Optimal. Leaf size=158 \[ \frac{\left (a b d+2 a c e+b^2 (-e)\right ) \tanh ^{-1}\left (\frac{2 a x+b}{\sqrt{b^2-4 a c}}\right )}{c \sqrt{b^2-4 a c} \left (a d^2-e (b d-c e)\right )}-\frac{e^2 \log (d+e x)}{d \left (a d^2-b d e+c e^2\right )}-\frac{(a d-b e) \log \left (a x^2+b x+c\right )}{2 c \left (a d^2-e (b d-c e)\right )}+\frac{\log (x)}{c d} \]
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Rubi [A] time = 0.270758, antiderivative size = 159, normalized size of antiderivative = 1.01, number of steps used = 7, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24, Rules used = {1569, 893, 634, 618, 206, 628} \[ \frac{\left (a b d+2 a c e+b^2 (-e)\right ) \tanh ^{-1}\left (\frac{2 a x+b}{\sqrt{b^2-4 a c}}\right )}{c \sqrt{b^2-4 a c} \left (a d^2-e (b d-c e)\right )}-\frac{e^2 \log (d+e x)}{d \left (a d^2-e (b d-c e)\right )}-\frac{(a d-b e) \log \left (a x^2+b x+c\right )}{2 c \left (a d^2-e (b d-c e)\right )}+\frac{\log (x)}{c d} \]
Antiderivative was successfully verified.
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Rule 1569
Rule 893
Rule 634
Rule 618
Rule 206
Rule 628
Rubi steps
\begin{align*} \int \frac{1}{\left (a+\frac{c}{x^2}+\frac{b}{x}\right ) x^3 (d+e x)} \, dx &=\int \frac{1}{x (d+e x) \left (c+b x+a x^2\right )} \, dx\\ &=\int \left (\frac{1}{c d x}+\frac{e^3}{d \left (-a d^2+e (b d-c e)\right ) (d+e x)}+\frac{b^2 e-a (b d+c e)-a (a d-b e) x}{c \left (a d^2-e (b d-c e)\right ) \left (c+b x+a x^2\right )}\right ) \, dx\\ &=\frac{\log (x)}{c d}-\frac{e^2 \log (d+e x)}{d \left (a d^2-e (b d-c e)\right )}+\frac{\int \frac{b^2 e-a (b d+c e)-a (a d-b e) x}{c+b x+a x^2} \, dx}{c \left (a d^2-b d e+c e^2\right )}\\ &=\frac{\log (x)}{c d}-\frac{e^2 \log (d+e x)}{d \left (a d^2-e (b d-c e)\right )}+\frac{\left (-a b d+b^2 e-2 a c e\right ) \int \frac{1}{c+b x+a x^2} \, dx}{2 c \left (a d^2-b d e+c e^2\right )}-\frac{(a d-b e) \int \frac{b+2 a x}{c+b x+a x^2} \, dx}{2 c \left (a d^2-e (b d-c e)\right )}\\ &=\frac{\log (x)}{c d}-\frac{e^2 \log (d+e x)}{d \left (a d^2-e (b d-c e)\right )}-\frac{(a d-b e) \log \left (c+b x+a x^2\right )}{2 c \left (a d^2-e (b d-c e)\right )}-\frac{\left (-a b 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 a x\right )}{c \left (a d^2-b d e+c e^2\right )}\\ &=\frac{\left (a b d-b^2 e+2 a c e\right ) \tanh ^{-1}\left (\frac{b+2 a x}{\sqrt{b^2-4 a c}}\right )}{c \sqrt{b^2-4 a c} \left (a d^2-b d e+c e^2\right )}+\frac{\log (x)}{c d}-\frac{e^2 \log (d+e x)}{d \left (a d^2-e (b d-c e)\right )}-\frac{(a d-b e) \log \left (c+b x+a x^2\right )}{2 c \left (a d^2-e (b d-c e)\right )}\\ \end{align*}
Mathematica [A] time = 0.188878, size = 152, normalized size = 0.96 \[ -\frac{\sqrt{4 a c-b^2} \left (-2 \log (x) \left (a d^2+e (c e-b d)\right )+d (a d-b e) \log (x (a x+b)+c)+2 c e^2 \log (d+e x)\right )+2 d \left (a b d+2 a c e+b^2 (-e)\right ) \tan ^{-1}\left (\frac{2 a x+b}{\sqrt{4 a c-b^2}}\right )}{2 c d \sqrt{4 a c-b^2} \left (a d^2+e (c e-b d)\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.008, size = 285, normalized size = 1.8 \begin{align*} -{\frac{{e}^{2}\ln \left ( ex+d \right ) }{d \left ( a{d}^{2}-bde+{e}^{2}c \right ) }}-{\frac{a\ln \left ( a{x}^{2}+bx+c \right ) d}{ \left ( 2\,a{d}^{2}-2\,bde+2\,{e}^{2}c \right ) c}}+{\frac{\ln \left ( a{x}^{2}+bx+c \right ) be}{ \left ( 2\,a{d}^{2}-2\,bde+2\,{e}^{2}c \right ) c}}-{\frac{abd}{ \left ( a{d}^{2}-bde+{e}^{2}c \right ) c}\arctan \left ({(2\,ax+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}-2\,{\frac{ae}{ \left ( a{d}^{2}-bde+{e}^{2}c \right ) \sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,ax+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }+{\frac{{b}^{2}e}{ \left ( a{d}^{2}-bde+{e}^{2}c \right ) c}\arctan \left ({(2\,ax+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}+{\frac{\ln \left ( x \right ) }{cd}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1042, size = 221, normalized size = 1.4 \begin{align*} -\frac{{\left (a d - b e\right )} \log \left (a x^{2} + b x + c\right )}{2 \,{\left (a c d^{2} - b c d e + c^{2} e^{2}\right )}} - \frac{e^{3} \log \left ({\left | x e + d \right |}\right )}{a d^{3} e - b d^{2} e^{2} + c d e^{3}} - \frac{{\left (a b d - b^{2} e + 2 \, a c e\right )} \arctan \left (\frac{2 \, a x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{{\left (a c d^{2} - b c d e + c^{2} e^{2}\right )} \sqrt{-b^{2} + 4 \, a c}} + \frac{\log \left ({\left | x \right |}\right )}{c d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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