Integrand size = 19, antiderivative size = 33 \[ \int \frac {1}{\sqrt [4]{-2-3 x} \sqrt [4]{5+2 x}} \, dx=-\frac {\sqrt {11} E\left (\left .\frac {1}{2} \arcsin \left (\frac {1}{11} (-19-12 x)\right )\right |2\right )}{\sqrt [4]{2} 3^{3/4}} \] Output:
1/6*11^(1/2)*EllipticE(sin(1/2*arcsin(19/11+12/11*x)),2^(1/2))*2^(3/4)*3^( 1/4)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.01 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.18 \[ \int \frac {1}{\sqrt [4]{-2-3 x} \sqrt [4]{5+2 x}} \, dx=-\frac {4 \left (-\frac {2}{3}-x\right )^{3/4} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {3}{4},\frac {7}{4},-\frac {2}{11} (2+3 x)\right )}{3 \sqrt [4]{11}} \] Input:
Integrate[1/((-2 - 3*x)^(1/4)*(5 + 2*x)^(1/4)),x]
Output:
(-4*(-2/3 - x)^(3/4)*Hypergeometric2F1[1/4, 3/4, 7/4, (-2*(2 + 3*x))/11])/ (3*11^(1/4))
Leaf count is larger than twice the leaf count of optimal. \(109\) vs. \(2(33)=66\).
Time = 0.23 (sec) , antiderivative size = 109, normalized size of antiderivative = 3.30, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.421, Rules used = {73, 27, 840, 842, 27, 858, 807, 226}
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.
\(\displaystyle \int \frac {1}{\sqrt [4]{-3 x-2} \sqrt [4]{2 x+5}} \, dx\) |
\(\Big \downarrow \) 73 |
\(\displaystyle -\frac {4}{3} \int \frac {\sqrt [4]{3} \sqrt {-3 x-2}}{\sqrt [4]{11-2 (-3 x-2)}}d\sqrt [4]{-3 x-2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {4 \int \frac {\sqrt {-3 x-2}}{\sqrt [4]{11-2 (-3 x-2)}}d\sqrt [4]{-3 x-2}}{3^{3/4}}\) |
\(\Big \downarrow \) 840 |
\(\displaystyle -\frac {4 \left (-\frac {11}{4} \int \frac {1}{\sqrt [4]{11-2 (-3 x-2)} \sqrt {-3 x-2}}d\sqrt [4]{-3 x-2}-\frac {(11-2 (-3 x-2))^{3/4}}{4 \sqrt [4]{-3 x-2}}\right )}{3^{3/4}}\) |
\(\Big \downarrow \) 842 |
\(\displaystyle -\frac {4 \left (-\frac {11 \sqrt [4]{2-\frac {11}{-3 x-2}} \sqrt [4]{-3 x-2} \int \frac {\sqrt [4]{2}}{\sqrt [4]{2-\frac {11}{-3 x-2}} (-3 x-2)^{3/4}}d\sqrt [4]{-3 x-2}}{4 \sqrt [4]{2} \sqrt [4]{11-2 (-3 x-2)}}-\frac {(11-2 (-3 x-2))^{3/4}}{4 \sqrt [4]{-3 x-2}}\right )}{3^{3/4}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {4 \left (-\frac {11 \sqrt [4]{2-\frac {11}{-3 x-2}} \sqrt [4]{-3 x-2} \int \frac {1}{\sqrt [4]{2-\frac {11}{-3 x-2}} (-3 x-2)^{3/4}}d\sqrt [4]{-3 x-2}}{4 \sqrt [4]{11-2 (-3 x-2)}}-\frac {(11-2 (-3 x-2))^{3/4}}{4 \sqrt [4]{-3 x-2}}\right )}{3^{3/4}}\) |
\(\Big \downarrow \) 858 |
\(\displaystyle -\frac {4 \left (\frac {11 \sqrt [4]{2-\frac {11}{-3 x-2}} \sqrt [4]{-3 x-2} \int \frac {1}{\sqrt [4]{2-11 (-3 x-2)} \sqrt [4]{-3 x-2}}d\frac {1}{\sqrt [4]{-3 x-2}}}{4 \sqrt [4]{11-2 (-3 x-2)}}-\frac {(11-2 (-3 x-2))^{3/4}}{4 \sqrt [4]{-3 x-2}}\right )}{3^{3/4}}\) |
\(\Big \downarrow \) 807 |
\(\displaystyle -\frac {4 \left (\frac {11 \sqrt [4]{2-\frac {11}{-3 x-2}} \sqrt [4]{-3 x-2} \int \frac {1}{\sqrt [4]{2-11 \sqrt {-3 x-2}}}d\sqrt {-3 x-2}}{8 \sqrt [4]{11-2 (-3 x-2)}}-\frac {(11-2 (-3 x-2))^{3/4}}{4 \sqrt [4]{-3 x-2}}\right )}{3^{3/4}}\) |
\(\Big \downarrow \) 226 |
\(\displaystyle -\frac {4 \left (\frac {\sqrt {11} \sqrt [4]{2-\frac {11}{-3 x-2}} \sqrt [4]{-3 x-2} E\left (\left .\frac {1}{2} \arcsin \left (\sqrt {\frac {11}{2}} \sqrt {-3 x-2}\right )\right |2\right )}{2\ 2^{3/4} \sqrt [4]{11-2 (-3 x-2)}}-\frac {(11-2 (-3 x-2))^{3/4}}{4 \sqrt [4]{-3 x-2}}\right )}{3^{3/4}}\) |
Input:
Int[1/((-2 - 3*x)^(1/4)*(5 + 2*x)^(1/4)),x]
Output:
(-4*(-1/4*(11 - 2*(-2 - 3*x))^(3/4)/(-2 - 3*x)^(1/4) + (Sqrt[11]*(2 - 11/( -2 - 3*x))^(1/4)*(-2 - 3*x)^(1/4)*EllipticE[ArcSin[Sqrt[11/2]*Sqrt[-2 - 3* x]]/2, 2])/(2*2^(3/4)*(11 - 2*(-2 - 3*x))^(1/4))))/3^(3/4)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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)^(-1/4), x_Symbol] :> Simp[(2/(a^(1/4)*Rt[-b/a, 2] ))*EllipticE[(1/2)*ArcSin[Rt[-b/a, 2]*x], 2], x] /; FreeQ[{a, b}, x] && GtQ [a, 0] && NegQ[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)^(1/4), x_Symbol] :> Simp[(a + b*x^4)^(3/4) /(2*b*x), x] + Simp[a/(2*b) Int[1/(x^2*(a + b*x^4)^(1/4)), x], x] /; Free Q[{a, b}, x] && NegQ[b/a]
Int[1/((x_)^2*((a_) + (b_.)*(x_)^4)^(1/4)), x_Symbol] :> Simp[x*((1 + a/(b* x^4))^(1/4)/(a + b*x^4)^(1/4)) Int[1/(x^3*(1 + a/(b*x^4))^(1/4)), x], x] /; FreeQ[{a, b}, x] && NegQ[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]
\[\int \frac {1}{\left (-2-3 x \right )^{\frac {1}{4}} \left (5+2 x \right )^{\frac {1}{4}}}d x\]
Input:
int(1/(-2-3*x)^(1/4)/(5+2*x)^(1/4),x)
Output:
int(1/(-2-3*x)^(1/4)/(5+2*x)^(1/4),x)
\[ \int \frac {1}{\sqrt [4]{-2-3 x} \sqrt [4]{5+2 x}} \, dx=\int { \frac {1}{{\left (2 \, x + 5\right )}^{\frac {1}{4}} {\left (-3 \, x - 2\right )}^{\frac {1}{4}}} \,d x } \] Input:
integrate(1/(-2-3*x)^(1/4)/(5+2*x)^(1/4),x, algorithm="fricas")
Output:
integral(-(2*x + 5)^(3/4)*(-3*x - 2)^(3/4)/(6*x^2 + 19*x + 10), x)
Result contains complex when optimal does not.
Time = 0.92 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.12 \[ \int \frac {1}{\sqrt [4]{-2-3 x} \sqrt [4]{5+2 x}} \, dx=- \frac {\left (-6\right )^{\frac {3}{4}} \sqrt {x + \frac {2}{3}} {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {1}{4} \\ \frac {1}{2} \end {matrix}\middle | {\frac {11 e^{i \pi }}{6 \left (x + \frac {2}{3}\right )}} \right )}}{3} \] Input:
integrate(1/(-2-3*x)**(1/4)/(5+2*x)**(1/4),x)
Output:
-(-6)**(3/4)*sqrt(x + 2/3)*hyper((-1/2, 1/4), (1/2,), 11*exp_polar(I*pi)/( 6*(x + 2/3)))/3
\[ \int \frac {1}{\sqrt [4]{-2-3 x} \sqrt [4]{5+2 x}} \, dx=\int { \frac {1}{{\left (2 \, x + 5\right )}^{\frac {1}{4}} {\left (-3 \, x - 2\right )}^{\frac {1}{4}}} \,d x } \] Input:
integrate(1/(-2-3*x)^(1/4)/(5+2*x)^(1/4),x, algorithm="maxima")
Output:
integrate(1/((2*x + 5)^(1/4)*(-3*x - 2)^(1/4)), x)
\[ \int \frac {1}{\sqrt [4]{-2-3 x} \sqrt [4]{5+2 x}} \, dx=\int { \frac {1}{{\left (2 \, x + 5\right )}^{\frac {1}{4}} {\left (-3 \, x - 2\right )}^{\frac {1}{4}}} \,d x } \] Input:
integrate(1/(-2-3*x)^(1/4)/(5+2*x)^(1/4),x, algorithm="giac")
Output:
integrate(1/((2*x + 5)^(1/4)*(-3*x - 2)^(1/4)), x)
Timed out. \[ \int \frac {1}{\sqrt [4]{-2-3 x} \sqrt [4]{5+2 x}} \, dx=\int \frac {1}{{\left (-3\,x-2\right )}^{1/4}\,{\left (2\,x+5\right )}^{1/4}} \,d x \] Input:
int(1/((- 3*x - 2)^(1/4)*(2*x + 5)^(1/4)),x)
Output:
int(1/((- 3*x - 2)^(1/4)*(2*x + 5)^(1/4)), x)
\[ \int \frac {1}{\sqrt [4]{-2-3 x} \sqrt [4]{5+2 x}} \, dx=\int \frac {1}{\left (2 x +5\right )^{\frac {1}{4}} \left (-3 x -2\right )^{\frac {1}{4}}}d x \] Input:
int(1/(-2-3*x)^(1/4)/(5+2*x)^(1/4),x)
Output:
int(1/((2*x + 5)**(1/4)*( - 3*x - 2)**(1/4)),x)