ -\frac{\sqrt{a+b x^2+c x^4} \left (-16 a A c-18 a b B+15 A b^2\right )}{48 a^3 x^2}+\frac{\left (8 a^2 B c-12 a A b c-6 a b^2 B+5 A b^3\right ) \tanh ^{-1}\left (\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right )}{32 a^{7/2}}+\frac{(5 A b-6 a B) \sqrt{a+b x^2+c x^4}}{24 a^2 x^4}-\frac{A \sqrt{a+b x^2+c x^4}}{6 a x^6} \]
Antiderivative was successfully verified.
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Rule 1251
Rule 834
Rule 806
Rule 724
Rule 206
Rubi steps
\begin{align*} \int \frac{A+B x^2}{x^7 \sqrt{a+b x^2+c x^4}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{A+B x}{x^4 \sqrt{a+b x+c x^2}} \, dx,x,x^2\right )\\ &=-\frac{A \sqrt{a+b x^2+c x^4}}{6 a x^6}-\frac{\operatorname{Subst}\left (\int \frac{\frac{1}{2} (5 A b-6 a B)+2 A c x}{x^3 \sqrt{a+b x+c x^2}} \, dx,x,x^2\right )}{6 a}\\ &=-\frac{A \sqrt{a+b x^2+c x^4}}{6 a x^6}+\frac{(5 A b-6 a 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 A b^2-18 a b B-16 a A c\right )+\frac{1}{2} (5 A b-6 a B) c x}{x^2 \sqrt{a+b x+c x^2}} \, dx,x,x^2\right )}{12 a^2}\\ &=-\frac{A \sqrt{a+b x^2+c x^4}}{6 a x^6}+\frac{(5 A b-6 a B) \sqrt{a+b x^2+c x^4}}{24 a^2 x^4}-\frac{\left (15 A b^2-18 a b B-16 a A c\right ) \sqrt{a+b x^2+c x^4}}{48 a^3 x^2}-\frac{\left (5 A b^3-6 a b^2 B-12 a A b c+8 a^2 B c\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x+c x^2}} \, dx,x,x^2\right )}{32 a^3}\\ &=-\frac{A \sqrt{a+b x^2+c x^4}}{6 a x^6}+\frac{(5 A b-6 a B) \sqrt{a+b x^2+c x^4}}{24 a^2 x^4}-\frac{\left (15 A b^2-18 a b B-16 a A c\right ) \sqrt{a+b x^2+c x^4}}{48 a^3 x^2}+\frac{\left (5 A b^3-6 a b^2 B-12 a A b c+8 a^2 B c\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{A \sqrt{a+b x^2+c x^4}}{6 a x^6}+\frac{(5 A b-6 a B) \sqrt{a+b x^2+c x^4}}{24 a^2 x^4}-\frac{\left (15 A b^2-18 a b B-16 a A c\right ) \sqrt{a+b x^2+c x^4}}{48 a^3 x^2}+\frac{\left (5 A b^3-6 a b^2 B-12 a A b c+8 a^2 B c\right ) \tanh ^{-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 = 0.115826, size = 148, normalized size = 0.84 \[ \frac{\sqrt{a+b x^2+c x^4} \left (-4 a^2 \left (2 A+3 B x^2\right )+2 a \left (5 A b x^2+8 A c x^4+9 b B x^4\right )-15 A b^2 x^4\right )}{48 a^3 x^6}+\frac{\left (8 a^2 B c-12 a A b c-6 a b^2 B+5 A b^3\right ) \tanh ^{-1}\left (\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right )}{32 a^{7/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.019, size = 311, normalized size = 1.8 \begin{align*} -{\frac{B}{4\,a{x}^{4}}\sqrt{c{x}^{4}+b{x}^{2}+a}}+{\frac{3\,bB}{8\,{a}^{2}{x}^{2}}\sqrt{c{x}^{4}+b{x}^{2}+a}}-{\frac{3\,{b}^{2}B}{16}\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 ){a}^{-{\frac{5}{2}}}}+{\frac{Bc}{4}\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 ){a}^{-{\frac{3}{2}}}}-{\frac{A}{6\,a{x}^{6}}\sqrt{c{x}^{4}+b{x}^{2}+a}}+{\frac{5\,Ab}{24\,{a}^{2}{x}^{4}}\sqrt{c{x}^{4}+b{x}^{2}+a}}-{\frac{5\,A{b}^{2}}{16\,{a}^{3}{x}^{2}}\sqrt{c{x}^{4}+b{x}^{2}+a}}+{\frac{5\,A{b}^{3}}{32}\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 ){a}^{-{\frac{7}{2}}}}-{\frac{3\,Abc}{8}\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 ){a}^{-{\frac{5}{2}}}}+{\frac{Ac}{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 = 3.14775, size = 779, normalized size = 4.4 \begin{align*} \left [\frac{3 \,{\left (6 \, B a b^{2} - 5 \, A b^{3} - 4 \,{\left (2 \, B a^{2} - 3 \, A a b\right )} 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 ({\left (18 \, B a^{2} b - 15 \, A a b^{2} + 16 \, A a^{2} c\right )} x^{4} - 8 \, A a^{3} - 2 \,{\left (6 \, B a^{3} - 5 \, A a^{2} b\right )} x^{2}\right )} \sqrt{c x^{4} + b x^{2} + a}}{192 \, a^{4} x^{6}}, \frac{3 \,{\left (6 \, B a b^{2} - 5 \, A b^{3} - 4 \,{\left (2 \, B a^{2} - 3 \, A a b\right )} 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 ({\left (18 \, B a^{2} b - 15 \, A a b^{2} + 16 \, A a^{2} c\right )} x^{4} - 8 \, A a^{3} - 2 \,{\left (6 \, B a^{3} - 5 \, A a^{2} b\right )} x^{2}\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{A + B x^{2}}{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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{B x^{2} + A}{\sqrt{c x^{4} + b x^{2} + a} x^{7}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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