Optimal. Leaf size=75 \[ \frac{3 A x}{8 a^2 \left (a+c x^2\right )}+\frac{3 A \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{8 a^{5/2} \sqrt{c}}-\frac{a B-A c x}{4 a c \left (a+c x^2\right )^2} \]
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Rubi [A] time = 0.0209181, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {639, 199, 205} \[ \frac{3 A x}{8 a^2 \left (a+c x^2\right )}+\frac{3 A \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{8 a^{5/2} \sqrt{c}}-\frac{a B-A c x}{4 a c \left (a+c x^2\right )^2} \]
Antiderivative was successfully verified.
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Rule 639
Rule 199
Rule 205
Rubi steps
\begin{align*} \int \frac{A+B x}{\left (a+c x^2\right )^3} \, dx &=-\frac{a B-A c x}{4 a c \left (a+c x^2\right )^2}+\frac{(3 A) \int \frac{1}{\left (a+c x^2\right )^2} \, dx}{4 a}\\ &=-\frac{a B-A c x}{4 a c \left (a+c x^2\right )^2}+\frac{3 A x}{8 a^2 \left (a+c x^2\right )}+\frac{(3 A) \int \frac{1}{a+c x^2} \, dx}{8 a^2}\\ &=-\frac{a B-A c x}{4 a c \left (a+c x^2\right )^2}+\frac{3 A x}{8 a^2 \left (a+c x^2\right )}+\frac{3 A \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{8 a^{5/2} \sqrt{c}}\\ \end{align*}
Mathematica [A] time = 0.0497051, size = 71, normalized size = 0.95 \[ \frac{\frac{\sqrt{a} \left (-2 a^2 B+5 a A c x+3 A c^2 x^3\right )}{\left (a+c x^2\right )^2}+3 A \sqrt{c} \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{8 a^{5/2} c} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 65, normalized size = 0.9 \begin{align*}{\frac{2\,Acx-2\,aB}{8\,ac \left ( c{x}^{2}+a \right ) ^{2}}}+{\frac{3\,Ax}{8\,{a}^{2} \left ( c{x}^{2}+a \right ) }}+{\frac{3\,A}{8\,{a}^{2}}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}} \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 = 1.92124, size = 455, normalized size = 6.07 \begin{align*} \left [\frac{6 \, A a c^{2} x^{3} + 10 \, A a^{2} c x - 4 \, B a^{3} - 3 \,{\left (A c^{2} x^{4} + 2 \, A a c x^{2} + A a^{2}\right )} \sqrt{-a c} \log \left (\frac{c x^{2} - 2 \, \sqrt{-a c} x - a}{c x^{2} + a}\right )}{16 \,{\left (a^{3} c^{3} x^{4} + 2 \, a^{4} c^{2} x^{2} + a^{5} c\right )}}, \frac{3 \, A a c^{2} x^{3} + 5 \, A a^{2} c x - 2 \, B a^{3} + 3 \,{\left (A c^{2} x^{4} + 2 \, A a c x^{2} + A a^{2}\right )} \sqrt{a c} \arctan \left (\frac{\sqrt{a c} x}{a}\right )}{8 \,{\left (a^{3} c^{3} x^{4} + 2 \, a^{4} c^{2} x^{2} + a^{5} c\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.06156, size = 124, normalized size = 1.65 \begin{align*} A \left (- \frac{3 \sqrt{- \frac{1}{a^{5} c}} \log{\left (- a^{3} \sqrt{- \frac{1}{a^{5} c}} + x \right )}}{16} + \frac{3 \sqrt{- \frac{1}{a^{5} c}} \log{\left (a^{3} \sqrt{- \frac{1}{a^{5} c}} + x \right )}}{16}\right ) + \frac{5 A a c x + 3 A c^{2} x^{3} - 2 B a^{2}}{8 a^{4} c + 16 a^{3} c^{2} x^{2} + 8 a^{2} c^{3} x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19553, size = 81, normalized size = 1.08 \begin{align*} \frac{3 \, A \arctan \left (\frac{c x}{\sqrt{a c}}\right )}{8 \, \sqrt{a c} a^{2}} + \frac{3 \, A c^{2} x^{3} + 5 \, A a c x - 2 \, B a^{2}}{8 \,{\left (c x^{2} + a\right )}^{2} a^{2} c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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