Optimal. Leaf size=260 \[ -\frac {3 (2 c d-b e) \left (c d^2-b d e+a e^2\right )^2 x}{e^6}+\frac {3 \left (c d^2-b d e+a e^2\right ) \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) (d+e x)^2}{2 e^7}-\frac {(2 c d-b e) \left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right ) (d+e x)^3}{3 e^7}+\frac {3 c \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) (d+e x)^4}{4 e^7}-\frac {3 c^2 (2 c d-b e) (d+e x)^5}{5 e^7}+\frac {c^3 (d+e x)^6}{6 e^7}+\frac {\left (c d^2-b d e+a e^2\right )^3 \log (d+e x)}{e^7} \]
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Rubi [A]
time = 0.22, 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 = {712}
\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 {\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} \end {gather*}
Antiderivative was successfully verified.
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Rule 712
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 = 0.10, size = 308, normalized size = 1.18 \begin {gather*} \frac {e x \left (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 )+10 b e^3 \left (18 a^2 e^2+9 a b e (-2 d+e x)+b^2 \left (6 d^2-3 d e x+2 e^2 x^2\right )\right )+15 c e^2 \left (6 a^2 e^2 (-2 d+e x)+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 )+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 )\right )+60 \left (c d^2+e (-b d+a e)\right )^3 \log (d+e x)}{60 e^7} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.75, size = 455, normalized size = 1.75
method | result | size |
norman | \(\frac {\left (3 a^{2} b \,e^{5}-3 d \,e^{4} a^{2} c -3 a \,b^{2} d \,e^{4}+6 a b c \,d^{2} e^{3}-3 d^{3} e^{2} c^{2} a +b^{3} d^{2} e^{3}-3 b^{2} c \,d^{3} e^{2}+3 b \,c^{2} d^{4} e -d^{5} c^{3}\right ) x}{e^{6}}+\frac {c^{3} x^{6}}{6 e}+\frac {\left (3 e^{4} a^{2} c +3 a \,b^{2} e^{4}-6 a b c d \,e^{3}+3 d^{2} e^{2} c^{2} a -b^{3} d \,e^{3}+3 b^{2} c \,d^{2} e^{2}-3 d^{3} e b \,c^{2}+d^{4} c^{3}\right ) x^{2}}{2 e^{5}}+\frac {\left (6 a b c \,e^{3}-3 d \,e^{2} c^{2} a +b^{3} e^{3}-3 b^{2} d \,e^{2} c +3 b \,c^{2} d^{2} e -c^{3} d^{3}\right ) x^{3}}{3 e^{4}}+\frac {c \left (3 a c \,e^{2}+3 b^{2} e^{2}-3 b c d e +c^{2} d^{2}\right ) x^{4}}{4 e^{3}}+\frac {c^{2} \left (3 b e -c d \right ) x^{5}}{5 e^{2}}+\frac {\left (e^{6} a^{3}-3 a^{2} b d \,e^{5}+3 e^{4} d^{2} a^{2} c +3 a \,b^{2} d^{2} e^{4}-6 a b c \,d^{3} e^{3}+3 d^{4} e^{2} c^{2} a -b^{3} d^{3} e^{3}+3 b^{2} c \,d^{4} e^{2}-3 b \,c^{2} d^{5} e +d^{6} c^{3}\right ) \ln \left (e x +d \right )}{e^{7}}\) | \(432\) |
default | \(\frac {\frac {c^{3} x^{6} e^{5}}{6}+\frac {\left (\left (b e -c d \right ) e^{4} c^{2}+2 c^{2} e^{5} b \right ) x^{5}}{5}+\frac {\left (2 \left (b e -c d \right ) e^{4} b c +c e \left (3 a c \,e^{4}+b^{2} e^{4}-d \,e^{3} b c +d^{2} e^{2} c^{2}\right )\right ) x^{4}}{4}+\frac {\left (\left (b e -c d \right ) \left (3 a c \,e^{4}+b^{2} e^{4}-d \,e^{3} b c +d^{2} e^{2} c^{2}\right )+c e \left (3 e^{4} a b -b^{2} d \,e^{3}+d^{2} e^{2} b c \right )\right ) x^{3}}{3}+\frac {\left (\left (b e -c d \right ) \left (3 e^{4} a b -b^{2} d \,e^{3}+d^{2} e^{2} b c \right )+c e \left (3 a^{2} e^{4}-3 d \,e^{3} a b +3 a c \,d^{2} e^{2}+b^{2} d^{2} e^{2}-2 d^{3} e b c +c^{2} d^{4}\right )\right ) x^{2}}{2}+\left (b e -c d \right ) \left (3 a^{2} e^{4}-3 d \,e^{3} a b +3 a c \,d^{2} e^{2}+b^{2} d^{2} e^{2}-2 d^{3} e b c +c^{2} d^{4}\right ) x}{e^{6}}+\frac {\left (e^{6} a^{3}-3 a^{2} b d \,e^{5}+3 e^{4} d^{2} a^{2} c +3 a \,b^{2} d^{2} e^{4}-6 a b c \,d^{3} e^{3}+3 d^{4} e^{2} c^{2} a -b^{3} d^{3} e^{3}+3 b^{2} c \,d^{4} e^{2}-3 b \,c^{2} d^{5} e +d^{6} c^{3}\right ) \ln \left (e x +d \right )}{e^{7}}\) | \(455\) |
risch | \(\frac {b^{3} x^{3}}{3 e}-\frac {3 x^{4} b \,c^{2} d}{4 e^{2}}+\frac {2 x^{3} a b c}{e}-\frac {x^{3} a \,c^{2} d}{e^{2}}-\frac {x^{3} b^{2} c d}{e^{2}}+\frac {x^{3} b \,c^{2} d^{2}}{e^{3}}+\frac {3 x^{2} b^{2} c \,d^{2}}{2 e^{3}}-\frac {3 x^{2} b \,c^{2} d^{3}}{2 e^{4}}+\frac {3 x^{2} c^{2} d^{2} a}{2 e^{3}}+\frac {x^{2} d^{4} c^{3}}{2 e^{5}}+\frac {3 a^{2} b x}{e}+\frac {b^{3} d^{2} x}{e^{3}}-\frac {c^{3} d^{5} x}{e^{6}}-\frac {x^{3} c^{3} d^{3}}{3 e^{4}}+\frac {3 x^{2} a \,b^{2}}{2 e}-\frac {x^{2} b^{3} d}{2 e^{2}}+\frac {3 x^{2} a^{2} c}{2 e}+\frac {3 x^{5} c^{2} b}{5 e}+\frac {3 x^{4} b^{2} c}{4 e}+\frac {3 x^{4} a \,c^{2}}{4 e}+\frac {x^{4} c^{3} d^{2}}{4 e^{3}}+\frac {6 a b c \,d^{2} x}{e^{3}}-\frac {6 \ln \left (e x +d \right ) a b c \,d^{3}}{e^{4}}-\frac {3 x^{2} a b c d}{e^{2}}+\frac {\ln \left (e x +d \right ) a^{3}}{e}+\frac {c^{3} x^{6}}{6 e}+\frac {3 \ln \left (e x +d \right ) d^{4} c^{2} a}{e^{5}}+\frac {3 \ln \left (e x +d \right ) b^{2} c \,d^{4}}{e^{5}}-\frac {3 \ln \left (e x +d \right ) b \,c^{2} d^{5}}{e^{6}}-\frac {\ln \left (e x +d \right ) b^{3} d^{3}}{e^{4}}+\frac {\ln \left (e x +d \right ) d^{6} c^{3}}{e^{7}}-\frac {c^{3} d \,x^{5}}{5 e^{2}}-\frac {3 \ln \left (e x +d \right ) a^{2} b d}{e^{2}}+\frac {3 \ln \left (e x +d \right ) d^{2} a^{2} c}{e^{3}}+\frac {3 \ln \left (e x +d \right ) a \,b^{2} d^{2}}{e^{3}}-\frac {3 b^{2} c \,d^{3} x}{e^{4}}-\frac {3 a^{2} c d x}{e^{2}}-\frac {3 a \,b^{2} d x}{e^{2}}-\frac {3 a \,c^{2} d^{3} x}{e^{4}}+\frac {3 b \,c^{2} d^{4} x}{e^{5}}\) | \(546\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 404, normalized size = 1.55 \begin {gather*} {\left (c^{3} d^{6} - 3 \, b c^{2} d^{5} e + 3 \, {\left (b^{2} c e^{2} + a c^{2} e^{2}\right )} d^{4} - 3 \, a^{2} b d e^{5} - {\left (b^{3} e^{3} + 6 \, a b c e^{3}\right )} d^{3} + a^{3} e^{6} + 3 \, {\left (a b^{2} e^{4} + a^{2} c e^{4}\right )} d^{2}\right )} e^{\left (-7\right )} \log \left (x e + d\right ) + \frac {1}{60} \, {\left (10 \, c^{3} x^{6} e^{5} - 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 \, b^{2} c e^{5} + 3 \, a c^{2} e^{5}\right )} x^{4} - 20 \, {\left (c^{3} d^{3} e^{2} - 3 \, b c^{2} d^{2} e^{3} - b^{3} e^{5} - 6 \, a b c e^{5} + 3 \, {\left (b^{2} c e^{4} + a c^{2} e^{4}\right )} d\right )} x^{3} + 30 \, {\left (c^{3} d^{4} e - 3 \, b c^{2} d^{3} e^{2} + 3 \, a b^{2} e^{5} + 3 \, a^{2} c e^{5} + 3 \, {\left (b^{2} c e^{3} + a c^{2} e^{3}\right )} d^{2} - {\left (b^{3} e^{4} + 6 \, a b c e^{4}\right )} d\right )} x^{2} - 60 \, {\left (c^{3} d^{5} - 3 \, b c^{2} d^{4} e + 3 \, {\left (b^{2} c e^{2} + a c^{2} e^{2}\right )} d^{3} - 3 \, a^{2} b e^{5} - {\left (b^{3} e^{3} + 6 \, a b c e^{3}\right )} d^{2} + 3 \, {\left (a b^{2} e^{4} + a^{2} c e^{4}\right )} d\right )} x\right )} e^{\left (-6\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.28, size = 388, normalized size = 1.49 \begin {gather*} -\frac {1}{60} \, {\left (60 \, c^{3} d^{5} x e - {\left (10 \, c^{3} x^{6} + 36 \, b c^{2} x^{5} + 45 \, {\left (b^{2} c + a c^{2}\right )} x^{4} + 180 \, a^{2} b x + 20 \, {\left (b^{3} + 6 \, a b c\right )} x^{3} + 90 \, {\left (a b^{2} + a^{2} c\right )} x^{2}\right )} e^{6} + 3 \, {\left (4 \, c^{3} d x^{5} + 15 \, b c^{2} d x^{4} + 20 \, {\left (b^{2} c + a c^{2}\right )} d x^{3} + 10 \, {\left (b^{3} + 6 \, a b c\right )} d x^{2} + 60 \, {\left (a b^{2} + a^{2} c\right )} d x\right )} e^{5} - 15 \, {\left (c^{3} d^{2} x^{4} + 4 \, b c^{2} d^{2} x^{3} + 6 \, {\left (b^{2} c + a c^{2}\right )} d^{2} x^{2} + 4 \, {\left (b^{3} + 6 \, a b c\right )} d^{2} x\right )} e^{4} + 10 \, {\left (2 \, c^{3} d^{3} x^{3} + 9 \, b c^{2} d^{3} x^{2} + 18 \, {\left (b^{2} c + a c^{2}\right )} d^{3} x\right )} e^{3} - 30 \, {\left (c^{3} d^{4} x^{2} + 6 \, b c^{2} d^{4} x\right )} e^{2} - 60 \, {\left (c^{3} d^{6} - 3 \, b c^{2} d^{5} e + 3 \, {\left (b^{2} c + a c^{2}\right )} d^{4} e^{2} - 3 \, a^{2} b d e^{5} - {\left (b^{3} + 6 \, a b c\right )} d^{3} e^{3} + a^{3} e^{6} + 3 \, {\left (a b^{2} + a^{2} c\right )} d^{2} e^{4}\right )} \log \left (x e + d\right )\right )} e^{\left (-7\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.51, size = 384, normalized size = 1.48 \begin {gather*} \frac {c^{3} x^{6}}{6 e} + x^{5} \cdot \left (\frac {3 b c^{2}}{5 e} - \frac {c^{3} d}{5 e^{2}}\right ) + x^{4} \cdot \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} \cdot \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} \cdot \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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Giac [A]
time = 1.38, 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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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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