Integrand size = 25, antiderivative size = 280 \[ \int \frac {x^3}{\left (a+\frac {c}{x^2}+\frac {b}{x}\right ) (d+e x)} \, dx=\frac {\left (a^2 d^2+b^2 e^2+a e (b d-c e)\right ) x}{a^3 e^3}-\frac {(a d+b e) x^2}{2 a^2 e^2}+\frac {x^3}{3 a e}+\frac {\left (b^5 d-5 a b^3 c d+5 a^2 b c^2 d-b^4 c e+4 a b^2 c^2 e-2 a^2 c^3 e\right ) \text {arctanh}\left (\frac {b+2 a x}{\sqrt {b^2-4 a c}}\right )}{a^4 \sqrt {b^2-4 a c} \left (a d^2-e (b d-c e)\right )}-\frac {d^5 \log (d+e x)}{e^4 \left (a d^2-e (b d-c e)\right )}+\frac {\left (b^4 d-3 a b^2 c d+a^2 c^2 d-b^3 c e+2 a b c^2 e\right ) \log \left (c+b x+a x^2\right )}{2 a^4 \left (a d^2-e (b d-c e)\right )} \]
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Time = 0.38 (sec) , antiderivative size = 280, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {1583, 1642, 648, 632, 212, 642} \[ \int \frac {x^3}{\left (a+\frac {c}{x^2}+\frac {b}{x}\right ) (d+e x)} \, dx=-\frac {x^2 (a d+b e)}{2 a^2 e^2}+\frac {\text {arctanh}\left (\frac {2 a x+b}{\sqrt {b^2-4 a c}}\right ) \left (5 a^2 b c^2 d-2 a^2 c^3 e-5 a b^3 c d+4 a b^2 c^2 e+b^5 d-b^4 c e\right )}{a^4 \sqrt {b^2-4 a c} \left (a d^2-e (b d-c e)\right )}+\frac {\left (a^2 c^2 d-3 a b^2 c d+2 a b c^2 e+b^4 d-b^3 c e\right ) \log \left (a x^2+b x+c\right )}{2 a^4 \left (a d^2-e (b d-c e)\right )}+\frac {x \left (a^2 d^2+a e (b d-c e)+b^2 e^2\right )}{a^3 e^3}-\frac {d^5 \log (d+e x)}{e^4 \left (a d^2-e (b d-c e)\right )}+\frac {x^3}{3 a e} \]
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Rule 212
Rule 632
Rule 642
Rule 648
Rule 1583
Rule 1642
Rubi steps \begin{align*} \text {integral}& = \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 ) \text {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*}
Time = 0.15 (sec) , antiderivative size = 283, normalized size of antiderivative = 1.01 \[ \int \frac {x^3}{\left (a+\frac {c}{x^2}+\frac {b}{x}\right ) (d+e x)} \, dx=\frac {\left (a^2 d^2+a b d e+b^2 e^2-a c e^2\right ) x}{a^3 e^3}-\frac {(a d+b e) x^2}{2 a^2 e^2}+\frac {x^3}{3 a e}+\frac {\left (b^5 d-5 a b^3 c d+5 a^2 b c^2 d-b^4 c e+4 a b^2 c^2 e-2 a^2 c^3 e\right ) \arctan \left (\frac {b+2 a x}{\sqrt {-b^2+4 a c}}\right )}{a^4 \sqrt {-b^2+4 a c} \left (-a d^2+b d e-c e^2\right )}-\frac {d^5 \log (d+e x)}{e^4 \left (a d^2-b d e+c e^2\right )}+\frac {\left (b^4 d-3 a b^2 c d+a^2 c^2 d-b^3 c e+2 a b c^2 e\right ) \log \left (c+b x+a x^2\right )}{2 a^4 \left (a d^2-b d e+c e^2\right )} \]
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Time = 0.67 (sec) , antiderivative size = 286, normalized size of antiderivative = 1.02
method | result | size |
default | \(\frac {\frac {1}{3} a^{2} e^{2} x^{3}-\frac {1}{2} a^{2} d e \,x^{2}-\frac {1}{2} a b \,e^{2} x^{2}+a^{2} d^{2} x +a b d e x -e^{2} a c x +b^{2} e^{2} x}{a^{3} e^{3}}+\frac {\frac {\left (a^{2} c^{2} d -3 a \,b^{2} c d +2 a b \,c^{2} e +d \,b^{4}-b^{3} c e \right ) \ln \left (a \,x^{2}+b x +c \right )}{2 a}+\frac {2 \left (-2 a b \,c^{2} d +a \,c^{3} e +b^{3} c d -b^{2} c^{2} e -\frac {\left (a^{2} c^{2} d -3 a \,b^{2} c d +2 a b \,c^{2} e +d \,b^{4}-b^{3} c e \right ) b}{2 a}\right ) \arctan \left (\frac {2 a x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}}{\left (a \,d^{2}-b d e +c \,e^{2}\right ) a^{3}}-\frac {d^{5} \ln \left (e x +d \right )}{e^{4} \left (a \,d^{2}-b d e +c \,e^{2}\right )}\) | \(286\) |
risch | \(\frac {x^{3}}{3 a e}-\frac {d \,x^{2}}{2 a \,e^{2}}-\frac {b \,x^{2}}{2 a^{2} e}+\frac {d^{2} x}{a \,e^{3}}+\frac {b d x}{a^{2} e^{2}}-\frac {c x}{a^{2} e}+\frac {b^{2} x}{a^{3} e}-\frac {d^{5} \ln \left (e x +d \right )}{e^{4} \left (a \,d^{2}-b d e +c \,e^{2}\right )}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (4 a^{3} c \,d^{2}-b^{2} d^{2} a^{2}-4 a^{2} b c d e +4 a^{2} c^{2} e^{2}+a \,b^{3} d e -a \,b^{2} c \,e^{2}\right ) \textit {\_Z}^{2}+\left (-4 c^{3} d \,a^{3} e^{3}+13 b^{2} c^{2} d \,e^{3} a^{2}-8 a^{2} b \,c^{3} e^{4}-7 d c \,b^{4} a \,e^{3}+6 a \,b^{3} c^{2} e^{4}+b^{6} d \,e^{3}-b^{5} c \,e^{4}\right ) \textit {\_Z} +a^{3} c^{5} e^{6}\right )}{\sum }\textit {\_R} \ln \left (\left (\left (-2 a^{3} d^{2} e +2 a^{2} b d \,e^{2}+6 a^{2} c \,e^{3}-2 a \,b^{2} e^{3}\right ) \textit {\_R}^{2}+\left (-2 a^{5} d^{5}+4 a^{4} c \,d^{3} e^{2}-a^{3} b^{2} d^{3} e^{2}+4 a^{3} b c \,d^{2} e^{3}-3 a^{3} c^{2} d \,e^{4}-a^{2} b^{3} d^{2} e^{3}+2 a^{2} b^{2} c d \,e^{4}-11 a^{2} b \,c^{2} e^{5}+10 a \,b^{3} c \,e^{5}-2 b^{5} e^{5}\right ) \textit {\_R} +a^{5} c^{2} d^{4} e^{3}-3 a^{4} b^{2} c \,d^{4} e^{3}-2 a^{4} b \,c^{2} d^{3} e^{4}-a^{4} c^{3} d^{2} e^{5}+a^{3} b^{4} d^{4} e^{3}+a^{3} b^{3} c \,d^{3} e^{4}+a^{3} b^{2} c^{2} d^{2} e^{5}+a^{3} b \,c^{3} d \,e^{6}+a^{3} c^{4} e^{7}\right ) x +\left (-a^{2} b \,d^{2} e +8 a^{2} c d \,e^{2}-a \,b^{2} d \,e^{2}-a b c \,e^{3}\right ) \textit {\_R}^{2}+\left (-a^{4} b \,d^{5}+4 a^{4} c \,d^{4} e -a^{3} b^{2} d^{4} e +4 a^{3} b c \,d^{3} e^{2}-4 a^{3} c^{2} d^{2} e^{3}-a^{2} b^{3} d^{3} e^{2}+5 a^{2} b^{2} c \,d^{2} e^{3}-10 a^{2} b \,c^{2} d \,e^{4}-a^{2} c^{3} e^{5}-a \,b^{4} d^{2} e^{3}+7 a \,b^{3} c d \,e^{4}+3 a \,b^{2} c^{2} e^{5}-b^{5} d \,e^{4}-b^{4} c \,e^{5}\right ) \textit {\_R} -2 a^{4} b \,c^{2} d^{4} e^{3}-a^{4} c^{3} d^{3} e^{4}+a^{3} b^{3} c \,d^{4} e^{3}+a^{3} b^{2} c^{2} d^{3} e^{4}+a^{3} b \,c^{3} d^{2} e^{5}+a^{3} c^{4} d \,e^{6}\right )}{a^{3} e^{3}}\) | \(861\) |
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Time = 18.16 (sec) , antiderivative size = 1027, normalized size of antiderivative = 3.67 \[ \int \frac {x^3}{\left (a+\frac {c}{x^2}+\frac {b}{x}\right ) (d+e x)} \, dx=\left [-\frac {6 \, {\left (a^{4} b^{2} - 4 \, a^{5} c\right )} d^{5} \log \left (e x + d\right ) - 2 \, {\left ({\left (a^{4} b^{2} - 4 \, a^{5} c\right )} d^{2} e^{3} - {\left (a^{3} b^{3} - 4 \, a^{4} b c\right )} d e^{4} + {\left (a^{3} b^{2} c - 4 \, a^{4} c^{2}\right )} e^{5}\right )} x^{3} + 3 \, {\left ({\left (a^{4} b^{2} - 4 \, a^{5} c\right )} d^{3} e^{2} - {\left (a^{2} b^{4} - 5 \, a^{3} b^{2} c + 4 \, a^{4} c^{2}\right )} d e^{4} + {\left (a^{2} b^{3} c - 4 \, a^{3} b c^{2}\right )} e^{5}\right )} x^{2} + 3 \, {\left ({\left (b^{5} - 5 \, a b^{3} c + 5 \, a^{2} b c^{2}\right )} d e^{4} - {\left (b^{4} c - 4 \, a b^{2} c^{2} + 2 \, a^{2} c^{3}\right )} e^{5}\right )} \sqrt {b^{2} - 4 \, a c} \log \left (\frac {2 \, a^{2} x^{2} + 2 \, a b x + b^{2} - 2 \, a c - \sqrt {b^{2} - 4 \, a c} {\left (2 \, a x + b\right )}}{a x^{2} + b x + c}\right ) - 6 \, {\left ({\left (a^{4} b^{2} - 4 \, a^{5} c\right )} d^{4} e - {\left (a b^{5} - 6 \, a^{2} b^{3} c + 8 \, a^{3} b c^{2}\right )} d e^{4} + {\left (a b^{4} c - 5 \, a^{2} b^{2} c^{2} + 4 \, a^{3} c^{3}\right )} e^{5}\right )} x - 3 \, {\left ({\left (b^{6} - 7 \, a b^{4} c + 13 \, a^{2} b^{2} c^{2} - 4 \, a^{3} c^{3}\right )} d e^{4} - {\left (b^{5} c - 6 \, a b^{3} c^{2} + 8 \, a^{2} b c^{3}\right )} e^{5}\right )} \log \left (a x^{2} + b x + c\right )}{6 \, {\left ({\left (a^{5} b^{2} - 4 \, a^{6} c\right )} d^{2} e^{4} - {\left (a^{4} b^{3} - 4 \, a^{5} b c\right )} d e^{5} + {\left (a^{4} b^{2} c - 4 \, a^{5} c^{2}\right )} e^{6}\right )}}, -\frac {6 \, {\left (a^{4} b^{2} - 4 \, a^{5} c\right )} d^{5} \log \left (e x + d\right ) - 2 \, {\left ({\left (a^{4} b^{2} - 4 \, a^{5} c\right )} d^{2} e^{3} - {\left (a^{3} b^{3} - 4 \, a^{4} b c\right )} d e^{4} + {\left (a^{3} b^{2} c - 4 \, a^{4} c^{2}\right )} e^{5}\right )} x^{3} + 3 \, {\left ({\left (a^{4} b^{2} - 4 \, a^{5} c\right )} d^{3} e^{2} - {\left (a^{2} b^{4} - 5 \, a^{3} b^{2} c + 4 \, a^{4} c^{2}\right )} d e^{4} + {\left (a^{2} b^{3} c - 4 \, a^{3} b c^{2}\right )} e^{5}\right )} x^{2} - 6 \, {\left ({\left (b^{5} - 5 \, a b^{3} c + 5 \, a^{2} b c^{2}\right )} d e^{4} - {\left (b^{4} c - 4 \, a b^{2} c^{2} + 2 \, a^{2} c^{3}\right )} e^{5}\right )} \sqrt {-b^{2} + 4 \, a c} \arctan \left (-\frac {\sqrt {-b^{2} + 4 \, a c} {\left (2 \, a x + b\right )}}{b^{2} - 4 \, a c}\right ) - 6 \, {\left ({\left (a^{4} b^{2} - 4 \, a^{5} c\right )} d^{4} e - {\left (a b^{5} - 6 \, a^{2} b^{3} c + 8 \, a^{3} b c^{2}\right )} d e^{4} + {\left (a b^{4} c - 5 \, a^{2} b^{2} c^{2} + 4 \, a^{3} c^{3}\right )} e^{5}\right )} x - 3 \, {\left ({\left (b^{6} - 7 \, a b^{4} c + 13 \, a^{2} b^{2} c^{2} - 4 \, a^{3} c^{3}\right )} d e^{4} - {\left (b^{5} c - 6 \, a b^{3} c^{2} + 8 \, a^{2} b c^{3}\right )} e^{5}\right )} \log \left (a x^{2} + b x + c\right )}{6 \, {\left ({\left (a^{5} b^{2} - 4 \, a^{6} c\right )} d^{2} e^{4} - {\left (a^{4} b^{3} - 4 \, a^{5} b c\right )} d e^{5} + {\left (a^{4} b^{2} c - 4 \, a^{5} c^{2}\right )} e^{6}\right )}}\right ] \]
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Timed out. \[ \int \frac {x^3}{\left (a+\frac {c}{x^2}+\frac {b}{x}\right ) (d+e x)} \, dx=\text {Timed out} \]
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Exception generated. \[ \int \frac {x^3}{\left (a+\frac {c}{x^2}+\frac {b}{x}\right ) (d+e x)} \, dx=\text {Exception raised: ValueError} \]
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Time = 0.34 (sec) , antiderivative size = 295, normalized size of antiderivative = 1.05 \[ \int \frac {x^3}{\left (a+\frac {c}{x^2}+\frac {b}{x}\right ) (d+e x)} \, dx=-\frac {d^{5} \log \left ({\left | e x + d \right |}\right )}{a d^{2} e^{4} - b d e^{5} + c e^{6}} + \frac {{\left (b^{4} d - 3 \, a b^{2} c d + a^{2} c^{2} d - b^{3} c e + 2 \, a b c^{2} e\right )} \log \left (a x^{2} + b x + c\right )}{2 \, {\left (a^{5} d^{2} - a^{4} b d e + a^{4} c e^{2}\right )}} - \frac {{\left (b^{5} d - 5 \, a b^{3} c d + 5 \, a^{2} b c^{2} d - b^{4} c e + 4 \, a b^{2} c^{2} e - 2 \, a^{2} c^{3} e\right )} \arctan \left (\frac {2 \, a x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{{\left (a^{5} d^{2} - a^{4} b d e + a^{4} c e^{2}\right )} \sqrt {-b^{2} + 4 \, a c}} + \frac {2 \, a^{2} e^{2} x^{3} - 3 \, a^{2} d e x^{2} - 3 \, a b e^{2} x^{2} + 6 \, a^{2} d^{2} x + 6 \, a b d e x + 6 \, b^{2} e^{2} x - 6 \, a c e^{2} x}{6 \, a^{3} e^{3}} \]
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Time = 11.49 (sec) , antiderivative size = 2490, normalized size of antiderivative = 8.89 \[ \int \frac {x^3}{\left (a+\frac {c}{x^2}+\frac {b}{x}\right ) (d+e x)} \, dx=\text {Too large to display} \]
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