Integrand size = 18, antiderivative size = 76 \[ \int \frac {1}{\sqrt {x} \left (1+\sqrt {x}+x\right )^{7/2}} \, dx=\frac {4 \left (1+2 \sqrt {x}\right )}{15 \left (1+\sqrt {x}+x\right )^{5/2}}+\frac {64 \left (1+2 \sqrt {x}\right )}{135 \left (1+\sqrt {x}+x\right )^{3/2}}+\frac {512 \left (1+2 \sqrt {x}\right )}{405 \sqrt {1+\sqrt {x}+x}} \] Output:
4/15*(1+2*x^(1/2))/(1+x^(1/2)+x)^(5/2)+64/135*(1+2*x^(1/2))/(1+x^(1/2)+x)^ (3/2)+512/405*(1+2*x^(1/2))/(1+x^(1/2)+x)^(1/2)
Time = 0.33 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.64 \[ \int \frac {1}{\sqrt {x} \left (1+\sqrt {x}+x\right )^{7/2}} \, dx=\frac {4 \left (1+2 \sqrt {x}\right ) \left (203+304 \sqrt {x}+432 x+256 x^{3/2}+128 x^2\right )}{405 \left (1+\sqrt {x}+x\right )^{5/2}} \] Input:
Integrate[1/(Sqrt[x]*(1 + Sqrt[x] + x)^(7/2)),x]
Output:
(4*(1 + 2*Sqrt[x])*(203 + 304*Sqrt[x] + 432*x + 256*x^(3/2) + 128*x^2))/(4 05*(1 + Sqrt[x] + x)^(5/2))
Time = 0.32 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.09, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {1690, 1089, 1089, 1088}
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.
\(\displaystyle \int \frac {1}{\sqrt {x} \left (x+\sqrt {x}+1\right )^{7/2}} \, dx\) |
\(\Big \downarrow \) 1690 |
\(\displaystyle 2 \int \frac {1}{\left (x+\sqrt {x}+1\right )^{7/2}}d\sqrt {x}\) |
\(\Big \downarrow \) 1089 |
\(\displaystyle 2 \left (\frac {16}{15} \int \frac {1}{\left (x+\sqrt {x}+1\right )^{5/2}}d\sqrt {x}+\frac {2 \left (2 \sqrt {x}+1\right )}{15 \left (x+\sqrt {x}+1\right )^{5/2}}\right )\) |
\(\Big \downarrow \) 1089 |
\(\displaystyle 2 \left (\frac {16}{15} \left (\frac {8}{9} \int \frac {1}{\left (x+\sqrt {x}+1\right )^{3/2}}d\sqrt {x}+\frac {2 \left (2 \sqrt {x}+1\right )}{9 \left (x+\sqrt {x}+1\right )^{3/2}}\right )+\frac {2 \left (2 \sqrt {x}+1\right )}{15 \left (x+\sqrt {x}+1\right )^{5/2}}\right )\) |
\(\Big \downarrow \) 1088 |
\(\displaystyle 2 \left (\frac {2 \left (2 \sqrt {x}+1\right )}{15 \left (x+\sqrt {x}+1\right )^{5/2}}+\frac {16}{15} \left (\frac {16 \left (2 \sqrt {x}+1\right )}{27 \sqrt {x+\sqrt {x}+1}}+\frac {2 \left (2 \sqrt {x}+1\right )}{9 \left (x+\sqrt {x}+1\right )^{3/2}}\right )\right )\) |
Input:
Int[1/(Sqrt[x]*(1 + Sqrt[x] + x)^(7/2)),x]
Output:
2*((2*(1 + 2*Sqrt[x]))/(15*(1 + Sqrt[x] + x)^(5/2)) + (16*((2*(1 + 2*Sqrt[ x]))/(9*(1 + Sqrt[x] + x)^(3/2)) + (16*(1 + 2*Sqrt[x]))/(27*Sqrt[1 + Sqrt[ x] + x])))/15)
Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-3/2), x_Symbol] :> Simp[-2*((b + 2*c*x)/((b^2 - 4*a*c)*Sqrt[a + b*x + c*x^2])), x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]
Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(b + 2*c*x) *((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c))), x] - Simp[2*c*((2*p + 3)/((p + 1)*(b^2 - 4*a*c))) Int[(a + b*x + c*x^2)^(p + 1), x], x] /; Fre eQ[{a, b, c}, x] && LtQ[p, -1] && (IntegerQ[4*p] || IntegerQ[3*p])
Int[(x_)^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol ] :> Simp[1/n Subst[Int[(a + b*x + c*x^2)^p, x], x, x^n], x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[n2, 2*n] && EqQ[Simplify[m - n + 1], 0]
Time = 0.02 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.70
method | result | size |
derivativedivides | \(\frac {\frac {4}{15}+\frac {8 \sqrt {x}}{15}}{\left (1+\sqrt {x}+x \right )^{\frac {5}{2}}}+\frac {\frac {64}{135}+\frac {128 \sqrt {x}}{135}}{\left (1+\sqrt {x}+x \right )^{\frac {3}{2}}}+\frac {\frac {512}{405}+\frac {1024 \sqrt {x}}{405}}{\sqrt {1+\sqrt {x}+x}}\) | \(53\) |
default | \(\frac {\frac {4}{15}+\frac {8 \sqrt {x}}{15}}{\left (1+\sqrt {x}+x \right )^{\frac {5}{2}}}+\frac {\frac {64}{135}+\frac {128 \sqrt {x}}{135}}{\left (1+\sqrt {x}+x \right )^{\frac {3}{2}}}+\frac {\frac {512}{405}+\frac {1024 \sqrt {x}}{405}}{\sqrt {1+\sqrt {x}+x}}\) | \(53\) |
Input:
int(1/x^(1/2)/(1+x^(1/2)+x)^(7/2),x,method=_RETURNVERBOSE)
Output:
4/15*(1+2*x^(1/2))/(1+x^(1/2)+x)^(5/2)+64/135*(1+2*x^(1/2))/(1+x^(1/2)+x)^ (3/2)+512/405*(1+2*x^(1/2))/(1+x^(1/2)+x)^(1/2)
Time = 0.12 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.25 \[ \int \frac {1}{\sqrt {x} \left (1+\sqrt {x}+x\right )^{7/2}} \, dx=-\frac {4 \, {\left (128 \, x^{5} + 272 \, x^{4} + 455 \, x^{3} + 232 \, x^{2} - {\left (256 \, x^{5} + 736 \, x^{4} + 1366 \, x^{3} + 1427 \, x^{2} + 839 \, x + 101\right )} \sqrt {x} - 128 \, x - 203\right )} \sqrt {x + \sqrt {x} + 1}}{405 \, {\left (x^{6} + 3 \, x^{5} + 6 \, x^{4} + 7 \, x^{3} + 6 \, x^{2} + 3 \, x + 1\right )}} \] Input:
integrate(1/x^(1/2)/(1+x^(1/2)+x)^(7/2),x, algorithm="fricas")
Output:
-4/405*(128*x^5 + 272*x^4 + 455*x^3 + 232*x^2 - (256*x^5 + 736*x^4 + 1366* x^3 + 1427*x^2 + 839*x + 101)*sqrt(x) - 128*x - 203)*sqrt(x + sqrt(x) + 1) /(x^6 + 3*x^5 + 6*x^4 + 7*x^3 + 6*x^2 + 3*x + 1)
\[ \int \frac {1}{\sqrt {x} \left (1+\sqrt {x}+x\right )^{7/2}} \, dx=\int \frac {1}{\sqrt {x} \left (\sqrt {x} + x + 1\right )^{\frac {7}{2}}}\, dx \] Input:
integrate(1/x**(1/2)/(1+x**(1/2)+x)**(7/2),x)
Output:
Integral(1/(sqrt(x)*(sqrt(x) + x + 1)**(7/2)), x)
\[ \int \frac {1}{\sqrt {x} \left (1+\sqrt {x}+x\right )^{7/2}} \, dx=\int { \frac {1}{{\left (x + \sqrt {x} + 1\right )}^{\frac {7}{2}} \sqrt {x}} \,d x } \] Input:
integrate(1/x^(1/2)/(1+x^(1/2)+x)^(7/2),x, algorithm="maxima")
Output:
integrate(1/((x + sqrt(x) + 1)^(7/2)*sqrt(x)), x)
Time = 0.12 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.59 \[ \int \frac {1}{\sqrt {x} \left (1+\sqrt {x}+x\right )^{7/2}} \, dx=\frac {4 \, {\left (2 \, {\left (8 \, {\left (2 \, {\left (4 \, \sqrt {x} {\left (2 \, \sqrt {x} + 5\right )} + 35\right )} \sqrt {x} + 65\right )} \sqrt {x} + 355\right )} \sqrt {x} + 203\right )}}{405 \, {\left (x + \sqrt {x} + 1\right )}^{\frac {5}{2}}} \] Input:
integrate(1/x^(1/2)/(1+x^(1/2)+x)^(7/2),x, algorithm="giac")
Output:
4/405*(2*(8*(2*(4*sqrt(x)*(2*sqrt(x) + 5) + 35)*sqrt(x) + 65)*sqrt(x) + 35 5)*sqrt(x) + 203)/(x + sqrt(x) + 1)^(5/2)
Timed out. \[ \int \frac {1}{\sqrt {x} \left (1+\sqrt {x}+x\right )^{7/2}} \, dx=\int \frac {1}{\sqrt {x}\,{\left (x+\sqrt {x}+1\right )}^{7/2}} \,d x \] Input:
int(1/(x^(1/2)*(x + x^(1/2) + 1)^(7/2)),x)
Output:
int(1/(x^(1/2)*(x + x^(1/2) + 1)^(7/2)), x)
Time = 0.18 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.68 \[ \int \frac {1}{\sqrt {x} \left (1+\sqrt {x}+x\right )^{7/2}} \, dx=\frac {1024 \sqrt {x}\, \sqrt {\sqrt {x}+x +1}\, x^{2}+4480 \sqrt {x}\, \sqrt {\sqrt {x}+x +1}\, x +2840 \sqrt {x}\, \sqrt {\sqrt {x}+x +1}+2560 \sqrt {\sqrt {x}+x +1}\, x^{2}+4160 \sqrt {\sqrt {x}+x +1}\, x +812 \sqrt {\sqrt {x}+x +1}-3072 \sqrt {x}\, x^{2}-7168 \sqrt {x}\, x -3072 \sqrt {x}-1024 x^{3}-6144 x^{2}-6144 x -1024}{1215 \sqrt {x}\, x^{2}+2835 \sqrt {x}\, x +1215 \sqrt {x}+405 x^{3}+2430 x^{2}+2430 x +405} \] Input:
int(1/x^(1/2)/(1+x^(1/2)+x)^(7/2),x)
Output:
(4*(256*sqrt(x)*sqrt(sqrt(x) + x + 1)*x**2 + 1120*sqrt(x)*sqrt(sqrt(x) + x + 1)*x + 710*sqrt(x)*sqrt(sqrt(x) + x + 1) + 640*sqrt(sqrt(x) + x + 1)*x* *2 + 1040*sqrt(sqrt(x) + x + 1)*x + 203*sqrt(sqrt(x) + x + 1) - 768*sqrt(x )*x**2 - 1792*sqrt(x)*x - 768*sqrt(x) - 256*x**3 - 1536*x**2 - 1536*x - 25 6))/(405*(3*sqrt(x)*x**2 + 7*sqrt(x)*x + 3*sqrt(x) + x**3 + 6*x**2 + 6*x + 1))