\(\int F^{c (a+b x)} \sqrt {d+e x} \, dx\) [42]

Optimal result

Integrand size = 19, antiderivative size = 105 \[ \int F^{c (a+b x)} \sqrt {d+e x} \, dx=-\frac {\sqrt {e} F^{c \left (a-\frac {b d}{e}\right )} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {b} \sqrt {c} \sqrt {d+e x} \sqrt {\log (F)}}{\sqrt {e}}\right )}{2 b^{3/2} c^{3/2} \log ^{\frac {3}{2}}(F)}+\frac {F^{c (a+b x)} \sqrt {d+e x}}{b c \log (F)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {2207, 2211, 2235} \[ \int F^{c (a+b x)} \sqrt {d+e x} \, dx=\frac {\sqrt {d+e x} F^{c (a+b x)}}{b c \log (F)}-\frac {\sqrt {\pi } \sqrt {e} F^{c \left (a-\frac {b d}{e}\right )} \text {erfi}\left (\frac {\sqrt {b} \sqrt {c} \sqrt {\log (F)} \sqrt {d+e x}}{\sqrt {e}}\right )}{2 b^{3/2} c^{3/2} \log ^{\frac {3}{2}}(F)} \]

[In]

Int[F^(c*(a + b*x))*Sqrt[d + e*x],x]

[Out]

-1/2*(Sqrt[e]*F^(c*(a - (b*d)/e))*Sqrt[Pi]*Erfi[(Sqrt[b]*Sqrt[c]*Sqrt[d + e*x]*Sqrt[Log[F]])/Sqrt[e]])/(b^(3/2
)*c^(3/2)*Log[F]^(3/2)) + (F^(c*(a + b*x))*Sqrt[d + e*x])/(b*c*Log[F])

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

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.60 \[ \int F^{c (a+b x)} \sqrt {d+e x} \, dx=-\frac {F^{c \left (a-\frac {b d}{e}\right )} (d+e x)^{3/2} \Gamma \left (\frac {3}{2},-\frac {b c (d+e x) \log (F)}{e}\right )}{e \left (-\frac {b c (d+e x) \log (F)}{e}\right )^{3/2}} \]

[In]

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

[Out]

-((F^(c*(a - (b*d)/e))*(d + e*x)^(3/2)*Gamma[3/2, -((b*c*(d + e*x)*Log[F])/e)])/(e*(-((b*c*(d + e*x)*Log[F])/e
))^(3/2)))

Maple [F]

\[\int F^{c \left (b x +a \right )} \sqrt {e x +d}d x\]

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.86 \[ \int F^{c (a+b x)} \sqrt {d+e x} \, dx=\frac {2 \, \sqrt {e x + d} F^{b c x + a c} b c \log \left (F\right ) + \frac {\sqrt {\pi } \sqrt {-\frac {b c \log \left (F\right )}{e}} e \operatorname {erf}\left (\sqrt {e x + d} \sqrt {-\frac {b c \log \left (F\right )}{e}}\right )}{F^{\frac {b c d - a c e}{e}}}}{2 \, b^{2} c^{2} \log \left (F\right )^{2}} \]

[In]

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

[Out]

1/2*(2*sqrt(e*x + d)*F^(b*c*x + a*c)*b*c*log(F) + sqrt(pi)*sqrt(-b*c*log(F)/e)*e*erf(sqrt(e*x + d)*sqrt(-b*c*l
og(F)/e))/F^((b*c*d - a*c*e)/e))/(b^2*c^2*log(F)^2)

Sympy [F]

\[ \int F^{c (a+b x)} \sqrt {d+e x} \, dx=\int F^{c \left (a + b x\right )} \sqrt {d + e x}\, dx \]

[In]

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

[Out]

Integral(F**(c*(a + b*x))*sqrt(d + e*x), x)

Maxima [F]

\[ \int F^{c (a+b x)} \sqrt {d+e x} \, dx=\int { \sqrt {e x + d} F^{{\left (b x + a\right )} c} \,d x } \]

[In]

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

[Out]

integrate(sqrt(e*x + d)*F^((b*x + a)*c), x)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 189 vs. \(2 (81) = 162\).

Time = 0.32 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.80 \[ \int F^{c (a+b x)} \sqrt {d+e x} \, dx=-\frac {\frac {2 \, \sqrt {\pi } d e \operatorname {erf}\left (-\frac {\sqrt {-b c e \log \left (F\right )} \sqrt {e x + d}}{e}\right ) e^{\left (-\frac {b c d \log \left (F\right ) - a c e \log \left (F\right )}{e}\right )}}{\sqrt {-b c e \log \left (F\right )}} - \frac {\sqrt {\pi } {\left (2 \, b c d \log \left (F\right ) + e\right )} e \operatorname {erf}\left (-\frac {\sqrt {-b c e \log \left (F\right )} \sqrt {e x + d}}{e}\right ) e^{\left (-\frac {b c d \log \left (F\right ) - a c e \log \left (F\right )}{e}\right )}}{\sqrt {-b c e \log \left (F\right )} b c \log \left (F\right )} - \frac {2 \, \sqrt {e x + d} e e^{\left (\frac {{\left (e x + d\right )} b c \log \left (F\right ) - b c d \log \left (F\right ) + a c e \log \left (F\right )}{e}\right )}}{b c \log \left (F\right )}}{2 \, e} \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int F^{c (a+b x)} \sqrt {d+e x} \, dx=\int F^{c\,\left (a+b\,x\right )}\,\sqrt {d+e\,x} \,d x \]

[In]

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

[Out]

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