Optimal. Leaf size=229 \[ -\frac {\left (-a^2 c^2 e+3 a b^2 c e-2 a b c^2 d+b^4 (-e)+b^3 c d\right ) \log \left (a+b x+c x^2\right )}{2 c^5}-\frac {\left (-5 a^2 b c^2 e+2 a^2 c^3 d+5 a b^3 c e-4 a b^2 c^2 d+b^5 (-e)+b^4 c d\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{c^5 \sqrt {b^2-4 a c}}-\frac {x^2 \left (a c e+b^2 (-e)+b c d\right )}{2 c^3}+\frac {x \left (2 a b c e-a c^2 d+b^3 (-e)+b^2 c d\right )}{c^4}+\frac {x^3 (c d-b e)}{3 c^2}+\frac {e x^4}{4 c} \]
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Rubi [A] time = 0.42, antiderivative size = 229, 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 (-a^2 c^2 e+3 a b^2 c e-2 a b c^2 d+b^3 c d+b^4 (-e)\right ) \log \left (a+b x+c x^2\right )}{2 c^5}-\frac {\left (-5 a^2 b c^2 e+2 a^2 c^3 d-4 a b^2 c^2 d+5 a b^3 c e+b^4 c d+b^5 (-e)\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 (a c e+b^2 (-e)+b c d\right )}{2 c^3}+\frac {x \left (2 a b c e-a c^2 d+b^2 c d+b^3 (-e)\right )}{c^4}+\frac {x^3 (c d-b e)}{3 c^2}+\frac {e x^4}{4 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^4 (d+e x)}{a+b x+c x^2} \, dx &=\int \left (\frac {b^2 c d-a c^2 d-b^3 e+2 a b c e}{c^4}-\frac {\left (b c d-b^2 e+a c e\right ) x}{c^3}+\frac {(c d-b e) x^2}{c^2}+\frac {e x^3}{c}-\frac {a \left (b^2 c d-a c^2 d-b^3 e+2 a b c e\right )+\left (b^3 c d-2 a b c^2 d-b^4 e+3 a b^2 c e-a^2 c^2 e\right ) x}{c^4 \left (a+b x+c x^2\right )}\right ) \, dx\\ &=\frac {\left (b^2 c d-a c^2 d-b^3 e+2 a b c e\right ) x}{c^4}-\frac {\left (b c d-b^2 e+a c e\right ) x^2}{2 c^3}+\frac {(c d-b e) x^3}{3 c^2}+\frac {e x^4}{4 c}-\frac {\int \frac {a \left (b^2 c d-a c^2 d-b^3 e+2 a b c e\right )+\left (b^3 c d-2 a b c^2 d-b^4 e+3 a b^2 c e-a^2 c^2 e\right ) x}{a+b x+c x^2} \, dx}{c^4}\\ &=\frac {\left (b^2 c d-a c^2 d-b^3 e+2 a b c e\right ) x}{c^4}-\frac {\left (b c d-b^2 e+a c e\right ) x^2}{2 c^3}+\frac {(c d-b e) x^3}{3 c^2}+\frac {e x^4}{4 c}-\frac {\left (b^3 c d-2 a b c^2 d-b^4 e+3 a b^2 c e-a^2 c^2 e\right ) \int \frac {b+2 c x}{a+b x+c x^2} \, dx}{2 c^5}+\frac {\left (b^4 c d-4 a b^2 c^2 d+2 a^2 c^3 d-b^5 e+5 a b^3 c e-5 a^2 b c^2 e\right ) \int \frac {1}{a+b x+c x^2} \, dx}{2 c^5}\\ &=\frac {\left (b^2 c d-a c^2 d-b^3 e+2 a b c e\right ) x}{c^4}-\frac {\left (b c d-b^2 e+a c e\right ) x^2}{2 c^3}+\frac {(c d-b e) x^3}{3 c^2}+\frac {e x^4}{4 c}-\frac {\left (b^3 c d-2 a b c^2 d-b^4 e+3 a b^2 c e-a^2 c^2 e\right ) \log \left (a+b x+c x^2\right )}{2 c^5}-\frac {\left (b^4 c d-4 a b^2 c^2 d+2 a^2 c^3 d-b^5 e+5 a b^3 c e-5 a^2 b 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^5}\\ &=\frac {\left (b^2 c d-a c^2 d-b^3 e+2 a b c e\right ) x}{c^4}-\frac {\left (b c d-b^2 e+a c e\right ) x^2}{2 c^3}+\frac {(c d-b e) x^3}{3 c^2}+\frac {e x^4}{4 c}-\frac {\left (b^4 c d-4 a b^2 c^2 d+2 a^2 c^3 d-b^5 e+5 a b^3 c e-5 a^2 b c^2 e\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^3 c d-2 a b c^2 d-b^4 e+3 a b^2 c e-a^2 c^2 e\right ) \log \left (a+b x+c x^2\right )}{2 c^5}\\ \end {align*}
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Mathematica [A] time = 0.14, size = 222, normalized size = 0.97 \begin {gather*} \frac {6 \left (a^2 c^2 e-3 a b^2 c e+2 a b c^2 d+b^4 e-b^3 c d\right ) \log (a+x (b+c x))+\frac {12 \left (-5 a^2 b c^2 e+2 a^2 c^3 d+5 a b^3 c e-4 a b^2 c^2 d+b^5 (-e)+b^4 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^2 x^2 \left (a c e+b^2 (-e)+b c d\right )-12 c x \left (-2 a b c e+a c^2 d+b^3 e-b^2 c d\right )+4 c^3 x^3 (c d-b e)+3 c^4 e x^4}{12 c^5} \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^4 (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 = 730, normalized size = 3.19 \begin {gather*} \left [\frac {3 \, {\left (b^{2} c^{4} - 4 \, a c^{5}\right )} e x^{4} + 4 \, {\left ({\left (b^{2} c^{4} - 4 \, a c^{5}\right )} d - {\left (b^{3} c^{3} - 4 \, a b c^{4}\right )} e\right )} x^{3} - 6 \, {\left ({\left (b^{3} c^{3} - 4 \, a b c^{4}\right )} d - {\left (b^{4} c^{2} - 5 \, a b^{2} c^{3} + 4 \, a^{2} c^{4}\right )} e\right )} x^{2} - 6 \, \sqrt {b^{2} - 4 \, a c} {\left ({\left (b^{4} c - 4 \, a b^{2} c^{2} + 2 \, a^{2} c^{3}\right )} d - {\left (b^{5} - 5 \, a b^{3} c + 5 \, a^{2} b 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 ) + 12 \, {\left ({\left (b^{4} c^{2} - 5 \, a b^{2} c^{3} + 4 \, a^{2} c^{4}\right )} d - {\left (b^{5} c - 6 \, a b^{3} c^{2} + 8 \, a^{2} b c^{3}\right )} e\right )} x - 6 \, {\left ({\left (b^{5} c - 6 \, a b^{3} c^{2} + 8 \, a^{2} b c^{3}\right )} d - {\left (b^{6} - 7 \, a b^{4} c + 13 \, a^{2} b^{2} c^{2} - 4 \, a^{3} c^{3}\right )} e\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 )} e x^{4} + 4 \, {\left ({\left (b^{2} c^{4} - 4 \, a c^{5}\right )} d - {\left (b^{3} c^{3} - 4 \, a b c^{4}\right )} e\right )} x^{3} - 6 \, {\left ({\left (b^{3} c^{3} - 4 \, a b c^{4}\right )} d - {\left (b^{4} c^{2} - 5 \, a b^{2} c^{3} + 4 \, a^{2} c^{4}\right )} e\right )} x^{2} - 12 \, \sqrt {-b^{2} + 4 \, a c} {\left ({\left (b^{4} c - 4 \, a b^{2} c^{2} + 2 \, a^{2} c^{3}\right )} d - {\left (b^{5} - 5 \, a b^{3} c + 5 \, a^{2} b 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 ) + 12 \, {\left ({\left (b^{4} c^{2} - 5 \, a b^{2} c^{3} + 4 \, a^{2} c^{4}\right )} d - {\left (b^{5} c - 6 \, a b^{3} c^{2} + 8 \, a^{2} b c^{3}\right )} e\right )} x - 6 \, {\left ({\left (b^{5} c - 6 \, a b^{3} c^{2} + 8 \, a^{2} b c^{3}\right )} d - {\left (b^{6} - 7 \, a b^{4} c + 13 \, a^{2} b^{2} c^{2} - 4 \, a^{3} c^{3}\right )} e\right )} \log \left (c x^{2} + b x + a\right )}{12 \, {\left (b^{2} c^{5} - 4 \, a c^{6}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 247, normalized size = 1.08 \begin {gather*} \frac {3 \, c^{3} x^{4} e + 4 \, c^{3} d x^{3} - 4 \, b c^{2} x^{3} e - 6 \, b c^{2} d x^{2} + 6 \, b^{2} c x^{2} e - 6 \, a c^{2} x^{2} e + 12 \, b^{2} c d x - 12 \, a c^{2} d x - 12 \, b^{3} x e + 24 \, a b c x e}{12 \, c^{4}} - \frac {{\left (b^{3} c d - 2 \, a b c^{2} d - b^{4} e + 3 \, a b^{2} c e - a^{2} c^{2} e\right )} \log \left (c x^{2} + b x + a\right )}{2 \, c^{5}} + \frac {{\left (b^{4} c d - 4 \, a b^{2} c^{2} d + 2 \, a^{2} c^{3} d - b^{5} e + 5 \, a b^{3} c e - 5 \, a^{2} b 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^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 445, normalized size = 1.94 \begin {gather*} \frac {e \,x^{4}}{4 c}-\frac {b e \,x^{3}}{3 c^{2}}+\frac {d \,x^{3}}{3 c}-\frac {5 a^{2} b e \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}\, c^{3}}+\frac {2 a^{2} d \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}\, c^{2}}+\frac {5 a \,b^{3} e \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}\, c^{4}}-\frac {4 a \,b^{2} d \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}\, c^{3}}-\frac {a e \,x^{2}}{2 c^{2}}-\frac {b^{5} e \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}\, c^{5}}+\frac {b^{4} d \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}\, c^{4}}+\frac {b^{2} e \,x^{2}}{2 c^{3}}-\frac {b d \,x^{2}}{2 c^{2}}+\frac {a^{2} e \ln \left (c \,x^{2}+b x +a \right )}{2 c^{3}}-\frac {3 a \,b^{2} e \ln \left (c \,x^{2}+b x +a \right )}{2 c^{4}}+\frac {a b d \ln \left (c \,x^{2}+b x +a \right )}{c^{3}}+\frac {2 a b e x}{c^{3}}-\frac {a d x}{c^{2}}+\frac {b^{4} e \ln \left (c \,x^{2}+b x +a \right )}{2 c^{5}}-\frac {b^{3} d \ln \left (c \,x^{2}+b x +a \right )}{2 c^{4}}-\frac {b^{3} e x}{c^{4}}+\frac {b^{2} d x}{c^{3}} \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 = 0.23, size = 302, normalized size = 1.32 \begin {gather*} x^3\,\left (\frac {d}{3\,c}-\frac {b\,e}{3\,c^2}\right )+x\,\left (\frac {b\,\left (\frac {b\,\left (\frac {d}{c}-\frac {b\,e}{c^2}\right )}{c}+\frac {a\,e}{c^2}\right )}{c}-\frac {a\,\left (\frac {d}{c}-\frac {b\,e}{c^2}\right )}{c}\right )-x^2\,\left (\frac {b\,\left (\frac {d}{c}-\frac {b\,e}{c^2}\right )}{2\,c}+\frac {a\,e}{2\,c^2}\right )+\frac {\ln \left (c\,x^2+b\,x+a\right )\,\left (4\,e\,a^3\,c^3-13\,e\,a^2\,b^2\,c^2+8\,d\,a^2\,b\,c^3+7\,e\,a\,b^4\,c-6\,d\,a\,b^3\,c^2-e\,b^6+d\,b^5\,c\right )}{2\,\left (4\,a\,c^6-b^2\,c^5\right )}+\frac {e\,x^4}{4\,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 (5\,e\,a^2\,b\,c^2-2\,d\,a^2\,c^3-5\,e\,a\,b^3\,c+4\,d\,a\,b^2\,c^2+e\,b^5-d\,b^4\,c\right )}{c^5\,\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 = 3.92, size = 1100, normalized size = 4.80 \begin {gather*} x^{3} \left (- \frac {b e}{3 c^{2}} + \frac {d}{3 c}\right ) + x^{2} \left (- \frac {a e}{2 c^{2}} + \frac {b^{2} e}{2 c^{3}} - \frac {b d}{2 c^{2}}\right ) + x \left (\frac {2 a b e}{c^{3}} - \frac {a d}{c^{2}} - \frac {b^{3} e}{c^{4}} + \frac {b^{2} d}{c^{3}}\right ) + \left (- \frac {\sqrt {- 4 a c + b^{2}} \left (5 a^{2} b c^{2} e - 2 a^{2} c^{3} d - 5 a b^{3} c e + 4 a b^{2} c^{2} d + b^{5} e - b^{4} c d\right )}{2 c^{5} \left (4 a c - b^{2}\right )} + \frac {a^{2} c^{2} e - 3 a b^{2} c e + 2 a b c^{2} d + b^{4} e - b^{3} c d}{2 c^{5}}\right ) \log {\left (x + \frac {2 a^{3} c^{2} e - 4 a^{2} b^{2} c e + 3 a^{2} b c^{2} d + a b^{4} e - a b^{3} c d - 4 a c^{5} \left (- \frac {\sqrt {- 4 a c + b^{2}} \left (5 a^{2} b c^{2} e - 2 a^{2} c^{3} d - 5 a b^{3} c e + 4 a b^{2} c^{2} d + b^{5} e - b^{4} c d\right )}{2 c^{5} \left (4 a c - b^{2}\right )} + \frac {a^{2} c^{2} e - 3 a b^{2} c e + 2 a b c^{2} d + b^{4} e - b^{3} c d}{2 c^{5}}\right ) + b^{2} c^{4} \left (- \frac {\sqrt {- 4 a c + b^{2}} \left (5 a^{2} b c^{2} e - 2 a^{2} c^{3} d - 5 a b^{3} c e + 4 a b^{2} c^{2} d + b^{5} e - b^{4} c d\right )}{2 c^{5} \left (4 a c - b^{2}\right )} + \frac {a^{2} c^{2} e - 3 a b^{2} c e + 2 a b c^{2} d + b^{4} e - b^{3} c d}{2 c^{5}}\right )}{5 a^{2} b c^{2} e - 2 a^{2} c^{3} d - 5 a b^{3} c e + 4 a b^{2} c^{2} d + b^{5} e - b^{4} c d} \right )} + \left (\frac {\sqrt {- 4 a c + b^{2}} \left (5 a^{2} b c^{2} e - 2 a^{2} c^{3} d - 5 a b^{3} c e + 4 a b^{2} c^{2} d + b^{5} e - b^{4} c d\right )}{2 c^{5} \left (4 a c - b^{2}\right )} + \frac {a^{2} c^{2} e - 3 a b^{2} c e + 2 a b c^{2} d + b^{4} e - b^{3} c d}{2 c^{5}}\right ) \log {\left (x + \frac {2 a^{3} c^{2} e - 4 a^{2} b^{2} c e + 3 a^{2} b c^{2} d + a b^{4} e - a b^{3} c d - 4 a c^{5} \left (\frac {\sqrt {- 4 a c + b^{2}} \left (5 a^{2} b c^{2} e - 2 a^{2} c^{3} d - 5 a b^{3} c e + 4 a b^{2} c^{2} d + b^{5} e - b^{4} c d\right )}{2 c^{5} \left (4 a c - b^{2}\right )} + \frac {a^{2} c^{2} e - 3 a b^{2} c e + 2 a b c^{2} d + b^{4} e - b^{3} c d}{2 c^{5}}\right ) + b^{2} c^{4} \left (\frac {\sqrt {- 4 a c + b^{2}} \left (5 a^{2} b c^{2} e - 2 a^{2} c^{3} d - 5 a b^{3} c e + 4 a b^{2} c^{2} d + b^{5} e - b^{4} c d\right )}{2 c^{5} \left (4 a c - b^{2}\right )} + \frac {a^{2} c^{2} e - 3 a b^{2} c e + 2 a b c^{2} d + b^{4} e - b^{3} c d}{2 c^{5}}\right )}{5 a^{2} b c^{2} e - 2 a^{2} c^{3} d - 5 a b^{3} c e + 4 a b^{2} c^{2} d + b^{5} e - b^{4} c d} \right )} + \frac {e x^{4}}{4 c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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