Optimal. Leaf size=149 \[ -\frac{\left (-2 a c d+b^2 d-b c e\right ) \tanh ^{-1}\left (\frac{2 a x+b}{\sqrt{b^2-4 a c}}\right )}{a \sqrt{b^2-4 a c} \left (a d^2-e (b d-c e)\right )}+\frac{d^2 \log (d+e x)}{e \left (a d^2-b d e+c e^2\right )}-\frac{(b d-c e) \log \left (a x^2+b x+c\right )}{2 a \left (a d^2-e (b d-c e)\right )} \]
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Rubi [A] time = 0.210117, antiderivative size = 149, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {1445, 1628, 634, 618, 206, 628} \[ -\frac{\left (-2 a c d+b^2 d-b c e\right ) \tanh ^{-1}\left (\frac{2 a x+b}{\sqrt{b^2-4 a c}}\right )}{a \sqrt{b^2-4 a c} \left (a d^2-e (b d-c e)\right )}+\frac{d^2 \log (d+e x)}{e \left (a d^2-b d e+c e^2\right )}-\frac{(b d-c e) \log \left (a x^2+b x+c\right )}{2 a \left (a d^2-e (b d-c e)\right )} \]
Antiderivative was successfully verified.
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Rule 1445
Rule 1628
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 ) (d+e x)} \, dx &=\int \frac{x^2}{(d+e x) \left (c+b x+a x^2\right )} \, dx\\ &=\int \left (\frac{d^2}{\left (a d^2-e (b d-c e)\right ) (d+e x)}+\frac{-c d-(b d-c e) x}{\left (a d^2-e (b d-c e)\right ) \left (c+b x+a x^2\right )}\right ) \, dx\\ &=\frac{d^2 \log (d+e x)}{e \left (a d^2-b d e+c e^2\right )}+\frac{\int \frac{-c d-(b d-c e) x}{c+b x+a x^2} \, dx}{a d^2-e (b d-c e)}\\ &=\frac{d^2 \log (d+e x)}{e \left (a d^2-b d e+c e^2\right )}-\frac{(b d-c e) \int \frac{b+2 a x}{c+b x+a x^2} \, dx}{2 a \left (a d^2-e (b d-c e)\right )}+\frac{\left (b^2 d-2 a c d-b c e\right ) \int \frac{1}{c+b x+a x^2} \, dx}{2 a \left (a d^2-e (b d-c e)\right )}\\ &=\frac{d^2 \log (d+e x)}{e \left (a d^2-b d e+c e^2\right )}-\frac{(b d-c e) \log \left (c+b x+a x^2\right )}{2 a \left (a d^2-e (b d-c e)\right )}-\frac{\left (b^2 d-2 a c d-b c e\right ) \operatorname{Subst}\left (\int \frac{1}{b^2-4 a c-x^2} \, dx,x,b+2 a x\right )}{a \left (a d^2-e (b d-c e)\right )}\\ &=-\frac{\left (b^2 d-2 a c d-b c e\right ) \tanh ^{-1}\left (\frac{b+2 a x}{\sqrt{b^2-4 a c}}\right )}{a \sqrt{b^2-4 a c} \left (a d^2-e (b d-c e)\right )}+\frac{d^2 \log (d+e x)}{e \left (a d^2-b d e+c e^2\right )}-\frac{(b d-c e) \log \left (c+b x+a x^2\right )}{2 a \left (a d^2-e (b d-c e)\right )}\\ \end{align*}
Mathematica [A] time = 0.127595, size = 132, normalized size = 0.89 \[ -\frac{\sqrt{4 a c-b^2} \left (e (b d-c e) \log (x (a x+b)+c)-2 a d^2 \log (d+e x)\right )+2 e \left (2 a c d+b^2 (-d)+b c e\right ) \tan ^{-1}\left (\frac{2 a x+b}{\sqrt{4 a c-b^2}}\right )}{2 a e \sqrt{4 a c-b^2} \left (a d^2+e (c e-b d)\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 275, normalized size = 1.9 \begin{align*}{\frac{{d}^{2}\ln \left ( ex+d \right ) }{e \left ( a{d}^{2}-bde+{e}^{2}c \right ) }}-{\frac{\ln \left ( a{x}^{2}+bx+c \right ) bd}{ \left ( 2\,a{d}^{2}-2\,bde+2\,{e}^{2}c \right ) a}}+{\frac{\ln \left ( a{x}^{2}+bx+c \right ) ce}{ \left ( 2\,a{d}^{2}-2\,bde+2\,{e}^{2}c \right ) a}}-2\,{\frac{cd}{ \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}d}{ \left ( a{d}^{2}-bde+{e}^{2}c \right ) a}\arctan \left ({(2\,ax+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}-{\frac{bce}{ \left ( a{d}^{2}-bde+{e}^{2}c \right ) a}\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 = 9.36673, size = 872, normalized size = 5.85 \begin{align*} \left [\frac{2 \,{\left (a b^{2} - 4 \, a^{2} c\right )} d^{2} \log \left (e x + d\right ) +{\left (b c e^{2} -{\left (b^{2} - 2 \, a c\right )} d e\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 ) -{\left ({\left (b^{3} - 4 \, a b c\right )} d e -{\left (b^{2} c - 4 \, a c^{2}\right )} e^{2}\right )} \log \left (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 )}}, \frac{2 \,{\left (a b^{2} - 4 \, a^{2} c\right )} d^{2} \log \left (e x + d\right ) + 2 \,{\left (b c e^{2} -{\left (b^{2} - 2 \, a c\right )} d e\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 ) -{\left ({\left (b^{3} - 4 \, a b c\right )} d e -{\left (b^{2} c - 4 \, a c^{2}\right )} e^{2}\right )} \log \left (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 )}}\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 = 1.11749, size = 201, normalized size = 1.35 \begin{align*} \frac{d^{2} \log \left ({\left | x e + d \right |}\right )}{a d^{2} e - b d e^{2} + c e^{3}} - \frac{{\left (b d - c e\right )} \log \left (a x^{2} + b x + c\right )}{2 \,{\left (a^{2} d^{2} - a b d e + a c e^{2}\right )}} + \frac{{\left (b^{2} d - 2 \, a c d - b c e\right )} \arctan \left (\frac{2 \, a x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{{\left (a^{2} d^{2} - a b d e + a 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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