3.1.64 \(\int (a+b e^x) \sqrt {c+d x} \, dx\) [64]

Optimal. Leaf size=71 \[ b e^x \sqrt {c+d x}+\frac {2 a (c+d x)^{3/2}}{3 d}-\frac {1}{2} b \sqrt {d} e^{-\frac {c}{d}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {c+d x}}{\sqrt {d}}\right ) \]

[Out]

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

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Rubi [A]
time = 0.06, antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {2214, 2207, 2211, 2235} \begin {gather*} \frac {2 a (c+d x)^{3/2}}{3 d}-\frac {1}{2} \sqrt {\pi } b \sqrt {d} e^{-\frac {c}{d}} \text {Erfi}\left (\frac {\sqrt {c+d x}}{\sqrt {d}}\right )+b e^x \sqrt {c+d x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*E^x)*Sqrt[c + d*x],x]

[Out]

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

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Mathematica [A]
time = 0.27, size = 71, normalized size = 1.00 \begin {gather*} b e^x \sqrt {c+d x}+\frac {2 a (c+d x)^{3/2}}{3 d}-\frac {1}{2} b \sqrt {d} e^{-\frac {c}{d}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {c+d x}}{\sqrt {d}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.06, size = 77, normalized size = 1.08

method	result	size
derivativedivides	\(\frac {\frac {2 \left (d x +c \right )^{\frac {3}{2}} a}{3}+2 b \,{\mathrm e}^{-\frac {c}{d}} \left (\frac {\sqrt {d x +c}\, {\mathrm e}^{\frac {d x +c}{d}} d}{2}-\frac {d \sqrt {\pi }\, \erf \left (\sqrt {-\frac {1}{d}}\, \sqrt {d x +c}\right )}{4 \sqrt {-\frac {1}{d}}}\right )}{d}\)	\(77\)
default	\(\frac {\frac {2 \left (d x +c \right )^{\frac {3}{2}} a}{3}+2 b \,{\mathrm e}^{-\frac {c}{d}} \left (\frac {\sqrt {d x +c}\, {\mathrm e}^{\frac {d x +c}{d}} d}{2}-\frac {d \sqrt {\pi }\, \erf \left (\sqrt {-\frac {1}{d}}\, \sqrt {d x +c}\right )}{4 \sqrt {-\frac {1}{d}}}\right )}{d}\)	\(77\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/d*(1/3*(d*x+c)^(3/2)*a+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))))

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Maxima [A]
time = 0.30, size = 82, normalized size = 1.15 \begin {gather*} \frac {4 \, {\left (d x + c\right )}^{\frac {3}{2}} a - 3 \, {\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 )} b}{6 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(4*(d*x + c)^(3/2)*a - 3*(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))*b)/d

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Fricas [A]
time = 0.47, size = 70, normalized size = 0.99 \begin {gather*} \frac {3 \, \sqrt {\pi } 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 )} + 2 \, {\left (2 \, a d x + 3 \, b d e^{x} + 2 \, a c\right )} \sqrt {d x + c}}{6 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]
time = 1.10, size = 85, normalized size = 1.20 \begin {gather*} \frac {2 a \left (c + d x\right )^{\frac {3}{2}}}{3 d} - \frac {\sqrt {\pi } b \sqrt {d} e^{- \frac {c}{d}} \operatorname {erfi}{\left (\frac {\sqrt {c + d x}}{d \sqrt {\frac {1}{d}}} \right )}}{2} + \frac {b \sqrt {c + d x} e^{- \frac {c}{d}} e^{\frac {c}{d} + x}}{\sqrt {d} \sqrt {\frac {1}{d}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 132 vs. \(2 (53) = 106\).
time = 2.19, size = 132, normalized size = 1.86 \begin {gather*} -\frac {\frac {6 \, \sqrt {\pi } b c d \operatorname {erf}\left (-\frac {\sqrt {d x + c} \sqrt {-d}}{d}\right ) e^{\left (-\frac {c}{d}\right )}}{\sqrt {-d}} - 12 \, \sqrt {d x + c} a c - 4 \, {\left ({\left (d x + c\right )}^{\frac {3}{2}} - 3 \, \sqrt {d x + c} c\right )} a - 3 \, {\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 )} b}{6 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*(6*sqrt(pi)*b*c*d*erf(-sqrt(d*x + c)*sqrt(-d)/d)*e^(-c/d)/sqrt(-d) - 12*sqrt(d*x + c)*a*c - 4*((d*x + c)^
(3/2) - 3*sqrt(d*x + c)*c)*a - 3*(sqrt(pi)*(2*c + d)*d*erf(-sqrt(d*x + c)*sqrt(-d)/d)*e^(-c/d)/sqrt(-d) + 2*sq
rt(d*x + c)*d*e^x)*b)/d

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \left (a+b\,{\mathrm {e}}^x\right )\,\sqrt {c+d\,x} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*exp(x))*(c + d*x)^(1/2),x)

[Out]

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

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