Optimal. Leaf size=72 \[ \frac{\left (a e^2+c d^2\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{2 a^{3/2} c^{3/2}}-\frac{(d+e x) (a e-c d x)}{2 a c \left (a+c x^2\right )} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0210555, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {723, 205} \[ \frac{\left (a e^2+c d^2\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{2 a^{3/2} c^{3/2}}-\frac{(d+e x) (a e-c d x)}{2 a c \left (a+c x^2\right )} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 723
Rule 205
Rubi steps
\begin{align*} \int \frac{(d+e x)^2}{\left (a+c x^2\right )^2} \, dx &=-\frac{(a e-c d x) (d+e x)}{2 a c \left (a+c x^2\right )}+\frac{\left (c d^2+a e^2\right ) \int \frac{1}{a+c x^2} \, dx}{2 a c}\\ &=-\frac{(a e-c d x) (d+e x)}{2 a c \left (a+c x^2\right )}+\frac{\left (c d^2+a e^2\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{2 a^{3/2} c^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0579996, size = 77, normalized size = 1.07 \[ \frac{\left (a e^2+c d^2\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{2 a^{3/2} c^{3/2}}+\frac{-2 a d e-a e^2 x+c d^2 x}{2 a c \left (a+c x^2\right )} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.05, size = 85, normalized size = 1.2 \begin{align*}{\frac{1}{c{x}^{2}+a} \left ( -{\frac{ \left ( a{e}^{2}-c{d}^{2} \right ) x}{2\,ac}}-{\frac{de}{c}} \right ) }+{\frac{{e}^{2}}{2\,c}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}+{\frac{{d}^{2}}{2\,a}\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 = 1.88351, size = 452, normalized size = 6.28 \begin{align*} \left [-\frac{4 \, a^{2} c d e +{\left (a c d^{2} + a^{2} e^{2} +{\left (c^{2} d^{2} + a c e^{2}\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 c^{2} d^{2} - a^{2} c e^{2}\right )} x}{4 \,{\left (a^{2} c^{3} x^{2} + a^{3} c^{2}\right )}}, -\frac{2 \, a^{2} c d e -{\left (a c d^{2} + a^{2} e^{2} +{\left (c^{2} d^{2} + a c e^{2}\right )} x^{2}\right )} \sqrt{a c} \arctan \left (\frac{\sqrt{a c} x}{a}\right ) -{\left (a c^{2} d^{2} - a^{2} c e^{2}\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 [B] time = 0.722852, size = 129, normalized size = 1.79 \begin{align*} - \frac{\sqrt{- \frac{1}{a^{3} c^{3}}} \left (a e^{2} + c d^{2}\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 e^{2} + c d^{2}\right ) \log{\left (a^{2} c \sqrt{- \frac{1}{a^{3} c^{3}}} + x \right )}}{4} - \frac{2 a d e + x \left (a e^{2} - c d^{2}\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 = 1.31404, size = 93, normalized size = 1.29 \begin{align*} \frac{{\left (c d^{2} + a e^{2}\right )} \arctan \left (\frac{c x}{\sqrt{a c}}\right )}{2 \, \sqrt{a c} a c} + \frac{c d^{2} x - a x e^{2} - 2 \, a d e}{2 \,{\left (c x^{2} + a\right )} a c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]