Optimal. Leaf size=183 \[ \frac{\left (a d (b d-4 c e)+b c e^2\right ) \tanh ^{-1}\left (\frac{2 a x+b}{\sqrt{b^2-4 a c}}\right )}{\sqrt{b^2-4 a c} \left (a d^2-e (b d-c e)\right )^2}+\frac{\left (a d^2-c e^2\right ) \log \left (a x^2+b x+c\right )}{2 \left (a d^2-e (b d-c e)\right )^2}+\frac{d}{(d+e x) \left (a d^2-b d e+c e^2\right )}-\frac{\left (a d^2-c e^2\right ) \log (d+e x)}{\left (a d^2-e (b d-c e)\right )^2} \]
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Rubi [A] time = 0.236279, antiderivative size = 183, normalized size of antiderivative = 1., 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, 800, 634, 618, 206, 628} \[ \frac{\left (a d (b d-4 c e)+b c e^2\right ) \tanh ^{-1}\left (\frac{2 a x+b}{\sqrt{b^2-4 a c}}\right )}{\sqrt{b^2-4 a c} \left (a d^2-e (b d-c e)\right )^2}+\frac{\left (a d^2-c e^2\right ) \log \left (a x^2+b x+c\right )}{2 \left (a d^2-e (b d-c e)\right )^2}+\frac{d}{(d+e x) \left (a d^2-b d e+c e^2\right )}-\frac{\left (a d^2-c e^2\right ) \log (d+e x)}{\left (a d^2-e (b d-c e)\right )^2} \]
Antiderivative was successfully verified.
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Rule 1569
Rule 800
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 (d+e x)^2} \, dx &=\int \frac{x}{(d+e x)^2 \left (c+b x+a x^2\right )} \, dx\\ &=\int \left (\frac{d e}{\left (-a d^2+e (b d-c e)\right ) (d+e x)^2}+\frac{e \left (-a d^2+c e^2\right )}{\left (a d^2-e (b d-c e)\right )^2 (d+e x)}+\frac{c e (2 a d-b e)+a \left (a d^2-c e^2\right ) x}{\left (a d^2-e (b d-c e)\right )^2 \left (c+b x+a x^2\right )}\right ) \, dx\\ &=\frac{d}{\left (a d^2-b d e+c e^2\right ) (d+e x)}-\frac{\left (a d^2-c e^2\right ) \log (d+e x)}{\left (a d^2-e (b d-c e)\right )^2}+\frac{\int \frac{c e (2 a d-b e)+a \left (a d^2-c e^2\right ) x}{c+b x+a x^2} \, dx}{\left (a d^2-e (b d-c e)\right )^2}\\ &=\frac{d}{\left (a d^2-b d e+c e^2\right ) (d+e x)}-\frac{\left (a d^2-c e^2\right ) \log (d+e x)}{\left (a d^2-e (b d-c e)\right )^2}+\frac{\left (a d^2-c e^2\right ) \int \frac{b+2 a x}{c+b x+a x^2} \, dx}{2 \left (a d^2-e (b d-c e)\right )^2}-\frac{\left (b c e^2+a d (b d-4 c e)\right ) \int \frac{1}{c+b x+a x^2} \, dx}{2 \left (a d^2-e (b d-c e)\right )^2}\\ &=\frac{d}{\left (a d^2-b d e+c e^2\right ) (d+e x)}-\frac{\left (a d^2-c e^2\right ) \log (d+e x)}{\left (a d^2-e (b d-c e)\right )^2}+\frac{\left (a d^2-c e^2\right ) \log \left (c+b x+a x^2\right )}{2 \left (a d^2-e (b d-c e)\right )^2}+\frac{\left (b c e^2+a d (b d-4 c e)\right ) \operatorname{Subst}\left (\int \frac{1}{b^2-4 a c-x^2} \, dx,x,b+2 a x\right )}{\left (a d^2-e (b d-c e)\right )^2}\\ &=\frac{d}{\left (a d^2-b d e+c e^2\right ) (d+e x)}+\frac{\left (b c e^2+a d (b d-4 c e)\right ) \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 )^2}-\frac{\left (a d^2-c e^2\right ) \log (d+e x)}{\left (a d^2-e (b d-c e)\right )^2}+\frac{\left (a d^2-c e^2\right ) \log \left (c+b x+a x^2\right )}{2 \left (a d^2-e (b d-c e)\right )^2}\\ \end{align*}
Mathematica [A] time = 0.262249, size = 148, normalized size = 0.81 \[ \frac{-\frac{2 \left (a d (b d-4 c e)+b c e^2\right ) \tan ^{-1}\left (\frac{2 a x+b}{\sqrt{4 a c-b^2}}\right )}{\sqrt{4 a c-b^2}}+\left (a d^2-c e^2\right ) \log (x (a x+b)+c)+\frac{2 d \left (a d^2+e (c e-b d)\right )}{d+e x}+\left (2 c e^2-2 a d^2\right ) \log (d+e x)}{2 \left (a d^2+e (c e-b d)\right )^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 328, normalized size = 1.8 \begin{align*}{\frac{d}{ \left ( a{d}^{2}-bde+{e}^{2}c \right ) \left ( ex+d \right ) }}-{\frac{\ln \left ( ex+d \right ) a{d}^{2}}{ \left ( a{d}^{2}-bde+{e}^{2}c \right ) ^{2}}}+{\frac{\ln \left ( ex+d \right ){e}^{2}c}{ \left ( a{d}^{2}-bde+{e}^{2}c \right ) ^{2}}}+{\frac{a\ln \left ( a{x}^{2}+bx+c \right ){d}^{2}}{2\, \left ( a{d}^{2}-bde+{e}^{2}c \right ) ^{2}}}-{\frac{\ln \left ( a{x}^{2}+bx+c \right ) c{e}^{2}}{2\, \left ( a{d}^{2}-bde+{e}^{2}c \right ) ^{2}}}-{\frac{ab{d}^{2}}{ \left ( a{d}^{2}-bde+{e}^{2}c \right ) ^{2}}\arctan \left ({(2\,ax+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}+4\,{\frac{acde}{ \left ( a{d}^{2}-bde+{e}^{2}c \right ) ^{2}\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,ax+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }-{\frac{bc{e}^{2}}{ \left ( a{d}^{2}-bde+{e}^{2}c \right ) ^{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 [B] time = 31.5593, size = 2217, normalized size = 12.11 \begin{align*} \text{result too large to display} \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 = 1.13598, size = 441, normalized size = 2.41 \begin{align*} \frac{1}{2} \,{\left (\frac{2 \,{\left (a b d^{2} e - 4 \, a c d e^{2} + b c e^{3}\right )} \arctan \left (-\frac{{\left (2 \, a d - \frac{2 \, a d^{2}}{x e + d} - b e + \frac{2 \, b d e}{x e + d} - \frac{2 \, c e^{2}}{x e + d}\right )} e^{\left (-1\right )}}{\sqrt{-b^{2} + 4 \, a c}}\right ) e^{\left (-2\right )}}{{\left (a^{2} d^{4} - 2 \, a b d^{3} e + b^{2} d^{2} e^{2} + 2 \, a c d^{2} e^{2} - 2 \, b c d e^{3} + c^{2} e^{4}\right )} \sqrt{-b^{2} + 4 \, a c}} + \frac{{\left (a d^{2} - c e^{2}\right )} \log \left (-a + \frac{2 \, a d}{x e + d} - \frac{a d^{2}}{{\left (x e + d\right )}^{2}} - \frac{b e}{x e + d} + \frac{b d e}{{\left (x e + d\right )}^{2}} - \frac{c e^{2}}{{\left (x e + d\right )}^{2}}\right )}{a^{2} d^{4} e - 2 \, a b d^{3} e^{2} + b^{2} d^{2} e^{3} + 2 \, a c d^{2} e^{3} - 2 \, b c d e^{4} + c^{2} e^{5}} + \frac{2 \, d e}{{\left (a d^{2} e^{2} - b d e^{3} + c e^{4}\right )}{\left (x e + d\right )}}\right )} e \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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