3.5.8 \(\int \frac {e^{\frac {e}{c+d x}}}{(a+b x)^3} \, dx\) [408]

Optimal. Leaf size=240 \[ \frac {d^2 e^{\frac {e}{c+d x}}}{2 b (b c-a d)^2}+\frac {d^2 e e^{\frac {e}{c+d x}}}{2 (b c-a d)^3}-\frac {e^{\frac {e}{c+d x}}}{2 b (a+b x)^2}+\frac {d e e^{\frac {e}{c+d x}}}{2 (b c-a d)^2 (a+b x)}+\frac {d^2 e e^{\frac {b e}{b c-a d}} \text {Ei}\left (-\frac {d e (a+b x)}{(b c-a d) (c+d x)}\right )}{(b c-a d)^3}+\frac {b d^2 e^2 e^{\frac {b e}{b c-a d}} \text {Ei}\left (-\frac {d e (a+b x)}{(b c-a d) (c+d x)}\right )}{2 (b c-a d)^4} \]

[Out]

1/2*d^2*exp(e/(d*x+c))/b/(-a*d+b*c)^2+1/2*d^2*e*exp(e/(d*x+c))/(-a*d+b*c)^3-1/2*exp(e/(d*x+c))/b/(b*x+a)^2+1/2
*d*e*exp(e/(d*x+c))/(-a*d+b*c)^2/(b*x+a)+d^2*e*exp(b*e/(-a*d+b*c))*Ei(-d*e*(b*x+a)/(-a*d+b*c)/(d*x+c))/(-a*d+b
*c)^3+1/2*b*d^2*e^2*exp(b*e/(-a*d+b*c))*Ei(-d*e*(b*x+a)/(-a*d+b*c)/(d*x+c))/(-a*d+b*c)^4

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Rubi [A]
time = 0.71, antiderivative size = 240, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 7, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {2255, 6874, 2254, 2241, 2260, 2209, 2240} \begin {gather*} \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}+\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 {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} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^(e/(c + d*x))/(a + b*x)^3,x]

[Out]

(d^2*E^(e/(c + d*x)))/(2*b*(b*c - a*d)^2) + (d^2*e*E^(e/(c + d*x)))/(2*(b*c - a*d)^3) - E^(e/(c + d*x))/(2*b*(
a + b*x)^2) + (d*e*E^(e/(c + d*x)))/(2*(b*c - a*d)^2*(a + b*x)) + (d^2*e*E^((b*e)/(b*c - a*d))*ExpIntegralEi[-
((d*e*(a + b*x))/((b*c - a*d)*(c + d*x)))])/(b*c - a*d)^3 + (b*d^2*e^2*E^((b*e)/(b*c - a*d))*ExpIntegralEi[-((
d*e*(a + b*x))/((b*c - a*d)*(c + d*x)))])/(2*(b*c - a*d)^4)

Rule 2209

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[(F^(g*(e - c*(f/d)))/d)*ExpInteg
ralEi[f*g*(c + d*x)*(Log[F]/d)], x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !TrueQ[$UseGamma]

Rule 2240

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Simp[(e + f*x)^n*(
F^(a + b*(c + d*x)^n)/(b*f*n*(c + d*x)^n*Log[F])), x] /; FreeQ[{F, a, b, c, d, e, f, n}, x] && EqQ[m, n - 1] &
& EqQ[d*e - c*f, 0]

Rule 2241

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> Simp[F^a*(ExpIntegralEi[
b*(c + d*x)^n*Log[F]]/(f*n)), x] /; FreeQ[{F, a, b, c, d, e, f, n}, x] && EqQ[d*e - c*f, 0]

Rule 2254

Int[(F_)^((a_.) + (b_.)/((c_.) + (d_.)*(x_)))/((e_.) + (f_.)*(x_)), x_Symbol] :> Dist[d/f, Int[F^(a + b/(c + d
*x))/(c + d*x), x], x] - Dist[(d*e - c*f)/f, Int[F^(a + b/(c + d*x))/((c + d*x)*(e + f*x)), x], x] /; FreeQ[{F
, a, b, c, d, e, f}, x] && NeQ[d*e - c*f, 0]

Rule 2255

Int[(F_)^((a_.) + (b_.)/((c_.) + (d_.)*(x_)))*((e_.) + (f_.)*(x_))^(m_), x_Symbol] :> Simp[(e + f*x)^(m + 1)*(
F^(a + b/(c + d*x))/(f*(m + 1))), x] + Dist[b*d*(Log[F]/(f*(m + 1))), Int[(e + f*x)^(m + 1)*(F^(a + b/(c + d*x
))/(c + d*x)^2), x], x] /; FreeQ[{F, a, b, c, d, e, f}, x] && NeQ[d*e - c*f, 0] && ILtQ[m, -1]

Rule 2260

Int[(F_)^((a_.) + (b_.)/((c_.) + (d_.)*(x_)))/(((e_.) + (f_.)*(x_))*((g_.) + (h_.)*(x_))), x_Symbol] :> Dist[-
d/(f*(d*g - c*h)), Subst[Int[F^(a - b*(h/(d*g - c*h)) + d*b*(x/(d*g - c*h)))/x, x], x, (g + h*x)/(c + d*x)], x
] /; FreeQ[{F, a, b, c, d, e, f}, x] && EqQ[d*e - c*f, 0]

Rule 6874

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin {align*} \int \frac {e^{\frac {e}{c+d x}}}{(a+b x)^3} \, dx &=-\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*}

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Mathematica [F]
time = 0.25, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {e^{\frac {e}{c+d x}}}{(a+b x)^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[E^(e/(c + d*x))/(a + b*x)^3,x]

[Out]

Integrate[E^(e/(c + d*x))/(a + b*x)^3, x]

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Maple [A]
time = 0.08, size = 240, normalized size = 1.00

method	result	size
derivativedivides	\(-\frac {e \left (\frac {d^{3} \left (-\frac {{\mathrm e}^{\frac {e}{d x +c}}}{\frac {e}{d x +c}+\frac {b e}{a d -c b}}-{\mathrm e}^{-\frac {b e}{a d -c b}} \expIntegral \left (1, -\frac {e}{d x +c}-\frac {b e}{a d -c b}\right )\right )}{\left (a d -c b \right )^{3}}-\frac {b e \,d^{3} \left (-\frac {{\mathrm e}^{\frac {e}{d x +c}}}{2 \left (\frac {e}{d x +c}+\frac {b e}{a d -c b}\right )^{2}}-\frac {{\mathrm e}^{\frac {e}{d x +c}}}{2 \left (\frac {e}{d x +c}+\frac {b e}{a d -c b}\right )}-\frac {{\mathrm e}^{-\frac {b e}{a d -c b}} \expIntegral \left (1, -\frac {e}{d x +c}-\frac {b e}{a d -c b}\right )}{2}\right )}{\left (a d -c b \right )^{4}}\right )}{d}\)	\(240\)
default	\(-\frac {e \left (\frac {d^{3} \left (-\frac {{\mathrm e}^{\frac {e}{d x +c}}}{\frac {e}{d x +c}+\frac {b e}{a d -c b}}-{\mathrm e}^{-\frac {b e}{a d -c b}} \expIntegral \left (1, -\frac {e}{d x +c}-\frac {b e}{a d -c b}\right )\right )}{\left (a d -c b \right )^{3}}-\frac {b e \,d^{3} \left (-\frac {{\mathrm e}^{\frac {e}{d x +c}}}{2 \left (\frac {e}{d x +c}+\frac {b e}{a d -c b}\right )^{2}}-\frac {{\mathrm e}^{\frac {e}{d x +c}}}{2 \left (\frac {e}{d x +c}+\frac {b e}{a d -c b}\right )}-\frac {{\mathrm e}^{-\frac {b e}{a d -c b}} \expIntegral \left (1, -\frac {e}{d x +c}-\frac {b e}{a d -c b}\right )}{2}\right )}{\left (a d -c b \right )^{4}}\right )}{d}\)	\(240\)
risch	\(\frac {e \,d^{2} {\mathrm e}^{\frac {e}{d x +c}}}{\left (a d -c b \right )^{3} \left (\frac {e}{d x +c}+\frac {b e}{a d -c b}\right )}+\frac {e \,d^{2} {\mathrm e}^{-\frac {b e}{a d -c b}} \expIntegral \left (1, -\frac {e}{d x +c}-\frac {b e}{a d -c b}\right )}{\left (a d -c b \right )^{3}}-\frac {e^{2} d^{2} b \,{\mathrm e}^{\frac {e}{d x +c}}}{2 \left (a d -c b \right )^{4} \left (\frac {e}{d x +c}+\frac {b e}{a d -c b}\right )^{2}}-\frac {e^{2} d^{2} b \,{\mathrm e}^{\frac {e}{d x +c}}}{2 \left (a d -c b \right )^{4} \left (\frac {e}{d x +c}+\frac {b e}{a d -c b}\right )}-\frac {e^{2} d^{2} b \,{\mathrm e}^{-\frac {b e}{a d -c b}} \expIntegral \left (1, -\frac {e}{d x +c}-\frac {b e}{a d -c b}\right )}{2 \left (a d -c b \right )^{4}}\)	\(278\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(e/(d*x+c))/(b*x+a)^3,x,method=_RETURNVERBOSE)

[Out]

-1/d*e*(d^3/(a*d-b*c)^3*(-exp(e/(d*x+c))/(e/(d*x+c)+b*e/(a*d-b*c))-exp(-b*e/(a*d-b*c))*Ei(1,-e/(d*x+c)-b*e/(a*
d-b*c)))-b*e/(a*d-b*c)^4*d^3*(-1/2*exp(e/(d*x+c))/(e/(d*x+c)+b*e/(a*d-b*c))^2-1/2*exp(e/(d*x+c))/(e/(d*x+c)+b*
e/(a*d-b*c))-1/2*exp(-b*e/(a*d-b*c))*Ei(1,-e/(d*x+c)-b*e/(a*d-b*c))))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(e/(d*x+c))/(b*x+a)^3,x, algorithm="maxima")

[Out]

integrate(e^(e/(d*x + c))/(b*x + a)^3, x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 513 vs. \(2 (234) = 468\).
time = 0.38, size = 513, normalized size = 2.14 \begin {gather*} \frac {{\left ({\left (b^{3} d^{2} x^{2} + 2 \, a b^{2} d^{2} x + a^{2} b d^{2}\right )} e^{2} + 2 \, {\left (a^{2} b c d^{2} - a^{3} d^{3} + {\left (b^{3} c d^{2} - a b^{2} d^{3}\right )} x^{2} + 2 \, {\left (a b^{2} c d^{2} - a^{2} b d^{3}\right )} x\right )} e\right )} {\rm Ei}\left (-\frac {{\left (b d x + a d\right )} 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}\right )} x^{2} - 2 \, {\left (a b^{2} c^{2} d^{2} - 2 \, a^{2} b c d^{3} + a^{3} d^{4}\right )} x - {\left (a b^{2} c^{2} d - a^{2} b c d^{2} + {\left (b^{3} c d^{2} - a b^{2} d^{3}\right )} x^{2} + {\left (b^{3} c^{2} d - a^{2} b d^{3}\right )} x\right )} e\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 )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(e/(d*x+c))/(b*x+a)^3,x, algorithm="fricas")

[Out]

1/2*(((b^3*d^2*x^2 + 2*a*b^2*d^2*x + a^2*b*d^2)*e^2 + 2*(a^2*b*c*d^2 - a^3*d^3 + (b^3*c*d^2 - a*b^2*d^3)*x^2 +
2*(a*b^2*c*d^2 - a^2*b*d^3)*x)*e)*Ei(-(b*d*x + a*d)*e/(b*c^2 - a*c*d + (b*c*d - a*d^2)*x))*e^(b*e/(b*c - a*d)
) - (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 - (b^3*c^2*d^2 - 2*a*b^2*c*d^3 + a^2*b*d^4)*x^2 -
2*(a*b^2*c^2*d^2 - 2*a^2*b*c*d^3 + a^3*d^4)*x - (a*b^2*c^2*d - a^2*b*c*d^2 + (b^3*c*d^2 - a*b^2*d^3)*x^2 + (b
^3*c^2*d - a^2*b*d^3)*x)*e)*e^(e/(d*x + c)))/(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 + (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)*x^2 + 2*(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)*x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {e^{\frac {e}{c + d x}}}{\left (a + b x\right )^{3}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(e/(d*x+c))/(b*x+a)**3,x)

[Out]

Integral(exp(e/(c + d*x))/(a + b*x)**3, x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1759 vs. \(2 (234) = 468\).
time = 2.56, size = 1759, normalized size = 7.33 \begin {gather*} \frac {{\left (2 \, b^{3} c d {\rm Ei}\left (-\frac {b e - \frac {b c e}{d x + c} + \frac {a d e}{d x + c}}{b c - a d}\right ) e^{\left (\frac {b e}{b c - a d} + 4\right )} - \frac {4 \, b^{3} c^{2} d {\rm Ei}\left (-\frac {b e - \frac {b c e}{d x + c} + \frac {a d e}{d x + c}}{b c - a d}\right ) e^{\left (\frac {b e}{b c - a d} + 4\right )}}{d x + c} + \frac {2 \, b^{3} c^{3} d {\rm Ei}\left (-\frac {b e - \frac {b c e}{d x + c} + \frac {a d e}{d x + c}}{b c - a d}\right ) e^{\left (\frac {b e}{b c - a d} + 4\right )}}{{\left (d x + c\right )}^{2}} - 2 \, a b^{2} d^{2} {\rm Ei}\left (-\frac {b e - \frac {b c e}{d x + c} + \frac {a d e}{d x + c}}{b c - a d}\right ) e^{\left (\frac {b e}{b c - a d} + 4\right )} + \frac {8 \, a b^{2} c d^{2} {\rm Ei}\left (-\frac {b e - \frac {b c e}{d x + c} + \frac {a d e}{d x + c}}{b c - a d}\right ) e^{\left (\frac {b e}{b c - a d} + 4\right )}}{d x + c} - \frac {6 \, a b^{2} c^{2} d^{2} {\rm Ei}\left (-\frac {b e - \frac {b c e}{d x + c} + \frac {a d e}{d x + c}}{b c - a d}\right ) e^{\left (\frac {b e}{b c - a d} + 4\right )}}{{\left (d x + c\right )}^{2}} - \frac {4 \, a^{2} b d^{3} {\rm Ei}\left (-\frac {b e - \frac {b c e}{d x + c} + \frac {a d e}{d x + c}}{b c - a d}\right ) e^{\left (\frac {b e}{b c - a d} + 4\right )}}{d x + c} + \frac {6 \, a^{2} b c d^{3} {\rm Ei}\left (-\frac {b e - \frac {b c e}{d x + c} + \frac {a d e}{d x + c}}{b c - a d}\right ) e^{\left (\frac {b e}{b c - a d} + 4\right )}}{{\left (d x + c\right )}^{2}} - \frac {2 \, a^{3} d^{4} {\rm Ei}\left (-\frac {b e - \frac {b c e}{d x + c} + \frac {a d e}{d x + c}}{b c - a d}\right ) e^{\left (\frac {b e}{b c - a d} + 4\right )}}{{\left (d x + c\right )}^{2}} + b^{3} c^{2} d e^{\left (\frac {e}{d x + c} + 3\right )} - \frac {2 \, b^{3} c^{3} d e^{\left (\frac {e}{d x + c} + 3\right )}}{d x + c} - 2 \, a b^{2} c d^{2} e^{\left (\frac {e}{d x + c} + 3\right )} + \frac {6 \, a b^{2} c^{2} d^{2} e^{\left (\frac {e}{d x + c} + 3\right )}}{d x + c} + a^{2} b d^{3} e^{\left (\frac {e}{d x + c} + 3\right )} - \frac {6 \, a^{2} b c d^{3} e^{\left (\frac {e}{d x + c} + 3\right )}}{d x + c} + \frac {2 \, a^{3} d^{4} e^{\left (\frac {e}{d x + c} + 3\right )}}{d x + c} + b^{3} d {\rm Ei}\left (-\frac {b e - \frac {b c e}{d x + c} + \frac {a d e}{d x + c}}{b c - a d}\right ) e^{\left (\frac {b e}{b c - a d} + 5\right )} - \frac {2 \, b^{3} c d {\rm Ei}\left (-\frac {b e - \frac {b c e}{d x + c} + \frac {a d e}{d x + c}}{b c - a d}\right ) e^{\left (\frac {b e}{b c - a d} + 5\right )}}{d x + c} + \frac {b^{3} c^{2} d {\rm Ei}\left (-\frac {b e - \frac {b c e}{d x + c} + \frac {a d e}{d x + c}}{b c - a d}\right ) e^{\left (\frac {b e}{b c - a d} + 5\right )}}{{\left (d x + c\right )}^{2}} + \frac {2 \, a b^{2} d^{2} {\rm Ei}\left (-\frac {b e - \frac {b c e}{d x + c} + \frac {a d e}{d x + c}}{b c - a d}\right ) e^{\left (\frac {b e}{b c - a d} + 5\right )}}{d x + c} - \frac {2 \, a b^{2} c d^{2} {\rm Ei}\left (-\frac {b e - \frac {b c e}{d x + c} + \frac {a d e}{d x + c}}{b c - a d}\right ) e^{\left (\frac {b e}{b c - a d} + 5\right )}}{{\left (d x + c\right )}^{2}} + \frac {a^{2} b d^{3} {\rm Ei}\left (-\frac {b e - \frac {b c e}{d x + c} + \frac {a d e}{d x + c}}{b c - a d}\right ) e^{\left (\frac {b e}{b c - a d} + 5\right )}}{{\left (d x + c\right )}^{2}} + b^{3} c d e^{\left (\frac {e}{d x + c} + 4\right )} - \frac {b^{3} c^{2} d e^{\left (\frac {e}{d x + c} + 4\right )}}{d x + c} - a b^{2} d^{2} e^{\left (\frac {e}{d x + c} + 4\right )} + \frac {2 \, a b^{2} c d^{2} e^{\left (\frac {e}{d x + c} + 4\right )}}{d x + c} - \frac {a^{2} b d^{3} e^{\left (\frac {e}{d x + c} + 4\right )}}{d x + c}\right )} d e^{\left (-1\right )}}{2 \, {\left (b^{6} c^{4} e^{2} - \frac {2 \, b^{6} c^{5} e^{2}}{d x + c} + \frac {b^{6} c^{6} e^{2}}{{\left (d x + c\right )}^{2}} - 4 \, a b^{5} c^{3} d e^{2} + \frac {10 \, a b^{5} c^{4} d e^{2}}{d x + c} - \frac {6 \, a b^{5} c^{5} d e^{2}}{{\left (d x + c\right )}^{2}} + 6 \, a^{2} b^{4} c^{2} d^{2} e^{2} - \frac {20 \, a^{2} b^{4} c^{3} d^{2} e^{2}}{d x + c} + \frac {15 \, a^{2} b^{4} c^{4} d^{2} e^{2}}{{\left (d x + c\right )}^{2}} - 4 \, a^{3} b^{3} c d^{3} e^{2} + \frac {20 \, a^{3} b^{3} c^{2} d^{3} e^{2}}{d x + c} - \frac {20 \, a^{3} b^{3} c^{3} d^{3} e^{2}}{{\left (d x + c\right )}^{2}} + a^{4} b^{2} d^{4} e^{2} - \frac {10 \, a^{4} b^{2} c d^{4} e^{2}}{d x + c} + \frac {15 \, a^{4} b^{2} c^{2} d^{4} e^{2}}{{\left (d x + c\right )}^{2}} + \frac {2 \, a^{5} b d^{5} e^{2}}{d x + c} - \frac {6 \, a^{5} b c d^{5} e^{2}}{{\left (d x + c\right )}^{2}} + \frac {a^{6} d^{6} e^{2}}{{\left (d x + c\right )}^{2}}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(e/(d*x+c))/(b*x+a)^3,x, algorithm="giac")

[Out]

1/2*(2*b^3*c*d*Ei(-(b*e - b*c*e/(d*x + c) + a*d*e/(d*x + c))/(b*c - a*d))*e^(b*e/(b*c - a*d) + 4) - 4*b^3*c^2*
d*Ei(-(b*e - b*c*e/(d*x + c) + a*d*e/(d*x + c))/(b*c - a*d))*e^(b*e/(b*c - a*d) + 4)/(d*x + c) + 2*b^3*c^3*d*E
i(-(b*e - b*c*e/(d*x + c) + a*d*e/(d*x + c))/(b*c - a*d))*e^(b*e/(b*c - a*d) + 4)/(d*x + c)^2 - 2*a*b^2*d^2*Ei
(-(b*e - b*c*e/(d*x + c) + a*d*e/(d*x + c))/(b*c - a*d))*e^(b*e/(b*c - a*d) + 4) + 8*a*b^2*c*d^2*Ei(-(b*e - b*
c*e/(d*x + c) + a*d*e/(d*x + c))/(b*c - a*d))*e^(b*e/(b*c - a*d) + 4)/(d*x + c) - 6*a*b^2*c^2*d^2*Ei(-(b*e - b
*c*e/(d*x + c) + a*d*e/(d*x + c))/(b*c - a*d))*e^(b*e/(b*c - a*d) + 4)/(d*x + c)^2 - 4*a^2*b*d^3*Ei(-(b*e - b*
c*e/(d*x + c) + a*d*e/(d*x + c))/(b*c - a*d))*e^(b*e/(b*c - a*d) + 4)/(d*x + c) + 6*a^2*b*c*d^3*Ei(-(b*e - b*c
*e/(d*x + c) + a*d*e/(d*x + c))/(b*c - a*d))*e^(b*e/(b*c - a*d) + 4)/(d*x + c)^2 - 2*a^3*d^4*Ei(-(b*e - b*c*e/
(d*x + c) + a*d*e/(d*x + c))/(b*c - a*d))*e^(b*e/(b*c - a*d) + 4)/(d*x + c)^2 + b^3*c^2*d*e^(e/(d*x + c) + 3)
- 2*b^3*c^3*d*e^(e/(d*x + c) + 3)/(d*x + c) - 2*a*b^2*c*d^2*e^(e/(d*x + c) + 3) + 6*a*b^2*c^2*d^2*e^(e/(d*x +
c) + 3)/(d*x + c) + a^2*b*d^3*e^(e/(d*x + c) + 3) - 6*a^2*b*c*d^3*e^(e/(d*x + c) + 3)/(d*x + c) + 2*a^3*d^4*e^
(e/(d*x + c) + 3)/(d*x + c) + b^3*d*Ei(-(b*e - b*c*e/(d*x + c) + a*d*e/(d*x + c))/(b*c - a*d))*e^(b*e/(b*c - a
*d) + 5) - 2*b^3*c*d*Ei(-(b*e - b*c*e/(d*x + c) + a*d*e/(d*x + c))/(b*c - a*d))*e^(b*e/(b*c - a*d) + 5)/(d*x +
c) + b^3*c^2*d*Ei(-(b*e - b*c*e/(d*x + c) + a*d*e/(d*x + c))/(b*c - a*d))*e^(b*e/(b*c - a*d) + 5)/(d*x + c)^2
+ 2*a*b^2*d^2*Ei(-(b*e - b*c*e/(d*x + c) + a*d*e/(d*x + c))/(b*c - a*d))*e^(b*e/(b*c - a*d) + 5)/(d*x + c) -
2*a*b^2*c*d^2*Ei(-(b*e - b*c*e/(d*x + c) + a*d*e/(d*x + c))/(b*c - a*d))*e^(b*e/(b*c - a*d) + 5)/(d*x + c)^2 +
a^2*b*d^3*Ei(-(b*e - b*c*e/(d*x + c) + a*d*e/(d*x + c))/(b*c - a*d))*e^(b*e/(b*c - a*d) + 5)/(d*x + c)^2 + b^
3*c*d*e^(e/(d*x + c) + 4) - b^3*c^2*d*e^(e/(d*x + c) + 4)/(d*x + c) - a*b^2*d^2*e^(e/(d*x + c) + 4) + 2*a*b^2*
c*d^2*e^(e/(d*x + c) + 4)/(d*x + c) - a^2*b*d^3*e^(e/(d*x + c) + 4)/(d*x + c))*d*e^(-1)/(b^6*c^4*e^2 - 2*b^6*c
^5*e^2/(d*x + c) + b^6*c^6*e^2/(d*x + c)^2 - 4*a*b^5*c^3*d*e^2 + 10*a*b^5*c^4*d*e^2/(d*x + c) - 6*a*b^5*c^5*d*
e^2/(d*x + c)^2 + 6*a^2*b^4*c^2*d^2*e^2 - 20*a^2*b^4*c^3*d^2*e^2/(d*x + c) + 15*a^2*b^4*c^4*d^2*e^2/(d*x + c)^
2 - 4*a^3*b^3*c*d^3*e^2 + 20*a^3*b^3*c^2*d^3*e^2/(d*x + c) - 20*a^3*b^3*c^3*d^3*e^2/(d*x + c)^2 + a^4*b^2*d^4*
e^2 - 10*a^4*b^2*c*d^4*e^2/(d*x + c) + 15*a^4*b^2*c^2*d^4*e^2/(d*x + c)^2 + 2*a^5*b*d^5*e^2/(d*x + c) - 6*a^5*
b*c*d^5*e^2/(d*x + c)^2 + a^6*d^6*e^2/(d*x + c)^2)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\mathrm {e}}^{\frac {e}{c+d\,x}}}{{\left (a+b\,x\right )}^3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(e/(c + d*x))/(a + b*x)^3,x)

[Out]

int(exp(e/(c + d*x))/(a + b*x)^3, x)

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