Optimal. Leaf size=218 \[ -\frac{\left (-2 a b c d+a c^2 e-b^2 c e+b^3 d\right ) \log \left (a x^2+b x+c\right )}{2 a^3 \left (a d^2-e (b d-c e)\right )}-\frac{\left (2 a^2 c^2 d-4 a b^2 c d+3 a b c^2 e-b^3 c e+b^4 d\right ) \tanh ^{-1}\left (\frac{2 a x+b}{\sqrt{b^2-4 a c}}\right )}{a^3 \sqrt{b^2-4 a c} \left (a d^2-e (b d-c e)\right )}-\frac{x (a d+b e)}{a^2 e^2}+\frac{d^4 \log (d+e x)}{e^3 \left (a d^2-e (b d-c e)\right )}+\frac{x^2}{2 a e} \]
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Rubi [A] time = 0.395218, antiderivative size = 218, 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, 1628, 634, 618, 206, 628} \[ -\frac{\left (-2 a b c d+a c^2 e-b^2 c e+b^3 d\right ) \log \left (a x^2+b x+c\right )}{2 a^3 \left (a d^2-e (b d-c e)\right )}-\frac{\left (2 a^2 c^2 d-4 a b^2 c d+3 a b c^2 e-b^3 c e+b^4 d\right ) \tanh ^{-1}\left (\frac{2 a x+b}{\sqrt{b^2-4 a c}}\right )}{a^3 \sqrt{b^2-4 a c} \left (a d^2-e (b d-c e)\right )}-\frac{x (a d+b e)}{a^2 e^2}+\frac{d^4 \log (d+e x)}{e^3 \left (a d^2-e (b d-c e)\right )}+\frac{x^2}{2 a e} \]
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^2}{\left (a+\frac{c}{x^2}+\frac{b}{x}\right ) (d+e x)} \, dx &=\int \frac{x^4}{(d+e x) \left (c+b x+a x^2\right )} \, dx\\ &=\int \left (\frac{-a d-b e}{a^2 e^2}+\frac{x}{a e}+\frac{d^4}{e^2 \left (a d^2-e (b d-c e)\right ) (d+e x)}+\frac{-c \left (b^2 d-a c d-b c e\right )-\left (b^3 d-2 a b c d-b^2 c e+a c^2 e\right ) x}{a^2 \left (a d^2-e (b d-c e)\right ) \left (c+b x+a x^2\right )}\right ) \, dx\\ &=-\frac{(a d+b e) x}{a^2 e^2}+\frac{x^2}{2 a e}+\frac{d^4 \log (d+e x)}{e^3 \left (a d^2-e (b d-c e)\right )}+\frac{\int \frac{-c \left (b^2 d-a c d-b c e\right )-\left (b^3 d-2 a b c d-b^2 c e+a c^2 e\right ) x}{c+b x+a x^2} \, dx}{a^2 \left (a d^2-e (b d-c e)\right )}\\ &=-\frac{(a d+b e) x}{a^2 e^2}+\frac{x^2}{2 a e}+\frac{d^4 \log (d+e x)}{e^3 \left (a d^2-e (b d-c e)\right )}-\frac{\left (b^3 d-2 a b c d-b^2 c e+a c^2 e\right ) \int \frac{b+2 a x}{c+b x+a x^2} \, dx}{2 a^3 \left (a d^2-e (b d-c e)\right )}+\frac{\left (b^4 d-4 a b^2 c d+2 a^2 c^2 d-b^3 c e+3 a b c^2 e\right ) \int \frac{1}{c+b x+a x^2} \, dx}{2 a^3 \left (a d^2-e (b d-c e)\right )}\\ &=-\frac{(a d+b e) x}{a^2 e^2}+\frac{x^2}{2 a e}+\frac{d^4 \log (d+e x)}{e^3 \left (a d^2-e (b d-c e)\right )}-\frac{\left (b^3 d-2 a b c d-b^2 c e+a c^2 e\right ) \log \left (c+b x+a x^2\right )}{2 a^3 \left (a d^2-e (b d-c e)\right )}-\frac{\left (b^4 d-4 a b^2 c d+2 a^2 c^2 d-b^3 c e+3 a b c^2 e\right ) \operatorname{Subst}\left (\int \frac{1}{b^2-4 a c-x^2} \, dx,x,b+2 a x\right )}{a^3 \left (a d^2-e (b d-c e)\right )}\\ &=-\frac{(a d+b e) x}{a^2 e^2}+\frac{x^2}{2 a e}-\frac{\left (b^4 d-4 a b^2 c d+2 a^2 c^2 d-b^3 c e+3 a b c^2 e\right ) \tanh ^{-1}\left (\frac{b+2 a x}{\sqrt{b^2-4 a c}}\right )}{a^3 \sqrt{b^2-4 a c} \left (a d^2-e (b d-c e)\right )}+\frac{d^4 \log (d+e x)}{e^3 \left (a d^2-e (b d-c e)\right )}-\frac{\left (b^3 d-2 a b c d-b^2 c e+a c^2 e\right ) \log \left (c+b x+a x^2\right )}{2 a^3 \left (a d^2-e (b d-c e)\right )}\\ \end{align*}
Mathematica [A] time = 0.18102, size = 218, normalized size = 1. \[ \frac{\left (2 a b c d-a c^2 e+b^2 c e+b^3 (-d)\right ) \log (x (a x+b)+c)}{2 a^3 \left (a d^2+e (c e-b d)\right )}+\frac{\left (2 a^2 c^2 d-4 a b^2 c d+3 a b c^2 e-b^3 c e+b^4 d\right ) \tan ^{-1}\left (\frac{2 a x+b}{\sqrt{4 a c-b^2}}\right )}{a^3 \sqrt{4 a c-b^2} \left (a d^2+e (c e-b d)\right )}-\frac{x (a d+b e)}{a^2 e^2}+\frac{d^4 \log (d+e x)}{e^3 \left (a d^2+e (c e-b d)\right )}+\frac{x^2}{2 a e} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.006, size = 512, normalized size = 2.4 \begin{align*}{\frac{{x}^{2}}{2\,ae}}-{\frac{dx}{a{e}^{2}}}-{\frac{bx}{{a}^{2}e}}+{\frac{{d}^{4}\ln \left ( ex+d \right ) }{{e}^{3} \left ( a{d}^{2}-bde+{e}^{2}c \right ) }}+{\frac{\ln \left ( a{x}^{2}+bx+c \right ) bcd}{ \left ( a{d}^{2}-bde+{e}^{2}c \right ){a}^{2}}}-{\frac{\ln \left ( a{x}^{2}+bx+c \right ){c}^{2}e}{ \left ( 2\,a{d}^{2}-2\,bde+2\,{e}^{2}c \right ){a}^{2}}}-{\frac{\ln \left ( a{x}^{2}+bx+c \right ){b}^{3}d}{ \left ( 2\,a{d}^{2}-2\,bde+2\,{e}^{2}c \right ){a}^{3}}}+{\frac{\ln \left ( a{x}^{2}+bx+c \right ){b}^{2}ce}{ \left ( 2\,a{d}^{2}-2\,bde+2\,{e}^{2}c \right ){a}^{3}}}+2\,{\frac{d{c}^{2}}{ \left ( a{d}^{2}-bde+{e}^{2}c \right ) a\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,ax+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }-4\,{\frac{{b}^{2}cd}{ \left ( a{d}^{2}-bde+{e}^{2}c \right ){a}^{2}\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,ax+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }+3\,{\frac{b{c}^{2}e}{ \left ( a{d}^{2}-bde+{e}^{2}c \right ){a}^{2}\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,ax+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }+{\frac{{b}^{4}d}{ \left ( a{d}^{2}-bde+{e}^{2}c \right ){a}^{3}}\arctan \left ({(2\,ax+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}-{\frac{{b}^{3}ce}{ \left ( a{d}^{2}-bde+{e}^{2}c \right ){a}^{3}}\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 [A] time = 84.0832, size = 1623, normalized size = 7.44 \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 = 1.11492, size = 302, normalized size = 1.39 \begin{align*} \frac{d^{4} \log \left ({\left | x e + d \right |}\right )}{a d^{2} e^{3} - b d e^{4} + c e^{5}} - \frac{{\left (b^{3} d - 2 \, a b c d - b^{2} c e + a c^{2} e\right )} \log \left (a x^{2} + b x + c\right )}{2 \,{\left (a^{4} d^{2} - a^{3} b d e + a^{3} c e^{2}\right )}} + \frac{{\left (b^{4} d - 4 \, a b^{2} c d + 2 \, a^{2} c^{2} d - b^{3} c e + 3 \, a b c^{2} e\right )} \arctan \left (\frac{2 \, a x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{{\left (a^{4} d^{2} - a^{3} b d e + a^{3} c e^{2}\right )} \sqrt{-b^{2} + 4 \, a c}} + \frac{{\left (a x^{2} e - 2 \, a d x - 2 \, b x e\right )} e^{\left (-2\right )}}{2 \, a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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