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

Optimal. Leaf size=72 \[ \frac{\sqrt{\pi } F^{c \left (a-\frac{b d}{e}\right )} \text{Erfi}\left (\frac{\sqrt{b} \sqrt{c} \sqrt{\log (F)} \sqrt{d+e x}}{\sqrt{e}}\right )}{\sqrt{b} \sqrt{c} \sqrt{e} \sqrt{\log (F)}} \]

[Out]

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

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Rubi [A]  time = 0.0838435, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ \frac{\sqrt{\pi } F^{c \left (a-\frac{b d}{e}\right )} \text{Erfi}\left (\frac{\sqrt{b} \sqrt{c} \sqrt{\log (F)} \sqrt{d+e x}}{\sqrt{e}}\right )}{\sqrt{b} \sqrt{c} \sqrt{e} \sqrt{\log (F)}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 11.9678, size = 70, normalized size = 0.97 \[ \frac{\sqrt{\pi } F^{\frac{c \left (a e - b d\right )}{e}} \operatorname{erfi}{\left (\frac{\sqrt{b} \sqrt{c} \sqrt{d + e x} \sqrt{\log{\left (F \right )}}}{\sqrt{e}} \right )}}{\sqrt{b} \sqrt{c} \sqrt{e} \sqrt{\log{\left (F \right )}}} \]

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]

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

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Mathematica [A]  time = 0.0762999, size = 70, normalized size = 0.97 \[ \frac{\sqrt{\pi } \sqrt{d+e x} F^{c \left (a-\frac{b d}{e}\right )} \left (\text{Erf}\left (\sqrt{-\frac{b c \log (F) (d+e x)}{e}}\right )-1\right )}{e \sqrt{-\frac{b c \log (F) (d+e x)}{e}}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.029, size = 0, normalized size = 0. \[ \int{{F}^{c \left ( bx+a \right ) }{\frac{1}{\sqrt{ex+d}}}}\, dx \]

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 = 0.861878, size = 70, normalized size = 0.97 \[ \frac{\sqrt{\pi } F^{a c - \frac{b c d}{e}} \operatorname{erf}\left (\sqrt{e x + d} \sqrt{-\frac{b c \log \left (F\right )}{e}}\right )}{\sqrt{-\frac{b c \log \left (F\right )}{e}} e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.258981, size = 76, normalized size = 1.06 \[ \frac{\sqrt{\pi } \operatorname{erf}\left (\sqrt{e x + d} \sqrt{-\frac{b c \log \left (F\right )}{e}}\right )}{\sqrt{-\frac{b c \log \left (F\right )}{e}} F^{\frac{b c d - a c e}{e}} e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

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 = 0.25674, size = 78, normalized size = 1.08 \[ -\frac{\sqrt{\pi } \operatorname{erf}\left (-\sqrt{-b c e{\rm ln}\left (F\right )} \sqrt{x e + d} e^{\left (-1\right )}\right ) e^{\left (-{\left (b c d{\rm ln}\left (F\right ) - a c e{\rm ln}\left (F\right )\right )} e^{\left (-1\right )}\right )}}{\sqrt{-b c e{\rm ln}\left (F\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-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))/sqrt(-b*c*e*ln(F))