Optimal. Leaf size=66 \[ -\frac {(2 c d-b e) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{c \sqrt {b^2-4 a c}}+\frac {e \log \left (a+b x+c x^2\right )}{2 c} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.02, antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {648, 632, 212,
642} \begin {gather*} \frac {e \log \left (a+b x+c x^2\right )}{2 c}-\frac {(2 c d-b e) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{c \sqrt {b^2-4 a c}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 212
Rule 632
Rule 642
Rule 648
Rubi steps
\begin {align*} \int \frac {d+e x}{a+b x+c x^2} \, dx &=\frac {e \int \frac {b+2 c x}{a+b x+c x^2} \, dx}{2 c}+\frac {(2 c d-b e) \int \frac {1}{a+b x+c x^2} \, dx}{2 c}\\ &=\frac {e \log \left (a+b x+c x^2\right )}{2 c}-\frac {(2 c d-b e) \text {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c x\right )}{c}\\ &=-\frac {(2 c d-b e) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{c \sqrt {b^2-4 a c}}+\frac {e \log \left (a+b x+c x^2\right )}{2 c}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.04, size = 66, normalized size = 1.00 \begin {gather*} \frac {-\frac {2 (-2 c d+b e) \tan ^{-1}\left (\frac {b+2 c x}{\sqrt {-b^2+4 a c}}\right )}{\sqrt {-b^2+4 a c}}+e \log (a+x (b+c x))}{2 c} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.73, size = 62, normalized size = 0.94
method | result | size |
default | \(\frac {e \ln \left (c \,x^{2}+b x +a \right )}{2 c}+\frac {2 \left (d -\frac {b e}{2 c}\right ) \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}\) | \(62\) |
risch | \(\frac {2 \ln \left (-4 a b c e +8 a \,c^{2} d +e \,b^{3}-2 b^{2} c d -2 \sqrt {-\left (b e -2 c d \right )^{2} \left (4 a c -b^{2}\right )}\, c x -\sqrt {-\left (b e -2 c d \right )^{2} \left (4 a c -b^{2}\right )}\, b \right ) a e}{4 a c -b^{2}}-\frac {\ln \left (-4 a b c e +8 a \,c^{2} d +e \,b^{3}-2 b^{2} c d -2 \sqrt {-\left (b e -2 c d \right )^{2} \left (4 a c -b^{2}\right )}\, c x -\sqrt {-\left (b e -2 c d \right )^{2} \left (4 a c -b^{2}\right )}\, b \right ) b^{2} e}{2 c \left (4 a c -b^{2}\right )}+\frac {\ln \left (-4 a b c e +8 a \,c^{2} d +e \,b^{3}-2 b^{2} c d -2 \sqrt {-\left (b e -2 c d \right )^{2} \left (4 a c -b^{2}\right )}\, c x -\sqrt {-\left (b e -2 c d \right )^{2} \left (4 a c -b^{2}\right )}\, b \right ) \sqrt {-\left (b e -2 c d \right )^{2} \left (4 a c -b^{2}\right )}}{2 c \left (4 a c -b^{2}\right )}+\frac {2 \ln \left (-4 a b c e +8 a \,c^{2} d +e \,b^{3}-2 b^{2} c d +2 \sqrt {-\left (b e -2 c d \right )^{2} \left (4 a c -b^{2}\right )}\, c x +\sqrt {-\left (b e -2 c d \right )^{2} \left (4 a c -b^{2}\right )}\, b \right ) a e}{4 a c -b^{2}}-\frac {\ln \left (-4 a b c e +8 a \,c^{2} d +e \,b^{3}-2 b^{2} c d +2 \sqrt {-\left (b e -2 c d \right )^{2} \left (4 a c -b^{2}\right )}\, c x +\sqrt {-\left (b e -2 c d \right )^{2} \left (4 a c -b^{2}\right )}\, b \right ) b^{2} e}{2 c \left (4 a c -b^{2}\right )}-\frac {\ln \left (-4 a b c e +8 a \,c^{2} d +e \,b^{3}-2 b^{2} c d +2 \sqrt {-\left (b e -2 c d \right )^{2} \left (4 a c -b^{2}\right )}\, c x +\sqrt {-\left (b e -2 c d \right )^{2} \left (4 a c -b^{2}\right )}\, b \right ) \sqrt {-\left (b e -2 c d \right )^{2} \left (4 a c -b^{2}\right )}}{2 c \left (4 a c -b^{2}\right )}\) | \(647\) |
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 = 2.49, size = 208, normalized size = 3.15 \begin {gather*} \left [\frac {{\left (b^{2} - 4 \, a c\right )} e \log \left (c x^{2} + b x + a\right ) - \sqrt {b^{2} - 4 \, a c} {\left (2 \, c d - b e\right )} \log \left (\frac {2 \, c^{2} x^{2} + 2 \, b c x + b^{2} - 2 \, a c + \sqrt {b^{2} - 4 \, a c} {\left (2 \, c x + b\right )}}{c x^{2} + b x + a}\right )}{2 \, {\left (b^{2} c - 4 \, a c^{2}\right )}}, \frac {{\left (b^{2} - 4 \, a c\right )} e \log \left (c x^{2} + b x + a\right ) - 2 \, \sqrt {-b^{2} + 4 \, a c} {\left (2 \, c d - b e\right )} \arctan \left (-\frac {\sqrt {-b^{2} + 4 \, a c} {\left (2 \, c x + b\right )}}{b^{2} - 4 \, a c}\right )}{2 \, {\left (b^{2} c - 4 \, a c^{2}\right )}}\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. 280 vs.
\(2 (58) = 116\).
time = 0.44, size = 280, normalized size = 4.24 \begin {gather*} \left (\frac {e}{2 c} - \frac {\sqrt {- 4 a c + b^{2}} \left (b e - 2 c d\right )}{2 c \left (4 a c - b^{2}\right )}\right ) \log {\left (x + \frac {- 4 a c \left (\frac {e}{2 c} - \frac {\sqrt {- 4 a c + b^{2}} \left (b e - 2 c d\right )}{2 c \left (4 a c - b^{2}\right )}\right ) + 2 a e + b^{2} \left (\frac {e}{2 c} - \frac {\sqrt {- 4 a c + b^{2}} \left (b e - 2 c d\right )}{2 c \left (4 a c - b^{2}\right )}\right ) - b d}{b e - 2 c d} \right )} + \left (\frac {e}{2 c} + \frac {\sqrt {- 4 a c + b^{2}} \left (b e - 2 c d\right )}{2 c \left (4 a c - b^{2}\right )}\right ) \log {\left (x + \frac {- 4 a c \left (\frac {e}{2 c} + \frac {\sqrt {- 4 a c + b^{2}} \left (b e - 2 c d\right )}{2 c \left (4 a c - b^{2}\right )}\right ) + 2 a e + b^{2} \left (\frac {e}{2 c} + \frac {\sqrt {- 4 a c + b^{2}} \left (b e - 2 c d\right )}{2 c \left (4 a c - b^{2}\right )}\right ) - b d}{b e - 2 c d} \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.72, size = 65, normalized size = 0.98 \begin {gather*} \frac {e \log \left (c x^{2} + b x + a\right )}{2 \, c} + \frac {{\left (2 \, c d - b e\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{\sqrt {-b^{2} + 4 \, a c} c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.12, size = 162, normalized size = 2.45 \begin {gather*} \frac {2\,d\,\mathrm {atan}\left (\frac {b}{\sqrt {4\,a\,c-b^2}}+\frac {2\,c\,x}{\sqrt {4\,a\,c-b^2}}\right )}{\sqrt {4\,a\,c-b^2}}-\frac {b^2\,e\,\ln \left (c\,x^2+b\,x+a\right )}{2\,\left (4\,a\,c^2-b^2\,c\right )}+\frac {2\,a\,c\,e\,\ln \left (c\,x^2+b\,x+a\right )}{4\,a\,c^2-b^2\,c}-\frac {b\,e\,\mathrm {atan}\left (\frac {b}{\sqrt {4\,a\,c-b^2}}+\frac {2\,c\,x}{\sqrt {4\,a\,c-b^2}}\right )}{c\,\sqrt {4\,a\,c-b^2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________