Optimal. Leaf size=81 \[ -\frac {\left (C \left (b^2-2 a c\right )+2 A c^2\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{c^2 \sqrt {b^2-4 a c}}-\frac {b C \log \left (a+b x+c x^2\right )}{2 c^2}+\frac {C x}{c} \]
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Rubi [A] time = 0.10, antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1657, 634, 618, 206, 628} \[ -\frac {\left (C \left (b^2-2 a c\right )+2 A c^2\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{c^2 \sqrt {b^2-4 a c}}-\frac {b C \log \left (a+b x+c x^2\right )}{2 c^2}+\frac {C x}{c} \]
Antiderivative was successfully verified.
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Rule 206
Rule 618
Rule 628
Rule 634
Rule 1657
Rubi steps
\begin {align*} \int \frac {A+C x^2}{a+b x+c x^2} \, dx &=\int \left (\frac {C}{c}+\frac {A c-a C-b C x}{c \left (a+b x+c x^2\right )}\right ) \, dx\\ &=\frac {C x}{c}+\frac {\int \frac {A c-a C-b C x}{a+b x+c x^2} \, dx}{c}\\ &=\frac {C x}{c}-\frac {(b C) \int \frac {b+2 c x}{a+b x+c x^2} \, dx}{2 c^2}+\frac {1}{2} \left (2 A+\frac {\left (b^2-2 a c\right ) C}{c^2}\right ) \int \frac {1}{a+b x+c x^2} \, dx\\ &=\frac {C x}{c}-\frac {b C \log \left (a+b x+c x^2\right )}{2 c^2}+\left (-2 A-\frac {\left (b^2-2 a c\right ) C}{c^2}\right ) \operatorname {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c x\right )\\ &=\frac {C x}{c}-\frac {\left (2 A+\frac {\left (b^2-2 a c\right ) C}{c^2}\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\sqrt {b^2-4 a c}}-\frac {b C \log \left (a+b x+c x^2\right )}{2 c^2}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 84, normalized size = 1.04 \[ \frac {\left (-2 a c C+2 A c^2+b^2 C\right ) \tan ^{-1}\left (\frac {b+2 c x}{\sqrt {4 a c-b^2}}\right )}{c^2 \sqrt {4 a c-b^2}}-\frac {b C \log \left (a+b x+c x^2\right )}{2 c^2}+\frac {C x}{c} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.77, size = 265, normalized size = 3.27 \[ \left [\frac {{\left (C b^{2} - 2 \, C a c + 2 \, A c^{2}\right )} \sqrt {b^{2} - 4 \, a c} \log \left (\frac {2 \, c^{2} x^{2} + 2 \, b c x + b^{2} - 2 \, a c - \sqrt {b^{2} - 4 \, a c} {\left (2 \, c x + b\right )}}{c x^{2} + b x + a}\right ) + 2 \, {\left (C b^{2} c - 4 \, C a c^{2}\right )} x - {\left (C b^{3} - 4 \, C a b c\right )} \log \left (c x^{2} + b x + a\right )}{2 \, {\left (b^{2} c^{2} - 4 \, a c^{3}\right )}}, -\frac {2 \, {\left (C b^{2} - 2 \, C a c + 2 \, A c^{2}\right )} \sqrt {-b^{2} + 4 \, a c} \arctan \left (-\frac {\sqrt {-b^{2} + 4 \, a c} {\left (2 \, c x + b\right )}}{b^{2} - 4 \, a c}\right ) - 2 \, {\left (C b^{2} c - 4 \, C a c^{2}\right )} x + {\left (C b^{3} - 4 \, C a b c\right )} \log \left (c x^{2} + b x + a\right )}{2 \, {\left (b^{2} c^{2} - 4 \, a c^{3}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 78, normalized size = 0.96 \[ \frac {C x}{c} - \frac {C b \log \left (c x^{2} + b x + a\right )}{2 \, c^{2}} + \frac {{\left (C b^{2} - 2 \, C a c + 2 \, A c^{2}\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{\sqrt {-b^{2} + 4 \, a c} c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 140, normalized size = 1.73 \[ \frac {2 A \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}-\frac {2 C a \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}\, c}+\frac {C \,b^{2} \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}\, c^{2}}-\frac {C b \ln \left (c \,x^{2}+b x +a \right )}{2 c^{2}}+\frac {C x}{c} \]
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 = 0.19, size = 224, normalized size = 2.77 \[ \frac {2\,A\,\mathrm {atan}\left (\frac {b}{\sqrt {4\,a\,c-b^2}}+\frac {2\,c\,x}{\sqrt {4\,a\,c-b^2}}\right )}{\sqrt {4\,a\,c-b^2}}+\frac {C\,x}{c}+\frac {C\,b^3\,\ln \left (c\,x^2+b\,x+a\right )}{2\,\left (4\,a\,c^3-b^2\,c^2\right )}-\frac {2\,C\,a\,\mathrm {atan}\left (\frac {b}{\sqrt {4\,a\,c-b^2}}+\frac {2\,c\,x}{\sqrt {4\,a\,c-b^2}}\right )}{c\,\sqrt {4\,a\,c-b^2}}+\frac {C\,b^2\,\mathrm {atan}\left (\frac {b}{\sqrt {4\,a\,c-b^2}}+\frac {2\,c\,x}{\sqrt {4\,a\,c-b^2}}\right )}{c^2\,\sqrt {4\,a\,c-b^2}}-\frac {2\,C\,a\,b\,c\,\ln \left (c\,x^2+b\,x+a\right )}{4\,a\,c^3-b^2\,c^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 1.21, size = 413, normalized size = 5.10 \[ \frac {C x}{c} + \left (- \frac {C b}{2 c^{2}} - \frac {\sqrt {- 4 a c + b^{2}} \left (- 2 A c^{2} + 2 C a c - C b^{2}\right )}{2 c^{2} \left (4 a c - b^{2}\right )}\right ) \log {\left (x + \frac {- A b c - C a b - 4 a c^{2} \left (- \frac {C b}{2 c^{2}} - \frac {\sqrt {- 4 a c + b^{2}} \left (- 2 A c^{2} + 2 C a c - C b^{2}\right )}{2 c^{2} \left (4 a c - b^{2}\right )}\right ) + b^{2} c \left (- \frac {C b}{2 c^{2}} - \frac {\sqrt {- 4 a c + b^{2}} \left (- 2 A c^{2} + 2 C a c - C b^{2}\right )}{2 c^{2} \left (4 a c - b^{2}\right )}\right )}{- 2 A c^{2} + 2 C a c - C b^{2}} \right )} + \left (- \frac {C b}{2 c^{2}} + \frac {\sqrt {- 4 a c + b^{2}} \left (- 2 A c^{2} + 2 C a c - C b^{2}\right )}{2 c^{2} \left (4 a c - b^{2}\right )}\right ) \log {\left (x + \frac {- A b c - C a b - 4 a c^{2} \left (- \frac {C b}{2 c^{2}} + \frac {\sqrt {- 4 a c + b^{2}} \left (- 2 A c^{2} + 2 C a c - C b^{2}\right )}{2 c^{2} \left (4 a c - b^{2}\right )}\right ) + b^{2} c \left (- \frac {C b}{2 c^{2}} + \frac {\sqrt {- 4 a c + b^{2}} \left (- 2 A c^{2} + 2 C a c - C b^{2}\right )}{2 c^{2} \left (4 a c - b^{2}\right )}\right )}{- 2 A c^{2} + 2 C a c - C b^{2}} \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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