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

Optimal. Leaf size=31 \[ \frac{F^{c \left (a-\frac{b d}{e}\right )} \text{ExpIntegralEi}\left (\frac{b c \log (F) (d+e x)}{e}\right )}{e} \]

[Out]

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

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Rubi [A]  time = 0.0404631, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059 \[ \frac{F^{c \left (a-\frac{b d}{e}\right )} \text{ExpIntegralEi}\left (\frac{b c \log (F) (d+e x)}{e}\right )}{e} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 5.15295, size = 27, normalized size = 0.87 \[ \frac{F^{\frac{c \left (a e - b d\right )}{e}} \operatorname{Ei}{\left (\frac{b c \left (d + e x\right ) \log{\left (F \right )}}{e} \right )}}{e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0161601, size = 31, normalized size = 1. \[ \frac{F^{c \left (a-\frac{b d}{e}\right )} \text{ExpIntegralEi}\left (\frac{b c \log (F) (d+e x)}{e}\right )}{e} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.034, size = 56, normalized size = 1.8 \[ -{\frac{1}{e}{F}^{{\frac{c \left ( ea-bd \right ) }{e}}}{\it Ei} \left ( 1,-bcx\ln \left ( F \right ) -\ln \left ( F \right ) ac-{\frac{-eac\ln \left ( F \right ) +\ln \left ( F \right ) bcd}{e}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(F^(c*(b*x+a))/(e*x+d),x)

[Out]

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

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Maxima [A]  time = 0.752654, size = 50, normalized size = 1.61 \[ -\frac{F^{a c} exp_integral_e\left (1, -\frac{{\left (e x + d\right )} b c \log \left (F\right )}{e}\right )}{F^{\frac{b c d}{e}} e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-F^(a*c)*exp_integral_e(1, -(e*x + d)*b*c*log(F)/e)/(F^(b*c*d/e)*e)

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Fricas [A]  time = 0.247157, size = 53, normalized size = 1.71 \[ \frac{{\rm Ei}\left (\frac{{\left (b c e x + b c d\right )} \log \left (F\right )}{e}\right )}{F^{\frac{b c d - a c e}{e}} e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Ei((b*c*e*x + b*c*d)*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. \[ \int \frac{F^{c \left (a + b x\right )}}{d + e x}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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