Optimal. Leaf size=250 \[ \frac{-2 a^2 d^2-2 a b c d+7 b^2 c^2}{6 a^3 c^2 x^3 (b c-a d)}-\frac{-2 a^2 b c d^2-2 a^3 d^3-2 a b^2 c^2 d+7 b^3 c^3}{2 a^4 c^3 x (b c-a d)}-\frac{b^{7/2} (7 b c-9 a d) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{9/2} (b c-a d)^2}-\frac{7 b c-2 a d}{10 a^2 c x^5 (b c-a d)}-\frac{d^{9/2} \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{c^{7/2} (b c-a d)^2}+\frac{b}{2 a x^5 \left (a+b x^2\right ) (b c-a d)} \]
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Rubi [A] time = 0.408762, antiderivative size = 250, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {472, 583, 522, 205} \[ \frac{-2 a^2 d^2-2 a b c d+7 b^2 c^2}{6 a^3 c^2 x^3 (b c-a d)}-\frac{-2 a^2 b c d^2-2 a^3 d^3-2 a b^2 c^2 d+7 b^3 c^3}{2 a^4 c^3 x (b c-a d)}-\frac{b^{7/2} (7 b c-9 a d) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{9/2} (b c-a d)^2}-\frac{7 b c-2 a d}{10 a^2 c x^5 (b c-a d)}-\frac{d^{9/2} \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{c^{7/2} (b c-a d)^2}+\frac{b}{2 a x^5 \left (a+b x^2\right ) (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 472
Rule 583
Rule 522
Rule 205
Rubi steps
\begin{align*} \int \frac{1}{x^6 \left (a+b x^2\right )^2 \left (c+d x^2\right )} \, dx &=\frac{b}{2 a (b c-a d) x^5 \left (a+b x^2\right )}-\frac{\int \frac{-7 b c+2 a d-7 b d x^2}{x^6 \left (a+b x^2\right ) \left (c+d x^2\right )} \, dx}{2 a (b c-a d)}\\ &=-\frac{7 b c-2 a d}{10 a^2 c (b c-a d) x^5}+\frac{b}{2 a (b c-a d) x^5 \left (a+b x^2\right )}+\frac{\int \frac{-5 \left (7 b^2 c^2-2 a b c d-2 a^2 d^2\right )-5 b d (7 b c-2 a d) x^2}{x^4 \left (a+b x^2\right ) \left (c+d x^2\right )} \, dx}{10 a^2 c (b c-a d)}\\ &=-\frac{7 b c-2 a d}{10 a^2 c (b c-a d) x^5}+\frac{7 b^2 c^2-2 a b c d-2 a^2 d^2}{6 a^3 c^2 (b c-a d) x^3}+\frac{b}{2 a (b c-a d) x^5 \left (a+b x^2\right )}-\frac{\int \frac{-15 \left (7 b^3 c^3-2 a b^2 c^2 d-2 a^2 b c d^2-2 a^3 d^3\right )-15 b d \left (7 b^2 c^2-2 a b c d-2 a^2 d^2\right ) x^2}{x^2 \left (a+b x^2\right ) \left (c+d x^2\right )} \, dx}{30 a^3 c^2 (b c-a d)}\\ &=-\frac{7 b c-2 a d}{10 a^2 c (b c-a d) x^5}+\frac{7 b^2 c^2-2 a b c d-2 a^2 d^2}{6 a^3 c^2 (b c-a d) x^3}-\frac{7 b^3 c^3-2 a b^2 c^2 d-2 a^2 b c d^2-2 a^3 d^3}{2 a^4 c^3 (b c-a d) x}+\frac{b}{2 a (b c-a d) x^5 \left (a+b x^2\right )}+\frac{\int \frac{-15 \left (7 b^4 c^4-2 a b^3 c^3 d-2 a^2 b^2 c^2 d^2-2 a^3 b c d^3-2 a^4 d^4\right )-15 b d \left (7 b^3 c^3-2 a b^2 c^2 d-2 a^2 b c d^2-2 a^3 d^3\right ) x^2}{\left (a+b x^2\right ) \left (c+d x^2\right )} \, dx}{30 a^4 c^3 (b c-a d)}\\ &=-\frac{7 b c-2 a d}{10 a^2 c (b c-a d) x^5}+\frac{7 b^2 c^2-2 a b c d-2 a^2 d^2}{6 a^3 c^2 (b c-a d) x^3}-\frac{7 b^3 c^3-2 a b^2 c^2 d-2 a^2 b c d^2-2 a^3 d^3}{2 a^4 c^3 (b c-a d) x}+\frac{b}{2 a (b c-a d) x^5 \left (a+b x^2\right )}-\frac{d^5 \int \frac{1}{c+d x^2} \, dx}{c^3 (b c-a d)^2}-\frac{\left (b^4 (7 b c-9 a d)\right ) \int \frac{1}{a+b x^2} \, dx}{2 a^4 (b c-a d)^2}\\ &=-\frac{7 b c-2 a d}{10 a^2 c (b c-a d) x^5}+\frac{7 b^2 c^2-2 a b c d-2 a^2 d^2}{6 a^3 c^2 (b c-a d) x^3}-\frac{7 b^3 c^3-2 a b^2 c^2 d-2 a^2 b c d^2-2 a^3 d^3}{2 a^4 c^3 (b c-a d) x}+\frac{b}{2 a (b c-a d) x^5 \left (a+b x^2\right )}-\frac{b^{7/2} (7 b c-9 a d) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{9/2} (b c-a d)^2}-\frac{d^{9/2} \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{c^{7/2} (b c-a d)^2}\\ \end{align*}
Mathematica [A] time = 0.304151, size = 179, normalized size = 0.72 \[ \frac{-a^2 d^2-2 a b c d-3 b^2 c^2}{a^4 c^3 x}+\frac{b^4 x}{2 a^4 \left (a+b x^2\right ) (a d-b c)}+\frac{b^{7/2} (9 a d-7 b c) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{9/2} (a d-b c)^2}+\frac{a d+2 b c}{3 a^3 c^2 x^3}-\frac{1}{5 a^2 c x^5}-\frac{d^{9/2} \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{c^{7/2} (b c-a d)^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.019, size = 234, normalized size = 0.9 \begin{align*} -{\frac{{d}^{5}}{{c}^{3} \left ( ad-bc \right ) ^{2}}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}-{\frac{1}{5\,{a}^{2}c{x}^{5}}}+{\frac{d}{3\,{a}^{2}{c}^{2}{x}^{3}}}+{\frac{2\,b}{3\,{a}^{3}c{x}^{3}}}-{\frac{{d}^{2}}{{a}^{2}{c}^{3}x}}-2\,{\frac{bd}{{a}^{3}{c}^{2}x}}-3\,{\frac{{b}^{2}}{{a}^{4}cx}}+{\frac{{b}^{4}xd}{2\,{a}^{3} \left ( ad-bc \right ) ^{2} \left ( b{x}^{2}+a \right ) }}-{\frac{{b}^{5}xc}{2\,{a}^{4} \left ( ad-bc \right ) ^{2} \left ( b{x}^{2}+a \right ) }}+{\frac{9\,{b}^{4}d}{2\,{a}^{3} \left ( ad-bc \right ) ^{2}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{7\,{b}^{5}c}{2\,{a}^{4} \left ( ad-bc \right ) ^{2}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \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 = 21.3758, size = 3002, normalized size = 12.01 \begin{align*} \text{result too large to display} \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 [A] time = 1.15089, size = 279, normalized size = 1.12 \begin{align*} -\frac{d^{5} \arctan \left (\frac{d x}{\sqrt{c d}}\right )}{{\left (b^{2} c^{5} - 2 \, a b c^{4} d + a^{2} c^{3} d^{2}\right )} \sqrt{c d}} - \frac{b^{4} x}{2 \,{\left (a^{4} b c - a^{5} d\right )}{\left (b x^{2} + a\right )}} - \frac{{\left (7 \, b^{5} c - 9 \, a b^{4} d\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{2 \,{\left (a^{4} b^{2} c^{2} - 2 \, a^{5} b c d + a^{6} d^{2}\right )} \sqrt{a b}} - \frac{45 \, b^{2} c^{2} x^{4} + 30 \, a b c d x^{4} + 15 \, a^{2} d^{2} x^{4} - 10 \, a b c^{2} x^{2} - 5 \, a^{2} c d x^{2} + 3 \, a^{2} c^{2}}{15 \, a^{4} c^{3} x^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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