\(\int f^{a+b x^3} \, dx\) [111]

Optimal result

Integrand size = 9, antiderivative size = 32 \[ \int f^{a+b x^3} \, dx=-\frac {f^a x \Gamma \left (\frac {1}{3},-b x^3 \log (f)\right )}{3 \sqrt [3]{-b x^3 \log (f)}} \]

[Out]

-1/3*f^a*x*GAMMA(1/3,-b*x^3*ln(f))/(-b*x^3*ln(f))^(1/3)

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2239} \[ \int f^{a+b x^3} \, dx=-\frac {x f^a \Gamma \left (\frac {1}{3},-b x^3 \log (f)\right )}{3 \sqrt [3]{-b x^3 \log (f)}} \]

[In]

Int[f^(a + b*x^3),x]

[Out]

-1/3*(f^a*x*Gamma[1/3, -(b*x^3*Log[f])])/(-(b*x^3*Log[f]))^(1/3)

Rule 2239

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_)), x_Symbol] :> Simp[(-F^a)*(c + d*x)*(Gamma[1/n, (-b)*(c + d
*x)^n*Log[F]]/(d*n*((-b)*(c + d*x)^n*Log[F])^(1/n))), x] /; FreeQ[{F, a, b, c, d, n}, x] &&  !IntegerQ[2/n]

Rubi steps \begin{align*} \text {integral}& = -\frac {f^a x \Gamma \left (\frac {1}{3},-b x^3 \log (f)\right )}{3 \sqrt [3]{-b x^3 \log (f)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00 \[ \int f^{a+b x^3} \, dx=-\frac {f^a x \Gamma \left (\frac {1}{3},-b x^3 \log (f)\right )}{3 \sqrt [3]{-b x^3 \log (f)}} \]

[In]

Integrate[f^(a + b*x^3),x]

[Out]

-1/3*(f^a*x*Gamma[1/3, -(b*x^3*Log[f])])/(-(b*x^3*Log[f]))^(1/3)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(77\) vs. \(2(26)=52\).

Time = 0.02 (sec) , antiderivative size = 78, normalized size of antiderivative = 2.44

method	result	size
meijerg	\(\frac {f^{a} \left (\frac {2 x \left (-b \right )^{\frac {1}{3}} \ln \left (f \right )^{\frac {1}{3}} \pi \sqrt {3}}{3 \Gamma \left (\frac {2}{3}\right ) \left (-b \,x^{3} \ln \left (f \right )\right )^{\frac {1}{3}}}-\frac {x \left (-b \right )^{\frac {1}{3}} \ln \left (f \right )^{\frac {1}{3}} \Gamma \left (\frac {1}{3}, -b \,x^{3} \ln \left (f \right )\right )}{\left (-b \,x^{3} \ln \left (f \right )\right )^{\frac {1}{3}}}\right )}{3 \left (-b \right )^{\frac {1}{3}} \ln \left (f \right )^{\frac {1}{3}}}\)	\(78\)

[In]

int(f^(b*x^3+a),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.08 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.91 \[ \int f^{a+b x^3} \, dx=\frac {\left (-b \log \left (f\right )\right )^{\frac {2}{3}} f^{a} \Gamma \left (\frac {1}{3}, -b x^{3} \log \left (f\right )\right )}{3 \, b \log \left (f\right )} \]

[In]

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

[Out]

1/3*(-b*log(f))^(2/3)*f^a*gamma(1/3, -b*x^3*log(f))/(b*log(f))

Sympy [F]

\[ \int f^{a+b x^3} \, dx=\int f^{a + b x^{3}}\, dx \]

[In]

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

[Out]

Integral(f**(a + b*x**3), x)

Maxima [A] (verification not implemented)

none

Time = 0.07 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.81 \[ \int f^{a+b x^3} \, dx=-\frac {f^{a} x \Gamma \left (\frac {1}{3}, -b x^{3} \log \left (f\right )\right )}{3 \, \left (-b x^{3} \log \left (f\right )\right )^{\frac {1}{3}}} \]

[In]

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

[Out]

-1/3*f^a*x*gamma(1/3, -b*x^3*log(f))/(-b*x^3*log(f))^(1/3)

Giac [F]

\[ \int f^{a+b x^3} \, dx=\int { f^{b x^{3} + a} \,d x } \]

[In]

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

[Out]

integrate(f^(b*x^3 + a), x)

Mupad [F(-1)]

Timed out. \[ \int f^{a+b x^3} \, dx=\int f^{b\,x^3+a} \,d x \]

[In]

int(f^(a + b*x^3),x)

[Out]

int(f^(a + b*x^3), x)