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

Optimal. Leaf size=35 \[ -\frac {x f^a \left (-b \log (f) x^n\right )^{-1/n} \Gamma \left (\frac {1}{n},-b x^n \log (f)\right )}{n} \]

[Out]

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

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Rubi [A]  time = 0.00, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2208} \[ -\frac {x f^a \left (-b \log (f) x^n\right )^{-1/n} \text {Gamma}\left (\frac {1}{n},-b \log (f) x^n\right )}{n} \]

Antiderivative was successfully verified.

[In]

Int[f^(a + b*x^n),x]

[Out]

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

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Mathematica [A]  time = 0.00, size = 35, normalized size = 1.00 \[ -\frac {x f^a \left (-b \log (f) x^n\right )^{-1/n} \Gamma \left (\frac {1}{n},-b x^n \log (f)\right )}{n} \]

Antiderivative was successfully verified.

[In]

Integrate[f^(a + b*x^n),x]

[Out]

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

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fricas [F]  time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left (f^{b x^{n} + a}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(f^(b*x^n + a), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(f^(b*x^n + a), x)

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maple [F]  time = 0.05, size = 0, normalized size = 0.00 \[ \int f^{b \,x^{n}+a}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.09, size = 35, normalized size = 1.00 \[ -\frac {f^{a} x \Gamma \left (\frac {1}{n}, -b x^{n} \log \relax (f)\right )}{\left (-b x^{n} \log \relax (f)\right )^{\left (\frac {1}{n}\right )} n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-f^a*x*gamma(1/n, -b*x^n*log(f))/((-b*x^n*log(f))^(1/n)*n)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \[ \int f^{a+b\,x^n} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(f^(a + b*x^n),x)

[Out]

int(f^(a + b*x^n), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \begin {cases} - \frac {b f^{a} f^{b x^{n}} n x x^{n} \log {\relax (f )}}{n + 1} + \frac {f^{a} f^{b x^{n}} n x}{n + 1} + \frac {f^{a} f^{b x^{n}} x}{n + 1} & \text {for}\: n \neq -1 \\\int f^{a + \frac {b}{x}}\, dx & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-b*f**a*f**(b*x**n)*n*x*x**n*log(f)/(n + 1) + f**a*f**(b*x**n)*n*x/(n + 1) + f**a*f**(b*x**n)*x/(n
+ 1), Ne(n, -1)), (Integral(f**(a + b/x), x), True))

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