3.250 \(\int f^{c (a+b x)^n} \, dx\)

Optimal. Leaf size=47 \[ -\frac {(a+b x) \left (-c \log (f) (a+b x)^n\right )^{-1/n} \Gamma \left (\frac {1}{n},-c (a+b x)^n \log (f)\right )}{b n} \]

[Out]

-(b*x+a)*GAMMA(1/n,-c*(b*x+a)^n*ln(f))/b/n/((-c*(b*x+a)^n*ln(f))^(1/n))

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Rubi [A]  time = 0.01, antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {2208} \[ -\frac {(a+b x) \left (-c \log (f) (a+b x)^n\right )^{-1/n} \text {Gamma}\left (\frac {1}{n},-c \log (f) (a+b x)^n\right )}{b n} \]

Antiderivative was successfully verified.

[In]

Int[f^(c*(a + b*x)^n),x]

[Out]

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

Rule 2208

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_)), x_Symbol] :> -Simp[(F^a*(c + d*x)*Gamma[1/n, -(b*(c + d*x)
^n*Log[F])])/(d*n*(-(b*(c + d*x)^n*Log[F]))^(1/n)), x] /; FreeQ[{F, a, b, c, d, n}, x] &&  !IntegerQ[2/n]

Rubi steps

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

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Mathematica [A]  time = 0.01, size = 47, normalized size = 1.00 \[ -\frac {(a+b x) \left (-c \log (f) (a+b x)^n\right )^{-1/n} \Gamma \left (\frac {1}{n},-c (a+b x)^n \log (f)\right )}{b n} \]

Antiderivative was successfully verified.

[In]

Integrate[f^(c*(a + b*x)^n),x]

[Out]

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

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fricas [F]  time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left (f^{{\left (b x + a\right )}^{n} c}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(f^((b*x + a)^n*c), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int f^{{\left (b x + a\right )}^{n} c}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(f^((b*x + a)^n*c), x)

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maple [F]  time = 0.04, size = 0, normalized size = 0.00 \[ \int f^{c \left (b x +a \right )^{n}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int f^{{\left (b x + a\right )}^{n} c}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(f^((b*x + a)^n*c), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int f^{c\,{\left (a+b\,x\right )}^n} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(f^(c*(a + b*x)^n),x)

[Out]

int(f^(c*(a + b*x)^n), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int f^{c \left (a + b x\right )^{n}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(f**(c*(a + b*x)**n), x)

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