Integrand size = 26, antiderivative size = 17 \[ \int \frac {9-9 \sqrt {x}+2 x}{\sqrt [3]{-3 \sqrt {x}+x}} \, dx=\frac {6}{5} \left (-3 \sqrt {x}+x\right )^{5/3} \] Output:
6/5*(-3*x^(1/2)+x)^(5/3)
Time = 0.06 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00 \[ \int \frac {9-9 \sqrt {x}+2 x}{\sqrt [3]{-3 \sqrt {x}+x}} \, dx=\frac {6}{5} \left (-3 \sqrt {x}+x\right )^{5/3} \] Input:
Integrate[(9 - 9*Sqrt[x] + 2*x)/(-3*Sqrt[x] + x)^(1/3),x]
Output:
(6*(-3*Sqrt[x] + x)^(5/3))/5
Time = 0.36 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2440, 2162, 25, 1104}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {2 x-9 \sqrt {x}+9}{\sqrt [3]{x-3 \sqrt {x}}} \, dx\) |
\(\Big \downarrow \) 2440 |
\(\displaystyle 2 \int \frac {\sqrt {x} \left (2 x-9 \sqrt {x}+9\right )}{\sqrt [3]{x-3 \sqrt {x}}}d\sqrt {x}\) |
\(\Big \downarrow \) 2162 |
\(\displaystyle 2 \int -\left (\left (3-2 \sqrt {x}\right ) \left (x-3 \sqrt {x}\right )^{2/3}\right )d\sqrt {x}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -2 \int \left (3-2 \sqrt {x}\right ) \left (x-3 \sqrt {x}\right )^{2/3}d\sqrt {x}\) |
\(\Big \downarrow \) 1104 |
\(\displaystyle \frac {6}{5} \left (x-3 \sqrt {x}\right )^{5/3}\) |
Input:
Int[(9 - 9*Sqrt[x] + 2*x)/(-3*Sqrt[x] + x)^(1/3),x]
Output:
(6*(-3*Sqrt[x] + x)^(5/3))/5
Int[((d_) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol ] :> Simp[d*((a + b*x + c*x^2)^(p + 1)/(b*(p + 1))), x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[2*c*d - b*e, 0]
Int[(Pq_)*((e_.)*(x_))^(m_.)*((b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] : > Simp[e Int[(e*x)^(m - 1)*PolynomialQuotient[Pq, b + c*x, x]*(b*x + c*x^ 2)^(p + 1), x], x] /; FreeQ[{b, c, e, m, p}, x] && PolyQ[Pq, x] && EqQ[Poly nomialRemainder[Pq, b + c*x, x], 0]
Int[(Pq_)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{d = Denominator[n]}, Simp[d Subst[Int[x^(d - 1)*(SubstFor[x^n, Pq, x] /. x - > x^(d*n))*(a*x^(d*j) + b*x^(d*n))^p, x], x, x^(1/d)], x]] /; FreeQ[{a, b, j, n, p}, x] && PolyQ[Pq, x^n] && !IntegerQ[p] && NeQ[n, j] && RationalQ[j , n] && IntegerQ[j/n] && LtQ[-1, n, 1]
Result contains higher order function than in optimal. Order 9 vs. order 2.
Time = 0.06 (sec) , antiderivative size = 125, normalized size of antiderivative = 7.35
method | result | size |
meijerg | \(\frac {18 \,3^{\frac {2}{3}} \left (-\operatorname {signum}\left (-1+\frac {\sqrt {x}}{3}\right )\right )^{\frac {1}{3}} x^{\frac {5}{6}} \operatorname {hypergeom}\left (\left [\frac {1}{3}, \frac {5}{3}\right ], \left [\frac {8}{3}\right ], \frac {\sqrt {x}}{3}\right )}{5 \operatorname {signum}\left (-1+\frac {\sqrt {x}}{3}\right )^{\frac {1}{3}}}-\frac {9 \,3^{\frac {2}{3}} \left (-\operatorname {signum}\left (-1+\frac {\sqrt {x}}{3}\right )\right )^{\frac {1}{3}} x^{\frac {4}{3}} \operatorname {hypergeom}\left (\left [\frac {1}{3}, \frac {8}{3}\right ], \left [\frac {11}{3}\right ], \frac {\sqrt {x}}{3}\right )}{4 \operatorname {signum}\left (-1+\frac {\sqrt {x}}{3}\right )^{\frac {1}{3}}}+\frac {4 \,3^{\frac {2}{3}} \left (-\operatorname {signum}\left (-1+\frac {\sqrt {x}}{3}\right )\right )^{\frac {1}{3}} x^{\frac {11}{6}} \operatorname {hypergeom}\left (\left [\frac {1}{3}, \frac {11}{3}\right ], \left [\frac {14}{3}\right ], \frac {\sqrt {x}}{3}\right )}{11 \operatorname {signum}\left (-1+\frac {\sqrt {x}}{3}\right )^{\frac {1}{3}}}\) | \(125\) |
Input:
int((9-9*x^(1/2)+2*x)/(-3*x^(1/2)+x)^(1/3),x,method=_RETURNVERBOSE)
Output:
18/5*3^(2/3)/signum(-1+1/3*x^(1/2))^(1/3)*(-signum(-1+1/3*x^(1/2)))^(1/3)* x^(5/6)*hypergeom([1/3,5/3],[8/3],1/3*x^(1/2))-9/4*3^(2/3)/signum(-1+1/3*x ^(1/2))^(1/3)*(-signum(-1+1/3*x^(1/2)))^(1/3)*x^(4/3)*hypergeom([1/3,8/3], [11/3],1/3*x^(1/2))+4/11*3^(2/3)/signum(-1+1/3*x^(1/2))^(1/3)*(-signum(-1+ 1/3*x^(1/2)))^(1/3)*x^(11/6)*hypergeom([1/3,11/3],[14/3],1/3*x^(1/2))
Time = 0.11 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.65 \[ \int \frac {9-9 \sqrt {x}+2 x}{\sqrt [3]{-3 \sqrt {x}+x}} \, dx=\frac {6}{5} \, {\left (x - 3 \, \sqrt {x}\right )}^{\frac {5}{3}} \] Input:
integrate((9-9*x^(1/2)+2*x)/(-3*x^(1/2)+x)^(1/3),x, algorithm="fricas")
Output:
6/5*(x - 3*sqrt(x))^(5/3)
\[ \int \frac {9-9 \sqrt {x}+2 x}{\sqrt [3]{-3 \sqrt {x}+x}} \, dx=\int \frac {- 9 \sqrt {x} + 2 x + 9}{\sqrt [3]{- 3 \sqrt {x} + x}}\, dx \] Input:
integrate((9-9*x**(1/2)+2*x)/(-3*x**(1/2)+x)**(1/3),x)
Output:
Integral((-9*sqrt(x) + 2*x + 9)/(-3*sqrt(x) + x)**(1/3), x)
\[ \int \frac {9-9 \sqrt {x}+2 x}{\sqrt [3]{-3 \sqrt {x}+x}} \, dx=\int { \frac {2 \, x - 9 \, \sqrt {x} + 9}{{\left (x - 3 \, \sqrt {x}\right )}^{\frac {1}{3}}} \,d x } \] Input:
integrate((9-9*x^(1/2)+2*x)/(-3*x^(1/2)+x)^(1/3),x, algorithm="maxima")
Output:
integrate((2*x - 9*sqrt(x) + 9)/(x - 3*sqrt(x))^(1/3), x)
Time = 0.92 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.65 \[ \int \frac {9-9 \sqrt {x}+2 x}{\sqrt [3]{-3 \sqrt {x}+x}} \, dx=\frac {6}{5} \, {\left (x - 3 \, \sqrt {x}\right )}^{\frac {5}{3}} \] Input:
integrate((9-9*x^(1/2)+2*x)/(-3*x^(1/2)+x)^(1/3),x, algorithm="giac")
Output:
6/5*(x - 3*sqrt(x))^(5/3)
Timed out. \[ \int \frac {9-9 \sqrt {x}+2 x}{\sqrt [3]{-3 \sqrt {x}+x}} \, dx=\int \frac {2\,x-9\,\sqrt {x}+9}{{\left (x-3\,\sqrt {x}\right )}^{1/3}} \,d x \] Input:
int((2*x - 9*x^(1/2) + 9)/(x - 3*x^(1/2))^(1/3),x)
Output:
int((2*x - 9*x^(1/2) + 9)/(x - 3*x^(1/2))^(1/3), x)
\[ \int \frac {9-9 \sqrt {x}+2 x}{\sqrt [3]{-3 \sqrt {x}+x}} \, dx =\text {Too large to display} \] Input:
int((9-9*x^(1/2)+2*x)/(-3*x^(1/2)+x)^(1/3),x)
Output:
- 2*int(x**3/(12*sqrt(x)*( - 3*sqrt(x) + x)**(1/3)*x + 108*sqrt(x)*( - 3* sqrt(x) + x)**(1/3) - ( - 3*sqrt(x) + x)**(1/3)*x**2 - 54*( - 3*sqrt(x) + x)**(1/3)*x - 81*( - 3*sqrt(x) + x)**(1/3)),x) - 225*int(x**2/(12*sqrt(x)* ( - 3*sqrt(x) + x)**(1/3)*x + 108*sqrt(x)*( - 3*sqrt(x) + x)**(1/3) - ( - 3*sqrt(x) + x)**(1/3)*x**2 - 54*( - 3*sqrt(x) + x)**(1/3)*x - 81*( - 3*sqr t(x) + x)**(1/3)),x) + 1701*int(sqrt(x)/(12*sqrt(x)*( - 3*sqrt(x) + x)**(1 /3)*x + 108*sqrt(x)*( - 3*sqrt(x) + x)**(1/3) - ( - 3*sqrt(x) + x)**(1/3)* x**2 - 54*( - 3*sqrt(x) + x)**(1/3)*x - 81*( - 3*sqrt(x) + x)**(1/3)),x) + 33*int((sqrt(x)*x**2)/(12*sqrt(x)*( - 3*sqrt(x) + x)**(1/3)*x + 108*sqrt( x)*( - 3*sqrt(x) + x)**(1/3) - ( - 3*sqrt(x) + x)**(1/3)*x**2 - 54*( - 3*s qrt(x) + x)**(1/3)*x - 81*( - 3*sqrt(x) + x)**(1/3)),x) + 810*int((sqrt(x) *x)/(12*sqrt(x)*( - 3*sqrt(x) + x)**(1/3)*x + 108*sqrt(x)*( - 3*sqrt(x) + x)**(1/3) - ( - 3*sqrt(x) + x)**(1/3)*x**2 - 54*( - 3*sqrt(x) + x)**(1/3)* x - 81*( - 3*sqrt(x) + x)**(1/3)),x) - 1620*int(x/(12*sqrt(x)*( - 3*sqrt(x ) + x)**(1/3)*x + 108*sqrt(x)*( - 3*sqrt(x) + x)**(1/3) - ( - 3*sqrt(x) + x)**(1/3)*x**2 - 54*( - 3*sqrt(x) + x)**(1/3)*x - 81*( - 3*sqrt(x) + x)**( 1/3)),x) - 729*int(1/(12*sqrt(x)*( - 3*sqrt(x) + x)**(1/3)*x + 108*sqrt(x) *( - 3*sqrt(x) + x)**(1/3) - ( - 3*sqrt(x) + x)**(1/3)*x**2 - 54*( - 3*sqr t(x) + x)**(1/3)*x - 81*( - 3*sqrt(x) + x)**(1/3)),x)