Optimal. Leaf size=85 \[ \frac {\left (2 a c e+b^2 (-e)+b c d\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{c^2 \sqrt {b^2-4 a c}}+\frac {(c d-b e) \log \left (a+b x+c x^2\right )}{2 c^2}+\frac {e x}{c} \]
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Rubi [A] time = 0.08, antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {773, 634, 618, 206, 628} \begin {gather*} \frac {\left (2 a c e+b^2 (-e)+b c d\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{c^2 \sqrt {b^2-4 a c}}+\frac {(c d-b e) \log \left (a+b x+c x^2\right )}{2 c^2}+\frac {e x}{c} \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 {x (d+e x)}{a+b x+c x^2} \, dx &=\frac {e x}{c}+\frac {\int \frac {-a e+(c d-b e) x}{a+b x+c x^2} \, dx}{c}\\ &=\frac {e x}{c}+\frac {(c d-b e) \int \frac {b+2 c x}{a+b x+c x^2} \, dx}{2 c^2}-\frac {\left (b c d-b^2 e+2 a c e\right ) \int \frac {1}{a+b x+c x^2} \, dx}{2 c^2}\\ &=\frac {e x}{c}+\frac {(c d-b e) \log \left (a+b x+c x^2\right )}{2 c^2}+\frac {\left (b c d-b^2 e+2 a c e\right ) \operatorname {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c x\right )}{c^2}\\ &=\frac {e x}{c}+\frac {\left (b c d-b^2 e+2 a c e\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{c^2 \sqrt {b^2-4 a c}}+\frac {(c d-b e) \log \left (a+b x+c x^2\right )}{2 c^2}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 86, normalized size = 1.01 \begin {gather*} \frac {\frac {2 \left (-2 a c e+b^2 e-b c d\right ) \tan ^{-1}\left (\frac {b+2 c x}{\sqrt {4 a c-b^2}}\right )}{\sqrt {4 a c-b^2}}+(c d-b e) \log (a+x (b+c x))+2 c e x}{2 c^2} \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 {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 = 0.43, size = 289, normalized size = 3.40 \begin {gather*} \left [\frac {2 \, {\left (b^{2} c - 4 \, a c^{2}\right )} e x + {\left (b c d - {\left (b^{2} - 2 \, a c\right )} e\right )} \sqrt {b^{2} - 4 \, a c} \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 ({\left (b^{2} c - 4 \, a c^{2}\right )} d - {\left (b^{3} - 4 \, a b c\right )} e\right )} \log \left (c x^{2} + b x + a\right )}{2 \, {\left (b^{2} c^{2} - 4 \, a c^{3}\right )}}, \frac {2 \, {\left (b^{2} c - 4 \, a c^{2}\right )} e x + 2 \, {\left (b c d - {\left (b^{2} - 2 \, a c\right )} e\right )} \sqrt {-b^{2} + 4 \, a c} \arctan \left (-\frac {\sqrt {-b^{2} + 4 \, a c} {\left (2 \, c x + b\right )}}{b^{2} - 4 \, a c}\right ) + {\left ({\left (b^{2} c - 4 \, a c^{2}\right )} d - {\left (b^{3} - 4 \, a b c\right )} e\right )} \log \left (c x^{2} + b x + a\right )}{2 \, {\left (b^{2} c^{2} - 4 \, a c^{3}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 88, normalized size = 1.04 \begin {gather*} \frac {x e}{c} + \frac {{\left (c d - b e\right )} \log \left (c x^{2} + b x + a\right )}{2 \, c^{2}} - \frac {{\left (b c d - b^{2} e + 2 \, 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^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.04, size = 161, normalized size = 1.89 \begin {gather*} -\frac {2 a e \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}\, c}+\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^{2}}-\frac {b d \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^{2}}+\frac {d \ln \left (c \,x^{2}+b x +a \right )}{2 c}+\frac {e x}{c} \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 = 1.34, size = 127, normalized size = 1.49 \begin {gather*} \frac {\ln \left (c\,x^2+b\,x+a\right )\,\left (e\,b^3-d\,b^2\,c-4\,a\,e\,b\,c+4\,a\,d\,c^2\right )}{2\,\left (4\,a\,c^3-b^2\,c^2\right )}+\frac {e\,x}{c}-\frac {\mathrm {atan}\left (\frac {b}{\sqrt {4\,a\,c-b^2}}+\frac {2\,c\,x}{\sqrt {4\,a\,c-b^2}}\right )\,\left (-e\,b^2+c\,d\,b+2\,a\,c\,e\right )}{c^2\,\sqrt {4\,a\,c-b^2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 1.35, size = 423, normalized size = 4.98 \begin {gather*} \left (- \frac {\sqrt {- 4 a c + b^{2}} \left (2 a c e - b^{2} e + b c d\right )}{2 c^{2} \left (4 a c - b^{2}\right )} - \frac {b e - c d}{2 c^{2}}\right ) \log {\left (x + \frac {- a b e - 4 a c^{2} \left (- \frac {\sqrt {- 4 a c + b^{2}} \left (2 a c e - b^{2} e + b c d\right )}{2 c^{2} \left (4 a c - b^{2}\right )} - \frac {b e - c d}{2 c^{2}}\right ) + 2 a c d + b^{2} c \left (- \frac {\sqrt {- 4 a c + b^{2}} \left (2 a c e - b^{2} e + b c d\right )}{2 c^{2} \left (4 a c - b^{2}\right )} - \frac {b e - c d}{2 c^{2}}\right )}{2 a c e - b^{2} e + b c d} \right )} + \left (\frac {\sqrt {- 4 a c + b^{2}} \left (2 a c e - b^{2} e + b c d\right )}{2 c^{2} \left (4 a c - b^{2}\right )} - \frac {b e - c d}{2 c^{2}}\right ) \log {\left (x + \frac {- a b e - 4 a c^{2} \left (\frac {\sqrt {- 4 a c + b^{2}} \left (2 a c e - b^{2} e + b c d\right )}{2 c^{2} \left (4 a c - b^{2}\right )} - \frac {b e - c d}{2 c^{2}}\right ) + 2 a c d + b^{2} c \left (\frac {\sqrt {- 4 a c + b^{2}} \left (2 a c e - b^{2} e + b c d\right )}{2 c^{2} \left (4 a c - b^{2}\right )} - \frac {b e - c d}{2 c^{2}}\right )}{2 a c e - b^{2} e + b c d} \right )} + \frac {e x}{c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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