Optimal. Leaf size=161 \[ \frac{x \left (7 a^2 d^2-6 a b c d+3 b^2 c^2\right )}{12 c^3 \left (c+d x^2\right )^2}+\frac{\left (35 a^2 d^2-30 a b c d+3 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{8 c^{9/2} \sqrt{d}}-\frac{a^2}{3 c x^3 \left (c+d x^2\right )^2}+\frac{x (3 b c-7 a d)^2}{24 c^4 \left (c+d x^2\right )}-\frac{a (6 b c-7 a d)}{3 c^4 x} \]
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Rubi [A] time = 0.191423, antiderivative size = 161, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {462, 456, 453, 205} \[ \frac{x \left (7 a^2 d^2-6 a b c d+3 b^2 c^2\right )}{12 c^3 \left (c+d x^2\right )^2}+\frac{\left (35 a^2 d^2-30 a b c d+3 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{8 c^{9/2} \sqrt{d}}-\frac{a^2}{3 c x^3 \left (c+d x^2\right )^2}+\frac{x (3 b c-7 a d)^2}{24 c^4 \left (c+d x^2\right )}-\frac{a (6 b c-7 a d)}{3 c^4 x} \]
Antiderivative was successfully verified.
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Rule 462
Rule 456
Rule 453
Rule 205
Rubi steps
\begin{align*} \int \frac{\left (a+b x^2\right )^2}{x^4 \left (c+d x^2\right )^3} \, dx &=-\frac{a^2}{3 c x^3 \left (c+d x^2\right )^2}+\frac{\int \frac{a (6 b c-7 a d)+3 b^2 c x^2}{x^2 \left (c+d x^2\right )^3} \, dx}{3 c}\\ &=-\frac{a^2}{3 c x^3 \left (c+d x^2\right )^2}+\frac{\left (3 b^2 c^2-6 a b c d+7 a^2 d^2\right ) x}{12 c^3 \left (c+d x^2\right )^2}-\frac{\int \frac{-\frac{4 a (6 b c-7 a d)}{c}-3 \left (3 b^2-\frac{6 a b d}{c}+\frac{7 a^2 d^2}{c^2}\right ) x^2}{x^2 \left (c+d x^2\right )^2} \, dx}{12 c}\\ &=-\frac{a^2}{3 c x^3 \left (c+d x^2\right )^2}+\frac{\left (3 b^2 c^2-6 a b c d+7 a^2 d^2\right ) x}{12 c^3 \left (c+d x^2\right )^2}+\frac{(3 b c-7 a d)^2 x}{24 c^4 \left (c+d x^2\right )}+\frac{\int \frac{\frac{8 a (6 b c-7 a d)}{c^2}+\frac{(3 b c-7 a d)^2 x^2}{c^3}}{x^2 \left (c+d x^2\right )} \, dx}{24 c}\\ &=-\frac{a (6 b c-7 a d)}{3 c^4 x}-\frac{a^2}{3 c x^3 \left (c+d x^2\right )^2}+\frac{\left (3 b^2 c^2-6 a b c d+7 a^2 d^2\right ) x}{12 c^3 \left (c+d x^2\right )^2}+\frac{(3 b c-7 a d)^2 x}{24 c^4 \left (c+d x^2\right )}+\frac{\left (3 b^2 c^2-30 a b c d+35 a^2 d^2\right ) \int \frac{1}{c+d x^2} \, dx}{8 c^4}\\ &=-\frac{a (6 b c-7 a d)}{3 c^4 x}-\frac{a^2}{3 c x^3 \left (c+d x^2\right )^2}+\frac{\left (3 b^2 c^2-6 a b c d+7 a^2 d^2\right ) x}{12 c^3 \left (c+d x^2\right )^2}+\frac{(3 b c-7 a d)^2 x}{24 c^4 \left (c+d x^2\right )}+\frac{\left (3 b^2 c^2-30 a b c d+35 a^2 d^2\right ) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{8 c^{9/2} \sqrt{d}}\\ \end{align*}
Mathematica [A] time = 0.0724567, size = 148, normalized size = 0.92 \[ \frac{x \left (11 a^2 d^2-14 a b c d+3 b^2 c^2\right )}{8 c^4 \left (c+d x^2\right )}+\frac{\left (35 a^2 d^2-30 a b c d+3 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{8 c^{9/2} \sqrt{d}}-\frac{a^2}{3 c^3 x^3}+\frac{x (b c-a d)^2}{4 c^3 \left (c+d x^2\right )^2}+\frac{a (3 a d-2 b c)}{c^4 x} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.014, size = 227, normalized size = 1.4 \begin{align*}{\frac{11\,{x}^{3}{a}^{2}{d}^{3}}{8\,{c}^{4} \left ( d{x}^{2}+c \right ) ^{2}}}-{\frac{7\,{x}^{3}ab{d}^{2}}{4\,{c}^{3} \left ( d{x}^{2}+c \right ) ^{2}}}+{\frac{3\,{b}^{2}d{x}^{3}}{8\,{c}^{2} \left ( d{x}^{2}+c \right ) ^{2}}}+{\frac{13\,{a}^{2}{d}^{2}x}{8\,{c}^{3} \left ( d{x}^{2}+c \right ) ^{2}}}-{\frac{9\,abdx}{4\,{c}^{2} \left ( d{x}^{2}+c \right ) ^{2}}}+{\frac{5\,{b}^{2}x}{8\,c \left ( d{x}^{2}+c \right ) ^{2}}}+{\frac{35\,{a}^{2}{d}^{2}}{8\,{c}^{4}}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}-{\frac{15\,abd}{4\,{c}^{3}}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}+{\frac{3\,{b}^{2}}{8\,{c}^{2}}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}-{\frac{{a}^{2}}{3\,{c}^{3}{x}^{3}}}+3\,{\frac{{a}^{2}d}{{c}^{4}x}}-2\,{\frac{ab}{{c}^{3}x}} \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 = 1.55762, size = 1131, normalized size = 7.02 \begin{align*} \left [-\frac{16 \, a^{2} c^{4} d - 6 \,{\left (3 \, b^{2} c^{3} d^{2} - 30 \, a b c^{2} d^{3} + 35 \, a^{2} c d^{4}\right )} x^{6} - 10 \,{\left (3 \, b^{2} c^{4} d - 30 \, a b c^{3} d^{2} + 35 \, a^{2} c^{2} d^{3}\right )} x^{4} + 16 \,{\left (6 \, a b c^{4} d - 7 \, a^{2} c^{3} d^{2}\right )} x^{2} + 3 \,{\left ({\left (3 \, b^{2} c^{2} d^{2} - 30 \, a b c d^{3} + 35 \, a^{2} d^{4}\right )} x^{7} + 2 \,{\left (3 \, b^{2} c^{3} d - 30 \, a b c^{2} d^{2} + 35 \, a^{2} c d^{3}\right )} x^{5} +{\left (3 \, b^{2} c^{4} - 30 \, a b c^{3} d + 35 \, a^{2} c^{2} d^{2}\right )} x^{3}\right )} \sqrt{-c d} \log \left (\frac{d x^{2} - 2 \, \sqrt{-c d} x - c}{d x^{2} + c}\right )}{48 \,{\left (c^{5} d^{3} x^{7} + 2 \, c^{6} d^{2} x^{5} + c^{7} d x^{3}\right )}}, -\frac{8 \, a^{2} c^{4} d - 3 \,{\left (3 \, b^{2} c^{3} d^{2} - 30 \, a b c^{2} d^{3} + 35 \, a^{2} c d^{4}\right )} x^{6} - 5 \,{\left (3 \, b^{2} c^{4} d - 30 \, a b c^{3} d^{2} + 35 \, a^{2} c^{2} d^{3}\right )} x^{4} + 8 \,{\left (6 \, a b c^{4} d - 7 \, a^{2} c^{3} d^{2}\right )} x^{2} - 3 \,{\left ({\left (3 \, b^{2} c^{2} d^{2} - 30 \, a b c d^{3} + 35 \, a^{2} d^{4}\right )} x^{7} + 2 \,{\left (3 \, b^{2} c^{3} d - 30 \, a b c^{2} d^{2} + 35 \, a^{2} c d^{3}\right )} x^{5} +{\left (3 \, b^{2} c^{4} - 30 \, a b c^{3} d + 35 \, a^{2} c^{2} d^{2}\right )} x^{3}\right )} \sqrt{c d} \arctan \left (\frac{\sqrt{c d} x}{c}\right )}{24 \,{\left (c^{5} d^{3} x^{7} + 2 \, c^{6} d^{2} x^{5} + c^{7} d x^{3}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.79279, size = 240, normalized size = 1.49 \begin{align*} - \frac{\sqrt{- \frac{1}{c^{9} d}} \left (35 a^{2} d^{2} - 30 a b c d + 3 b^{2} c^{2}\right ) \log{\left (- c^{5} \sqrt{- \frac{1}{c^{9} d}} + x \right )}}{16} + \frac{\sqrt{- \frac{1}{c^{9} d}} \left (35 a^{2} d^{2} - 30 a b c d + 3 b^{2} c^{2}\right ) \log{\left (c^{5} \sqrt{- \frac{1}{c^{9} d}} + x \right )}}{16} + \frac{- 8 a^{2} c^{3} + x^{6} \left (105 a^{2} d^{3} - 90 a b c d^{2} + 9 b^{2} c^{2} d\right ) + x^{4} \left (175 a^{2} c d^{2} - 150 a b c^{2} d + 15 b^{2} c^{3}\right ) + x^{2} \left (56 a^{2} c^{2} d - 48 a b c^{3}\right )}{24 c^{6} x^{3} + 48 c^{5} d x^{5} + 24 c^{4} d^{2} x^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18574, size = 204, normalized size = 1.27 \begin{align*} \frac{{\left (3 \, b^{2} c^{2} - 30 \, a b c d + 35 \, a^{2} d^{2}\right )} \arctan \left (\frac{d x}{\sqrt{c d}}\right )}{8 \, \sqrt{c d} c^{4}} + \frac{3 \, b^{2} c^{2} d x^{3} - 14 \, a b c d^{2} x^{3} + 11 \, a^{2} d^{3} x^{3} + 5 \, b^{2} c^{3} x - 18 \, a b c^{2} d x + 13 \, a^{2} c d^{2} x}{8 \,{\left (d x^{2} + c\right )}^{2} c^{4}} - \frac{6 \, a b c x^{2} - 9 \, a^{2} d x^{2} + a^{2} c}{3 \, c^{4} x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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