Integrand size = 21, antiderivative size = 112 \[ \int \frac {1}{x^{5/2} \sqrt {b \sqrt {x}+a x}} \, dx=-\frac {4 \sqrt {b \sqrt {x}+a x}}{7 b x^2}+\frac {24 a \sqrt {b \sqrt {x}+a x}}{35 b^2 x^{3/2}}-\frac {32 a^2 \sqrt {b \sqrt {x}+a x}}{35 b^3 x}+\frac {64 a^3 \sqrt {b \sqrt {x}+a x}}{35 b^4 \sqrt {x}} \]
-4/7*(b*x^(1/2)+a*x)^(1/2)/b/x^2+24/35*a*(b*x^(1/2)+a*x)^(1/2)/b^2/x^(3/2) -32/35*a^2*(b*x^(1/2)+a*x)^(1/2)/b^3/x+64/35*a^3*(b*x^(1/2)+a*x)^(1/2)/b^4 /x^(1/2)
Time = 0.18 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.53 \[ \int \frac {1}{x^{5/2} \sqrt {b \sqrt {x}+a x}} \, dx=-\frac {4 \sqrt {b \sqrt {x}+a x} \left (5 b^3-6 a b^2 \sqrt {x}+8 a^2 b x-16 a^3 x^{3/2}\right )}{35 b^4 x^2} \]
(-4*Sqrt[b*Sqrt[x] + a*x]*(5*b^3 - 6*a*b^2*Sqrt[x] + 8*a^2*b*x - 16*a^3*x^ (3/2)))/(35*b^4*x^2)
Time = 0.27 (sec) , antiderivative size = 124, 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 = {1922, 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.
\(\displaystyle \int \frac {1}{x^{5/2} \sqrt {a x+b \sqrt {x}}} \, dx\) |
\(\Big \downarrow \) 1922 |
\(\displaystyle -\frac {6 a \int \frac {1}{x^2 \sqrt {\sqrt {x} b+a x}}dx}{7 b}-\frac {4 \sqrt {a x+b \sqrt {x}}}{7 b x^2}\) |
\(\Big \downarrow \) 1922 |
\(\displaystyle -\frac {6 a \left (-\frac {4 a \int \frac {1}{x^{3/2} \sqrt {\sqrt {x} b+a x}}dx}{5 b}-\frac {4 \sqrt {a x+b \sqrt {x}}}{5 b x^{3/2}}\right )}{7 b}-\frac {4 \sqrt {a x+b \sqrt {x}}}{7 b x^2}\) |
\(\Big \downarrow \) 1922 |
\(\displaystyle -\frac {6 a \left (-\frac {4 a \left (-\frac {2 a \int \frac {1}{x \sqrt {\sqrt {x} b+a x}}dx}{3 b}-\frac {4 \sqrt {a x+b \sqrt {x}}}{3 b x}\right )}{5 b}-\frac {4 \sqrt {a x+b \sqrt {x}}}{5 b x^{3/2}}\right )}{7 b}-\frac {4 \sqrt {a x+b \sqrt {x}}}{7 b x^2}\) |
\(\Big \downarrow \) 1920 |
\(\displaystyle -\frac {6 a \left (-\frac {4 a \left (\frac {8 a \sqrt {a x+b \sqrt {x}}}{3 b^2 \sqrt {x}}-\frac {4 \sqrt {a x+b \sqrt {x}}}{3 b x}\right )}{5 b}-\frac {4 \sqrt {a x+b \sqrt {x}}}{5 b x^{3/2}}\right )}{7 b}-\frac {4 \sqrt {a x+b \sqrt {x}}}{7 b x^2}\) |
(-4*Sqrt[b*Sqrt[x] + a*x])/(7*b*x^2) - (6*a*((-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)))/(7*b)
3.2.22.3.1 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*(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 = 2.13 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.83
method | result | size |
derivativedivides | \(-\frac {4 \sqrt {b \sqrt {x}+a x}}{7 b \,x^{2}}-\frac {12 a \left (-\frac {2 \sqrt {b \sqrt {x}+a x}}{5 b \,x^{\frac {3}{2}}}-\frac {4 a \left (-\frac {2 \sqrt {b \sqrt {x}+a x}}{3 b x}+\frac {4 a \sqrt {b \sqrt {x}+a x}}{3 b^{2} \sqrt {x}}\right )}{5 b}\right )}{7 b}\) | \(93\) |
default | \(-\frac {\sqrt {b \sqrt {x}+a x}\, \left (70 x^{\frac {9}{2}} \sqrt {b \sqrt {x}+a x}\, a^{\frac {9}{2}}+70 x^{\frac {9}{2}} a^{\frac {9}{2}} \sqrt {\sqrt {x}\, \left (a \sqrt {x}+b \right )}-140 x^{\frac {7}{2}} \left (b \sqrt {x}+a x \right )^{\frac {3}{2}} a^{\frac {7}{2}}+35 x^{\frac {9}{2}} \ln \left (\frac {2 \sqrt {b \sqrt {x}+a x}\, \sqrt {a}+2 a \sqrt {x}+b}{2 \sqrt {a}}\right ) a^{4} b -35 x^{\frac {9}{2}} \ln \left (\frac {2 a \sqrt {x}+2 \sqrt {\sqrt {x}\, \left (a \sqrt {x}+b \right )}\, \sqrt {a}+b}{2 \sqrt {a}}\right ) a^{4} b -44 x^{\frac {5}{2}} \left (b \sqrt {x}+a x \right )^{\frac {3}{2}} a^{\frac {3}{2}} b^{2}+76 a^{\frac {5}{2}} \left (b \sqrt {x}+a x \right )^{\frac {3}{2}} b \,x^{3}+20 \left (b \sqrt {x}+a x \right )^{\frac {3}{2}} \sqrt {a}\, b^{3} x^{2}\right )}{35 \sqrt {\sqrt {x}\, \left (a \sqrt {x}+b \right )}\, b^{5} x^{\frac {9}{2}} \sqrt {a}}\) | \(240\) |
-4/7*(b*x^(1/2)+a*x)^(1/2)/b/x^2-12/7*a/b*(-2/5*(b*x^(1/2)+a*x)^(1/2)/b/x^ (3/2)-4/5*a/b*(-2/3*(b*x^(1/2)+a*x)^(1/2)/b/x+4/3*a*(b*x^(1/2)+a*x)^(1/2)/ b^2/x^(1/2)))
Time = 0.34 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.45 \[ \int \frac {1}{x^{5/2} \sqrt {b \sqrt {x}+a x}} \, dx=-\frac {4 \, {\left (8 \, a^{2} b x + 5 \, b^{3} - 2 \, {\left (8 \, a^{3} x + 3 \, a b^{2}\right )} \sqrt {x}\right )} \sqrt {a x + b \sqrt {x}}}{35 \, b^{4} x^{2}} \]
\[ \int \frac {1}{x^{5/2} \sqrt {b \sqrt {x}+a x}} \, dx=\int \frac {1}{x^{\frac {5}{2}} \sqrt {a x + b \sqrt {x}}}\, dx \]
\[ \int \frac {1}{x^{5/2} \sqrt {b \sqrt {x}+a x}} \, dx=\int { \frac {1}{\sqrt {a x + b \sqrt {x}} x^{\frac {5}{2}}} \,d x } \]
Time = 0.31 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.03 \[ \int \frac {1}{x^{5/2} \sqrt {b \sqrt {x}+a x}} \, dx=\frac {4 \, {\left (70 \, a^{\frac {3}{2}} {\left (\sqrt {a} \sqrt {x} - \sqrt {a x + b \sqrt {x}}\right )}^{3} + 84 \, a b {\left (\sqrt {a} \sqrt {x} - \sqrt {a x + b \sqrt {x}}\right )}^{2} + 35 \, \sqrt {a} b^{2} {\left (\sqrt {a} \sqrt {x} - \sqrt {a x + b \sqrt {x}}\right )} + 5 \, b^{3}\right )}}{35 \, {\left (\sqrt {a} \sqrt {x} - \sqrt {a x + b \sqrt {x}}\right )}^{7}} \]
4/35*(70*a^(3/2)*(sqrt(a)*sqrt(x) - sqrt(a*x + b*sqrt(x)))^3 + 84*a*b*(sqr t(a)*sqrt(x) - sqrt(a*x + b*sqrt(x)))^2 + 35*sqrt(a)*b^2*(sqrt(a)*sqrt(x) - sqrt(a*x + b*sqrt(x))) + 5*b^3)/(sqrt(a)*sqrt(x) - sqrt(a*x + b*sqrt(x)) )^7
Timed out. \[ \int \frac {1}{x^{5/2} \sqrt {b \sqrt {x}+a x}} \, dx=\int \frac {1}{x^{5/2}\,\sqrt {a\,x+b\,\sqrt {x}}} \,d x \]