3.182 \(\int \frac {f^{a+b x^n}}{x^4} \, dx\)

Optimal. Leaf size=39 \[ -\frac {f^a \left (-b \log (f) x^n\right )^{3/n} \Gamma \left (-\frac {3}{n},-b x^n \log (f)\right )}{n x^3} \]

[Out]

-f^a*GAMMA(-3/n,-b*x^n*ln(f))*(-b*x^n*ln(f))^(3/n)/n/x^3

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Rubi [A]  time = 0.02, antiderivative size = 39, normalized size of antiderivative = 1.00, 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 = {2218} \[ -\frac {f^a \left (-b \log (f) x^n\right )^{3/n} \text {Gamma}\left (-\frac {3}{n},-b \log (f) x^n\right )}{n x^3} \]

Antiderivative was successfully verified.

[In]

Int[f^(a + b*x^n)/x^4,x]

[Out]

-((f^a*Gamma[-3/n, -(b*x^n*Log[f])]*(-(b*x^n*Log[f]))^(3/n))/(n*x^3))

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 \frac {f^{a+b x^n}}{x^4} \, dx &=-\frac {f^a \Gamma \left (-\frac {3}{n},-b x^n \log (f)\right ) \left (-b x^n \log (f)\right )^{3/n}}{n x^3}\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 39, normalized size = 1.00 \[ -\frac {f^a \left (-b \log (f) x^n\right )^{3/n} \Gamma \left (-\frac {3}{n},-b x^n \log (f)\right )}{n x^3} \]

Antiderivative was successfully verified.

[In]

Integrate[f^(a + b*x^n)/x^4,x]

[Out]

-((f^a*Gamma[-3/n, -(b*x^n*Log[f])]*(-(b*x^n*Log[f]))^(3/n))/(n*x^3))

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fricas [F]  time = 0.40, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {f^{b x^{n} + a}}{x^{4}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(a+b*x^n)/x^4,x, algorithm="fricas")

[Out]

integral(f^(b*x^n + a)/x^4, x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {f^{b x^{n} + a}}{x^{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(a+b*x^n)/x^4,x, algorithm="giac")

[Out]

integrate(f^(b*x^n + a)/x^4, x)

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maple [F]  time = 0.04, size = 0, normalized size = 0.00 \[ \int \frac {f^{b \,x^{n}+a}}{x^{4}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(f^(b*x^n+a)/x^4,x)

[Out]

int(f^(b*x^n+a)/x^4,x)

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maxima [A]  time = 1.14, size = 39, normalized size = 1.00 \[ -\frac {\left (-b x^{n} \log \relax (f)\right )^{\frac {3}{n}} f^{a} \Gamma \left (-\frac {3}{n}, -b x^{n} \log \relax (f)\right )}{n x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(a+b*x^n)/x^4,x, algorithm="maxima")

[Out]

-(-b*x^n*log(f))^(3/n)*f^a*gamma(-3/n, -b*x^n*log(f))/(n*x^3)

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mupad [B]  time = 3.48, size = 52, normalized size = 1.33 \[ -\frac {f^a\,{\mathrm {e}}^{\frac {b\,x^n\,\ln \relax (f)}{2}}\,{\mathrm {M}}_{\frac {3}{2\,n}+\frac {1}{2},-\frac {3}{2\,n}}\left (b\,x^n\,\ln \relax (f)\right )\,{\left (b\,x^n\,\ln \relax (f)\right )}^{\frac {3}{2\,n}-\frac {1}{2}}}{3\,x^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(f^(a + b*x^n)/x^4,x)

[Out]

-(f^a*exp((b*x^n*log(f))/2)*whittakerM(3/(2*n) + 1/2, -3/(2*n), b*x^n*log(f))*(b*x^n*log(f))^(3/(2*n) - 1/2))/
(3*x^3)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \begin {cases} \frac {b f^{a} f^{b x^{n}} n x^{n} \log {\relax (f )}}{3 n x^{3} - 9 x^{3}} - \frac {f^{a} f^{b x^{n}} n}{3 n x^{3} - 9 x^{3}} + \frac {3 f^{a} f^{b x^{n}}}{3 n x^{3} - 9 x^{3}} & \text {for}\: n \neq 3 \\\int \frac {f^{a + b x^{3}}}{x^{4}}\, dx & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f**(a+b*x**n)/x**4,x)

[Out]

Piecewise((b*f**a*f**(b*x**n)*n*x**n*log(f)/(3*n*x**3 - 9*x**3) - f**a*f**(b*x**n)*n/(3*n*x**3 - 9*x**3) + 3*f
**a*f**(b*x**n)/(3*n*x**3 - 9*x**3), Ne(n, 3)), (Integral(f**(a + b*x**3)/x**4, x), True))

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