3.1.66 \(\int (a+b e^x)^3 \sqrt {c+d x} \, dx\) [66]

Optimal. Leaf size=224 \[ 3 a^2 b e^x \sqrt {c+d x}+\frac {3}{2} a b^2 e^{2 x} \sqrt {c+d x}+\frac {1}{3} b^3 e^{3 x} \sqrt {c+d x}+\frac {2 a^3 (c+d x)^{3/2}}{3 d}-\frac {3}{2} a^2 b \sqrt {d} e^{-\frac {c}{d}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {c+d x}}{\sqrt {d}}\right )-\frac {3}{4} a b^2 \sqrt {d} e^{-\frac {2 c}{d}} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {c+d x}}{\sqrt {d}}\right )-\frac {1}{6} b^3 \sqrt {d} e^{-\frac {3 c}{d}} \sqrt {\frac {\pi }{3}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {c+d x}}{\sqrt {d}}\right ) \]

[Out]

2/3*a^3*(d*x+c)^(3/2)/d-1/18*b^3*erfi(3^(1/2)*(d*x+c)^(1/2)/d^(1/2))*d^(1/2)*3^(1/2)*Pi^(1/2)/exp(3*c/d)-3/8*a
*b^2*erfi(2^(1/2)*(d*x+c)^(1/2)/d^(1/2))*d^(1/2)*2^(1/2)*Pi^(1/2)/exp(2*c/d)-3/2*a^2*b*erfi((d*x+c)^(1/2)/d^(1
/2))*d^(1/2)*Pi^(1/2)/exp(c/d)+3*a^2*b*exp(x)*(d*x+c)^(1/2)+3/2*a*b^2*exp(2*x)*(d*x+c)^(1/2)+1/3*b^3*exp(3*x)*
(d*x+c)^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.21, antiderivative size = 224, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {2214, 2207, 2211, 2235} \begin {gather*} \frac {2 a^3 (c+d x)^{3/2}}{3 d}-\frac {3}{2} \sqrt {\pi } a^2 b \sqrt {d} e^{-\frac {c}{d}} \text {Erfi}\left (\frac {\sqrt {c+d x}}{\sqrt {d}}\right )+3 a^2 b e^x \sqrt {c+d x}-\frac {3}{4} \sqrt {\frac {\pi }{2}} a b^2 \sqrt {d} e^{-\frac {2 c}{d}} \text {Erfi}\left (\frac {\sqrt {2} \sqrt {c+d x}}{\sqrt {d}}\right )+\frac {3}{2} a b^2 e^{2 x} \sqrt {c+d x}-\frac {1}{6} \sqrt {\frac {\pi }{3}} b^3 \sqrt {d} e^{-\frac {3 c}{d}} \text {Erfi}\left (\frac {\sqrt {3} \sqrt {c+d x}}{\sqrt {d}}\right )+\frac {1}{3} b^3 e^{3 x} \sqrt {c+d x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*E^x)^3*Sqrt[c + d*x],x]

[Out]

3*a^2*b*E^x*Sqrt[c + d*x] + (3*a*b^2*E^(2*x)*Sqrt[c + d*x])/2 + (b^3*E^(3*x)*Sqrt[c + d*x])/3 + (2*a^3*(c + d*
x)^(3/2))/(3*d) - (3*a^2*b*Sqrt[d]*Sqrt[Pi]*Erfi[Sqrt[c + d*x]/Sqrt[d]])/(2*E^(c/d)) - (3*a*b^2*Sqrt[d]*Sqrt[P
i/2]*Erfi[(Sqrt[2]*Sqrt[c + d*x])/Sqrt[d]])/(4*E^((2*c)/d)) - (b^3*Sqrt[d]*Sqrt[Pi/3]*Erfi[(Sqrt[3]*Sqrt[c + d
*x])/Sqrt[d]])/(6*E^((3*c)/d))

Rule 2207

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(c + d*x)^m*
((b*F^(g*(e + f*x)))^n/(f*g*n*Log[F])), x] - Dist[d*(m/(f*g*n*Log[F])), Int[(c + d*x)^(m - 1)*(b*F^(g*(e + f*x
)))^n, x], x] /; FreeQ[{F, b, c, d, e, f, g, n}, x] && GtQ[m, 0] && IntegerQ[2*m] &&  !TrueQ[$UseGamma]

Rule 2211

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[2/d, Subst[Int[F^(g*(e - c*(
f/d)) + f*g*(x^2/d)), x], x, Sqrt[c + d*x]], x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !TrueQ[$UseGamma]

Rule 2214

Int[((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.))^(p_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> In
t[ExpandIntegrand[(c + d*x)^m, (a + b*(F^(g*(e + f*x)))^n)^p, x], x] /; FreeQ[{F, a, b, c, d, e, f, g, m, n},
x] && IGtQ[p, 0]

Rule 2235

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 \left (a+b e^x\right )^3 \sqrt {c+d x} \, dx &=\int \left (a^3 \sqrt {c+d x}+3 a^2 b e^x \sqrt {c+d x}+3 a b^2 e^{2 x} \sqrt {c+d x}+b^3 e^{3 x} \sqrt {c+d x}\right ) \, dx\\ &=\frac {2 a^3 (c+d x)^{3/2}}{3 d}+\left (3 a^2 b\right ) \int e^x \sqrt {c+d x} \, dx+\left (3 a b^2\right ) \int e^{2 x} \sqrt {c+d x} \, dx+b^3 \int e^{3 x} \sqrt {c+d x} \, dx\\ &=3 a^2 b e^x \sqrt {c+d x}+\frac {3}{2} a b^2 e^{2 x} \sqrt {c+d x}+\frac {1}{3} b^3 e^{3 x} \sqrt {c+d x}+\frac {2 a^3 (c+d x)^{3/2}}{3 d}-\frac {1}{2} \left (3 a^2 b d\right ) \int \frac {e^x}{\sqrt {c+d x}} \, dx-\frac {1}{4} \left (3 a b^2 d\right ) \int \frac {e^{2 x}}{\sqrt {c+d x}} \, dx-\frac {1}{6} \left (b^3 d\right ) \int \frac {e^{3 x}}{\sqrt {c+d x}} \, dx\\ &=3 a^2 b e^x \sqrt {c+d x}+\frac {3}{2} a b^2 e^{2 x} \sqrt {c+d x}+\frac {1}{3} b^3 e^{3 x} \sqrt {c+d x}+\frac {2 a^3 (c+d x)^{3/2}}{3 d}-\left (3 a^2 b\right ) \text {Subst}\left (\int e^{-\frac {c}{d}+\frac {x^2}{d}} \, dx,x,\sqrt {c+d x}\right )-\frac {1}{2} \left (3 a b^2\right ) \text {Subst}\left (\int e^{-\frac {2 c}{d}+\frac {2 x^2}{d}} \, dx,x,\sqrt {c+d x}\right )-\frac {1}{3} b^3 \text {Subst}\left (\int e^{-\frac {3 c}{d}+\frac {3 x^2}{d}} \, dx,x,\sqrt {c+d x}\right )\\ &=3 a^2 b e^x \sqrt {c+d x}+\frac {3}{2} a b^2 e^{2 x} \sqrt {c+d x}+\frac {1}{3} b^3 e^{3 x} \sqrt {c+d x}+\frac {2 a^3 (c+d x)^{3/2}}{3 d}-\frac {3}{2} a^2 b \sqrt {d} e^{-\frac {c}{d}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {c+d x}}{\sqrt {d}}\right )-\frac {3}{4} a b^2 \sqrt {d} e^{-\frac {2 c}{d}} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {c+d x}}{\sqrt {d}}\right )-\frac {1}{6} b^3 \sqrt {d} e^{-\frac {3 c}{d}} \sqrt {\frac {\pi }{3}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {c+d x}}{\sqrt {d}}\right )\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 1.43, size = 299, normalized size = 1.33 \begin {gather*} \frac {24 a^3 d \left (-\frac {c+d x}{d}\right )^{5/2}+54 a^2 b e^{-\frac {c}{d}} (c+d x) \left (\sqrt {\pi }+2 e^{\frac {c}{d}+x} \sqrt {-\frac {c+d x}{d}}-\sqrt {\pi } \text {erf}\left (\sqrt {-\frac {c+d x}{d}}\right )\right )+\frac {27 a b^2 e^{-\frac {2 c}{d}} (c+d x) \left (\sqrt {\pi }+2 \sqrt {2} e^{2 \left (\frac {c}{d}+x\right )} \sqrt {-\frac {c+d x}{d}}-\sqrt {\pi } \text {erf}\left (\sqrt {2} \sqrt {-\frac {c+d x}{d}}\right )\right )}{\sqrt {2}}+2 \sqrt {3} b^3 e^{-\frac {3 c}{d}} (c+d x) \left (\sqrt {\pi }+2 \sqrt {3} e^{3 \left (\frac {c}{d}+x\right )} \sqrt {-\frac {c+d x}{d}}-\sqrt {\pi } \text {erf}\left (\sqrt {3} \sqrt {-\frac {c+d x}{d}}\right )\right )}{36 \sqrt {c+d x} \sqrt {-\frac {c+d x}{d}}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(24*a^3*d*(-((c + d*x)/d))^(5/2) + (54*a^2*b*(c + d*x)*(Sqrt[Pi] + 2*E^(c/d + x)*Sqrt[-((c + d*x)/d)] - Sqrt[P
i]*Erf[Sqrt[-((c + d*x)/d)]]))/E^(c/d) + (27*a*b^2*(c + d*x)*(Sqrt[Pi] + 2*Sqrt[2]*E^(2*(c/d + x))*Sqrt[-((c +
d*x)/d)] - Sqrt[Pi]*Erf[Sqrt[2]*Sqrt[-((c + d*x)/d)]]))/(Sqrt[2]*E^((2*c)/d)) + (2*Sqrt[3]*b^3*(c + d*x)*(Sqr
t[Pi] + 2*Sqrt[3]*E^(3*(c/d + x))*Sqrt[-((c + d*x)/d)] - Sqrt[Pi]*Erf[Sqrt[3]*Sqrt[-((c + d*x)/d)]]))/E^((3*c)
/d))/(36*Sqrt[c + d*x]*Sqrt[-((c + d*x)/d)])

________________________________________________________________________________________

Maple [A]
time = 0.05, size = 211, normalized size = 0.94

method	result	size
derivativedivides	\(\frac {\frac {2 \left (d x +c \right )^{\frac {3}{2}} a^{3}}{3}+2 b^{3} {\mathrm e}^{-\frac {3 c}{d}} \left (\frac {d \sqrt {d x +c}\, {\mathrm e}^{\frac {3 d x +3 c}{d}}}{6}-\frac {d \sqrt {\pi }\, \erf \left (\sqrt {-\frac {3}{d}}\, \sqrt {d x +c}\right )}{12 \sqrt {-\frac {3}{d}}}\right )+6 a \,b^{2} {\mathrm e}^{-\frac {2 c}{d}} \left (\frac {d \sqrt {d x +c}\, {\mathrm e}^{\frac {2 d x +2 c}{d}}}{4}-\frac {d \sqrt {\pi }\, \erf \left (\sqrt {-\frac {2}{d}}\, \sqrt {d x +c}\right )}{8 \sqrt {-\frac {2}{d}}}\right )+6 a^{2} b \,{\mathrm e}^{-\frac {c}{d}} \left (\frac {\sqrt {d x +c}\, {\mathrm e}^{\frac {d x +c}{d}} d}{2}-\frac {d \sqrt {\pi }\, \erf \left (\sqrt {-\frac {1}{d}}\, \sqrt {d x +c}\right )}{4 \sqrt {-\frac {1}{d}}}\right )}{d}\)	\(211\)
default	\(\frac {\frac {2 \left (d x +c \right )^{\frac {3}{2}} a^{3}}{3}+2 b^{3} {\mathrm e}^{-\frac {3 c}{d}} \left (\frac {d \sqrt {d x +c}\, {\mathrm e}^{\frac {3 d x +3 c}{d}}}{6}-\frac {d \sqrt {\pi }\, \erf \left (\sqrt {-\frac {3}{d}}\, \sqrt {d x +c}\right )}{12 \sqrt {-\frac {3}{d}}}\right )+6 a \,b^{2} {\mathrm e}^{-\frac {2 c}{d}} \left (\frac {d \sqrt {d x +c}\, {\mathrm e}^{\frac {2 d x +2 c}{d}}}{4}-\frac {d \sqrt {\pi }\, \erf \left (\sqrt {-\frac {2}{d}}\, \sqrt {d x +c}\right )}{8 \sqrt {-\frac {2}{d}}}\right )+6 a^{2} b \,{\mathrm e}^{-\frac {c}{d}} \left (\frac {\sqrt {d x +c}\, {\mathrm e}^{\frac {d x +c}{d}} d}{2}-\frac {d \sqrt {\pi }\, \erf \left (\sqrt {-\frac {1}{d}}\, \sqrt {d x +c}\right )}{4 \sqrt {-\frac {1}{d}}}\right )}{d}\)	\(211\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*exp(x))^3*(d*x+c)^(1/2),x,method=_RETURNVERBOSE)

[Out]

2/d*(1/3*(d*x+c)^(3/2)*a^3+b^3/exp(1/d*c)^3*(1/6*d*(d*x+c)^(1/2)*exp(3/d*(d*x+c))-1/12*d*Pi^(1/2)/(-3/d)^(1/2)
*erf((-3/d)^(1/2)*(d*x+c)^(1/2)))+3*a*b^2/exp(1/d*c)^2*(1/4*d*(d*x+c)^(1/2)*exp(2/d*(d*x+c))-1/8*d*Pi^(1/2)/(-
2/d)^(1/2)*erf((-2/d)^(1/2)*(d*x+c)^(1/2)))+3*a^2*b/exp(1/d*c)*(1/2*(d*x+c)^(1/2)*exp(1/d*(d*x+c))*d-1/4*d*Pi^
(1/2)/(-1/d)^(1/2)*erf((-1/d)^(1/2)*(d*x+c)^(1/2))))

________________________________________________________________________________________

Maxima [A]
time = 0.52, size = 238, normalized size = 1.06 \begin {gather*} \frac {48 \, {\left (d x + c\right )}^{\frac {3}{2}} a^{3} - 108 \, {\left (\frac {\sqrt {\pi } d \operatorname {erf}\left (\sqrt {d x + c} \sqrt {-\frac {1}{d}}\right ) e^{\left (-\frac {c}{d}\right )}}{\sqrt {-\frac {1}{d}}} - 2 \, \sqrt {d x + c} d e^{\left (\frac {d x + c}{d} - \frac {c}{d}\right )}\right )} a^{2} b - 27 \, {\left (\frac {\sqrt {2} \sqrt {\pi } d \operatorname {erf}\left (\sqrt {2} \sqrt {d x + c} \sqrt {-\frac {1}{d}}\right ) e^{\left (-\frac {2 \, c}{d}\right )}}{\sqrt {-\frac {1}{d}}} - 4 \, \sqrt {d x + c} d e^{\left (\frac {2 \, {\left (d x + c\right )}}{d} - \frac {2 \, c}{d}\right )}\right )} a b^{2} - 4 \, {\left (\frac {\sqrt {3} \sqrt {\pi } d \operatorname {erf}\left (\sqrt {3} \sqrt {d x + c} \sqrt {-\frac {1}{d}}\right ) e^{\left (-\frac {3 \, c}{d}\right )}}{\sqrt {-\frac {1}{d}}} - 6 \, \sqrt {d x + c} d e^{\left (\frac {3 \, {\left (d x + c\right )}}{d} - \frac {3 \, c}{d}\right )}\right )} b^{3}}{72 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*exp(x))^3*(d*x+c)^(1/2),x, algorithm="maxima")

[Out]

1/72*(48*(d*x + c)^(3/2)*a^3 - 108*(sqrt(pi)*d*erf(sqrt(d*x + c)*sqrt(-1/d))*e^(-c/d)/sqrt(-1/d) - 2*sqrt(d*x
+ c)*d*e^((d*x + c)/d - c/d))*a^2*b - 27*(sqrt(2)*sqrt(pi)*d*erf(sqrt(2)*sqrt(d*x + c)*sqrt(-1/d))*e^(-2*c/d)/
sqrt(-1/d) - 4*sqrt(d*x + c)*d*e^(2*(d*x + c)/d - 2*c/d))*a*b^2 - 4*(sqrt(3)*sqrt(pi)*d*erf(sqrt(3)*sqrt(d*x +
c)*sqrt(-1/d))*e^(-3*c/d)/sqrt(-1/d) - 6*sqrt(d*x + c)*d*e^(3*(d*x + c)/d - 3*c/d))*b^3)/d

________________________________________________________________________________________

Fricas [A]
time = 0.39, size = 196, normalized size = 0.88 \begin {gather*} \frac {27 \, \sqrt {2} \sqrt {\pi } a b^{2} d^{2} \sqrt {-\frac {1}{d}} \operatorname {erf}\left (\sqrt {2} \sqrt {d x + c} \sqrt {-\frac {1}{d}}\right ) e^{\left (-\frac {2 \, c}{d}\right )} + 4 \, \sqrt {3} \sqrt {\pi } b^{3} d^{2} \sqrt {-\frac {1}{d}} \operatorname {erf}\left (\sqrt {3} \sqrt {d x + c} \sqrt {-\frac {1}{d}}\right ) e^{\left (-\frac {3 \, c}{d}\right )} + 108 \, \sqrt {\pi } a^{2} b d^{2} \sqrt {-\frac {1}{d}} \operatorname {erf}\left (\sqrt {d x + c} \sqrt {-\frac {1}{d}}\right ) e^{\left (-\frac {c}{d}\right )} + 12 \, {\left (4 \, a^{3} d x + 2 \, b^{3} d e^{\left (3 \, x\right )} + 9 \, a b^{2} d e^{\left (2 \, x\right )} + 18 \, a^{2} b d e^{x} + 4 \, a^{3} c\right )} \sqrt {d x + c}}{72 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*exp(x))^3*(d*x+c)^(1/2),x, algorithm="fricas")

[Out]

1/72*(27*sqrt(2)*sqrt(pi)*a*b^2*d^2*sqrt(-1/d)*erf(sqrt(2)*sqrt(d*x + c)*sqrt(-1/d))*e^(-2*c/d) + 4*sqrt(3)*sq
rt(pi)*b^3*d^2*sqrt(-1/d)*erf(sqrt(3)*sqrt(d*x + c)*sqrt(-1/d))*e^(-3*c/d) + 108*sqrt(pi)*a^2*b*d^2*sqrt(-1/d)
*erf(sqrt(d*x + c)*sqrt(-1/d))*e^(-c/d) + 12*(4*a^3*d*x + 2*b^3*d*e^(3*x) + 9*a*b^2*d*e^(2*x) + 18*a^2*b*d*e^x
+ 4*a^3*c)*sqrt(d*x + c))/d

________________________________________________________________________________________

Sympy [A]
time = 2.45, size = 291, normalized size = 1.30 \begin {gather*} \frac {2 a^{3} \left (c + d x\right )^{\frac {3}{2}}}{3 d} - \frac {3 \sqrt {\pi } a^{2} b \sqrt {d} e^{- \frac {c}{d}} \operatorname {erfi}{\left (\frac {\sqrt {c + d x}}{d \sqrt {\frac {1}{d}}} \right )}}{2} + \frac {3 a^{2} b \sqrt {c + d x} e^{- \frac {c}{d}} e^{\frac {c}{d} + x}}{\sqrt {d} \sqrt {\frac {1}{d}}} - \frac {3 \sqrt {2} \sqrt {\pi } a b^{2} \sqrt {d} e^{- \frac {2 c}{d}} \operatorname {erfi}{\left (\frac {\sqrt {2} \sqrt {c + d x}}{d \sqrt {\frac {1}{d}}} \right )}}{8} + \frac {3 a b^{2} \sqrt {c + d x} e^{- \frac {2 c}{d}} e^{\frac {2 c}{d} + 2 x}}{2 \sqrt {d} \sqrt {\frac {1}{d}}} - \frac {\sqrt {3} \sqrt {\pi } b^{3} \sqrt {d} e^{- \frac {3 c}{d}} \operatorname {erfi}{\left (\frac {\sqrt {3} \sqrt {c + d x}}{d \sqrt {\frac {1}{d}}} \right )}}{18} + \frac {b^{3} \sqrt {c + d x} e^{- \frac {3 c}{d}} e^{\frac {3 c}{d} + 3 x}}{3 \sqrt {d} \sqrt {\frac {1}{d}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*exp(x))**3*(d*x+c)**(1/2),x)

[Out]

2*a**3*(c + d*x)**(3/2)/(3*d) - 3*sqrt(pi)*a**2*b*sqrt(d)*exp(-c/d)*erfi(sqrt(c + d*x)/(d*sqrt(1/d)))/2 + 3*a*
*2*b*sqrt(c + d*x)*exp(-c/d)*exp(c/d + x)/(sqrt(d)*sqrt(1/d)) - 3*sqrt(2)*sqrt(pi)*a*b**2*sqrt(d)*exp(-2*c/d)*
erfi(sqrt(2)*sqrt(c + d*x)/(d*sqrt(1/d)))/8 + 3*a*b**2*sqrt(c + d*x)*exp(-2*c/d)*exp(2*c/d + 2*x)/(2*sqrt(d)*s
qrt(1/d)) - sqrt(3)*sqrt(pi)*b**3*sqrt(d)*exp(-3*c/d)*erfi(sqrt(3)*sqrt(c + d*x)/(d*sqrt(1/d)))/18 + b**3*sqrt
(c + d*x)*exp(-3*c/d)*exp(3*c/d + 3*x)/(3*sqrt(d)*sqrt(1/d))

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 368 vs. \(2 (168) = 336\).
time = 1.99, size = 368, normalized size = 1.64 \begin {gather*} -\frac {\frac {108 \, \sqrt {2} \sqrt {\pi } a b^{2} c d \operatorname {erf}\left (-\frac {\sqrt {2} \sqrt {d x + c} \sqrt {-d}}{d}\right ) e^{\left (-\frac {2 \, c}{d}\right )}}{\sqrt {-d}} + \frac {24 \, \sqrt {3} \sqrt {\pi } b^{3} c d \operatorname {erf}\left (-\frac {\sqrt {3} \sqrt {d x + c} \sqrt {-d}}{d}\right ) e^{\left (-\frac {3 \, c}{d}\right )}}{\sqrt {-d}} + \frac {216 \, \sqrt {\pi } a^{2} b c d \operatorname {erf}\left (-\frac {\sqrt {d x + c} \sqrt {-d}}{d}\right ) e^{\left (-\frac {c}{d}\right )}}{\sqrt {-d}} - 144 \, \sqrt {d x + c} a^{3} c - 48 \, {\left ({\left (d x + c\right )}^{\frac {3}{2}} - 3 \, \sqrt {d x + c} c\right )} a^{3} - 108 \, {\left (\frac {\sqrt {\pi } {\left (2 \, c + d\right )} d \operatorname {erf}\left (-\frac {\sqrt {d x + c} \sqrt {-d}}{d}\right ) e^{\left (-\frac {c}{d}\right )}}{\sqrt {-d}} + 2 \, \sqrt {d x + c} d e^{x}\right )} a^{2} b - 27 \, {\left (\frac {\sqrt {2} \sqrt {\pi } {\left (4 \, c + d\right )} d \operatorname {erf}\left (-\frac {\sqrt {2} \sqrt {d x + c} \sqrt {-d}}{d}\right ) e^{\left (-\frac {2 \, c}{d}\right )}}{\sqrt {-d}} + 4 \, \sqrt {d x + c} d e^{\left (2 \, x\right )}\right )} a b^{2} - 4 \, {\left (\frac {\sqrt {3} \sqrt {\pi } {\left (6 \, c + d\right )} d \operatorname {erf}\left (-\frac {\sqrt {3} \sqrt {d x + c} \sqrt {-d}}{d}\right ) e^{\left (-\frac {3 \, c}{d}\right )}}{\sqrt {-d}} + 6 \, \sqrt {d x + c} d e^{\left (3 \, x\right )}\right )} b^{3}}{72 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*exp(x))^3*(d*x+c)^(1/2),x, algorithm="giac")

[Out]

-1/72*(108*sqrt(2)*sqrt(pi)*a*b^2*c*d*erf(-sqrt(2)*sqrt(d*x + c)*sqrt(-d)/d)*e^(-2*c/d)/sqrt(-d) + 24*sqrt(3)*
sqrt(pi)*b^3*c*d*erf(-sqrt(3)*sqrt(d*x + c)*sqrt(-d)/d)*e^(-3*c/d)/sqrt(-d) + 216*sqrt(pi)*a^2*b*c*d*erf(-sqrt
(d*x + c)*sqrt(-d)/d)*e^(-c/d)/sqrt(-d) - 144*sqrt(d*x + c)*a^3*c - 48*((d*x + c)^(3/2) - 3*sqrt(d*x + c)*c)*a
^3 - 108*(sqrt(pi)*(2*c + d)*d*erf(-sqrt(d*x + c)*sqrt(-d)/d)*e^(-c/d)/sqrt(-d) + 2*sqrt(d*x + c)*d*e^x)*a^2*b
- 27*(sqrt(2)*sqrt(pi)*(4*c + d)*d*erf(-sqrt(2)*sqrt(d*x + c)*sqrt(-d)/d)*e^(-2*c/d)/sqrt(-d) + 4*sqrt(d*x +
c)*d*e^(2*x))*a*b^2 - 4*(sqrt(3)*sqrt(pi)*(6*c + d)*d*erf(-sqrt(3)*sqrt(d*x + c)*sqrt(-d)/d)*e^(-3*c/d)/sqrt(-
d) + 6*sqrt(d*x + c)*d*e^(3*x))*b^3)/d

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*exp(x))^3*(c + d*x)^(1/2),x)

[Out]

int((a + b*exp(x))^3*(c + d*x)^(1/2), x)

________________________________________________________________________________________