Integrand size = 13, antiderivative size = 94 \[ \int x \sqrt {x+x^{3/2}} \, dx=-\frac {32}{99} \left (x+x^{3/2}\right )^{3/2}+\frac {512 \left (x+x^{3/2}\right )^{3/2}}{3465 x^{3/2}}-\frac {256 \left (x+x^{3/2}\right )^{3/2}}{1155 x}+\frac {64 \left (x+x^{3/2}\right )^{3/2}}{231 \sqrt {x}}+\frac {4}{11} \sqrt {x} \left (x+x^{3/2}\right )^{3/2} \] Output:
-32/99*(x+x^(3/2))^(3/2)+512/3465*(x+x^(3/2))^(3/2)/x^(3/2)-256/1155*(x+x^ (3/2))^(3/2)/x+64/231*(x+x^(3/2))^(3/2)/x^(1/2)+4/11*x^(1/2)*(x+x^(3/2))^( 3/2)
Time = 0.05 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.54 \[ \int x \sqrt {x+x^{3/2}} \, dx=\frac {4 \sqrt {x+x^{3/2}} \left (128-64 \sqrt {x}+48 x-40 x^{3/2}+35 x^2+315 x^{5/2}\right )}{3465 \sqrt {x}} \] Input:
Integrate[x*Sqrt[x + x^(3/2)],x]
Output:
(4*Sqrt[x + x^(3/2)]*(128 - 64*Sqrt[x] + 48*x - 40*x^(3/2) + 35*x^2 + 315* x^(5/2)))/(3465*Sqrt[x])
Time = 0.40 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.16, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {1922, 1922, 1908, 1922, 1920}
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.
\(\displaystyle \int x \sqrt {x^{3/2}+x} \, dx\) |
\(\Big \downarrow \) 1922 |
\(\displaystyle \frac {4}{11} \sqrt {x} \left (x^{3/2}+x\right )^{3/2}-\frac {8}{11} \int \sqrt {x} \sqrt {x^{3/2}+x}dx\) |
\(\Big \downarrow \) 1922 |
\(\displaystyle \frac {4}{11} \sqrt {x} \left (x^{3/2}+x\right )^{3/2}-\frac {8}{11} \left (\frac {4}{9} \left (x^{3/2}+x\right )^{3/2}-\frac {2}{3} \int \sqrt {x^{3/2}+x}dx\right )\) |
\(\Big \downarrow \) 1908 |
\(\displaystyle \frac {4}{11} \sqrt {x} \left (x^{3/2}+x\right )^{3/2}-\frac {8}{11} \left (\frac {4}{9} \left (x^{3/2}+x\right )^{3/2}-\frac {2}{3} \left (\frac {4 \left (x^{3/2}+x\right )^{3/2}}{7 \sqrt {x}}-\frac {4}{7} \int \frac {\sqrt {x^{3/2}+x}}{\sqrt {x}}dx\right )\right )\) |
\(\Big \downarrow \) 1922 |
\(\displaystyle \frac {4}{11} \sqrt {x} \left (x^{3/2}+x\right )^{3/2}-\frac {8}{11} \left (\frac {4}{9} \left (x^{3/2}+x\right )^{3/2}-\frac {2}{3} \left (\frac {4 \left (x^{3/2}+x\right )^{3/2}}{7 \sqrt {x}}-\frac {4}{7} \left (\frac {4 \left (x^{3/2}+x\right )^{3/2}}{5 x}-\frac {2}{5} \int \frac {\sqrt {x^{3/2}+x}}{x}dx\right )\right )\right )\) |
\(\Big \downarrow \) 1920 |
\(\displaystyle \frac {4}{11} \sqrt {x} \left (x^{3/2}+x\right )^{3/2}-\frac {8}{11} \left (\frac {4}{9} \left (x^{3/2}+x\right )^{3/2}-\frac {2}{3} \left (\frac {4 \left (x^{3/2}+x\right )^{3/2}}{7 \sqrt {x}}-\frac {4}{7} \left (\frac {4 \left (x^{3/2}+x\right )^{3/2}}{5 x}-\frac {8 \left (x^{3/2}+x\right )^{3/2}}{15 x^{3/2}}\right )\right )\right )\) |
Input:
Int[x*Sqrt[x + x^(3/2)],x]
Output:
(4*Sqrt[x]*(x + x^(3/2))^(3/2))/11 - (8*((4*(x + x^(3/2))^(3/2))/9 - (2*(( 4*(x + x^(3/2))^(3/2))/(7*Sqrt[x]) - (4*((-8*(x + x^(3/2))^(3/2))/(15*x^(3 /2)) + (4*(x + x^(3/2))^(3/2))/(5*x)))/7))/3))/11
Int[((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Simp[(a*x^j + b*x^n)^(p + 1)/(a*(j*p + 1)*x^(j - 1)), x] - Simp[b*((n*p + n - j + 1)/(a*( j*p + 1))) Int[x^(n - j)*(a*x^j + b*x^n)^p, x], x] /; FreeQ[{a, b, j, n, p}, x] && !IntegerQ[p] && NeQ[n, j] && ILtQ[Simplify[(n*p + n - j + 1)/(n - j)], 0] && NeQ[j*p + 1, 0]
Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol ] :> Simp[(-c^(j - 1))*(c*x)^(m - j + 1)*((a*x^j + b*x^n)^(p + 1)/(a*(n - j )*(p + 1))), x] /; FreeQ[{a, b, c, j, m, n, p}, x] && !IntegerQ[p] && NeQ[ n, j] && EqQ[m + n*p + n - j + 1, 0] && (IntegerQ[j] || GtQ[c, 0])
Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol ] :> Simp[c^(j - 1)*(c*x)^(m - j + 1)*((a*x^j + b*x^n)^(p + 1)/(a*(m + j*p + 1))), x] - Simp[b*((m + n*p + n - j + 1)/(a*c^(n - j)*(m + j*p + 1))) I nt[(c*x)^(m + n - j)*(a*x^j + b*x^n)^p, x], x] /; FreeQ[{a, b, c, j, m, n, p}, x] && !IntegerQ[p] && NeQ[n, j] && ILtQ[Simplify[(m + n*p + n - j + 1) /(n - j)], 0] && NeQ[m + j*p + 1, 0] && (IntegersQ[j, n] || GtQ[c, 0])
Time = 0.04 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.40
method | result | size |
derivativedivides | \(\frac {4 \sqrt {x +x^{\frac {3}{2}}}\, \left (1+\sqrt {x}\right ) \left (315 x^{2}-280 x^{\frac {3}{2}}+240 x -192 \sqrt {x}+128\right )}{3465 \sqrt {x}}\) | \(38\) |
default | \(\frac {4 \sqrt {x +x^{\frac {3}{2}}}\, \left (1+\sqrt {x}\right ) \left (315 x^{2}-280 x^{\frac {3}{2}}+240 x -192 \sqrt {x}+128\right )}{3465 \sqrt {x}}\) | \(38\) |
meijerg | \(-\frac {\frac {512 \sqrt {\pi }}{3465}-\frac {4 \sqrt {\pi }\, \left (1+\sqrt {x}\right )^{\frac {3}{2}} \left (315 x^{2}-280 x^{\frac {3}{2}}+240 x -192 \sqrt {x}+128\right )}{3465}}{\sqrt {\pi }}\) | \(44\) |
Input:
int(x*(x+x^(3/2))^(1/2),x,method=_RETURNVERBOSE)
Output:
4/3465*(x+x^(3/2))^(1/2)*(1+x^(1/2))*(315*x^2-280*x^(3/2)+240*x-192*x^(1/2 )+128)/x^(1/2)
Time = 0.11 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.43 \[ \int x \sqrt {x+x^{3/2}} \, dx=\frac {4 \, {\left (315 \, x^{3} - 40 \, x^{2} + {\left (35 \, x^{2} + 48 \, x + 128\right )} \sqrt {x} - 64 \, x\right )} \sqrt {x^{\frac {3}{2}} + x}}{3465 \, x} \] Input:
integrate(x*(x+x^(3/2))^(1/2),x, algorithm="fricas")
Output:
4/3465*(315*x^3 - 40*x^2 + (35*x^2 + 48*x + 128)*sqrt(x) - 64*x)*sqrt(x^(3 /2) + x)/x
\[ \int x \sqrt {x+x^{3/2}} \, dx=\int x \sqrt {x^{\frac {3}{2}} + x}\, dx \] Input:
integrate(x*(x+x**(3/2))**(1/2),x)
Output:
Integral(x*sqrt(x**(3/2) + x), x)
\[ \int x \sqrt {x+x^{3/2}} \, dx=\int { \sqrt {x^{\frac {3}{2}} + x} x \,d x } \] Input:
integrate(x*(x+x^(3/2))^(1/2),x, algorithm="maxima")
Output:
integrate(sqrt(x^(3/2) + x)*x, x)
Time = 0.12 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.54 \[ \int x \sqrt {x+x^{3/2}} \, dx=\frac {4}{3465} \, {\left (315 \, {\left (\sqrt {x} + 1\right )}^{\frac {11}{2}} - 1540 \, {\left (\sqrt {x} + 1\right )}^{\frac {9}{2}} + 2970 \, {\left (\sqrt {x} + 1\right )}^{\frac {7}{2}} - 2772 \, {\left (\sqrt {x} + 1\right )}^{\frac {5}{2}} + 1155 \, {\left (\sqrt {x} + 1\right )}^{\frac {3}{2}} - 128\right )} \mathrm {sgn}\left (x\right ) \] Input:
integrate(x*(x+x^(3/2))^(1/2),x, algorithm="giac")
Output:
4/3465*(315*(sqrt(x) + 1)^(11/2) - 1540*(sqrt(x) + 1)^(9/2) + 2970*(sqrt(x ) + 1)^(7/2) - 2772*(sqrt(x) + 1)^(5/2) + 1155*(sqrt(x) + 1)^(3/2) - 128)* sgn(x)
Timed out. \[ \int x \sqrt {x+x^{3/2}} \, dx=\int x\,\sqrt {x+x^{3/2}} \,d x \] Input:
int(x*(x + x^(3/2))^(1/2),x)
Output:
int(x*(x + x^(3/2))^(1/2), x)
Time = 0.18 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.35 \[ \int x \sqrt {x+x^{3/2}} \, dx=\frac {4 \sqrt {\sqrt {x}+1}\, \left (315 \sqrt {x}\, x^{2}-40 \sqrt {x}\, x -64 \sqrt {x}+35 x^{2}+48 x +128\right )}{3465} \] Input:
int(x*(x+x^(3/2))^(1/2),x)
Output:
(4*sqrt(sqrt(x) + 1)*(315*sqrt(x)*x**2 - 40*sqrt(x)*x - 64*sqrt(x) + 35*x* *2 + 48*x + 128))/3465