3.1.24 \(\int F^{c (a+b x)} (d+e x)^{-m} \, dx\) [24]

Optimal. Leaf size=69 \[ \frac {F^{c \left (a-\frac {b d}{e}\right )} (d+e x)^{-m} \Gamma \left (1-m,-\frac {b c (d+e x) \log (F)}{e}\right ) \left (-\frac {b c (d+e x) \log (F)}{e}\right )^m}{b c \log (F)} \]

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {2212} \begin {gather*} \frac {(d+e x)^{-m} F^{c \left (a-\frac {b d}{e}\right )} \left (-\frac {b c \log (F) (d+e x)}{e}\right )^m \text {Gamma}\left (1-m,-\frac {b c \log (F) (d+e x)}{e}\right )}{b c \log (F)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[F^(c*(a + b*x))/(d + e*x)^m,x]

[Out]

(F^(c*(a - (b*d)/e))*Gamma[1 - m, -((b*c*(d + e*x)*Log[F])/e)]*(-((b*c*(d + e*x)*Log[F])/e))^m)/(b*c*(d + e*x)
^m*Log[F])

Rule 2212

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))*((c_.) + (d_.)*(x_))^(m_), x_Symbol] :> Simp[(-F^(g*(e - c*(f/d))))*((c
+ d*x)^FracPart[m]/(d*((-f)*g*(Log[F]/d))^(IntPart[m] + 1)*((-f)*g*Log[F]*((c + d*x)/d))^FracPart[m]))*Gamma[m
+ 1, ((-f)*g*(Log[F]/d))*(c + d*x)], x] /; FreeQ[{F, c, d, e, f, g, m}, x] &&  !IntegerQ[m]

Rubi steps

\begin {align*} \int F^{c (a+b x)} (d+e x)^{-m} \, dx &=\frac {F^{c \left (a-\frac {b d}{e}\right )} (d+e x)^{-m} \Gamma \left (1-m,-\frac {b c (d+e x) \log (F)}{e}\right ) \left (-\frac {b c (d+e x) \log (F)}{e}\right )^m}{b c \log (F)}\\ \end {align*}

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Mathematica [A]
time = 0.03, size = 69, normalized size = 1.00 \begin {gather*} \frac {F^{c \left (a-\frac {b d}{e}\right )} (d+e x)^{-m} \Gamma \left (1-m,-\frac {b c (d+e x) \log (F)}{e}\right ) \left (-\frac {b c (d+e x) \log (F)}{e}\right )^m}{b c \log (F)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(F^(c*(a - (b*d)/e))*Gamma[1 - m, -((b*c*(d + e*x)*Log[F])/e)]*(-((b*c*(d + e*x)*Log[F])/e))^m)/(b*c*(d + e*x)
^m*Log[F])

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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int F^{c \left (b x +a \right )} \left (e x +d \right )^{-m}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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)^m),x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.11, size = 67, normalized size = 0.97 \begin {gather*} \frac {e^{\left ({\left (m e \log \left (-b c e^{\left (-1\right )} \log \left (F\right )\right ) - {\left (b c d - a c e\right )} \log \left (F\right )\right )} e^{\left (-1\right )}\right )} \Gamma \left (-m + 1, -{\left (b c x e + b c d\right )} e^{\left (-1\right )} \log \left (F\right )\right )}{b c \log \left (F\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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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)^m),x, algorithm="giac")

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {F^{c\,\left (a+b\,x\right )}}{{\left (d+e\,x\right )}^m} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(F^(c*(a + b*x))/(d + e*x)^m,x)

[Out]

int(F^(c*(a + b*x))/(d + e*x)^m, x)

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