Optimal. Leaf size=112 \[ \frac{b^{5/2} \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{2 a^{5/2} (b c-a d)}+\frac{a d+b c}{2 a^2 c^2 x^2}-\frac{d^{5/2} \tan ^{-1}\left (\frac{\sqrt{d} x^2}{\sqrt{c}}\right )}{2 c^{5/2} (b c-a d)}-\frac{1}{6 a c x^6} \]
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Rubi [A] time = 0.218099, antiderivative size = 112, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227, Rules used = {465, 480, 583, 522, 205} \[ \frac{b^{5/2} \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{2 a^{5/2} (b c-a d)}+\frac{a d+b c}{2 a^2 c^2 x^2}-\frac{d^{5/2} \tan ^{-1}\left (\frac{\sqrt{d} x^2}{\sqrt{c}}\right )}{2 c^{5/2} (b c-a d)}-\frac{1}{6 a c x^6} \]
Antiderivative was successfully verified.
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Rule 465
Rule 480
Rule 583
Rule 522
Rule 205
Rubi steps
\begin{align*} \int \frac{1}{x^7 \left (a+b x^4\right ) \left (c+d x^4\right )} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{x^4 \left (a+b x^2\right ) \left (c+d x^2\right )} \, dx,x,x^2\right )\\ &=-\frac{1}{6 a c x^6}+\frac{\operatorname{Subst}\left (\int \frac{-3 (b c+a d)-3 b d x^2}{x^2 \left (a+b x^2\right ) \left (c+d x^2\right )} \, dx,x,x^2\right )}{6 a c}\\ &=-\frac{1}{6 a c x^6}+\frac{b c+a d}{2 a^2 c^2 x^2}-\frac{\operatorname{Subst}\left (\int \frac{-3 \left (b^2 c^2+a b c d+a^2 d^2\right )-3 b d (b c+a d) x^2}{\left (a+b x^2\right ) \left (c+d x^2\right )} \, dx,x,x^2\right )}{6 a^2 c^2}\\ &=-\frac{1}{6 a c x^6}+\frac{b c+a d}{2 a^2 c^2 x^2}+\frac{b^3 \operatorname{Subst}\left (\int \frac{1}{a+b x^2} \, dx,x,x^2\right )}{2 a^2 (b c-a d)}-\frac{d^3 \operatorname{Subst}\left (\int \frac{1}{c+d x^2} \, dx,x,x^2\right )}{2 c^2 (b c-a d)}\\ &=-\frac{1}{6 a c x^6}+\frac{b c+a d}{2 a^2 c^2 x^2}+\frac{b^{5/2} \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{2 a^{5/2} (b c-a d)}-\frac{d^{5/2} \tan ^{-1}\left (\frac{\sqrt{d} x^2}{\sqrt{c}}\right )}{2 c^{5/2} (b c-a d)}\\ \end{align*}
Mathematica [A] time = 0.213169, size = 193, normalized size = 1.72 \[ \frac{-\frac{3 b^2 x^4}{a^2}+\frac{3 b^{5/2} x^6 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{a^{5/2}}+\frac{3 b^{5/2} x^6 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{a^{5/2}}+\frac{b}{a}+\frac{3 d^2 x^4}{c^2}-\frac{3 d^{5/2} x^6 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )}{c^{5/2}}-\frac{3 d^{5/2} x^6 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} x}{\sqrt [4]{c}}+1\right )}{c^{5/2}}-\frac{d}{c}}{6 x^6 (a d-b c)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.012, size = 105, normalized size = 0.9 \begin{align*}{\frac{{d}^{3}}{2\,{c}^{2} \left ( ad-bc \right ) }\arctan \left ({{x}^{2}d{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}-{\frac{{b}^{3}}{2\,{a}^{2} \left ( ad-bc \right ) }\arctan \left ({b{x}^{2}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{1}{6\,ac{x}^{6}}}+{\frac{d}{2\,{c}^{2}a{x}^{2}}}+{\frac{b}{2\,{a}^{2}c{x}^{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 = 13.4613, size = 1191, normalized size = 10.63 \begin{align*} \left [-\frac{3 \, b^{2} c^{2} x^{6} \sqrt{-\frac{b}{a}} \log \left (\frac{b x^{4} - 2 \, a x^{2} \sqrt{-\frac{b}{a}} - a}{b x^{4} + a}\right ) + 3 \, a^{2} d^{2} x^{6} \sqrt{-\frac{d}{c}} \log \left (\frac{d x^{4} + 2 \, c x^{2} \sqrt{-\frac{d}{c}} - c}{d x^{4} + c}\right ) - 6 \,{\left (b^{2} c^{2} - a^{2} d^{2}\right )} x^{4} + 2 \, a b c^{2} - 2 \, a^{2} c d}{12 \,{\left (a^{2} b c^{3} - a^{3} c^{2} d\right )} x^{6}}, \frac{6 \, a^{2} d^{2} x^{6} \sqrt{\frac{d}{c}} \arctan \left (\frac{c \sqrt{\frac{d}{c}}}{d x^{2}}\right ) - 3 \, b^{2} c^{2} x^{6} \sqrt{-\frac{b}{a}} \log \left (\frac{b x^{4} - 2 \, a x^{2} \sqrt{-\frac{b}{a}} - a}{b x^{4} + a}\right ) + 6 \,{\left (b^{2} c^{2} - a^{2} d^{2}\right )} x^{4} - 2 \, a b c^{2} + 2 \, a^{2} c d}{12 \,{\left (a^{2} b c^{3} - a^{3} c^{2} d\right )} x^{6}}, -\frac{6 \, b^{2} c^{2} x^{6} \sqrt{\frac{b}{a}} \arctan \left (\frac{a \sqrt{\frac{b}{a}}}{b x^{2}}\right ) + 3 \, a^{2} d^{2} x^{6} \sqrt{-\frac{d}{c}} \log \left (\frac{d x^{4} + 2 \, c x^{2} \sqrt{-\frac{d}{c}} - c}{d x^{4} + c}\right ) - 6 \,{\left (b^{2} c^{2} - a^{2} d^{2}\right )} x^{4} + 2 \, a b c^{2} - 2 \, a^{2} c d}{12 \,{\left (a^{2} b c^{3} - a^{3} c^{2} d\right )} x^{6}}, -\frac{3 \, b^{2} c^{2} x^{6} \sqrt{\frac{b}{a}} \arctan \left (\frac{a \sqrt{\frac{b}{a}}}{b x^{2}}\right ) - 3 \, a^{2} d^{2} x^{6} \sqrt{\frac{d}{c}} \arctan \left (\frac{c \sqrt{\frac{d}{c}}}{d x^{2}}\right ) - 3 \,{\left (b^{2} c^{2} - a^{2} d^{2}\right )} x^{4} + a b c^{2} - a^{2} c d}{6 \,{\left (a^{2} b c^{3} - a^{3} c^{2} d\right )} x^{6}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.69473, size = 736, normalized size = 6.57 \begin{align*} -\frac{{\left (\sqrt{c d} a^{2} b^{3} c^{3} d x^{4}{\left | d \right |} + \sqrt{c d} a^{3} b^{2} c^{2} d^{2} x^{4}{\left | d \right |} + \sqrt{c d} a^{2} b^{3} c^{4}{\left | d \right |} + \sqrt{c d} a^{3} b^{2} c^{3} d{\left | d \right |} + \sqrt{c d} a^{4} b c^{2} d^{2}{\left | d \right |}\right )} \arctan \left (\frac{2 \, x^{2}}{\sqrt{\frac{2 \, a^{2} b c^{3} + 2 \, a^{3} c^{2} d + \sqrt{-16 \, a^{5} b c^{5} d + 4 \,{\left (a^{2} b c^{3} + a^{3} c^{2} d\right )}^{2}}}{a^{2} b c^{2} d}}}\right )}{a^{2} b c^{3} d{\left | a^{2} b c^{3} - a^{3} c^{2} d \right |} + a^{3} c^{2} d^{2}{\left | a^{2} b c^{3} - a^{3} c^{2} d \right |} +{\left (a^{2} b c^{3} - a^{3} c^{2} d\right )}^{2} d} + \frac{{\left (\sqrt{a b} a^{2} b^{2} c^{3} d^{2} x^{4}{\left | b \right |} + \sqrt{a b} a^{3} b c^{2} d^{3} x^{4}{\left | b \right |} + \sqrt{a b} a^{2} b^{2} c^{4} d{\left | b \right |} + \sqrt{a b} a^{3} b c^{3} d^{2}{\left | b \right |} + \sqrt{a b} a^{4} c^{2} d^{3}{\left | b \right |}\right )} \arctan \left (\frac{2 \, x^{2}}{\sqrt{\frac{2 \, a^{2} b c^{3} + 2 \, a^{3} c^{2} d - \sqrt{-16 \, a^{5} b c^{5} d + 4 \,{\left (a^{2} b c^{3} + a^{3} c^{2} d\right )}^{2}}}{a^{2} b c^{2} d}}}\right )}{a^{2} b^{2} c^{3}{\left | a^{2} b c^{3} - a^{3} c^{2} d \right |} + a^{3} b c^{2} d{\left | a^{2} b c^{3} - a^{3} c^{2} d \right |} -{\left (a^{2} b c^{3} - a^{3} c^{2} d\right )}^{2} b} + \frac{3 \, b c x^{4} + 3 \, a d x^{4} - a c}{6 \, a^{2} c^{2} x^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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