Integrand size = 19, antiderivative size = 255 \[ \int e^{\frac {e}{c+d x}} (a+b x)^2 \, dx=\frac {(b c-a d)^2 e^{\frac {e}{c+d x}} (c+d x)}{d^3}-\frac {b (b c-a d) e e^{\frac {e}{c+d x}} (c+d x)}{d^3}+\frac {b^2 e^2 e^{\frac {e}{c+d x}} (c+d x)}{6 d^3}-\frac {b (b c-a d) e^{\frac {e}{c+d x}} (c+d x)^2}{d^3}+\frac {b^2 e e^{\frac {e}{c+d x}} (c+d x)^2}{6 d^3}+\frac {b^2 e^{\frac {e}{c+d x}} (c+d x)^3}{3 d^3}-\frac {(b c-a d)^2 e \operatorname {ExpIntegralEi}\left (\frac {e}{c+d x}\right )}{d^3}+\frac {b (b c-a d) e^2 \operatorname {ExpIntegralEi}\left (\frac {e}{c+d x}\right )}{d^3}-\frac {b^2 e^3 \operatorname {ExpIntegralEi}\left (\frac {e}{c+d x}\right )}{6 d^3} \]
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Time = 0.16 (sec) , antiderivative size = 255, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {2258, 2237, 2241, 2245} \[ \int e^{\frac {e}{c+d x}} (a+b x)^2 \, dx=\frac {b e^2 (b c-a d) \operatorname {ExpIntegralEi}\left (\frac {e}{c+d x}\right )}{d^3}-\frac {e (b c-a d)^2 \operatorname {ExpIntegralEi}\left (\frac {e}{c+d x}\right )}{d^3}-\frac {b e (c+d x) (b c-a d) e^{\frac {e}{c+d x}}}{d^3}-\frac {b (c+d x)^2 (b c-a d) e^{\frac {e}{c+d x}}}{d^3}+\frac {(c+d x) (b c-a d)^2 e^{\frac {e}{c+d x}}}{d^3}-\frac {b^2 e^3 \operatorname {ExpIntegralEi}\left (\frac {e}{c+d x}\right )}{6 d^3}+\frac {b^2 e^2 (c+d x) e^{\frac {e}{c+d x}}}{6 d^3}+\frac {b^2 e (c+d x)^2 e^{\frac {e}{c+d x}}}{6 d^3}+\frac {b^2 (c+d x)^3 e^{\frac {e}{c+d x}}}{3 d^3} \]
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Rule 2237
Rule 2241
Rule 2245
Rule 2258
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {(-b c+a d)^2 e^{\frac {e}{c+d x}}}{d^2}-\frac {2 b (b c-a d) e^{\frac {e}{c+d x}} (c+d x)}{d^2}+\frac {b^2 e^{\frac {e}{c+d x}} (c+d x)^2}{d^2}\right ) \, dx \\ & = \frac {b^2 \int e^{\frac {e}{c+d x}} (c+d x)^2 \, dx}{d^2}-\frac {(2 b (b c-a d)) \int e^{\frac {e}{c+d x}} (c+d x) \, dx}{d^2}+\frac {(b c-a d)^2 \int e^{\frac {e}{c+d x}} \, dx}{d^2} \\ & = \frac {(b c-a d)^2 e^{\frac {e}{c+d x}} (c+d x)}{d^3}-\frac {b (b c-a d) e^{\frac {e}{c+d x}} (c+d x)^2}{d^3}+\frac {b^2 e^{\frac {e}{c+d x}} (c+d x)^3}{3 d^3}+\frac {\left (b^2 e\right ) \int e^{\frac {e}{c+d x}} (c+d x) \, dx}{3 d^2}-\frac {(b (b c-a d) e) \int e^{\frac {e}{c+d x}} \, dx}{d^2}+\frac {\left ((b c-a d)^2 e\right ) \int \frac {e^{\frac {e}{c+d x}}}{c+d x} \, dx}{d^2} \\ & = \frac {(b c-a d)^2 e^{\frac {e}{c+d x}} (c+d x)}{d^3}-\frac {b (b c-a d) e e^{\frac {e}{c+d x}} (c+d x)}{d^3}-\frac {b (b c-a d) e^{\frac {e}{c+d x}} (c+d x)^2}{d^3}+\frac {b^2 e e^{\frac {e}{c+d x}} (c+d x)^2}{6 d^3}+\frac {b^2 e^{\frac {e}{c+d x}} (c+d x)^3}{3 d^3}-\frac {(b c-a d)^2 e \text {Ei}\left (\frac {e}{c+d x}\right )}{d^3}+\frac {\left (b^2 e^2\right ) \int e^{\frac {e}{c+d x}} \, dx}{6 d^2}-\frac {\left (b (b c-a d) e^2\right ) \int \frac {e^{\frac {e}{c+d x}}}{c+d x} \, dx}{d^2} \\ & = \frac {(b c-a d)^2 e^{\frac {e}{c+d x}} (c+d x)}{d^3}-\frac {b (b c-a d) e e^{\frac {e}{c+d x}} (c+d x)}{d^3}+\frac {b^2 e^2 e^{\frac {e}{c+d x}} (c+d x)}{6 d^3}-\frac {b (b c-a d) e^{\frac {e}{c+d x}} (c+d x)^2}{d^3}+\frac {b^2 e e^{\frac {e}{c+d x}} (c+d x)^2}{6 d^3}+\frac {b^2 e^{\frac {e}{c+d x}} (c+d x)^3}{3 d^3}-\frac {(b c-a d)^2 e \text {Ei}\left (\frac {e}{c+d x}\right )}{d^3}+\frac {b (b c-a d) e^2 \text {Ei}\left (\frac {e}{c+d x}\right )}{d^3}+\frac {\left (b^2 e^3\right ) \int \frac {e^{\frac {e}{c+d x}}}{c+d x} \, dx}{6 d^2} \\ & = \frac {(b c-a d)^2 e^{\frac {e}{c+d x}} (c+d x)}{d^3}-\frac {b (b c-a d) e e^{\frac {e}{c+d x}} (c+d x)}{d^3}+\frac {b^2 e^2 e^{\frac {e}{c+d x}} (c+d x)}{6 d^3}-\frac {b (b c-a d) e^{\frac {e}{c+d x}} (c+d x)^2}{d^3}+\frac {b^2 e e^{\frac {e}{c+d x}} (c+d x)^2}{6 d^3}+\frac {b^2 e^{\frac {e}{c+d x}} (c+d x)^3}{3 d^3}-\frac {(b c-a d)^2 e \text {Ei}\left (\frac {e}{c+d x}\right )}{d^3}+\frac {b (b c-a d) e^2 \text {Ei}\left (\frac {e}{c+d x}\right )}{d^3}-\frac {b^2 e^3 \text {Ei}\left (\frac {e}{c+d x}\right )}{6 d^3} \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 170, normalized size of antiderivative = 0.67 \[ \int e^{\frac {e}{c+d x}} (a+b x)^2 \, dx=\frac {c \left (6 a^2 d^2+6 a b d (-c+e)+b^2 \left (2 c^2-5 c e+e^2\right )\right ) e^{\frac {e}{c+d x}}}{6 d^3}+\frac {d e^{\frac {e}{c+d x}} x \left (6 a^2 d^2+6 a b d (e+d x)+b^2 \left (-4 c e+e^2+d e x+2 d^2 x^2\right )\right )-e \left (6 a^2 d^2+6 a b d (-2 c+e)+b^2 \left (6 c^2-6 c e+e^2\right )\right ) \operatorname {ExpIntegralEi}\left (\frac {e}{c+d x}\right )}{6 d^3} \]
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Time = 0.32 (sec) , antiderivative size = 356, normalized size of antiderivative = 1.40
method | result | size |
derivativedivides | \(-\frac {e \left (a^{2} \left (-\frac {\left (d x +c \right ) {\mathrm e}^{\frac {e}{d x +c}}}{e}-\operatorname {Ei}_{1}\left (-\frac {e}{d x +c}\right )\right )+\frac {b^{2} e^{2} \left (-\frac {\left (d x +c \right )^{3} {\mathrm e}^{\frac {e}{d x +c}}}{3 e^{3}}-\frac {{\mathrm e}^{\frac {e}{d x +c}} \left (d x +c \right )^{2}}{6 e^{2}}-\frac {\left (d x +c \right ) {\mathrm e}^{\frac {e}{d x +c}}}{6 e}-\frac {\operatorname {Ei}_{1}\left (-\frac {e}{d x +c}\right )}{6}\right )}{d^{2}}+\frac {b^{2} c^{2} \left (-\frac {\left (d x +c \right ) {\mathrm e}^{\frac {e}{d x +c}}}{e}-\operatorname {Ei}_{1}\left (-\frac {e}{d x +c}\right )\right )}{d^{2}}+\frac {2 b e a \left (-\frac {{\mathrm e}^{\frac {e}{d x +c}} \left (d x +c \right )^{2}}{2 e^{2}}-\frac {\left (d x +c \right ) {\mathrm e}^{\frac {e}{d x +c}}}{2 e}-\frac {\operatorname {Ei}_{1}\left (-\frac {e}{d x +c}\right )}{2}\right )}{d}-\frac {2 b^{2} e c \left (-\frac {{\mathrm e}^{\frac {e}{d x +c}} \left (d x +c \right )^{2}}{2 e^{2}}-\frac {\left (d x +c \right ) {\mathrm e}^{\frac {e}{d x +c}}}{2 e}-\frac {\operatorname {Ei}_{1}\left (-\frac {e}{d x +c}\right )}{2}\right )}{d^{2}}-\frac {2 b c a \left (-\frac {\left (d x +c \right ) {\mathrm e}^{\frac {e}{d x +c}}}{e}-\operatorname {Ei}_{1}\left (-\frac {e}{d x +c}\right )\right )}{d}\right )}{d}\) | \(356\) |
default | \(-\frac {e \left (a^{2} \left (-\frac {\left (d x +c \right ) {\mathrm e}^{\frac {e}{d x +c}}}{e}-\operatorname {Ei}_{1}\left (-\frac {e}{d x +c}\right )\right )+\frac {b^{2} e^{2} \left (-\frac {\left (d x +c \right )^{3} {\mathrm e}^{\frac {e}{d x +c}}}{3 e^{3}}-\frac {{\mathrm e}^{\frac {e}{d x +c}} \left (d x +c \right )^{2}}{6 e^{2}}-\frac {\left (d x +c \right ) {\mathrm e}^{\frac {e}{d x +c}}}{6 e}-\frac {\operatorname {Ei}_{1}\left (-\frac {e}{d x +c}\right )}{6}\right )}{d^{2}}+\frac {b^{2} c^{2} \left (-\frac {\left (d x +c \right ) {\mathrm e}^{\frac {e}{d x +c}}}{e}-\operatorname {Ei}_{1}\left (-\frac {e}{d x +c}\right )\right )}{d^{2}}+\frac {2 b e a \left (-\frac {{\mathrm e}^{\frac {e}{d x +c}} \left (d x +c \right )^{2}}{2 e^{2}}-\frac {\left (d x +c \right ) {\mathrm e}^{\frac {e}{d x +c}}}{2 e}-\frac {\operatorname {Ei}_{1}\left (-\frac {e}{d x +c}\right )}{2}\right )}{d}-\frac {2 b^{2} e c \left (-\frac {{\mathrm e}^{\frac {e}{d x +c}} \left (d x +c \right )^{2}}{2 e^{2}}-\frac {\left (d x +c \right ) {\mathrm e}^{\frac {e}{d x +c}}}{2 e}-\frac {\operatorname {Ei}_{1}\left (-\frac {e}{d x +c}\right )}{2}\right )}{d^{2}}-\frac {2 b c a \left (-\frac {\left (d x +c \right ) {\mathrm e}^{\frac {e}{d x +c}}}{e}-\operatorname {Ei}_{1}\left (-\frac {e}{d x +c}\right )\right )}{d}\right )}{d}\) | \(356\) |
risch | \(a b \,{\mathrm e}^{\frac {e}{d x +c}} x^{2}-\frac {a b \,{\mathrm e}^{\frac {e}{d x +c}} c^{2}}{d^{2}}+\frac {a^{2} {\mathrm e}^{\frac {e}{d x +c}} c}{d}+\frac {e \,b^{2} {\mathrm e}^{\frac {e}{d x +c}} x^{2}}{6 d}+a^{2} {\mathrm e}^{\frac {e}{d x +c}} x +\frac {b^{2} {\mathrm e}^{\frac {e}{d x +c}} x^{3}}{3}+\frac {b^{2} {\mathrm e}^{\frac {e}{d x +c}} c^{3}}{3 d^{3}}-\frac {2 e \,b^{2} {\mathrm e}^{\frac {e}{d x +c}} c x}{3 d^{2}}+\frac {e^{2} b^{2} {\mathrm e}^{\frac {e}{d x +c}} x}{6 d^{2}}+\frac {e^{2} b^{2} {\mathrm e}^{\frac {e}{d x +c}} c}{6 d^{3}}+\frac {e a b \,{\mathrm e}^{\frac {e}{d x +c}} c}{d^{2}}+\frac {e^{2} a b \,\operatorname {Ei}_{1}\left (-\frac {e}{d x +c}\right )}{d^{2}}-\frac {e^{2} c \,b^{2} \operatorname {Ei}_{1}\left (-\frac {e}{d x +c}\right )}{d^{3}}-\frac {2 e a b c \,\operatorname {Ei}_{1}\left (-\frac {e}{d x +c}\right )}{d^{2}}+\frac {e a b \,{\mathrm e}^{\frac {e}{d x +c}} x}{d}+\frac {e \,b^{2} c^{2} \operatorname {Ei}_{1}\left (-\frac {e}{d x +c}\right )}{d^{3}}+\frac {e \,a^{2} \operatorname {Ei}_{1}\left (-\frac {e}{d x +c}\right )}{d}+\frac {e^{3} b^{2} \operatorname {Ei}_{1}\left (-\frac {e}{d x +c}\right )}{6 d^{3}}-\frac {5 e \,b^{2} {\mathrm e}^{\frac {e}{d x +c}} c^{2}}{6 d^{3}}\) | \(387\) |
parts | \(b^{2} {\mathrm e}^{\frac {e}{d x +c}} x^{3}+2 a b \,{\mathrm e}^{\frac {e}{d x +c}} x^{2}+a^{2} {\mathrm e}^{\frac {e}{d x +c}} x +\frac {b^{2} {\mathrm e}^{\frac {e}{d x +c}} c \,x^{2}}{d}+\frac {2 a b \,{\mathrm e}^{\frac {e}{d x +c}} c x}{d}+\frac {a^{2} {\mathrm e}^{\frac {e}{d x +c}} c}{d}+\frac {e \,\operatorname {Ei}_{1}\left (-\frac {e}{d x +c}\right ) b^{2} x^{2}}{d}+\frac {2 e \,\operatorname {Ei}_{1}\left (-\frac {e}{d x +c}\right ) a b x}{d}+\frac {e \,a^{2} \operatorname {Ei}_{1}\left (-\frac {e}{d x +c}\right )}{d}-\frac {4 x^{3} {\mathrm e}^{\frac {e}{d x +c}} b^{2} c^{2} d^{3}+6 x^{2} {\mathrm e}^{\frac {e}{d x +c}} a b \,c^{2} d^{3}+6 x^{2} {\mathrm e}^{\frac {e}{d x +c}} b^{2} c^{3} d^{2}-x^{2} {\mathrm e}^{\frac {e}{d x +c}} b^{2} c^{2} d^{2} e +6 x^{2} \operatorname {Ei}_{1}\left (-\frac {e}{d x +c}\right ) b^{2} c^{2} d^{2} e +12 x \,{\mathrm e}^{\frac {e}{d x +c}} a b \,c^{3} d^{2}-6 x \,{\mathrm e}^{\frac {e}{d x +c}} a b \,c^{2} d^{2} e +4 x \,{\mathrm e}^{\frac {e}{d x +c}} b^{2} c^{3} d e -x \,{\mathrm e}^{\frac {e}{d x +c}} b^{2} c^{2} d \,e^{2}+12 x \,\operatorname {Ei}_{1}\left (-\frac {e}{d x +c}\right ) a b \,c^{2} d^{2} e +6 \,{\mathrm e}^{\frac {e}{d x +c}} a b \,c^{4} d -6 \,{\mathrm e}^{\frac {e}{d x +c}} a b \,c^{3} d e -2 \,{\mathrm e}^{\frac {e}{d x +c}} b^{2} c^{5}+5 \,{\mathrm e}^{\frac {e}{d x +c}} b^{2} c^{4} e -{\mathrm e}^{\frac {e}{d x +c}} b^{2} c^{3} e^{2}+12 \,\operatorname {Ei}_{1}\left (-\frac {e}{d x +c}\right ) a b \,c^{3} d e -6 \,\operatorname {Ei}_{1}\left (-\frac {e}{d x +c}\right ) a b \,c^{2} d \,e^{2}-6 \,\operatorname {Ei}_{1}\left (-\frac {e}{d x +c}\right ) b^{2} c^{4} e +6 \,\operatorname {Ei}_{1}\left (-\frac {e}{d x +c}\right ) b^{2} c^{3} e^{2}-\operatorname {Ei}_{1}\left (-\frac {e}{d x +c}\right ) b^{2} c^{2} e^{3}}{6 c^{2} d^{3}}\) | \(622\) |
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Time = 0.31 (sec) , antiderivative size = 197, normalized size of antiderivative = 0.77 \[ \int e^{\frac {e}{c+d x}} (a+b x)^2 \, dx=-\frac {{\left (b^{2} e^{3} - 6 \, {\left (b^{2} c - a b d\right )} e^{2} + 6 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} e\right )} {\rm Ei}\left (\frac {e}{d x + c}\right ) - {\left (2 \, b^{2} d^{3} x^{3} + 2 \, b^{2} c^{3} - 6 \, a b c^{2} d + 6 \, a^{2} c d^{2} + b^{2} c e^{2} + {\left (6 \, a b d^{3} + b^{2} d^{2} e\right )} x^{2} - {\left (5 \, b^{2} c^{2} - 6 \, a b c d\right )} e + {\left (6 \, a^{2} d^{3} + b^{2} d e^{2} - 2 \, {\left (2 \, b^{2} c d - 3 \, a b d^{2}\right )} e\right )} x\right )} e^{\left (\frac {e}{d x + c}\right )}}{6 \, d^{3}} \]
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\[ \int e^{\frac {e}{c+d x}} (a+b x)^2 \, dx=\int \left (a + b x\right )^{2} e^{\frac {e}{c + d x}}\, dx \]
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\[ \int e^{\frac {e}{c+d x}} (a+b x)^2 \, dx=\int { {\left (b x + a\right )}^{2} e^{\left (\frac {e}{d x + c}\right )} \,d x } \]
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Time = 0.44 (sec) , antiderivative size = 425, normalized size of antiderivative = 1.67 \[ \int e^{\frac {e}{c+d x}} (a+b x)^2 \, dx=-\frac {{\left (\frac {6 \, b^{2} c^{2} e^{5} {\rm Ei}\left (\frac {e}{d x + c}\right )}{{\left (d x + c\right )}^{3}} - \frac {12 \, a b c d e^{5} {\rm Ei}\left (\frac {e}{d x + c}\right )}{{\left (d x + c\right )}^{3}} + \frac {6 \, a^{2} d^{2} e^{5} {\rm Ei}\left (\frac {e}{d x + c}\right )}{{\left (d x + c\right )}^{3}} - \frac {6 \, b^{2} c e^{6} {\rm Ei}\left (\frac {e}{d x + c}\right )}{{\left (d x + c\right )}^{3}} + \frac {6 \, a b d e^{6} {\rm Ei}\left (\frac {e}{d x + c}\right )}{{\left (d x + c\right )}^{3}} + \frac {b^{2} e^{7} {\rm Ei}\left (\frac {e}{d x + c}\right )}{{\left (d x + c\right )}^{3}} - 2 \, b^{2} e^{4} e^{\left (\frac {e}{d x + c}\right )} + \frac {6 \, b^{2} c e^{4} e^{\left (\frac {e}{d x + c}\right )}}{d x + c} - \frac {6 \, b^{2} c^{2} e^{4} e^{\left (\frac {e}{d x + c}\right )}}{{\left (d x + c\right )}^{2}} - \frac {6 \, a b d e^{4} e^{\left (\frac {e}{d x + c}\right )}}{d x + c} + \frac {12 \, a b c d e^{4} e^{\left (\frac {e}{d x + c}\right )}}{{\left (d x + c\right )}^{2}} - \frac {6 \, a^{2} d^{2} e^{4} e^{\left (\frac {e}{d x + c}\right )}}{{\left (d x + c\right )}^{2}} - \frac {b^{2} e^{5} e^{\left (\frac {e}{d x + c}\right )}}{d x + c} + \frac {6 \, b^{2} c e^{5} e^{\left (\frac {e}{d x + c}\right )}}{{\left (d x + c\right )}^{2}} - \frac {6 \, a b d e^{5} e^{\left (\frac {e}{d x + c}\right )}}{{\left (d x + c\right )}^{2}} - \frac {b^{2} e^{6} e^{\left (\frac {e}{d x + c}\right )}}{{\left (d x + c\right )}^{2}}\right )} {\left (d x + c\right )}^{3}}{6 \, d^{3} e^{4}} \]
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Time = 0.73 (sec) , antiderivative size = 306, normalized size of antiderivative = 1.20 \[ \int e^{\frac {e}{c+d x}} (a+b x)^2 \, dx=\frac {x\,{\mathrm {e}}^{\frac {e}{c+d\,x}}\,\left (2\,a^2\,c+\frac {\frac {b^2\,c^3}{3}-d\,\left (a\,b\,c^2-2\,a\,b\,c\,e\right )+\frac {b^2\,c\,e^2}{3}-\frac {3\,b^2\,c^2\,e}{2}}{d^2}\right )+\frac {{\mathrm {e}}^{\frac {e}{c+d\,x}}\,\left (\frac {b^2\,c^4}{3}-d\,\left (a\,b\,c^3-a\,b\,c^2\,e\right )-\frac {5\,b^2\,c^3\,e}{6}+a^2\,c^2\,d^2+\frac {b^2\,c^2\,e^2}{6}\right )}{d^3}+x^2\,{\mathrm {e}}^{\frac {e}{c+d\,x}}\,\left (\frac {\frac {b^2\,e^2}{6}-\frac {b^2\,c\,e}{2}}{d}+a^2\,d+a\,b\,c+a\,b\,e\right )+\frac {b^2\,d\,x^4\,{\mathrm {e}}^{\frac {e}{c+d\,x}}}{3}+\frac {b\,x^3\,{\mathrm {e}}^{\frac {e}{c+d\,x}}\,\left (6\,a\,d+2\,b\,c+b\,e\right )}{6}}{c+d\,x}-\frac {\mathrm {ei}\left (\frac {e}{c+d\,x}\right )\,\left (\frac {b^2\,e^3}{6}+d\,\left (a\,b\,e^2-2\,a\,b\,c\,e\right )+a^2\,d^2\,e-b^2\,c\,e^2+b^2\,c^2\,e\right )}{d^3} \]
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