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

Optimal. Leaf size=39 \[ -\frac {x^2 f^a \left (-b \log (f) x^n\right )^{-2/n} \Gamma \left (\frac {2}{n},-b x^n \log (f)\right )}{n} \]

[Out]

-f^a*x^2*GAMMA(2/n,-b*x^n*ln(f))/n/((-b*x^n*ln(f))^(2/n))

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Rubi [A]  time = 0.01, antiderivative size = 39, 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 = {2218} \[ -\frac {x^2 f^a \left (-b \log (f) x^n\right )^{-2/n} \text {Gamma}\left (\frac {2}{n},-b \log (f) x^n\right )}{n} \]

Antiderivative was successfully verified.

[In]

Int[f^(a + b*x^n)*x,x]

[Out]

-((f^a*x^2*Gamma[2/n, -(b*x^n*Log[f])])/(n*(-(b*x^n*Log[f]))^(2/n)))

Rule 2218

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> -Simp[(F^a*(e + f*
x)^(m + 1)*Gamma[(m + 1)/n, -(b*(c + d*x)^n*Log[F])])/(f*n*(-(b*(c + d*x)^n*Log[F]))^((m + 1)/n)), x] /; FreeQ
[{F, a, b, c, d, e, f, m, n}, x] && EqQ[d*e - c*f, 0]

Rubi steps

\begin {align*} \int f^{a+b x^n} x \, dx &=-\frac {f^a x^2 \Gamma \left (\frac {2}{n},-b x^n \log (f)\right ) \left (-b x^n \log (f)\right )^{-2/n}}{n}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 39, normalized size = 1.00 \[ -\frac {x^2 f^a \left (-b \log (f) x^n\right )^{-2/n} \Gamma \left (\frac {2}{n},-b x^n \log (f)\right )}{n} \]

Antiderivative was successfully verified.

[In]

Integrate[f^(a + b*x^n)*x,x]

[Out]

-((f^a*x^2*Gamma[2/n, -(b*x^n*Log[f])])/(n*(-(b*x^n*Log[f]))^(2/n)))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(f^(b*x^n + a)*x, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(f^(b*x^n + a)*x, x)

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maple [F]  time = 0.06, size = 0, normalized size = 0.00 \[ \int x \,f^{b \,x^{n}+a}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.08, size = 41, normalized size = 1.05 \[ -\frac {f^{a} x^{2} \Gamma \left (\frac {2}{n}, -b x^{n} \log \relax (f)\right )}{\left (-b x^{n} \log \relax (f)\right )^{\frac {2}{n}} n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-f^a*x^2*gamma(2/n, -b*x^n*log(f))/((-b*x^n*log(f))^(2/n)*n)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(f^(a + b*x^n)*x,x)

[Out]

int(f^(a + b*x^n)*x, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \begin {cases} - \frac {b f^{a} f^{b x^{n}} n x^{2} x^{n} \log {\relax (f )}}{2 n + 4} + \frac {f^{a} f^{b x^{n}} n x^{2}}{2 n + 4} + \frac {2 f^{a} f^{b x^{n}} x^{2}}{2 n + 4} & \text {for}\: n \neq -2 \\\int f^{a + \frac {b}{x^{2}}} x\, dx & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-b*f**a*f**(b*x**n)*n*x**2*x**n*log(f)/(2*n + 4) + f**a*f**(b*x**n)*n*x**2/(2*n + 4) + 2*f**a*f**(b
*x**n)*x**2/(2*n + 4), Ne(n, -2)), (Integral(f**(a + b/x**2)*x, x), True))

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