Optimal. Leaf size=124 \[ \frac{(b d-2 c e) \tanh ^{-1}\left (\frac{2 a x+b}{\sqrt{b^2-4 a c}}\right )}{\sqrt{b^2-4 a c} \left (a d^2-e (b d-c e)\right )}+\frac{d \log \left (a x^2+b x+c\right )}{2 \left (a d^2-e (b d-c e)\right )}-\frac{d \log (d+e x)}{a d^2-e (b d-c e)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.14495, antiderivative size = 124, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24, Rules used = {1569, 800, 634, 618, 206, 628} \[ \frac{(b d-2 c e) \tanh ^{-1}\left (\frac{2 a x+b}{\sqrt{b^2-4 a c}}\right )}{\sqrt{b^2-4 a c} \left (a d^2-e (b d-c e)\right )}+\frac{d \log \left (a x^2+b x+c\right )}{2 \left (a d^2-e (b d-c e)\right )}-\frac{d \log (d+e x)}{a d^2-e (b d-c e)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 1569
Rule 800
Rule 634
Rule 618
Rule 206
Rule 628
Rubi steps
\begin{align*} \int \frac{1}{\left (a+\frac{c}{x^2}+\frac{b}{x}\right ) x (d+e x)} \, dx &=\int \frac{x}{(d+e x) \left (c+b x+a x^2\right )} \, dx\\ &=\int \left (\frac{d e}{\left (-a d^2+e (b d-c e)\right ) (d+e x)}+\frac{c e+a d x}{\left (a d^2-e (b d-c e)\right ) \left (c+b x+a x^2\right )}\right ) \, dx\\ &=-\frac{d \log (d+e x)}{a d^2-b d e+c e^2}+\frac{\int \frac{c e+a d x}{c+b x+a x^2} \, dx}{a d^2-e (b d-c e)}\\ &=-\frac{d \log (d+e x)}{a d^2-b d e+c e^2}+\frac{d \int \frac{b+2 a x}{c+b x+a x^2} \, dx}{2 \left (a d^2-b d e+c e^2\right )}+\frac{(-b d+2 c e) \int \frac{1}{c+b x+a x^2} \, dx}{2 \left (a d^2-e (b d-c e)\right )}\\ &=-\frac{d \log (d+e x)}{a d^2-b d e+c e^2}+\frac{d \log \left (c+b x+a x^2\right )}{2 \left (a d^2-b d e+c e^2\right )}+\frac{(b d-2 c e) \operatorname{Subst}\left (\int \frac{1}{b^2-4 a c-x^2} \, dx,x,b+2 a x\right )}{a d^2-e (b d-c e)}\\ &=\frac{(b d-2 c e) \tanh ^{-1}\left (\frac{b+2 a x}{\sqrt{b^2-4 a c}}\right )}{\sqrt{b^2-4 a c} \left (a d^2-e (b d-c e)\right )}-\frac{d \log (d+e x)}{a d^2-b d e+c e^2}+\frac{d \log \left (c+b x+a x^2\right )}{2 \left (a d^2-b d e+c e^2\right )}\\ \end{align*}
Mathematica [A] time = 0.0789739, size = 107, normalized size = 0.86 \[ \frac{d \sqrt{4 a c-b^2} (2 \log (d+e x)-\log (x (a x+b)+c))+2 (b d-2 c e) \tan ^{-1}\left (\frac{2 a x+b}{\sqrt{4 a c-b^2}}\right )}{2 \sqrt{4 a c-b^2} \left (e (b d-c e)-a d^2\right )} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.003, size = 169, normalized size = 1.4 \begin{align*} -{\frac{d\ln \left ( ex+d \right ) }{a{d}^{2}-bde+{e}^{2}c}}+{\frac{d\ln \left ( a{x}^{2}+bx+c \right ) }{2\,a{d}^{2}-2\,bde+2\,{e}^{2}c}}-{\frac{bd}{a{d}^{2}-bde+{e}^{2}c}\arctan \left ({(2\,ax+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}+2\,{\frac{ce}{ \left ( a{d}^{2}-bde+{e}^{2}c \right ) \sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,ax+b}{\sqrt{4\,ac-{b}^{2}}}} \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 = 3.2561, size = 694, normalized size = 5.6 \begin{align*} \left [\frac{{\left (b^{2} - 4 \, a c\right )} d \log \left (a x^{2} + b x + c\right ) - 2 \,{\left (b^{2} - 4 \, a c\right )} d \log \left (e x + d\right ) - \sqrt{b^{2} - 4 \, a c}{\left (b d - 2 \, c e\right )} \log \left (\frac{2 \, a^{2} x^{2} + 2 \, a b x + b^{2} - 2 \, a c - \sqrt{b^{2} - 4 \, a c}{\left (2 \, a x + b\right )}}{a x^{2} + b x + c}\right )}{2 \,{\left ({\left (a b^{2} - 4 \, a^{2} c\right )} d^{2} -{\left (b^{3} - 4 \, a b c\right )} d e +{\left (b^{2} c - 4 \, a c^{2}\right )} e^{2}\right )}}, \frac{{\left (b^{2} - 4 \, a c\right )} d \log \left (a x^{2} + b x + c\right ) - 2 \,{\left (b^{2} - 4 \, a c\right )} d \log \left (e x + d\right ) + 2 \, \sqrt{-b^{2} + 4 \, a c}{\left (b d - 2 \, c e\right )} \arctan \left (-\frac{\sqrt{-b^{2} + 4 \, a c}{\left (2 \, a x + b\right )}}{b^{2} - 4 \, a c}\right )}{2 \,{\left ({\left (a b^{2} - 4 \, a^{2} c\right )} d^{2} -{\left (b^{3} - 4 \, a b c\right )} d e +{\left (b^{2} c - 4 \, a c^{2}\right )} e^{2}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.10904, size = 171, normalized size = 1.38 \begin{align*} -\frac{d e \log \left ({\left | x e + d \right |}\right )}{a d^{2} e - b d e^{2} + c e^{3}} + \frac{d \log \left (a x^{2} + b x + c\right )}{2 \,{\left (a d^{2} - b d e + c e^{2}\right )}} - \frac{{\left (b d - 2 \, c e\right )} \arctan \left (\frac{2 \, a x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{{\left (a d^{2} - b d e + c e^{2}\right )} \sqrt{-b^{2} + 4 \, a c}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]