\(\int e^{\frac {e}{(c+d x)^3}} (a+b x) \, dx\) [418]

Optimal result

Integrand size = 17, antiderivative size = 92 \[ \int e^{\frac {e}{(c+d x)^3}} (a+b x) \, dx=\frac {b \left (-\frac {e}{(c+d x)^3}\right )^{2/3} (c+d x)^2 \Gamma \left (-\frac {2}{3},-\frac {e}{(c+d x)^3}\right )}{3 d^2}-\frac {(b c-a d) \sqrt [3]{-\frac {e}{(c+d x)^3}} (c+d x) \Gamma \left (-\frac {1}{3},-\frac {e}{(c+d x)^3}\right )}{3 d^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {2258, 2239, 2250} \[ \int e^{\frac {e}{(c+d x)^3}} (a+b x) \, dx=\frac {b (c+d x)^2 \left (-\frac {e}{(c+d x)^3}\right )^{2/3} \Gamma \left (-\frac {2}{3},-\frac {e}{(c+d x)^3}\right )}{3 d^2}-\frac {(c+d x) (b c-a d) \sqrt [3]{-\frac {e}{(c+d x)^3}} \Gamma \left (-\frac {1}{3},-\frac {e}{(c+d x)^3}\right )}{3 d^2} \]

[In]

Int[E^(e/(c + d*x)^3)*(a + b*x),x]

[Out]

(b*(-(e/(c + d*x)^3))^(2/3)*(c + d*x)^2*Gamma[-2/3, -(e/(c + d*x)^3)])/(3*d^2) - ((b*c - a*d)*(-(e/(c + d*x)^3
))^(1/3)*(c + d*x)*Gamma[-1/3, -(e/(c + d*x)^3)])/(3*d^2)

Rule 2239

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_)), x_Symbol] :> Simp[(-F^a)*(c + d*x)*(Gamma[1/n, (-b)*(c + d
*x)^n*Log[F]]/(d*n*((-b)*(c + d*x)^n*Log[F])^(1/n))), x] /; FreeQ[{F, a, b, c, d, n}, x] &&  !IntegerQ[2/n]

Rule 2250

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

Rule 2258

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*(u_), x_Symbol] :> Int[ExpandLinearProduct[F^(a + b*(c + d*
x)^n), u, c, d, x], x] /; FreeQ[{F, a, b, c, d, n}, x] && PolynomialQ[u, x]

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {(-b c+a d) e^{\frac {e}{(c+d x)^3}}}{d}+\frac {b e^{\frac {e}{(c+d x)^3}} (c+d x)}{d}\right ) \, dx \\ & = \frac {b \int e^{\frac {e}{(c+d x)^3}} (c+d x) \, dx}{d}+\frac {(-b c+a d) \int e^{\frac {e}{(c+d x)^3}} \, dx}{d} \\ & = \frac {b \left (-\frac {e}{(c+d x)^3}\right )^{2/3} (c+d x)^2 \Gamma \left (-\frac {2}{3},-\frac {e}{(c+d x)^3}\right )}{3 d^2}-\frac {(b c-a d) \sqrt [3]{-\frac {e}{(c+d x)^3}} (c+d x) \Gamma \left (-\frac {1}{3},-\frac {e}{(c+d x)^3}\right )}{3 d^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.92 \[ \int e^{\frac {e}{(c+d x)^3}} (a+b x) \, dx=\frac {(c+d x) \left (b \left (-\frac {e}{(c+d x)^3}\right )^{2/3} (c+d x) \Gamma \left (-\frac {2}{3},-\frac {e}{(c+d x)^3}\right )+(-b c+a d) \sqrt [3]{-\frac {e}{(c+d x)^3}} \Gamma \left (-\frac {1}{3},-\frac {e}{(c+d x)^3}\right )\right )}{3 d^2} \]

[In]

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

[Out]

((c + d*x)*(b*(-(e/(c + d*x)^3))^(2/3)*(c + d*x)*Gamma[-2/3, -(e/(c + d*x)^3)] + (-(b*c) + a*d)*(-(e/(c + d*x)
^3))^(1/3)*Gamma[-1/3, -(e/(c + d*x)^3)]))/(3*d^2)

Maple [F]

\[\int {\mathrm e}^{\frac {e}{\left (d x +c \right )^{3}}} \left (b x +a \right )d x\]

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 169 vs. \(2 (80) = 160\).

Time = 0.09 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.84 \[ \int e^{\frac {e}{(c+d x)^3}} (a+b x) \, dx=-\frac {b d^{2} \left (-\frac {e}{d^{3}}\right )^{\frac {2}{3}} \Gamma \left (\frac {1}{3}, -\frac {e}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\right ) - 2 \, {\left (b c d - a d^{2}\right )} \left (-\frac {e}{d^{3}}\right )^{\frac {1}{3}} \Gamma \left (\frac {2}{3}, -\frac {e}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\right ) - {\left (b d^{2} x^{2} + 2 \, a d^{2} x - b c^{2} + 2 \, a c d\right )} e^{\left (\frac {e}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\right )}}{2 \, d^{2}} \]

[In]

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

[Out]

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

Sympy [F]

\[ \int e^{\frac {e}{(c+d x)^3}} (a+b x) \, dx=\int \left (a + b x\right ) e^{\frac {e}{c^{3} + 3 c^{2} d x + 3 c d^{2} x^{2} + d^{3} x^{3}}}\, dx \]

[In]

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

[Out]

Integral((a + b*x)*exp(e/(c**3 + 3*c**2*d*x + 3*c*d**2*x**2 + d**3*x**3)), x)

Maxima [F]

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

[In]

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

[Out]

1/2*(b*x^2 + 2*a*x)*e^(e/(d^3*x^3 + 3*c*d^2*x^2 + 3*c^2*d*x + c^3)) + integrate(3/2*(b*d*e*x^2 + 2*a*d*e*x)*e^
(e/(d^3*x^3 + 3*c*d^2*x^2 + 3*c^2*d*x + c^3))/(d^4*x^4 + 4*c*d^3*x^3 + 6*c^2*d^2*x^2 + 4*c^3*d*x + c^4), x)

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int e^{\frac {e}{(c+d x)^3}} (a+b x) \, dx=\int {\mathrm {e}}^{\frac {e}{{\left (c+d\,x\right )}^3}}\,\left (a+b\,x\right ) \,d x \]

[In]

int(exp(e/(c + d*x)^3)*(a + b*x),x)

[Out]

int(exp(e/(c + d*x)^3)*(a + b*x), x)