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

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

[Out]

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

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Rubi [A]  time = 0.19, antiderivative size = 206, normalized size of antiderivative = 1.00, 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 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 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]

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 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 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]

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*}

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Mathematica [A]  time = 0.19, size = 195, normalized size = 0.95 \[ \frac {3 b^2 e (b c-a d) \text {Ei}\left (\frac {e}{(c+d x)^3}\right )-3 b^2 (c+d x)^3 (b c-a d) e^{\frac {e}{(c+d x)^3}}+3 b (c+d x)^2 (b c-a d)^2 \left (-\frac {e}{(c+d x)^3}\right )^{2/3} \Gamma \left (-\frac {2}{3},-\frac {e}{(c+d x)^3}\right )-(c+d x) (b c-a d)^3 \sqrt [3]{-\frac {e}{(c+d x)^3}} \Gamma \left (-\frac {1}{3},-\frac {e}{(c+d x)^3}\right )+b^3 (c+d x)^4 \left (-\frac {e}{(c+d x)^3}\right )^{4/3} \Gamma \left (-\frac {4}{3},-\frac {e}{(c+d x)^3}\right )}{3 d^4} \]

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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fricas [A]  time = 0.49, size = 349, normalized size = 1.69 \[ \frac {4 \, {\left (b^{3} c - a b^{2} d\right )} e {\rm Ei}\left (\frac {e}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\right ) - 6 \, {\left (b^{3} c^{2} d^{2} - 2 \, a b^{2} c d^{3} + a^{2} b d^{4}\right )} \left (-\frac {e}{d^{3}}\right )^{\frac {2}{3}} \Gamma \left (\frac {1}{3}, -\frac {e}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\right ) + {\left (4 \, b^{3} c^{3} d - 12 \, a b^{2} c^{2} d^{2} + 12 \, a^{2} b c d^{3} - 4 \, a^{3} d^{4} - 3 \, b^{3} d e\right )} \left (-\frac {e}{d^{3}}\right )^{\frac {1}{3}} \Gamma \left (\frac {2}{3}, -\frac {e}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\right ) + {\left (b^{3} d^{4} x^{4} + 4 \, a b^{2} d^{4} x^{3} + 6 \, a^{2} b d^{4} x^{2} - b^{3} c^{4} + 4 \, a b^{2} c^{3} d - 6 \, a^{2} b c^{2} d^{2} + 4 \, a^{3} c d^{3} + 3 \, b^{3} c e + {\left (4 \, a^{3} d^{4} + 3 \, b^{3} d e\right )} x\right )} e^{\left (\frac {e}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\right )}}{4 \, d^{4}} \]

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

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)

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maple [F]  time = 0.13, size = 0, normalized size = 0.00 \[ \int \left (b x +a \right )^{3} {\mathrm e}^{\frac {e}{\left (d x +c \right )^{3}}}\, dx \]

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 = 0.00, size = 0, normalized size = 0.00 \[ \frac {{\left (b^{3} d^{3} x^{4} + 4 \, a b^{2} d^{3} x^{3} + 6 \, a^{2} b d^{3} x^{2} + {\left (4 \, a^{3} d^{3} + 3 \, b^{3} e\right )} x\right )} e^{\left (\frac {e}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\right )}}{4 \, d^{3}} + \int -\frac {3 \, {\left (b^{3} c^{4} e + 4 \, {\left (b^{3} c d^{3} e - a b^{2} d^{4} e\right )} x^{3} + 6 \, {\left (b^{3} c^{2} d^{2} e - a^{2} b d^{4} e\right )} x^{2} - {\left (4 \, a^{3} d^{4} e - {\left (4 \, c^{3} d e - 3 \, d e^{2}\right )} b^{3}\right )} x\right )} e^{\left (\frac {e}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\right )}}{4 \, {\left (d^{7} x^{4} + 4 \, c d^{6} x^{3} + 6 \, c^{2} d^{5} x^{2} + 4 \, c^{3} d^{4} x + c^{4} d^{3}\right )}}\,{d x} \]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(e/(c + d*x)^3)*(a + b*x)^3,x)

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a + b x\right )^{3} e^{\frac {e}{c^{3} + 3 c^{2} d x + 3 c d^{2} x^{2} + d^{3} x^{3}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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