Optimal. Leaf size=89 \[ \frac {\left (b^2-a c\right ) \log \left (a+b x+c x^2\right )}{2 c^3}+\frac {b \left (b^2-3 a c\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{c^3 \sqrt {b^2-4 a c}}-\frac {b x}{c^2}+\frac {x^2}{2 c} \]
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Rubi [A] time = 0.09, antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {1585, 701, 634, 618, 206, 628} \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 206
Rule 618
Rule 628
Rule 634
Rule 701
Rule 1585
Rubi steps
\begin {align*} \int \frac {x^5}{a x^2+b x^3+c x^4} \, 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 ) \operatorname {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 = 0.11, size = 84, normalized size = 0.94 \begin {gather*} \frac {\left (b^2-a c\right ) \log (a+x (b+c x))-\frac {2 b \left (b^2-3 a c\right ) \tan ^{-1}\left (\frac {b+2 c x}{\sqrt {4 a c-b^2}}\right )}{\sqrt {4 a c-b^2}}+c x (c x-2 b)}{2 c^3} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^5}{a x^2+b x^3+c x^4} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.86, 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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giac [A] time = 0.45, 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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maple [A] time = 0.00, size = 132, normalized size = 1.48 \begin {gather*} \frac {3 a b \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}\, c^{2}}-\frac {b^{3} \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}\, c^{3}}+\frac {x^{2}}{2 c}-\frac {a \ln \left (c \,x^{2}+b x +a \right )}{2 c^{2}}+\frac {b^{2} \ln \left (c \,x^{2}+b x +a \right )}{2 c^{3}}-\frac {b x}{c^{2}} \end {gather*}
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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mupad [B] time = 0.14, 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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sympy [B] time = 0.84, size = 381, normalized size = 4.28 \begin {gather*} - \frac {b x}{c^{2}} + \left (- \frac {b \sqrt {- 4 a c + b^{2}} \left (3 a c - b^{2}\right )}{2 c^{3} \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}} \left (3 a c - b^{2}\right )}{2 c^{3} \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}} \left (3 a c - b^{2}\right )}{2 c^{3} \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}} \left (3 a c - b^{2}\right )}{2 c^{3} \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}} \left (3 a c - b^{2}\right )}{2 c^{3} \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}} \left (3 a c - b^{2}\right )}{2 c^{3} \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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