3.242 \(\int f^{c (a+b x)} x^m \, dx\)

Optimal. Leaf size=41 \[ \frac{x^m f^{a c} (-b c x \log (f))^{-m} \text{Gamma}(m+1,-b c x \log (f))}{b c \log (f)} \]

[Out]

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

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Rubi [A]  time = 0.0217245, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {2181} \[ \frac{x^m f^{a c} (-b c x \log (f))^{-m} \text{Gamma}(m+1,-b c x \log (f))}{b c \log (f)} \]

Antiderivative was successfully verified.

[In]

Int[f^(c*(a + b*x))*x^m,x]

[Out]

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

Rule 2181

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

Rubi steps

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

Mathematica [A]  time = 0.0078306, size = 36, normalized size = 0.88 \[ x^{m+1} \left (-f^{a c}\right ) (-b c x \log (f))^{-m-1} \text{Gamma}(m+1,-b c x \log (f)) \]

Antiderivative was successfully verified.

[In]

Integrate[f^(c*(a + b*x))*x^m,x]

[Out]

-(f^(a*c)*x^(1 + m)*Gamma[1 + m, -(b*c*x*Log[f])]*(-(b*c*x*Log[f]))^(-1 - m))

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Maple [B]  time = 0.036, size = 117, normalized size = 2.9 \begin{align*} -{\frac{{f}^{ac} \left ( -bc \right ) ^{-m} \left ( \ln \left ( f \right ) \right ) ^{-m-1} \left ({x}^{m} \left ( -bc \right ) ^{m} \left ( \ln \left ( f \right ) \right ) ^{m}m\Gamma \left ( m \right ) \left ( -bcx\ln \left ( f \right ) \right ) ^{-m}-{x}^{m} \left ( -bc \right ) ^{m} \left ( \ln \left ( f \right ) \right ) ^{m}{{\rm e}^{bcx\ln \left ( f \right ) }}-{x}^{m} \left ( -bc \right ) ^{m} \left ( \ln \left ( f \right ) \right ) ^{m}m \left ( -bcx\ln \left ( f \right ) \right ) ^{-m}\Gamma \left ( m,-bcx\ln \left ( f \right ) \right ) \right ) }{bc}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(f^(c*(b*x+a))*x^m,x)

[Out]

-f^(a*c)*(-b*c)^(-m)*ln(f)^(-m-1)/b/c*(x^m*(-b*c)^m*ln(f)^m*m*GAMMA(m)*(-b*c*x*ln(f))^(-m)-x^m*(-b*c)^m*ln(f)^
m*exp(b*c*x*ln(f))-x^m*(-b*c)^m*ln(f)^m*m*(-b*c*x*ln(f))^(-m)*GAMMA(m,-b*c*x*ln(f)))

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Maxima [A]  time = 1.26006, size = 49, normalized size = 1.2 \begin{align*} -\left (-b c x \log \left (f\right )\right )^{-m - 1} f^{a c} x^{m + 1} \Gamma \left (m + 1, -b c x \log \left (f\right )\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(c*(b*x+a))*x^m,x, algorithm="maxima")

[Out]

-(-b*c*x*log(f))^(-m - 1)*f^(a*c)*x^(m + 1)*gamma(m + 1, -b*c*x*log(f))

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Fricas [A]  time = 1.5318, size = 105, normalized size = 2.56 \begin{align*} \frac{e^{\left (a c \log \left (f\right ) - m \log \left (-b c \log \left (f\right )\right )\right )} \Gamma \left (m + 1, -b c x \log \left (f\right )\right )}{b c \log \left (f\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(c*(b*x+a))*x^m,x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int f^{c \left (a + b x\right )} x^{m}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f**(c*(b*x+a))*x**m,x)

[Out]

Integral(f**(c*(a + b*x))*x**m, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int f^{{\left (b x + a\right )} c} x^{m}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(c*(b*x+a))*x^m,x, algorithm="giac")

[Out]

integrate(f^((b*x + a)*c)*x^m, x)