3.45 \(\int \frac{F^{c (a+b x)}}{(d+e x)^{5/2}} \, dx\)

Optimal. Leaf size=130 \[ \frac{4 \sqrt{\pi } b^{3/2} c^{3/2} \log ^{\frac{3}{2}}(F) F^{c \left (a-\frac{b d}{e}\right )} \text{Erfi}\left (\frac{\sqrt{b} \sqrt{c} \sqrt{\log (F)} \sqrt{d+e x}}{\sqrt{e}}\right )}{3 e^{5/2}}-\frac{4 b c \log (F) F^{c (a+b x)}}{3 e^2 \sqrt{d+e x}}-\frac{2 F^{c (a+b x)}}{3 e (d+e x)^{3/2}} \]

[Out]

(-2*F^(c*(a + b*x)))/(3*e*(d + e*x)^(3/2)) - (4*b*c*F^(c*(a + b*x))*Log[F])/(3*e
^2*Sqrt[d + e*x]) + (4*b^(3/2)*c^(3/2)*F^(c*(a - (b*d)/e))*Sqrt[Pi]*Erfi[(Sqrt[b
]*Sqrt[c]*Sqrt[d + e*x]*Sqrt[Log[F]])/Sqrt[e]]*Log[F]^(3/2))/(3*e^(5/2))

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Rubi [A]  time = 0.201524, antiderivative size = 130, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158 \[ \frac{4 \sqrt{\pi } b^{3/2} c^{3/2} \log ^{\frac{3}{2}}(F) F^{c \left (a-\frac{b d}{e}\right )} \text{Erfi}\left (\frac{\sqrt{b} \sqrt{c} \sqrt{\log (F)} \sqrt{d+e x}}{\sqrt{e}}\right )}{3 e^{5/2}}-\frac{4 b c \log (F) F^{c (a+b x)}}{3 e^2 \sqrt{d+e x}}-\frac{2 F^{c (a+b x)}}{3 e (d+e x)^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[F^(c*(a + b*x))/(d + e*x)^(5/2),x]

[Out]

(-2*F^(c*(a + b*x)))/(3*e*(d + e*x)^(3/2)) - (4*b*c*F^(c*(a + b*x))*Log[F])/(3*e
^2*Sqrt[d + e*x]) + (4*b^(3/2)*c^(3/2)*F^(c*(a - (b*d)/e))*Sqrt[Pi]*Erfi[(Sqrt[b
]*Sqrt[c]*Sqrt[d + e*x]*Sqrt[Log[F]])/Sqrt[e]]*Log[F]^(3/2))/(3*e^(5/2))

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Rubi in Sympy [A]  time = 28.4258, size = 126, normalized size = 0.97 \[ - \frac{4 F^{c \left (a + b x\right )} b c \log{\left (F \right )}}{3 e^{2} \sqrt{d + e x}} - \frac{2 F^{c \left (a + b x\right )}}{3 e \left (d + e x\right )^{\frac{3}{2}}} + \frac{4 \sqrt{\pi } F^{\frac{c \left (a e - b d\right )}{e}} b^{\frac{3}{2}} c^{\frac{3}{2}} \log{\left (F \right )}^{\frac{3}{2}} \operatorname{erfi}{\left (\frac{\sqrt{b} \sqrt{c} \sqrt{d + e x} \sqrt{\log{\left (F \right )}}}{\sqrt{e}} \right )}}{3 e^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(F**(c*(b*x+a))/(e*x+d)**(5/2),x)

[Out]

-4*F**(c*(a + b*x))*b*c*log(F)/(3*e**2*sqrt(d + e*x)) - 2*F**(c*(a + b*x))/(3*e*
(d + e*x)**(3/2)) + 4*sqrt(pi)*F**(c*(a*e - b*d)/e)*b**(3/2)*c**(3/2)*log(F)**(3
/2)*erfi(sqrt(b)*sqrt(c)*sqrt(d + e*x)*sqrt(log(F))/sqrt(e))/(3*e**(5/2))

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Mathematica [A]  time = 0.428486, size = 140, normalized size = 1.08 \[ -\frac{2 F^{c \left (a-\frac{b d}{e}\right )} \left (-2 \sqrt{\pi } e \left (-\frac{b c \log (F) (d+e x)}{e}\right )^{3/2} \text{Erf}\left (\sqrt{-\frac{b c \log (F) (d+e x)}{e}}\right )+e F^{\frac{b c (d+e x)}{e}}+2 b c \log (F) (d+e x) \left (F^{\frac{b c (d+e x)}{e}}-\sqrt{\pi } \sqrt{-\frac{b c \log (F) (d+e x)}{e}}\right )\right )}{3 e^2 (d+e x)^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[F^(c*(a + b*x))/(d + e*x)^(5/2),x]

[Out]

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

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Maple [F]  time = 0.026, size = 0, normalized size = 0. \[ \int{{F}^{c \left ( bx+a \right ) } \left ( ex+d \right ) ^{-{\frac{5}{2}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(F^(c*(b*x+a))/(e*x+d)^(5/2),x)

[Out]

int(F^(c*(b*x+a))/(e*x+d)^(5/2),x)

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Maxima [A]  time = 0.883238, size = 81, normalized size = 0.62 \[ -\frac{\left (-\frac{{\left (e x + d\right )} b c \log \left (F\right )}{e}\right )^{\frac{3}{2}} F^{a c} \Gamma \left (-\frac{3}{2}, -\frac{{\left (e x + d\right )} b c \log \left (F\right )}{e}\right )}{{\left (e x + d\right )}^{\frac{3}{2}} F^{\frac{b c d}{e}} e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(F^((b*x + a)*c)/(e*x + d)^(5/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.254141, size = 203, normalized size = 1.56 \[ \frac{2 \,{\left (\frac{2 \, \sqrt{\pi }{\left (b^{2} c^{2} e x + b^{2} c^{2} d\right )} \sqrt{e x + d} \operatorname{erf}\left (\sqrt{e x + d} \sqrt{-\frac{b c \log \left (F\right )}{e}}\right ) \log \left (F\right )^{2}}{F^{\frac{b c d - a c e}{e}}} -{\left (e^{2} + 2 \,{\left (b c e^{2} x + b c d e\right )} \log \left (F\right )\right )} \sqrt{-\frac{b c \log \left (F\right )}{e}} F^{b c x + a c}\right )}}{3 \,{\left (e^{4} x + d e^{3}\right )} \sqrt{e x + d} \sqrt{-\frac{b c \log \left (F\right )}{e}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(F^((b*x + a)*c)/(e*x + d)^(5/2),x, algorithm="fricas")

[Out]

2/3*(2*sqrt(pi)*(b^2*c^2*e*x + b^2*c^2*d)*sqrt(e*x + d)*erf(sqrt(e*x + d)*sqrt(-
b*c*log(F)/e))*log(F)^2/F^((b*c*d - a*c*e)/e) - (e^2 + 2*(b*c*e^2*x + b*c*d*e)*l
og(F))*sqrt(-b*c*log(F)/e)*F^(b*c*x + a*c))/((e^4*x + d*e^3)*sqrt(e*x + d)*sqrt(
-b*c*log(F)/e))

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(F**(c*(b*x+a))/(e*x+d)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{F^{{\left (b x + a\right )} c}}{{\left (e x + d\right )}^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(F^((b*x + a)*c)/(e*x + d)^(5/2),x, algorithm="giac")

[Out]

integrate(F^((b*x + a)*c)/(e*x + d)^(5/2), x)