3.71 \(\int f^{a+b x^2} x^9 \, dx\)

Optimal. Leaf size=65 \[ \frac{f^{a+b x^2} \left (b^4 x^8 \log ^4(f)-4 b^3 x^6 \log ^3(f)+12 b^2 x^4 \log ^2(f)-24 b x^2 \log (f)+24\right )}{2 b^5 \log ^5(f)} \]

[Out]

(f^(a + b*x^2)*(24 - 24*b*x^2*Log[f] + 12*b^2*x^4*Log[f]^2 - 4*b^3*x^6*Log[f]^3 + b^4*x^8*Log[f]^4))/(2*b^5*Lo
g[f]^5)

________________________________________________________________________________________

Rubi [C]  time = 0.0240763, antiderivative size = 24, normalized size of antiderivative = 0.37, 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} \[ \frac{f^a \text{Gamma}\left (5,-b x^2 \log (f)\right )}{2 b^5 \log ^5(f)} \]

Antiderivative was successfully verified.

[In]

Int[f^(a + b*x^2)*x^9,x]

[Out]

(f^a*Gamma[5, -(b*x^2*Log[f])])/(2*b^5*Log[f]^5)

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^2} x^9 \, dx &=\frac{f^a \Gamma \left (5,-b x^2 \log (f)\right )}{2 b^5 \log ^5(f)}\\ \end{align*}

Mathematica [C]  time = 0.0028788, size = 24, normalized size = 0.37 \[ \frac{f^a \text{Gamma}\left (5,-b x^2 \log (f)\right )}{2 b^5 \log ^5(f)} \]

Antiderivative was successfully verified.

[In]

Integrate[f^(a + b*x^2)*x^9,x]

[Out]

(f^a*Gamma[5, -(b*x^2*Log[f])])/(2*b^5*Log[f]^5)

________________________________________________________________________________________

Maple [A]  time = 0.008, size = 64, normalized size = 1. \begin{align*}{\frac{{f}^{b{x}^{2}+a} \left ( 24-24\,b{x}^{2}\ln \left ( f \right ) +12\,{b}^{2}{x}^{4} \left ( \ln \left ( f \right ) \right ) ^{2}-4\,{b}^{3}{x}^{6} \left ( \ln \left ( f \right ) \right ) ^{3}+{b}^{4}{x}^{8} \left ( \ln \left ( f \right ) \right ) ^{4} \right ) }{2\,{b}^{5} \left ( \ln \left ( f \right ) \right ) ^{5}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(f^(b*x^2+a)*x^9,x)

[Out]

1/2*f^(b*x^2+a)*(24-24*b*x^2*ln(f)+12*b^2*x^4*ln(f)^2-4*b^3*x^6*ln(f)^3+b^4*x^8*ln(f)^4)/b^5/ln(f)^5

________________________________________________________________________________________

Maxima [A]  time = 1.12678, size = 104, normalized size = 1.6 \begin{align*} \frac{{\left (b^{4} f^{a} x^{8} \log \left (f\right )^{4} - 4 \, b^{3} f^{a} x^{6} \log \left (f\right )^{3} + 12 \, b^{2} f^{a} x^{4} \log \left (f\right )^{2} - 24 \, b f^{a} x^{2} \log \left (f\right ) + 24 \, f^{a}\right )} f^{b x^{2}}}{2 \, b^{5} \log \left (f\right )^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(b*x^2+a)*x^9,x, algorithm="maxima")

[Out]

1/2*(b^4*f^a*x^8*log(f)^4 - 4*b^3*f^a*x^6*log(f)^3 + 12*b^2*f^a*x^4*log(f)^2 - 24*b*f^a*x^2*log(f) + 24*f^a)*f
^(b*x^2)/(b^5*log(f)^5)

________________________________________________________________________________________

Fricas [A]  time = 1.528, size = 161, normalized size = 2.48 \begin{align*} \frac{{\left (b^{4} x^{8} \log \left (f\right )^{4} - 4 \, b^{3} x^{6} \log \left (f\right )^{3} + 12 \, b^{2} x^{4} \log \left (f\right )^{2} - 24 \, b x^{2} \log \left (f\right ) + 24\right )} f^{b x^{2} + a}}{2 \, b^{5} \log \left (f\right )^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(b*x^2+a)*x^9,x, algorithm="fricas")

[Out]

1/2*(b^4*x^8*log(f)^4 - 4*b^3*x^6*log(f)^3 + 12*b^2*x^4*log(f)^2 - 24*b*x^2*log(f) + 24)*f^(b*x^2 + a)/(b^5*lo
g(f)^5)

________________________________________________________________________________________

Sympy [A]  time = 0.253196, size = 82, normalized size = 1.26 \begin{align*} \begin{cases} \frac{f^{a + b x^{2}} \left (b^{4} x^{8} \log{\left (f \right )}^{4} - 4 b^{3} x^{6} \log{\left (f \right )}^{3} + 12 b^{2} x^{4} \log{\left (f \right )}^{2} - 24 b x^{2} \log{\left (f \right )} + 24\right )}{2 b^{5} \log{\left (f \right )}^{5}} & \text{for}\: 2 b^{5} \log{\left (f \right )}^{5} \neq 0 \\\frac{x^{10}}{10} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f**(b*x**2+a)*x**9,x)

[Out]

Piecewise((f**(a + b*x**2)*(b**4*x**8*log(f)**4 - 4*b**3*x**6*log(f)**3 + 12*b**2*x**4*log(f)**2 - 24*b*x**2*l
og(f) + 24)/(2*b**5*log(f)**5), Ne(2*b**5*log(f)**5, 0)), (x**10/10, True))

________________________________________________________________________________________

Giac [A]  time = 1.25853, size = 90, normalized size = 1.38 \begin{align*} \frac{{\left (b^{4} x^{8} \log \left (f\right )^{4} - 4 \, b^{3} x^{6} \log \left (f\right )^{3} + 12 \, b^{2} x^{4} \log \left (f\right )^{2} - 24 \, b x^{2} \log \left (f\right ) + 24\right )} e^{\left (b x^{2} \log \left (f\right ) + a \log \left (f\right )\right )}}{2 \, b^{5} \log \left (f\right )^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(b*x^2+a)*x^9,x, algorithm="giac")

[Out]

1/2*(b^4*x^8*log(f)^4 - 4*b^3*x^6*log(f)^3 + 12*b^2*x^4*log(f)^2 - 24*b*x^2*log(f) + 24)*e^(b*x^2*log(f) + a*l
og(f))/(b^5*log(f)^5)