Optimal. Leaf size=113 \[ \frac{\left (a e^2+3 c d^2\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{8 a^{5/2} c^{3/2}}-\frac{2 a d e-x \left (a e^2+3 c d^2\right )}{8 a^2 c \left (a+c x^2\right )}-\frac{(d+e x) (a e-c d x)}{4 a c \left (a+c x^2\right )^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0493133, antiderivative size = 113, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {739, 639, 205} \[ \frac{\left (a e^2+3 c d^2\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{8 a^{5/2} c^{3/2}}-\frac{2 a d e-x \left (a e^2+3 c d^2\right )}{8 a^2 c \left (a+c x^2\right )}-\frac{(d+e x) (a e-c d x)}{4 a c \left (a+c x^2\right )^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 739
Rule 639
Rule 205
Rubi steps
\begin{align*} \int \frac{(d+e x)^2}{\left (a+c x^2\right )^3} \, dx &=-\frac{(a e-c d x) (d+e x)}{4 a c \left (a+c x^2\right )^2}+\frac{\int \frac{3 c d^2+a e^2+2 c d e x}{\left (a+c x^2\right )^2} \, dx}{4 a c}\\ &=-\frac{(a e-c d x) (d+e x)}{4 a c \left (a+c x^2\right )^2}-\frac{2 a d e-\left (3 c d^2+a e^2\right ) x}{8 a^2 c \left (a+c x^2\right )}+\frac{\left (3 c d^2+a e^2\right ) \int \frac{1}{a+c x^2} \, dx}{8 a^2 c}\\ &=-\frac{(a e-c d x) (d+e x)}{4 a c \left (a+c x^2\right )^2}-\frac{2 a d e-\left (3 c d^2+a e^2\right ) x}{8 a^2 c \left (a+c x^2\right )}+\frac{\left (3 c d^2+a e^2\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{8 a^{5/2} c^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0703977, size = 101, normalized size = 0.89 \[ \frac{-a^2 e (4 d+e x)+a c x \left (5 d^2+e^2 x^2\right )+3 c^2 d^2 x^3}{8 a^2 c \left (a+c x^2\right )^2}+\frac{\left (a e^2+3 c d^2\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{8 a^{5/2} c^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.047, size = 108, normalized size = 1. \begin{align*}{\frac{1}{ \left ( c{x}^{2}+a \right ) ^{2}} \left ({\frac{ \left ( a{e}^{2}+3\,c{d}^{2} \right ){x}^{3}}{8\,{a}^{2}}}-{\frac{ \left ( a{e}^{2}-5\,c{d}^{2} \right ) x}{8\,ac}}-{\frac{de}{2\,c}} \right ) }+{\frac{{e}^{2}}{8\,ac}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}+{\frac{3\,{d}^{2}}{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.
[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 = 1.98568, size = 716, normalized size = 6.34 \begin{align*} \left [-\frac{8 \, a^{3} c d e - 2 \,{\left (3 \, a c^{3} d^{2} + a^{2} c^{2} e^{2}\right )} x^{3} +{\left (3 \, a^{2} c d^{2} + a^{3} e^{2} +{\left (3 \, c^{3} d^{2} + a c^{2} e^{2}\right )} x^{4} + 2 \,{\left (3 \, a c^{2} d^{2} + a^{2} 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 (5 \, a^{2} c^{2} d^{2} - a^{3} c e^{2}\right )} x}{16 \,{\left (a^{3} c^{4} x^{4} + 2 \, a^{4} c^{3} x^{2} + a^{5} c^{2}\right )}}, -\frac{4 \, a^{3} c d e -{\left (3 \, a c^{3} d^{2} + a^{2} c^{2} e^{2}\right )} x^{3} -{\left (3 \, a^{2} c d^{2} + a^{3} e^{2} +{\left (3 \, c^{3} d^{2} + a c^{2} e^{2}\right )} x^{4} + 2 \,{\left (3 \, a c^{2} d^{2} + a^{2} c e^{2}\right )} x^{2}\right )} \sqrt{a c} \arctan \left (\frac{\sqrt{a c} x}{a}\right ) -{\left (5 \, a^{2} c^{2} d^{2} - a^{3} c e^{2}\right )} x}{8 \,{\left (a^{3} c^{4} x^{4} + 2 \, a^{4} c^{3} x^{2} + a^{5} c^{2}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 1.11785, size = 172, normalized size = 1.52 \begin{align*} - \frac{\sqrt{- \frac{1}{a^{5} c^{3}}} \left (a e^{2} + 3 c d^{2}\right ) \log{\left (- a^{3} c \sqrt{- \frac{1}{a^{5} c^{3}}} + x \right )}}{16} + \frac{\sqrt{- \frac{1}{a^{5} c^{3}}} \left (a e^{2} + 3 c d^{2}\right ) \log{\left (a^{3} c \sqrt{- \frac{1}{a^{5} c^{3}}} + x \right )}}{16} + \frac{- 4 a^{2} d e + x^{3} \left (a c e^{2} + 3 c^{2} d^{2}\right ) + x \left (- a^{2} e^{2} + 5 a c d^{2}\right )}{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.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.25419, size = 128, normalized size = 1.13 \begin{align*} \frac{{\left (3 \, c d^{2} + a e^{2}\right )} \arctan \left (\frac{c x}{\sqrt{a c}}\right )}{8 \, \sqrt{a c} a^{2} c} + \frac{3 \, c^{2} d^{2} x^{3} + a c x^{3} e^{2} + 5 \, a c d^{2} x - a^{2} x e^{2} - 4 \, a^{2} d e}{8 \,{\left (c x^{2} + a\right )}^{2} a^{2} c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]