3.96 \(\int f^{a+b x^3} x^{17} \, dx\)

Optimal. Leaf size=78 \[ -\frac{f^{a+b x^3} \left (-b^5 x^{15} \log ^5(f)+5 b^4 x^{12} \log ^4(f)-20 b^3 x^9 \log ^3(f)+60 b^2 x^6 \log ^2(f)-120 b x^3 \log (f)+120\right )}{3 b^6 \log ^6(f)} \]

[Out]

-(f^(a + b*x^3)*(120 - 120*b*x^3*Log[f] + 60*b^2*x^6*Log[f]^2 - 20*b^3*x^9*Log[f]^3 + 5*b^4*x^12*Log[f]^4 - b^
5*x^15*Log[f]^5))/(3*b^6*Log[f]^6)

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Rubi [C]  time = 0.0224891, antiderivative size = 24, normalized size of antiderivative = 0.31, 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} \[ -\frac{f^a \text{Gamma}\left (6,-b x^3 \log (f)\right )}{3 b^6 \log ^6(f)} \]

Antiderivative was successfully verified.

[In]

Int[f^(a + b*x^3)*x^17,x]

[Out]

-(f^a*Gamma[6, -(b*x^3*Log[f])])/(3*b^6*Log[f]^6)

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^3} x^{17} \, dx &=-\frac{f^a \Gamma \left (6,-b x^3 \log (f)\right )}{3 b^6 \log ^6(f)}\\ \end{align*}

Mathematica [C]  time = 0.0028409, size = 24, normalized size = 0.31 \[ -\frac{f^a \text{Gamma}\left (6,-b x^3 \log (f)\right )}{3 b^6 \log ^6(f)} \]

Antiderivative was successfully verified.

[In]

Integrate[f^(a + b*x^3)*x^17,x]

[Out]

-(f^a*Gamma[6, -(b*x^3*Log[f])])/(3*b^6*Log[f]^6)

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Maple [A]  time = 0.013, size = 76, normalized size = 1. \begin{align*}{\frac{ \left ({b}^{5}{x}^{15} \left ( \ln \left ( f \right ) \right ) ^{5}-5\,{b}^{4}{x}^{12} \left ( \ln \left ( f \right ) \right ) ^{4}+20\,{b}^{3}{x}^{9} \left ( \ln \left ( f \right ) \right ) ^{3}-60\,{b}^{2}{x}^{6} \left ( \ln \left ( f \right ) \right ) ^{2}+120\,b{x}^{3}\ln \left ( f \right ) -120 \right ){f}^{b{x}^{3}+a}}{3\,{b}^{6} \left ( \ln \left ( f \right ) \right ) ^{6}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(f^(b*x^3+a)*x^17,x)

[Out]

1/3*(b^5*x^15*ln(f)^5-5*b^4*x^12*ln(f)^4+20*b^3*x^9*ln(f)^3-60*b^2*x^6*ln(f)^2+120*b*x^3*ln(f)-120)*f^(b*x^3+a
)/b^6/ln(f)^6

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Maxima [A]  time = 1.2323, size = 124, normalized size = 1.59 \begin{align*} \frac{{\left (b^{5} f^{a} x^{15} \log \left (f\right )^{5} - 5 \, b^{4} f^{a} x^{12} \log \left (f\right )^{4} + 20 \, b^{3} f^{a} x^{9} \log \left (f\right )^{3} - 60 \, b^{2} f^{a} x^{6} \log \left (f\right )^{2} + 120 \, b f^{a} x^{3} \log \left (f\right ) - 120 \, f^{a}\right )} f^{b x^{3}}}{3 \, b^{6} \log \left (f\right )^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(b*x^3+a)*x^17,x, algorithm="maxima")

[Out]

1/3*(b^5*f^a*x^15*log(f)^5 - 5*b^4*f^a*x^12*log(f)^4 + 20*b^3*f^a*x^9*log(f)^3 - 60*b^2*f^a*x^6*log(f)^2 + 120
*b*f^a*x^3*log(f) - 120*f^a)*f^(b*x^3)/(b^6*log(f)^6)

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Fricas [A]  time = 1.73179, size = 196, normalized size = 2.51 \begin{align*} \frac{{\left (b^{5} x^{15} \log \left (f\right )^{5} - 5 \, b^{4} x^{12} \log \left (f\right )^{4} + 20 \, b^{3} x^{9} \log \left (f\right )^{3} - 60 \, b^{2} x^{6} \log \left (f\right )^{2} + 120 \, b x^{3} \log \left (f\right ) - 120\right )} f^{b x^{3} + a}}{3 \, b^{6} \log \left (f\right )^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(b*x^3+a)*x^17,x, algorithm="fricas")

[Out]

1/3*(b^5*x^15*log(f)^5 - 5*b^4*x^12*log(f)^4 + 20*b^3*x^9*log(f)^3 - 60*b^2*x^6*log(f)^2 + 120*b*x^3*log(f) -
120)*f^(b*x^3 + a)/(b^6*log(f)^6)

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Sympy [A]  time = 0.161398, size = 95, normalized size = 1.22 \begin{align*} \begin{cases} \frac{f^{a + b x^{3}} \left (b^{5} x^{15} \log{\left (f \right )}^{5} - 5 b^{4} x^{12} \log{\left (f \right )}^{4} + 20 b^{3} x^{9} \log{\left (f \right )}^{3} - 60 b^{2} x^{6} \log{\left (f \right )}^{2} + 120 b x^{3} \log{\left (f \right )} - 120\right )}{3 b^{6} \log{\left (f \right )}^{6}} & \text{for}\: 3 b^{6} \log{\left (f \right )}^{6} \neq 0 \\\frac{x^{18}}{18} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f**(b*x**3+a)*x**17,x)

[Out]

Piecewise((f**(a + b*x**3)*(b**5*x**15*log(f)**5 - 5*b**4*x**12*log(f)**4 + 20*b**3*x**9*log(f)**3 - 60*b**2*x
**6*log(f)**2 + 120*b*x**3*log(f) - 120)/(3*b**6*log(f)**6), Ne(3*b**6*log(f)**6, 0)), (x**18/18, True))

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(b*x^3+a)*x^17,x, algorithm="giac")

[Out]

Exception raised: TypeError