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

Optimal. Leaf size=206 \[ \frac{b (c+d x)^2 (b c-a d)^2 \left (-\frac{e}{(c+d x)^3}\right )^{2/3} \text{Gamma}\left (-\frac{2}{3},-\frac{e}{(c+d x)^3}\right )}{d^4}-\frac{(c+d x) (b c-a d)^3 \sqrt [3]{-\frac{e}{(c+d x)^3}} \text{Gamma}\left (-\frac{1}{3},-\frac{e}{(c+d x)^3}\right )}{3 d^4}+\frac{b^3 (c+d x)^4 \left (-\frac{e}{(c+d x)^3}\right )^{4/3} \text{Gamma}\left (-\frac{4}{3},-\frac{e}{(c+d x)^3}\right )}{3 d^4}+\frac{b^2 e (b c-a d) \text{Ei}\left (\frac{e}{(c+d x)^3}\right )}{d^4}-\frac{b^2 (c+d x)^3 (b c-a d) e^{\frac{e}{(c+d x)^3}}}{d^4} \]

[Out]

-((b^2*(b*c - a*d)*E^(e/(c + d*x)^3)*(c + d*x)^3)/d^4) + (b^2*(b*c - a*d)*e*ExpIntegralEi[e/(c + d*x)^3])/d^4
+ (b^3*(-(e/(c + d*x)^3))^(4/3)*(c + d*x)^4*Gamma[-4/3, -(e/(c + d*x)^3)])/(3*d^4) + (b*(b*c - a*d)^2*(-(e/(c
+ d*x)^3))^(2/3)*(c + d*x)^2*Gamma[-2/3, -(e/(c + d*x)^3)])/d^4 - ((b*c - a*d)^3*(-(e/(c + d*x)^3))^(1/3)*(c +
d*x)*Gamma[-1/3, -(e/(c + d*x)^3)])/(3*d^4)

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Rubi [A]  time = 0.189264, antiderivative size = 206, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263, Rules used = {2226, 2208, 2218, 2214, 2210} \[ \frac{b (c+d x)^2 (b c-a d)^2 \left (-\frac{e}{(c+d x)^3}\right )^{2/3} \text{Gamma}\left (-\frac{2}{3},-\frac{e}{(c+d x)^3}\right )}{d^4}-\frac{(c+d x) (b c-a d)^3 \sqrt [3]{-\frac{e}{(c+d x)^3}} \text{Gamma}\left (-\frac{1}{3},-\frac{e}{(c+d x)^3}\right )}{3 d^4}+\frac{b^3 (c+d x)^4 \left (-\frac{e}{(c+d x)^3}\right )^{4/3} \text{Gamma}\left (-\frac{4}{3},-\frac{e}{(c+d x)^3}\right )}{3 d^4}+\frac{b^2 e (b c-a d) \text{Ei}\left (\frac{e}{(c+d x)^3}\right )}{d^4}-\frac{b^2 (c+d x)^3 (b c-a d) e^{\frac{e}{(c+d x)^3}}}{d^4} \]

Antiderivative was successfully verified.

[In]

Int[E^(e/(c + d*x)^3)*(a + b*x)^3,x]

[Out]

-((b^2*(b*c - a*d)*E^(e/(c + d*x)^3)*(c + d*x)^3)/d^4) + (b^2*(b*c - a*d)*e*ExpIntegralEi[e/(c + d*x)^3])/d^4
+ (b^3*(-(e/(c + d*x)^3))^(4/3)*(c + d*x)^4*Gamma[-4/3, -(e/(c + d*x)^3)])/(3*d^4) + (b*(b*c - a*d)^2*(-(e/(c
+ d*x)^3))^(2/3)*(c + d*x)^2*Gamma[-2/3, -(e/(c + d*x)^3)])/d^4 - ((b*c - a*d)^3*(-(e/(c + d*x)^3))^(1/3)*(c +
d*x)*Gamma[-1/3, -(e/(c + d*x)^3)])/(3*d^4)

Rule 2226

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]

Rule 2208

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 2218

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

Rule 2214

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^(m
+ 1)*F^(a + b*(c + d*x)^n))/(d*(m + 1)), x] - Dist[(b*n*Log[F])/(m + 1), Int[(c + d*x)^(m + n)*F^(a + b*(c +
d*x)^n), x], x] /; FreeQ[{F, a, b, c, d}, x] && IntegerQ[(2*(m + 1))/n] && LtQ[-4, (m + 1)/n, 5] && IntegerQ[n
] && ((GtQ[n, 0] && LtQ[m, -1]) || (GtQ[-n, 0] && LeQ[-n, m + 1]))

Rule 2210

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]

Rubi steps

\begin{align*} \int e^{\frac{e}{(c+d x)^3}} (a+b x)^3 \, dx &=\int \left (\frac{(-b c+a d)^3 e^{\frac{e}{(c+d x)^3}}}{d^3}+\frac{3 b (b c-a d)^2 e^{\frac{e}{(c+d x)^3}} (c+d x)}{d^3}-\frac{3 b^2 (b c-a d) e^{\frac{e}{(c+d x)^3}} (c+d x)^2}{d^3}+\frac{b^3 e^{\frac{e}{(c+d x)^3}} (c+d x)^3}{d^3}\right ) \, dx\\ &=\frac{b^3 \int e^{\frac{e}{(c+d x)^3}} (c+d x)^3 \, dx}{d^3}-\frac{\left (3 b^2 (b c-a d)\right ) \int e^{\frac{e}{(c+d x)^3}} (c+d x)^2 \, dx}{d^3}+\frac{\left (3 b (b c-a d)^2\right ) \int e^{\frac{e}{(c+d x)^3}} (c+d x) \, dx}{d^3}-\frac{(b c-a d)^3 \int e^{\frac{e}{(c+d x)^3}} \, dx}{d^3}\\ &=-\frac{b^2 (b c-a d) e^{\frac{e}{(c+d x)^3}} (c+d x)^3}{d^4}+\frac{b^3 \left (-\frac{e}{(c+d x)^3}\right )^{4/3} (c+d x)^4 \Gamma \left (-\frac{4}{3},-\frac{e}{(c+d x)^3}\right )}{3 d^4}+\frac{b (b c-a d)^2 \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 )}{d^4}-\frac{(b c-a d)^3 \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^4}-\frac{\left (3 b^2 (b c-a d) e\right ) \int \frac{e^{\frac{e}{(c+d x)^3}}}{c+d x} \, dx}{d^3}\\ &=-\frac{b^2 (b c-a d) e^{\frac{e}{(c+d x)^3}} (c+d x)^3}{d^4}+\frac{b^2 (b c-a d) e \text{Ei}\left (\frac{e}{(c+d x)^3}\right )}{d^4}+\frac{b^3 \left (-\frac{e}{(c+d x)^3}\right )^{4/3} (c+d x)^4 \Gamma \left (-\frac{4}{3},-\frac{e}{(c+d x)^3}\right )}{3 d^4}+\frac{b (b c-a d)^2 \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 )}{d^4}-\frac{(b c-a d)^3 \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^4}\\ \end{align*}

Mathematica [A]  time = 0.145976, size = 195, normalized size = 0.95 \[ \frac{3 b (c+d x)^2 (b c-a d)^2 \left (-\frac{e}{(c+d x)^3}\right )^{2/3} \text{Gamma}\left (-\frac{2}{3},-\frac{e}{(c+d x)^3}\right )-(c+d x) (b c-a d)^3 \sqrt [3]{-\frac{e}{(c+d x)^3}} \text{Gamma}\left (-\frac{1}{3},-\frac{e}{(c+d x)^3}\right )+b^3 (c+d x)^4 \left (-\frac{e}{(c+d x)^3}\right )^{4/3} \text{Gamma}\left (-\frac{4}{3},-\frac{e}{(c+d x)^3}\right )+3 b^2 e (b c-a d) \text{Ei}\left (\frac{e}{(c+d x)^3}\right )-3 b^2 (c+d x)^3 (b c-a d) e^{\frac{e}{(c+d x)^3}}}{3 d^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*b^2*(b*c - a*d)*E^(e/(c + d*x)^3)*(c + d*x)^3 + 3*b^2*(b*c - a*d)*e*ExpIntegralEi[e/(c + d*x)^3] + b^3*(-(
e/(c + d*x)^3))^(4/3)*(c + d*x)^4*Gamma[-4/3, -(e/(c + d*x)^3)] + 3*b*(b*c - a*d)^2*(-(e/(c + d*x)^3))^(2/3)*(
c + d*x)^2*Gamma[-2/3, -(e/(c + d*x)^3)] - (b*c - a*d)^3*(-(e/(c + d*x)^3))^(1/3)*(c + d*x)*Gamma[-1/3, -(e/(c
+ d*x)^3)])/(3*d^4)

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Maple [F]  time = 0.098, size = 0, normalized size = 0. \begin{align*} \int{{\rm e}^{{\frac{e}{ \left ( dx+c \right ) ^{3}}}}} \left ( bx+a \right ) ^{3}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{{\left (b^{3} d^{3} x^{4} + 4 \, a b^{2} d^{3} x^{3} + 6 \, a^{2} b d^{3} x^{2} +{\left (4 \, a^{3} d^{3} + 3 \, b^{3} e\right )} x\right )} e^{\left (\frac{e}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\right )}}{4 \, d^{3}} + \int -\frac{3 \,{\left (b^{3} c^{4} e + 4 \,{\left (b^{3} c d^{3} e - a b^{2} d^{4} e\right )} x^{3} + 6 \,{\left (b^{3} c^{2} d^{2} e - a^{2} b d^{4} e\right )} x^{2} -{\left (4 \, a^{3} d^{4} e -{\left (4 \, c^{3} d e - 3 \, d e^{2}\right )} b^{3}\right )} x\right )} e^{\left (\frac{e}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\right )}}{4 \,{\left (d^{7} x^{4} + 4 \, c d^{6} x^{3} + 6 \, c^{2} d^{5} x^{2} + 4 \, c^{3} d^{4} x + c^{4} d^{3}\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(b^3*d^3*x^4 + 4*a*b^2*d^3*x^3 + 6*a^2*b*d^3*x^2 + (4*a^3*d^3 + 3*b^3*e)*x)*e^(e/(d^3*x^3 + 3*c*d^2*x^2 +
3*c^2*d*x + c^3))/d^3 + integrate(-3/4*(b^3*c^4*e + 4*(b^3*c*d^3*e - a*b^2*d^4*e)*x^3 + 6*(b^3*c^2*d^2*e - a^2
*b*d^4*e)*x^2 - (4*a^3*d^4*e - (4*c^3*d*e - 3*d*e^2)*b^3)*x)*e^(e/(d^3*x^3 + 3*c*d^2*x^2 + 3*c^2*d*x + c^3))/(
d^7*x^4 + 4*c*d^6*x^3 + 6*c^2*d^5*x^2 + 4*c^3*d^4*x + c^4*d^3), x)

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Fricas [A]  time = 1.64772, size = 736, normalized size = 3.57 \begin{align*} \frac{4 \,{\left (b^{3} c - a b^{2} d\right )} e{\rm Ei}\left (\frac{e}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\right ) - 6 \,{\left (b^{3} c^{2} d^{2} - 2 \, a b^{2} c d^{3} + a^{2} b d^{4}\right )} \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 ) +{\left (4 \, b^{3} c^{3} d - 12 \, a b^{2} c^{2} d^{2} + 12 \, a^{2} b c d^{3} - 4 \, a^{3} d^{4} - 3 \, b^{3} d e\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^{3} d^{4} x^{4} + 4 \, a b^{2} d^{4} x^{3} + 6 \, a^{2} b d^{4} x^{2} - b^{3} c^{4} + 4 \, a b^{2} c^{3} d - 6 \, a^{2} b c^{2} d^{2} + 4 \, a^{3} c d^{3} + 3 \, b^{3} c e +{\left (4 \, a^{3} d^{4} + 3 \, b^{3} d e\right )} x\right )} e^{\left (\frac{e}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\right )}}{4 \, d^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(4*(b^3*c - a*b^2*d)*e*Ei(e/(d^3*x^3 + 3*c*d^2*x^2 + 3*c^2*d*x + c^3)) - 6*(b^3*c^2*d^2 - 2*a*b^2*c*d^3 +
a^2*b*d^4)*(-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)) + (4*b^3*c^3*d - 12*a*b^2*c
^2*d^2 + 12*a^2*b*c*d^3 - 4*a^3*d^4 - 3*b^3*d*e)*(-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^3*d^4*x^4 + 4*a*b^2*d^4*x^3 + 6*a^2*b*d^4*x^2 - b^3*c^4 + 4*a*b^2*c^3*d - 6*a^2*b*c^2*d^2 + 4*
a^3*c*d^3 + 3*b^3*c*e + (4*a^3*d^4 + 3*b^3*d*e)*x)*e^(e/(d^3*x^3 + 3*c*d^2*x^2 + 3*c^2*d*x + c^3)))/d^4

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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