Optimal. Leaf size=86 \[ -\frac{e \log \left (a+c x^2\right )}{2 \left (a e^2+c d^2\right )}+\frac{e \log (d+e x)}{a e^2+c d^2}+\frac{\sqrt{c} d \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{\sqrt{a} \left (a e^2+c d^2\right )} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0380176, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.294, Rules used = {706, 31, 635, 205, 260} \[ -\frac{e \log \left (a+c x^2\right )}{2 \left (a e^2+c d^2\right )}+\frac{e \log (d+e x)}{a e^2+c d^2}+\frac{\sqrt{c} d \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{\sqrt{a} \left (a e^2+c d^2\right )} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 706
Rule 31
Rule 635
Rule 205
Rule 260
Rubi steps
\begin{align*} \int \frac{1}{(d+e x) \left (a+c x^2\right )} \, dx &=\frac{\int \frac{c d-c e x}{a+c x^2} \, dx}{c d^2+a e^2}+\frac{e^2 \int \frac{1}{d+e x} \, dx}{c d^2+a e^2}\\ &=\frac{e \log (d+e x)}{c d^2+a e^2}+\frac{(c d) \int \frac{1}{a+c x^2} \, dx}{c d^2+a e^2}-\frac{(c e) \int \frac{x}{a+c x^2} \, dx}{c d^2+a e^2}\\ &=\frac{\sqrt{c} d \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{\sqrt{a} \left (c d^2+a e^2\right )}+\frac{e \log (d+e x)}{c d^2+a e^2}-\frac{e \log \left (a+c x^2\right )}{2 \left (c d^2+a e^2\right )}\\ \end{align*}
Mathematica [A] time = 0.0354208, size = 63, normalized size = 0.73 \[ \frac{\frac{2 \sqrt{c} d \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{\sqrt{a}}-e \log \left (a+c x^2\right )+2 e \log (d+e x)}{2 a e^2+2 c d^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.049, size = 77, normalized size = 0.9 \begin{align*} -{\frac{e\ln \left ( c{x}^{2}+a \right ) }{2\,a{e}^{2}+2\,c{d}^{2}}}+{\frac{cd}{a{e}^{2}+c{d}^{2}}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}+{\frac{e\ln \left ( ex+d \right ) }{a{e}^{2}+c{d}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.14763, size = 298, normalized size = 3.47 \begin{align*} \left [\frac{d \sqrt{-\frac{c}{a}} \log \left (\frac{c x^{2} + 2 \, a x \sqrt{-\frac{c}{a}} - a}{c x^{2} + a}\right ) - e \log \left (c x^{2} + a\right ) + 2 \, e \log \left (e x + d\right )}{2 \,{\left (c d^{2} + a e^{2}\right )}}, \frac{2 \, d \sqrt{\frac{c}{a}} \arctan \left (x \sqrt{\frac{c}{a}}\right ) - e \log \left (c x^{2} + a\right ) + 2 \, e \log \left (e x + d\right )}{2 \,{\left (c d^{2} + a e^{2}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] time = 3.80207, size = 1134, normalized size = 13.19 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.1845, size = 107, normalized size = 1.24 \begin{align*} \frac{c d \arctan \left (\frac{c x}{\sqrt{a c}}\right )}{{\left (c d^{2} + a e^{2}\right )} \sqrt{a c}} - \frac{e \log \left (c x^{2} + a\right )}{2 \,{\left (c d^{2} + a e^{2}\right )}} + \frac{e^{2} \log \left ({\left | x e + d \right |}\right )}{c d^{2} e + a e^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]