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

Optimal. Leaf size=46 \[ \frac{(c+d x) F^{a+\frac{b}{c+d x}}}{d}-\frac{b F^a \log (F) \text{ExpIntegralEi}\left (\frac{b \log (F)}{c+d x}\right )}{d} \]

[Out]

(F^(a + b/(c + d*x))*(c + d*x))/d - (b*F^a*ExpIntegralEi[(b*Log[F])/(c + d*x)]*L
og[F])/d

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Rubi [A]  time = 0.0856284, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{(c+d x) F^{a+\frac{b}{c+d x}}}{d}-\frac{b F^a \log (F) \text{ExpIntegralEi}\left (\frac{b \log (F)}{c+d x}\right )}{d} \]

Antiderivative was successfully verified.

[In]  Int[F^(a + b/(c + d*x)),x]

[Out]

(F^(a + b/(c + d*x))*(c + d*x))/d - (b*F^a*ExpIntegralEi[(b*Log[F])/(c + d*x)]*L
og[F])/d

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Rubi in Sympy [A]  time = 6.16692, size = 37, normalized size = 0.8 \[ - \frac{F^{a} b \log{\left (F \right )} \operatorname{Ei}{\left (\frac{b \log{\left (F \right )}}{c + d x} \right )}}{d} + \frac{F^{a + \frac{b}{c + d x}} \left (c + d x\right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-F**a*b*log(F)*Ei(b*log(F)/(c + d*x))/d + F**(a + b/(c + d*x))*(c + d*x)/d

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Mathematica [A]  time = 0.0259141, size = 42, normalized size = 0.91 \[ \frac{F^a \left ((c+d x) F^{\frac{b}{c+d x}}-b \log (F) \text{ExpIntegralEi}\left (\frac{b \log (F)}{c+d x}\right )\right )}{d} \]

Antiderivative was successfully verified.

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

[Out]

(F^a*(F^(b/(c + d*x))*(c + d*x) - b*ExpIntegralEi[(b*Log[F])/(c + d*x)]*Log[F]))
/d

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Maple [A]  time = 0.023, size = 71, normalized size = 1.5 \[{F}^{{\frac{xda+ac+b}{dx+c}}}x+{\frac{c}{d}{F}^{{\frac{xda+ac+b}{dx+c}}}}+{\frac{b\ln \left ( F \right ){F}^{a}}{d}{\it Ei} \left ( 1,-{\frac{b\ln \left ( F \right ) }{dx+c}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(F^(a+b/(d*x+c)),x)

[Out]

F^((a*d*x+a*c+b)/(d*x+c))*x+1/d*F^((a*d*x+a*c+b)/(d*x+c))*c+b/d*ln(F)*F^a*Ei(1,-
b*ln(F)/(d*x+c))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.251727, size = 69, normalized size = 1.5 \[ -\frac{F^{a} b{\rm Ei}\left (\frac{b \log \left (F\right )}{d x + c}\right ) \log \left (F\right ) -{\left (d x + c\right )} F^{\frac{a d x + a c + b}{d x + c}}}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(F^a*b*Ei(b*log(F)/(d*x + c))*log(F) - (d*x + c)*F^((a*d*x + a*c + b)/(d*x + c)
))/d

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(F**(a+b/(d*x+c)),x)

[Out]

Integral(F**(a + b/(c + d*x)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int F^{a + \frac{b}{d x + c}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(F^(a + b/(d*x + c)), x)