Optimal. Leaf size=38 \[ \frac {2 \tan ^{-1}\left (\frac {c+2 b x}{\sqrt {4 a b-c^2}}\right )}{\sqrt {4 a b-c^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.02, antiderivative size = 38, normalized size of antiderivative = 1.00, 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 = {632, 210}
\begin {gather*} \frac {2 \text {ArcTan}\left (\frac {2 b x+c}{\sqrt {4 a b-c^2}}\right )}{\sqrt {4 a b-c^2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 210
Rule 632
Rubi steps
\begin {align*} \int \frac {1}{a+c x+b x^2} \, dx &=-\left (2 \text {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 = 0.01, size = 38, normalized size = 1.00 \begin {gather*} \frac {2 \tan ^{-1}\left (\frac {c+2 b x}{\sqrt {4 a b-c^2}}\right )}{\sqrt {4 a b-c^2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.64, size = 35, normalized size = 0.92
method | result | size |
default | \(\frac {2 \arctan \left (\frac {2 b x +c}{\sqrt {4 a b -c^{2}}}\right )}{\sqrt {4 a b -c^{2}}}\) | \(35\) |
risch | \(-\frac {\ln \left (2 b x +\sqrt {-4 a b +c^{2}}+c \right )}{\sqrt {-4 a b +c^{2}}}+\frac {\ln \left (-2 b x +\sqrt {-4 a b +c^{2}}-c \right )}{\sqrt {-4 a b +c^{2}}}\) | \(61\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 1.40, size = 113, normalized size = 2.97 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 124 vs.
\(2 (32) = 64\).
time = 0.09, size = 124, normalized size = 3.26 \begin {gather*} - \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 1.30, size = 34, normalized size = 0.89 \begin {gather*} \frac {2 \, \arctan \left (\frac {2 \, b x + c}{\sqrt {4 \, a b - c^{2}}}\right )}{\sqrt {4 \, a b - c^{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.23, size = 46, normalized size = 1.21 \begin {gather*} \frac {2\,\mathrm {atan}\left (\frac {c}{\sqrt {4\,a\,b-c^2}}+\frac {2\,b\,x}{\sqrt {4\,a\,b-c^2}}\right )}{\sqrt {4\,a\,b-c^2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________