Optimal. Leaf size=70 \[ -\frac {e \sqrt {b^2-4 a c} \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{c}+\frac {(2 c d-b e) \log \left (a+b x+c x^2\right )}{2 c}+2 e x \]
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Rubi [A] time = 0.09, antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {773, 634, 618, 206, 628} \begin {gather*} -\frac {e \sqrt {b^2-4 a c} \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{c}+\frac {(2 c d-b e) \log \left (a+b x+c x^2\right )}{2 c}+2 e x \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 618
Rule 628
Rule 634
Rule 773
Rubi steps
\begin {align*} \int \frac {(b+2 c x) (d+e x)}{a+b x+c x^2} \, dx &=2 e x+\frac {\int \frac {b c d-2 a c e+\left (2 c^2 d-b c e\right ) x}{a+b x+c x^2} \, dx}{c}\\ &=2 e x+\frac {\left (\left (b^2-4 a c\right ) e\right ) \int \frac {1}{a+b x+c x^2} \, dx}{2 c}+\frac {(2 c d-b e) \int \frac {b+2 c x}{a+b x+c x^2} \, dx}{2 c}\\ &=2 e x+\frac {(2 c d-b e) \log \left (a+b x+c x^2\right )}{2 c}-\frac {\left (\left (b^2-4 a c\right ) e\right ) \operatorname {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c x\right )}{c}\\ &=2 e x-\frac {\sqrt {b^2-4 a c} e \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{c}+\frac {(2 c d-b e) \log \left (a+b x+c x^2\right )}{2 c}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 72, normalized size = 1.03 \begin {gather*} \frac {-2 e \sqrt {4 a c-b^2} \tan ^{-1}\left (\frac {b+2 c x}{\sqrt {4 a c-b^2}}\right )+(2 c d-b e) \log (a+x (b+c x))+4 c e x}{2 c} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(b+2 c x) (d+e x)}{a+b x+c x^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.42, size = 176, normalized size = 2.51 \begin {gather*} \left [\frac {4 \, c e x + \sqrt {b^{2} - 4 \, a c} e \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 ) + {\left (2 \, c d - b e\right )} \log \left (c x^{2} + b x + a\right )}{2 \, c}, \frac {4 \, c e x - 2 \, \sqrt {-b^{2} + 4 \, a c} e \arctan \left (-\frac {\sqrt {-b^{2} + 4 \, a c} {\left (2 \, c x + b\right )}}{b^{2} - 4 \, a c}\right ) + {\left (2 \, c d - b e\right )} \log \left (c x^{2} + b x + a\right )}{2 \, c}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 81, normalized size = 1.16 \begin {gather*} 2 \, x e + \frac {{\left (2 \, c d - b e\right )} \log \left (c x^{2} + b x + a\right )}{2 \, c} + \frac {{\left (b^{2} e - 4 \, a c 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.
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maple [A] time = 0.06, size = 113, normalized size = 1.61 \begin {gather*} -\frac {4 a e \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}+\frac {b^{2} e \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}\, c}-\frac {b e \ln \left (c \,x^{2}+b x +a \right )}{2 c}+d \ln \left (c \,x^{2}+b x +a \right )+2 e x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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mupad [B] time = 0.32, size = 129, normalized size = 1.84 \begin {gather*} \ln \left (b\,\sqrt {b^2-4\,a\,c}-4\,a\,c+b^2+2\,c\,x\,\sqrt {b^2-4\,a\,c}\right )\,\left (d-\frac {\frac {b\,e}{2}+\frac {e\,\sqrt {b^2-4\,a\,c}}{2}}{c}\right )+2\,e\,x+\ln \left (4\,a\,c+b\,\sqrt {b^2-4\,a\,c}-b^2+2\,c\,x\,\sqrt {b^2-4\,a\,c}\right )\,\left (d-\frac {\frac {b\,e}{2}-\frac {e\,\sqrt {b^2-4\,a\,c}}{2}}{c}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.71, size = 134, normalized size = 1.91 \begin {gather*} 2 e x + \left (- \frac {e \sqrt {- 4 a c + b^{2}}}{2 c} - \frac {b e - 2 c d}{2 c}\right ) \log {\left (x + \frac {d + \frac {e \sqrt {- 4 a c + b^{2}}}{2 c} + \frac {b e - 2 c d}{2 c}}{e} \right )} + \left (\frac {e \sqrt {- 4 a c + b^{2}}}{2 c} - \frac {b e - 2 c d}{2 c}\right ) \log {\left (x + \frac {d - \frac {e \sqrt {- 4 a c + b^{2}}}{2 c} + \frac {b e - 2 c d}{2 c}}{e} \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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