Optimal. Leaf size=248 \[ -\frac{\left (a^2 d^2-a e (2 b d+c e)+b^2 e^2\right ) \log \left (a x^2+b x+c\right )}{2 c \left (a d^2-e (b d-c e)\right )^2}+\frac{\left (a^2 d (b d+4 c e)-a b e (2 b d+3 c e)+b^3 e^2\right ) \tanh ^{-1}\left (\frac{2 a x+b}{\sqrt{b^2-4 a c}}\right )}{c \sqrt{b^2-4 a c} \left (a d^2-e (b d-c e)\right )^2}+\frac{e^2}{d (d+e x) \left (a d^2-b d e+c e^2\right )}-\frac{e^2 \log (d+e x) \left (3 a d^2-e (2 b d-c e)\right )}{d^2 \left (a d^2-e (b d-c e)\right )^2}+\frac{\log (x)}{c d^2} \]
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Rubi [A] time = 0.408682, antiderivative size = 249, 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, 893, 634, 618, 206, 628} \[ -\frac{\left (a^2 d^2-a e (2 b d+c e)+b^2 e^2\right ) \log \left (a x^2+b x+c\right )}{2 c \left (a d^2-e (b d-c e)\right )^2}+\frac{\left (a^2 d (b d+4 c e)-a b e (2 b d+3 c e)+b^3 e^2\right ) \tanh ^{-1}\left (\frac{2 a x+b}{\sqrt{b^2-4 a c}}\right )}{c \sqrt{b^2-4 a c} \left (a d^2-e (b d-c e)\right )^2}+\frac{e^2}{d (d+e x) \left (a d^2-e (b d-c e)\right )}-\frac{e^2 \log (d+e x) \left (3 a d^2-e (2 b d-c e)\right )}{d^2 \left (a d^2-e (b d-c e)\right )^2}+\frac{\log (x)}{c d^2} \]
Antiderivative was successfully verified.
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Rule 1569
Rule 893
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^3 (d+e x)^2} \, dx &=\int \frac{1}{x (d+e x)^2 \left (c+b x+a x^2\right )} \, dx\\ &=\int \left (\frac{1}{c d^2 x}+\frac{e^3}{d \left (-a d^2+e (b d-c e)\right ) (d+e x)^2}+\frac{e^3 \left (-3 a d^2+e (2 b d-c e)\right )}{d^2 \left (a d^2-e (b d-c e)\right )^2 (d+e x)}+\frac{-(a d-b e) \left (a b d-b^2 e+2 a c e\right )-a \left (a^2 d^2+b^2 e^2-a e (2 b d+c e)\right ) x}{c \left (a d^2-e (b d-c e)\right )^2 \left (c+b x+a x^2\right )}\right ) \, dx\\ &=\frac{e^2}{d \left (a d^2-e (b d-c e)\right ) (d+e x)}+\frac{\log (x)}{c d^2}-\frac{e^2 \left (3 a d^2-e (2 b d-c e)\right ) \log (d+e x)}{d^2 \left (a d^2-e (b d-c e)\right )^2}+\frac{\int \frac{-(a d-b e) \left (a b d-b^2 e+2 a c e\right )-a \left (a^2 d^2+b^2 e^2-a e (2 b d+c e)\right ) x}{c+b x+a x^2} \, dx}{c \left (a d^2-e (b d-c e)\right )^2}\\ &=\frac{e^2}{d \left (a d^2-e (b d-c e)\right ) (d+e x)}+\frac{\log (x)}{c d^2}-\frac{e^2 \left (3 a d^2-e (2 b d-c e)\right ) \log (d+e x)}{d^2 \left (a d^2-e (b d-c e)\right )^2}-\frac{\left (a^2 d^2+b^2 e^2-a e (2 b d+c e)\right ) \int \frac{b+2 a x}{c+b x+a x^2} \, dx}{2 c \left (a d^2-e (b d-c e)\right )^2}-\frac{\left (b^3 e^2-a b e (2 b d+3 c e)+a^2 d (b d+4 c e)\right ) \int \frac{1}{c+b x+a x^2} \, dx}{2 c \left (a d^2-e (b d-c e)\right )^2}\\ &=\frac{e^2}{d \left (a d^2-e (b d-c e)\right ) (d+e x)}+\frac{\log (x)}{c d^2}-\frac{e^2 \left (3 a d^2-e (2 b d-c e)\right ) \log (d+e x)}{d^2 \left (a d^2-e (b d-c e)\right )^2}-\frac{\left (a^2 d^2+b^2 e^2-a e (2 b d+c e)\right ) \log \left (c+b x+a x^2\right )}{2 c \left (a d^2-e (b d-c e)\right )^2}+\frac{\left (b^3 e^2-a b e (2 b d+3 c e)+a^2 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 )}{c \left (a d^2-e (b d-c e)\right )^2}\\ &=\frac{e^2}{d \left (a d^2-e (b d-c e)\right ) (d+e x)}+\frac{\left (b^3 e^2-a b e (2 b d+3 c e)+a^2 d (b d+4 c e)\right ) \tanh ^{-1}\left (\frac{b+2 a x}{\sqrt{b^2-4 a c}}\right )}{c \sqrt{b^2-4 a c} \left (a d^2-e (b d-c e)\right )^2}+\frac{\log (x)}{c d^2}-\frac{e^2 \left (3 a d^2-e (2 b d-c e)\right ) \log (d+e x)}{d^2 \left (a d^2-e (b d-c e)\right )^2}-\frac{\left (a^2 d^2+b^2 e^2-a e (2 b d+c e)\right ) \log \left (c+b x+a x^2\right )}{2 c \left (a d^2-e (b d-c e)\right )^2}\\ \end{align*}
Mathematica [A] time = 0.284886, size = 246, normalized size = 0.99 \[ \frac{\left (-a^2 d^2+a e (2 b d+c e)-b^2 e^2\right ) \log (x (a x+b)+c)}{2 c \left (a d^2+e (c e-b d)\right )^2}-\frac{\left (a^2 d (b d+4 c e)-a b e (2 b d+3 c e)+b^3 e^2\right ) \tan ^{-1}\left (\frac{2 a x+b}{\sqrt{4 a c-b^2}}\right )}{c \sqrt{4 a c-b^2} \left (a d^2+e (c e-b d)\right )^2}+\frac{e^2}{d (d+e x) \left (a d^2+e (c e-b d)\right )}-\frac{e^2 \log (d+e x) \left (3 a d^2+e (c e-2 b d)\right )}{\left (a d^3+d e (c e-b d)\right )^2}+\frac{\log (x)}{c d^2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.013, size = 589, normalized size = 2.4 \begin{align*}{\frac{{e}^{2}}{d \left ( a{d}^{2}-bde+{e}^{2}c \right ) \left ( ex+d \right ) }}-3\,{\frac{{e}^{2}\ln \left ( ex+d \right ) a}{ \left ( a{d}^{2}-bde+{e}^{2}c \right ) ^{2}}}+2\,{\frac{{e}^{3}\ln \left ( ex+d \right ) b}{d \left ( a{d}^{2}-bde+{e}^{2}c \right ) ^{2}}}-{\frac{{e}^{4}\ln \left ( ex+d \right ) c}{ \left ( a{d}^{2}-bde+{e}^{2}c \right ) ^{2}{d}^{2}}}-{\frac{{a}^{2}\ln \left ( a{x}^{2}+bx+c \right ){d}^{2}}{2\, \left ( a{d}^{2}-bde+{e}^{2}c \right ) ^{2}c}}+{\frac{a\ln \left ( a{x}^{2}+bx+c \right ) bde}{ \left ( a{d}^{2}-bde+{e}^{2}c \right ) ^{2}c}}+{\frac{a\ln \left ( a{x}^{2}+bx+c \right ){e}^{2}}{2\, \left ( a{d}^{2}-bde+{e}^{2}c \right ) ^{2}}}-{\frac{\ln \left ( a{x}^{2}+bx+c \right ){b}^{2}{e}^{2}}{2\, \left ( a{d}^{2}-bde+{e}^{2}c \right ) ^{2}c}}-{\frac{{a}^{2}b{d}^{2}}{ \left ( a{d}^{2}-bde+{e}^{2}c \right ) ^{2}c}\arctan \left ({(2\,ax+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}-4\,{\frac{{a}^{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 ) }+2\,{\frac{a{b}^{2}de}{ \left ( a{d}^{2}-bde+{e}^{2}c \right ) ^{2}c\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,ax+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }+3\,{\frac{ab{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 ) }-{\frac{{b}^{3}{e}^{2}}{ \left ( a{d}^{2}-bde+{e}^{2}c \right ) ^{2}c}\arctan \left ({(2\,ax+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}+{\frac{\ln \left ( x \right ) }{c{d}^{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 [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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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 = 1.13596, size = 528, normalized size = 2.13 \begin{align*} -\frac{{\left (a^{2} b d^{2} e^{2} - 2 \, a b^{2} d e^{3} + 4 \, a^{2} c d e^{3} + b^{3} e^{4} - 3 \, a b c 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} c d^{4} - 2 \, a b c d^{3} e + b^{2} c d^{2} e^{2} + 2 \, a c^{2} d^{2} e^{2} - 2 \, b c^{2} d e^{3} + c^{3} e^{4}\right )} \sqrt{-b^{2} + 4 \, a c}} - \frac{{\left (a^{2} d^{2} - 2 \, a b d e + b^{2} e^{2} - a 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 )}{2 \,{\left (a^{2} c d^{4} - 2 \, a b c d^{3} e + b^{2} c d^{2} e^{2} + 2 \, a c^{2} d^{2} e^{2} - 2 \, b c^{2} d e^{3} + c^{3} e^{4}\right )}} + \frac{e^{5}}{{\left (a d^{3} e^{3} - b d^{2} e^{4} + c d e^{5}\right )}{\left (x e + d\right )}} + \frac{\log \left ({\left | -\frac{d}{x e + d} + 1 \right |}\right )}{c d^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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