Integrand size = 13, antiderivative size = 75 \[ \int \sqrt {x} (1+x)^{5/2} \, dx=\frac {5}{64} \sqrt {x} \sqrt {1+x}+\frac {5}{32} x^{3/2} \sqrt {1+x}+\frac {5}{24} x^{3/2} (1+x)^{3/2}+\frac {1}{4} x^{3/2} (1+x)^{5/2}-\frac {5 \text {arcsinh}\left (\sqrt {x}\right )}{64} \]
5/24*x^(3/2)*(1+x)^(3/2)+1/4*x^(3/2)*(1+x)^(5/2)-5/64*arcsinh(x^(1/2))+5/3 2*x^(3/2)*(1+x)^(1/2)+5/64*x^(1/2)*(1+x)^(1/2)
Time = 0.09 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.69 \[ \int \sqrt {x} (1+x)^{5/2} \, dx=\frac {1}{192} \sqrt {x} \sqrt {1+x} \left (15+118 x+136 x^2+48 x^3\right )+\frac {5}{64} \log \left (-\sqrt {x}+\sqrt {1+x}\right ) \]
(Sqrt[x]*Sqrt[1 + x]*(15 + 118*x + 136*x^2 + 48*x^3))/192 + (5*Log[-Sqrt[x ] + Sqrt[1 + x]])/64
Time = 0.16 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.13, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {60, 60, 60, 60, 63, 222}
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.
\(\displaystyle \int \sqrt {x} (x+1)^{5/2} \, dx\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {5}{8} \int \sqrt {x} (x+1)^{3/2}dx+\frac {1}{4} x^{3/2} (x+1)^{5/2}\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {5}{8} \left (\frac {1}{2} \int \sqrt {x} \sqrt {x+1}dx+\frac {1}{3} x^{3/2} (x+1)^{3/2}\right )+\frac {1}{4} x^{3/2} (x+1)^{5/2}\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {5}{8} \left (\frac {1}{2} \left (\frac {1}{4} \int \frac {\sqrt {x}}{\sqrt {x+1}}dx+\frac {1}{2} \sqrt {x+1} x^{3/2}\right )+\frac {1}{3} x^{3/2} (x+1)^{3/2}\right )+\frac {1}{4} x^{3/2} (x+1)^{5/2}\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {5}{8} \left (\frac {1}{2} \left (\frac {1}{4} \left (\sqrt {x} \sqrt {x+1}-\frac {1}{2} \int \frac {1}{\sqrt {x} \sqrt {x+1}}dx\right )+\frac {1}{2} \sqrt {x+1} x^{3/2}\right )+\frac {1}{3} x^{3/2} (x+1)^{3/2}\right )+\frac {1}{4} x^{3/2} (x+1)^{5/2}\) |
\(\Big \downarrow \) 63 |
\(\displaystyle \frac {5}{8} \left (\frac {1}{2} \left (\frac {1}{4} \left (\sqrt {x} \sqrt {x+1}-\int \frac {1}{\sqrt {x+1}}d\sqrt {x}\right )+\frac {1}{2} \sqrt {x+1} x^{3/2}\right )+\frac {1}{3} x^{3/2} (x+1)^{3/2}\right )+\frac {1}{4} x^{3/2} (x+1)^{5/2}\) |
\(\Big \downarrow \) 222 |
\(\displaystyle \frac {5}{8} \left (\frac {1}{2} \left (\frac {1}{4} \left (\sqrt {x} \sqrt {x+1}-\text {arcsinh}\left (\sqrt {x}\right )\right )+\frac {1}{2} \sqrt {x+1} x^{3/2}\right )+\frac {1}{3} x^{3/2} (x+1)^{3/2}\right )+\frac {1}{4} x^{3/2} (x+1)^{5/2}\) |
(x^(3/2)*(1 + x)^(5/2))/4 + (5*((x^(3/2)*(1 + x)^(3/2))/3 + ((x^(3/2)*Sqrt [1 + x])/2 + (Sqrt[x]*Sqrt[1 + x] - ArcSinh[Sqrt[x]])/4)/2))/8
3.1.38.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( b*(m + n + 1))) Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !Integer Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && IntLinear Q[a, b, c, d, m, n, x]
Int[1/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[2/b S ubst[Int[1/Sqrt[c + d*(x^2/b)], x], x, Sqrt[b*x]], x] /; FreeQ[{b, c, d}, x ] && GtQ[c, 0]
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[Rt[b, 2]*(x/Sqrt [a])]/Rt[b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && PosQ[b]
Time = 0.12 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.59
method | result | size |
meijerg | \(-\frac {15 \left (-\frac {\sqrt {\pi }\, \sqrt {x}\, \left (48 x^{3}+136 x^{2}+118 x +15\right ) \sqrt {1+x}}{360}+\frac {\sqrt {\pi }\, \operatorname {arcsinh}\left (\sqrt {x}\right )}{24}\right )}{8 \sqrt {\pi }}\) | \(44\) |
risch | \(\frac {\left (48 x^{3}+136 x^{2}+118 x +15\right ) \sqrt {x}\, \sqrt {1+x}}{192}-\frac {5 \sqrt {x \left (1+x \right )}\, \ln \left (x +\frac {1}{2}+\sqrt {x^{2}+x}\right )}{128 \sqrt {1+x}\, \sqrt {x}}\) | \(55\) |
default | \(\frac {\sqrt {x}\, \left (1+x \right )^{\frac {7}{2}}}{4}-\frac {\sqrt {x}\, \left (1+x \right )^{\frac {5}{2}}}{24}-\frac {5 \sqrt {x}\, \left (1+x \right )^{\frac {3}{2}}}{96}-\frac {5 \sqrt {x}\, \sqrt {1+x}}{64}-\frac {5 \sqrt {x \left (1+x \right )}\, \ln \left (x +\frac {1}{2}+\sqrt {x^{2}+x}\right )}{128 \sqrt {1+x}\, \sqrt {x}}\) | \(70\) |
-15/8/Pi^(1/2)*(-1/360*Pi^(1/2)*x^(1/2)*(48*x^3+136*x^2+118*x+15)*(1+x)^(1 /2)+1/24*Pi^(1/2)*arcsinh(x^(1/2)))
Time = 0.24 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.59 \[ \int \sqrt {x} (1+x)^{5/2} \, dx=\frac {1}{192} \, {\left (48 \, x^{3} + 136 \, x^{2} + 118 \, x + 15\right )} \sqrt {x + 1} \sqrt {x} + \frac {5}{128} \, \log \left (2 \, \sqrt {x + 1} \sqrt {x} - 2 \, x - 1\right ) \]
1/192*(48*x^3 + 136*x^2 + 118*x + 15)*sqrt(x + 1)*sqrt(x) + 5/128*log(2*sq rt(x + 1)*sqrt(x) - 2*x - 1)
Result contains complex when optimal does not.
Time = 18.65 (sec) , antiderivative size = 190, normalized size of antiderivative = 2.53 \[ \int \sqrt {x} (1+x)^{5/2} \, dx=\begin {cases} - \frac {5 \operatorname {acosh}{\left (\sqrt {x + 1} \right )}}{64} + \frac {\left (x + 1\right )^{\frac {9}{2}}}{4 \sqrt {x}} - \frac {7 \left (x + 1\right )^{\frac {7}{2}}}{24 \sqrt {x}} - \frac {\left (x + 1\right )^{\frac {5}{2}}}{96 \sqrt {x}} - \frac {5 \left (x + 1\right )^{\frac {3}{2}}}{192 \sqrt {x}} + \frac {5 \sqrt {x + 1}}{64 \sqrt {x}} & \text {for}\: \left |{x + 1}\right | > 1 \\\frac {5 i \operatorname {asin}{\left (\sqrt {x + 1} \right )}}{64} - \frac {i \left (x + 1\right )^{\frac {9}{2}}}{4 \sqrt {- x}} + \frac {7 i \left (x + 1\right )^{\frac {7}{2}}}{24 \sqrt {- x}} + \frac {i \left (x + 1\right )^{\frac {5}{2}}}{96 \sqrt {- x}} + \frac {5 i \left (x + 1\right )^{\frac {3}{2}}}{192 \sqrt {- x}} - \frac {5 i \sqrt {x + 1}}{64 \sqrt {- x}} & \text {otherwise} \end {cases} \]
Piecewise((-5*acosh(sqrt(x + 1))/64 + (x + 1)**(9/2)/(4*sqrt(x)) - 7*(x + 1)**(7/2)/(24*sqrt(x)) - (x + 1)**(5/2)/(96*sqrt(x)) - 5*(x + 1)**(3/2)/(1 92*sqrt(x)) + 5*sqrt(x + 1)/(64*sqrt(x)), Abs(x + 1) > 1), (5*I*asin(sqrt( x + 1))/64 - I*(x + 1)**(9/2)/(4*sqrt(-x)) + 7*I*(x + 1)**(7/2)/(24*sqrt(- x)) + I*(x + 1)**(5/2)/(96*sqrt(-x)) + 5*I*(x + 1)**(3/2)/(192*sqrt(-x)) - 5*I*sqrt(x + 1)/(64*sqrt(-x)), True))
Leaf count of result is larger than twice the leaf count of optimal. 113 vs. \(2 (47) = 94\).
Time = 0.18 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.51 \[ \int \sqrt {x} (1+x)^{5/2} \, dx=\frac {\frac {15 \, {\left (x + 1\right )}^{\frac {7}{2}}}{x^{\frac {7}{2}}} + \frac {73 \, {\left (x + 1\right )}^{\frac {5}{2}}}{x^{\frac {5}{2}}} - \frac {55 \, {\left (x + 1\right )}^{\frac {3}{2}}}{x^{\frac {3}{2}}} + \frac {15 \, \sqrt {x + 1}}{\sqrt {x}}}{192 \, {\left (\frac {{\left (x + 1\right )}^{4}}{x^{4}} - \frac {4 \, {\left (x + 1\right )}^{3}}{x^{3}} + \frac {6 \, {\left (x + 1\right )}^{2}}{x^{2}} - \frac {4 \, {\left (x + 1\right )}}{x} + 1\right )}} - \frac {5}{128} \, \log \left (\frac {\sqrt {x + 1}}{\sqrt {x}} + 1\right ) + \frac {5}{128} \, \log \left (\frac {\sqrt {x + 1}}{\sqrt {x}} - 1\right ) \]
1/192*(15*(x + 1)^(7/2)/x^(7/2) + 73*(x + 1)^(5/2)/x^(5/2) - 55*(x + 1)^(3 /2)/x^(3/2) + 15*sqrt(x + 1)/sqrt(x))/((x + 1)^4/x^4 - 4*(x + 1)^3/x^3 + 6 *(x + 1)^2/x^2 - 4*(x + 1)/x + 1) - 5/128*log(sqrt(x + 1)/sqrt(x) + 1) + 5 /128*log(sqrt(x + 1)/sqrt(x) - 1)
Time = 0.29 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.20 \[ \int \sqrt {x} (1+x)^{5/2} \, dx=\frac {1}{192} \, {\left (2 \, {\left (4 \, {\left (6 \, x - 19\right )} {\left (x + 1\right )} + 163\right )} {\left (x + 1\right )} - 279\right )} \sqrt {x + 1} \sqrt {x} + \frac {1}{8} \, {\left (2 \, {\left (4 \, x - 9\right )} {\left (x + 1\right )} + 33\right )} \sqrt {x + 1} \sqrt {x} + \frac {3}{4} \, {\left (2 \, x - 3\right )} \sqrt {x + 1} \sqrt {x} + \sqrt {x + 1} \sqrt {x} + \frac {5}{64} \, \log \left (\sqrt {x + 1} - \sqrt {x}\right ) \]
1/192*(2*(4*(6*x - 19)*(x + 1) + 163)*(x + 1) - 279)*sqrt(x + 1)*sqrt(x) + 1/8*(2*(4*x - 9)*(x + 1) + 33)*sqrt(x + 1)*sqrt(x) + 3/4*(2*x - 3)*sqrt(x + 1)*sqrt(x) + sqrt(x + 1)*sqrt(x) + 5/64*log(sqrt(x + 1) - sqrt(x))
Timed out. \[ \int \sqrt {x} (1+x)^{5/2} \, dx=\int \sqrt {x}\,{\left (x+1\right )}^{5/2} \,d x \]