Integrand size = 19, antiderivative size = 320 \[ \int e^{\frac {e}{c+d x}} (a+b x)^3 \, dx=-\frac {(b c-a d)^3 e^{\frac {e}{c+d x}} (c+d x)}{d^4}+\frac {3 b (b c-a d)^2 e e^{\frac {e}{c+d x}} (c+d x)}{2 d^4}-\frac {b^2 (b c-a d) e^2 e^{\frac {e}{c+d x}} (c+d x)}{2 d^4}+\frac {3 b (b c-a d)^2 e^{\frac {e}{c+d x}} (c+d x)^2}{2 d^4}-\frac {b^2 (b c-a d) e e^{\frac {e}{c+d x}} (c+d x)^2}{2 d^4}-\frac {b^2 (b c-a d) e^{\frac {e}{c+d x}} (c+d x)^3}{d^4}+\frac {(b c-a d)^3 e \operatorname {ExpIntegralEi}\left (\frac {e}{c+d x}\right )}{d^4}-\frac {3 b (b c-a d)^2 e^2 \operatorname {ExpIntegralEi}\left (\frac {e}{c+d x}\right )}{2 d^4}+\frac {b^2 (b c-a d) e^3 \operatorname {ExpIntegralEi}\left (\frac {e}{c+d x}\right )}{2 d^4}+\frac {b^3 e^4 \Gamma \left (-4,-\frac {e}{c+d x}\right )}{d^4} \]
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Time = 0.20 (sec) , antiderivative size = 320, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {2258, 2237, 2241, 2245, 2250} \[ \int e^{\frac {e}{c+d x}} (a+b x)^3 \, dx=\frac {b^2 e^3 (b c-a d) \operatorname {ExpIntegralEi}\left (\frac {e}{c+d x}\right )}{2 d^4}-\frac {b^2 e^2 (c+d x) (b c-a d) e^{\frac {e}{c+d x}}}{2 d^4}-\frac {b^2 e (c+d x)^2 (b c-a d) e^{\frac {e}{c+d x}}}{2 d^4}-\frac {b^2 (c+d x)^3 (b c-a d) e^{\frac {e}{c+d x}}}{d^4}-\frac {3 b e^2 (b c-a d)^2 \operatorname {ExpIntegralEi}\left (\frac {e}{c+d x}\right )}{2 d^4}+\frac {e (b c-a d)^3 \operatorname {ExpIntegralEi}\left (\frac {e}{c+d x}\right )}{d^4}+\frac {3 b e (c+d x) (b c-a d)^2 e^{\frac {e}{c+d x}}}{2 d^4}+\frac {3 b (c+d x)^2 (b c-a d)^2 e^{\frac {e}{c+d x}}}{2 d^4}-\frac {(c+d x) (b c-a d)^3 e^{\frac {e}{c+d x}}}{d^4}+\frac {b^3 e^4 \Gamma \left (-4,-\frac {e}{c+d x}\right )}{d^4} \]
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Rule 2237
Rule 2241
Rule 2245
Rule 2250
Rule 2258
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {(-b c+a d)^3 e^{\frac {e}{c+d x}}}{d^3}+\frac {3 b (b c-a d)^2 e^{\frac {e}{c+d x}} (c+d x)}{d^3}-\frac {3 b^2 (b c-a d) e^{\frac {e}{c+d x}} (c+d x)^2}{d^3}+\frac {b^3 e^{\frac {e}{c+d x}} (c+d x)^3}{d^3}\right ) \, dx \\ & = \frac {b^3 \int e^{\frac {e}{c+d x}} (c+d x)^3 \, dx}{d^3}-\frac {\left (3 b^2 (b c-a d)\right ) \int e^{\frac {e}{c+d x}} (c+d x)^2 \, dx}{d^3}+\frac {\left (3 b (b c-a d)^2\right ) \int e^{\frac {e}{c+d x}} (c+d x) \, dx}{d^3}-\frac {(b c-a d)^3 \int e^{\frac {e}{c+d x}} \, dx}{d^3} \\ & = -\frac {(b c-a d)^3 e^{\frac {e}{c+d x}} (c+d x)}{d^4}+\frac {3 b (b c-a d)^2 e^{\frac {e}{c+d x}} (c+d x)^2}{2 d^4}-\frac {b^2 (b c-a d) e^{\frac {e}{c+d x}} (c+d x)^3}{d^4}+\frac {b^3 e^4 \Gamma \left (-4,-\frac {e}{c+d x}\right )}{d^4}-\frac {\left (b^2 (b c-a d) e\right ) \int e^{\frac {e}{c+d x}} (c+d x) \, dx}{d^3}+\frac {\left (3 b (b c-a d)^2 e\right ) \int e^{\frac {e}{c+d x}} \, dx}{2 d^3}-\frac {\left ((b c-a d)^3 e\right ) \int \frac {e^{\frac {e}{c+d x}}}{c+d x} \, dx}{d^3} \\ & = -\frac {(b c-a d)^3 e^{\frac {e}{c+d x}} (c+d x)}{d^4}+\frac {3 b (b c-a d)^2 e e^{\frac {e}{c+d x}} (c+d x)}{2 d^4}+\frac {3 b (b c-a d)^2 e^{\frac {e}{c+d x}} (c+d x)^2}{2 d^4}-\frac {b^2 (b c-a d) e e^{\frac {e}{c+d x}} (c+d x)^2}{2 d^4}-\frac {b^2 (b c-a d) e^{\frac {e}{c+d x}} (c+d x)^3}{d^4}+\frac {(b c-a d)^3 e \text {Ei}\left (\frac {e}{c+d x}\right )}{d^4}+\frac {b^3 e^4 \Gamma \left (-4,-\frac {e}{c+d x}\right )}{d^4}-\frac {\left (b^2 (b c-a d) e^2\right ) \int e^{\frac {e}{c+d x}} \, dx}{2 d^3}+\frac {\left (3 b (b c-a d)^2 e^2\right ) \int \frac {e^{\frac {e}{c+d x}}}{c+d x} \, dx}{2 d^3} \\ & = -\frac {(b c-a d)^3 e^{\frac {e}{c+d x}} (c+d x)}{d^4}+\frac {3 b (b c-a d)^2 e e^{\frac {e}{c+d x}} (c+d x)}{2 d^4}-\frac {b^2 (b c-a d) e^2 e^{\frac {e}{c+d x}} (c+d x)}{2 d^4}+\frac {3 b (b c-a d)^2 e^{\frac {e}{c+d x}} (c+d x)^2}{2 d^4}-\frac {b^2 (b c-a d) e e^{\frac {e}{c+d x}} (c+d x)^2}{2 d^4}-\frac {b^2 (b c-a d) e^{\frac {e}{c+d x}} (c+d x)^3}{d^4}+\frac {(b c-a d)^3 e \text {Ei}\left (\frac {e}{c+d x}\right )}{d^4}-\frac {3 b (b c-a d)^2 e^2 \text {Ei}\left (\frac {e}{c+d x}\right )}{2 d^4}+\frac {b^3 e^4 \Gamma \left (-4,-\frac {e}{c+d x}\right )}{d^4}-\frac {\left (b^2 (b c-a d) e^3\right ) \int \frac {e^{\frac {e}{c+d x}}}{c+d x} \, dx}{2 d^3} \\ & = -\frac {(b c-a d)^3 e^{\frac {e}{c+d x}} (c+d x)}{d^4}+\frac {3 b (b c-a d)^2 e e^{\frac {e}{c+d x}} (c+d x)}{2 d^4}-\frac {b^2 (b c-a d) e^2 e^{\frac {e}{c+d x}} (c+d x)}{2 d^4}+\frac {3 b (b c-a d)^2 e^{\frac {e}{c+d x}} (c+d x)^2}{2 d^4}-\frac {b^2 (b c-a d) e e^{\frac {e}{c+d x}} (c+d x)^2}{2 d^4}-\frac {b^2 (b c-a d) e^{\frac {e}{c+d x}} (c+d x)^3}{d^4}+\frac {(b c-a d)^3 e \text {Ei}\left (\frac {e}{c+d x}\right )}{d^4}-\frac {3 b (b c-a d)^2 e^2 \text {Ei}\left (\frac {e}{c+d x}\right )}{2 d^4}+\frac {b^2 (b c-a d) e^3 \text {Ei}\left (\frac {e}{c+d x}\right )}{2 d^4}+\frac {b^3 e^4 \Gamma \left (-4,-\frac {e}{c+d x}\right )}{d^4} \\ \end{align*}
Time = 0.16 (sec) , antiderivative size = 292, normalized size of antiderivative = 0.91 \[ \int e^{\frac {e}{c+d x}} (a+b x)^3 \, dx=-\frac {c \left (-24 a^3 d^3+36 a^2 b d^2 (c-e)-12 a b^2 d \left (2 c^2-5 c e+e^2\right )+b^3 \left (6 c^3-26 c^2 e+11 c e^2-e^3\right )\right ) e^{\frac {e}{c+d x}}}{24 d^4}+\frac {d e^{\frac {e}{c+d x}} x \left (24 a^3 d^3+36 a^2 b d^2 (e+d x)+12 a b^2 d \left (-4 c e+e^2+d e x+2 d^2 x^2\right )+b^3 \left (18 c^2 e+e^3+d e^2 x+2 d^2 e x^2+6 d^3 x^3-2 c e (5 e+3 d x)\right )\right )-e \left (24 a^3 d^3+36 a^2 b d^2 (-2 c+e)+12 a b^2 d \left (6 c^2-6 c e+e^2\right )+b^3 \left (-24 c^3+36 c^2 e-12 c e^2+e^3\right )\right ) \operatorname {ExpIntegralEi}\left (\frac {e}{c+d x}\right )}{24 d^4} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(681\) vs. \(2(306)=612\).
Time = 0.36 (sec) , antiderivative size = 682, normalized size of antiderivative = 2.13
method | result | size |
derivativedivides | \(-\frac {e \left (a^{3} \left (-\frac {\left (d x +c \right ) {\mathrm e}^{\frac {e}{d x +c}}}{e}-\operatorname {Ei}_{1}\left (-\frac {e}{d x +c}\right )\right )+\frac {b^{3} e^{3} \left (-\frac {\left (d x +c \right )^{4} {\mathrm e}^{\frac {e}{d x +c}}}{4 e^{4}}-\frac {\left (d x +c \right )^{3} {\mathrm e}^{\frac {e}{d x +c}}}{12 e^{3}}-\frac {{\mathrm e}^{\frac {e}{d x +c}} \left (d x +c \right )^{2}}{24 e^{2}}-\frac {\left (d x +c \right ) {\mathrm e}^{\frac {e}{d x +c}}}{24 e}-\frac {\operatorname {Ei}_{1}\left (-\frac {e}{d x +c}\right )}{24}\right )}{d^{3}}-\frac {b^{3} c^{3} \left (-\frac {\left (d x +c \right ) {\mathrm e}^{\frac {e}{d x +c}}}{e}-\operatorname {Ei}_{1}\left (-\frac {e}{d x +c}\right )\right )}{d^{3}}+\frac {3 b^{2} e^{2} a \left (-\frac {\left (d x +c \right )^{3} {\mathrm e}^{\frac {e}{d x +c}}}{3 e^{3}}-\frac {{\mathrm e}^{\frac {e}{d x +c}} \left (d x +c \right )^{2}}{6 e^{2}}-\frac {\left (d x +c \right ) {\mathrm e}^{\frac {e}{d x +c}}}{6 e}-\frac {\operatorname {Ei}_{1}\left (-\frac {e}{d x +c}\right )}{6}\right )}{d^{2}}-\frac {3 b^{3} e^{2} c \left (-\frac {\left (d x +c \right )^{3} {\mathrm e}^{\frac {e}{d x +c}}}{3 e^{3}}-\frac {{\mathrm e}^{\frac {e}{d x +c}} \left (d x +c \right )^{2}}{6 e^{2}}-\frac {\left (d x +c \right ) {\mathrm e}^{\frac {e}{d x +c}}}{6 e}-\frac {\operatorname {Ei}_{1}\left (-\frac {e}{d x +c}\right )}{6}\right )}{d^{3}}+\frac {3 b e \,a^{2} \left (-\frac {{\mathrm e}^{\frac {e}{d x +c}} \left (d x +c \right )^{2}}{2 e^{2}}-\frac {\left (d x +c \right ) {\mathrm e}^{\frac {e}{d x +c}}}{2 e}-\frac {\operatorname {Ei}_{1}\left (-\frac {e}{d x +c}\right )}{2}\right )}{d}+\frac {3 b^{3} e \,c^{2} \left (-\frac {{\mathrm e}^{\frac {e}{d x +c}} \left (d x +c \right )^{2}}{2 e^{2}}-\frac {\left (d x +c \right ) {\mathrm e}^{\frac {e}{d x +c}}}{2 e}-\frac {\operatorname {Ei}_{1}\left (-\frac {e}{d x +c}\right )}{2}\right )}{d^{3}}-\frac {3 b c \,a^{2} \left (-\frac {\left (d x +c \right ) {\mathrm e}^{\frac {e}{d x +c}}}{e}-\operatorname {Ei}_{1}\left (-\frac {e}{d x +c}\right )\right )}{d}+\frac {3 b^{2} c^{2} a \left (-\frac {\left (d x +c \right ) {\mathrm e}^{\frac {e}{d x +c}}}{e}-\operatorname {Ei}_{1}\left (-\frac {e}{d x +c}\right )\right )}{d^{2}}-\frac {6 b^{2} e c a \left (-\frac {{\mathrm e}^{\frac {e}{d x +c}} \left (d x +c \right )^{2}}{2 e^{2}}-\frac {\left (d x +c \right ) {\mathrm e}^{\frac {e}{d x +c}}}{2 e}-\frac {\operatorname {Ei}_{1}\left (-\frac {e}{d x +c}\right )}{2}\right )}{d^{2}}\right )}{d}\) | \(682\) |
default | \(-\frac {e \left (a^{3} \left (-\frac {\left (d x +c \right ) {\mathrm e}^{\frac {e}{d x +c}}}{e}-\operatorname {Ei}_{1}\left (-\frac {e}{d x +c}\right )\right )+\frac {b^{3} e^{3} \left (-\frac {\left (d x +c \right )^{4} {\mathrm e}^{\frac {e}{d x +c}}}{4 e^{4}}-\frac {\left (d x +c \right )^{3} {\mathrm e}^{\frac {e}{d x +c}}}{12 e^{3}}-\frac {{\mathrm e}^{\frac {e}{d x +c}} \left (d x +c \right )^{2}}{24 e^{2}}-\frac {\left (d x +c \right ) {\mathrm e}^{\frac {e}{d x +c}}}{24 e}-\frac {\operatorname {Ei}_{1}\left (-\frac {e}{d x +c}\right )}{24}\right )}{d^{3}}-\frac {b^{3} c^{3} \left (-\frac {\left (d x +c \right ) {\mathrm e}^{\frac {e}{d x +c}}}{e}-\operatorname {Ei}_{1}\left (-\frac {e}{d x +c}\right )\right )}{d^{3}}+\frac {3 b^{2} e^{2} a \left (-\frac {\left (d x +c \right )^{3} {\mathrm e}^{\frac {e}{d x +c}}}{3 e^{3}}-\frac {{\mathrm e}^{\frac {e}{d x +c}} \left (d x +c \right )^{2}}{6 e^{2}}-\frac {\left (d x +c \right ) {\mathrm e}^{\frac {e}{d x +c}}}{6 e}-\frac {\operatorname {Ei}_{1}\left (-\frac {e}{d x +c}\right )}{6}\right )}{d^{2}}-\frac {3 b^{3} e^{2} c \left (-\frac {\left (d x +c \right )^{3} {\mathrm e}^{\frac {e}{d x +c}}}{3 e^{3}}-\frac {{\mathrm e}^{\frac {e}{d x +c}} \left (d x +c \right )^{2}}{6 e^{2}}-\frac {\left (d x +c \right ) {\mathrm e}^{\frac {e}{d x +c}}}{6 e}-\frac {\operatorname {Ei}_{1}\left (-\frac {e}{d x +c}\right )}{6}\right )}{d^{3}}+\frac {3 b e \,a^{2} \left (-\frac {{\mathrm e}^{\frac {e}{d x +c}} \left (d x +c \right )^{2}}{2 e^{2}}-\frac {\left (d x +c \right ) {\mathrm e}^{\frac {e}{d x +c}}}{2 e}-\frac {\operatorname {Ei}_{1}\left (-\frac {e}{d x +c}\right )}{2}\right )}{d}+\frac {3 b^{3} e \,c^{2} \left (-\frac {{\mathrm e}^{\frac {e}{d x +c}} \left (d x +c \right )^{2}}{2 e^{2}}-\frac {\left (d x +c \right ) {\mathrm e}^{\frac {e}{d x +c}}}{2 e}-\frac {\operatorname {Ei}_{1}\left (-\frac {e}{d x +c}\right )}{2}\right )}{d^{3}}-\frac {3 b c \,a^{2} \left (-\frac {\left (d x +c \right ) {\mathrm e}^{\frac {e}{d x +c}}}{e}-\operatorname {Ei}_{1}\left (-\frac {e}{d x +c}\right )\right )}{d}+\frac {3 b^{2} c^{2} a \left (-\frac {\left (d x +c \right ) {\mathrm e}^{\frac {e}{d x +c}}}{e}-\operatorname {Ei}_{1}\left (-\frac {e}{d x +c}\right )\right )}{d^{2}}-\frac {6 b^{2} e c a \left (-\frac {{\mathrm e}^{\frac {e}{d x +c}} \left (d x +c \right )^{2}}{2 e^{2}}-\frac {\left (d x +c \right ) {\mathrm e}^{\frac {e}{d x +c}}}{2 e}-\frac {\operatorname {Ei}_{1}\left (-\frac {e}{d x +c}\right )}{2}\right )}{d^{2}}\right )}{d}\) | \(682\) |
risch | \(\frac {{\mathrm e}^{\frac {e}{d x +c}} a^{3} c}{d}-\frac {{\mathrm e}^{\frac {e}{d x +c}} b^{3} c^{4}}{4 d^{4}}+\frac {\operatorname {Ei}_{1}\left (-\frac {e}{d x +c}\right ) a^{3} e}{d}+\frac {\operatorname {Ei}_{1}\left (-\frac {e}{d x +c}\right ) b^{3} e^{4}}{24 d^{4}}+{\mathrm e}^{\frac {e}{d x +c}} a \,b^{2} x^{3}+\frac {3 \,{\mathrm e}^{\frac {e}{d x +c}} a^{2} b \,x^{2}}{2}-\frac {2 \,{\mathrm e}^{\frac {e}{d x +c}} a \,b^{2} c e x}{d^{2}}+\frac {\operatorname {Ei}_{1}\left (-\frac {e}{d x +c}\right ) a \,b^{2} e^{3}}{2 d^{3}}-\frac {\operatorname {Ei}_{1}\left (-\frac {e}{d x +c}\right ) b^{3} c^{3} e}{d^{4}}+\frac {3 \,\operatorname {Ei}_{1}\left (-\frac {e}{d x +c}\right ) b^{3} c^{2} e^{2}}{2 d^{4}}-\frac {\operatorname {Ei}_{1}\left (-\frac {e}{d x +c}\right ) b^{3} c \,e^{3}}{2 d^{4}}+\frac {{\mathrm e}^{\frac {e}{d x +c}} b^{3} e \,x^{3}}{12 d}+\frac {{\mathrm e}^{\frac {e}{d x +c}} b^{3} e^{2} x^{2}}{24 d^{2}}+\frac {{\mathrm e}^{\frac {e}{d x +c}} b^{3} e^{3} x}{24 d^{3}}-\frac {3 \,{\mathrm e}^{\frac {e}{d x +c}} a^{2} b \,c^{2}}{2 d^{2}}+\frac {{\mathrm e}^{\frac {e}{d x +c}} a \,b^{2} c^{3}}{d^{3}}+\frac {13 \,{\mathrm e}^{\frac {e}{d x +c}} b^{3} c^{3} e}{12 d^{4}}-\frac {11 \,{\mathrm e}^{\frac {e}{d x +c}} b^{3} c^{2} e^{2}}{24 d^{4}}+\frac {{\mathrm e}^{\frac {e}{d x +c}} b^{3} c \,e^{3}}{24 d^{4}}+\frac {3 \,\operatorname {Ei}_{1}\left (-\frac {e}{d x +c}\right ) a^{2} b \,e^{2}}{2 d^{2}}+\frac {{\mathrm e}^{\frac {e}{d x +c}} b^{3} x^{4}}{4}+{\mathrm e}^{\frac {e}{d x +c}} a^{3} x -\frac {5 \,{\mathrm e}^{\frac {e}{d x +c}} b^{3} c \,e^{2} x}{12 d^{3}}+\frac {3 \,{\mathrm e}^{\frac {e}{d x +c}} a^{2} b c e}{2 d^{2}}-\frac {5 \,{\mathrm e}^{\frac {e}{d x +c}} a \,b^{2} c^{2} e}{2 d^{3}}+\frac {{\mathrm e}^{\frac {e}{d x +c}} a \,b^{2} c \,e^{2}}{2 d^{3}}-\frac {3 \,\operatorname {Ei}_{1}\left (-\frac {e}{d x +c}\right ) a^{2} b c e}{d^{2}}+\frac {3 \,\operatorname {Ei}_{1}\left (-\frac {e}{d x +c}\right ) a \,b^{2} c^{2} e}{d^{3}}-\frac {3 \,\operatorname {Ei}_{1}\left (-\frac {e}{d x +c}\right ) a \,b^{2} c \,e^{2}}{d^{3}}+\frac {{\mathrm e}^{\frac {e}{d x +c}} a \,b^{2} e \,x^{2}}{2 d}-\frac {{\mathrm e}^{\frac {e}{d x +c}} b^{3} c e \,x^{2}}{4 d^{2}}+\frac {3 \,{\mathrm e}^{\frac {e}{d x +c}} a^{2} b e x}{2 d}+\frac {{\mathrm e}^{\frac {e}{d x +c}} a \,b^{2} e^{2} x}{2 d^{2}}+\frac {3 \,{\mathrm e}^{\frac {e}{d x +c}} b^{3} c^{2} e x}{4 d^{3}}\) | \(753\) |
parts | \(\text {Expression too large to display}\) | \(1129\) |
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Time = 0.33 (sec) , antiderivative size = 377, normalized size of antiderivative = 1.18 \[ \int e^{\frac {e}{c+d x}} (a+b x)^3 \, dx=-\frac {{\left (b^{3} e^{4} - 12 \, {\left (b^{3} c - a b^{2} d\right )} e^{3} + 36 \, {\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} e^{2} - 24 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} e\right )} {\rm Ei}\left (\frac {e}{d x + c}\right ) - {\left (6 \, b^{3} d^{4} x^{4} - 6 \, b^{3} c^{4} + 24 \, a b^{2} c^{3} d - 36 \, a^{2} b c^{2} d^{2} + 24 \, a^{3} c d^{3} + b^{3} c e^{3} + 2 \, {\left (12 \, a b^{2} d^{4} + b^{3} d^{3} e\right )} x^{3} - {\left (11 \, b^{3} c^{2} - 12 \, a b^{2} c d\right )} e^{2} + {\left (36 \, a^{2} b d^{4} + b^{3} d^{2} e^{2} - 6 \, {\left (b^{3} c d^{2} - 2 \, a b^{2} d^{3}\right )} e\right )} x^{2} + 2 \, {\left (13 \, b^{3} c^{3} - 30 \, a b^{2} c^{2} d + 18 \, a^{2} b c d^{2}\right )} e + {\left (24 \, a^{3} d^{4} + b^{3} d e^{3} - 2 \, {\left (5 \, b^{3} c d - 6 \, a b^{2} d^{2}\right )} e^{2} + 6 \, {\left (3 \, b^{3} c^{2} d - 8 \, a b^{2} c d^{2} + 6 \, a^{2} b d^{3}\right )} e\right )} x\right )} e^{\left (\frac {e}{d x + c}\right )}}{24 \, d^{4}} \]
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\[ \int e^{\frac {e}{c+d x}} (a+b x)^3 \, dx=\int \left (a + b x\right )^{3} e^{\frac {e}{c + d x}}\, dx \]
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\[ \int e^{\frac {e}{c+d x}} (a+b x)^3 \, dx=\int { {\left (b x + a\right )}^{3} e^{\left (\frac {e}{d x + c}\right )} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 831 vs. \(2 (306) = 612\).
Time = 0.33 (sec) , antiderivative size = 831, normalized size of antiderivative = 2.60 \[ \int e^{\frac {e}{c+d x}} (a+b x)^3 \, dx=\frac {{\left (\frac {24 \, b^{3} c^{3} e^{6} {\rm Ei}\left (\frac {e}{d x + c}\right )}{{\left (d x + c\right )}^{4}} - \frac {72 \, a b^{2} c^{2} d e^{6} {\rm Ei}\left (\frac {e}{d x + c}\right )}{{\left (d x + c\right )}^{4}} + \frac {72 \, a^{2} b c d^{2} e^{6} {\rm Ei}\left (\frac {e}{d x + c}\right )}{{\left (d x + c\right )}^{4}} - \frac {24 \, a^{3} d^{3} e^{6} {\rm Ei}\left (\frac {e}{d x + c}\right )}{{\left (d x + c\right )}^{4}} - \frac {36 \, b^{3} c^{2} e^{7} {\rm Ei}\left (\frac {e}{d x + c}\right )}{{\left (d x + c\right )}^{4}} + \frac {72 \, a b^{2} c d e^{7} {\rm Ei}\left (\frac {e}{d x + c}\right )}{{\left (d x + c\right )}^{4}} - \frac {36 \, a^{2} b d^{2} e^{7} {\rm Ei}\left (\frac {e}{d x + c}\right )}{{\left (d x + c\right )}^{4}} + \frac {12 \, b^{3} c e^{8} {\rm Ei}\left (\frac {e}{d x + c}\right )}{{\left (d x + c\right )}^{4}} - \frac {12 \, a b^{2} d e^{8} {\rm Ei}\left (\frac {e}{d x + c}\right )}{{\left (d x + c\right )}^{4}} - \frac {b^{3} e^{9} {\rm Ei}\left (\frac {e}{d x + c}\right )}{{\left (d x + c\right )}^{4}} + 6 \, b^{3} e^{5} e^{\left (\frac {e}{d x + c}\right )} - \frac {24 \, b^{3} c e^{5} e^{\left (\frac {e}{d x + c}\right )}}{d x + c} + \frac {36 \, b^{3} c^{2} e^{5} e^{\left (\frac {e}{d x + c}\right )}}{{\left (d x + c\right )}^{2}} - \frac {24 \, b^{3} c^{3} e^{5} e^{\left (\frac {e}{d x + c}\right )}}{{\left (d x + c\right )}^{3}} + \frac {24 \, a b^{2} d e^{5} e^{\left (\frac {e}{d x + c}\right )}}{d x + c} - \frac {72 \, a b^{2} c d e^{5} e^{\left (\frac {e}{d x + c}\right )}}{{\left (d x + c\right )}^{2}} + \frac {72 \, a b^{2} c^{2} d e^{5} e^{\left (\frac {e}{d x + c}\right )}}{{\left (d x + c\right )}^{3}} + \frac {36 \, a^{2} b d^{2} e^{5} e^{\left (\frac {e}{d x + c}\right )}}{{\left (d x + c\right )}^{2}} - \frac {72 \, a^{2} b c d^{2} e^{5} e^{\left (\frac {e}{d x + c}\right )}}{{\left (d x + c\right )}^{3}} + \frac {24 \, a^{3} d^{3} e^{5} e^{\left (\frac {e}{d x + c}\right )}}{{\left (d x + c\right )}^{3}} + \frac {2 \, b^{3} e^{6} e^{\left (\frac {e}{d x + c}\right )}}{d x + c} - \frac {12 \, b^{3} c e^{6} e^{\left (\frac {e}{d x + c}\right )}}{{\left (d x + c\right )}^{2}} + \frac {36 \, b^{3} c^{2} e^{6} e^{\left (\frac {e}{d x + c}\right )}}{{\left (d x + c\right )}^{3}} + \frac {12 \, a b^{2} d e^{6} e^{\left (\frac {e}{d x + c}\right )}}{{\left (d x + c\right )}^{2}} - \frac {72 \, a b^{2} c d e^{6} e^{\left (\frac {e}{d x + c}\right )}}{{\left (d x + c\right )}^{3}} + \frac {36 \, a^{2} b d^{2} e^{6} e^{\left (\frac {e}{d x + c}\right )}}{{\left (d x + c\right )}^{3}} + \frac {b^{3} e^{7} e^{\left (\frac {e}{d x + c}\right )}}{{\left (d x + c\right )}^{2}} - \frac {12 \, b^{3} c e^{7} e^{\left (\frac {e}{d x + c}\right )}}{{\left (d x + c\right )}^{3}} + \frac {12 \, a b^{2} d e^{7} e^{\left (\frac {e}{d x + c}\right )}}{{\left (d x + c\right )}^{3}} + \frac {b^{3} e^{8} e^{\left (\frac {e}{d x + c}\right )}}{{\left (d x + c\right )}^{3}}\right )} {\left (d x + c\right )}^{4}}{24 \, d^{4} e^{5}} \]
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Timed out. \[ \int e^{\frac {e}{c+d x}} (a+b x)^3 \, dx=\int {\mathrm {e}}^{\frac {e}{c+d\,x}}\,{\left (a+b\,x\right )}^3 \,d x \]
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