Optimal. Leaf size=79 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right ) (a B e+A c d)}{2 a^{3/2} c^{3/2}}-\frac{a (A e+B d)-x (A c d-a B e)}{2 a c \left (a+c x^2\right )} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0320804, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {778, 205} \[ \frac{\tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right ) (a B e+A c d)}{2 a^{3/2} c^{3/2}}-\frac{a (A e+B d)-x (A c d-a B e)}{2 a c \left (a+c x^2\right )} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 778
Rule 205
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 c \left (a+c x^2\right )}+\frac{(A c d+a B e) \int \frac{1}{a+c x^2} \, dx}{2 a c}\\ &=-\frac{a (B d+A e)-(A c d-a B e) x}{2 a c \left (a+c x^2\right )}+\frac{(A c d+a B e) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{2 a^{3/2} c^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0772702, size = 78, normalized size = 0.99 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right ) (a B e+A c d)}{2 a^{3/2} c^{3/2}}+\frac{-a A e-a B d-a B e x+A c d x}{2 a c \left (a+c x^2\right )} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.008, size = 86, normalized size = 1.1 \begin{align*}{\frac{1}{c{x}^{2}+a} \left ({\frac{ \left ( Acd-aBe \right ) x}{2\,ac}}-{\frac{Ae+Bd}{2\,c}} \right ) }+{\frac{Ad}{2\,a}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}+{\frac{Be}{2\,c}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}} \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 = 1.79371, size = 485, normalized size = 6.14 \begin{align*} \left [-\frac{2 \, B a^{2} c d + 2 \, A a^{2} c e +{\left (A a c d + B a^{2} e +{\left (A c^{2} d + B a c e\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^{2} d - B a^{2} c e\right )} x}{4 \,{\left (a^{2} c^{3} x^{2} + a^{3} c^{2}\right )}}, -\frac{B a^{2} c d + A a^{2} c e -{\left (A a c d + B a^{2} e +{\left (A c^{2} d + B a c e\right )} x^{2}\right )} \sqrt{a c} \arctan \left (\frac{\sqrt{a c} x}{a}\right ) -{\left (A a c^{2} d - B a^{2} c e\right )} x}{2 \,{\left (a^{2} c^{3} x^{2} + a^{3} c^{2}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 1.1791, size = 133, normalized size = 1.68 \begin{align*} - \frac{\sqrt{- \frac{1}{a^{3} c^{3}}} \left (A c d + B a e\right ) \log{\left (- a^{2} c \sqrt{- \frac{1}{a^{3} c^{3}}} + x \right )}}{4} + \frac{\sqrt{- \frac{1}{a^{3} c^{3}}} \left (A c d + B a e\right ) \log{\left (a^{2} c \sqrt{- \frac{1}{a^{3} c^{3}}} + x \right )}}{4} - \frac{A a e + B a d + x \left (- A c d + B a e\right )}{2 a^{2} c + 2 a c^{2} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.18991, size = 100, normalized size = 1.27 \begin{align*} \frac{{\left (A c d + B a e\right )} \arctan \left (\frac{c x}{\sqrt{a c}}\right )}{2 \, \sqrt{a c} a c} + \frac{A c d x - B a x e - B a d - A a e}{2 \,{\left (c x^{2} + a\right )} a c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]