Optimal. Leaf size=169 \[ \frac {\left (-2 a^2 c^2 e+4 a b^2 c e-3 a b c^2 d+b^4 (-e)+b^3 c d\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{c^4 \sqrt {b^2-4 a c}}-\frac {x \left (a c e+b^2 (-e)+b c d\right )}{c^3}+\frac {\left (2 a b c e-a c^2 d+b^3 (-e)+b^2 c d\right ) \log \left (a+b x+c x^2\right )}{2 c^4}+\frac {x^2 (c d-b e)}{2 c^2}+\frac {e x^3}{3 c} \]
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Rubi [A] time = 0.24, antiderivative size = 169, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {800, 634, 618, 206, 628} \begin {gather*} \frac {\left (-2 a^2 c^2 e+4 a b^2 c e-3 a b c^2 d+b^3 c d+b^4 (-e)\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{c^4 \sqrt {b^2-4 a c}}+\frac {\left (2 a b c e-a c^2 d+b^2 c d+b^3 (-e)\right ) \log \left (a+b x+c x^2\right )}{2 c^4}-\frac {x \left (a c e+b^2 (-e)+b c d\right )}{c^3}+\frac {x^2 (c d-b e)}{2 c^2}+\frac {e x^3}{3 c} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 618
Rule 628
Rule 634
Rule 800
Rubi steps
\begin {align*} \int \frac {x^3 (d+e x)}{a+b x+c x^2} \, dx &=\int \left (-\frac {b c d-b^2 e+a c e}{c^3}+\frac {(c d-b e) x}{c^2}+\frac {e x^2}{c}+\frac {a \left (b c d-b^2 e+a c e\right )+\left (b^2 c d-a c^2 d-b^3 e+2 a b c e\right ) x}{c^3 \left (a+b x+c x^2\right )}\right ) \, dx\\ &=-\frac {\left (b c d-b^2 e+a c e\right ) x}{c^3}+\frac {(c d-b e) x^2}{2 c^2}+\frac {e x^3}{3 c}+\frac {\int \frac {a \left (b c d-b^2 e+a c e\right )+\left (b^2 c d-a c^2 d-b^3 e+2 a b c e\right ) x}{a+b x+c x^2} \, dx}{c^3}\\ &=-\frac {\left (b c d-b^2 e+a c e\right ) x}{c^3}+\frac {(c d-b e) x^2}{2 c^2}+\frac {e x^3}{3 c}+\frac {\left (b^2 c d-a c^2 d-b^3 e+2 a b c e\right ) \int \frac {b+2 c x}{a+b x+c x^2} \, dx}{2 c^4}-\frac {\left (b^3 c d-3 a b c^2 d-b^4 e+4 a b^2 c e-2 a^2 c^2 e\right ) \int \frac {1}{a+b x+c x^2} \, dx}{2 c^4}\\ &=-\frac {\left (b c d-b^2 e+a c e\right ) x}{c^3}+\frac {(c d-b e) x^2}{2 c^2}+\frac {e x^3}{3 c}+\frac {\left (b^2 c d-a c^2 d-b^3 e+2 a b c e\right ) \log \left (a+b x+c x^2\right )}{2 c^4}+\frac {\left (b^3 c d-3 a b c^2 d-b^4 e+4 a b^2 c e-2 a^2 c^2 e\right ) \operatorname {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c x\right )}{c^4}\\ &=-\frac {\left (b c d-b^2 e+a c e\right ) x}{c^3}+\frac {(c d-b e) x^2}{2 c^2}+\frac {e x^3}{3 c}+\frac {\left (b^3 c d-3 a b c^2 d-b^4 e+4 a b^2 c e-2 a^2 c^2 e\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{c^4 \sqrt {b^2-4 a c}}+\frac {\left (b^2 c d-a c^2 d-b^3 e+2 a b c e\right ) \log \left (a+b x+c x^2\right )}{2 c^4}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 165, normalized size = 0.98 \begin {gather*} \frac {\frac {6 \left (2 a^2 c^2 e-4 a b^2 c e+3 a b c^2 d+b^4 e-b^3 c d\right ) \tan ^{-1}\left (\frac {b+2 c x}{\sqrt {4 a c-b^2}}\right )}{\sqrt {4 a c-b^2}}-6 c x \left (a c e+b^2 (-e)+b c d\right )-3 \left (-2 a b c e+a c^2 d+b^3 e-b^2 c d\right ) \log (a+x (b+c x))+3 c^2 x^2 (c d-b e)+2 c^3 e x^3}{6 c^4} \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^3 (d+e x)}{a+b x+c x^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.45, size = 563, normalized size = 3.33 \begin {gather*} \left [\frac {2 \, {\left (b^{2} c^{3} - 4 \, a c^{4}\right )} e x^{3} + 3 \, {\left ({\left (b^{2} c^{3} - 4 \, a c^{4}\right )} d - {\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} e\right )} x^{2} - 3 \, \sqrt {b^{2} - 4 \, a c} {\left ({\left (b^{3} c - 3 \, a b c^{2}\right )} d - {\left (b^{4} - 4 \, a b^{2} c + 2 \, a^{2} c^{2}\right )} e\right )} \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 ) - 6 \, {\left ({\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} d - {\left (b^{4} c - 5 \, a b^{2} c^{2} + 4 \, a^{2} c^{3}\right )} e\right )} x + 3 \, {\left ({\left (b^{4} c - 5 \, a b^{2} c^{2} + 4 \, a^{2} c^{3}\right )} d - {\left (b^{5} - 6 \, a b^{3} c + 8 \, a^{2} b c^{2}\right )} e\right )} \log \left (c x^{2} + b x + a\right )}{6 \, {\left (b^{2} c^{4} - 4 \, a c^{5}\right )}}, \frac {2 \, {\left (b^{2} c^{3} - 4 \, a c^{4}\right )} e x^{3} + 3 \, {\left ({\left (b^{2} c^{3} - 4 \, a c^{4}\right )} d - {\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} e\right )} x^{2} + 6 \, \sqrt {-b^{2} + 4 \, a c} {\left ({\left (b^{3} c - 3 \, a b c^{2}\right )} d - {\left (b^{4} - 4 \, a b^{2} c + 2 \, a^{2} c^{2}\right )} e\right )} \arctan \left (-\frac {\sqrt {-b^{2} + 4 \, a c} {\left (2 \, c x + b\right )}}{b^{2} - 4 \, a c}\right ) - 6 \, {\left ({\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} d - {\left (b^{4} c - 5 \, a b^{2} c^{2} + 4 \, a^{2} c^{3}\right )} e\right )} x + 3 \, {\left ({\left (b^{4} c - 5 \, a b^{2} c^{2} + 4 \, a^{2} c^{3}\right )} d - {\left (b^{5} - 6 \, a b^{3} c + 8 \, a^{2} b c^{2}\right )} e\right )} \log \left (c x^{2} + b x + a\right )}{6 \, {\left (b^{2} c^{4} - 4 \, a c^{5}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 178, normalized size = 1.05 \begin {gather*} \frac {2 \, c^{2} x^{3} e + 3 \, c^{2} d x^{2} - 3 \, b c x^{2} e - 6 \, b c d x + 6 \, b^{2} x e - 6 \, a c x e}{6 \, c^{3}} + \frac {{\left (b^{2} c d - a c^{2} d - b^{3} e + 2 \, a b c e\right )} \log \left (c x^{2} + b x + a\right )}{2 \, c^{4}} - \frac {{\left (b^{3} c d - 3 \, a b c^{2} d - b^{4} e + 4 \, a b^{2} c e - 2 \, a^{2} c^{2} e\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{\sqrt {-b^{2} + 4 \, a c} c^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 335, normalized size = 1.98 \begin {gather*} \frac {e \,x^{3}}{3 c}+\frac {2 a^{2} e \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}\, c^{2}}-\frac {4 a \,b^{2} e \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}\, c^{3}}+\frac {3 a b d \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}\, c^{2}}+\frac {b^{4} e \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}\, c^{4}}-\frac {b^{3} d \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}\, c^{3}}-\frac {b e \,x^{2}}{2 c^{2}}+\frac {d \,x^{2}}{2 c}+\frac {a b e \ln \left (c \,x^{2}+b x +a \right )}{c^{3}}-\frac {a d \ln \left (c \,x^{2}+b x +a \right )}{2 c^{2}}-\frac {a e x}{c^{2}}-\frac {b^{3} e \ln \left (c \,x^{2}+b x +a \right )}{2 c^{4}}+\frac {b^{2} d \ln \left (c \,x^{2}+b x +a \right )}{2 c^{3}}+\frac {b^{2} e x}{c^{3}}-\frac {b d 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 = 1.30, size = 221, normalized size = 1.31 \begin {gather*} x^2\,\left (\frac {d}{2\,c}-\frac {b\,e}{2\,c^2}\right )-x\,\left (\frac {b\,\left (\frac {d}{c}-\frac {b\,e}{c^2}\right )}{c}+\frac {a\,e}{c^2}\right )+\frac {\ln \left (c\,x^2+b\,x+a\right )\,\left (8\,e\,a^2\,b\,c^2-4\,d\,a^2\,c^3-6\,e\,a\,b^3\,c+5\,d\,a\,b^2\,c^2+e\,b^5-d\,b^4\,c\right )}{2\,\left (4\,a\,c^5-b^2\,c^4\right )}+\frac {e\,x^3}{3\,c}+\frac {\mathrm {atan}\left (\frac {b}{\sqrt {4\,a\,c-b^2}}+\frac {2\,c\,x}{\sqrt {4\,a\,c-b^2}}\right )\,\left (2\,e\,a^2\,c^2-4\,e\,a\,b^2\,c+3\,d\,a\,b\,c^2+e\,b^4-d\,b^3\,c\right )}{c^4\,\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 = 2.96, size = 840, normalized size = 4.97 \begin {gather*} x^{2} \left (- \frac {b e}{2 c^{2}} + \frac {d}{2 c}\right ) + x \left (- \frac {a e}{c^{2}} + \frac {b^{2} e}{c^{3}} - \frac {b d}{c^{2}}\right ) + \left (- \frac {\sqrt {- 4 a c + b^{2}} \left (2 a^{2} c^{2} e - 4 a b^{2} c e + 3 a b c^{2} d + b^{4} e - b^{3} c d\right )}{2 c^{4} \left (4 a c - b^{2}\right )} + \frac {2 a b c e - a c^{2} d - b^{3} e + b^{2} c d}{2 c^{4}}\right ) \log {\left (x + \frac {- 3 a^{2} b c e + 2 a^{2} c^{2} d + a b^{3} e - a b^{2} c d + 4 a c^{4} \left (- \frac {\sqrt {- 4 a c + b^{2}} \left (2 a^{2} c^{2} e - 4 a b^{2} c e + 3 a b c^{2} d + b^{4} e - b^{3} c d\right )}{2 c^{4} \left (4 a c - b^{2}\right )} + \frac {2 a b c e - a c^{2} d - b^{3} e + b^{2} c d}{2 c^{4}}\right ) - b^{2} c^{3} \left (- \frac {\sqrt {- 4 a c + b^{2}} \left (2 a^{2} c^{2} e - 4 a b^{2} c e + 3 a b c^{2} d + b^{4} e - b^{3} c d\right )}{2 c^{4} \left (4 a c - b^{2}\right )} + \frac {2 a b c e - a c^{2} d - b^{3} e + b^{2} c d}{2 c^{4}}\right )}{2 a^{2} c^{2} e - 4 a b^{2} c e + 3 a b c^{2} d + b^{4} e - b^{3} c d} \right )} + \left (\frac {\sqrt {- 4 a c + b^{2}} \left (2 a^{2} c^{2} e - 4 a b^{2} c e + 3 a b c^{2} d + b^{4} e - b^{3} c d\right )}{2 c^{4} \left (4 a c - b^{2}\right )} + \frac {2 a b c e - a c^{2} d - b^{3} e + b^{2} c d}{2 c^{4}}\right ) \log {\left (x + \frac {- 3 a^{2} b c e + 2 a^{2} c^{2} d + a b^{3} e - a b^{2} c d + 4 a c^{4} \left (\frac {\sqrt {- 4 a c + b^{2}} \left (2 a^{2} c^{2} e - 4 a b^{2} c e + 3 a b c^{2} d + b^{4} e - b^{3} c d\right )}{2 c^{4} \left (4 a c - b^{2}\right )} + \frac {2 a b c e - a c^{2} d - b^{3} e + b^{2} c d}{2 c^{4}}\right ) - b^{2} c^{3} \left (\frac {\sqrt {- 4 a c + b^{2}} \left (2 a^{2} c^{2} e - 4 a b^{2} c e + 3 a b c^{2} d + b^{4} e - b^{3} c d\right )}{2 c^{4} \left (4 a c - b^{2}\right )} + \frac {2 a b c e - a c^{2} d - b^{3} e + b^{2} c d}{2 c^{4}}\right )}{2 a^{2} c^{2} e - 4 a b^{2} c e + 3 a b c^{2} d + b^{4} e - b^{3} c d} \right )} + \frac {e x^{3}}{3 c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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