Optimal. Leaf size=194 \[ -\frac{\left (-2 c \left (a d^2-c e^2\right )+b^2 d^2-2 b c d e\right ) \tanh ^{-1}\left (\frac{2 a x+b}{\sqrt{b^2-4 a c}}\right )}{\sqrt{b^2-4 a c} \left (a d^2-e (b d-c e)\right )^2}-\frac{d^2}{e (d+e x) \left (a d^2-b d e+c e^2\right )}-\frac{d (b d-2 c e) \log \left (a x^2+b x+c\right )}{2 \left (a d^2-e (b d-c e)\right )^2}+\frac{d (b d-2 c e) \log (d+e x)}{\left (a d^2-e (b d-c e)\right )^2} \]
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Rubi [A] time = 0.306283, antiderivative size = 194, 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} \[ -\frac{\left (-2 c \left (a d^2-c e^2\right )+b^2 d^2-2 b c d e\right ) \tanh ^{-1}\left (\frac{2 a x+b}{\sqrt{b^2-4 a c}}\right )}{\sqrt{b^2-4 a c} \left (a d^2-e (b d-c e)\right )^2}-\frac{d^2}{e (d+e x) \left (a d^2-b d e+c e^2\right )}-\frac{d (b d-2 c e) \log \left (a x^2+b x+c\right )}{2 \left (a d^2-e (b d-c e)\right )^2}+\frac{d (b d-2 c e) \log (d+e x)}{\left (a d^2-e (b d-c e)\right )^2} \]
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)^2} \, dx &=\int \frac{x^2}{(d+e x)^2 \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)^2}+\frac{d e (b d-2 c e)}{\left (a d^2-e (b d-c e)\right )^2 (d+e x)}+\frac{-c \left (a d^2-c e^2\right )-a d (b d-2 c e) x}{\left (a d^2-e (b d-c e)\right )^2 \left (c+b x+a x^2\right )}\right ) \, dx\\ &=-\frac{d^2}{e \left (a d^2-b d e+c e^2\right ) (d+e x)}+\frac{d (b d-2 c e) \log (d+e x)}{\left (a d^2-e (b d-c e)\right )^2}+\frac{\int \frac{-c \left (a d^2-c e^2\right )-a d (b d-2 c e) x}{c+b x+a x^2} \, dx}{\left (a d^2-e (b d-c e)\right )^2}\\ &=-\frac{d^2}{e \left (a d^2-b d e+c e^2\right ) (d+e x)}+\frac{d (b d-2 c e) \log (d+e x)}{\left (a d^2-e (b d-c e)\right )^2}-\frac{(d (b d-2 c e)) \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^2 d^2-2 b c d e-2 c \left (a d^2-c e^2\right )\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^2}{e \left (a d^2-b d e+c e^2\right ) (d+e x)}+\frac{d (b d-2 c e) \log (d+e x)}{\left (a d^2-e (b d-c e)\right )^2}-\frac{d (b d-2 c e) \log \left (c+b x+a x^2\right )}{2 \left (a d^2-e (b d-c e)\right )^2}-\frac{\left (b^2 d^2-2 b c d e-2 c \left (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 )}{\left (a d^2-e (b d-c e)\right )^2}\\ &=-\frac{d^2}{e \left (a d^2-b d e+c e^2\right ) (d+e x)}-\frac{\left (b^2 d^2-2 b c d e-2 c \left (a d^2-c e^2\right )\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{d (b d-2 c e) \log (d+e x)}{\left (a d^2-e (b d-c e)\right )^2}-\frac{d (b d-2 c e) \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 = 0.239581, size = 159, normalized size = 0.82 \[ \frac{\frac{2 \left (2 c \left (c e^2-a d^2\right )+b^2 d^2-2 b c d e\right ) \tan ^{-1}\left (\frac{2 a x+b}{\sqrt{4 a c-b^2}}\right )}{\sqrt{4 a c-b^2}}-\frac{2 d^2 \left (a d^2+e (c e-b d)\right )}{e (d+e x)}-d (b d-2 c e) \log (x (a x+b)+c)+2 d (b d-2 c e) \log (d+e x)}{2 \left (a d^2+e (c e-b d)\right )^2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.008, size = 389, normalized size = 2. \begin{align*} -{\frac{{d}^{2}}{ \left ( a{d}^{2}-bde+{e}^{2}c \right ) e \left ( ex+d \right ) }}+{\frac{{d}^{2}\ln \left ( ex+d \right ) b}{ \left ( a{d}^{2}-bde+{e}^{2}c \right ) ^{2}}}-2\,{\frac{d\ln \left ( ex+d \right ) ce}{ \left ( a{d}^{2}-bde+{e}^{2}c \right ) ^{2}}}-{\frac{\ln \left ( a{x}^{2}+bx+c \right ) b{d}^{2}}{2\, \left ( a{d}^{2}-bde+{e}^{2}c \right ) ^{2}}}+{\frac{\ln \left ( a{x}^{2}+bx+c \right ) cde}{ \left ( a{d}^{2}-bde+{e}^{2}c \right ) ^{2}}}-2\,{\frac{ac{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 ) }+{\frac{{b}^{2}{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}}}}}-2\,{\frac{bcde}{ \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 ) }+2\,{\frac{{c}^{2}{e}^{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 ) } \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 = 36.9767, size = 2365, normalized size = 12.19 \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 = 1.1218, size = 455, normalized size = 2.35 \begin{align*} -\frac{{\left (b^{2} d^{2} e^{2} - 2 \, a c d^{2} e^{2} - 2 \, b c d e^{3} + 2 \, c^{2} e^{4}\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{d^{2} e}{{\left (a d^{2} e^{2} - b d e^{3} + c e^{4}\right )}{\left (x e + d\right )}} - \frac{{\left (b d^{2} - 2 \, c d e\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 )}{2 \,{\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 )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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