Integrand size = 21, antiderivative size = 107 \[ \int \frac {1}{x^{3/2} \left (b \sqrt {x}+a x\right )^{3/2}} \, dx=\frac {4}{b x \sqrt {b \sqrt {x}+a x}}-\frac {24 \sqrt {b \sqrt {x}+a x}}{5 b^2 x^{3/2}}+\frac {32 a \sqrt {b \sqrt {x}+a x}}{5 b^3 x}-\frac {64 a^2 \sqrt {b \sqrt {x}+a x}}{5 b^4 \sqrt {x}} \]
4/b/x/(b*x^(1/2)+a*x)^(1/2)-24/5*(b*x^(1/2)+a*x)^(1/2)/b^2/x^(3/2)+32/5*a* (b*x^(1/2)+a*x)^(1/2)/b^3/x-64/5*a^2*(b*x^(1/2)+a*x)^(1/2)/b^4/x^(1/2)
Time = 0.25 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.65 \[ \int \frac {1}{x^{3/2} \left (b \sqrt {x}+a x\right )^{3/2}} \, dx=-\frac {4 \sqrt {b \sqrt {x}+a x} \left (b^3-2 a b^2 \sqrt {x}+8 a^2 b x+16 a^3 x^{3/2}\right )}{5 b^4 \left (b+a \sqrt {x}\right ) x^{3/2}} \]
(-4*Sqrt[b*Sqrt[x] + a*x]*(b^3 - 2*a*b^2*Sqrt[x] + 8*a^2*b*x + 16*a^3*x^(3 /2)))/(5*b^4*(b + a*Sqrt[x])*x^(3/2))
Time = 0.28 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.11, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {1921, 1922, 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.
\(\displaystyle \int \frac {1}{x^{3/2} \left (a x+b \sqrt {x}\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 1921 |
\(\displaystyle \frac {6 \int \frac {1}{x^2 \sqrt {\sqrt {x} b+a x}}dx}{b}+\frac {4}{b x \sqrt {a x+b \sqrt {x}}}\) |
\(\Big \downarrow \) 1922 |
\(\displaystyle \frac {6 \left (-\frac {4 a \int \frac {1}{x^{3/2} \sqrt {\sqrt {x} b+a x}}dx}{5 b}-\frac {4 \sqrt {a x+b \sqrt {x}}}{5 b x^{3/2}}\right )}{b}+\frac {4}{b x \sqrt {a x+b \sqrt {x}}}\) |
\(\Big \downarrow \) 1922 |
\(\displaystyle \frac {6 \left (-\frac {4 a \left (-\frac {2 a \int \frac {1}{x \sqrt {\sqrt {x} b+a x}}dx}{3 b}-\frac {4 \sqrt {a x+b \sqrt {x}}}{3 b x}\right )}{5 b}-\frac {4 \sqrt {a x+b \sqrt {x}}}{5 b x^{3/2}}\right )}{b}+\frac {4}{b x \sqrt {a x+b \sqrt {x}}}\) |
\(\Big \downarrow \) 1920 |
\(\displaystyle \frac {6 \left (-\frac {4 a \left (\frac {8 a \sqrt {a x+b \sqrt {x}}}{3 b^2 \sqrt {x}}-\frac {4 \sqrt {a x+b \sqrt {x}}}{3 b x}\right )}{5 b}-\frac {4 \sqrt {a x+b \sqrt {x}}}{5 b x^{3/2}}\right )}{b}+\frac {4}{b x \sqrt {a x+b \sqrt {x}}}\) |
4/(b*x*Sqrt[b*Sqrt[x] + a*x]) + (6*((-4*Sqrt[b*Sqrt[x] + a*x])/(5*b*x^(3/2 )) - (4*a*((-4*Sqrt[b*Sqrt[x] + a*x])/(3*b*x) + (8*a*Sqrt[b*Sqrt[x] + a*x] )/(3*b^2*Sqrt[x])))/(5*b)))/b
3.2.28.3.1 Defintions of rubi rules used
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*(n - j )*(p + 1))), x] + Simp[c^j*((m + n*p + n - j + 1)/(a*(n - j)*(p + 1))) In t[(c*x)^(m - j)*(a*x^j + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, c, j, m, n} , x] && !IntegerQ[p] && NeQ[n, j] && ILtQ[Simplify[(m + n*p + n - j + 1)/( n - j)], 0] && LtQ[p, -1] && (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 = 2.17 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.67
method | result | size |
derivativedivides | \(-\frac {4}{5 b x \sqrt {b \sqrt {x}+a x}}-\frac {12 a \left (-\frac {2}{3 b \sqrt {x}\, \sqrt {b \sqrt {x}+a x}}+\frac {8 a \left (b +2 a \sqrt {x}\right )}{3 b^{3} \sqrt {b \sqrt {x}+a x}}\right )}{5 b}\) | \(72\) |
default | \(\frac {2 \sqrt {b \sqrt {x}+a x}\, \left (10 x^{\frac {9}{2}} \sqrt {b \sqrt {x}+a x}\, a^{\frac {11}{2}}+10 x^{\frac {9}{2}} a^{\frac {11}{2}} \sqrt {\sqrt {x}\, \left (a \sqrt {x}+b \right )}-30 x^{\frac {7}{2}} \left (b \sqrt {x}+a x \right )^{\frac {3}{2}} a^{\frac {9}{2}}+10 x^{\frac {7}{2}} a^{\frac {9}{2}} \left (\sqrt {x}\, \left (a \sqrt {x}+b \right )\right )^{\frac {3}{2}}+10 x^{\frac {7}{2}} \sqrt {b \sqrt {x}+a x}\, a^{\frac {7}{2}} b^{2}+10 x^{\frac {7}{2}} a^{\frac {7}{2}} \sqrt {\sqrt {x}\, \left (a \sqrt {x}+b \right )}\, b^{2}+5 x^{\frac {9}{2}} \ln \left (\frac {2 \sqrt {b \sqrt {x}+a x}\, \sqrt {a}+2 a \sqrt {x}+b}{2 \sqrt {a}}\right ) a^{5} b -5 x^{\frac {9}{2}} \ln \left (\frac {2 a \sqrt {x}+2 \sqrt {\sqrt {x}\, \left (a \sqrt {x}+b \right )}\, \sqrt {a}+b}{2 \sqrt {a}}\right ) a^{5} b -16 x^{\frac {5}{2}} \left (b \sqrt {x}+a x \right )^{\frac {3}{2}} a^{\frac {5}{2}} b^{2}+20 x^{4} \sqrt {b \sqrt {x}+a x}\, a^{\frac {9}{2}} b +20 x^{4} a^{\frac {9}{2}} \sqrt {\sqrt {x}\, \left (a \sqrt {x}+b \right )}\, b +5 x^{\frac {7}{2}} \ln \left (\frac {2 \sqrt {b \sqrt {x}+a x}\, \sqrt {a}+2 a \sqrt {x}+b}{2 \sqrt {a}}\right ) a^{3} b^{3}-5 x^{\frac {7}{2}} \ln \left (\frac {2 a \sqrt {x}+2 \sqrt {\sqrt {x}\, \left (a \sqrt {x}+b \right )}\, \sqrt {a}+b}{2 \sqrt {a}}\right ) a^{3} b^{3}-52 x^{3} \left (b \sqrt {x}+a x \right )^{\frac {3}{2}} a^{\frac {7}{2}} b -2 x^{\frac {3}{2}} \left (b \sqrt {x}+a x \right )^{\frac {3}{2}} \sqrt {a}\, b^{4}+4 x^{2} \left (b \sqrt {x}+a x \right )^{\frac {3}{2}} a^{\frac {3}{2}} b^{3}+10 x^{4} \ln \left (\frac {2 \sqrt {b \sqrt {x}+a x}\, \sqrt {a}+2 a \sqrt {x}+b}{2 \sqrt {a}}\right ) a^{4} b^{2}-10 x^{4} \ln \left (\frac {2 a \sqrt {x}+2 \sqrt {\sqrt {x}\, \left (a \sqrt {x}+b \right )}\, \sqrt {a}+b}{2 \sqrt {a}}\right ) a^{4} b^{2}\right )}{5 \sqrt {\sqrt {x}\, \left (a \sqrt {x}+b \right )}\, b^{5} x^{\frac {7}{2}} \sqrt {a}\, \left (a \sqrt {x}+b \right )^{2}}\) | \(548\) |
-4/5/b/x/(b*x^(1/2)+a*x)^(1/2)-12/5*a/b*(-2/3/b/x^(1/2)/(b*x^(1/2)+a*x)^(1 /2)+8/3*a/b^3*(b+2*a*x^(1/2))/(b*x^(1/2)+a*x)^(1/2))
Time = 0.49 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.74 \[ \int \frac {1}{x^{3/2} \left (b \sqrt {x}+a x\right )^{3/2}} \, dx=\frac {4 \, {\left (8 \, a^{3} b x^{2} - 3 \, a b^{3} x - {\left (16 \, a^{4} x^{2} - 10 \, a^{2} b^{2} x - b^{4}\right )} \sqrt {x}\right )} \sqrt {a x + b \sqrt {x}}}{5 \, {\left (a^{2} b^{4} x^{3} - b^{6} x^{2}\right )}} \]
4/5*(8*a^3*b*x^2 - 3*a*b^3*x - (16*a^4*x^2 - 10*a^2*b^2*x - b^4)*sqrt(x))* sqrt(a*x + b*sqrt(x))/(a^2*b^4*x^3 - b^6*x^2)
\[ \int \frac {1}{x^{3/2} \left (b \sqrt {x}+a x\right )^{3/2}} \, dx=\int \frac {1}{x^{\frac {3}{2}} \left (a x + b \sqrt {x}\right )^{\frac {3}{2}}}\, dx \]
\[ \int \frac {1}{x^{3/2} \left (b \sqrt {x}+a x\right )^{3/2}} \, dx=\int { \frac {1}{{\left (a x + b \sqrt {x}\right )}^{\frac {3}{2}} x^{\frac {3}{2}}} \,d x } \]
\[ \int \frac {1}{x^{3/2} \left (b \sqrt {x}+a x\right )^{3/2}} \, dx=\int { \frac {1}{{\left (a x + b \sqrt {x}\right )}^{\frac {3}{2}} x^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {1}{x^{3/2} \left (b \sqrt {x}+a x\right )^{3/2}} \, dx=\int \frac {1}{x^{3/2}\,{\left (a\,x+b\,\sqrt {x}\right )}^{3/2}} \,d x \]