Optimal. Leaf size=252 \[ \frac{\left (a^2 c d-a b (b d+2 c e)+b^3 e\right ) \log \left (a x^2+b x+c\right )}{2 c^3 \left (a d^2-e (b d-c e)\right )}-\frac{\left (a^2 c (3 b d+2 c e)-a b^2 (b d+4 c e)+b^4 e\right ) \tanh ^{-1}\left (\frac{2 a x+b}{\sqrt{b^2-4 a c}}\right )}{c^3 \sqrt{b^2-4 a c} \left (a d^2-e (b d-c e)\right )}+\frac{\log (x) \left (-c \left (a d^2-c e^2\right )+b^2 d^2+b c d e\right )}{c^3 d^3}-\frac{e^4 \log (d+e x)}{d^3 \left (a d^2-e (b d-c e)\right )}+\frac{b d+c e}{c^2 d^2 x}-\frac{1}{2 c d x^2} \]
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Rubi [A] time = 0.428431, antiderivative size = 252, 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} \[ \frac{\left (a^2 c d-a b (b d+2 c e)+b^3 e\right ) \log \left (a x^2+b x+c\right )}{2 c^3 \left (a d^2-e (b d-c e)\right )}-\frac{\left (a^2 c (3 b d+2 c e)-a b^2 (b d+4 c e)+b^4 e\right ) \tanh ^{-1}\left (\frac{2 a x+b}{\sqrt{b^2-4 a c}}\right )}{c^3 \sqrt{b^2-4 a c} \left (a d^2-e (b d-c e)\right )}+\frac{\log (x) \left (-c \left (a d^2-c e^2\right )+b^2 d^2+b c d e\right )}{c^3 d^3}-\frac{e^4 \log (d+e x)}{d^3 \left (a d^2-e (b d-c e)\right )}+\frac{b d+c e}{c^2 d^2 x}-\frac{1}{2 c d x^2} \]
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^5 (d+e x)} \, dx &=\int \frac{1}{x^3 (d+e x) \left (c+b x+a x^2\right )} \, dx\\ &=\int \left (\frac{1}{c d x^3}+\frac{-b d-c e}{c^2 d^2 x^2}+\frac{b^2 d^2+b c d e-c \left (a d^2-c e^2\right )}{c^3 d^3 x}+\frac{e^5}{d^3 \left (-a d^2+e (b d-c e)\right ) (d+e x)}+\frac{b^4 e+a^2 c (2 b d+c e)-a b^2 (b d+3 c e)+a \left (a^2 c d+b^3 e-a b (b d+2 c e)\right ) x}{c^3 \left (a d^2-e (b d-c e)\right ) \left (c+b x+a x^2\right )}\right ) \, dx\\ &=-\frac{1}{2 c d x^2}+\frac{b d+c e}{c^2 d^2 x}+\frac{\left (b^2 d^2+b c d e-c \left (a d^2-c e^2\right )\right ) \log (x)}{c^3 d^3}-\frac{e^4 \log (d+e x)}{d^3 \left (a d^2-e (b d-c e)\right )}+\frac{\int \frac{b^4 e+a^2 c (2 b d+c e)-a b^2 (b d+3 c e)+a \left (a^2 c d+b^3 e-a b (b d+2 c e)\right ) x}{c+b x+a x^2} \, dx}{c^3 \left (a d^2-e (b d-c e)\right )}\\ &=-\frac{1}{2 c d x^2}+\frac{b d+c e}{c^2 d^2 x}+\frac{\left (b^2 d^2+b c d e-c \left (a d^2-c e^2\right )\right ) \log (x)}{c^3 d^3}-\frac{e^4 \log (d+e x)}{d^3 \left (a d^2-e (b d-c e)\right )}+\frac{\left (a^2 c d+b^3 e-a b (b d+2 c e)\right ) \int \frac{b+2 a x}{c+b x+a x^2} \, dx}{2 c^3 \left (a d^2-e (b d-c e)\right )}+\frac{\left (b^4 e+a^2 c (3 b d+2 c e)-a b^2 (b d+4 c e)\right ) \int \frac{1}{c+b x+a x^2} \, dx}{2 c^3 \left (a d^2-e (b d-c e)\right )}\\ &=-\frac{1}{2 c d x^2}+\frac{b d+c e}{c^2 d^2 x}+\frac{\left (b^2 d^2+b c d e-c \left (a d^2-c e^2\right )\right ) \log (x)}{c^3 d^3}-\frac{e^4 \log (d+e x)}{d^3 \left (a d^2-e (b d-c e)\right )}+\frac{\left (a^2 c d+b^3 e-a b (b d+2 c e)\right ) \log \left (c+b x+a x^2\right )}{2 c^3 \left (a d^2-e (b d-c e)\right )}-\frac{\left (b^4 e+a^2 c (3 b d+2 c e)-a b^2 (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^3 \left (a d^2-e (b d-c e)\right )}\\ &=-\frac{1}{2 c d x^2}+\frac{b d+c e}{c^2 d^2 x}-\frac{\left (b^4 e+a^2 c (3 b d+2 c e)-a b^2 (b d+4 c e)\right ) \tanh ^{-1}\left (\frac{b+2 a x}{\sqrt{b^2-4 a c}}\right )}{c^3 \sqrt{b^2-4 a c} \left (a d^2-e (b d-c e)\right )}+\frac{\left (b^2 d^2+b c d e-c \left (a d^2-c e^2\right )\right ) \log (x)}{c^3 d^3}-\frac{e^4 \log (d+e x)}{d^3 \left (a d^2-e (b d-c e)\right )}+\frac{\left (a^2 c d+b^3 e-a b (b d+2 c e)\right ) \log \left (c+b x+a x^2\right )}{2 c^3 \left (a d^2-e (b d-c e)\right )}\\ \end{align*}
Mathematica [A] time = 0.233504, size = 252, normalized size = 1. \[ \frac{\left (a^2 c d-a b (b d+2 c e)+b^3 e\right ) \log (x (a x+b)+c)}{2 c^3 \left (a d^2+e (c e-b d)\right )}-\frac{\left (a^2 c (3 b d+2 c e)-a b^2 (b d+4 c e)+b^4 e\right ) \tan ^{-1}\left (\frac{2 a x+b}{\sqrt{4 a c-b^2}}\right )}{c^3 \sqrt{4 a c-b^2} \left (e (b d-c e)-a d^2\right )}+\frac{\log (x) \left (c \left (c e^2-a d^2\right )+b^2 d^2+b c d e\right )}{c^3 d^3}-\frac{e^4 \log (d+e x)}{a d^5+d^3 e (c e-b d)}+\frac{b d+c e}{c^2 d^2 x}-\frac{1}{2 c d x^2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.012, size = 562, normalized size = 2.2 \begin{align*} -{\frac{{e}^{4}\ln \left ( ex+d \right ) }{{d}^{3} \left ( a{d}^{2}-bde+{e}^{2}c \right ) }}+{\frac{{a}^{2}\ln \left ( a{x}^{2}+bx+c \right ) d}{ \left ( 2\,a{d}^{2}-2\,bde+2\,{e}^{2}c \right ){c}^{2}}}-{\frac{a\ln \left ( a{x}^{2}+bx+c \right ){b}^{2}d}{ \left ( 2\,a{d}^{2}-2\,bde+2\,{e}^{2}c \right ){c}^{3}}}-{\frac{a\ln \left ( a{x}^{2}+bx+c \right ) be}{ \left ( a{d}^{2}-bde+{e}^{2}c \right ){c}^{2}}}+{\frac{\ln \left ( a{x}^{2}+bx+c \right ){b}^{3}e}{ \left ( 2\,a{d}^{2}-2\,bde+2\,{e}^{2}c \right ){c}^{3}}}+3\,{\frac{{a}^{2}bd}{ \left ( a{d}^{2}-bde+{e}^{2}c \right ){c}^{2}\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,ax+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }+2\,{\frac{{a}^{2}e}{ \left ( a{d}^{2}-bde+{e}^{2}c \right ) c\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,ax+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }-{\frac{a{b}^{3}d}{ \left ( a{d}^{2}-bde+{e}^{2}c \right ){c}^{3}}\arctan \left ({(2\,ax+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}-4\,{\frac{a{b}^{2}e}{ \left ( a{d}^{2}-bde+{e}^{2}c \right ){c}^{2}\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,ax+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }+{\frac{{b}^{4}e}{ \left ( a{d}^{2}-bde+{e}^{2}c \right ){c}^{3}}\arctan \left ({(2\,ax+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}-{\frac{1}{2\,cd{x}^{2}}}+{\frac{b}{{c}^{2}dx}}+{\frac{e}{c{d}^{2}x}}-{\frac{\ln \left ( x \right ) a}{{c}^{2}d}}+{\frac{\ln \left ( x \right ){b}^{2}}{{c}^{3}d}}+{\frac{b\ln \left ( x \right ) e}{{c}^{2}{d}^{2}}}+{\frac{\ln \left ( x \right ){e}^{2}}{c{d}^{3}}} \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 = 1.09894, size = 377, normalized size = 1.5 \begin{align*} -\frac{{\left (a b^{2} d - a^{2} c d - b^{3} e + 2 \, a b c e\right )} \log \left (a x^{2} + b x + c\right )}{2 \,{\left (a c^{3} d^{2} - b c^{3} d e + c^{4} e^{2}\right )}} - \frac{e^{5} \log \left ({\left | x e + d \right |}\right )}{a d^{5} e - b d^{4} e^{2} + c d^{3} e^{3}} - \frac{{\left (a b^{3} d - 3 \, a^{2} b c d - b^{4} e + 4 \, a b^{2} c e - 2 \, a^{2} c^{2} e\right )} \arctan \left (\frac{2 \, a x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{{\left (a c^{3} d^{2} - b c^{3} d e + c^{4} e^{2}\right )} \sqrt{-b^{2} + 4 \, a c}} + \frac{{\left (b^{2} d^{2} - a c d^{2} + b c d e + c^{2} e^{2}\right )} \log \left ({\left | x \right |}\right )}{c^{3} d^{3}} - \frac{c^{2} d^{2} - 2 \,{\left (b c d^{2} + c^{2} d e\right )} x}{2 \, c^{3} d^{3} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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