Optimal. Leaf size=147 \[ \frac{c \left (2 a^2 d^2-9 a b c d+5 b^2 c^2\right )}{2 a^3 b x}+\frac{(b c-a d)^2 (a d+5 b c) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{7/2} b^{3/2}}-\frac{c^2 (5 b c-3 a d)}{6 a^2 b x^3}+\frac{\left (c+d x^2\right )^2 (b c-a d)}{2 a b x^3 \left (a+b x^2\right )} \]
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Rubi [A] time = 0.140974, antiderivative size = 147, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {468, 570, 205} \[ \frac{c \left (2 a^2 d^2-9 a b c d+5 b^2 c^2\right )}{2 a^3 b x}+\frac{(b c-a d)^2 (a d+5 b c) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{7/2} b^{3/2}}-\frac{c^2 (5 b c-3 a d)}{6 a^2 b x^3}+\frac{\left (c+d x^2\right )^2 (b c-a d)}{2 a b x^3 \left (a+b x^2\right )} \]
Antiderivative was successfully verified.
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Rule 468
Rule 570
Rule 205
Rubi steps
\begin{align*} \int \frac{\left (c+d x^2\right )^3}{x^4 \left (a+b x^2\right )^2} \, dx &=\frac{(b c-a d) \left (c+d x^2\right )^2}{2 a b x^3 \left (a+b x^2\right )}-\frac{\int \frac{\left (c+d x^2\right ) \left (-c (5 b c-3 a d)-d (b c+a d) x^2\right )}{x^4 \left (a+b x^2\right )} \, dx}{2 a b}\\ &=\frac{(b c-a d) \left (c+d x^2\right )^2}{2 a b x^3 \left (a+b x^2\right )}-\frac{\int \left (\frac{c^2 (-5 b c+3 a d)}{a x^4}+\frac{c \left (5 b^2 c^2-9 a b c d+2 a^2 d^2\right )}{a^2 x^2}-\frac{(-b c+a d)^2 (5 b c+a d)}{a^2 \left (a+b x^2\right )}\right ) \, dx}{2 a b}\\ &=-\frac{c^2 (5 b c-3 a d)}{6 a^2 b x^3}+\frac{c \left (5 b^2 c^2-9 a b c d+2 a^2 d^2\right )}{2 a^3 b x}+\frac{(b c-a d) \left (c+d x^2\right )^2}{2 a b x^3 \left (a+b x^2\right )}+\frac{\left ((b c-a d)^2 (5 b c+a d)\right ) \int \frac{1}{a+b x^2} \, dx}{2 a^3 b}\\ &=-\frac{c^2 (5 b c-3 a d)}{6 a^2 b x^3}+\frac{c \left (5 b^2 c^2-9 a b c d+2 a^2 d^2\right )}{2 a^3 b x}+\frac{(b c-a d) \left (c+d x^2\right )^2}{2 a b x^3 \left (a+b x^2\right )}+\frac{(b c-a d)^2 (5 b c+a d) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{7/2} b^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0622429, size = 109, normalized size = 0.74 \[ \frac{(a d-b c)^2 (a d+5 b c) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{7/2} b^{3/2}}-\frac{c^2 (3 a d-2 b c)}{a^3 x}-\frac{x (a d-b c)^3}{2 a^3 b \left (a+b x^2\right )}-\frac{c^3}{3 a^2 x^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.011, size = 209, normalized size = 1.4 \begin{align*} -{\frac{{c}^{3}}{3\,{a}^{2}{x}^{3}}}-3\,{\frac{{c}^{2}d}{{a}^{2}x}}+2\,{\frac{{c}^{3}b}{{a}^{3}x}}-{\frac{x{d}^{3}}{2\,b \left ( b{x}^{2}+a \right ) }}+{\frac{3\,cx{d}^{2}}{2\,a \left ( b{x}^{2}+a \right ) }}-{\frac{3\,bx{c}^{2}d}{2\,{a}^{2} \left ( b{x}^{2}+a \right ) }}+{\frac{{b}^{2}{c}^{3}x}{2\,{a}^{3} \left ( b{x}^{2}+a \right ) }}+{\frac{{d}^{3}}{2\,b}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{3\,c{d}^{2}}{2\,a}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{9\,b{c}^{2}d}{2\,{a}^{2}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{5\,{b}^{2}{c}^{3}}{2\,{a}^{3}}\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 = 1.68138, size = 921, normalized size = 6.27 \begin{align*} \left [-\frac{4 \, a^{3} b^{2} c^{3} - 6 \,{\left (5 \, a b^{4} c^{3} - 9 \, a^{2} b^{3} c^{2} d + 3 \, a^{3} b^{2} c d^{2} - a^{4} b d^{3}\right )} x^{4} - 4 \,{\left (5 \, a^{2} b^{3} c^{3} - 9 \, a^{3} b^{2} c^{2} d\right )} x^{2} + 3 \,{\left ({\left (5 \, b^{4} c^{3} - 9 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} + a^{3} b d^{3}\right )} x^{5} +{\left (5 \, a b^{3} c^{3} - 9 \, a^{2} b^{2} c^{2} d + 3 \, a^{3} b c d^{2} + a^{4} d^{3}\right )} x^{3}\right )} \sqrt{-a b} \log \left (\frac{b x^{2} - 2 \, \sqrt{-a b} x - a}{b x^{2} + a}\right )}{12 \,{\left (a^{4} b^{3} x^{5} + a^{5} b^{2} x^{3}\right )}}, -\frac{2 \, a^{3} b^{2} c^{3} - 3 \,{\left (5 \, a b^{4} c^{3} - 9 \, a^{2} b^{3} c^{2} d + 3 \, a^{3} b^{2} c d^{2} - a^{4} b d^{3}\right )} x^{4} - 2 \,{\left (5 \, a^{2} b^{3} c^{3} - 9 \, a^{3} b^{2} c^{2} d\right )} x^{2} - 3 \,{\left ({\left (5 \, b^{4} c^{3} - 9 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} + a^{3} b d^{3}\right )} x^{5} +{\left (5 \, a b^{3} c^{3} - 9 \, a^{2} b^{2} c^{2} d + 3 \, a^{3} b c d^{2} + a^{4} d^{3}\right )} x^{3}\right )} \sqrt{a b} \arctan \left (\frac{\sqrt{a b} x}{a}\right )}{6 \,{\left (a^{4} b^{3} x^{5} + a^{5} b^{2} x^{3}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 2.2596, size = 321, normalized size = 2.18 \begin{align*} - \frac{\sqrt{- \frac{1}{a^{7} b^{3}}} \left (a d - b c\right )^{2} \left (a d + 5 b c\right ) \log{\left (- \frac{a^{4} b \sqrt{- \frac{1}{a^{7} b^{3}}} \left (a d - b c\right )^{2} \left (a d + 5 b c\right )}{a^{3} d^{3} + 3 a^{2} b c d^{2} - 9 a b^{2} c^{2} d + 5 b^{3} c^{3}} + x \right )}}{4} + \frac{\sqrt{- \frac{1}{a^{7} b^{3}}} \left (a d - b c\right )^{2} \left (a d + 5 b c\right ) \log{\left (\frac{a^{4} b \sqrt{- \frac{1}{a^{7} b^{3}}} \left (a d - b c\right )^{2} \left (a d + 5 b c\right )}{a^{3} d^{3} + 3 a^{2} b c d^{2} - 9 a b^{2} c^{2} d + 5 b^{3} c^{3}} + x \right )}}{4} - \frac{2 a^{2} b c^{3} + x^{4} \left (3 a^{3} d^{3} - 9 a^{2} b c d^{2} + 27 a b^{2} c^{2} d - 15 b^{3} c^{3}\right ) + x^{2} \left (18 a^{2} b c^{2} d - 10 a b^{2} c^{3}\right )}{6 a^{4} b x^{3} + 6 a^{3} b^{2} x^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15098, size = 203, normalized size = 1.38 \begin{align*} \frac{{\left (5 \, b^{3} c^{3} - 9 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} + a^{3} d^{3}\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{2 \, \sqrt{a b} a^{3} b} + \frac{b^{3} c^{3} x - 3 \, a b^{2} c^{2} d x + 3 \, a^{2} b c d^{2} x - a^{3} d^{3} x}{2 \,{\left (b x^{2} + a\right )} a^{3} b} + \frac{6 \, b c^{3} x^{2} - 9 \, a c^{2} d x^{2} - a c^{3}}{3 \, a^{3} x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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