Integrand size = 19, antiderivative size = 39 \[ \int \frac {1}{\sqrt [4]{2-x} \sqrt [4]{3-x}} \, dx=-\frac {2 (2-x)^{3/4}}{\sqrt [4]{3-x}}-2 E\left (\left .\frac {1}{2} \arcsin \left (\frac {1}{\sqrt {3-x}}\right )\right |2\right ) \] Output:
-2*(2-x)^(3/4)/(3-x)^(1/4)-2*EllipticE(sin(1/2*arcsin(1/(3-x)^(1/2))),2^(1 /2))
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.01 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.67 \[ \int \frac {1}{\sqrt [4]{2-x} \sqrt [4]{3-x}} \, dx=-\frac {4}{3} (2-x)^{3/4} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {3}{4},\frac {7}{4},-2+x\right ) \] Input:
Integrate[1/((2 - x)^(1/4)*(3 - x)^(1/4)),x]
Output:
(-4*(2 - x)^(3/4)*Hypergeometric2F1[1/4, 3/4, 7/4, -2 + x])/3
Time = 0.35 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.95, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {73, 839, 813, 858, 807, 212}
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.
\(\displaystyle \int \frac {1}{\sqrt [4]{2-x} \sqrt [4]{3-x}} \, dx\) |
\(\Big \downarrow \) 73 |
\(\displaystyle -4 \int \frac {\sqrt {2-x}}{\sqrt [4]{3-x}}d\sqrt [4]{2-x}\) |
\(\Big \downarrow \) 839 |
\(\displaystyle -4 \left (\frac {(2-x)^{3/4}}{2 \sqrt [4]{3-x}}-\frac {1}{2} \int \frac {\sqrt {2-x}}{(3-x)^{5/4}}d\sqrt [4]{2-x}\right )\) |
\(\Big \downarrow \) 813 |
\(\displaystyle -4 \left (\frac {(2-x)^{3/4}}{2 \sqrt [4]{3-x}}-\frac {\sqrt [4]{\frac {1}{2-x}+1} \sqrt [4]{2-x} \int \frac {1}{\left (1+\frac {1}{2-x}\right )^{5/4} (2-x)^{3/4}}d\sqrt [4]{2-x}}{2 \sqrt [4]{3-x}}\right )\) |
\(\Big \downarrow \) 858 |
\(\displaystyle -4 \left (\frac {\sqrt [4]{\frac {1}{2-x}+1} \sqrt [4]{2-x} \int \frac {1}{\sqrt [4]{2-x} (3-x)^{5/4}}d\frac {1}{\sqrt [4]{2-x}}}{2 \sqrt [4]{3-x}}+\frac {(2-x)^{3/4}}{2 \sqrt [4]{3-x}}\right )\) |
\(\Big \downarrow \) 807 |
\(\displaystyle -4 \left (\frac {\sqrt [4]{\frac {1}{2-x}+1} \sqrt [4]{2-x} \int \frac {1}{\left (\sqrt {2-x}+1\right )^{5/4}}d\sqrt {2-x}}{4 \sqrt [4]{3-x}}+\frac {(2-x)^{3/4}}{2 \sqrt [4]{3-x}}\right )\) |
\(\Big \downarrow \) 212 |
\(\displaystyle -4 \left (\frac {\sqrt [4]{\frac {1}{2-x}+1} \sqrt [4]{2-x} E\left (\left .\frac {1}{2} \arctan \left (\sqrt {2-x}\right )\right |2\right )}{2 \sqrt [4]{3-x}}+\frac {(2-x)^{3/4}}{2 \sqrt [4]{3-x}}\right )\) |
Input:
Int[1/((2 - x)^(1/4)*(3 - x)^(1/4)),x]
Output:
-4*((2 - x)^(3/4)/(2*(3 - x)^(1/4)) + ((1 + (2 - x)^(-1))^(1/4)*(2 - x)^(1 /4)*EllipticE[ArcTan[Sqrt[2 - x]]/2, 2])/(2*(3 - x)^(1/4)))
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_) + (b_.)*(x_)^2)^(-5/4), x_Symbol] :> Simp[(2/(a^(5/4)*Rt[b/a, 2]) )*EllipticE[(1/2)*ArcTan[Rt[b/a, 2]*x], 2], x] /; FreeQ[{a, b}, x] && GtQ[a , 0] && PosQ[b/a]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Simp[1/k Subst[Int[x^((m + 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]
Int[(x_)^2/((a_) + (b_.)*(x_)^4)^(5/4), x_Symbol] :> Simp[x*((1 + a/(b*x^4) )^(1/4)/(b*(a + b*x^4)^(1/4))) Int[1/(x^3*(1 + a/(b*x^4))^(5/4)), x], x] /; FreeQ[{a, b}, x] && PosQ[b/a]
Int[(x_)^2/((a_) + (b_.)*(x_)^4)^(1/4), x_Symbol] :> Simp[x^3/(2*(a + b*x^4 )^(1/4)), x] - Simp[a/2 Int[x^2/(a + b*x^4)^(5/4), x], x] /; FreeQ[{a, b} , x] && PosQ[b/a]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Subst[Int[(a + b/x^n)^p/x^(m + 2), x], x, 1/x] /; FreeQ[{a, b, p}, x] && ILtQ[n, 0] && Int egerQ[m]
\[\int \frac {1}{\left (2-x \right )^{\frac {1}{4}} \left (3-x \right )^{\frac {1}{4}}}d x\]
Input:
int(1/(2-x)^(1/4)/(3-x)^(1/4),x)
Output:
int(1/(2-x)^(1/4)/(3-x)^(1/4),x)
\[ \int \frac {1}{\sqrt [4]{2-x} \sqrt [4]{3-x}} \, dx=\int { \frac {1}{{\left (-x + 3\right )}^{\frac {1}{4}} {\left (-x + 2\right )}^{\frac {1}{4}}} \,d x } \] Input:
integrate(1/(2-x)^(1/4)/(3-x)^(1/4),x, algorithm="fricas")
Output:
integral((-x + 3)^(3/4)*(-x + 2)^(3/4)/(x^2 - 5*x + 6), x)
Result contains complex when optimal does not.
Time = 0.75 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.92 \[ \int \frac {1}{\sqrt [4]{2-x} \sqrt [4]{3-x}} \, dx=- 2 \sqrt [4]{-1} \sqrt {x - 2} e^{\frac {i \pi }{4}} {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {1}{4} \\ \frac {1}{2} \end {matrix}\middle | {\frac {1}{x - 2}} \right )} \] Input:
integrate(1/(2-x)**(1/4)/(3-x)**(1/4),x)
Output:
-2*(-1)**(1/4)*sqrt(x - 2)*exp(I*pi/4)*hyper((-1/2, 1/4), (1/2,), 1/(x - 2 ))
\[ \int \frac {1}{\sqrt [4]{2-x} \sqrt [4]{3-x}} \, dx=\int { \frac {1}{{\left (-x + 3\right )}^{\frac {1}{4}} {\left (-x + 2\right )}^{\frac {1}{4}}} \,d x } \] Input:
integrate(1/(2-x)^(1/4)/(3-x)^(1/4),x, algorithm="maxima")
Output:
integrate(1/((-x + 3)^(1/4)*(-x + 2)^(1/4)), x)
\[ \int \frac {1}{\sqrt [4]{2-x} \sqrt [4]{3-x}} \, dx=\int { \frac {1}{{\left (-x + 3\right )}^{\frac {1}{4}} {\left (-x + 2\right )}^{\frac {1}{4}}} \,d x } \] Input:
integrate(1/(2-x)^(1/4)/(3-x)^(1/4),x, algorithm="giac")
Output:
integrate(1/((-x + 3)^(1/4)*(-x + 2)^(1/4)), x)
Timed out. \[ \int \frac {1}{\sqrt [4]{2-x} \sqrt [4]{3-x}} \, dx=\int \frac {1}{{\left (2-x\right )}^{1/4}\,{\left (3-x\right )}^{1/4}} \,d x \] Input:
int(1/((2 - x)^(1/4)*(3 - x)^(1/4)),x)
Output:
int(1/((2 - x)^(1/4)*(3 - x)^(1/4)), x)
\[ \int \frac {1}{\sqrt [4]{2-x} \sqrt [4]{3-x}} \, dx=\int \frac {1}{\left (-x +2\right )^{\frac {1}{4}} \left (-x +3\right )^{\frac {1}{4}}}d x \] Input:
int(1/(2-x)^(1/4)/(3-x)^(1/4),x)
Output:
int(1/(( - x + 2)**(1/4)*( - x + 3)**(1/4)),x)