Optimal. Leaf size=146 \[ \frac {\tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right ) (-2 a B e+A b e-2 A c d+b B d)}{\sqrt {b^2-4 a c} \left (a e^2-b d e+c d^2\right )}+\frac {(B d-A e) \log \left (a+b x+c x^2\right )}{2 \left (a e^2-b d e+c d^2\right )}-\frac {(B d-A e) \log (d+e x)}{a e^2-b d e+c d^2} \]
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Rubi [A] time = 0.22, antiderivative size = 146, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {800, 634, 618, 206, 628} \begin {gather*} \frac {\tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right ) (-2 a B e+A b e-2 A c d+b B d)}{\sqrt {b^2-4 a c} \left (a e^2-b d e+c d^2\right )}+\frac {(B d-A e) \log \left (a+b x+c x^2\right )}{2 \left (a e^2-b d e+c d^2\right )}-\frac {(B d-A e) \log (d+e x)}{a e^2-b d e+c d^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 618
Rule 628
Rule 634
Rule 800
Rubi steps
\begin {align*} \int \frac {A+B x}{(d+e x) \left (a+b x+c x^2\right )} \, dx &=\int \left (\frac {e (-B d+A e)}{\left (c d^2-b d e+a e^2\right ) (d+e x)}+\frac {A c d-A b e+a B e+c (B d-A e) x}{\left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )}\right ) \, dx\\ &=-\frac {(B d-A e) \log (d+e x)}{c d^2-b d e+a e^2}+\frac {\int \frac {A c d-A b e+a B e+c (B d-A e) x}{a+b x+c x^2} \, dx}{c d^2-b d e+a e^2}\\ &=-\frac {(B d-A e) \log (d+e x)}{c d^2-b d e+a e^2}+\frac {(B d-A 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 {(b B d-2 A c d+A b e-2 a B e) \int \frac {1}{a+b x+c x^2} \, dx}{2 \left (c d^2-b d e+a e^2\right )}\\ &=-\frac {(B d-A e) \log (d+e x)}{c d^2-b d e+a e^2}+\frac {(B d-A e) \log \left (a+b x+c x^2\right )}{2 \left (c d^2-b d e+a e^2\right )}+\frac {(b B d-2 A c d+A b e-2 a B e) \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 {(b B d-2 A c d+A b e-2 a B e) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\sqrt {b^2-4 a c} \left (c d^2-b d e+a e^2\right )}-\frac {(B d-A e) \log (d+e x)}{c d^2-b d e+a e^2}+\frac {(B d-A 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 = 125, normalized size = 0.86 \begin {gather*} \frac {\sqrt {4 a c-b^2} (B d-A e) (2 \log (d+e x)-\log (a+x (b+c x)))+2 \tan ^{-1}\left (\frac {b+2 c x}{\sqrt {4 a c-b^2}}\right ) (-2 a B e+A b e-2 A c d+b B d)}{2 \sqrt {4 a c-b^2} \left (e (b d-a e)-c d^2\right )} \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 {A+B x}{(d+e x) \left (a+b x+c x^2\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 4.81, size = 409, normalized size = 2.80 \begin {gather*} \left [\frac {\sqrt {b^{2} - 4 \, a c} {\left ({\left (B b - 2 \, A c\right )} d - {\left (2 \, B a - A b\right )} 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 ) + {\left ({\left (B b^{2} - 4 \, B a c\right )} d - {\left (A b^{2} - 4 \, A a c\right )} e\right )} \log \left (c x^{2} + b x + a\right ) - 2 \, {\left ({\left (B b^{2} - 4 \, B a c\right )} d - {\left (A b^{2} - 4 \, A a c\right )} e\right )} \log \left (e x + d\right )}{2 \, {\left ({\left (b^{2} c - 4 \, a c^{2}\right )} d^{2} - {\left (b^{3} - 4 \, a b c\right )} d e + {\left (a b^{2} - 4 \, a^{2} c\right )} e^{2}\right )}}, \frac {2 \, \sqrt {-b^{2} + 4 \, a c} {\left ({\left (B b - 2 \, A c\right )} d - {\left (2 \, B a - A b\right )} e\right )} \arctan \left (-\frac {\sqrt {-b^{2} + 4 \, a c} {\left (2 \, c x + b\right )}}{b^{2} - 4 \, a c}\right ) + {\left ({\left (B b^{2} - 4 \, B a c\right )} d - {\left (A b^{2} - 4 \, A a c\right )} e\right )} \log \left (c x^{2} + b x + a\right ) - 2 \, {\left ({\left (B b^{2} - 4 \, B a c\right )} d - {\left (A b^{2} - 4 \, A a c\right )} e\right )} \log \left (e x + d\right )}{2 \, {\left ({\left (b^{2} c - 4 \, a c^{2}\right )} d^{2} - {\left (b^{3} - 4 \, a b c\right )} d e + {\left (a b^{2} - 4 \, a^{2} c\right )} e^{2}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 155, normalized size = 1.06 \begin {gather*} \frac {{\left (B d - A 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 (B d e - A 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 b d - 2 \, A c d - 2 \, B a e + A b 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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 343, normalized size = 2.35 \begin {gather*} -\frac {A b 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 {2 A c d \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 {2 B a 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 b d \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 {A e \ln \left (e x +d \right )}{a \,e^{2}-b d e +c \,d^{2}}-\frac {A e \ln \left (c \,x^{2}+b x +a \right )}{2 \left (a \,e^{2}-b d e +c \,d^{2}\right )}-\frac {B d \ln \left (e x +d \right )}{a \,e^{2}-b d e +c \,d^{2}}+\frac {B d \ln \left (c \,x^{2}+b x +a \right )}{2 a \,e^{2}-2 b d e +2 c \,d^{2}} \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 = 8.70, size = 1027, normalized size = 7.03 \begin {gather*} \frac {\ln \left (d+e\,x\right )\,\left (A\,e-B\,d\right )}{c\,d^2-b\,d\,e+a\,e^2}-\frac {\ln \left (B^2\,c\,e\,x-\frac {\left (B\,a\,c\,e^2-A\,b\,c\,e^2-A\,c^2\,d\,e+c\,e\,x\,\left (B\,b\,e-3\,A\,c\,e+B\,c\,d\right )+B\,b\,c\,d\,e+\frac {c\,e\,\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 )\,\left (\frac {A\,b^2\,e}{2}-\frac {B\,b^2\,d}{2}+\frac {A\,b\,e\,\sqrt {b^2-4\,a\,c}}{2}-A\,c\,d\,\sqrt {b^2-4\,a\,c}-B\,a\,e\,\sqrt {b^2-4\,a\,c}+\frac {B\,b\,d\,\sqrt {b^2-4\,a\,c}}{2}-2\,A\,a\,c\,e+2\,B\,a\,c\,d\right )}{\left (4\,a\,c-b^2\right )\,\left (c\,d^2-b\,d\,e+a\,e^2\right )}\right )\,\left (\frac {A\,b^2\,e}{2}-\frac {B\,b^2\,d}{2}+\frac {A\,b\,e\,\sqrt {b^2-4\,a\,c}}{2}-A\,c\,d\,\sqrt {b^2-4\,a\,c}-B\,a\,e\,\sqrt {b^2-4\,a\,c}+\frac {B\,b\,d\,\sqrt {b^2-4\,a\,c}}{2}-2\,A\,a\,c\,e+2\,B\,a\,c\,d\right )}{\left (4\,a\,c-b^2\right )\,\left (c\,d^2-b\,d\,e+a\,e^2\right )}+A\,B\,c\,e\right )\,\left (b\,\left (\frac {A\,e\,\sqrt {b^2-4\,a\,c}}{2}+\frac {B\,d\,\sqrt {b^2-4\,a\,c}}{2}\right )+b^2\,\left (\frac {A\,e}{2}-\frac {B\,d}{2}\right )-A\,c\,d\,\sqrt {b^2-4\,a\,c}-B\,a\,e\,\sqrt {b^2-4\,a\,c}-2\,A\,a\,c\,e+2\,B\,a\,c\,d\right )}{-4\,a^2\,c\,e^2+a\,b^2\,e^2+4\,a\,b\,c\,d\,e-4\,a\,c^2\,d^2-b^3\,d\,e+b^2\,c\,d^2}-\frac {\ln \left (B^2\,c\,e\,x-\frac {\left (B\,a\,c\,e^2-A\,b\,c\,e^2-A\,c^2\,d\,e+c\,e\,x\,\left (B\,b\,e-3\,A\,c\,e+B\,c\,d\right )+B\,b\,c\,d\,e+\frac {c\,e\,\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 )\,\left (\frac {A\,b^2\,e}{2}-\frac {B\,b^2\,d}{2}-\frac {A\,b\,e\,\sqrt {b^2-4\,a\,c}}{2}+A\,c\,d\,\sqrt {b^2-4\,a\,c}+B\,a\,e\,\sqrt {b^2-4\,a\,c}-\frac {B\,b\,d\,\sqrt {b^2-4\,a\,c}}{2}-2\,A\,a\,c\,e+2\,B\,a\,c\,d\right )}{\left (4\,a\,c-b^2\right )\,\left (c\,d^2-b\,d\,e+a\,e^2\right )}\right )\,\left (\frac {A\,b^2\,e}{2}-\frac {B\,b^2\,d}{2}-\frac {A\,b\,e\,\sqrt {b^2-4\,a\,c}}{2}+A\,c\,d\,\sqrt {b^2-4\,a\,c}+B\,a\,e\,\sqrt {b^2-4\,a\,c}-\frac {B\,b\,d\,\sqrt {b^2-4\,a\,c}}{2}-2\,A\,a\,c\,e+2\,B\,a\,c\,d\right )}{\left (4\,a\,c-b^2\right )\,\left (c\,d^2-b\,d\,e+a\,e^2\right )}+A\,B\,c\,e\right )\,\left (b^2\,\left (\frac {A\,e}{2}-\frac {B\,d}{2}\right )-b\,\left (\frac {A\,e\,\sqrt {b^2-4\,a\,c}}{2}+\frac {B\,d\,\sqrt {b^2-4\,a\,c}}{2}\right )+A\,c\,d\,\sqrt {b^2-4\,a\,c}+B\,a\,e\,\sqrt {b^2-4\,a\,c}-2\,A\,a\,c\,e+2\,B\,a\,c\,d\right )}{-4\,a^2\,c\,e^2+a\,b^2\,e^2+4\,a\,b\,c\,d\,e-4\,a\,c^2\,d^2-b^3\,d\,e+b^2\,c\,d^2} \end {gather*}
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 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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