Optimal. Leaf size=107 \[ -\frac{b^2 x (2 b c-3 a d)}{d^3}+\frac{(a d+5 b c) (b c-a d)^2 \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{2 c^{3/2} d^{7/2}}-\frac{x (b c-a d)^3}{2 c d^3 \left (c+d x^2\right )}+\frac{b^3 x^3}{3 d^2} \]
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Rubi [A] time = 0.0961711, antiderivative size = 107, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {390, 385, 205} \[ -\frac{b^2 x (2 b c-3 a d)}{d^3}+\frac{(a d+5 b c) (b c-a d)^2 \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{2 c^{3/2} d^{7/2}}-\frac{x (b c-a d)^3}{2 c d^3 \left (c+d x^2\right )}+\frac{b^3 x^3}{3 d^2} \]
Antiderivative was successfully verified.
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Rule 390
Rule 385
Rule 205
Rubi steps
\begin{align*} \int \frac{\left (a+b x^2\right )^3}{\left (c+d x^2\right )^2} \, dx &=\int \left (-\frac{b^2 (2 b c-3 a d)}{d^3}+\frac{b^3 x^2}{d^2}+\frac{(b c-a d)^2 (2 b c+a d)+3 b d (b c-a d)^2 x^2}{d^3 \left (c+d x^2\right )^2}\right ) \, dx\\ &=-\frac{b^2 (2 b c-3 a d) x}{d^3}+\frac{b^3 x^3}{3 d^2}+\frac{\int \frac{(b c-a d)^2 (2 b c+a d)+3 b d (b c-a d)^2 x^2}{\left (c+d x^2\right )^2} \, dx}{d^3}\\ &=-\frac{b^2 (2 b c-3 a d) x}{d^3}+\frac{b^3 x^3}{3 d^2}-\frac{(b c-a d)^3 x}{2 c d^3 \left (c+d x^2\right )}+\frac{\left ((b c-a d)^2 (5 b c+a d)\right ) \int \frac{1}{c+d x^2} \, dx}{2 c d^3}\\ &=-\frac{b^2 (2 b c-3 a d) x}{d^3}+\frac{b^3 x^3}{3 d^2}-\frac{(b c-a d)^3 x}{2 c d^3 \left (c+d x^2\right )}+\frac{(b c-a d)^2 (5 b c+a d) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{2 c^{3/2} d^{7/2}}\\ \end{align*}
Mathematica [A] time = 0.0582583, size = 107, normalized size = 1. \[ -\frac{b^2 x (2 b c-3 a d)}{d^3}+\frac{(a d+5 b c) (b c-a d)^2 \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{2 c^{3/2} d^{7/2}}-\frac{x (b c-a d)^3}{2 c d^3 \left (c+d x^2\right )}+\frac{b^3 x^3}{3 d^2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.01, size = 205, normalized size = 1.9 \begin{align*}{\frac{{b}^{3}{x}^{3}}{3\,{d}^{2}}}+3\,{\frac{a{b}^{2}x}{{d}^{2}}}-2\,{\frac{{b}^{3}xc}{{d}^{3}}}+{\frac{x{a}^{3}}{2\,c \left ( d{x}^{2}+c \right ) }}-{\frac{3\,{a}^{2}bx}{2\,d \left ( d{x}^{2}+c \right ) }}+{\frac{3\,acx{b}^{2}}{2\,{d}^{2} \left ( d{x}^{2}+c \right ) }}-{\frac{{c}^{2}x{b}^{3}}{2\,{d}^{3} \left ( d{x}^{2}+c \right ) }}+{\frac{{a}^{3}}{2\,c}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}+{\frac{3\,{a}^{2}b}{2\,d}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}-{\frac{9\,a{b}^{2}c}{2\,{d}^{2}}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}+{\frac{5\,{b}^{3}{c}^{2}}{2\,{d}^{3}}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}} \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 [B] time = 1.79082, size = 896, normalized size = 8.37 \begin{align*} \left [\frac{4 \, b^{3} c^{2} d^{3} x^{5} - 4 \,{\left (5 \, b^{3} c^{3} d^{2} - 9 \, a b^{2} c^{2} d^{3}\right )} x^{3} - 3 \,{\left (5 \, b^{3} c^{4} - 9 \, a b^{2} c^{3} d + 3 \, a^{2} b c^{2} d^{2} + a^{3} c d^{3} +{\left (5 \, b^{3} c^{3} d - 9 \, a b^{2} c^{2} d^{2} + 3 \, a^{2} b c d^{3} + a^{3} d^{4}\right )} x^{2}\right )} \sqrt{-c d} \log \left (\frac{d x^{2} - 2 \, \sqrt{-c d} x - c}{d x^{2} + c}\right ) - 6 \,{\left (5 \, b^{3} c^{4} d - 9 \, a b^{2} c^{3} d^{2} + 3 \, a^{2} b c^{2} d^{3} - a^{3} c d^{4}\right )} x}{12 \,{\left (c^{2} d^{5} x^{2} + c^{3} d^{4}\right )}}, \frac{2 \, b^{3} c^{2} d^{3} x^{5} - 2 \,{\left (5 \, b^{3} c^{3} d^{2} - 9 \, a b^{2} c^{2} d^{3}\right )} x^{3} + 3 \,{\left (5 \, b^{3} c^{4} - 9 \, a b^{2} c^{3} d + 3 \, a^{2} b c^{2} d^{2} + a^{3} c d^{3} +{\left (5 \, b^{3} c^{3} d - 9 \, a b^{2} c^{2} d^{2} + 3 \, a^{2} b c d^{3} + a^{3} d^{4}\right )} x^{2}\right )} \sqrt{c d} \arctan \left (\frac{\sqrt{c d} x}{c}\right ) - 3 \,{\left (5 \, b^{3} c^{4} d - 9 \, a b^{2} c^{3} d^{2} + 3 \, a^{2} b c^{2} d^{3} - a^{3} c d^{4}\right )} x}{6 \,{\left (c^{2} d^{5} x^{2} + c^{3} d^{4}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 1.21979, size = 313, normalized size = 2.93 \begin{align*} \frac{b^{3} x^{3}}{3 d^{2}} + \frac{x \left (a^{3} d^{3} - 3 a^{2} b c d^{2} + 3 a b^{2} c^{2} d - b^{3} c^{3}\right )}{2 c^{2} d^{3} + 2 c d^{4} x^{2}} - \frac{\sqrt{- \frac{1}{c^{3} d^{7}}} \left (a d - b c\right )^{2} \left (a d + 5 b c\right ) \log{\left (- \frac{c^{2} d^{3} \sqrt{- \frac{1}{c^{3} d^{7}}} \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}{c^{3} d^{7}}} \left (a d - b c\right )^{2} \left (a d + 5 b c\right ) \log{\left (\frac{c^{2} d^{3} \sqrt{- \frac{1}{c^{3} d^{7}}} \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{x \left (3 a b^{2} d - 2 b^{3} c\right )}{d^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.63629, size = 205, normalized size = 1.92 \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{d x}{\sqrt{c d}}\right )}{2 \, \sqrt{c d} c d^{3}} - \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 (d x^{2} + c\right )} c d^{3}} + \frac{b^{3} d^{4} x^{3} - 6 \, b^{3} c d^{3} x + 9 \, a b^{2} d^{4} x}{3 \, d^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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