Optimal. Leaf size=176 \[ \frac {x}{a e}+\frac {\left (b^3 d-3 a b c d-b^2 c e+2 a c^2 e\right ) \tanh ^{-1}\left (\frac {b+2 a x}{\sqrt {b^2-4 a c}}\right )}{a^2 \sqrt {b^2-4 a c} \left (a d^2-e (b d-c e)\right )}-\frac {d^3 \log (d+e x)}{e^2 \left (a d^2-e (b d-c e)\right )}+\frac {\left (b^2 d-a c d-b c e\right ) \log \left (c+b x+a x^2\right )}{2 a^2 \left (a d^2-e (b d-c e)\right )} \]
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Rubi [A]
time = 0.19, antiderivative size = 176, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {1583, 1642,
648, 632, 212, 642} \begin {gather*} \frac {\left (-a c d+b^2 d-b c e\right ) \log \left (a x^2+b x+c\right )}{2 a^2 \left (a d^2-e (b d-c e)\right )}+\frac {\left (-3 a b c d+2 a c^2 e+b^3 d-b^2 c e\right ) \tanh ^{-1}\left (\frac {2 a x+b}{\sqrt {b^2-4 a c}}\right )}{a^2 \sqrt {b^2-4 a c} \left (a d^2-e (b d-c e)\right )}-\frac {d^3 \log (d+e x)}{e^2 \left (a d^2-e (b d-c e)\right )}+\frac {x}{a e} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 632
Rule 642
Rule 648
Rule 1583
Rule 1642
Rubi steps
\begin {align*} \int \frac {x}{\left (a+\frac {c}{x^2}+\frac {b}{x}\right ) (d+e x)} \, dx &=\int \frac {x^3}{(d+e x) \left (c+b x+a x^2\right )} \, dx\\ &=\int \left (\frac {1}{a e}+\frac {d^3}{e \left (-a d^2+e (b d-c e)\right ) (d+e x)}+\frac {c (b d-c e)+\left (b^2 d-a c d-b c e\right ) x}{a \left (a d^2-e (b d-c e)\right ) \left (c+b x+a x^2\right )}\right ) \, dx\\ &=\frac {x}{a e}-\frac {d^3 \log (d+e x)}{e^2 \left (a d^2-e (b d-c e)\right )}+\frac {\int \frac {c (b d-c e)+\left (b^2 d-a c d-b c e\right ) x}{c+b x+a x^2} \, dx}{a \left (a d^2-b d e+c e^2\right )}\\ &=\frac {x}{a e}-\frac {d^3 \log (d+e x)}{e^2 \left (a d^2-e (b d-c e)\right )}+\frac {\left (b^2 d-a c d-b c e\right ) \int \frac {b+2 a x}{c+b x+a x^2} \, dx}{2 a^2 \left (a d^2-e (b d-c e)\right )}-\frac {\left (b^3 d-3 a b c d-b^2 c e+2 a c^2 e\right ) \int \frac {1}{c+b x+a x^2} \, dx}{2 a^2 \left (a d^2-e (b d-c e)\right )}\\ &=\frac {x}{a e}-\frac {d^3 \log (d+e x)}{e^2 \left (a d^2-e (b d-c e)\right )}+\frac {\left (b^2 d-a c d-b c e\right ) \log \left (c+b x+a x^2\right )}{2 a^2 \left (a d^2-e (b d-c e)\right )}+\frac {\left (b^3 d-3 a b c d-b^2 c e+2 a c^2 e\right ) \text {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 a x\right )}{a^2 \left (a d^2-e (b d-c e)\right )}\\ &=\frac {x}{a e}+\frac {\left (b^3 d-3 a b c d-b^2 c e+2 a c^2 e\right ) \tanh ^{-1}\left (\frac {b+2 a x}{\sqrt {b^2-4 a c}}\right )}{a^2 \sqrt {b^2-4 a c} \left (a d^2-e (b d-c e)\right )}-\frac {d^3 \log (d+e x)}{e^2 \left (a d^2-e (b d-c e)\right )}+\frac {\left (b^2 d-a c d-b c e\right ) \log \left (c+b x+a x^2\right )}{2 a^2 \left (a d^2-e (b d-c e)\right )}\\ \end {align*}
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Mathematica [A]
time = 0.13, size = 178, normalized size = 1.01 \begin {gather*} \frac {x}{a e}+\frac {\left (b^3 d-3 a b c d-b^2 c e+2 a c^2 e\right ) \tan ^{-1}\left (\frac {b+2 a x}{\sqrt {-b^2+4 a c}}\right )}{a^2 \sqrt {-b^2+4 a c} \left (-a d^2+b d e-c e^2\right )}-\frac {d^3 \log (d+e x)}{e^2 \left (a d^2-b d e+c e^2\right )}+\frac {\left (b^2 d-a c d-b c e\right ) \log \left (c+b x+a x^2\right )}{2 a^2 \left (a d^2-b d e+c e^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.25, size = 164, normalized size = 0.93
method | result | size |
default | \(\frac {x}{a e}-\frac {d^{3} \ln \left (e x +d \right )}{e^{2} \left (a \,d^{2}-d e b +c \,e^{2}\right )}+\frac {\frac {\left (-a c d +b^{2} d -b c e \right ) \ln \left (a \,x^{2}+b x +c \right )}{2 a}+\frac {2 \left (b c d -c^{2} e -\frac {\left (-a c d +b^{2} d -b c e \right ) b}{2 a}\right ) \arctan \left (\frac {2 a x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}}{\left (a \,d^{2}-d e b +c \,e^{2}\right ) a}\) | \(164\) |
risch | \(\frac {x}{a e}-\frac {d^{3} \ln \left (e x +d \right )}{e^{2} \left (a \,d^{2}-d e b +c \,e^{2}\right )}+\frac {\munderset {\textit {\_R} =\RootOf \left (\left (4 a^{3} c \,d^{2}-b^{2} d^{2} a^{2}-4 a^{2} b c d e +4 e^{2} c^{2} a^{2}+a \,b^{3} d e -a \,b^{2} c \,e^{2}\right ) \textit {\_Z}^{2}+\left (4 a^{2} c^{2} d e -5 a \,b^{2} c d e +4 c^{2} e^{2} a b +d e \,b^{4}-c \,e^{2} b^{3}\right ) \textit {\_Z} +a \,c^{3} e^{2}\right )}{\sum }\textit {\_R} \ln \left (\left (\left (-2 a^{3} d^{2} e +2 a^{2} b d \,e^{2}+6 e^{3} c \,a^{2}-2 a \,b^{2} e^{3}\right ) \textit {\_R}^{2}+\left (-2 a^{3} d^{3}+3 a^{2} c d \,e^{2}+6 a b c \,e^{3}-2 b^{3} e^{3}\right ) \textit {\_R} -a^{2} c \,d^{2} e +a \,b^{2} d^{2} e +a b c d \,e^{2}+a \,c^{2} e^{3}\right ) x +\left (-a^{2} b \,d^{2} e +8 a^{2} c d \,e^{2}-a \,b^{2} d \,e^{2}-a b c \,e^{3}\right ) \textit {\_R}^{2}+\left (-a^{2} b \,d^{3}+4 a^{2} c \,d^{2} e -a \,b^{2} d^{2} e +5 a b c d \,e^{2}+a \,c^{2} e^{3}-b^{3} d \,e^{2}-b^{2} c \,e^{3}\right ) \textit {\_R} +a b c \,d^{2} e +a \,c^{2} d \,e^{2}\right )}{a e}\) | \(419\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.31, size = 581, normalized size = 3.30 \begin {gather*} \left [\frac {2 \, {\left (a^{2} b^{2} - 4 \, a^{3} c\right )} d^{2} x e - 2 \, {\left (a^{2} b^{2} - 4 \, a^{3} c\right )} d^{3} \log \left (x e + d\right ) - 2 \, {\left (a b^{3} - 4 \, a^{2} b c\right )} d x e^{2} + 2 \, {\left (a b^{2} c - 4 \, a^{2} c^{2}\right )} x e^{3} + {\left ({\left (b^{3} - 3 \, a b c\right )} d e^{2} - {\left (b^{2} c - 2 \, a c^{2}\right )} e^{3}\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 ) + {\left ({\left (b^{4} - 5 \, a b^{2} c + 4 \, a^{2} c^{2}\right )} d e^{2} - {\left (b^{3} c - 4 \, a b c^{2}\right )} e^{3}\right )} \log \left (a x^{2} + b x + c\right )}{2 \, {\left ({\left (a^{3} b^{2} - 4 \, a^{4} c\right )} d^{2} e^{2} - {\left (a^{2} b^{3} - 4 \, a^{3} b c\right )} d e^{3} + {\left (a^{2} b^{2} c - 4 \, a^{3} c^{2}\right )} e^{4}\right )}}, \frac {2 \, {\left (a^{2} b^{2} - 4 \, a^{3} c\right )} d^{2} x e - 2 \, {\left (a^{2} b^{2} - 4 \, a^{3} c\right )} d^{3} \log \left (x e + d\right ) - 2 \, {\left (a b^{3} - 4 \, a^{2} b c\right )} d x e^{2} + 2 \, {\left (a b^{2} c - 4 \, a^{2} c^{2}\right )} x e^{3} + 2 \, {\left ({\left (b^{3} - 3 \, a b c\right )} d e^{2} - {\left (b^{2} c - 2 \, a c^{2}\right )} e^{3}\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 ) + {\left ({\left (b^{4} - 5 \, a b^{2} c + 4 \, a^{2} c^{2}\right )} d e^{2} - {\left (b^{3} c - 4 \, a b c^{2}\right )} e^{3}\right )} \log \left (a x^{2} + b x + c\right )}{2 \, {\left ({\left (a^{3} b^{2} - 4 \, a^{4} c\right )} d^{2} e^{2} - {\left (a^{2} b^{3} - 4 \, a^{3} b c\right )} d e^{3} + {\left (a^{2} b^{2} c - 4 \, a^{3} c^{2}\right )} e^{4}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.26, size = 185, normalized size = 1.05 \begin {gather*} -\frac {d^{3} \log \left ({\left | x e + d \right |}\right )}{a d^{2} e^{2} - b d e^{3} + c e^{4}} + \frac {x e^{\left (-1\right )}}{a} + \frac {{\left (b^{2} d - a c d - b c e\right )} \log \left (a x^{2} + b x + c\right )}{2 \, {\left (a^{3} d^{2} - a^{2} b d e + a^{2} c e^{2}\right )}} - \frac {{\left (b^{3} d - 3 \, a b c d - b^{2} c e + 2 \, a c^{2} e\right )} \arctan \left (\frac {2 \, a x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{{\left (a^{3} d^{2} - a^{2} b d e + a^{2} c e^{2}\right )} \sqrt {-b^{2} + 4 \, a c}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.34, size = 1367, normalized size = 7.77 \begin {gather*} \frac {x}{a\,e}-\frac {\ln \left (c^3\,e^5\,\sqrt {b^2-4\,a\,c}-b\,c^3\,e^5-4\,a^3\,c\,d^5+a^2\,b^2\,d^5+b^4\,d^3\,e^2+3\,b^2\,c^2\,d\,e^4-3\,b^3\,c\,d^2\,e^3-b^3\,d^3\,e^2\,\sqrt {b^2-4\,a\,c}+6\,a^2\,c^2\,d^3\,e^2-6\,a\,c^3\,d\,e^4-2\,a\,c^3\,e^5\,x-a^2\,b\,d^5\,\sqrt {b^2-4\,a\,c}-2\,a^3\,d^5\,x\,\sqrt {b^2-4\,a\,c}-8\,a^3\,c\,d^4\,e\,x+4\,a^2\,c\,d^4\,e\,\sqrt {b^2-4\,a\,c}-3\,b\,c^2\,d\,e^4\,\sqrt {b^2-4\,a\,c}+9\,a\,b\,c^2\,d^2\,e^3-5\,a\,b^2\,c\,d^3\,e^2+2\,a^2\,b^2\,d^4\,e\,x-3\,a\,c^2\,d^2\,e^3\,\sqrt {b^2-4\,a\,c}+3\,b^2\,c\,d^2\,e^3\,\sqrt {b^2-4\,a\,c}+6\,a^2\,c^2\,d^2\,e^3\,x-2\,a\,b^2\,d^3\,e^2\,x\,\sqrt {b^2-4\,a\,c}+3\,a^2\,c\,d^3\,e^2\,x\,\sqrt {b^2-4\,a\,c}+3\,a\,b\,c^2\,d\,e^4\,x+a\,b\,c\,d^3\,e^2\,\sqrt {b^2-4\,a\,c}+2\,a^2\,b\,d^4\,e\,x\,\sqrt {b^2-4\,a\,c}-3\,a\,c^2\,d\,e^4\,x\,\sqrt {b^2-4\,a\,c}-3\,a\,b^2\,c\,d^2\,e^3\,x+a^2\,b\,c\,d^3\,e^2\,x+3\,a\,b\,c\,d^2\,e^3\,x\,\sqrt {b^2-4\,a\,c}\right )\,\left (b^4\,d-b^3\,d\,\sqrt {b^2-4\,a\,c}+4\,a^2\,c^2\,d-b^3\,c\,e-5\,a\,b^2\,c\,d+4\,a\,b\,c^2\,e-2\,a\,c^2\,e\,\sqrt {b^2-4\,a\,c}+b^2\,c\,e\,\sqrt {b^2-4\,a\,c}+3\,a\,b\,c\,d\,\sqrt {b^2-4\,a\,c}\right )}{2\,\left (4\,a^4\,c\,d^2-a^3\,b^2\,d^2-4\,a^3\,b\,c\,d\,e+4\,a^3\,c^2\,e^2+a^2\,b^3\,d\,e-a^2\,b^2\,c\,e^2\right )}-\frac {\ln \left (a^2\,b^2\,d^5-b\,c^3\,e^5-c^3\,e^5\,\sqrt {b^2-4\,a\,c}-4\,a^3\,c\,d^5+b^4\,d^3\,e^2+3\,b^2\,c^2\,d\,e^4-3\,b^3\,c\,d^2\,e^3+b^3\,d^3\,e^2\,\sqrt {b^2-4\,a\,c}+6\,a^2\,c^2\,d^3\,e^2-6\,a\,c^3\,d\,e^4-2\,a\,c^3\,e^5\,x+a^2\,b\,d^5\,\sqrt {b^2-4\,a\,c}+2\,a^3\,d^5\,x\,\sqrt {b^2-4\,a\,c}-8\,a^3\,c\,d^4\,e\,x-4\,a^2\,c\,d^4\,e\,\sqrt {b^2-4\,a\,c}+3\,b\,c^2\,d\,e^4\,\sqrt {b^2-4\,a\,c}+9\,a\,b\,c^2\,d^2\,e^3-5\,a\,b^2\,c\,d^3\,e^2+2\,a^2\,b^2\,d^4\,e\,x+3\,a\,c^2\,d^2\,e^3\,\sqrt {b^2-4\,a\,c}-3\,b^2\,c\,d^2\,e^3\,\sqrt {b^2-4\,a\,c}+6\,a^2\,c^2\,d^2\,e^3\,x+2\,a\,b^2\,d^3\,e^2\,x\,\sqrt {b^2-4\,a\,c}-3\,a^2\,c\,d^3\,e^2\,x\,\sqrt {b^2-4\,a\,c}+3\,a\,b\,c^2\,d\,e^4\,x-a\,b\,c\,d^3\,e^2\,\sqrt {b^2-4\,a\,c}-2\,a^2\,b\,d^4\,e\,x\,\sqrt {b^2-4\,a\,c}+3\,a\,c^2\,d\,e^4\,x\,\sqrt {b^2-4\,a\,c}-3\,a\,b^2\,c\,d^2\,e^3\,x+a^2\,b\,c\,d^3\,e^2\,x-3\,a\,b\,c\,d^2\,e^3\,x\,\sqrt {b^2-4\,a\,c}\right )\,\left (b^4\,d+b^3\,d\,\sqrt {b^2-4\,a\,c}+4\,a^2\,c^2\,d-b^3\,c\,e-5\,a\,b^2\,c\,d+4\,a\,b\,c^2\,e+2\,a\,c^2\,e\,\sqrt {b^2-4\,a\,c}-b^2\,c\,e\,\sqrt {b^2-4\,a\,c}-3\,a\,b\,c\,d\,\sqrt {b^2-4\,a\,c}\right )}{2\,\left (4\,a^4\,c\,d^2-a^3\,b^2\,d^2-4\,a^3\,b\,c\,d\,e+4\,a^3\,c^2\,e^2+a^2\,b^3\,d\,e-a^2\,b^2\,c\,e^2\right )}-\frac {d^3\,\ln \left (d+e\,x\right )}{a\,d^2\,e^2-b\,d\,e^3+c\,e^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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