3.1.7 \(\int \frac {F^{c (a+b x)}}{d+e x} \, dx\) [7]

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

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {2209} \begin {gather*} \frac {F^{c \left (a-\frac {b d}{e}\right )} \text {Ei}\left (\frac {b c (d+e x) \log (F)}{e}\right )}{e} \end {gather*}

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

Rule 2209

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[(F^(g*(e - c*(f/d)))/d)*ExpInteg
ralEi[f*g*(c + d*x)*(Log[F]/d)], x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !TrueQ[$UseGamma]

Rubi steps

\begin {align*} \int \frac {F^{c (a+b x)}}{d+e x} \, dx &=\frac {F^{c \left (a-\frac {b d}{e}\right )} \text {Ei}\left (\frac {b c (d+e x) \log (F)}{e}\right )}{e}\\ \end {align*}

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Mathematica [A]
time = 0.05, size = 31, normalized size = 1.00 \begin {gather*} \frac {F^{c \left (a-\frac {b d}{e}\right )} \text {Ei}\left (\frac {b c (d+e x) \log (F)}{e}\right )}{e} \end {gather*}

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 = 0.07, size = 56, normalized size = 1.81

method	result	size
risch	\(-\frac {F^{\frac {c \left (a e -b d \right )}{e}} \expIntegral \left (1, -b c x \ln \left (F \right )-c a \ln \left (F \right )-\frac {-\ln \left (F \right ) a c e +\ln \left (F \right ) b c d}{e}\right )}{e}\)	\(56\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(F^(c*(b*x+a))/(e*x+d),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.37, size = 38, normalized size = 1.23 \begin {gather*} \frac {{\rm Ei}\left ({\left (b c x e + b c d\right )} e^{\left (-1\right )} \log \left (F\right )\right ) e^{\left (-1\right )}}{F^{{\left (b c d - a c e\right )} e^{\left (-1\right )}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {F^{c \left (a + b x\right )}}{d + e x}\, dx \end {gather*}

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 [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {F^{c\,\left (a+b\,x\right )}}{d+e\,x} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(F^(c*(a + b*x))/(d + e*x),x)

[Out]

int(F^(c*(a + b*x))/(d + e*x), x)

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