Optimal. Leaf size=246 \[ \frac{\left (-b c \left (3 a d^2-c e^2\right )+4 a c^2 d e-2 b^2 c d e+b^3 d^2\right ) \tanh ^{-1}\left (\frac{2 a x+b}{\sqrt{b^2-4 a c}}\right )}{a \sqrt{b^2-4 a c} \left (a d^2-e (b d-c e)\right )^2}+\frac{\left (-c \left (a d^2-c e^2\right )+b^2 d^2-2 b c d e\right ) \log \left (a x^2+b x+c\right )}{2 a \left (a d^2-e (b d-c e)\right )^2}+\frac{d^3}{e^2 (d+e x) \left (a d^2-e (b d-c e)\right )}+\frac{d^2 \log (d+e x) \left (a d^2-e (2 b d-3 c e)\right )}{e^2 \left (a d^2-e (b d-c e)\right )^2} \]
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Rubi [A] time = 0.395346, antiderivative size = 246, 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} \[ \frac{\left (-b c \left (3 a d^2-c e^2\right )+4 a c^2 d e-2 b^2 c d e+b^3 d^2\right ) \tanh ^{-1}\left (\frac{2 a x+b}{\sqrt{b^2-4 a c}}\right )}{a \sqrt{b^2-4 a c} \left (a d^2-e (b d-c e)\right )^2}+\frac{\left (-c \left (a d^2-c e^2\right )+b^2 d^2-2 b c d e\right ) \log \left (a x^2+b x+c\right )}{2 a \left (a d^2-e (b d-c e)\right )^2}+\frac{d^3}{e^2 (d+e x) \left (a d^2-e (b d-c e)\right )}+\frac{d^2 \log (d+e x) \left (a d^2-e (2 b d-3 c e)\right )}{e^2 \left (a d^2-e (b d-c e)\right )^2} \]
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)^2} \, dx &=\int \frac{x^3}{(d+e x)^2 \left (c+b x+a x^2\right )} \, dx\\ &=\int \left (\frac{d^3}{e \left (-a d^2+e (b d-c e)\right ) (d+e x)^2}+\frac{d^2 \left (a d^2-e (2 b d-3 c e)\right )}{e \left (a d^2-e (b d-c e)\right )^2 (d+e x)}+\frac{c d (b d-2 c e)+\left (b^2 d^2-2 b c d e-c \left (a d^2-c e^2\right )\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^3}{e^2 \left (a d^2-e (b d-c e)\right ) (d+e x)}+\frac{d^2 \left (a d^2-e (2 b d-3 c e)\right ) \log (d+e x)}{e^2 \left (a d^2-e (b d-c e)\right )^2}+\frac{\int \frac{c d (b d-2 c e)+\left (b^2 d^2-2 b c d e-c \left (a d^2-c e^2\right )\right ) x}{c+b x+a x^2} \, dx}{\left (a d^2-e (b d-c e)\right )^2}\\ &=\frac{d^3}{e^2 \left (a d^2-e (b d-c e)\right ) (d+e x)}+\frac{d^2 \left (a d^2-e (2 b d-3 c e)\right ) \log (d+e x)}{e^2 \left (a d^2-e (b d-c e)\right )^2}+\frac{\left (b^2 d^2-2 b c d e-c \left (a d^2-c e^2\right )\right ) \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 )^2}-\frac{\left (b^3 d^2-2 b^2 c d e+4 a c^2 d e-b c \left (3 a d^2-c e^2\right )\right ) \int \frac{1}{c+b x+a x^2} \, dx}{2 a \left (a d^2-e (b d-c e)\right )^2}\\ &=\frac{d^3}{e^2 \left (a d^2-e (b d-c e)\right ) (d+e x)}+\frac{d^2 \left (a d^2-e (2 b d-3 c e)\right ) \log (d+e x)}{e^2 \left (a d^2-e (b d-c e)\right )^2}+\frac{\left (b^2 d^2-2 b c d e-c \left (a d^2-c e^2\right )\right ) \log \left (c+b x+a x^2\right )}{2 a \left (a d^2-e (b d-c e)\right )^2}+\frac{\left (b^3 d^2-2 b^2 c d e+4 a c^2 d e-b c \left (3 a d^2-c e^2\right )\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 )^2}\\ &=\frac{d^3}{e^2 \left (a d^2-e (b d-c e)\right ) (d+e x)}+\frac{\left (b^3 d^2-2 b^2 c d e+4 a c^2 d e-b c \left (3 a d^2-c e^2\right )\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 )^2}+\frac{d^2 \left (a d^2-e (2 b d-3 c e)\right ) \log (d+e x)}{e^2 \left (a d^2-e (b d-c e)\right )^2}+\frac{\left (b^2 d^2-2 b c d e-c \left (a d^2-c e^2\right )\right ) \log \left (c+b x+a x^2\right )}{2 a \left (a d^2-e (b d-c e)\right )^2}\\ \end{align*}
Mathematica [A] time = 0.248042, size = 207, normalized size = 0.84 \[ \frac{-\frac{2 \left (b c \left (c e^2-3 a d^2\right )+4 a c^2 d e-2 b^2 c d e+b^3 d^2\right ) \tan ^{-1}\left (\frac{2 a x+b}{\sqrt{4 a c-b^2}}\right )}{a \sqrt{4 a c-b^2}}+\frac{\left (c \left (c e^2-a d^2\right )+b^2 d^2-2 b c d e\right ) \log (x (a x+b)+c)}{a}+\frac{2 d^3 \left (a d^2+e (c e-b d)\right )}{e^2 (d+e x)}+\frac{2 \log (d+e x) \left (a d^4+d^2 e (3 c e-2 b d)\right )}{e^2}}{2 \left (a d^2+e (c e-b d)\right )^2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.007, size = 580, normalized size = 2.4 \begin{align*}{\frac{{d}^{4}\ln \left ( ex+d \right ) a}{{e}^{2} \left ( a{d}^{2}-bde+{e}^{2}c \right ) ^{2}}}-2\,{\frac{{d}^{3}\ln \left ( ex+d \right ) b}{ \left ( a{d}^{2}-bde+{e}^{2}c \right ) ^{2}e}}+3\,{\frac{{d}^{2}\ln \left ( ex+d \right ) c}{ \left ( a{d}^{2}-bde+{e}^{2}c \right ) ^{2}}}+{\frac{{d}^{3}}{{e}^{2} \left ( a{d}^{2}-bde+{e}^{2}c \right ) \left ( ex+d \right ) }}-{\frac{\ln \left ( a{x}^{2}+bx+c \right ) c{d}^{2}}{2\, \left ( a{d}^{2}-bde+{e}^{2}c \right ) ^{2}}}+{\frac{\ln \left ( a{x}^{2}+bx+c \right ){b}^{2}{d}^{2}}{2\, \left ( a{d}^{2}-bde+{e}^{2}c \right ) ^{2}a}}-{\frac{\ln \left ( a{x}^{2}+bx+c \right ) bcde}{ \left ( a{d}^{2}-bde+{e}^{2}c \right ) ^{2}a}}+{\frac{\ln \left ( a{x}^{2}+bx+c \right ){c}^{2}{e}^{2}}{2\, \left ( a{d}^{2}-bde+{e}^{2}c \right ) ^{2}a}}+3\,{\frac{bc{d}^{2}}{ \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 ) }-4\,{\frac{{c}^{2}de}{ \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{{b}^{3}{d}^{2}}{ \left ( a{d}^{2}-bde+{e}^{2}c \right ) ^{2}a}\arctan \left ({(2\,ax+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}+2\,{\frac{{b}^{2}cde}{ \left ( a{d}^{2}-bde+{e}^{2}c \right ) ^{2}\sqrt{4\,ac-{b}^{2}}a}\arctan \left ({\frac{2\,ax+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }-{\frac{b{c}^{2}{e}^{2}}{ \left ( a{d}^{2}-bde+{e}^{2}c \right ) ^{2}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 [B] time = 110.084, size = 3005, normalized size = 12.22 \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 = 1.11668, size = 556, normalized size = 2.26 \begin{align*} \frac{1}{2} \,{\left (\frac{2 \, d^{3} e^{2}}{{\left (a d^{2} e^{3} - b d e^{4} + c e^{5}\right )}{\left (x e + d\right )}} + \frac{2 \,{\left (b^{3} d^{2} e^{3} - 3 \, a b c d^{2} e^{3} - 2 \, b^{2} c d e^{4} + 4 \, a c^{2} d e^{4} + b c^{2} e^{5}\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^{3} d^{4} - 2 \, a^{2} b d^{3} e + a b^{2} d^{2} e^{2} + 2 \, a^{2} c d^{2} e^{2} - 2 \, a b c d e^{3} + a c^{2} e^{4}\right )} \sqrt{-b^{2} + 4 \, a c}} + \frac{{\left (b^{2} d^{2} e - a c d^{2} e - 2 \, b c d e^{2} + c^{2} e^{3}\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^{3} d^{4} - 2 \, a^{2} b d^{3} e + a b^{2} d^{2} e^{2} + 2 \, a^{2} c d^{2} e^{2} - 2 \, a b c d e^{3} + a c^{2} e^{4}} - \frac{2 \, e^{\left (-1\right )} \log \left (\frac{{\left | x e + d \right |} e^{\left (-1\right )}}{{\left (x e + d\right )}^{2}}\right )}{a}\right )} e^{\left (-1\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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