Optimal. Leaf size=195 \[ -\frac{\tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right ) \left (a B e \left (c d^2-a e^2\right )-A c d \left (3 a e^2+c d^2\right )\right )}{2 a^{3/2} \sqrt{c} \left (a e^2+c d^2\right )^2}-\frac{a (B d-A e)-x (a B e+A c d)}{2 a \left (a+c x^2\right ) \left (a e^2+c d^2\right )}+\frac{e^2 \log \left (a+c x^2\right ) (B d-A e)}{2 \left (a e^2+c d^2\right )^2}-\frac{e^2 (B d-A e) \log (d+e x)}{\left (a e^2+c d^2\right )^2} \]
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Rubi [A] time = 0.280877, antiderivative size = 195, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227, Rules used = {823, 801, 635, 205, 260} \[ -\frac{\tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right ) \left (a B e \left (c d^2-a e^2\right )-A c d \left (3 a e^2+c d^2\right )\right )}{2 a^{3/2} \sqrt{c} \left (a e^2+c d^2\right )^2}-\frac{a (B d-A e)-x (a B e+A c d)}{2 a \left (a+c x^2\right ) \left (a e^2+c d^2\right )}+\frac{e^2 \log \left (a+c x^2\right ) (B d-A e)}{2 \left (a e^2+c d^2\right )^2}-\frac{e^2 (B d-A e) \log (d+e x)}{\left (a e^2+c d^2\right )^2} \]
Antiderivative was successfully verified.
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Rule 823
Rule 801
Rule 635
Rule 205
Rule 260
Rubi steps
\begin{align*} \int \frac{A+B x}{(d+e x) \left (a+c x^2\right )^2} \, dx &=-\frac{a (B d-A e)-(A c d+a B e) x}{2 a \left (c d^2+a e^2\right ) \left (a+c x^2\right )}-\frac{\int \frac{c \left (a B d e-A \left (c d^2+2 a e^2\right )\right )-c e (A c d+a B e) x}{(d+e x) \left (a+c x^2\right )} \, dx}{2 a c \left (c d^2+a e^2\right )}\\ &=-\frac{a (B d-A e)-(A c d+a B e) x}{2 a \left (c d^2+a e^2\right ) \left (a+c x^2\right )}-\frac{\int \left (-\frac{2 a c e^3 (-B d+A e)}{\left (c d^2+a e^2\right ) (d+e x)}+\frac{c \left (a B e \left (c d^2-a e^2\right )-A c d \left (c d^2+3 a e^2\right )-2 a c e^2 (B d-A e) x\right )}{\left (c d^2+a e^2\right ) \left (a+c x^2\right )}\right ) \, dx}{2 a c \left (c d^2+a e^2\right )}\\ &=-\frac{a (B d-A e)-(A c d+a B e) x}{2 a \left (c d^2+a e^2\right ) \left (a+c x^2\right )}-\frac{e^2 (B d-A e) \log (d+e x)}{\left (c d^2+a e^2\right )^2}-\frac{\int \frac{a B e \left (c d^2-a e^2\right )-A c d \left (c d^2+3 a e^2\right )-2 a c e^2 (B d-A e) x}{a+c x^2} \, dx}{2 a \left (c d^2+a e^2\right )^2}\\ &=-\frac{a (B d-A e)-(A c d+a B e) x}{2 a \left (c d^2+a e^2\right ) \left (a+c x^2\right )}-\frac{e^2 (B d-A e) \log (d+e x)}{\left (c d^2+a e^2\right )^2}+\frac{\left (c e^2 (B d-A e)\right ) \int \frac{x}{a+c x^2} \, dx}{\left (c d^2+a e^2\right )^2}-\frac{\left (a B e \left (c d^2-a e^2\right )-A c d \left (c d^2+3 a e^2\right )\right ) \int \frac{1}{a+c x^2} \, dx}{2 a \left (c d^2+a e^2\right )^2}\\ &=-\frac{a (B d-A e)-(A c d+a B e) x}{2 a \left (c d^2+a e^2\right ) \left (a+c x^2\right )}-\frac{\left (a B e \left (c d^2-a e^2\right )-A c d \left (c d^2+3 a e^2\right )\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{2 a^{3/2} \sqrt{c} \left (c d^2+a e^2\right )^2}-\frac{e^2 (B d-A e) \log (d+e x)}{\left (c d^2+a e^2\right )^2}+\frac{e^2 (B d-A e) \log \left (a+c x^2\right )}{2 \left (c d^2+a e^2\right )^2}\\ \end{align*}
Mathematica [A] time = 0.184453, size = 158, normalized size = 0.81 \[ \frac{\frac{\tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right ) \left (A c d \left (3 a e^2+c d^2\right )+a B e \left (a e^2-c d^2\right )\right )}{a^{3/2} \sqrt{c}}+\frac{\left (a e^2+c d^2\right ) (a (A e-B d+B e x)+A c d x)}{a \left (a+c x^2\right )}+e^2 \log \left (a+c x^2\right ) (B d-A e)+2 e^2 (A e-B d) \log (d+e x)}{2 \left (a e^2+c d^2\right )^2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.024, size = 495, normalized size = 2.5 \begin{align*}{\frac{{e}^{3}\ln \left ( ex+d \right ) A}{ \left ( a{e}^{2}+c{d}^{2} \right ) ^{2}}}-{\frac{{e}^{2}\ln \left ( ex+d \right ) Bd}{ \left ( a{e}^{2}+c{d}^{2} \right ) ^{2}}}+{\frac{Acxd{e}^{2}}{2\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2} \left ( c{x}^{2}+a \right ) }}+{\frac{xA{d}^{3}{c}^{2}}{2\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2} \left ( c{x}^{2}+a \right ) a}}+{\frac{aBx{e}^{3}}{2\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2} \left ( c{x}^{2}+a \right ) }}+{\frac{Bcx{d}^{2}e}{2\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2} \left ( c{x}^{2}+a \right ) }}+{\frac{aA{e}^{3}}{2\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2} \left ( c{x}^{2}+a \right ) }}+{\frac{Ac{d}^{2}e}{2\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2} \left ( c{x}^{2}+a \right ) }}-{\frac{aBd{e}^{2}}{2\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2} \left ( c{x}^{2}+a \right ) }}-{\frac{Bc{d}^{3}}{2\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2} \left ( c{x}^{2}+a \right ) }}-{\frac{\ln \left ( c{x}^{2}+a \right ) A{e}^{3}}{2\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2}}}+{\frac{\ln \left ( c{x}^{2}+a \right ) Bd{e}^{2}}{2\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2}}}+{\frac{3\,Acd{e}^{2}}{2\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2}}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}+{\frac{A{d}^{3}{c}^{2}}{2\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2}a}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}+{\frac{B{e}^{3}a}{2\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2}}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}-{\frac{Bc{d}^{2}e}{2\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{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 [B] time = 110.401, size = 1600, normalized size = 8.21 \begin{align*} \left [-\frac{2 \, B a^{2} c^{2} d^{3} - 2 \, A a^{2} c^{2} d^{2} e + 2 \, B a^{3} c d e^{2} - 2 \, A a^{3} c e^{3} +{\left (A a c^{2} d^{3} - B a^{2} c d^{2} e + 3 \, A a^{2} c d e^{2} + B a^{3} e^{3} +{\left (A c^{3} d^{3} - B a c^{2} d^{2} e + 3 \, A a c^{2} d e^{2} + B a^{2} c e^{3}\right )} x^{2}\right )} \sqrt{-a c} \log \left (\frac{c x^{2} - 2 \, \sqrt{-a c} x - a}{c x^{2} + a}\right ) - 2 \,{\left (A a c^{3} d^{3} + B a^{2} c^{2} d^{2} e + A a^{2} c^{2} d e^{2} + B a^{3} c e^{3}\right )} x - 2 \,{\left (B a^{3} c d e^{2} - A a^{3} c e^{3} +{\left (B a^{2} c^{2} d e^{2} - A a^{2} c^{2} e^{3}\right )} x^{2}\right )} \log \left (c x^{2} + a\right ) + 4 \,{\left (B a^{3} c d e^{2} - A a^{3} c e^{3} +{\left (B a^{2} c^{2} d e^{2} - A a^{2} c^{2} e^{3}\right )} x^{2}\right )} \log \left (e x + d\right )}{4 \,{\left (a^{3} c^{3} d^{4} + 2 \, a^{4} c^{2} d^{2} e^{2} + a^{5} c e^{4} +{\left (a^{2} c^{4} d^{4} + 2 \, a^{3} c^{3} d^{2} e^{2} + a^{4} c^{2} e^{4}\right )} x^{2}\right )}}, -\frac{B a^{2} c^{2} d^{3} - A a^{2} c^{2} d^{2} e + B a^{3} c d e^{2} - A a^{3} c e^{3} -{\left (A a c^{2} d^{3} - B a^{2} c d^{2} e + 3 \, A a^{2} c d e^{2} + B a^{3} e^{3} +{\left (A c^{3} d^{3} - B a c^{2} d^{2} e + 3 \, A a c^{2} d e^{2} + B a^{2} c e^{3}\right )} x^{2}\right )} \sqrt{a c} \arctan \left (\frac{\sqrt{a c} x}{a}\right ) -{\left (A a c^{3} d^{3} + B a^{2} c^{2} d^{2} e + A a^{2} c^{2} d e^{2} + B a^{3} c e^{3}\right )} x -{\left (B a^{3} c d e^{2} - A a^{3} c e^{3} +{\left (B a^{2} c^{2} d e^{2} - A a^{2} c^{2} e^{3}\right )} x^{2}\right )} \log \left (c x^{2} + a\right ) + 2 \,{\left (B a^{3} c d e^{2} - A a^{3} c e^{3} +{\left (B a^{2} c^{2} d e^{2} - A a^{2} c^{2} e^{3}\right )} x^{2}\right )} \log \left (e x + d\right )}{2 \,{\left (a^{3} c^{3} d^{4} + 2 \, a^{4} c^{2} d^{2} e^{2} + a^{5} c e^{4} +{\left (a^{2} c^{4} d^{4} + 2 \, a^{3} c^{3} d^{2} e^{2} + a^{4} c^{2} e^{4}\right )} x^{2}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18533, size = 362, normalized size = 1.86 \begin{align*} \frac{{\left (B d e^{2} - A e^{3}\right )} \log \left (c x^{2} + a\right )}{2 \,{\left (c^{2} d^{4} + 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )}} - \frac{{\left (B d e^{3} - A e^{4}\right )} \log \left ({\left | x e + d \right |}\right )}{c^{2} d^{4} e + 2 \, a c d^{2} e^{3} + a^{2} e^{5}} + \frac{{\left (A c^{2} d^{3} - B a c d^{2} e + 3 \, A a c d e^{2} + B a^{2} e^{3}\right )} \arctan \left (\frac{c x}{\sqrt{a c}}\right )}{2 \,{\left (a c^{2} d^{4} + 2 \, a^{2} c d^{2} e^{2} + a^{3} e^{4}\right )} \sqrt{a c}} - \frac{B a c d^{3} - A a c d^{2} e + B a^{2} d e^{2} - A a^{2} e^{3} -{\left (A c^{2} d^{3} + B a c d^{2} e + A a c d e^{2} + B a^{2} e^{3}\right )} x}{2 \,{\left (c d^{2} + a e^{2}\right )}^{2}{\left (c x^{2} + a\right )} a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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