Optimal. Leaf size=147 \[ \frac {\left (a^2 c^2-3 a b^2 c+b^4\right ) \log \left (a+b x+c x^2\right )}{2 c^5}+\frac {b \left (5 a^2 c^2-5 a b^2 c+b^4\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{c^5 \sqrt {b^2-4 a c}}-\frac {b x \left (b^2-2 a c\right )}{c^4}+\frac {x^2 \left (b^2-a c\right )}{2 c^3}-\frac {b x^3}{3 c^2}+\frac {x^4}{4 c} \]
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Rubi [A] time = 0.14, antiderivative size = 147, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {1354, 701, 634, 618, 206, 628} \[ \frac {\left (a^2 c^2-3 a b^2 c+b^4\right ) \log \left (a+b x+c x^2\right )}{2 c^5}+\frac {b \left (5 a^2 c^2-5 a b^2 c+b^4\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{c^5 \sqrt {b^2-4 a c}}+\frac {x^2 \left (b^2-a c\right )}{2 c^3}-\frac {b x \left (b^2-2 a c\right )}{c^4}-\frac {b x^3}{3 c^2}+\frac {x^4}{4 c} \]
Antiderivative was successfully verified.
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Rule 206
Rule 618
Rule 628
Rule 634
Rule 701
Rule 1354
Rubi steps
\begin {align*} \int \frac {x^3}{c+\frac {a}{x^2}+\frac {b}{x}} \, dx &=\int \frac {x^5}{a+b x+c x^2} \, dx\\ &=\int \left (-\frac {b \left (b^2-2 a c\right )}{c^4}+\frac {\left (b^2-a c\right ) x}{c^3}-\frac {b x^2}{c^2}+\frac {x^3}{c}+\frac {a b \left (b^2-2 a c\right )+\left (b^4-3 a b^2 c+a^2 c^2\right ) x}{c^4 \left (a+b x+c x^2\right )}\right ) \, dx\\ &=-\frac {b \left (b^2-2 a c\right ) x}{c^4}+\frac {\left (b^2-a c\right ) x^2}{2 c^3}-\frac {b x^3}{3 c^2}+\frac {x^4}{4 c}+\frac {\int \frac {a b \left (b^2-2 a c\right )+\left (b^4-3 a b^2 c+a^2 c^2\right ) x}{a+b x+c x^2} \, dx}{c^4}\\ &=-\frac {b \left (b^2-2 a c\right ) x}{c^4}+\frac {\left (b^2-a c\right ) x^2}{2 c^3}-\frac {b x^3}{3 c^2}+\frac {x^4}{4 c}+\frac {\left (b^4-3 a b^2 c+a^2 c^2\right ) \int \frac {b+2 c x}{a+b x+c x^2} \, dx}{2 c^5}-\frac {\left (b \left (b^4-5 a b^2 c+5 a^2 c^2\right )\right ) \int \frac {1}{a+b x+c x^2} \, dx}{2 c^5}\\ &=-\frac {b \left (b^2-2 a c\right ) x}{c^4}+\frac {\left (b^2-a c\right ) x^2}{2 c^3}-\frac {b x^3}{3 c^2}+\frac {x^4}{4 c}+\frac {\left (b^4-3 a b^2 c+a^2 c^2\right ) \log \left (a+b x+c x^2\right )}{2 c^5}+\frac {\left (b \left (b^4-5 a b^2 c+5 a^2 c^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c x\right )}{c^5}\\ &=-\frac {b \left (b^2-2 a c\right ) x}{c^4}+\frac {\left (b^2-a c\right ) x^2}{2 c^3}-\frac {b x^3}{3 c^2}+\frac {x^4}{4 c}+\frac {b \left (b^4-5 a b^2 c+5 a^2 c^2\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{c^5 \sqrt {b^2-4 a c}}+\frac {\left (b^4-3 a b^2 c+a^2 c^2\right ) \log \left (a+b x+c x^2\right )}{2 c^5}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 140, normalized size = 0.95 \[ \frac {6 \left (a^2 c^2-3 a b^2 c+b^4\right ) \log (a+x (b+c x))-\frac {12 b \left (5 a^2 c^2-5 a b^2 c+b^4\right ) \tan ^{-1}\left (\frac {b+2 c x}{\sqrt {4 a c-b^2}}\right )}{\sqrt {4 a c-b^2}}+c x \left (-4 b c \left (c x^2-6 a\right )+3 c^2 x \left (c x^2-2 a\right )-12 b^3+6 b^2 c x\right )}{12 c^5} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.90, size = 466, normalized size = 3.17 \[ \left [\frac {3 \, {\left (b^{2} c^{4} - 4 \, a c^{5}\right )} x^{4} - 4 \, {\left (b^{3} c^{3} - 4 \, a b c^{4}\right )} x^{3} + 6 \, {\left (b^{4} c^{2} - 5 \, a b^{2} c^{3} + 4 \, a^{2} c^{4}\right )} x^{2} + 6 \, {\left (b^{5} - 5 \, a b^{3} c + 5 \, a^{2} b c^{2}\right )} \sqrt {b^{2} - 4 \, a c} \log \left (\frac {2 \, c^{2} x^{2} + 2 \, b c x + b^{2} - 2 \, a c + \sqrt {b^{2} - 4 \, a c} {\left (2 \, c x + b\right )}}{c x^{2} + b x + a}\right ) - 12 \, {\left (b^{5} c - 6 \, a b^{3} c^{2} + 8 \, a^{2} b c^{3}\right )} x + 6 \, {\left (b^{6} - 7 \, a b^{4} c + 13 \, a^{2} b^{2} c^{2} - 4 \, a^{3} c^{3}\right )} \log \left (c x^{2} + b x + a\right )}{12 \, {\left (b^{2} c^{5} - 4 \, a c^{6}\right )}}, \frac {3 \, {\left (b^{2} c^{4} - 4 \, a c^{5}\right )} x^{4} - 4 \, {\left (b^{3} c^{3} - 4 \, a b c^{4}\right )} x^{3} + 6 \, {\left (b^{4} c^{2} - 5 \, a b^{2} c^{3} + 4 \, a^{2} c^{4}\right )} x^{2} + 12 \, {\left (b^{5} - 5 \, a b^{3} c + 5 \, a^{2} b c^{2}\right )} \sqrt {-b^{2} + 4 \, a c} \arctan \left (-\frac {\sqrt {-b^{2} + 4 \, a c} {\left (2 \, c x + b\right )}}{b^{2} - 4 \, a c}\right ) - 12 \, {\left (b^{5} c - 6 \, a b^{3} c^{2} + 8 \, a^{2} b c^{3}\right )} x + 6 \, {\left (b^{6} - 7 \, a b^{4} c + 13 \, a^{2} b^{2} c^{2} - 4 \, a^{3} c^{3}\right )} \log \left (c x^{2} + b x + a\right )}{12 \, {\left (b^{2} c^{5} - 4 \, a c^{6}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.29, size = 145, normalized size = 0.99 \[ \frac {3 \, c^{3} x^{4} - 4 \, b c^{2} x^{3} + 6 \, b^{2} c x^{2} - 6 \, a c^{2} x^{2} - 12 \, b^{3} x + 24 \, a b c x}{12 \, c^{4}} + \frac {{\left (b^{4} - 3 \, a b^{2} c + a^{2} c^{2}\right )} \log \left (c x^{2} + b x + a\right )}{2 \, c^{5}} - \frac {{\left (b^{5} - 5 \, a b^{3} c + 5 \, a^{2} b c^{2}\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{\sqrt {-b^{2} + 4 \, a c} c^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 236, normalized size = 1.61 \[ \frac {x^{4}}{4 c}-\frac {b \,x^{3}}{3 c^{2}}-\frac {5 a^{2} b \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}\, c^{3}}+\frac {5 a \,b^{3} \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}\, c^{4}}-\frac {a \,x^{2}}{2 c^{2}}-\frac {b^{5} \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}\, c^{5}}+\frac {b^{2} x^{2}}{2 c^{3}}+\frac {a^{2} \ln \left (c \,x^{2}+b x +a \right )}{2 c^{3}}-\frac {3 a \,b^{2} \ln \left (c \,x^{2}+b x +a \right )}{2 c^{4}}+\frac {2 a b x}{c^{3}}+\frac {b^{4} \ln \left (c \,x^{2}+b x +a \right )}{2 c^{5}}-\frac {b^{3} x}{c^{4}} \]
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 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.16, size = 183, normalized size = 1.24 \[ x\,\left (\frac {b\,\left (\frac {a}{c^2}-\frac {b^2}{c^3}\right )}{c}+\frac {a\,b}{c^3}\right )+\frac {x^4}{4\,c}-x^2\,\left (\frac {a}{2\,c^2}-\frac {b^2}{2\,c^3}\right )-\frac {\ln \left (c\,x^2+b\,x+a\right )\,\left (-4\,a^3\,c^3+13\,a^2\,b^2\,c^2-7\,a\,b^4\,c+b^6\right )}{2\,\left (4\,a\,c^6-b^2\,c^5\right )}-\frac {b\,x^3}{3\,c^2}-\frac {b\,\mathrm {atan}\left (\frac {b+2\,c\,x}{\sqrt {4\,a\,c-b^2}}\right )\,\left (5\,a^2\,c^2-5\,a\,b^2\,c+b^4\right )}{c^5\,\sqrt {4\,a\,c-b^2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 1.25, size = 605, normalized size = 4.12 \[ - \frac {b x^{3}}{3 c^{2}} + x^{2} \left (- \frac {a}{2 c^{2}} + \frac {b^{2}}{2 c^{3}}\right ) + x \left (\frac {2 a b}{c^{3}} - \frac {b^{3}}{c^{4}}\right ) + \left (- \frac {b \sqrt {- 4 a c + b^{2}} \left (5 a^{2} c^{2} - 5 a b^{2} c + b^{4}\right )}{2 c^{5} \left (4 a c - b^{2}\right )} + \frac {a^{2} c^{2} - 3 a b^{2} c + b^{4}}{2 c^{5}}\right ) \log {\left (x + \frac {2 a^{3} c^{2} - 4 a^{2} b^{2} c + a b^{4} - 4 a c^{5} \left (- \frac {b \sqrt {- 4 a c + b^{2}} \left (5 a^{2} c^{2} - 5 a b^{2} c + b^{4}\right )}{2 c^{5} \left (4 a c - b^{2}\right )} + \frac {a^{2} c^{2} - 3 a b^{2} c + b^{4}}{2 c^{5}}\right ) + b^{2} c^{4} \left (- \frac {b \sqrt {- 4 a c + b^{2}} \left (5 a^{2} c^{2} - 5 a b^{2} c + b^{4}\right )}{2 c^{5} \left (4 a c - b^{2}\right )} + \frac {a^{2} c^{2} - 3 a b^{2} c + b^{4}}{2 c^{5}}\right )}{5 a^{2} b c^{2} - 5 a b^{3} c + b^{5}} \right )} + \left (\frac {b \sqrt {- 4 a c + b^{2}} \left (5 a^{2} c^{2} - 5 a b^{2} c + b^{4}\right )}{2 c^{5} \left (4 a c - b^{2}\right )} + \frac {a^{2} c^{2} - 3 a b^{2} c + b^{4}}{2 c^{5}}\right ) \log {\left (x + \frac {2 a^{3} c^{2} - 4 a^{2} b^{2} c + a b^{4} - 4 a c^{5} \left (\frac {b \sqrt {- 4 a c + b^{2}} \left (5 a^{2} c^{2} - 5 a b^{2} c + b^{4}\right )}{2 c^{5} \left (4 a c - b^{2}\right )} + \frac {a^{2} c^{2} - 3 a b^{2} c + b^{4}}{2 c^{5}}\right ) + b^{2} c^{4} \left (\frac {b \sqrt {- 4 a c + b^{2}} \left (5 a^{2} c^{2} - 5 a b^{2} c + b^{4}\right )}{2 c^{5} \left (4 a c - b^{2}\right )} + \frac {a^{2} c^{2} - 3 a b^{2} c + b^{4}}{2 c^{5}}\right )}{5 a^{2} b c^{2} - 5 a b^{3} c + b^{5}} \right )} + \frac {x^{4}}{4 c} \]
Verification of antiderivative is not currently implemented for this CAS.
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