Optimal. Leaf size=108 \[ \frac{\log \left (a+c x^2\right ) \left (-a B e^2+2 A c d e+B c d^2\right )}{2 c^2}+\frac{\tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right ) \left (-a A e^2-2 a B d e+A c d^2\right )}{\sqrt{a} c^{3/2}}+\frac{e x (A e+2 B d)}{c}+\frac{B e^2 x^2}{2 c} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0946352, antiderivative size = 108, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {801, 635, 205, 260} \[ \frac{\log \left (a+c x^2\right ) \left (-a B e^2+2 A c d e+B c d^2\right )}{2 c^2}+\frac{\tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right ) \left (-a A e^2-2 a B d e+A c d^2\right )}{\sqrt{a} c^{3/2}}+\frac{e x (A e+2 B d)}{c}+\frac{B e^2 x^2}{2 c} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 801
Rule 635
Rule 205
Rule 260
Rubi steps
\begin{align*} \int \frac{(A+B x) (d+e x)^2}{a+c x^2} \, dx &=\int \left (\frac{e (2 B d+A e)}{c}+\frac{B e^2 x}{c}+\frac{A c d^2-2 a B d e-a A e^2+\left (B c d^2+2 A c d e-a B e^2\right ) x}{c \left (a+c x^2\right )}\right ) \, dx\\ &=\frac{e (2 B d+A e) x}{c}+\frac{B e^2 x^2}{2 c}+\frac{\int \frac{A c d^2-2 a B d e-a A e^2+\left (B c d^2+2 A c d e-a B e^2\right ) x}{a+c x^2} \, dx}{c}\\ &=\frac{e (2 B d+A e) x}{c}+\frac{B e^2 x^2}{2 c}+\frac{\left (A c d^2-2 a B d e-a A e^2\right ) \int \frac{1}{a+c x^2} \, dx}{c}+\frac{\left (B c d^2+2 A c d e-a B e^2\right ) \int \frac{x}{a+c x^2} \, dx}{c}\\ &=\frac{e (2 B d+A e) x}{c}+\frac{B e^2 x^2}{2 c}+\frac{\left (A c d^2-2 a B d e-a A e^2\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{\sqrt{a} c^{3/2}}+\frac{\left (B c d^2+2 A c d e-a B e^2\right ) \log \left (a+c x^2\right )}{2 c^2}\\ \end{align*}
Mathematica [A] time = 0.0970936, size = 99, normalized size = 0.92 \[ \frac{\log \left (a+c x^2\right ) \left (-a B e^2+2 A c d e+B c d^2\right )-\frac{2 \sqrt{c} \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right ) \left (a A e^2+2 a B d e-A c d^2\right )}{\sqrt{a}}+c e x (2 A e+4 B d+B e x)}{2 c^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.006, size = 148, normalized size = 1.4 \begin{align*}{\frac{B{e}^{2}{x}^{2}}{2\,c}}+{\frac{A{e}^{2}x}{c}}+2\,{\frac{Bdex}{c}}+{\frac{\ln \left ( c{x}^{2}+a \right ) Ade}{c}}-{\frac{\ln \left ( c{x}^{2}+a \right ) aB{e}^{2}}{2\,{c}^{2}}}+{\frac{\ln \left ( c{x}^{2}+a \right ) B{d}^{2}}{2\,c}}-{\frac{Aa{e}^{2}}{c}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}+{A{d}^{2}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}-2\,{\frac{aBde}{c\sqrt{ac}}\arctan \left ({\frac{cx}{\sqrt{ac}}} \right ) } \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 = 1.97686, size = 533, normalized size = 4.94 \begin{align*} \left [\frac{B a c e^{2} x^{2} +{\left (A c d^{2} - 2 \, B a d e - A a e^{2}\right )} \sqrt{-a c} \log \left (\frac{c x^{2} + 2 \, \sqrt{-a c} x - a}{c x^{2} + a}\right ) + 2 \,{\left (2 \, B a c d e + A a c e^{2}\right )} x +{\left (B a c d^{2} + 2 \, A a c d e - B a^{2} e^{2}\right )} \log \left (c x^{2} + a\right )}{2 \, a c^{2}}, \frac{B a c e^{2} x^{2} + 2 \,{\left (A c d^{2} - 2 \, B a d e - A a e^{2}\right )} \sqrt{a c} \arctan \left (\frac{\sqrt{a c} x}{a}\right ) + 2 \,{\left (2 \, B a c d e + A a c e^{2}\right )} x +{\left (B a c d^{2} + 2 \, A a c d e - B a^{2} e^{2}\right )} \log \left (c x^{2} + a\right )}{2 \, a c^{2}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] time = 3.14249, size = 423, normalized size = 3.92 \begin{align*} \frac{B e^{2} x^{2}}{2 c} + \left (- \frac{- 2 A c d e + B a e^{2} - B c d^{2}}{2 c^{2}} - \frac{\sqrt{- a c^{5}} \left (A a e^{2} - A c d^{2} + 2 B a d e\right )}{2 a c^{4}}\right ) \log{\left (x + \frac{2 A a c d e - B a^{2} e^{2} + B a c d^{2} - 2 a c^{2} \left (- \frac{- 2 A c d e + B a e^{2} - B c d^{2}}{2 c^{2}} - \frac{\sqrt{- a c^{5}} \left (A a e^{2} - A c d^{2} + 2 B a d e\right )}{2 a c^{4}}\right )}{A a c e^{2} - A c^{2} d^{2} + 2 B a c d e} \right )} + \left (- \frac{- 2 A c d e + B a e^{2} - B c d^{2}}{2 c^{2}} + \frac{\sqrt{- a c^{5}} \left (A a e^{2} - A c d^{2} + 2 B a d e\right )}{2 a c^{4}}\right ) \log{\left (x + \frac{2 A a c d e - B a^{2} e^{2} + B a c d^{2} - 2 a c^{2} \left (- \frac{- 2 A c d e + B a e^{2} - B c d^{2}}{2 c^{2}} + \frac{\sqrt{- a c^{5}} \left (A a e^{2} - A c d^{2} + 2 B a d e\right )}{2 a c^{4}}\right )}{A a c e^{2} - A c^{2} d^{2} + 2 B a c d e} \right )} + \frac{x \left (A e^{2} + 2 B d e\right )}{c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.13471, size = 136, normalized size = 1.26 \begin{align*} \frac{{\left (A c d^{2} - 2 \, B a d e - A a e^{2}\right )} \arctan \left (\frac{c x}{\sqrt{a c}}\right )}{\sqrt{a c} c} + \frac{{\left (B c d^{2} + 2 \, A c d e - B a e^{2}\right )} \log \left (c x^{2} + a\right )}{2 \, c^{2}} + \frac{B c x^{2} e^{2} + 4 \, B c d x e + 2 \, A c x e^{2}}{2 \, c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]