Optimal. Leaf size=176 \[ \frac{\left (-3 a b c d+2 a c^2 e-b^2 c e+b^3 d\right ) \tanh ^{-1}\left (\frac{2 a x+b}{\sqrt{b^2-4 a c}}\right )}{a^2 \sqrt{b^2-4 a c} \left (a d^2-e (b d-c e)\right )}+\frac{\left (-a c d+b^2 d-b c e\right ) \log \left (a x^2+b x+c\right )}{2 a^2 \left (a d^2-e (b d-c e)\right )}-\frac{d^3 \log (d+e x)}{e^2 \left (a d^2-e (b d-c e)\right )}+\frac{x}{a e} \]
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Rubi [A] time = 0.285269, antiderivative size = 176, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.261, Rules used = {1569, 1628, 634, 618, 206, 628} \[ \frac{\left (-3 a b c d+2 a c^2 e-b^2 c e+b^3 d\right ) \tanh ^{-1}\left (\frac{2 a x+b}{\sqrt{b^2-4 a c}}\right )}{a^2 \sqrt{b^2-4 a c} \left (a d^2-e (b d-c e)\right )}+\frac{\left (-a c d+b^2 d-b c e\right ) \log \left (a x^2+b x+c\right )}{2 a^2 \left (a d^2-e (b d-c e)\right )}-\frac{d^3 \log (d+e x)}{e^2 \left (a d^2-e (b d-c e)\right )}+\frac{x}{a e} \]
Antiderivative was successfully verified.
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Rule 1569
Rule 1628
Rule 634
Rule 618
Rule 206
Rule 628
Rubi steps
\begin{align*} \int \frac{x}{\left (a+\frac{c}{x^2}+\frac{b}{x}\right ) (d+e x)} \, dx &=\int \frac{x^3}{(d+e x) \left (c+b x+a x^2\right )} \, dx\\ &=\int \left (\frac{1}{a e}+\frac{d^3}{e \left (-a d^2+e (b d-c e)\right ) (d+e x)}+\frac{c (b d-c e)+\left (b^2 d-a c d-b c e\right ) x}{a \left (a d^2-e (b d-c e)\right ) \left (c+b x+a x^2\right )}\right ) \, dx\\ &=\frac{x}{a e}-\frac{d^3 \log (d+e x)}{e^2 \left (a d^2-e (b d-c e)\right )}+\frac{\int \frac{c (b d-c e)+\left (b^2 d-a c d-b c e\right ) x}{c+b x+a x^2} \, dx}{a \left (a d^2-b d e+c e^2\right )}\\ &=\frac{x}{a e}-\frac{d^3 \log (d+e x)}{e^2 \left (a d^2-e (b d-c e)\right )}+\frac{\left (b^2 d-a c d-b c e\right ) \int \frac{b+2 a x}{c+b x+a x^2} \, dx}{2 a^2 \left (a d^2-e (b d-c e)\right )}-\frac{\left (b^3 d-3 a b c d-b^2 c e+2 a c^2 e\right ) \int \frac{1}{c+b x+a x^2} \, dx}{2 a^2 \left (a d^2-e (b d-c e)\right )}\\ &=\frac{x}{a e}-\frac{d^3 \log (d+e x)}{e^2 \left (a d^2-e (b d-c e)\right )}+\frac{\left (b^2 d-a c d-b c e\right ) \log \left (c+b x+a x^2\right )}{2 a^2 \left (a d^2-e (b d-c e)\right )}+\frac{\left (b^3 d-3 a b c d-b^2 c e+2 a c^2 e\right ) \operatorname{Subst}\left (\int \frac{1}{b^2-4 a c-x^2} \, dx,x,b+2 a x\right )}{a^2 \left (a d^2-e (b d-c e)\right )}\\ &=\frac{x}{a e}+\frac{\left (b^3 d-3 a b c d-b^2 c e+2 a c^2 e\right ) \tanh ^{-1}\left (\frac{b+2 a x}{\sqrt{b^2-4 a c}}\right )}{a^2 \sqrt{b^2-4 a c} \left (a d^2-e (b d-c e)\right )}-\frac{d^3 \log (d+e x)}{e^2 \left (a d^2-e (b d-c e)\right )}+\frac{\left (b^2 d-a c d-b c e\right ) \log \left (c+b x+a x^2\right )}{2 a^2 \left (a d^2-e (b d-c e)\right )}\\ \end{align*}
Mathematica [A] time = 0.187473, size = 178, normalized size = 1.01 \[ \frac{\left (-3 a b c d+2 a c^2 e-b^2 c e+b^3 d\right ) \tan ^{-1}\left (\frac{2 a x+b}{\sqrt{4 a c-b^2}}\right )}{a^2 \sqrt{4 a c-b^2} \left (-a d^2+b d e-c e^2\right )}+\frac{\left (-a c d+b^2 d-b c e\right ) \log \left (a x^2+b x+c\right )}{2 a^2 \left (a d^2-b d e+c e^2\right )}-\frac{d^3 \log (d+e x)}{e^2 \left (a d^2-b d e+c e^2\right )}+\frac{x}{a e} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.007, size = 388, normalized size = 2.2 \begin{align*}{\frac{x}{ae}}-{\frac{{d}^{3}\ln \left ( ex+d \right ) }{{e}^{2} \left ( a{d}^{2}-bde+{e}^{2}c \right ) }}-{\frac{\ln \left ( a{x}^{2}+bx+c \right ) cd}{ \left ( 2\,a{d}^{2}-2\,bde+2\,{e}^{2}c \right ) a}}+{\frac{\ln \left ( a{x}^{2}+bx+c \right ){b}^{2}d}{ \left ( 2\,a{d}^{2}-2\,bde+2\,{e}^{2}c \right ){a}^{2}}}-{\frac{\ln \left ( a{x}^{2}+bx+c \right ) bce}{ \left ( 2\,a{d}^{2}-2\,bde+2\,{e}^{2}c \right ){a}^{2}}}+3\,{\frac{bcd}{ \left ( a{d}^{2}-bde+{e}^{2}c \right ) a\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,ax+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }-2\,{\frac{{c}^{2}e}{ \left ( a{d}^{2}-bde+{e}^{2}c \right ) a\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,ax+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }-{\frac{{b}^{3}d}{ \left ( a{d}^{2}-bde+{e}^{2}c \right ){a}^{2}}\arctan \left ({(2\,ax+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}+{\frac{{b}^{2}ce}{ \left ( a{d}^{2}-bde+{e}^{2}c \right ){a}^{2}}\arctan \left ({(2\,ax+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \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 [A] time = 30.6945, size = 1237, normalized size = 7.03 \begin{align*} \left [-\frac{2 \,{\left (a^{2} b^{2} - 4 \, a^{3} c\right )} d^{3} \log \left (e x + d\right ) -{\left ({\left (b^{3} - 3 \, a b c\right )} d e^{2} -{\left (b^{2} c - 2 \, a c^{2}\right )} e^{3}\right )} \sqrt{b^{2} - 4 \, a c} \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^{2} b^{2} - 4 \, a^{3} c\right )} d^{2} e -{\left (a b^{3} - 4 \, a^{2} b c\right )} d e^{2} +{\left (a b^{2} c - 4 \, a^{2} c^{2}\right )} e^{3}\right )} x -{\left ({\left (b^{4} - 5 \, a b^{2} c + 4 \, a^{2} c^{2}\right )} d e^{2} -{\left (b^{3} c - 4 \, a b c^{2}\right )} e^{3}\right )} \log \left (a x^{2} + b x + c\right )}{2 \,{\left ({\left (a^{3} b^{2} - 4 \, a^{4} c\right )} d^{2} e^{2} -{\left (a^{2} b^{3} - 4 \, a^{3} b c\right )} d e^{3} +{\left (a^{2} b^{2} c - 4 \, a^{3} c^{2}\right )} e^{4}\right )}}, -\frac{2 \,{\left (a^{2} b^{2} - 4 \, a^{3} c\right )} d^{3} \log \left (e x + d\right ) - 2 \,{\left ({\left (b^{3} - 3 \, a b c\right )} d e^{2} -{\left (b^{2} c - 2 \, a c^{2}\right )} e^{3}\right )} \sqrt{-b^{2} + 4 \, a c} \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^{2} b^{2} - 4 \, a^{3} c\right )} d^{2} e -{\left (a b^{3} - 4 \, a^{2} b c\right )} d e^{2} +{\left (a b^{2} c - 4 \, a^{2} c^{2}\right )} e^{3}\right )} x -{\left ({\left (b^{4} - 5 \, a b^{2} c + 4 \, a^{2} c^{2}\right )} d e^{2} -{\left (b^{3} c - 4 \, a b c^{2}\right )} e^{3}\right )} \log \left (a x^{2} + b x + c\right )}{2 \,{\left ({\left (a^{3} b^{2} - 4 \, a^{4} c\right )} d^{2} e^{2} -{\left (a^{2} b^{3} - 4 \, a^{3} b c\right )} d e^{3} +{\left (a^{2} b^{2} c - 4 \, a^{3} c^{2}\right )} e^{4}\right )}}\right ] \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 = 1.09625, size = 250, normalized size = 1.42 \begin{align*} -\frac{d^{3} \log \left ({\left | x e + d \right |}\right )}{a d^{2} e^{2} - b d e^{3} + c e^{4}} + \frac{x e^{\left (-1\right )}}{a} + \frac{{\left (b^{2} d - a c d - b c e\right )} \log \left (a x^{2} + b x + c\right )}{2 \,{\left (a^{3} d^{2} - a^{2} b d e + a^{2} c e^{2}\right )}} - \frac{{\left (b^{3} d - 3 \, a b c d - b^{2} c e + 2 \, a c^{2} e\right )} \arctan \left (\frac{2 \, a x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{{\left (a^{3} d^{2} - a^{2} b d e + a^{2} c e^{2}\right )} \sqrt{-b^{2} + 4 \, a c}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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