Optimal. Leaf size=38 \[ \frac{2 \tan ^{-1}\left (\frac{2 b x+c}{\sqrt{4 a b-c^2}}\right )}{\sqrt{4 a b-c^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0340968, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {618, 204} \[ \frac{2 \tan ^{-1}\left (\frac{2 b x+c}{\sqrt{4 a b-c^2}}\right )}{\sqrt{4 a b-c^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 618
Rule 204
Rubi steps
\begin{align*} \int \frac{1}{a+c x+b x^2} \, dx &=-\left (2 \operatorname{Subst}\left (\int \frac{1}{-4 a b+c^2-x^2} \, dx,x,c+2 b x\right )\right )\\ &=\frac{2 \tan ^{-1}\left (\frac{c+2 b x}{\sqrt{4 a b-c^2}}\right )}{\sqrt{4 a b-c^2}}\\ \end{align*}
Mathematica [A] time = 0.0111292, size = 38, normalized size = 1. \[ \frac{2 \tan ^{-1}\left (\frac{2 b x+c}{\sqrt{4 a b-c^2}}\right )}{\sqrt{4 a b-c^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.223, size = 35, normalized size = 0.9 \begin{align*} 2\,{\frac{1}{\sqrt{4\,ab-{c}^{2}}}\arctan \left ({\frac{2\,bx+c}{\sqrt{4\,ab-{c}^{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 = 2.09181, size = 259, normalized size = 6.82 \begin{align*} \left [-\frac{\sqrt{-4 \, a b + c^{2}} \log \left (\frac{2 \, b^{2} x^{2} + 2 \, b c x - 2 \, a b + c^{2} - \sqrt{-4 \, a b + c^{2}}{\left (2 \, b x + c\right )}}{b x^{2} + c x + a}\right )}{4 \, a b - c^{2}}, -\frac{2 \, \arctan \left (-\frac{2 \, b x + c}{\sqrt{4 \, a b - c^{2}}}\right )}{\sqrt{4 \, a b - c^{2}}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] time = 0.290812, size = 124, normalized size = 3.26 \begin{align*} - \sqrt{- \frac{1}{4 a b - c^{2}}} \log{\left (x + \frac{- 4 a b \sqrt{- \frac{1}{4 a b - c^{2}}} + c^{2} \sqrt{- \frac{1}{4 a b - c^{2}}} + c}{2 b} \right )} + \sqrt{- \frac{1}{4 a b - c^{2}}} \log{\left (x + \frac{4 a b \sqrt{- \frac{1}{4 a b - c^{2}}} - c^{2} \sqrt{- \frac{1}{4 a b - c^{2}}} + c}{2 b} \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.18351, size = 46, normalized size = 1.21 \begin{align*} \frac{2 \, \arctan \left (\frac{2 \, b x + c}{\sqrt{4 \, a b - c^{2}}}\right )}{\sqrt{4 \, a b - c^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]