Integrand size = 19, antiderivative size = 240 \[ \int \frac {e^{\frac {e}{c+d x}}}{(a+b x)^3} \, dx=\frac {d^2 e^{\frac {e}{c+d x}}}{2 b (b c-a d)^2}+\frac {d^2 e e^{\frac {e}{c+d x}}}{2 (b c-a d)^3}-\frac {e^{\frac {e}{c+d x}}}{2 b (a+b x)^2}+\frac {d e e^{\frac {e}{c+d x}}}{2 (b c-a d)^2 (a+b x)}+\frac {d^2 e e^{\frac {b e}{b c-a d}} \operatorname {ExpIntegralEi}\left (-\frac {d e (a+b x)}{(b c-a d) (c+d x)}\right )}{(b c-a d)^3}+\frac {b d^2 e^2 e^{\frac {b e}{b c-a d}} \operatorname {ExpIntegralEi}\left (-\frac {d e (a+b x)}{(b c-a d) (c+d x)}\right )}{2 (b c-a d)^4} \]
[Out]
Time = 0.64 (sec) , antiderivative size = 240, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {2255, 6874, 2254, 2241, 2260, 2209, 2240} \[ \int \frac {e^{\frac {e}{c+d x}}}{(a+b x)^3} \, dx=\frac {b d^2 e^2 e^{\frac {b e}{b c-a d}} \operatorname {ExpIntegralEi}\left (-\frac {d e (a+b x)}{(b c-a d) (c+d x)}\right )}{2 (b c-a d)^4}+\frac {d^2 e e^{\frac {b e}{b c-a d}} \operatorname {ExpIntegralEi}\left (-\frac {d e (a+b x)}{(b c-a d) (c+d x)}\right )}{(b c-a d)^3}+\frac {d^2 e e^{\frac {e}{c+d x}}}{2 (b c-a d)^3}+\frac {d^2 e^{\frac {e}{c+d x}}}{2 b (b c-a d)^2}+\frac {d e e^{\frac {e}{c+d x}}}{2 (a+b x) (b c-a d)^2}-\frac {e^{\frac {e}{c+d x}}}{2 b (a+b x)^2} \]
[In]
[Out]
Rule 2209
Rule 2240
Rule 2241
Rule 2254
Rule 2255
Rule 2260
Rule 6874
Rubi steps \begin{align*} \text {integral}& = -\frac {e^{\frac {e}{c+d x}}}{2 b (a+b x)^2}-\frac {(d e) \int \frac {e^{\frac {e}{c+d x}}}{(a+b x)^2 (c+d x)^2} \, dx}{2 b} \\ & = -\frac {e^{\frac {e}{c+d x}}}{2 b (a+b x)^2}-\frac {(d e) \int \left (\frac {b^2 e^{\frac {e}{c+d x}}}{(b c-a d)^2 (a+b x)^2}-\frac {2 b^2 d e^{\frac {e}{c+d x}}}{(b c-a d)^3 (a+b x)}+\frac {d^2 e^{\frac {e}{c+d x}}}{(b c-a d)^2 (c+d x)^2}+\frac {2 b d^2 e^{\frac {e}{c+d x}}}{(b c-a d)^3 (c+d x)}\right ) \, dx}{2 b} \\ & = -\frac {e^{\frac {e}{c+d x}}}{2 b (a+b x)^2}+\frac {\left (b d^2 e\right ) \int \frac {e^{\frac {e}{c+d x}}}{a+b x} \, dx}{(b c-a d)^3}-\frac {\left (d^3 e\right ) \int \frac {e^{\frac {e}{c+d x}}}{c+d x} \, dx}{(b c-a d)^3}-\frac {(b d e) \int \frac {e^{\frac {e}{c+d x}}}{(a+b x)^2} \, dx}{2 (b c-a d)^2}-\frac {\left (d^3 e\right ) \int \frac {e^{\frac {e}{c+d x}}}{(c+d x)^2} \, dx}{2 b (b c-a d)^2} \\ & = \frac {d^2 e^{\frac {e}{c+d x}}}{2 b (b c-a d)^2}-\frac {e^{\frac {e}{c+d x}}}{2 b (a+b x)^2}+\frac {d e e^{\frac {e}{c+d x}}}{2 (b c-a d)^2 (a+b x)}+\frac {d^2 e \text {Ei}\left (\frac {e}{c+d x}\right )}{(b c-a d)^3}+\frac {\left (d^3 e\right ) \int \frac {e^{\frac {e}{c+d x}}}{c+d x} \, dx}{(b c-a d)^3}+\frac {\left (d^2 e\right ) \int \frac {e^{\frac {e}{c+d x}}}{(a+b x) (c+d x)} \, dx}{(b c-a d)^2}+\frac {\left (d^2 e^2\right ) \int \frac {e^{\frac {e}{c+d x}}}{(a+b x) (c+d x)^2} \, dx}{2 (b c-a d)^2} \\ & = \frac {d^2 e^{\frac {e}{c+d x}}}{2 b (b c-a d)^2}-\frac {e^{\frac {e}{c+d x}}}{2 b (a+b x)^2}+\frac {d e e^{\frac {e}{c+d x}}}{2 (b c-a d)^2 (a+b x)}+\frac {\left (d^2 e\right ) \text {Subst}\left (\int \frac {\exp \left (-\frac {b e}{-b c+a d}+\frac {d e x}{-b c+a d}\right )}{x} \, dx,x,\frac {a+b x}{c+d x}\right )}{(b c-a d)^3}+\frac {\left (d^2 e^2\right ) \int \left (\frac {b^2 e^{\frac {e}{c+d x}}}{(b c-a d)^2 (a+b x)}-\frac {d e^{\frac {e}{c+d x}}}{(b c-a d) (c+d x)^2}-\frac {b d e^{\frac {e}{c+d x}}}{(b c-a d)^2 (c+d x)}\right ) \, dx}{2 (b c-a d)^2} \\ & = \frac {d^2 e^{\frac {e}{c+d x}}}{2 b (b c-a d)^2}-\frac {e^{\frac {e}{c+d x}}}{2 b (a+b x)^2}+\frac {d e e^{\frac {e}{c+d x}}}{2 (b c-a d)^2 (a+b x)}+\frac {d^2 e e^{\frac {b e}{b c-a d}} \text {Ei}\left (-\frac {d e (a+b x)}{(b c-a d) (c+d x)}\right )}{(b c-a d)^3}+\frac {\left (b^2 d^2 e^2\right ) \int \frac {e^{\frac {e}{c+d x}}}{a+b x} \, dx}{2 (b c-a d)^4}-\frac {\left (b d^3 e^2\right ) \int \frac {e^{\frac {e}{c+d x}}}{c+d x} \, dx}{2 (b c-a d)^4}-\frac {\left (d^3 e^2\right ) \int \frac {e^{\frac {e}{c+d x}}}{(c+d x)^2} \, dx}{2 (b c-a d)^3} \\ & = \frac {d^2 e^{\frac {e}{c+d x}}}{2 b (b c-a d)^2}+\frac {d^2 e e^{\frac {e}{c+d x}}}{2 (b c-a d)^3}-\frac {e^{\frac {e}{c+d x}}}{2 b (a+b x)^2}+\frac {d e e^{\frac {e}{c+d x}}}{2 (b c-a d)^2 (a+b x)}+\frac {b d^2 e^2 \text {Ei}\left (\frac {e}{c+d x}\right )}{2 (b c-a d)^4}+\frac {d^2 e e^{\frac {b e}{b c-a d}} \text {Ei}\left (-\frac {d e (a+b x)}{(b c-a d) (c+d x)}\right )}{(b c-a d)^3}+\frac {\left (b d^3 e^2\right ) \int \frac {e^{\frac {e}{c+d x}}}{c+d x} \, dx}{2 (b c-a d)^4}+\frac {\left (b d^2 e^2\right ) \int \frac {e^{\frac {e}{c+d x}}}{(a+b x) (c+d x)} \, dx}{2 (b c-a d)^3} \\ & = \frac {d^2 e^{\frac {e}{c+d x}}}{2 b (b c-a d)^2}+\frac {d^2 e e^{\frac {e}{c+d x}}}{2 (b c-a d)^3}-\frac {e^{\frac {e}{c+d x}}}{2 b (a+b x)^2}+\frac {d e e^{\frac {e}{c+d x}}}{2 (b c-a d)^2 (a+b x)}+\frac {d^2 e e^{\frac {b e}{b c-a d}} \text {Ei}\left (-\frac {d e (a+b x)}{(b c-a d) (c+d x)}\right )}{(b c-a d)^3}+\frac {\left (b d^2 e^2\right ) \text {Subst}\left (\int \frac {\exp \left (-\frac {b e}{-b c+a d}+\frac {d e x}{-b c+a d}\right )}{x} \, dx,x,\frac {a+b x}{c+d x}\right )}{2 (b c-a d)^4} \\ & = \frac {d^2 e^{\frac {e}{c+d x}}}{2 b (b c-a d)^2}+\frac {d^2 e e^{\frac {e}{c+d x}}}{2 (b c-a d)^3}-\frac {e^{\frac {e}{c+d x}}}{2 b (a+b x)^2}+\frac {d e e^{\frac {e}{c+d x}}}{2 (b c-a d)^2 (a+b x)}+\frac {d^2 e e^{\frac {b e}{b c-a d}} \text {Ei}\left (-\frac {d e (a+b x)}{(b c-a d) (c+d x)}\right )}{(b c-a d)^3}+\frac {b d^2 e^2 e^{\frac {b e}{b c-a d}} \text {Ei}\left (-\frac {d e (a+b x)}{(b c-a d) (c+d x)}\right )}{2 (b c-a d)^4} \\ \end{align*}
\[ \int \frac {e^{\frac {e}{c+d x}}}{(a+b x)^3} \, dx=\int \frac {e^{\frac {e}{c+d x}}}{(a+b x)^3} \, dx \]
[In]
[Out]
Time = 0.38 (sec) , antiderivative size = 240, normalized size of antiderivative = 1.00
method | result | size |
derivativedivides | \(-\frac {e \left (\frac {d^{3} \left (-\frac {{\mathrm e}^{\frac {e}{d x +c}}}{\frac {e}{d x +c}+\frac {b e}{a d -c b}}-{\mathrm e}^{-\frac {b e}{a d -c b}} \operatorname {Ei}_{1}\left (-\frac {e}{d x +c}-\frac {b e}{a d -c b}\right )\right )}{\left (a d -c b \right )^{3}}-\frac {b e \,d^{3} \left (-\frac {{\mathrm e}^{\frac {e}{d x +c}}}{2 \left (\frac {e}{d x +c}+\frac {b e}{a d -c b}\right )^{2}}-\frac {{\mathrm e}^{\frac {e}{d x +c}}}{2 \left (\frac {e}{d x +c}+\frac {b e}{a d -c b}\right )}-\frac {{\mathrm e}^{-\frac {b e}{a d -c b}} \operatorname {Ei}_{1}\left (-\frac {e}{d x +c}-\frac {b e}{a d -c b}\right )}{2}\right )}{\left (a d -c b \right )^{4}}\right )}{d}\) | \(240\) |
default | \(-\frac {e \left (\frac {d^{3} \left (-\frac {{\mathrm e}^{\frac {e}{d x +c}}}{\frac {e}{d x +c}+\frac {b e}{a d -c b}}-{\mathrm e}^{-\frac {b e}{a d -c b}} \operatorname {Ei}_{1}\left (-\frac {e}{d x +c}-\frac {b e}{a d -c b}\right )\right )}{\left (a d -c b \right )^{3}}-\frac {b e \,d^{3} \left (-\frac {{\mathrm e}^{\frac {e}{d x +c}}}{2 \left (\frac {e}{d x +c}+\frac {b e}{a d -c b}\right )^{2}}-\frac {{\mathrm e}^{\frac {e}{d x +c}}}{2 \left (\frac {e}{d x +c}+\frac {b e}{a d -c b}\right )}-\frac {{\mathrm e}^{-\frac {b e}{a d -c b}} \operatorname {Ei}_{1}\left (-\frac {e}{d x +c}-\frac {b e}{a d -c b}\right )}{2}\right )}{\left (a d -c b \right )^{4}}\right )}{d}\) | \(240\) |
risch | \(\frac {e \,d^{2} {\mathrm e}^{\frac {e}{d x +c}}}{\left (a d -c b \right )^{3} \left (\frac {e}{d x +c}+\frac {b e}{a d -c b}\right )}+\frac {e \,d^{2} {\mathrm e}^{-\frac {b e}{a d -c b}} \operatorname {Ei}_{1}\left (-\frac {e}{d x +c}-\frac {b e}{a d -c b}\right )}{\left (a d -c b \right )^{3}}-\frac {e^{2} d^{2} b \,{\mathrm e}^{\frac {e}{d x +c}}}{2 \left (a d -c b \right )^{4} \left (\frac {e}{d x +c}+\frac {b e}{a d -c b}\right )^{2}}-\frac {e^{2} d^{2} b \,{\mathrm e}^{\frac {e}{d x +c}}}{2 \left (a d -c b \right )^{4} \left (\frac {e}{d x +c}+\frac {b e}{a d -c b}\right )}-\frac {e^{2} d^{2} b \,{\mathrm e}^{-\frac {b e}{a d -c b}} \operatorname {Ei}_{1}\left (-\frac {e}{d x +c}-\frac {b e}{a d -c b}\right )}{2 \left (a d -c b \right )^{4}}\) | \(278\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 517 vs. \(2 (224) = 448\).
Time = 0.29 (sec) , antiderivative size = 517, normalized size of antiderivative = 2.15 \[ \int \frac {e^{\frac {e}{c+d x}}}{(a+b x)^3} \, dx=\frac {{\left (a^{2} b d^{2} e^{2} + {\left (b^{3} d^{2} e^{2} + 2 \, {\left (b^{3} c d^{2} - a b^{2} d^{3}\right )} e\right )} x^{2} + 2 \, {\left (a^{2} b c d^{2} - a^{3} d^{3}\right )} e + 2 \, {\left (a b^{2} d^{2} e^{2} + 2 \, {\left (a b^{2} c d^{2} - a^{2} b d^{3}\right )} e\right )} x\right )} {\rm Ei}\left (-\frac {b d e x + a d e}{b c^{2} - a c d + {\left (b c d - a d^{2}\right )} x}\right ) e^{\left (\frac {b e}{b c - a d}\right )} - {\left (b^{3} c^{4} - 4 \, a b^{2} c^{3} d + 5 \, a^{2} b c^{2} d^{2} - 2 \, a^{3} c d^{3} - {\left (b^{3} c^{2} d^{2} - 2 \, a b^{2} c d^{3} + a^{2} b d^{4} + {\left (b^{3} c d^{2} - a b^{2} d^{3}\right )} e\right )} x^{2} - {\left (a b^{2} c^{2} d - a^{2} b c d^{2}\right )} e - {\left (2 \, a b^{2} c^{2} d^{2} - 4 \, a^{2} b c d^{3} + 2 \, a^{3} d^{4} + {\left (b^{3} c^{2} d - a^{2} b d^{3}\right )} e\right )} x\right )} e^{\left (\frac {e}{d x + c}\right )}}{2 \, {\left (a^{2} b^{4} c^{4} - 4 \, a^{3} b^{3} c^{3} d + 6 \, a^{4} b^{2} c^{2} d^{2} - 4 \, a^{5} b c d^{3} + a^{6} d^{4} + {\left (b^{6} c^{4} - 4 \, a b^{5} c^{3} d + 6 \, a^{2} b^{4} c^{2} d^{2} - 4 \, a^{3} b^{3} c d^{3} + a^{4} b^{2} d^{4}\right )} x^{2} + 2 \, {\left (a b^{5} c^{4} - 4 \, a^{2} b^{4} c^{3} d + 6 \, a^{3} b^{3} c^{2} d^{2} - 4 \, a^{4} b^{2} c d^{3} + a^{5} b d^{4}\right )} x\right )}} \]
[In]
[Out]
\[ \int \frac {e^{\frac {e}{c+d x}}}{(a+b x)^3} \, dx=\int \frac {e^{\frac {e}{c + d x}}}{\left (a + b x\right )^{3}}\, dx \]
[In]
[Out]
\[ \int \frac {e^{\frac {e}{c+d x}}}{(a+b x)^3} \, dx=\int { \frac {e^{\left (\frac {e}{d x + c}\right )}}{{\left (b x + a\right )}^{3}} \,d x } \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 1733 vs. \(2 (224) = 448\).
Time = 0.34 (sec) , antiderivative size = 1733, normalized size of antiderivative = 7.22 \[ \int \frac {e^{\frac {e}{c+d x}}}{(a+b x)^3} \, dx=\text {Too large to display} \]
[In]
[Out]
Timed out. \[ \int \frac {e^{\frac {e}{c+d x}}}{(a+b x)^3} \, dx=\int \frac {{\mathrm {e}}^{\frac {e}{c+d\,x}}}{{\left (a+b\,x\right )}^3} \,d x \]
[In]
[Out]