Integrand size = 19, antiderivative size = 105 \[ \int F^{c (a+b x)} \sqrt {d+e x} \, dx=-\frac {\sqrt {e} F^{c \left (a-\frac {b d}{e}\right )} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {b} \sqrt {c} \sqrt {d+e x} \sqrt {\log (F)}}{\sqrt {e}}\right )}{2 b^{3/2} c^{3/2} \log ^{\frac {3}{2}}(F)}+\frac {F^{c (a+b x)} \sqrt {d+e x}}{b c \log (F)} \] Output:
-1/2*e^(1/2)*F^(c*(a-b*d/e))*Pi^(1/2)*erfi(b^(1/2)*c^(1/2)*(e*x+d)^(1/2)*l n(F)^(1/2)/e^(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)
Time = 0.19 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.60 \[ \int F^{c (a+b x)} \sqrt {d+e x} \, dx=-\frac {F^{c \left (a-\frac {b d}{e}\right )} (d+e x)^{3/2} \Gamma \left (\frac {3}{2},-\frac {b c (d+e x) \log (F)}{e}\right )}{e \left (-\frac {b c (d+e x) \log (F)}{e}\right )^{3/2}} \] Input:
Integrate[F^(c*(a + b*x))*Sqrt[d + e*x],x]
Output:
-((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)))
Time = 0.48 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {2607, 2611, 2633}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \sqrt {d+e x} F^{c (a+b x)} \, dx\) |
\(\Big \downarrow \) 2607 |
\(\displaystyle \frac {\sqrt {d+e x} F^{c (a+b x)}}{b c \log (F)}-\frac {e \int \frac {F^{c (a+b x)}}{\sqrt {d+e x}}dx}{2 b c \log (F)}\) |
\(\Big \downarrow \) 2611 |
\(\displaystyle \frac {\sqrt {d+e x} F^{c (a+b x)}}{b c \log (F)}-\frac {\int F^{c \left (a-\frac {b d}{e}\right )+\frac {b c (d+e x)}{e}}d\sqrt {d+e x}}{b c \log (F)}\) |
\(\Big \downarrow \) 2633 |
\(\displaystyle \frac {\sqrt {d+e x} F^{c (a+b x)}}{b c \log (F)}-\frac {\sqrt {\pi } \sqrt {e} F^{c \left (a-\frac {b d}{e}\right )} \text {erfi}\left (\frac {\sqrt {b} \sqrt {c} \sqrt {\log (F)} \sqrt {d+e x}}{\sqrt {e}}\right )}{2 b^{3/2} c^{3/2} \log ^{\frac {3}{2}}(F)}\) |
Input:
Int[F^(c*(a + b*x))*Sqrt[d + e*x],x]
Output:
-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])
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] - Simp[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[$UseGamma]
Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] : > Simp[2/d Subst[Int[F^(g*(e - c*(f/d)) + f*g*(x^2/d)), x], x, Sqrt[c + d *x]], x] /; FreeQ[{F, c, d, e, f, g}, x] && !TrueQ[$UseGamma]
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]
\[\int F^{c \left (b x +a \right )} \sqrt {e x +d}d x\]
Input:
int(F^(c*(b*x+a))*(e*x+d)^(1/2),x)
Output:
int(F^(c*(b*x+a))*(e*x+d)^(1/2),x)
Time = 0.11 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.86 \[ \int F^{c (a+b x)} \sqrt {d+e x} \, dx=\frac {2 \, \sqrt {e x + d} F^{b c x + a c} b c \log \left (F\right ) + \frac {\sqrt {\pi } \sqrt {-\frac {b c \log \left (F\right )}{e}} e \operatorname {erf}\left (\sqrt {e x + d} \sqrt {-\frac {b c \log \left (F\right )}{e}}\right )}{F^{\frac {b c d - a c e}{e}}}}{2 \, b^{2} c^{2} \log \left (F\right )^{2}} \] Input:
integrate(F^((b*x+a)*c)*(e*x+d)^(1/2),x, algorithm="fricas")
Output:
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*log(F)/e))/F^((b*c*d - a*c*e)/e))/(b^2* c^2*log(F)^2)
\[ \int F^{c (a+b x)} \sqrt {d+e x} \, dx=\int F^{c \left (a + b x\right )} \sqrt {d + e x}\, dx \] Input:
integrate(F**((b*x+a)*c)*(e*x+d)**(1/2),x)
Output:
Integral(F**(c*(a + b*x))*sqrt(d + e*x), x)
\[ \int F^{c (a+b x)} \sqrt {d+e x} \, dx=\int { \sqrt {e x + d} F^{{\left (b x + a\right )} c} \,d x } \] Input:
integrate(F^((b*x+a)*c)*(e*x+d)^(1/2),x, algorithm="maxima")
Output:
integrate(sqrt(e*x + d)*F^((b*x + a)*c), x)
Leaf count of result is larger than twice the leaf count of optimal. 189 vs. \(2 (81) = 162\).
Time = 0.13 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.80 \[ \int F^{c (a+b x)} \sqrt {d+e x} \, dx=-\frac {\frac {2 \, \sqrt {\pi } d e \operatorname {erf}\left (-\frac {\sqrt {-b c e \log \left (F\right )} \sqrt {e x + d}}{e}\right ) e^{\left (-\frac {b c d \log \left (F\right ) - a c e \log \left (F\right )}{e}\right )}}{\sqrt {-b c e \log \left (F\right )}} - \frac {\sqrt {\pi } {\left (2 \, b c d \log \left (F\right ) + e\right )} e \operatorname {erf}\left (-\frac {\sqrt {-b c e \log \left (F\right )} \sqrt {e x + d}}{e}\right ) e^{\left (-\frac {b c d \log \left (F\right ) - a c e \log \left (F\right )}{e}\right )}}{\sqrt {-b c e \log \left (F\right )} b c \log \left (F\right )} - \frac {2 \, \sqrt {e x + d} e e^{\left (\frac {{\left (e x + d\right )} b c \log \left (F\right ) - b c d \log \left (F\right ) + a c e \log \left (F\right )}{e}\right )}}{b c \log \left (F\right )}}{2 \, e} \] Input:
integrate(F^((b*x+a)*c)*(e*x+d)^(1/2),x, algorithm="giac")
Output:
-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
Timed out. \[ \int F^{c (a+b x)} \sqrt {d+e x} \, dx=\int F^{c\,\left (a+b\,x\right )}\,\sqrt {d+e\,x} \,d x \] Input:
int(F^(c*(a + b*x))*(d + e*x)^(1/2),x)
Output:
int(F^(c*(a + b*x))*(d + e*x)^(1/2), x)
\[ \int F^{c (a+b x)} \sqrt {d+e x} \, dx=f^{a c} \left (\int f^{b c x} \sqrt {e x +d}d x \right ) \] Input:
int(F^((b*x+a)*c)*(e*x+d)^(1/2),x)
Output:
f**(a*c)*int(f**(b*c*x)*sqrt(d + e*x),x)