Optimal. Leaf size=92 \[ \frac{(3 a d+b c) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{8 c^{5/2} d^{3/2}}+\frac{x (3 a d+b c)}{8 c^2 d \left (c+d x^2\right )}-\frac{x (b c-a d)}{4 c d \left (c+d x^2\right )^2} \]
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Rubi [A] time = 0.0306019, antiderivative size = 92, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {385, 199, 205} \[ \frac{(3 a d+b c) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{8 c^{5/2} d^{3/2}}+\frac{x (3 a d+b c)}{8 c^2 d \left (c+d x^2\right )}-\frac{x (b c-a d)}{4 c d \left (c+d x^2\right )^2} \]
Antiderivative was successfully verified.
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Rule 385
Rule 199
Rule 205
Rubi steps
\begin{align*} \int \frac{a+b x^2}{\left (c+d x^2\right )^3} \, dx &=-\frac{(b c-a d) x}{4 c d \left (c+d x^2\right )^2}+\frac{(b c+3 a d) \int \frac{1}{\left (c+d x^2\right )^2} \, dx}{4 c d}\\ &=-\frac{(b c-a d) x}{4 c d \left (c+d x^2\right )^2}+\frac{(b c+3 a d) x}{8 c^2 d \left (c+d x^2\right )}+\frac{(b c+3 a d) \int \frac{1}{c+d x^2} \, dx}{8 c^2 d}\\ &=-\frac{(b c-a d) x}{4 c d \left (c+d x^2\right )^2}+\frac{(b c+3 a d) x}{8 c^2 d \left (c+d x^2\right )}+\frac{(b c+3 a d) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{8 c^{5/2} d^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0575071, size = 82, normalized size = 0.89 \[ \frac{(3 a d+b c) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{8 c^{5/2} d^{3/2}}+\frac{x \left (a d \left (5 c+3 d x^2\right )+b c \left (d x^2-c\right )\right )}{8 c^2 d \left (c+d x^2\right )^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 90, normalized size = 1. \begin{align*}{\frac{1}{ \left ( d{x}^{2}+c \right ) ^{2}} \left ({\frac{ \left ( 3\,ad+bc \right ){x}^{3}}{8\,{c}^{2}}}+{\frac{ \left ( 5\,ad-bc \right ) x}{8\,cd}} \right ) }+{\frac{3\,a}{8\,{c}^{2}}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}+{\frac{b}{8\,cd}\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 [A] time = 1.77139, size = 621, normalized size = 6.75 \begin{align*} \left [\frac{2 \,{\left (b c^{2} d^{2} + 3 \, a c d^{3}\right )} x^{3} -{\left ({\left (b c d^{2} + 3 \, a d^{3}\right )} x^{4} + b c^{3} + 3 \, a c^{2} d + 2 \,{\left (b c^{2} d + 3 \, a c d^{2}\right )} x^{2}\right )} \sqrt{-c d} \log \left (\frac{d x^{2} - 2 \, \sqrt{-c d} x - c}{d x^{2} + c}\right ) - 2 \,{\left (b c^{3} d - 5 \, a c^{2} d^{2}\right )} x}{16 \,{\left (c^{3} d^{4} x^{4} + 2 \, c^{4} d^{3} x^{2} + c^{5} d^{2}\right )}}, \frac{{\left (b c^{2} d^{2} + 3 \, a c d^{3}\right )} x^{3} +{\left ({\left (b c d^{2} + 3 \, a d^{3}\right )} x^{4} + b c^{3} + 3 \, a c^{2} d + 2 \,{\left (b c^{2} d + 3 \, a c d^{2}\right )} x^{2}\right )} \sqrt{c d} \arctan \left (\frac{\sqrt{c d} x}{c}\right ) -{\left (b c^{3} d - 5 \, a c^{2} d^{2}\right )} x}{8 \,{\left (c^{3} d^{4} x^{4} + 2 \, c^{4} d^{3} x^{2} + c^{5} d^{2}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.689287, size = 150, normalized size = 1.63 \begin{align*} - \frac{\sqrt{- \frac{1}{c^{5} d^{3}}} \left (3 a d + b c\right ) \log{\left (- c^{3} d \sqrt{- \frac{1}{c^{5} d^{3}}} + x \right )}}{16} + \frac{\sqrt{- \frac{1}{c^{5} d^{3}}} \left (3 a d + b c\right ) \log{\left (c^{3} d \sqrt{- \frac{1}{c^{5} d^{3}}} + x \right )}}{16} + \frac{x^{3} \left (3 a d^{2} + b c d\right ) + x \left (5 a c d - b c^{2}\right )}{8 c^{4} d + 16 c^{3} d^{2} x^{2} + 8 c^{2} d^{3} x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.29951, size = 105, normalized size = 1.14 \begin{align*} \frac{{\left (b c + 3 \, a d\right )} \arctan \left (\frac{d x}{\sqrt{c d}}\right )}{8 \, \sqrt{c d} c^{2} d} + \frac{b c d x^{3} + 3 \, a d^{2} x^{3} - b c^{2} x + 5 \, a c d x}{8 \,{\left (d x^{2} + c\right )}^{2} c^{2} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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