3.2.76 \(\int f^{a+b x^n} x^2 \, dx\) [176]

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

[Out]

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

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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 = {2250} \begin {gather*} -\frac {x^3 f^a \left (-b \log (f) x^n\right )^{-3/n} \text {Gamma}\left (\frac {3}{n},-b \log (f) x^n\right )}{n} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[f^(a + b*x^n)*x^2,x]

[Out]

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

Rule 2250

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Simp[(-F^a)*((e +
f*x)^(m + 1)/(f*n*((-b)*(c + d*x)^n*Log[F])^((m + 1)/n)))*Gamma[(m + 1)/n, (-b)*(c + d*x)^n*Log[F]], x] /; Fre
eQ[{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^2 \, dx &=-\frac {f^a x^3 \Gamma \left (\frac {3}{n},-b x^n \log (f)\right ) \left (-b x^n \log (f)\right )^{-3/n}}{n}\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 39, normalized size = 1.00 \begin {gather*} -\frac {f^a x^3 \Gamma \left (\frac {3}{n},-b x^n \log (f)\right ) \left (-b x^n \log (f)\right )^{-3/n}}{n} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[f^(a + b*x^n)*x^2,x]

[Out]

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

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.04, size = 212, normalized size = 5.44

method	result	size
meijerg	\(\frac {f^{a} \left (-b \right )^{-\frac {3}{n}} \ln \left (f \right )^{-\frac {3}{n}} \left (\frac {n \,x^{3} \left (-b \right )^{\frac {3}{n}} \ln \left (f \right )^{\frac {3}{n}} \left (\ln \left (f \right ) x^{n} b n +n +3\right ) \Gamma \left (1-\frac {3}{n}\right ) \Gamma \left (\frac {3+n}{n}+1\right ) L_{-\frac {3}{n}}^{\left (\frac {3+n}{n}\right )}\left (b \,x^{n} \ln \left (f \right )\right )}{3 \left (3+n \right ) \Gamma \left (-\frac {3}{n}+\frac {3+n}{n}+1\right )}-\frac {n^{2} x^{3+n} \left (-b \right )^{\frac {3}{n}} \ln \left (f \right )^{1+\frac {3}{n}} b L_{-\frac {3}{n}}^{\left (\frac {3+n}{n}+1\right )}\left (b \,x^{n} \ln \left (f \right )\right ) \Gamma \left (1-\frac {3}{n}\right ) \Gamma \left (\frac {3+n}{n}+1\right )}{3 \left (3+n \right ) \Gamma \left (-\frac {3}{n}+\frac {3+n}{n}+1\right )}\right )}{n}\)	\(212\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(f^(a+b*x^n)*x^2,x,method=_RETURNVERBOSE)

[Out]

f^a*(-b)^(-3/n)*ln(f)^(-3/n)/n*(1/3*n*x^3*(-b)^(3/n)*ln(f)^(3/n)*(ln(f)*x^n*b*n+n+3)/(3+n)/GAMMA(-3/n+(3+n)/n+
1)*GAMMA(1-3/n)*GAMMA((3+n)/n+1)*LaguerreL(-3/n,(3+n)/n,b*x^n*ln(f))-1/3*n^2*x^(3+n)*(-b)^(3/n)*ln(f)^(1+3/n)*
b/(3+n)*LaguerreL(-3/n,(3+n)/n+1,b*x^n*ln(f))*GAMMA(1-3/n)*GAMMA((3+n)/n+1)/GAMMA(-3/n+(3+n)/n+1))

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Maxima [A]
time = 0.07, size = 41, normalized size = 1.05 \begin {gather*} -\frac {f^{a} x^{3} \Gamma \left (\frac {3}{n}, -b x^{n} \log \left (f\right )\right )}{\left (-b x^{n} \log \left (f\right )\right )^{\frac {3}{n}} n} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(a+b*x^n)*x^2,x, algorithm="maxima")

[Out]

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

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(a+b*x^n)*x^2,x, algorithm="fricas")

[Out]

integral(f^(b*x^n + a)*x^2, x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \begin {cases} - \frac {b f^{a} f^{b x^{n}} n x^{3} x^{n} \log {\left (f \right )}}{3 n + 9} + \frac {f^{a} f^{b x^{n}} n x^{3}}{3 n + 9} + \frac {3 f^{a} f^{b x^{n}} x^{3}}{3 n + 9} & \text {for}\: n \neq -3 \\\int f^{a + \frac {b}{x^{3}}} x^{2}\, dx & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f**(a+b*x**n)*x**2,x)

[Out]

Piecewise((-b*f**a*f**(b*x**n)*n*x**3*x**n*log(f)/(3*n + 9) + f**a*f**(b*x**n)*n*x**3/(3*n + 9) + 3*f**a*f**(b
*x**n)*x**3/(3*n + 9), Ne(n, -3)), (Integral(f**(a + b/x**3)*x**2, x), True))

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(a+b*x^n)*x^2,x, algorithm="giac")

[Out]

integrate(f^(b*x^n + a)*x^2, x)

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Mupad [B]
time = 3.51, size = 54, normalized size = 1.38 \begin {gather*} \frac {f^a\,x^3\,{\mathrm {e}}^{\frac {b\,x^n\,\ln \left (f\right )}{2}}\,{\mathrm {M}}_{\frac {1}{2}-\frac {3}{2\,n},\frac {3}{2\,n}}\left (b\,x^n\,\ln \left (f\right )\right )}{3\,{\left (b\,x^n\,\ln \left (f\right )\right )}^{\frac {3}{2\,n}+\frac {1}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(f^(a + b*x^n)*x^2,x)

[Out]

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

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