Optimal. Leaf size=189 \[ \frac{-2 a^2 d^2-2 a b c d+5 b^2 c^2}{2 a^3 c^2 x (b c-a d)}+\frac{b^{5/2} (5 b c-7 a d) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{7/2} (b c-a d)^2}-\frac{5 b c-2 a d}{6 a^2 c x^3 (b c-a d)}+\frac{d^{7/2} \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{c^{5/2} (b c-a d)^2}+\frac{b}{2 a x^3 \left (a+b x^2\right ) (b c-a d)} \]
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Rubi [A] time = 0.277482, antiderivative size = 189, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {472, 583, 522, 205} \[ \frac{-2 a^2 d^2-2 a b c d+5 b^2 c^2}{2 a^3 c^2 x (b c-a d)}+\frac{b^{5/2} (5 b c-7 a d) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{7/2} (b c-a d)^2}-\frac{5 b c-2 a d}{6 a^2 c x^3 (b c-a d)}+\frac{d^{7/2} \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{c^{5/2} (b c-a d)^2}+\frac{b}{2 a x^3 \left (a+b x^2\right ) (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 472
Rule 583
Rule 522
Rule 205
Rubi steps
\begin{align*} \int \frac{1}{x^4 \left (a+b x^2\right )^2 \left (c+d x^2\right )} \, dx &=\frac{b}{2 a (b c-a d) x^3 \left (a+b x^2\right )}-\frac{\int \frac{-5 b c+2 a d-5 b d x^2}{x^4 \left (a+b x^2\right ) \left (c+d x^2\right )} \, dx}{2 a (b c-a d)}\\ &=-\frac{5 b c-2 a d}{6 a^2 c (b c-a d) x^3}+\frac{b}{2 a (b c-a d) x^3 \left (a+b x^2\right )}+\frac{\int \frac{-3 \left (5 b^2 c^2-2 a b c d-2 a^2 d^2\right )-3 b d (5 b c-2 a d) x^2}{x^2 \left (a+b x^2\right ) \left (c+d x^2\right )} \, dx}{6 a^2 c (b c-a d)}\\ &=-\frac{5 b c-2 a d}{6 a^2 c (b c-a d) x^3}+\frac{5 b^2 c^2-2 a b c d-2 a^2 d^2}{2 a^3 c^2 (b c-a d) x}+\frac{b}{2 a (b c-a d) x^3 \left (a+b x^2\right )}-\frac{\int \frac{-3 \left (5 b^3 c^3-2 a b^2 c^2 d-2 a^2 b c d^2-2 a^3 d^3\right )-3 b d \left (5 b^2 c^2-2 a b c d-2 a^2 d^2\right ) x^2}{\left (a+b x^2\right ) \left (c+d x^2\right )} \, dx}{6 a^3 c^2 (b c-a d)}\\ &=-\frac{5 b c-2 a d}{6 a^2 c (b c-a d) x^3}+\frac{5 b^2 c^2-2 a b c d-2 a^2 d^2}{2 a^3 c^2 (b c-a d) x}+\frac{b}{2 a (b c-a d) x^3 \left (a+b x^2\right )}+\frac{d^4 \int \frac{1}{c+d x^2} \, dx}{c^2 (b c-a d)^2}+\frac{\left (b^3 (5 b c-7 a d)\right ) \int \frac{1}{a+b x^2} \, dx}{2 a^3 (b c-a d)^2}\\ &=-\frac{5 b c-2 a d}{6 a^2 c (b c-a d) x^3}+\frac{5 b^2 c^2-2 a b c d-2 a^2 d^2}{2 a^3 c^2 (b c-a d) x}+\frac{b}{2 a (b c-a d) x^3 \left (a+b x^2\right )}+\frac{b^{5/2} (5 b c-7 a d) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{7/2} (b c-a d)^2}+\frac{d^{7/2} \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{c^{5/2} (b c-a d)^2}\\ \end{align*}
Mathematica [A] time = 0.270661, size = 142, normalized size = 0.75 \[ -\frac{b^3 x}{2 a^3 \left (a+b x^2\right ) (a d-b c)}-\frac{b^{5/2} (7 a d-5 b c) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{7/2} (a d-b c)^2}+\frac{a d+2 b c}{a^3 c^2 x}-\frac{1}{3 a^2 c x^3}+\frac{d^{7/2} \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{c^{5/2} (b c-a d)^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.015, size = 191, normalized size = 1. \begin{align*}{\frac{{d}^{4}}{{c}^{2} \left ( ad-bc \right ) ^{2}}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}-{\frac{1}{3\,{a}^{2}c{x}^{3}}}+{\frac{d}{{a}^{2}{c}^{2}x}}+2\,{\frac{b}{{a}^{3}cx}}-{\frac{{b}^{3}xd}{2\,{a}^{2} \left ( ad-bc \right ) ^{2} \left ( b{x}^{2}+a \right ) }}+{\frac{{b}^{4}xc}{2\,{a}^{3} \left ( ad-bc \right ) ^{2} \left ( b{x}^{2}+a \right ) }}-{\frac{7\,{b}^{3}d}{2\,{a}^{2} \left ( ad-bc \right ) ^{2}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{5\,{b}^{4}c}{2\,{a}^{3} \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 = 7.85329, size = 2558, normalized size = 13.53 \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 = 1.16141, size = 223, normalized size = 1.18 \begin{align*} \frac{d^{4} \arctan \left (\frac{d x}{\sqrt{c d}}\right )}{{\left (b^{2} c^{4} - 2 \, a b c^{3} d + a^{2} c^{2} d^{2}\right )} \sqrt{c d}} + \frac{b^{3} x}{2 \,{\left (a^{3} b c - a^{4} d\right )}{\left (b x^{2} + a\right )}} + \frac{{\left (5 \, b^{4} c - 7 \, a b^{3} d\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{2 \,{\left (a^{3} b^{2} c^{2} - 2 \, a^{4} b c d + a^{5} d^{2}\right )} \sqrt{a b}} + \frac{6 \, b c x^{2} + 3 \, a d x^{2} - a c}{3 \, a^{3} c^{2} x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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