Optimal. Leaf size=158 \[ \frac {\left (a b d-b^2 e+2 a c e\right ) \tanh ^{-1}\left (\frac {b+2 a x}{\sqrt {b^2-4 a c}}\right )}{c \sqrt {b^2-4 a c} \left (a d^2-e (b d-c e)\right )}+\frac {\log (x)}{c d}-\frac {e^2 \log (d+e x)}{d \left (a d^2-b d e+c e^2\right )}-\frac {(a d-b e) \log \left (c+b x+a x^2\right )}{2 c \left (a d^2-e (b d-c e)\right )} \]
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Rubi [A]
time = 0.17, antiderivative size = 159, normalized size of antiderivative = 1.01, number of steps
used = 7, number of rules used = 6, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {1583, 907, 648,
632, 212, 642} \begin {gather*} \frac {\left (a b d+2 a c e+b^2 (-e)\right ) \tanh ^{-1}\left (\frac {2 a x+b}{\sqrt {b^2-4 a c}}\right )}{c \sqrt {b^2-4 a c} \left (a d^2-e (b d-c e)\right )}-\frac {e^2 \log (d+e x)}{d \left (a d^2-e (b d-c e)\right )}-\frac {(a d-b e) \log \left (a x^2+b x+c\right )}{2 c \left (a d^2-e (b d-c e)\right )}+\frac {\log (x)}{c d} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 632
Rule 642
Rule 648
Rule 907
Rule 1583
Rubi steps
\begin {align*} \int \frac {1}{\left (a+\frac {c}{x^2}+\frac {b}{x}\right ) x^3 (d+e x)} \, dx &=\int \frac {1}{x (d+e x) \left (c+b x+a x^2\right )} \, dx\\ &=\int \left (\frac {1}{c d x}+\frac {e^3}{d \left (-a d^2+e (b d-c e)\right ) (d+e x)}+\frac {b^2 e-a (b d+c e)-a (a d-b e) x}{c \left (a d^2-e (b d-c e)\right ) \left (c+b x+a x^2\right )}\right ) \, dx\\ &=\frac {\log (x)}{c d}-\frac {e^2 \log (d+e x)}{d \left (a d^2-e (b d-c e)\right )}+\frac {\int \frac {b^2 e-a (b d+c e)-a (a d-b e) x}{c+b x+a x^2} \, dx}{c \left (a d^2-b d e+c e^2\right )}\\ &=\frac {\log (x)}{c d}-\frac {e^2 \log (d+e x)}{d \left (a d^2-e (b d-c e)\right )}+\frac {\left (-a b d+b^2 e-2 a c e\right ) \int \frac {1}{c+b x+a x^2} \, dx}{2 c \left (a d^2-b d e+c e^2\right )}-\frac {(a d-b e) \int \frac {b+2 a x}{c+b x+a x^2} \, dx}{2 c \left (a d^2-e (b d-c e)\right )}\\ &=\frac {\log (x)}{c d}-\frac {e^2 \log (d+e x)}{d \left (a d^2-e (b d-c e)\right )}-\frac {(a d-b e) \log \left (c+b x+a x^2\right )}{2 c \left (a d^2-e (b d-c e)\right )}-\frac {\left (-a b d+b^2 e-2 a c e\right ) \text {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 a x\right )}{c \left (a d^2-b d e+c e^2\right )}\\ &=\frac {\left (a b d-b^2 e+2 a c e\right ) \tanh ^{-1}\left (\frac {b+2 a x}{\sqrt {b^2-4 a c}}\right )}{c \sqrt {b^2-4 a c} \left (a d^2-b d e+c e^2\right )}+\frac {\log (x)}{c d}-\frac {e^2 \log (d+e x)}{d \left (a d^2-e (b d-c e)\right )}-\frac {(a d-b e) \log \left (c+b x+a x^2\right )}{2 c \left (a d^2-e (b d-c e)\right )}\\ \end {align*}
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Mathematica [A]
time = 0.13, size = 152, normalized size = 0.96 \begin {gather*} -\frac {2 d \left (a b d-b^2 e+2 a c e\right ) \tan ^{-1}\left (\frac {b+2 a x}{\sqrt {-b^2+4 a c}}\right )+\sqrt {-b^2+4 a c} \left (-2 \left (a d^2+e (-b d+c e)\right ) \log (x)+2 c e^2 \log (d+e x)+d (a d-b e) \log (c+x (b+a x))\right )}{2 c \sqrt {-b^2+4 a c} d \left (a d^2+e (-b d+c e)\right )} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.20, size = 160, normalized size = 1.01
method | result | size |
default | \(-\frac {e^{2} \ln \left (e x +d \right )}{d \left (a \,d^{2}-d e b +c \,e^{2}\right )}+\frac {\frac {\left (-d \,a^{2}+a b e \right ) \ln \left (a \,x^{2}+b x +c \right )}{2 a}+\frac {2 \left (-a b d -a c e +b^{2} e -\frac {\left (-d \,a^{2}+a b e \right ) b}{2 a}\right ) \arctan \left (\frac {2 a x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}}{\left (a \,d^{2}-d e b +c \,e^{2}\right ) c}+\frac {\ln \left (x \right )}{c d}\) | \(160\) |
risch | \(-\frac {e^{2} \ln \left (-e x -d \right )}{d \left (a \,d^{2}-d e b +c \,e^{2}\right )}+\frac {\ln \left (-x \right )}{c d}+\left (\munderset {\textit {\_R} =\RootOf \left (\left (4 a^{2} c^{2} d^{2}-a \,b^{2} c \,d^{2}-4 e d \,c^{2} b a +4 a \,c^{3} e^{2}+b^{3} c d e -b^{2} c^{2} e^{2}\right ) \textit {\_Z}^{2}+\left (4 a^{2} c d -a \,b^{2} d -4 a b c e +b^{3} e \right ) \textit {\_Z} +a^{2}\right )}{\sum }\textit {\_R} \ln \left (\left (\left (-6 d^{4} c \,a^{3}+2 a^{2} b^{2} d^{4}+14 a^{2} b c \,d^{3} e -6 a^{2} c^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e -6 a \,b^{2} c \,d^{2} e^{2}+8 a b \,c^{2} d \,e^{3}-8 a \,c^{3} e^{4}+2 b^{4} d^{2} e^{2}-2 b^{3} c d \,e^{3}+2 b^{2} c^{2} e^{4}\right ) \textit {\_R}^{2}+\left (-3 a^{3} d^{3}+3 a^{2} b \,d^{2} e -12 a^{2} c d \,e^{2}+3 a \,b^{2} d \,e^{2}+12 a b c \,e^{3}-3 b^{3} e^{3}\right ) \textit {\_R} -4 e^{2} a^{2}\right ) x +\left (a^{2} b c \,d^{4}+4 a^{2} c^{2} d^{3} e -2 a \,b^{2} c \,d^{3} e -3 a b \,c^{2} d^{2} e^{2}-4 a \,c^{3} d \,e^{3}+b^{3} c \,d^{2} e^{2}+b^{2} c^{2} d \,e^{3}\right ) \textit {\_R}^{2}+\left (-a^{2} b \,d^{3}-3 a^{2} c \,d^{2} e +2 a \,b^{2} d^{2} e +4 a b c d \,e^{2}+4 a \,c^{2} e^{3}-b^{3} d \,e^{2}-b^{2} c \,e^{3}\right ) \textit {\_R} -d e \,a^{2}-a b \,e^{2}\right )\right )\) | \(507\) |
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 = 177.70, size = 508, normalized size = 3.22 \begin {gather*} \left [-\frac {2 \, {\left (b^{2} c - 4 \, a c^{2}\right )} e^{2} \log \left (x e + d\right ) - {\left (a b d^{2} - {\left (b^{2} - 2 \, a c\right )} d e\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 (a b^{2} - 4 \, a^{2} c\right )} d^{2} - {\left (b^{3} - 4 \, a b c\right )} d e\right )} \log \left (a x^{2} + b x + c\right ) - 2 \, {\left ({\left (a b^{2} - 4 \, a^{2} c\right )} d^{2} - {\left (b^{3} - 4 \, a b c\right )} d e + {\left (b^{2} c - 4 \, a c^{2}\right )} e^{2}\right )} \log \left (x\right )}{2 \, {\left ({\left (a b^{2} c - 4 \, a^{2} c^{2}\right )} d^{3} - {\left (b^{3} c - 4 \, a b c^{2}\right )} d^{2} e + {\left (b^{2} c^{2} - 4 \, a c^{3}\right )} d e^{2}\right )}}, -\frac {2 \, {\left (b^{2} c - 4 \, a c^{2}\right )} e^{2} \log \left (x e + d\right ) - 2 \, {\left (a b d^{2} - {\left (b^{2} - 2 \, a c\right )} d e\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 (a b^{2} - 4 \, a^{2} c\right )} d^{2} - {\left (b^{3} - 4 \, a b c\right )} d e\right )} \log \left (a x^{2} + b x + c\right ) - 2 \, {\left ({\left (a b^{2} - 4 \, a^{2} c\right )} d^{2} - {\left (b^{3} - 4 \, a b c\right )} d e + {\left (b^{2} c - 4 \, a c^{2}\right )} e^{2}\right )} \log \left (x\right )}{2 \, {\left ({\left (a b^{2} c - 4 \, a^{2} c^{2}\right )} d^{3} - {\left (b^{3} c - 4 \, a b c^{2}\right )} d^{2} e + {\left (b^{2} c^{2} - 4 \, a c^{3}\right )} d e^{2}\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 = 4.42, size = 164, normalized size = 1.04 \begin {gather*} -\frac {{\left (a d - b e\right )} \log \left (a x^{2} + b x + c\right )}{2 \, {\left (a c d^{2} - b c d e + c^{2} e^{2}\right )}} - \frac {e^{3} \log \left ({\left | x e + d \right |}\right )}{a d^{3} e - b d^{2} e^{2} + c d e^{3}} - \frac {{\left (a b d - b^{2} e + 2 \, a c e\right )} \arctan \left (\frac {2 \, a x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{{\left (a c d^{2} - b c d e + c^{2} e^{2}\right )} \sqrt {-b^{2} + 4 \, a c}} + \frac {\log \left ({\left | x \right |}\right )}{c d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 5.40, size = 2399, normalized size = 15.18 \begin {gather*} \frac {\ln \left (b^3\,c^3\,e^5-6\,a^4\,c^2\,d^5+2\,a^3\,b^2\,c\,d^5+8\,a^2\,c^4\,d\,e^4-b^4\,c^2\,d\,e^4-2\,b^5\,c\,d^2\,e^3+2\,a^3\,b^3\,d^5\,x+8\,a^2\,c^4\,e^5\,x+b^4\,c^2\,e^5\,x-2\,b^6\,d^2\,e^3\,x+b^2\,c^3\,e^5\,\sqrt {b^2-4\,a\,c}+18\,a^3\,c^3\,d^3\,e^2-4\,a\,b\,c^4\,e^5-4\,a\,c^4\,e^5\,\sqrt {b^2-4\,a\,c}-5\,a^2\,c^3\,d^2\,e^3\,\sqrt {b^2-4\,a\,c}-7\,a^4\,b\,c\,d^5\,x-b^5\,c\,d\,e^4\,x-27\,a^2\,b^2\,c^2\,d^3\,e^2+2\,a^3\,b\,c\,d^5\,\sqrt {b^2-4\,a\,c}-3\,a^4\,c\,d^5\,x\,\sqrt {b^2-4\,a\,c}+2\,a\,b^2\,c^3\,d\,e^4+6\,a\,b^4\,c\,d^3\,e^2-6\,a^2\,b^3\,c\,d^4\,e+21\,a^3\,b\,c^2\,d^4\,e-6\,a\,b^2\,c^3\,e^5\,x+6\,a\,b^5\,d^3\,e^2\,x-6\,a^2\,b^4\,d^4\,e\,x-14\,a^4\,c^2\,d^4\,e\,x+7\,a^3\,c^2\,d^4\,e\,\sqrt {b^2-4\,a\,c}-b^3\,c^2\,d\,e^4\,\sqrt {b^2-4\,a\,c}-2\,b^4\,c\,d^2\,e^3\,\sqrt {b^2-4\,a\,c}+2\,a^3\,b^2\,d^5\,x\,\sqrt {b^2-4\,a\,c}+b^3\,c^2\,e^5\,x\,\sqrt {b^2-4\,a\,c}-2\,b^5\,d^2\,e^3\,x\,\sqrt {b^2-4\,a\,c}+13\,a\,b^3\,c^2\,d^2\,e^3-21\,a^2\,b\,c^3\,d^2\,e^3+10\,a^3\,c^3\,d^2\,e^3\,x+6\,a\,b^3\,c\,d^3\,e^2\,\sqrt {b^2-4\,a\,c}-6\,a^2\,b^2\,c\,d^4\,e\,\sqrt {b^2-4\,a\,c}+6\,a\,b^4\,d^3\,e^2\,x\,\sqrt {b^2-4\,a\,c}-6\,a^2\,b^3\,d^4\,e\,x\,\sqrt {b^2-4\,a\,c}+4\,a^2\,c^3\,d\,e^4\,x\,\sqrt {b^2-4\,a\,c}-32\,a^2\,b^3\,c\,d^3\,e^2\,x+35\,a^3\,b\,c^2\,d^3\,e^2\,x+7\,a\,b^2\,c^2\,d^2\,e^3\,\sqrt {b^2-4\,a\,c}-13\,a^2\,b\,c^2\,d^3\,e^2\,\sqrt {b^2-4\,a\,c}+9\,a^3\,c^2\,d^3\,e^2\,x\,\sqrt {b^2-4\,a\,c}-27\,a^2\,b^2\,c^2\,d^2\,e^3\,x+4\,a\,b\,c^3\,d\,e^4\,\sqrt {b^2-4\,a\,c}-4\,a\,b\,c^3\,e^5\,x\,\sqrt {b^2-4\,a\,c}-b^4\,c\,d\,e^4\,x\,\sqrt {b^2-4\,a\,c}+5\,a\,b^3\,c^2\,d\,e^4\,x+14\,a\,b^4\,c\,d^2\,e^3\,x-4\,a^2\,b\,c^3\,d\,e^4\,x+26\,a^3\,b^2\,c\,d^4\,e\,x+14\,a^3\,b\,c\,d^4\,e\,x\,\sqrt {b^2-4\,a\,c}+3\,a\,b^2\,c^2\,d\,e^4\,x\,\sqrt {b^2-4\,a\,c}+10\,a\,b^3\,c\,d^2\,e^3\,x\,\sqrt {b^2-4\,a\,c}-13\,a^2\,b\,c^2\,d^2\,e^3\,x\,\sqrt {b^2-4\,a\,c}-20\,a^2\,b^2\,c\,d^3\,e^2\,x\,\sqrt {b^2-4\,a\,c}\right )\,\left (d\,\left (\frac {a\,b^2}{2}-2\,a^2\,c+\frac {a\,b\,\sqrt {b^2-4\,a\,c}}{2}\right )-\frac {b^3\,e}{2}-\frac {b^2\,e\,\sqrt {b^2-4\,a\,c}}{2}+a\,c\,e\,\sqrt {b^2-4\,a\,c}+2\,a\,b\,c\,e\right )}{4\,a^2\,c^2\,d^2-a\,b^2\,c\,d^2-4\,a\,b\,c^2\,d\,e+4\,a\,c^3\,e^2+b^3\,c\,d\,e-b^2\,c^2\,e^2}-\frac {\ln \left (6\,a^4\,c^2\,d^5-b^3\,c^3\,e^5-2\,a^3\,b^2\,c\,d^5-8\,a^2\,c^4\,d\,e^4+b^4\,c^2\,d\,e^4+2\,b^5\,c\,d^2\,e^3-2\,a^3\,b^3\,d^5\,x-8\,a^2\,c^4\,e^5\,x-b^4\,c^2\,e^5\,x+2\,b^6\,d^2\,e^3\,x+b^2\,c^3\,e^5\,\sqrt {b^2-4\,a\,c}-18\,a^3\,c^3\,d^3\,e^2+4\,a\,b\,c^4\,e^5-4\,a\,c^4\,e^5\,\sqrt {b^2-4\,a\,c}-5\,a^2\,c^3\,d^2\,e^3\,\sqrt {b^2-4\,a\,c}+7\,a^4\,b\,c\,d^5\,x+b^5\,c\,d\,e^4\,x+27\,a^2\,b^2\,c^2\,d^3\,e^2+2\,a^3\,b\,c\,d^5\,\sqrt {b^2-4\,a\,c}-3\,a^4\,c\,d^5\,x\,\sqrt {b^2-4\,a\,c}-2\,a\,b^2\,c^3\,d\,e^4-6\,a\,b^4\,c\,d^3\,e^2+6\,a^2\,b^3\,c\,d^4\,e-21\,a^3\,b\,c^2\,d^4\,e+6\,a\,b^2\,c^3\,e^5\,x-6\,a\,b^5\,d^3\,e^2\,x+6\,a^2\,b^4\,d^4\,e\,x+14\,a^4\,c^2\,d^4\,e\,x+7\,a^3\,c^2\,d^4\,e\,\sqrt {b^2-4\,a\,c}-b^3\,c^2\,d\,e^4\,\sqrt {b^2-4\,a\,c}-2\,b^4\,c\,d^2\,e^3\,\sqrt {b^2-4\,a\,c}+2\,a^3\,b^2\,d^5\,x\,\sqrt {b^2-4\,a\,c}+b^3\,c^2\,e^5\,x\,\sqrt {b^2-4\,a\,c}-2\,b^5\,d^2\,e^3\,x\,\sqrt {b^2-4\,a\,c}-13\,a\,b^3\,c^2\,d^2\,e^3+21\,a^2\,b\,c^3\,d^2\,e^3-10\,a^3\,c^3\,d^2\,e^3\,x+6\,a\,b^3\,c\,d^3\,e^2\,\sqrt {b^2-4\,a\,c}-6\,a^2\,b^2\,c\,d^4\,e\,\sqrt {b^2-4\,a\,c}+6\,a\,b^4\,d^3\,e^2\,x\,\sqrt {b^2-4\,a\,c}-6\,a^2\,b^3\,d^4\,e\,x\,\sqrt {b^2-4\,a\,c}+4\,a^2\,c^3\,d\,e^4\,x\,\sqrt {b^2-4\,a\,c}+32\,a^2\,b^3\,c\,d^3\,e^2\,x-35\,a^3\,b\,c^2\,d^3\,e^2\,x+7\,a\,b^2\,c^2\,d^2\,e^3\,\sqrt {b^2-4\,a\,c}-13\,a^2\,b\,c^2\,d^3\,e^2\,\sqrt {b^2-4\,a\,c}+9\,a^3\,c^2\,d^3\,e^2\,x\,\sqrt {b^2-4\,a\,c}+27\,a^2\,b^2\,c^2\,d^2\,e^3\,x+4\,a\,b\,c^3\,d\,e^4\,\sqrt {b^2-4\,a\,c}-4\,a\,b\,c^3\,e^5\,x\,\sqrt {b^2-4\,a\,c}-b^4\,c\,d\,e^4\,x\,\sqrt {b^2-4\,a\,c}-5\,a\,b^3\,c^2\,d\,e^4\,x-14\,a\,b^4\,c\,d^2\,e^3\,x+4\,a^2\,b\,c^3\,d\,e^4\,x-26\,a^3\,b^2\,c\,d^4\,e\,x+14\,a^3\,b\,c\,d^4\,e\,x\,\sqrt {b^2-4\,a\,c}+3\,a\,b^2\,c^2\,d\,e^4\,x\,\sqrt {b^2-4\,a\,c}+10\,a\,b^3\,c\,d^2\,e^3\,x\,\sqrt {b^2-4\,a\,c}-13\,a^2\,b\,c^2\,d^2\,e^3\,x\,\sqrt {b^2-4\,a\,c}-20\,a^2\,b^2\,c\,d^3\,e^2\,x\,\sqrt {b^2-4\,a\,c}\right )\,\left (\frac {b^3\,e}{2}+d\,\left (2\,a^2\,c-\frac {a\,b^2}{2}+\frac {a\,b\,\sqrt {b^2-4\,a\,c}}{2}\right )-\frac {b^2\,e\,\sqrt {b^2-4\,a\,c}}{2}+a\,c\,e\,\sqrt {b^2-4\,a\,c}-2\,a\,b\,c\,e\right )}{4\,a^2\,c^2\,d^2-a\,b^2\,c\,d^2-4\,a\,b\,c^2\,d\,e+4\,a\,c^3\,e^2+b^3\,c\,d\,e-b^2\,c^2\,e^2}-\frac {e^2\,\ln \left (d+e\,x\right )}{a\,d^3-b\,d^2\,e+c\,d\,e^2}+\frac {\ln \left (x\right )}{c\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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