Integrand size = 24, antiderivative size = 41 \[ \int \frac {\sqrt {-1-2 x} \sqrt [3]{-x}}{2+3 x} \, dx=-\frac {1}{3} 2^{2/3} (-1-2 x)^{3/2} \operatorname {AppellF1}\left (\frac {3}{2},-\frac {1}{3},1,\frac {5}{2},1+2 x,-3 (1+2 x)\right ) \] Output:
-1/3*2^(2/3)*(-1-2*x)^(3/2)*AppellF1(3/2,1,-1/3,5/2,-3-6*x,1+2*x)
Leaf count is larger than twice the leaf count of optimal. \(86\) vs. \(2(41)=82\).
Time = 6.71 (sec) , antiderivative size = 86, normalized size of antiderivative = 2.10 \[ \int \frac {\sqrt {-1-2 x} \sqrt [3]{-x}}{2+3 x} \, dx=\frac {\sqrt [3]{-x} \left (-16-32 x+16 \sqrt {1+2 x} \operatorname {AppellF1}\left (\frac {1}{3},\frac {1}{2},1,\frac {4}{3},-2 x,-\frac {3 x}{2}\right )+11 x \sqrt {1+2 x} \operatorname {AppellF1}\left (\frac {4}{3},\frac {1}{2},1,\frac {7}{3},-2 x,-\frac {3 x}{2}\right )\right )}{40 \sqrt {-1-2 x}} \] Input:
Integrate[(Sqrt[-1 - 2*x]*(-x)^(1/3))/(2 + 3*x),x]
Output:
((-x)^(1/3)*(-16 - 32*x + 16*Sqrt[1 + 2*x]*AppellF1[1/3, 1/2, 1, 4/3, -2*x , (-3*x)/2] + 11*x*Sqrt[1 + 2*x]*AppellF1[4/3, 1/2, 1, 7/3, -2*x, (-3*x)/2 ]))/(40*Sqrt[-1 - 2*x])
Time = 0.19 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.17, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {148, 1013, 1012}
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.
\(\displaystyle \int \frac {\sqrt {-2 x-1} \sqrt [3]{-x}}{3 x+2} \, dx\) |
\(\Big \downarrow \) 148 |
\(\displaystyle -3 \int -\frac {\sqrt {-2 x-1} x}{3 x+2}d\sqrt [3]{-x}\) |
\(\Big \downarrow \) 1013 |
\(\displaystyle -\frac {3 \sqrt {-2 x-1} \int -\frac {x \sqrt {2 x+1}}{3 x+2}d\sqrt [3]{-x}}{\sqrt {2 x+1}}\) |
\(\Big \downarrow \) 1012 |
\(\displaystyle -\frac {3 \sqrt {-2 x-1} (-x)^{4/3} \operatorname {AppellF1}\left (\frac {4}{3},1,-\frac {1}{2},\frac {7}{3},-\frac {3 x}{2},-2 x\right )}{8 \sqrt {2 x+1}}\) |
Input:
Int[(Sqrt[-1 - 2*x]*(-x)^(1/3))/(2 + 3*x),x]
Output:
(-3*Sqrt[-1 - 2*x]*(-x)^(4/3)*AppellF1[4/3, 1, -1/2, 7/3, (-3*x)/2, -2*x]) /(8*Sqrt[1 + 2*x])
Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_))^(p_.), x_] :> With[{k = Denominator[m]}, Simp[k/b Subst[Int[x^(k*(m + 1) - 1)*(c + d*(x^k/b))^n*(e + f*(x^k/b))^p, x], x, (b*x)^(1/k)], x]] /; FreeQ[{b, c, d, e, f, n, p}, x] && FractionQ[m] && IntegerQ[p]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_ ))^(q_), x_Symbol] :> Simp[a^p*c^q*((e*x)^(m + 1)/(e*(m + 1)))*AppellF1[(m + 1)/n, -p, -q, 1 + (m + 1)/n, (-b)*(x^n/a), (-d)*(x^n/c)], x] /; FreeQ[{a, b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, n - 1] && (IntegerQ[p] || GtQ[a, 0]) && (IntegerQ[q] || GtQ[c, 0])
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_ ))^(q_), x_Symbol] :> Simp[a^IntPart[p]*((a + b*x^n)^FracPart[p]/(1 + b*(x^ n/a))^FracPart[p]) Int[(e*x)^m*(1 + b*(x^n/a))^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] & & NeQ[m, n - 1] && !(IntegerQ[p] || GtQ[a, 0])
\[\int \frac {\sqrt {-1-2 x}\, \left (-x \right )^{\frac {1}{3}}}{2+3 x}d x\]
Input:
int((-1-2*x)^(1/2)*(-x)^(1/3)/(2+3*x),x)
Output:
int((-1-2*x)^(1/2)*(-x)^(1/3)/(2+3*x),x)
Timed out. \[ \int \frac {\sqrt {-1-2 x} \sqrt [3]{-x}}{2+3 x} \, dx=\text {Timed out} \] Input:
integrate((-1-2*x)^(1/2)*(-x)^(1/3)/(2+3*x),x, algorithm="fricas")
Output:
Timed out
\[ \int \frac {\sqrt {-1-2 x} \sqrt [3]{-x}}{2+3 x} \, dx=\int \frac {\sqrt [3]{- x} \sqrt {- 2 x - 1}}{3 x + 2}\, dx \] Input:
integrate((-1-2*x)**(1/2)*(-x)**(1/3)/(2+3*x),x)
Output:
Integral((-x)**(1/3)*sqrt(-2*x - 1)/(3*x + 2), x)
\[ \int \frac {\sqrt {-1-2 x} \sqrt [3]{-x}}{2+3 x} \, dx=\int { \frac {\left (-x\right )^{\frac {1}{3}} \sqrt {-2 \, x - 1}}{3 \, x + 2} \,d x } \] Input:
integrate((-1-2*x)^(1/2)*(-x)^(1/3)/(2+3*x),x, algorithm="maxima")
Output:
integrate((-x)^(1/3)*sqrt(-2*x - 1)/(3*x + 2), x)
\[ \int \frac {\sqrt {-1-2 x} \sqrt [3]{-x}}{2+3 x} \, dx=\int { \frac {\left (-x\right )^{\frac {1}{3}} \sqrt {-2 \, x - 1}}{3 \, x + 2} \,d x } \] Input:
integrate((-1-2*x)^(1/2)*(-x)^(1/3)/(2+3*x),x, algorithm="giac")
Output:
integrate((-x)^(1/3)*sqrt(-2*x - 1)/(3*x + 2), x)
Timed out. \[ \int \frac {\sqrt {-1-2 x} \sqrt [3]{-x}}{2+3 x} \, dx=\int \frac {{\left (-x\right )}^{1/3}\,\sqrt {-2\,x-1}}{3\,x+2} \,d x \] Input:
int(((-x)^(1/3)*(- 2*x - 1)^(1/2))/(3*x + 2),x)
Output:
int(((-x)^(1/3)*(- 2*x - 1)^(1/2))/(3*x + 2), x)
\[ \int \frac {\sqrt {-1-2 x} \sqrt [3]{-x}}{2+3 x} \, dx=-\frac {3 x^{\frac {1}{3}} \sqrt {-2 x -1}}{13}-\frac {11 \left (\int \frac {x^{\frac {4}{3}} \sqrt {-2 x -1}}{6 x^{2}+7 x +2}d x \right )}{13}+\frac {2 \left (\int \frac {x^{\frac {1}{3}} \sqrt {-2 x -1}}{6 x^{3}+7 x^{2}+2 x}d x \right )}{13} \] Input:
int((-1-2*x)^(1/2)*(-x)^(1/3)/(2+3*x),x)
Output:
( - 3*x**(1/3)*sqrt( - 2*x - 1) - 11*int((x**(1/3)*sqrt( - 2*x - 1)*x)/(6* x**2 + 7*x + 2),x) + 2*int((x**(1/3)*sqrt( - 2*x - 1))/(6*x**3 + 7*x**2 + 2*x),x))/13