3.44 \(\int \frac{F^{c (a+b x)}}{(d+e x)^{3/2}} \, dx\)

Optimal. Leaf size=97 \[ \frac{2 \sqrt{\pi } \sqrt{b} \sqrt{c} \sqrt{\log (F)} F^{c \left (a-\frac{b d}{e}\right )} \text{Erfi}\left (\frac{\sqrt{b} \sqrt{c} \sqrt{\log (F)} \sqrt{d+e x}}{\sqrt{e}}\right )}{e^{3/2}}-\frac{2 F^{c (a+b x)}}{e \sqrt{d+e x}} \]

[Out]

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

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Rubi [A]  time = 0.146311, antiderivative size = 97, 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 \[ \frac{2 \sqrt{\pi } \sqrt{b} \sqrt{c} \sqrt{\log (F)} F^{c \left (a-\frac{b d}{e}\right )} \text{Erfi}\left (\frac{\sqrt{b} \sqrt{c} \sqrt{\log (F)} \sqrt{d+e x}}{\sqrt{e}}\right )}{e^{3/2}}-\frac{2 F^{c (a+b x)}}{e \sqrt{d+e x}} \]

Antiderivative was successfully verified.

[In]  Int[F^(c*(a + b*x))/(d + e*x)^(3/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 19.558, size = 92, normalized size = 0.95 \[ - \frac{2 F^{c \left (a + b x\right )}}{e \sqrt{d + e x}} + \frac{2 \sqrt{\pi } F^{\frac{c \left (a e - b d\right )}{e}} \sqrt{b} \sqrt{c} \sqrt{\log{\left (F \right )}} \operatorname{erfi}{\left (\frac{\sqrt{b} \sqrt{c} \sqrt{d + e x} \sqrt{\log{\left (F \right )}}}{\sqrt{e}} \right )}}{e^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(F**(c*(b*x+a))/(e*x+d)**(3/2),x)

[Out]

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

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Mathematica [A]  time = 0.150211, size = 110, normalized size = 1.13 \[ -\frac{F^{a c-\frac{b c d}{e}} \sqrt{-\frac{b c \log (F) (d+e x)}{e}} \left (\frac{2 F^{\frac{b c (d+e x)}{e}}}{\sqrt{-\frac{b c \log (F) (d+e x)}{e}}}-2 \sqrt{\pi } \left (1-\text{Erf}\left (\sqrt{-\frac{b c \log (F) (d+e x)}{e}}\right )\right )\right )}{e \sqrt{d+e x}} \]

Antiderivative was successfully verified.

[In]  Integrate[F^(c*(a + b*x))/(d + e*x)^(3/2),x]

[Out]

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

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Maple [F]  time = 0.026, size = 0, normalized size = 0. \[ \int{{F}^{c \left ( bx+a \right ) } \left ( ex+d \right ) ^{-{\frac{3}{2}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(F^(c*(b*x+a))/(e*x+d)^(3/2),x)

[Out]

int(F^(c*(b*x+a))/(e*x+d)^(3/2),x)

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Maxima [A]  time = 0.886623, size = 81, normalized size = 0.84 \[ -\frac{\sqrt{-\frac{{\left (e x + d\right )} b c \log \left (F\right )}{e}} F^{a c} \Gamma \left (-\frac{1}{2}, -\frac{{\left (e x + d\right )} b c \log \left (F\right )}{e}\right )}{\sqrt{e x + d} F^{\frac{b c d}{e}} e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(F^((b*x + a)*c)/(e*x + d)^(3/2),x, algorithm="maxima")

[Out]

-sqrt(-(e*x + d)*b*c*log(F)/e)*F^(a*c)*gamma(-1/2, -(e*x + d)*b*c*log(F)/e)/(sqr
t(e*x + d)*F^(b*c*d/e)*e)

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Fricas [A]  time = 0.259238, size = 136, normalized size = 1.4 \[ \frac{2 \,{\left (\frac{\sqrt{\pi } \sqrt{e x + d} b c \operatorname{erf}\left (\sqrt{e x + d} \sqrt{-\frac{b c \log \left (F\right )}{e}}\right ) \log \left (F\right )}{F^{\frac{b c d - a c e}{e}}} - \sqrt{-\frac{b c \log \left (F\right )}{e}} F^{b c x + a c} e\right )}}{\sqrt{e x + d} \sqrt{-\frac{b c \log \left (F\right )}{e}} e^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(F^((b*x + a)*c)/(e*x + d)^(3/2),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{F^{c \left (a + b x\right )}}{\left (d + e x\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(F**(c*(b*x+a))/(e*x+d)**(3/2),x)

[Out]

Integral(F**(c*(a + b*x))/(d + e*x)**(3/2), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{F^{{\left (b x + a\right )} c}}{{\left (e x + d\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(F^((b*x + a)*c)/(e*x + d)^(3/2),x, algorithm="giac")

[Out]

integrate(F^((b*x + a)*c)/(e*x + d)^(3/2), x)