Optimal. Leaf size=89 \[ -\frac {b x}{c^2}+\frac {x^2}{2 c}+\frac {b \left (b^2-3 a c\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{c^3 \sqrt {b^2-4 a c}}+\frac {\left (b^2-a c\right ) \log \left (a+b x+c x^2\right )}{2 c^3} \]
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Rubi [A]
time = 0.06, antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 6, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {1368, 715, 648,
632, 212, 642} \begin {gather*} \frac {\left (b^2-a c\right ) \log \left (a+b x+c x^2\right )}{2 c^3}+\frac {b \left (b^2-3 a c\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{c^3 \sqrt {b^2-4 a c}}-\frac {b x}{c^2}+\frac {x^2}{2 c} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 632
Rule 642
Rule 648
Rule 715
Rule 1368
Rubi steps
\begin {align*} \int \frac {x}{c+\frac {a}{x^2}+\frac {b}{x}} \, dx &=\int \frac {x^3}{a+b x+c x^2} \, dx\\ &=\int \left (-\frac {b}{c^2}+\frac {x}{c}+\frac {a b+\left (b^2-a c\right ) x}{c^2 \left (a+b x+c x^2\right )}\right ) \, dx\\ &=-\frac {b x}{c^2}+\frac {x^2}{2 c}+\frac {\int \frac {a b+\left (b^2-a c\right ) x}{a+b x+c x^2} \, dx}{c^2}\\ &=-\frac {b x}{c^2}+\frac {x^2}{2 c}-\frac {\left (b \left (b^2-3 a c\right )\right ) \int \frac {1}{a+b x+c x^2} \, dx}{2 c^3}+\frac {\left (b^2-a c\right ) \int \frac {b+2 c x}{a+b x+c x^2} \, dx}{2 c^3}\\ &=-\frac {b x}{c^2}+\frac {x^2}{2 c}+\frac {\left (b^2-a c\right ) \log \left (a+b x+c x^2\right )}{2 c^3}+\frac {\left (b \left (b^2-3 a c\right )\right ) \text {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c x\right )}{c^3}\\ &=-\frac {b x}{c^2}+\frac {x^2}{2 c}+\frac {b \left (b^2-3 a c\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{c^3 \sqrt {b^2-4 a c}}+\frac {\left (b^2-a c\right ) \log \left (a+b x+c x^2\right )}{2 c^3}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 84, normalized size = 0.94 \begin {gather*} \frac {c x (-2 b+c x)-\frac {2 b \left (b^2-3 a c\right ) \tan ^{-1}\left (\frac {b+2 c x}{\sqrt {-b^2+4 a c}}\right )}{\sqrt {-b^2+4 a c}}+\left (b^2-a c\right ) \log (a+x (b+c x))}{2 c^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.04, size = 98, normalized size = 1.10
method | result | size |
default | \(-\frac {-\frac {1}{2} c \,x^{2}+b x}{c^{2}}+\frac {\frac {\left (-a c +b^{2}\right ) \ln \left (c \,x^{2}+b x +a \right )}{2 c}+\frac {2 \left (a b -\frac {\left (-a c +b^{2}\right ) b}{2 c}\right ) \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}}{c^{2}}\) | \(98\) |
risch | \(\frac {x^{2}}{2 c}-\frac {b x}{c^{2}}-\frac {2 \ln \left (12 a^{2} b \,c^{2}-7 a \,b^{3} c +b^{5}-2 \sqrt {-b^{2} \left (4 a c -b^{2}\right ) \left (3 a c -b^{2}\right )^{2}}\, c x -\sqrt {-b^{2} \left (4 a c -b^{2}\right ) \left (3 a c -b^{2}\right )^{2}}\, b \right ) a^{2}}{c \left (4 a c -b^{2}\right )}+\frac {5 \ln \left (12 a^{2} b \,c^{2}-7 a \,b^{3} c +b^{5}-2 \sqrt {-b^{2} \left (4 a c -b^{2}\right ) \left (3 a c -b^{2}\right )^{2}}\, c x -\sqrt {-b^{2} \left (4 a c -b^{2}\right ) \left (3 a c -b^{2}\right )^{2}}\, b \right ) a \,b^{2}}{2 c^{2} \left (4 a c -b^{2}\right )}-\frac {\ln \left (12 a^{2} b \,c^{2}-7 a \,b^{3} c +b^{5}-2 \sqrt {-b^{2} \left (4 a c -b^{2}\right ) \left (3 a c -b^{2}\right )^{2}}\, c x -\sqrt {-b^{2} \left (4 a c -b^{2}\right ) \left (3 a c -b^{2}\right )^{2}}\, b \right ) b^{4}}{2 c^{3} \left (4 a c -b^{2}\right )}+\frac {\ln \left (12 a^{2} b \,c^{2}-7 a \,b^{3} c +b^{5}-2 \sqrt {-b^{2} \left (4 a c -b^{2}\right ) \left (3 a c -b^{2}\right )^{2}}\, c x -\sqrt {-b^{2} \left (4 a c -b^{2}\right ) \left (3 a c -b^{2}\right )^{2}}\, b \right ) \sqrt {-b^{2} \left (4 a c -b^{2}\right ) \left (3 a c -b^{2}\right )^{2}}}{2 c^{3} \left (4 a c -b^{2}\right )}-\frac {2 \ln \left (12 a^{2} b \,c^{2}-7 a \,b^{3} c +b^{5}+2 \sqrt {-b^{2} \left (4 a c -b^{2}\right ) \left (3 a c -b^{2}\right )^{2}}\, c x +\sqrt {-b^{2} \left (4 a c -b^{2}\right ) \left (3 a c -b^{2}\right )^{2}}\, b \right ) a^{2}}{c \left (4 a c -b^{2}\right )}+\frac {5 \ln \left (12 a^{2} b \,c^{2}-7 a \,b^{3} c +b^{5}+2 \sqrt {-b^{2} \left (4 a c -b^{2}\right ) \left (3 a c -b^{2}\right )^{2}}\, c x +\sqrt {-b^{2} \left (4 a c -b^{2}\right ) \left (3 a c -b^{2}\right )^{2}}\, b \right ) a \,b^{2}}{2 c^{2} \left (4 a c -b^{2}\right )}-\frac {\ln \left (12 a^{2} b \,c^{2}-7 a \,b^{3} c +b^{5}+2 \sqrt {-b^{2} \left (4 a c -b^{2}\right ) \left (3 a c -b^{2}\right )^{2}}\, c x +\sqrt {-b^{2} \left (4 a c -b^{2}\right ) \left (3 a c -b^{2}\right )^{2}}\, b \right ) b^{4}}{2 c^{3} \left (4 a c -b^{2}\right )}-\frac {\ln \left (12 a^{2} b \,c^{2}-7 a \,b^{3} c +b^{5}+2 \sqrt {-b^{2} \left (4 a c -b^{2}\right ) \left (3 a c -b^{2}\right )^{2}}\, c x +\sqrt {-b^{2} \left (4 a c -b^{2}\right ) \left (3 a c -b^{2}\right )^{2}}\, b \right ) \sqrt {-b^{2} \left (4 a c -b^{2}\right ) \left (3 a c -b^{2}\right )^{2}}}{2 c^{3} \left (4 a c -b^{2}\right )}\) | \(915\) |
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 = 0.36, size = 297, normalized size = 3.34 \begin {gather*} \left [\frac {{\left (b^{2} c^{2} - 4 \, a c^{3}\right )} x^{2} - {\left (b^{3} - 3 \, a b c\right )} \sqrt {b^{2} - 4 \, a c} \log \left (\frac {2 \, c^{2} x^{2} + 2 \, b c x + b^{2} - 2 \, a c - \sqrt {b^{2} - 4 \, a c} {\left (2 \, c x + b\right )}}{c x^{2} + b x + a}\right ) - 2 \, {\left (b^{3} c - 4 \, a b c^{2}\right )} x + {\left (b^{4} - 5 \, a b^{2} c + 4 \, a^{2} c^{2}\right )} \log \left (c x^{2} + b x + a\right )}{2 \, {\left (b^{2} c^{3} - 4 \, a c^{4}\right )}}, \frac {{\left (b^{2} c^{2} - 4 \, a c^{3}\right )} x^{2} + 2 \, {\left (b^{3} - 3 \, a b c\right )} \sqrt {-b^{2} + 4 \, a c} \arctan \left (-\frac {\sqrt {-b^{2} + 4 \, a c} {\left (2 \, c x + b\right )}}{b^{2} - 4 \, a c}\right ) - 2 \, {\left (b^{3} c - 4 \, a b c^{2}\right )} x + {\left (b^{4} - 5 \, a b^{2} c + 4 \, a^{2} c^{2}\right )} \log \left (c x^{2} + b x + a\right )}{2 \, {\left (b^{2} c^{3} - 4 \, a c^{4}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 381 vs.
\(2 (83) = 166\).
time = 0.45, size = 381, normalized size = 4.28 \begin {gather*} - \frac {b x}{c^{2}} + \left (- \frac {b \sqrt {- 4 a c + b^{2}} \cdot \left (3 a c - b^{2}\right )}{2 c^{3} \cdot \left (4 a c - b^{2}\right )} - \frac {a c - b^{2}}{2 c^{3}}\right ) \log {\left (x + \frac {2 a^{2} c - a b^{2} + 4 a c^{3} \left (- \frac {b \sqrt {- 4 a c + b^{2}} \cdot \left (3 a c - b^{2}\right )}{2 c^{3} \cdot \left (4 a c - b^{2}\right )} - \frac {a c - b^{2}}{2 c^{3}}\right ) - b^{2} c^{2} \left (- \frac {b \sqrt {- 4 a c + b^{2}} \cdot \left (3 a c - b^{2}\right )}{2 c^{3} \cdot \left (4 a c - b^{2}\right )} - \frac {a c - b^{2}}{2 c^{3}}\right )}{3 a b c - b^{3}} \right )} + \left (\frac {b \sqrt {- 4 a c + b^{2}} \cdot \left (3 a c - b^{2}\right )}{2 c^{3} \cdot \left (4 a c - b^{2}\right )} - \frac {a c - b^{2}}{2 c^{3}}\right ) \log {\left (x + \frac {2 a^{2} c - a b^{2} + 4 a c^{3} \left (\frac {b \sqrt {- 4 a c + b^{2}} \cdot \left (3 a c - b^{2}\right )}{2 c^{3} \cdot \left (4 a c - b^{2}\right )} - \frac {a c - b^{2}}{2 c^{3}}\right ) - b^{2} c^{2} \left (\frac {b \sqrt {- 4 a c + b^{2}} \cdot \left (3 a c - b^{2}\right )}{2 c^{3} \cdot \left (4 a c - b^{2}\right )} - \frac {a c - b^{2}}{2 c^{3}}\right )}{3 a b c - b^{3}} \right )} + \frac {x^{2}}{2 c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.56, size = 86, normalized size = 0.97 \begin {gather*} \frac {c x^{2} - 2 \, b x}{2 \, c^{2}} + \frac {{\left (b^{2} - a c\right )} \log \left (c x^{2} + b x + a\right )}{2 \, c^{3}} - \frac {{\left (b^{3} - 3 \, a b c\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{\sqrt {-b^{2} + 4 \, a c} c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.13, size = 112, normalized size = 1.26 \begin {gather*} \frac {x^2}{2\,c}-\frac {\ln \left (c\,x^2+b\,x+a\right )\,\left (4\,a^2\,c^2-5\,a\,b^2\,c+b^4\right )}{2\,\left (4\,a\,c^4-b^2\,c^3\right )}-\frac {b\,x}{c^2}+\frac {b\,\mathrm {atan}\left (\frac {b+2\,c\,x}{\sqrt {4\,a\,c-b^2}}\right )\,\left (3\,a\,c-b^2\right )}{c^3\,\sqrt {4\,a\,c-b^2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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