Optimal. Leaf size=280 \[ \frac{\left (a^2 c^2 d-3 a b^2 c d+2 a b c^2 e-b^3 c e+b^4 d\right ) \log \left (a x^2+b x+c\right )}{2 a^4 \left (a d^2-e (b d-c e)\right )}+\frac{\left (5 a^2 b c^2 d-2 a^2 c^3 e+4 a b^2 c^2 e-5 a b^3 c d-b^4 c e+b^5 d\right ) \tanh ^{-1}\left (\frac{2 a x+b}{\sqrt{b^2-4 a c}}\right )}{a^4 \sqrt{b^2-4 a c} \left (a d^2-e (b d-c e)\right )}+\frac{x \left (a^2 d^2+a e (b d-c e)+b^2 e^2\right )}{a^3 e^3}-\frac{x^2 (a d+b e)}{2 a^2 e^2}-\frac{d^5 \log (d+e x)}{e^4 \left (a d^2-e (b d-c e)\right )}+\frac{x^3}{3 a e} \]
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Rubi [A] time = 0.59737, antiderivative size = 280, 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} \[ \frac{\left (a^2 c^2 d-3 a b^2 c d+2 a b c^2 e-b^3 c e+b^4 d\right ) \log \left (a x^2+b x+c\right )}{2 a^4 \left (a d^2-e (b d-c e)\right )}+\frac{\left (5 a^2 b c^2 d-2 a^2 c^3 e+4 a b^2 c^2 e-5 a b^3 c d-b^4 c e+b^5 d\right ) \tanh ^{-1}\left (\frac{2 a x+b}{\sqrt{b^2-4 a c}}\right )}{a^4 \sqrt{b^2-4 a c} \left (a d^2-e (b d-c e)\right )}+\frac{x \left (a^2 d^2+a e (b d-c e)+b^2 e^2\right )}{a^3 e^3}-\frac{x^2 (a d+b e)}{2 a^2 e^2}-\frac{d^5 \log (d+e x)}{e^4 \left (a d^2-e (b d-c e)\right )}+\frac{x^3}{3 a e} \]
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^3}{\left (a+\frac{c}{x^2}+\frac{b}{x}\right ) (d+e x)} \, dx &=\int \frac{x^5}{(d+e x) \left (c+b x+a x^2\right )} \, dx\\ &=\int \left (\frac{a^2 d^2+b^2 e^2+a e (b d-c e)}{a^3 e^3}-\frac{(a d+b e) x}{a^2 e^2}+\frac{x^2}{a e}+\frac{d^5}{e^3 \left (-a d^2+e (b d-c e)\right ) (d+e x)}+\frac{c \left (b^3 d-2 a b c d-b^2 c e+a c^2 e\right )+\left (b^4 d-3 a b^2 c d+a^2 c^2 d-b^3 c e+2 a b c^2 e\right ) x}{a^3 \left (a d^2-e (b d-c e)\right ) \left (c+b x+a x^2\right )}\right ) \, dx\\ &=\frac{\left (a^2 d^2+b^2 e^2+a e (b d-c e)\right ) x}{a^3 e^3}-\frac{(a d+b e) x^2}{2 a^2 e^2}+\frac{x^3}{3 a e}-\frac{d^5 \log (d+e x)}{e^4 \left (a d^2-e (b d-c e)\right )}+\frac{\int \frac{c \left (b^3 d-2 a b c d-b^2 c e+a c^2 e\right )+\left (b^4 d-3 a b^2 c d+a^2 c^2 d-b^3 c e+2 a b c^2 e\right ) x}{c+b x+a x^2} \, dx}{a^3 \left (a d^2-e (b d-c e)\right )}\\ &=\frac{\left (a^2 d^2+b^2 e^2+a e (b d-c e)\right ) x}{a^3 e^3}-\frac{(a d+b e) x^2}{2 a^2 e^2}+\frac{x^3}{3 a e}-\frac{d^5 \log (d+e x)}{e^4 \left (a d^2-e (b d-c e)\right )}+\frac{\left (b^4 d-3 a b^2 c d+a^2 c^2 d-b^3 c e+2 a b c^2 e\right ) \int \frac{b+2 a x}{c+b x+a x^2} \, dx}{2 a^4 \left (a d^2-e (b d-c e)\right )}-\frac{\left (b^5 d-5 a b^3 c d+5 a^2 b c^2 d-b^4 c e+4 a b^2 c^2 e-2 a^2 c^3 e\right ) \int \frac{1}{c+b x+a x^2} \, dx}{2 a^4 \left (a d^2-e (b d-c e)\right )}\\ &=\frac{\left (a^2 d^2+b^2 e^2+a e (b d-c e)\right ) x}{a^3 e^3}-\frac{(a d+b e) x^2}{2 a^2 e^2}+\frac{x^3}{3 a e}-\frac{d^5 \log (d+e x)}{e^4 \left (a d^2-e (b d-c e)\right )}+\frac{\left (b^4 d-3 a b^2 c d+a^2 c^2 d-b^3 c e+2 a b c^2 e\right ) \log \left (c+b x+a x^2\right )}{2 a^4 \left (a d^2-e (b d-c e)\right )}+\frac{\left (b^5 d-5 a b^3 c d+5 a^2 b c^2 d-b^4 c e+4 a b^2 c^2 e-2 a^2 c^3 e\right ) \operatorname{Subst}\left (\int \frac{1}{b^2-4 a c-x^2} \, dx,x,b+2 a x\right )}{a^4 \left (a d^2-e (b d-c e)\right )}\\ &=\frac{\left (a^2 d^2+b^2 e^2+a e (b d-c e)\right ) x}{a^3 e^3}-\frac{(a d+b e) x^2}{2 a^2 e^2}+\frac{x^3}{3 a e}+\frac{\left (b^5 d-5 a b^3 c d+5 a^2 b c^2 d-b^4 c e+4 a b^2 c^2 e-2 a^2 c^3 e\right ) \tanh ^{-1}\left (\frac{b+2 a x}{\sqrt{b^2-4 a c}}\right )}{a^4 \sqrt{b^2-4 a c} \left (a d^2-e (b d-c e)\right )}-\frac{d^5 \log (d+e x)}{e^4 \left (a d^2-e (b d-c e)\right )}+\frac{\left (b^4 d-3 a b^2 c d+a^2 c^2 d-b^3 c e+2 a b c^2 e\right ) \log \left (c+b x+a x^2\right )}{2 a^4 \left (a d^2-e (b d-c e)\right )}\\ \end{align*}
Mathematica [A] time = 0.241123, size = 283, normalized size = 1.01 \[ \frac{\left (a^2 c^2 d-3 a b^2 c d+2 a b c^2 e-b^3 c e+b^4 d\right ) \log \left (a x^2+b x+c\right )}{2 a^4 \left (a d^2-b d e+c e^2\right )}+\frac{\left (5 a^2 b c^2 d-2 a^2 c^3 e+4 a b^2 c^2 e-5 a b^3 c d-b^4 c e+b^5 d\right ) \tan ^{-1}\left (\frac{2 a x+b}{\sqrt{4 a c-b^2}}\right )}{a^4 \sqrt{4 a c-b^2} \left (-a d^2+b d e-c e^2\right )}+\frac{x \left (a^2 d^2+a b d e-a c e^2+b^2 e^2\right )}{a^3 e^3}-\frac{x^2 (a d+b e)}{2 a^2 e^2}-\frac{d^5 \log (d+e x)}{e^4 \left (a d^2-b d e+c e^2\right )}+\frac{x^3}{3 a e} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.014, size = 662, normalized size = 2.4 \begin{align*}{\frac{{x}^{3}}{3\,ae}}-{\frac{{x}^{2}d}{2\,a{e}^{2}}}-{\frac{{x}^{2}b}{2\,e{a}^{2}}}+{\frac{{d}^{2}x}{{e}^{3}a}}+{\frac{bdx}{{a}^{2}{e}^{2}}}-{\frac{cx}{e{a}^{2}}}+{\frac{{b}^{2}x}{e{a}^{3}}}-{\frac{{d}^{5}\ln \left ( ex+d \right ) }{{e}^{4} \left ( a{d}^{2}-bde+{e}^{2}c \right ) }}+{\frac{\ln \left ( a{x}^{2}+bx+c \right ){c}^{2}d}{ \left ( 2\,a{d}^{2}-2\,bde+2\,{e}^{2}c \right ){a}^{2}}}-{\frac{3\,\ln \left ( a{x}^{2}+bx+c \right ){b}^{2}cd}{ \left ( 2\,a{d}^{2}-2\,bde+2\,{e}^{2}c \right ){a}^{3}}}+{\frac{\ln \left ( a{x}^{2}+bx+c \right ) b{c}^{2}e}{ \left ( a{d}^{2}-bde+{e}^{2}c \right ){a}^{3}}}+{\frac{\ln \left ( a{x}^{2}+bx+c \right ){b}^{4}d}{ \left ( 2\,a{d}^{2}-2\,bde+2\,{e}^{2}c \right ){a}^{4}}}-{\frac{\ln \left ( a{x}^{2}+bx+c \right ){b}^{3}ce}{ \left ( 2\,a{d}^{2}-2\,bde+2\,{e}^{2}c \right ){a}^{4}}}-5\,{\frac{d{c}^{2}b}{ \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 ) }+2\,{\frac{e{c}^{3}}{ \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 ) }+5\,{\frac{{b}^{3}cd}{ \left ( a{d}^{2}-bde+{e}^{2}c \right ){a}^{3}\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,ax+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }-4\,{\frac{{b}^{2}{c}^{2}e}{ \left ( a{d}^{2}-bde+{e}^{2}c \right ){a}^{3}\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,ax+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }-{\frac{{b}^{5}d}{ \left ( a{d}^{2}-bde+{e}^{2}c \right ){a}^{4}}\arctan \left ({(2\,ax+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}+{\frac{{b}^{4}ce}{ \left ( a{d}^{2}-bde+{e}^{2}c \right ){a}^{4}}\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 = 137.062, size = 2079, normalized size = 7.42 \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 = 1.10794, size = 398, normalized size = 1.42 \begin{align*} -\frac{d^{5} \log \left ({\left | x e + d \right |}\right )}{a d^{2} e^{4} - b d e^{5} + c e^{6}} + \frac{{\left (b^{4} d - 3 \, a b^{2} c d + a^{2} c^{2} d - b^{3} c e + 2 \, a b c^{2} e\right )} \log \left (a x^{2} + b x + c\right )}{2 \,{\left (a^{5} d^{2} - a^{4} b d e + a^{4} c e^{2}\right )}} - \frac{{\left (b^{5} d - 5 \, a b^{3} c d + 5 \, a^{2} b c^{2} d - b^{4} c e + 4 \, a b^{2} c^{2} e - 2 \, a^{2} c^{3} e\right )} \arctan \left (\frac{2 \, a x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{{\left (a^{5} d^{2} - a^{4} b d e + a^{4} c e^{2}\right )} \sqrt{-b^{2} + 4 \, a c}} + \frac{{\left (2 \, a^{2} x^{3} e^{2} - 3 \, a^{2} d x^{2} e + 6 \, a^{2} d^{2} x - 3 \, a b x^{2} e^{2} + 6 \, a b d x e + 6 \, b^{2} x e^{2} - 6 \, a c x e^{2}\right )} e^{\left (-3\right )}}{6 \, a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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