Optimal. Leaf size=116 \[ \frac{3 x \left (\frac{c^2}{a^2}-\frac{d^2}{b^2}\right )}{8 \left (a+b x^2\right )}+\frac{\left (3 a^2 d^2+2 a b c d+3 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 a^{5/2} b^{5/2}}+\frac{x \left (c+d x^2\right ) (b c-a d)}{4 a b \left (a+b x^2\right )^2} \]
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Rubi [A] time = 0.0769482, antiderivative size = 116, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {413, 385, 205} \[ \frac{3 x \left (\frac{c^2}{a^2}-\frac{d^2}{b^2}\right )}{8 \left (a+b x^2\right )}+\frac{\left (3 a^2 d^2+2 a b c d+3 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 a^{5/2} b^{5/2}}+\frac{x \left (c+d x^2\right ) (b c-a d)}{4 a b \left (a+b x^2\right )^2} \]
Antiderivative was successfully verified.
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Rule 413
Rule 385
Rule 205
Rubi steps
\begin{align*} \int \frac{\left (c+d x^2\right )^2}{\left (a+b x^2\right )^3} \, dx &=\frac{(b c-a d) x \left (c+d x^2\right )}{4 a b \left (a+b x^2\right )^2}+\frac{\int \frac{c (3 b c+a d)+d (b c+3 a d) x^2}{\left (a+b x^2\right )^2} \, dx}{4 a b}\\ &=\frac{3 \left (\frac{c^2}{a^2}-\frac{d^2}{b^2}\right ) x}{8 \left (a+b x^2\right )}+\frac{(b c-a d) x \left (c+d x^2\right )}{4 a b \left (a+b x^2\right )^2}+\frac{\left (3 b^2 c^2+2 a b c d+3 a^2 d^2\right ) \int \frac{1}{a+b x^2} \, dx}{8 a^2 b^2}\\ &=\frac{3 \left (\frac{c^2}{a^2}-\frac{d^2}{b^2}\right ) x}{8 \left (a+b x^2\right )}+\frac{(b c-a d) x \left (c+d x^2\right )}{4 a b \left (a+b x^2\right )^2}+\frac{\left (3 b^2 c^2+2 a b c d+3 a^2 d^2\right ) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 a^{5/2} b^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.0945662, size = 124, normalized size = 1.07 \[ \frac{x \left (-a^2 b d \left (2 c+5 d x^2\right )-3 a^3 d^2+a b^2 c \left (5 c+2 d x^2\right )+3 b^3 c^2 x^2\right )}{8 a^2 b^2 \left (a+b x^2\right )^2}+\frac{\left (3 a^2 d^2+2 a b c d+3 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 a^{5/2} b^{5/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.008, size = 147, normalized size = 1.3 \begin{align*}{\frac{1}{ \left ( b{x}^{2}+a \right ) ^{2}} \left ( -{\frac{ \left ( 5\,{a}^{2}{d}^{2}-2\,abcd-3\,{b}^{2}{c}^{2} \right ){x}^{3}}{8\,{a}^{2}b}}-{\frac{ \left ( 3\,{a}^{2}{d}^{2}+2\,abcd-5\,{b}^{2}{c}^{2} \right ) x}{8\,a{b}^{2}}} \right ) }+{\frac{3\,{d}^{2}}{8\,{b}^{2}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{cd}{4\,ab}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{3\,{c}^{2}}{8\,{a}^{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 [B] time = 1.87662, size = 918, normalized size = 7.91 \begin{align*} \left [\frac{2 \,{\left (3 \, a b^{4} c^{2} + 2 \, a^{2} b^{3} c d - 5 \, a^{3} b^{2} d^{2}\right )} x^{3} -{\left (3 \, a^{2} b^{2} c^{2} + 2 \, a^{3} b c d + 3 \, a^{4} d^{2} +{\left (3 \, b^{4} c^{2} + 2 \, a b^{3} c d + 3 \, a^{2} b^{2} d^{2}\right )} x^{4} + 2 \,{\left (3 \, a b^{3} c^{2} + 2 \, a^{2} b^{2} c d + 3 \, a^{3} b d^{2}\right )} x^{2}\right )} \sqrt{-a b} \log \left (\frac{b x^{2} - 2 \, \sqrt{-a b} x - a}{b x^{2} + a}\right ) + 2 \,{\left (5 \, a^{2} b^{3} c^{2} - 2 \, a^{3} b^{2} c d - 3 \, a^{4} b d^{2}\right )} x}{16 \,{\left (a^{3} b^{5} x^{4} + 2 \, a^{4} b^{4} x^{2} + a^{5} b^{3}\right )}}, \frac{{\left (3 \, a b^{4} c^{2} + 2 \, a^{2} b^{3} c d - 5 \, a^{3} b^{2} d^{2}\right )} x^{3} +{\left (3 \, a^{2} b^{2} c^{2} + 2 \, a^{3} b c d + 3 \, a^{4} d^{2} +{\left (3 \, b^{4} c^{2} + 2 \, a b^{3} c d + 3 \, a^{2} b^{2} d^{2}\right )} x^{4} + 2 \,{\left (3 \, a b^{3} c^{2} + 2 \, a^{2} b^{2} c d + 3 \, a^{3} b d^{2}\right )} x^{2}\right )} \sqrt{a b} \arctan \left (\frac{\sqrt{a b} x}{a}\right ) +{\left (5 \, a^{2} b^{3} c^{2} - 2 \, a^{3} b^{2} c d - 3 \, a^{4} b d^{2}\right )} x}{8 \,{\left (a^{3} b^{5} x^{4} + 2 \, a^{4} b^{4} x^{2} + a^{5} b^{3}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 1.19055, size = 223, normalized size = 1.92 \begin{align*} - \frac{\sqrt{- \frac{1}{a^{5} b^{5}}} \left (3 a^{2} d^{2} + 2 a b c d + 3 b^{2} c^{2}\right ) \log{\left (- a^{3} b^{2} \sqrt{- \frac{1}{a^{5} b^{5}}} + x \right )}}{16} + \frac{\sqrt{- \frac{1}{a^{5} b^{5}}} \left (3 a^{2} d^{2} + 2 a b c d + 3 b^{2} c^{2}\right ) \log{\left (a^{3} b^{2} \sqrt{- \frac{1}{a^{5} b^{5}}} + x \right )}}{16} - \frac{x^{3} \left (5 a^{2} b d^{2} - 2 a b^{2} c d - 3 b^{3} c^{2}\right ) + x \left (3 a^{3} d^{2} + 2 a^{2} b c d - 5 a b^{2} c^{2}\right )}{8 a^{4} b^{2} + 16 a^{3} b^{3} x^{2} + 8 a^{2} b^{4} x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18004, size = 170, normalized size = 1.47 \begin{align*} \frac{{\left (3 \, b^{2} c^{2} + 2 \, a b c d + 3 \, a^{2} d^{2}\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{8 \, \sqrt{a b} a^{2} b^{2}} + \frac{3 \, b^{3} c^{2} x^{3} + 2 \, a b^{2} c d x^{3} - 5 \, a^{2} b d^{2} x^{3} + 5 \, a b^{2} c^{2} x - 2 \, a^{2} b c d x - 3 \, a^{3} d^{2} x}{8 \,{\left (b x^{2} + a\right )}^{2} a^{2} b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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