Optimal. Leaf size=130 \[ \frac {e \sqrt {b^2-4 a c} \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{a e^2-b d e+c d^2}+\frac {(2 c d-b e) \log \left (a+b x+c x^2\right )}{2 \left (a e^2-b d e+c d^2\right )}-\frac {(2 c d-b e) \log (d+e x)}{a e^2-b d e+c d^2} \]
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Rubi [A] time = 0.18, antiderivative size = 130, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {800, 634, 618, 206, 628} \[ \frac {e \sqrt {b^2-4 a c} \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{a e^2-b d e+c d^2}+\frac {(2 c d-b e) \log \left (a+b x+c x^2\right )}{2 \left (a e^2-b d e+c d^2\right )}-\frac {(2 c d-b e) \log (d+e x)}{a e^2-b d e+c d^2} \]
Antiderivative was successfully verified.
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Rule 206
Rule 618
Rule 628
Rule 634
Rule 800
Rubi steps
\begin {align*} \int \frac {b+2 c x}{(d+e x) \left (a+b x+c x^2\right )} \, dx &=\int \left (\frac {e (-2 c d+b e)}{\left (c d^2-b d e+a e^2\right ) (d+e x)}+\frac {b c d-b^2 e+2 a c e+c (2 c d-b e) x}{\left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )}\right ) \, dx\\ &=-\frac {(2 c d-b e) \log (d+e x)}{c d^2-b d e+a e^2}+\frac {\int \frac {b c d-b^2 e+2 a c e+c (2 c d-b e) x}{a+b x+c x^2} \, dx}{c d^2-b d e+a e^2}\\ &=-\frac {(2 c d-b e) \log (d+e x)}{c d^2-b d e+a e^2}-\frac {\left (\left (b^2-4 a c\right ) e\right ) \int \frac {1}{a+b x+c x^2} \, dx}{2 \left (c d^2-b d e+a e^2\right )}+\frac {(2 c d-b e) \int \frac {b+2 c x}{a+b x+c x^2} \, dx}{2 \left (c d^2-b d e+a e^2\right )}\\ &=-\frac {(2 c d-b e) \log (d+e x)}{c d^2-b d e+a e^2}+\frac {(2 c d-b e) \log \left (a+b x+c x^2\right )}{2 \left (c d^2-b d e+a e^2\right )}+\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 d^2-b d e+a e^2}\\ &=\frac {\sqrt {b^2-4 a c} e \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{c d^2-b d e+a e^2}-\frac {(2 c d-b e) \log (d+e x)}{c d^2-b d e+a e^2}+\frac {(2 c d-b e) \log \left (a+b x+c x^2\right )}{2 \left (c d^2-b d e+a e^2\right )}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 116, normalized size = 0.89 \[ \frac {\sqrt {4 a c-b^2} (2 c d-b e) (2 \log (d+e x)-\log (a+x (b+c x)))+2 e \left (b^2-4 a c\right ) \tan ^{-1}\left (\frac {b+2 c x}{\sqrt {4 a c-b^2}}\right )}{2 \sqrt {4 a c-b^2} \left (e (b d-a e)-c d^2\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.82, size = 229, normalized size = 1.76 \[ \left [\frac {\sqrt {b^{2} - 4 \, a c} e \log \left (\frac {2 \, c^{2} x^{2} + 2 \, b c x + b^{2} - 2 \, a c + \sqrt {b^{2} - 4 \, a c} {\left (2 \, c x + b\right )}}{c x^{2} + b x + a}\right ) + {\left (2 \, c d - b e\right )} \log \left (c x^{2} + b x + a\right ) - 2 \, {\left (2 \, c d - b e\right )} \log \left (e x + d\right )}{2 \, {\left (c d^{2} - b d e + a e^{2}\right )}}, \frac {2 \, \sqrt {-b^{2} + 4 \, a c} e \arctan \left (-\frac {\sqrt {-b^{2} + 4 \, a c} {\left (2 \, c x + b\right )}}{b^{2} - 4 \, a c}\right ) + {\left (2 \, c d - b e\right )} \log \left (c x^{2} + b x + a\right ) - 2 \, {\left (2 \, c d - b e\right )} \log \left (e x + d\right )}{2 \, {\left (c d^{2} - b d e + a e^{2}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 149, normalized size = 1.15 \[ \frac {{\left (2 \, c d - b e\right )} \log \left (c x^{2} + b x + a\right )}{2 \, {\left (c d^{2} - b d e + a e^{2}\right )}} - \frac {{\left (2 \, c d e - b e^{2}\right )} \log \left ({\left | x e + d \right |}\right )}{c d^{2} e - b d e^{2} + a e^{3}} - \frac {{\left (b^{2} e - 4 \, a c e\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{{\left (c d^{2} - b d e + a e^{2}\right )} \sqrt {-b^{2} + 4 \, a c}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 233, normalized size = 1.79 \[ \frac {4 a c e \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\left (a \,e^{2}-b d e +c \,d^{2}\right ) \sqrt {4 a c -b^{2}}}-\frac {b^{2} e \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\left (a \,e^{2}-b d e +c \,d^{2}\right ) \sqrt {4 a c -b^{2}}}+\frac {b e \ln \left (e x +d \right )}{a \,e^{2}-b d e +c \,d^{2}}-\frac {b e \ln \left (c \,x^{2}+b x +a \right )}{2 \left (a \,e^{2}-b d e +c \,d^{2}\right )}-\frac {2 c d \ln \left (e x +d \right )}{a \,e^{2}-b d e +c \,d^{2}}+\frac {c d \ln \left (c \,x^{2}+b x +a \right )}{a \,e^{2}-b d e +c \,d^{2}} \]
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 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.38, size = 515, normalized size = 3.96 \[ \frac {\ln \left (2\,b\,c^2\,e-\frac {\left (c\,d-\frac {b\,e}{2}+\frac {e\,\sqrt {b^2-4\,a\,c}}{2}\right )\,\left (2\,a\,c^2\,e^2-b^2\,c\,e^2+b\,c^2\,d\,e-c^2\,e\,x\,\left (b\,e-2\,c\,d\right )+\frac {c\,e\,\left (c\,d-\frac {b\,e}{2}+\frac {e\,\sqrt {b^2-4\,a\,c}}{2}\right )\,\left (b^2\,d\,e+2\,x\,b^2\,e^2+b\,c\,d^2-2\,x\,b\,c\,d\,e+a\,b\,e^2+2\,x\,c^2\,d^2-8\,a\,c\,d\,e-6\,a\,x\,c\,e^2\right )}{c\,d^2-b\,d\,e+a\,e^2}\right )}{c\,d^2-b\,d\,e+a\,e^2}+4\,c^3\,e\,x\right )\,\left (c\,d-e\,\left (\frac {b}{2}-\frac {\sqrt {b^2-4\,a\,c}}{2}\right )\right )}{c\,d^2-b\,d\,e+a\,e^2}+\frac {\ln \left (2\,b\,c^2\,e-\frac {\left (\frac {b\,e}{2}-c\,d+\frac {e\,\sqrt {b^2-4\,a\,c}}{2}\right )\,\left (b^2\,c\,e^2-2\,a\,c^2\,e^2-b\,c^2\,d\,e+c^2\,e\,x\,\left (b\,e-2\,c\,d\right )+\frac {c\,e\,\left (\frac {b\,e}{2}-c\,d+\frac {e\,\sqrt {b^2-4\,a\,c}}{2}\right )\,\left (b^2\,d\,e+2\,x\,b^2\,e^2+b\,c\,d^2-2\,x\,b\,c\,d\,e+a\,b\,e^2+2\,x\,c^2\,d^2-8\,a\,c\,d\,e-6\,a\,x\,c\,e^2\right )}{c\,d^2-b\,d\,e+a\,e^2}\right )}{c\,d^2-b\,d\,e+a\,e^2}+4\,c^3\,e\,x\right )\,\left (c\,d-e\,\left (\frac {b}{2}+\frac {\sqrt {b^2-4\,a\,c}}{2}\right )\right )}{c\,d^2-b\,d\,e+a\,e^2}+\frac {\ln \left (d+e\,x\right )\,\left (b\,e-2\,c\,d\right )}{c\,d^2-b\,d\,e+a\,e^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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