Optimal. Leaf size=123 \[ -\frac {(2 a d-b e) \tanh ^{-1}\left (\frac {2 a x+b}{\sqrt {b^2-4 a c}}\right )}{\sqrt {b^2-4 a c} \left (a d^2-e (b d-c e)\right )}-\frac {e \log \left (a x^2+b x+c\right )}{2 \left (a d^2-b d e+c e^2\right )}+\frac {e \log (d+e x)}{a d^2-b d e+c e^2} \]
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Rubi [A] time = 0.11, antiderivative size = 123, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {1569, 705, 31, 634, 618, 206, 628} \begin {gather*} -\frac {(2 a d-b e) \tanh ^{-1}\left (\frac {2 a x+b}{\sqrt {b^2-4 a c}}\right )}{\sqrt {b^2-4 a c} \left (a d^2-e (b d-c e)\right )}-\frac {e \log \left (a x^2+b x+c\right )}{2 \left (a d^2-b d e+c e^2\right )}+\frac {e \log (d+e x)}{a d^2-b d e+c e^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 206
Rule 618
Rule 628
Rule 634
Rule 705
Rule 1569
Rubi steps
\begin {align*} \int \frac {1}{\left (a+\frac {c}{x^2}+\frac {b}{x}\right ) x^2 (d+e x)} \, dx &=\int \frac {1}{(d+e x) \left (c+b x+a x^2\right )} \, dx\\ &=\frac {e^2 \int \frac {1}{d+e x} \, dx}{a d^2-b d e+c e^2}+\frac {\int \frac {a d-b e-a e x}{c+b x+a x^2} \, dx}{a d^2-e (b d-c e)}\\ &=\frac {e \log (d+e x)}{a d^2-b d e+c e^2}-\frac {e \int \frac {b+2 a x}{c+b x+a x^2} \, dx}{2 \left (a d^2-b d e+c e^2\right )}+\frac {(2 a d-b e) \int \frac {1}{c+b x+a x^2} \, dx}{2 \left (a d^2-e (b d-c e)\right )}\\ &=\frac {e \log (d+e x)}{a d^2-b d e+c e^2}-\frac {e \log \left (c+b x+a x^2\right )}{2 \left (a d^2-b d e+c e^2\right )}-\frac {(2 a d-b e) \operatorname {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 a x\right )}{a d^2-e (b d-c e)}\\ &=-\frac {(2 a d-b e) \tanh ^{-1}\left (\frac {b+2 a x}{\sqrt {b^2-4 a c}}\right )}{\sqrt {b^2-4 a c} \left (a d^2-e (b d-c e)\right )}+\frac {e \log (d+e x)}{a d^2-b d e+c e^2}-\frac {e \log \left (c+b x+a x^2\right )}{2 \left (a d^2-b d e+c e^2\right )}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 105, normalized size = 0.85 \begin {gather*} \frac {e \sqrt {4 a c-b^2} (\log (x (a x+b)+c)-2 \log (d+e x))+(2 b e-4 a d) \tan ^{-1}\left (\frac {2 a x+b}{\sqrt {4 a c-b^2}}\right )}{2 \sqrt {4 a c-b^2} \left (e (b d-c e)-a 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 {1}{\left (a+\frac {c}{x^2}+\frac {b}{x}\right ) x^2 (d+e x)} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 2.05, size = 305, normalized size = 2.48 \begin {gather*} \left [-\frac {{\left (b^{2} - 4 \, a c\right )} e \log \left (a x^{2} + b x + c\right ) - 2 \, {\left (b^{2} - 4 \, a c\right )} e \log \left (e x + d\right ) + \sqrt {b^{2} - 4 \, a c} {\left (2 \, a d - b e\right )} \log \left (\frac {2 \, a^{2} x^{2} + 2 \, a b x + b^{2} - 2 \, a c + \sqrt {b^{2} - 4 \, a c} {\left (2 \, a x + b\right )}}{a x^{2} + b x + c}\right )}{2 \, {\left ({\left (a b^{2} - 4 \, a^{2} c\right )} d^{2} - {\left (b^{3} - 4 \, a b c\right )} d e + {\left (b^{2} c - 4 \, a c^{2}\right )} e^{2}\right )}}, -\frac {{\left (b^{2} - 4 \, a c\right )} e \log \left (a x^{2} + b x + c\right ) - 2 \, {\left (b^{2} - 4 \, a c\right )} e \log \left (e x + d\right ) + 2 \, \sqrt {-b^{2} + 4 \, a c} {\left (2 \, a d - b e\right )} \arctan \left (-\frac {\sqrt {-b^{2} + 4 \, a c} {\left (2 \, a x + b\right )}}{b^{2} - 4 \, a c}\right )}{2 \, {\left ({\left (a b^{2} - 4 \, a^{2} c\right )} d^{2} - {\left (b^{3} - 4 \, a b c\right )} d e + {\left (b^{2} c - 4 \, a c^{2}\right )} e^{2}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.34, size = 126, normalized size = 1.02 \begin {gather*} -\frac {e \log \left (a x^{2} + b x + c\right )}{2 \, {\left (a d^{2} - b d e + c e^{2}\right )}} + \frac {e^{2} \log \left ({\left | x e + d \right |}\right )}{a d^{2} e - b d e^{2} + c e^{3}} + \frac {{\left (2 \, a d - b e\right )} \arctan \left (\frac {2 \, a x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{{\left (a d^{2} - b d e + c 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 [A] time = 0.01, size = 168, normalized size = 1.37 \begin {gather*} \frac {2 a d \arctan \left (\frac {2 a x +b}{\sqrt {4 a c -b^{2}}}\right )}{\left (a \,d^{2}-d e b +c \,e^{2}\right ) \sqrt {4 a c -b^{2}}}-\frac {b e \arctan \left (\frac {2 a x +b}{\sqrt {4 a c -b^{2}}}\right )}{\left (a \,d^{2}-d e b +c \,e^{2}\right ) \sqrt {4 a c -b^{2}}}+\frac {e \ln \left (e x +d \right )}{a \,d^{2}-d e b +c \,e^{2}}-\frac {e \ln \left (a \,x^{2}+b x +c \right )}{2 \left (a \,d^{2}-d e b +c \,e^{2}\right )} \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 = 3.82, size = 521, normalized size = 4.24 \begin {gather*} \frac {\ln \left (3\,a^2\,e^2\,x+a\,b\,e^2+a^2\,d\,e-\frac {a\,e\,\left (\frac {b^2\,e}{2}-2\,a\,c\,e+a\,d\,\sqrt {b^2-4\,a\,c}-\frac {b\,e\,\sqrt {b^2-4\,a\,c}}{2}\right )\,\left (2\,x\,a^2\,d^2+a\,b\,d^2-2\,x\,a\,b\,d\,e-8\,c\,a\,d\,e-6\,c\,x\,a\,e^2+b^2\,d\,e+2\,x\,b^2\,e^2+c\,b\,e^2\right )}{\left (4\,a\,c-b^2\right )\,\left (a\,d^2-b\,d\,e+c\,e^2\right )}\right )\,\left (e\,\left (2\,a\,c+\frac {b\,\sqrt {b^2-4\,a\,c}}{2}-\frac {b^2}{2}\right )-a\,d\,\sqrt {b^2-4\,a\,c}\right )}{-4\,a^2\,c\,d^2+a\,b^2\,d^2+4\,a\,b\,c\,d\,e-4\,a\,c^2\,e^2-b^3\,d\,e+b^2\,c\,e^2}-\frac {\ln \left (3\,a^2\,e^2\,x+a\,b\,e^2+a^2\,d\,e-\frac {a\,e\,\left (\frac {b^2\,e}{2}-2\,a\,c\,e-a\,d\,\sqrt {b^2-4\,a\,c}+\frac {b\,e\,\sqrt {b^2-4\,a\,c}}{2}\right )\,\left (2\,x\,a^2\,d^2+a\,b\,d^2-2\,x\,a\,b\,d\,e-8\,c\,a\,d\,e-6\,c\,x\,a\,e^2+b^2\,d\,e+2\,x\,b^2\,e^2+c\,b\,e^2\right )}{\left (4\,a\,c-b^2\right )\,\left (a\,d^2-b\,d\,e+c\,e^2\right )}\right )\,\left (e\,\left (\frac {b\,\sqrt {b^2-4\,a\,c}}{2}-2\,a\,c+\frac {b^2}{2}\right )-a\,d\,\sqrt {b^2-4\,a\,c}\right )}{-4\,a^2\,c\,d^2+a\,b^2\,d^2+4\,a\,b\,c\,d\,e-4\,a\,c^2\,e^2-b^3\,d\,e+b^2\,c\,e^2}+\frac {e\,\ln \left (d+e\,x\right )}{a\,d^2-b\,d\,e+c\,e^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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