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

Optimal. Leaf size=297 \[ -\frac{\sqrt{\pi } e^{-\frac{(b c-a d)^2}{4 b d}} (a d+b c)^3 \text{Erfi}\left (\frac{a d+b c+2 b d x}{2 \sqrt{b} \sqrt{d}}\right )}{16 b^{7/2} d^{7/2}}+\frac{3 \sqrt{\pi } e^{-\frac{(b c-a d)^2}{4 b d}} (a d+b c) \text{Erfi}\left (\frac{a d+b c+2 b d x}{2 \sqrt{b} \sqrt{d}}\right )}{8 b^{5/2} d^{5/2}}+\frac{(a d+b c)^2 e^{x (a d+b c)+a c+b d x^2}}{8 b^3 d^3}-\frac{x (a d+b c) e^{x (a d+b c)+a c+b d x^2}}{4 b^2 d^2}-\frac{e^{x (a d+b c)+a c+b d x^2}}{2 b^2 d^2}+\frac{x^2 e^{x (a d+b c)+a c+b d x^2}}{2 b d} \]

[Out]

-E^(a*c + (b*c + a*d)*x + b*d*x^2)/(2*b^2*d^2) + ((b*c + a*d)^2*E^(a*c + (b*c + a*d)*x + b*d*x^2))/(8*b^3*d^3)
- ((b*c + a*d)*E^(a*c + (b*c + a*d)*x + b*d*x^2)*x)/(4*b^2*d^2) + (E^(a*c + (b*c + a*d)*x + b*d*x^2)*x^2)/(2*
b*d) + (3*(b*c + a*d)*Sqrt[Pi]*Erfi[(b*c + a*d + 2*b*d*x)/(2*Sqrt[b]*Sqrt[d])])/(8*b^(5/2)*d^(5/2)*E^((b*c - a
*d)^2/(4*b*d))) - ((b*c + a*d)^3*Sqrt[Pi]*Erfi[(b*c + a*d + 2*b*d*x)/(2*Sqrt[b]*Sqrt[d])])/(16*b^(7/2)*d^(7/2)
*E^((b*c - a*d)^2/(4*b*d)))

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Rubi [A]  time = 0.63586, antiderivative size = 297, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 5, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.294, Rules used = {2244, 2241, 2240, 2234, 2204} \[ -\frac{\sqrt{\pi } e^{-\frac{(b c-a d)^2}{4 b d}} (a d+b c)^3 \text{Erfi}\left (\frac{a d+b c+2 b d x}{2 \sqrt{b} \sqrt{d}}\right )}{16 b^{7/2} d^{7/2}}+\frac{3 \sqrt{\pi } e^{-\frac{(b c-a d)^2}{4 b d}} (a d+b c) \text{Erfi}\left (\frac{a d+b c+2 b d x}{2 \sqrt{b} \sqrt{d}}\right )}{8 b^{5/2} d^{5/2}}+\frac{(a d+b c)^2 e^{x (a d+b c)+a c+b d x^2}}{8 b^3 d^3}-\frac{x (a d+b c) e^{x (a d+b c)+a c+b d x^2}}{4 b^2 d^2}-\frac{e^{x (a d+b c)+a c+b d x^2}}{2 b^2 d^2}+\frac{x^2 e^{x (a d+b c)+a c+b d x^2}}{2 b d} \]

Antiderivative was successfully verified.

[In]

Int[E^((a + b*x)*(c + d*x))*x^3,x]

[Out]

-E^(a*c + (b*c + a*d)*x + b*d*x^2)/(2*b^2*d^2) + ((b*c + a*d)^2*E^(a*c + (b*c + a*d)*x + b*d*x^2))/(8*b^3*d^3)
- ((b*c + a*d)*E^(a*c + (b*c + a*d)*x + b*d*x^2)*x)/(4*b^2*d^2) + (E^(a*c + (b*c + a*d)*x + b*d*x^2)*x^2)/(2*
b*d) + (3*(b*c + a*d)*Sqrt[Pi]*Erfi[(b*c + a*d + 2*b*d*x)/(2*Sqrt[b]*Sqrt[d])])/(8*b^(5/2)*d^(5/2)*E^((b*c - a
*d)^2/(4*b*d))) - ((b*c + a*d)^3*Sqrt[Pi]*Erfi[(b*c + a*d + 2*b*d*x)/(2*Sqrt[b]*Sqrt[d])])/(16*b^(7/2)*d^(7/2)
*E^((b*c - a*d)^2/(4*b*d)))

Rule 2244

Int[(F_)^(v_)*(u_)^(m_.), x_Symbol] :> Int[ExpandToSum[u, x]^m*F^ExpandToSum[v, x], x] /; FreeQ[{F, m}, x] &&
LinearQ[u, x] && QuadraticQ[v, x] &&  !(LinearMatchQ[u, x] && QuadraticMatchQ[v, x])

Rule 2241

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

Rule 2240

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

Rule 2234

Int[(F_)^((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[F^(a - b^2/(4*c)), Int[F^((b + 2*c*x)^2/(4*c))
, x], x] /; FreeQ[{F, a, b, c}, x]

Rule 2204

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[(F^a*Sqrt[Pi]*Erfi[(c + d*x)*Rt[b*Log[F], 2
]])/(2*d*Rt[b*Log[F], 2]), x] /; FreeQ[{F, a, b, c, d}, x] && PosQ[b]

Rubi steps

\begin{align*} \int e^{(a+b x) (c+d x)} x^3 \, dx &=\int e^{a c+(b c+a d) x+b d x^2} x^3 \, dx\\ &=\frac{e^{a c+(b c+a d) x+b d x^2} x^2}{2 b d}-\frac{\int e^{a c+(b c+a d) x+b d x^2} x \, dx}{b d}-\frac{(b c+a d) \int e^{a c+(b c+a d) x+b d x^2} x^2 \, dx}{2 b d}\\ &=-\frac{e^{a c+(b c+a d) x+b d x^2}}{2 b^2 d^2}-\frac{(b c+a d) e^{a c+(b c+a d) x+b d x^2} x}{4 b^2 d^2}+\frac{e^{a c+(b c+a d) x+b d x^2} x^2}{2 b d}+\frac{(b c+a d) \int e^{a c+(b c+a d) x+b d x^2} \, dx}{4 b^2 d^2}+\frac{(b c+a d) \int e^{a c+(b c+a d) x+b d x^2} \, dx}{2 b^2 d^2}+\frac{(b c+a d)^2 \int e^{a c+(b c+a d) x+b d x^2} x \, dx}{4 b^2 d^2}\\ &=-\frac{e^{a c+(b c+a d) x+b d x^2}}{2 b^2 d^2}+\frac{(b c+a d)^2 e^{a c+(b c+a d) x+b d x^2}}{8 b^3 d^3}-\frac{(b c+a d) e^{a c+(b c+a d) x+b d x^2} x}{4 b^2 d^2}+\frac{e^{a c+(b c+a d) x+b d x^2} x^2}{2 b d}-\frac{(b c+a d)^3 \int e^{a c+(b c+a d) x+b d x^2} \, dx}{8 b^3 d^3}+\frac{\left ((b c+a d) e^{-\frac{(b c-a d)^2}{4 b d}}\right ) \int e^{\frac{(b c+a d+2 b d x)^2}{4 b d}} \, dx}{4 b^2 d^2}+\frac{\left ((b c+a d) e^{-\frac{(b c-a d)^2}{4 b d}}\right ) \int e^{\frac{(b c+a d+2 b d x)^2}{4 b d}} \, dx}{2 b^2 d^2}\\ &=-\frac{e^{a c+(b c+a d) x+b d x^2}}{2 b^2 d^2}+\frac{(b c+a d)^2 e^{a c+(b c+a d) x+b d x^2}}{8 b^3 d^3}-\frac{(b c+a d) e^{a c+(b c+a d) x+b d x^2} x}{4 b^2 d^2}+\frac{e^{a c+(b c+a d) x+b d x^2} x^2}{2 b d}+\frac{3 (b c+a d) e^{-\frac{(b c-a d)^2}{4 b d}} \sqrt{\pi } \text{erfi}\left (\frac{b c+a d+2 b d x}{2 \sqrt{b} \sqrt{d}}\right )}{8 b^{5/2} d^{5/2}}-\frac{\left ((b c+a d)^3 e^{-\frac{(b c-a d)^2}{4 b d}}\right ) \int e^{\frac{(b c+a d+2 b d x)^2}{4 b d}} \, dx}{8 b^3 d^3}\\ &=-\frac{e^{a c+(b c+a d) x+b d x^2}}{2 b^2 d^2}+\frac{(b c+a d)^2 e^{a c+(b c+a d) x+b d x^2}}{8 b^3 d^3}-\frac{(b c+a d) e^{a c+(b c+a d) x+b d x^2} x}{4 b^2 d^2}+\frac{e^{a c+(b c+a d) x+b d x^2} x^2}{2 b d}+\frac{3 (b c+a d) e^{-\frac{(b c-a d)^2}{4 b d}} \sqrt{\pi } \text{erfi}\left (\frac{b c+a d+2 b d x}{2 \sqrt{b} \sqrt{d}}\right )}{8 b^{5/2} d^{5/2}}-\frac{(b c+a d)^3 e^{-\frac{(b c-a d)^2}{4 b d}} \sqrt{\pi } \text{erfi}\left (\frac{b c+a d+2 b d x}{2 \sqrt{b} \sqrt{d}}\right )}{16 b^{7/2} d^{7/2}}\\ \end{align*}

Mathematica [A]  time = 0.354298, size = 191, normalized size = 0.64 \[ \frac{e^{-\frac{(b c-a d)^2}{4 b d}} \left (2 \sqrt{b} \sqrt{d} e^{\frac{(a d+b (c+2 d x))^2}{4 b d}} \left (a^2 d^2-2 b d (-a c+a d x+2)+b^2 \left (c^2-2 c d x+4 d^2 x^2\right )\right )-\sqrt{\pi } \left (a^3 d^3+3 b^2 c d (a c-2)+3 a b d^2 (a c-2)+b^3 c^3\right ) \text{Erfi}\left (\frac{a d+b (c+2 d x)}{2 \sqrt{b} \sqrt{d}}\right )\right )}{16 b^{7/2} d^{7/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[b]*Sqrt[d]*E^((a*d + b*(c + 2*d*x))^2/(4*b*d))*(a^2*d^2 - 2*b*d*(2 - a*c + a*d*x) + b^2*(c^2 - 2*c*d*x
+ 4*d^2*x^2)) - (b^3*c^3 + 3*b^2*c*(-2 + a*c)*d + 3*a*b*(-2 + a*c)*d^2 + a^3*d^3)*Sqrt[Pi]*Erfi[(a*d + b*(c +
2*d*x))/(2*Sqrt[b]*Sqrt[d])])/(16*b^(7/2)*d^(7/2)*E^((b*c - a*d)^2/(4*b*d)))

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Maple [A]  time = 0.011, size = 368, normalized size = 1.2 \begin{align*}{\frac{{{\rm e}^{ac+ \left ( ad+bc \right ) x+bd{x}^{2}}}{x}^{2}}{2\,bd}}-{\frac{ad+bc}{2\,bd} \left ({\frac{{{\rm e}^{ac+ \left ( ad+bc \right ) x+bd{x}^{2}}}x}{2\,bd}}-{\frac{ad+bc}{2\,bd} \left ({\frac{{{\rm e}^{ac+ \left ( ad+bc \right ) x+bd{x}^{2}}}}{2\,bd}}+{\frac{ \left ( ad+bc \right ) \sqrt{\pi }}{4\,bd}{{\rm e}^{ac-{\frac{ \left ( ad+bc \right ) ^{2}}{4\,bd}}}}{\it Erf} \left ( -\sqrt{-bd}x+{\frac{ad+bc}{2}{\frac{1}{\sqrt{-bd}}}} \right ){\frac{1}{\sqrt{-bd}}}} \right ) }+{\frac{\sqrt{\pi }}{4\,bd}{{\rm e}^{ac-{\frac{ \left ( ad+bc \right ) ^{2}}{4\,bd}}}}{\it Erf} \left ( -\sqrt{-bd}x+{\frac{ad+bc}{2}{\frac{1}{\sqrt{-bd}}}} \right ){\frac{1}{\sqrt{-bd}}}} \right ) }-{\frac{1}{bd} \left ({\frac{{{\rm e}^{ac+ \left ( ad+bc \right ) x+bd{x}^{2}}}}{2\,bd}}+{\frac{ \left ( ad+bc \right ) \sqrt{\pi }}{4\,bd}{{\rm e}^{ac-{\frac{ \left ( ad+bc \right ) ^{2}}{4\,bd}}}}{\it Erf} \left ( -\sqrt{-bd}x+{\frac{ad+bc}{2}{\frac{1}{\sqrt{-bd}}}} \right ){\frac{1}{\sqrt{-bd}}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*exp(a*c+(a*d+b*c)*x+b*d*x^2)*x^2/b/d-1/2*(a*d+b*c)/b/d*(1/2*exp(a*c+(a*d+b*c)*x+b*d*x^2)*x/b/d-1/2*(a*d+b*
c)/b/d*(1/2*exp(a*c+(a*d+b*c)*x+b*d*x^2)/b/d+1/4*(a*d+b*c)/b/d*Pi^(1/2)*exp(a*c-1/4*(a*d+b*c)^2/b/d)/(-b*d)^(1
/2)*erf(-(-b*d)^(1/2)*x+1/2*(a*d+b*c)/(-b*d)^(1/2)))+1/4/b/d*Pi^(1/2)*exp(a*c-1/4*(a*d+b*c)^2/b/d)/(-b*d)^(1/2
)*erf(-(-b*d)^(1/2)*x+1/2*(a*d+b*c)/(-b*d)^(1/2)))-1/b/d*(1/2*exp(a*c+(a*d+b*c)*x+b*d*x^2)/b/d+1/4*(a*d+b*c)/b
/d*Pi^(1/2)*exp(a*c-1/4*(a*d+b*c)^2/b/d)/(-b*d)^(1/2)*erf(-(-b*d)^(1/2)*x+1/2*(a*d+b*c)/(-b*d)^(1/2)))

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Maxima [A]  time = 1.19841, size = 360, normalized size = 1.21 \begin{align*} -\frac{{\left (\frac{\sqrt{\pi }{\left (2 \, b d x + b c + a d\right )}{\left (b c + a d\right )}^{3}{\left (\operatorname{erf}\left (\frac{1}{2} \, \sqrt{-\frac{{\left (2 \, b d x + b c + a d\right )}^{2}}{b d}}\right ) - 1\right )}}{\left (b d\right )^{\frac{7}{2}} \sqrt{-\frac{{\left (2 \, b d x + b c + a d\right )}^{2}}{b d}}} - \frac{6 \,{\left (b c + a d\right )}^{2} b d e^{\left (\frac{{\left (2 \, b d x + b c + a d\right )}^{2}}{4 \, b d}\right )}}{\left (b d\right )^{\frac{7}{2}}} + \frac{8 \, b^{2} d^{2} \Gamma \left (2, -\frac{{\left (2 \, b d x + b c + a d\right )}^{2}}{4 \, b d}\right )}{\left (b d\right )^{\frac{7}{2}}} - \frac{12 \,{\left (2 \, b d x + b c + a d\right )}^{3}{\left (b c + a d\right )} \Gamma \left (\frac{3}{2}, -\frac{{\left (2 \, b d x + b c + a d\right )}^{2}}{4 \, b d}\right )}{\left (b d\right )^{\frac{7}{2}} \left (-\frac{{\left (2 \, b d x + b c + a d\right )}^{2}}{b d}\right )^{\frac{3}{2}}}\right )} e^{\left (a c - \frac{{\left (b c + a d\right )}^{2}}{4 \, b d}\right )}}{16 \, \sqrt{b d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/16*(sqrt(pi)*(2*b*d*x + b*c + a*d)*(b*c + a*d)^3*(erf(1/2*sqrt(-(2*b*d*x + b*c + a*d)^2/(b*d))) - 1)/((b*d)
^(7/2)*sqrt(-(2*b*d*x + b*c + a*d)^2/(b*d))) - 6*(b*c + a*d)^2*b*d*e^(1/4*(2*b*d*x + b*c + a*d)^2/(b*d))/(b*d)
^(7/2) + 8*b^2*d^2*gamma(2, -1/4*(2*b*d*x + b*c + a*d)^2/(b*d))/(b*d)^(7/2) - 12*(2*b*d*x + b*c + a*d)^3*(b*c
+ a*d)*gamma(3/2, -1/4*(2*b*d*x + b*c + a*d)^2/(b*d))/((b*d)^(7/2)*(-(2*b*d*x + b*c + a*d)^2/(b*d))^(3/2)))*e^
(a*c - 1/4*(b*c + a*d)^2/(b*d))/sqrt(b*d)

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Fricas [A]  time = 1.51075, size = 460, normalized size = 1.55 \begin{align*} \frac{\sqrt{\pi }{\left (b^{3} c^{3} + a^{3} d^{3} + 3 \,{\left (a^{2} b c - 2 \, a b\right )} d^{2} + 3 \,{\left (a b^{2} c^{2} - 2 \, b^{2} c\right )} d\right )} \sqrt{-b d} \operatorname{erf}\left (\frac{{\left (2 \, b d x + b c + a d\right )} \sqrt{-b d}}{2 \, b d}\right ) e^{\left (-\frac{b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}}{4 \, b d}\right )} + 2 \,{\left (4 \, b^{3} d^{3} x^{2} + b^{3} c^{2} d + a^{2} b d^{3} + 2 \,{\left (a b^{2} c - 2 \, b^{2}\right )} d^{2} - 2 \,{\left (b^{3} c d^{2} + a b^{2} d^{3}\right )} x\right )} e^{\left (b d x^{2} + a c +{\left (b c + a d\right )} x\right )}}{16 \, b^{4} d^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16*(sqrt(pi)*(b^3*c^3 + a^3*d^3 + 3*(a^2*b*c - 2*a*b)*d^2 + 3*(a*b^2*c^2 - 2*b^2*c)*d)*sqrt(-b*d)*erf(1/2*(2
*b*d*x + b*c + a*d)*sqrt(-b*d)/(b*d))*e^(-1/4*(b^2*c^2 - 2*a*b*c*d + a^2*d^2)/(b*d)) + 2*(4*b^3*d^3*x^2 + b^3*
c^2*d + a^2*b*d^3 + 2*(a*b^2*c - 2*b^2)*d^2 - 2*(b^3*c*d^2 + a*b^2*d^3)*x)*e^(b*d*x^2 + a*c + (b*c + a*d)*x))/
(b^4*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((b*x+a)*(d*x+c))*x**3,x)

[Out]

Timed out

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Giac [A]  time = 1.16414, size = 338, normalized size = 1.14 \begin{align*} \frac{\frac{\sqrt{\pi }{\left (b^{3} c^{3} + 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} + a^{3} d^{3} - 6 \, b^{2} c d - 6 \, a b d^{2}\right )} \operatorname{erf}\left (-\frac{1}{2} \, \sqrt{-b d}{\left (2 \, x + \frac{b c + a d}{b d}\right )}\right ) e^{\left (-\frac{b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}}{4 \, b d}\right )}}{\sqrt{-b d}} + 2 \,{\left (b^{2} d^{2}{\left (2 \, x + \frac{b c + a d}{b d}\right )}^{2} - 3 \, b^{2} c d{\left (2 \, x + \frac{b c + a d}{b d}\right )} - 3 \, a b d^{2}{\left (2 \, x + \frac{b c + a d}{b d}\right )} + 3 \, b^{2} c^{2} + 6 \, a b c d + 3 \, a^{2} d^{2} - 4 \, b d\right )} e^{\left (b d x^{2} + b c x + a d x + a c\right )}}{16 \, b^{3} d^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16*(sqrt(pi)*(b^3*c^3 + 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 + a^3*d^3 - 6*b^2*c*d - 6*a*b*d^2)*erf(-1/2*sqrt(-b*d)
*(2*x + (b*c + a*d)/(b*d)))*e^(-1/4*(b^2*c^2 - 2*a*b*c*d + a^2*d^2)/(b*d))/sqrt(-b*d) + 2*(b^2*d^2*(2*x + (b*c
+ a*d)/(b*d))^2 - 3*b^2*c*d*(2*x + (b*c + a*d)/(b*d)) - 3*a*b*d^2*(2*x + (b*c + a*d)/(b*d)) + 3*b^2*c^2 + 6*a
*b*c*d + 3*a^2*d^2 - 4*b*d)*e^(b*d*x^2 + b*c*x + a*d*x + a*c))/(b^3*d^3)