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

Optimal. Leaf size=105 \[ \frac{\sqrt{d+e x} F^{c (a+b x)}}{b c \log (F)}-\frac{\sqrt{\pi } \sqrt{e} F^{c \left (a-\frac{b d}{e}\right )} \text{Erfi}\left (\frac{\sqrt{b} \sqrt{c} \sqrt{\log (F)} \sqrt{d+e x}}{\sqrt{e}}\right )}{2 b^{3/2} c^{3/2} \log ^{\frac{3}{2}}(F)} \]

[Out]

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

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Rubi [A]  time = 0.137317, antiderivative size = 105, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158 \[ \frac{\sqrt{d+e x} F^{c (a+b x)}}{b c \log (F)}-\frac{\sqrt{\pi } \sqrt{e} F^{c \left (a-\frac{b d}{e}\right )} \text{Erfi}\left (\frac{\sqrt{b} \sqrt{c} \sqrt{\log (F)} \sqrt{d+e x}}{\sqrt{e}}\right )}{2 b^{3/2} c^{3/2} \log ^{\frac{3}{2}}(F)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 20.5824, size = 95, normalized size = 0.9 \[ \frac{F^{c \left (a + b x\right )} \sqrt{d + e x}}{b c \log{\left (F \right )}} - \frac{\sqrt{\pi } F^{\frac{c \left (a e - b d\right )}{e}} \sqrt{e} \operatorname{erfi}{\left (\frac{\sqrt{b} \sqrt{c} \sqrt{d + e x} \sqrt{\log{\left (F \right )}}}{\sqrt{e}} \right )}}{2 b^{\frac{3}{2}} c^{\frac{3}{2}} \log{\left (F \right )}^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(F**(c*(b*x+a))*(e*x+d)**(1/2),x)

[Out]

F**(c*(a + b*x))*sqrt(d + e*x)/(b*c*log(F)) - sqrt(pi)*F**(c*(a*e - b*d)/e)*sqrt
(e)*erfi(sqrt(b)*sqrt(c)*sqrt(d + e*x)*sqrt(log(F))/sqrt(e))/(2*b**(3/2)*c**(3/2
)*log(F)**(3/2))

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Mathematica [A]  time = 0.18703, size = 108, normalized size = 1.03 \[ -\frac{(d+e x)^{3/2} F^{c \left (a-\frac{b d}{e}\right )} \left (F^{\frac{b c (d+e x)}{e}} \sqrt{-\frac{b c \log (F) (d+e x)}{e}}-\frac{1}{2} \sqrt{\pi } \left (\text{Erf}\left (\sqrt{-\frac{b c \log (F) (d+e x)}{e}}\right )-1\right )\right )}{e \left (-\frac{b c \log (F) (d+e x)}{e}\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.024, size = 0, normalized size = 0. \[ \int{F}^{c \left ( bx+a \right ) }\sqrt{ex+d}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Maxima [A]  time = 0.85139, size = 153, normalized size = 1.46 \[ \frac{F^{a c}{\left (\frac{2 \, \sqrt{e x + d} F^{\frac{{\left (e x + d\right )} b c}{e}} e}{F^{\frac{b c d}{e}} b c \log \left (F\right )} - \frac{\sqrt{\pi } e \operatorname{erf}\left (\sqrt{e x + d} \sqrt{-\frac{b c \log \left (F\right )}{e}}\right )}{\sqrt{-\frac{b c \log \left (F\right )}{e}} F^{\frac{b c d}{e}} b c \log \left (F\right )}\right )}}{2 \, e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(e*x + d)*F^((b*x + a)*c),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.258327, size = 131, normalized size = 1.25 \[ \frac{2 \, \sqrt{e x + d} \sqrt{-\frac{b c \log \left (F\right )}{e}} F^{b c x + a c} - \frac{\sqrt{\pi } \operatorname{erf}\left (\sqrt{e x + d} \sqrt{-\frac{b c \log \left (F\right )}{e}}\right )}{F^{\frac{b c d - a c e}{e}}}}{2 \, \sqrt{-\frac{b c \log \left (F\right )}{e}} b c \log \left (F\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(e*x + d)*F^((b*x + a)*c),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int F^{c \left (a + b x\right )} \sqrt{d + e x}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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GIAC/XCAS [A]  time = 0.251082, size = 170, normalized size = 1.62 \[ \frac{1}{2} \,{\left (\frac{\sqrt{\pi } \operatorname{erf}\left (-\sqrt{-b c e{\rm ln}\left (F\right )} \sqrt{x e + d} e^{\left (-1\right )}\right ) e^{\left (-{\left (b c d{\rm ln}\left (F\right ) - a c e{\rm ln}\left (F\right )\right )} e^{\left (-1\right )} + 2\right )}}{\sqrt{-b c e{\rm ln}\left (F\right )} b c{\rm ln}\left (F\right )} + \frac{2 \, \sqrt{x e + d} e^{\left ({\left ({\left (x e + d\right )} b c{\rm ln}\left (F\right ) - b c d{\rm ln}\left (F\right ) + a c e{\rm ln}\left (F\right )\right )} e^{\left (-1\right )} + 1\right )}}{b c{\rm ln}\left (F\right )}\right )} e^{\left (-1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(e*x + d)*F^((b*x + a)*c),x, algorithm="giac")

[Out]

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