Optimal. Leaf size=260 \[ \frac {3 c (d+e x)^4 \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{4 e^7}-\frac {(d+e x)^3 (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{3 e^7}+\frac {3 (d+e x)^2 \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{2 e^7}+\frac {\log (d+e x) \left (a e^2-b d e+c d^2\right )^3}{e^7}-\frac {3 x (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{e^6}-\frac {3 c^2 (d+e x)^5 (2 c d-b e)}{5 e^7}+\frac {c^3 (d+e x)^6}{6 e^7} \]
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Rubi [A] time = 0.32, antiderivative size = 260, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {698} \begin {gather*} \frac {3 c (d+e x)^4 \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{4 e^7}-\frac {(d+e x)^3 (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{3 e^7}+\frac {3 (d+e x)^2 \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{2 e^7}-\frac {3 x (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{e^6}+\frac {\log (d+e x) \left (a e^2-b d e+c d^2\right )^3}{e^7}-\frac {3 c^2 (d+e x)^5 (2 c d-b e)}{5 e^7}+\frac {c^3 (d+e x)^6}{6 e^7} \end {gather*}
Antiderivative was successfully verified.
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Rule 698
Rubi steps
\begin {align*} \int \frac {\left (a+b x+c x^2\right )^3}{d+e x} \, dx &=\int \left (\frac {3 (-2 c d+b e) \left (c d^2-b d e+a e^2\right )^2}{e^6}+\frac {\left (c d^2-b d e+a e^2\right )^3}{e^6 (d+e x)}+\frac {3 \left (c d^2-b d e+a e^2\right ) \left (5 c^2 d^2-5 b c d e+b^2 e^2+a c e^2\right ) (d+e x)}{e^6}+\frac {(2 c d-b e) \left (-10 c^2 d^2-b^2 e^2+2 c e (5 b d-3 a e)\right ) (d+e x)^2}{e^6}+\frac {3 c \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) (d+e x)^3}{e^6}-\frac {3 c^2 (2 c d-b e) (d+e x)^4}{e^6}+\frac {c^3 (d+e x)^5}{e^6}\right ) \, dx\\ &=-\frac {3 (2 c d-b e) \left (c d^2-b d e+a e^2\right )^2 x}{e^6}+\frac {3 \left (c d^2-b d e+a e^2\right ) \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) (d+e x)^2}{2 e^7}-\frac {(2 c d-b e) \left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right ) (d+e x)^3}{3 e^7}+\frac {3 c \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) (d+e x)^4}{4 e^7}-\frac {3 c^2 (2 c d-b e) (d+e x)^5}{5 e^7}+\frac {c^3 (d+e x)^6}{6 e^7}+\frac {\left (c d^2-b d e+a e^2\right )^3 \log (d+e x)}{e^7}\\ \end {align*}
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Mathematica [A] time = 0.16, size = 308, normalized size = 1.18 \begin {gather*} \frac {e x \left (15 c e^2 \left (6 a^2 e^2 (e x-2 d)+4 a b e \left (6 d^2-3 d e x+2 e^2 x^2\right )+b^2 \left (-12 d^3+6 d^2 e x-4 d e^2 x^2+3 e^3 x^3\right )\right )+10 b e^3 \left (18 a^2 e^2+9 a b e (e x-2 d)+b^2 \left (6 d^2-3 d e x+2 e^2 x^2\right )\right )+3 c^2 e \left (5 a e \left (-12 d^3+6 d^2 e x-4 d e^2 x^2+3 e^3 x^3\right )+b \left (60 d^4-30 d^3 e x+20 d^2 e^2 x^2-15 d e^3 x^3+12 e^4 x^4\right )\right )+c^3 \left (-60 d^5+30 d^4 e x-20 d^3 e^2 x^2+15 d^2 e^3 x^3-12 d e^4 x^4+10 e^5 x^5\right )\right )+60 \log (d+e x) \left (e (a e-b d)+c d^2\right )^3}{60 e^7} \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 {\left (a+b x+c x^2\right )^3}{d+e x} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.40, size = 403, normalized size = 1.55 \begin {gather*} \frac {10 \, c^{3} e^{6} x^{6} - 12 \, {\left (c^{3} d e^{5} - 3 \, b c^{2} e^{6}\right )} x^{5} + 15 \, {\left (c^{3} d^{2} e^{4} - 3 \, b c^{2} d e^{5} + 3 \, {\left (b^{2} c + a c^{2}\right )} e^{6}\right )} x^{4} - 20 \, {\left (c^{3} d^{3} e^{3} - 3 \, b c^{2} d^{2} e^{4} + 3 \, {\left (b^{2} c + a c^{2}\right )} d e^{5} - {\left (b^{3} + 6 \, a b c\right )} e^{6}\right )} x^{3} + 30 \, {\left (c^{3} d^{4} e^{2} - 3 \, b c^{2} d^{3} e^{3} + 3 \, {\left (b^{2} c + a c^{2}\right )} d^{2} e^{4} - {\left (b^{3} + 6 \, a b c\right )} d e^{5} + 3 \, {\left (a b^{2} + a^{2} c\right )} e^{6}\right )} x^{2} - 60 \, {\left (c^{3} d^{5} e - 3 \, b c^{2} d^{4} e^{2} - 3 \, a^{2} b e^{6} + 3 \, {\left (b^{2} c + a c^{2}\right )} d^{3} e^{3} - {\left (b^{3} + 6 \, a b c\right )} d^{2} e^{4} + 3 \, {\left (a b^{2} + a^{2} c\right )} d e^{5}\right )} x + 60 \, {\left (c^{3} d^{6} - 3 \, b c^{2} d^{5} e - 3 \, a^{2} b d e^{5} + a^{3} e^{6} + 3 \, {\left (b^{2} c + a c^{2}\right )} d^{4} e^{2} - {\left (b^{3} + 6 \, a b c\right )} d^{3} e^{3} + 3 \, {\left (a b^{2} + a^{2} c\right )} d^{2} e^{4}\right )} \log \left (e x + d\right )}{60 \, e^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 460, normalized size = 1.77 \begin {gather*} {\left (c^{3} d^{6} - 3 \, b c^{2} d^{5} e + 3 \, b^{2} c d^{4} e^{2} + 3 \, a c^{2} d^{4} e^{2} - b^{3} d^{3} e^{3} - 6 \, a b c d^{3} e^{3} + 3 \, a b^{2} d^{2} e^{4} + 3 \, a^{2} c d^{2} e^{4} - 3 \, a^{2} b d e^{5} + a^{3} e^{6}\right )} e^{\left (-7\right )} \log \left ({\left | x e + d \right |}\right ) + \frac {1}{60} \, {\left (10 \, c^{3} x^{6} e^{5} - 12 \, c^{3} d x^{5} e^{4} + 15 \, c^{3} d^{2} x^{4} e^{3} - 20 \, c^{3} d^{3} x^{3} e^{2} + 30 \, c^{3} d^{4} x^{2} e - 60 \, c^{3} d^{5} x + 36 \, b c^{2} x^{5} e^{5} - 45 \, b c^{2} d x^{4} e^{4} + 60 \, b c^{2} d^{2} x^{3} e^{3} - 90 \, b c^{2} d^{3} x^{2} e^{2} + 180 \, b c^{2} d^{4} x e + 45 \, b^{2} c x^{4} e^{5} + 45 \, a c^{2} x^{4} e^{5} - 60 \, b^{2} c d x^{3} e^{4} - 60 \, a c^{2} d x^{3} e^{4} + 90 \, b^{2} c d^{2} x^{2} e^{3} + 90 \, a c^{2} d^{2} x^{2} e^{3} - 180 \, b^{2} c d^{3} x e^{2} - 180 \, a c^{2} d^{3} x e^{2} + 20 \, b^{3} x^{3} e^{5} + 120 \, a b c x^{3} e^{5} - 30 \, b^{3} d x^{2} e^{4} - 180 \, a b c d x^{2} e^{4} + 60 \, b^{3} d^{2} x e^{3} + 360 \, a b c d^{2} x e^{3} + 90 \, a b^{2} x^{2} e^{5} + 90 \, a^{2} c x^{2} e^{5} - 180 \, a b^{2} d x e^{4} - 180 \, a^{2} c d x e^{4} + 180 \, a^{2} b x e^{5}\right )} e^{\left (-6\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.04, size = 546, normalized size = 2.10 \begin {gather*} \frac {c^{3} x^{6}}{6 e}+\frac {3 b \,c^{2} x^{5}}{5 e}-\frac {c^{3} d \,x^{5}}{5 e^{2}}+\frac {3 a \,c^{2} x^{4}}{4 e}+\frac {3 b^{2} c \,x^{4}}{4 e}-\frac {3 b \,c^{2} d \,x^{4}}{4 e^{2}}+\frac {c^{3} d^{2} x^{4}}{4 e^{3}}+\frac {2 a b c \,x^{3}}{e}-\frac {a \,c^{2} d \,x^{3}}{e^{2}}+\frac {b^{3} x^{3}}{3 e}-\frac {b^{2} c d \,x^{3}}{e^{2}}+\frac {b \,c^{2} d^{2} x^{3}}{e^{3}}-\frac {c^{3} d^{3} x^{3}}{3 e^{4}}+\frac {3 a^{2} c \,x^{2}}{2 e}+\frac {3 a \,b^{2} x^{2}}{2 e}-\frac {3 a b c d \,x^{2}}{e^{2}}+\frac {3 a \,c^{2} d^{2} x^{2}}{2 e^{3}}-\frac {b^{3} d \,x^{2}}{2 e^{2}}+\frac {3 b^{2} c \,d^{2} x^{2}}{2 e^{3}}-\frac {3 b \,c^{2} d^{3} x^{2}}{2 e^{4}}+\frac {c^{3} d^{4} x^{2}}{2 e^{5}}+\frac {a^{3} \ln \left (e x +d \right )}{e}-\frac {3 a^{2} b d \ln \left (e x +d \right )}{e^{2}}+\frac {3 a^{2} b x}{e}+\frac {3 a^{2} c \,d^{2} \ln \left (e x +d \right )}{e^{3}}-\frac {3 a^{2} c d x}{e^{2}}+\frac {3 a \,b^{2} d^{2} \ln \left (e x +d \right )}{e^{3}}-\frac {3 a \,b^{2} d x}{e^{2}}-\frac {6 a b c \,d^{3} \ln \left (e x +d \right )}{e^{4}}+\frac {6 a b c \,d^{2} x}{e^{3}}+\frac {3 a \,c^{2} d^{4} \ln \left (e x +d \right )}{e^{5}}-\frac {3 a \,c^{2} d^{3} x}{e^{4}}-\frac {b^{3} d^{3} \ln \left (e x +d \right )}{e^{4}}+\frac {b^{3} d^{2} x}{e^{3}}+\frac {3 b^{2} c \,d^{4} \ln \left (e x +d \right )}{e^{5}}-\frac {3 b^{2} c \,d^{3} x}{e^{4}}-\frac {3 b \,c^{2} d^{5} \ln \left (e x +d \right )}{e^{6}}+\frac {3 b \,c^{2} d^{4} x}{e^{5}}+\frac {c^{3} d^{6} \ln \left (e x +d \right )}{e^{7}}-\frac {c^{3} d^{5} x}{e^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.12, size = 401, normalized size = 1.54 \begin {gather*} \frac {10 \, c^{3} e^{5} x^{6} - 12 \, {\left (c^{3} d e^{4} - 3 \, b c^{2} e^{5}\right )} x^{5} + 15 \, {\left (c^{3} d^{2} e^{3} - 3 \, b c^{2} d e^{4} + 3 \, {\left (b^{2} c + a c^{2}\right )} e^{5}\right )} x^{4} - 20 \, {\left (c^{3} d^{3} e^{2} - 3 \, b c^{2} d^{2} e^{3} + 3 \, {\left (b^{2} c + a c^{2}\right )} d e^{4} - {\left (b^{3} + 6 \, a b c\right )} e^{5}\right )} x^{3} + 30 \, {\left (c^{3} d^{4} e - 3 \, b c^{2} d^{3} e^{2} + 3 \, {\left (b^{2} c + a c^{2}\right )} d^{2} e^{3} - {\left (b^{3} + 6 \, a b c\right )} d e^{4} + 3 \, {\left (a b^{2} + a^{2} c\right )} e^{5}\right )} x^{2} - 60 \, {\left (c^{3} d^{5} - 3 \, b c^{2} d^{4} e - 3 \, a^{2} b e^{5} + 3 \, {\left (b^{2} c + a c^{2}\right )} d^{3} e^{2} - {\left (b^{3} + 6 \, a b c\right )} d^{2} e^{3} + 3 \, {\left (a b^{2} + a^{2} c\right )} d e^{4}\right )} x}{60 \, e^{6}} + \frac {{\left (c^{3} d^{6} - 3 \, b c^{2} d^{5} e - 3 \, a^{2} b d e^{5} + a^{3} e^{6} + 3 \, {\left (b^{2} c + a c^{2}\right )} d^{4} e^{2} - {\left (b^{3} + 6 \, a b c\right )} d^{3} e^{3} + 3 \, {\left (a b^{2} + a^{2} c\right )} d^{2} e^{4}\right )} \log \left (e x + d\right )}{e^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.68, size = 433, normalized size = 1.67 \begin {gather*} x\,\left (\frac {3\,a^2\,b}{e}-\frac {d\,\left (\frac {3\,a\,\left (b^2+a\,c\right )}{e}-\frac {d\,\left (\frac {b^3+6\,a\,c\,b}{e}+\frac {d\,\left (\frac {d\,\left (\frac {3\,b\,c^2}{e}-\frac {c^3\,d}{e^2}\right )}{e}-\frac {3\,c\,\left (b^2+a\,c\right )}{e}\right )}{e}\right )}{e}\right )}{e}\right )+x^5\,\left (\frac {3\,b\,c^2}{5\,e}-\frac {c^3\,d}{5\,e^2}\right )+x^2\,\left (\frac {3\,a\,\left (b^2+a\,c\right )}{2\,e}-\frac {d\,\left (\frac {b^3+6\,a\,c\,b}{e}+\frac {d\,\left (\frac {d\,\left (\frac {3\,b\,c^2}{e}-\frac {c^3\,d}{e^2}\right )}{e}-\frac {3\,c\,\left (b^2+a\,c\right )}{e}\right )}{e}\right )}{2\,e}\right )+x^3\,\left (\frac {b^3+6\,a\,c\,b}{3\,e}+\frac {d\,\left (\frac {d\,\left (\frac {3\,b\,c^2}{e}-\frac {c^3\,d}{e^2}\right )}{e}-\frac {3\,c\,\left (b^2+a\,c\right )}{e}\right )}{3\,e}\right )-x^4\,\left (\frac {d\,\left (\frac {3\,b\,c^2}{e}-\frac {c^3\,d}{e^2}\right )}{4\,e}-\frac {3\,c\,\left (b^2+a\,c\right )}{4\,e}\right )+\frac {c^3\,x^6}{6\,e}+\frac {\ln \left (d+e\,x\right )\,\left (a^3\,e^6-3\,a^2\,b\,d\,e^5+3\,a^2\,c\,d^2\,e^4+3\,a\,b^2\,d^2\,e^4-6\,a\,b\,c\,d^3\,e^3+3\,a\,c^2\,d^4\,e^2-b^3\,d^3\,e^3+3\,b^2\,c\,d^4\,e^2-3\,b\,c^2\,d^5\,e+c^3\,d^6\right )}{e^7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.87, size = 384, normalized size = 1.48 \begin {gather*} \frac {c^{3} x^{6}}{6 e} + x^{5} \left (\frac {3 b c^{2}}{5 e} - \frac {c^{3} d}{5 e^{2}}\right ) + x^{4} \left (\frac {3 a c^{2}}{4 e} + \frac {3 b^{2} c}{4 e} - \frac {3 b c^{2} d}{4 e^{2}} + \frac {c^{3} d^{2}}{4 e^{3}}\right ) + x^{3} \left (\frac {2 a b c}{e} - \frac {a c^{2} d}{e^{2}} + \frac {b^{3}}{3 e} - \frac {b^{2} c d}{e^{2}} + \frac {b c^{2} d^{2}}{e^{3}} - \frac {c^{3} d^{3}}{3 e^{4}}\right ) + x^{2} \left (\frac {3 a^{2} c}{2 e} + \frac {3 a b^{2}}{2 e} - \frac {3 a b c d}{e^{2}} + \frac {3 a c^{2} d^{2}}{2 e^{3}} - \frac {b^{3} d}{2 e^{2}} + \frac {3 b^{2} c d^{2}}{2 e^{3}} - \frac {3 b c^{2} d^{3}}{2 e^{4}} + \frac {c^{3} d^{4}}{2 e^{5}}\right ) + x \left (\frac {3 a^{2} b}{e} - \frac {3 a^{2} c d}{e^{2}} - \frac {3 a b^{2} d}{e^{2}} + \frac {6 a b c d^{2}}{e^{3}} - \frac {3 a c^{2} d^{3}}{e^{4}} + \frac {b^{3} d^{2}}{e^{3}} - \frac {3 b^{2} c d^{3}}{e^{4}} + \frac {3 b c^{2} d^{4}}{e^{5}} - \frac {c^{3} d^{5}}{e^{6}}\right ) + \frac {\left (a e^{2} - b d e + c d^{2}\right )^{3} \log {\left (d + e x \right )}}{e^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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