Integrand size = 19, antiderivative size = 200 \[ \int \frac {1}{x^4 \sqrt {b \sqrt {x}+a x}} \, dx=-\frac {4 \sqrt {b \sqrt {x}+a x}}{13 b x^{7/2}}+\frac {48 a \sqrt {b \sqrt {x}+a x}}{143 b^2 x^3}-\frac {160 a^2 \sqrt {b \sqrt {x}+a x}}{429 b^3 x^{5/2}}+\frac {1280 a^3 \sqrt {b \sqrt {x}+a x}}{3003 b^4 x^2}-\frac {512 a^4 \sqrt {b \sqrt {x}+a x}}{1001 b^5 x^{3/2}}+\frac {2048 a^5 \sqrt {b \sqrt {x}+a x}}{3003 b^6 x}-\frac {4096 a^6 \sqrt {b \sqrt {x}+a x}}{3003 b^7 \sqrt {x}} \] Output:
-4/13*(b*x^(1/2)+a*x)^(1/2)/b/x^(7/2)+48/143*a*(b*x^(1/2)+a*x)^(1/2)/b^2/x ^3-160/429*a^2*(b*x^(1/2)+a*x)^(1/2)/b^3/x^(5/2)+1280/3003*a^3*(b*x^(1/2)+ a*x)^(1/2)/b^4/x^2-512/1001*a^4*(b*x^(1/2)+a*x)^(1/2)/b^5/x^(3/2)+2048/300 3*a^5*(b*x^(1/2)+a*x)^(1/2)/b^6/x-4096/3003*a^6*(b*x^(1/2)+a*x)^(1/2)/b^7/ x^(1/2)
Time = 0.16 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.48 \[ \int \frac {1}{x^4 \sqrt {b \sqrt {x}+a x}} \, dx=-\frac {4 \sqrt {b \sqrt {x}+a x} \left (231 b^6-252 a b^5 \sqrt {x}+280 a^2 b^4 x-320 a^3 b^3 x^{3/2}+384 a^4 b^2 x^2-512 a^5 b x^{5/2}+1024 a^6 x^3\right )}{3003 b^7 x^{7/2}} \] Input:
Integrate[1/(x^4*Sqrt[b*Sqrt[x] + a*x]),x]
Output:
(-4*Sqrt[b*Sqrt[x] + a*x]*(231*b^6 - 252*a*b^5*Sqrt[x] + 280*a^2*b^4*x - 3 20*a^3*b^3*x^(3/2) + 384*a^4*b^2*x^2 - 512*a^5*b*x^(5/2) + 1024*a^6*x^3))/ (3003*b^7*x^(7/2))
Time = 0.43 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.15, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {1922, 1922, 1922, 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.
| \(\displaystyle \int \frac {1}{x^4 \sqrt {a x+b \sqrt {x}}} \, dx\) |
| \(\Big \downarrow \) 1922 |
| \(\displaystyle -\frac {12 a \int \frac {1}{x^{7/2} \sqrt {\sqrt {x} b+a x}}dx}{13 b}-\frac {4 \sqrt {a x+b \sqrt {x}}}{13 b x^{7/2}}\) |
| \(\Big \downarrow \) 1922 |
| \(\displaystyle -\frac {12 a \left (-\frac {10 a \int \frac {1}{x^3 \sqrt {\sqrt {x} b+a x}}dx}{11 b}-\frac {4 \sqrt {a x+b \sqrt {x}}}{11 b x^3}\right )}{13 b}-\frac {4 \sqrt {a x+b \sqrt {x}}}{13 b x^{7/2}}\) |
| \(\Big \downarrow \) 1922 |
| \(\displaystyle -\frac {12 a \left (-\frac {10 a \left (-\frac {8 a \int \frac {1}{x^{5/2} \sqrt {\sqrt {x} b+a x}}dx}{9 b}-\frac {4 \sqrt {a x+b \sqrt {x}}}{9 b x^{5/2}}\right )}{11 b}-\frac {4 \sqrt {a x+b \sqrt {x}}}{11 b x^3}\right )}{13 b}-\frac {4 \sqrt {a x+b \sqrt {x}}}{13 b x^{7/2}}\) |
| \(\Big \downarrow \) 1922 |
| \(\displaystyle -\frac {12 a \left (-\frac {10 a \left (-\frac {8 a \left (-\frac {6 a \int \frac {1}{x^2 \sqrt {\sqrt {x} b+a x}}dx}{7 b}-\frac {4 \sqrt {a x+b \sqrt {x}}}{7 b x^2}\right )}{9 b}-\frac {4 \sqrt {a x+b \sqrt {x}}}{9 b x^{5/2}}\right )}{11 b}-\frac {4 \sqrt {a x+b \sqrt {x}}}{11 b x^3}\right )}{13 b}-\frac {4 \sqrt {a x+b \sqrt {x}}}{13 b x^{7/2}}\) |
| \(\Big \downarrow \) 1922 |
| \(\displaystyle -\frac {12 a \left (-\frac {10 a \left (-\frac {8 a \left (-\frac {6 a \left (-\frac {4 a \int \frac {1}{x^{3/2} \sqrt {\sqrt {x} b+a x}}dx}{5 b}-\frac {4 \sqrt {a x+b \sqrt {x}}}{5 b x^{3/2}}\right )}{7 b}-\frac {4 \sqrt {a x+b \sqrt {x}}}{7 b x^2}\right )}{9 b}-\frac {4 \sqrt {a x+b \sqrt {x}}}{9 b x^{5/2}}\right )}{11 b}-\frac {4 \sqrt {a x+b \sqrt {x}}}{11 b x^3}\right )}{13 b}-\frac {4 \sqrt {a x+b \sqrt {x}}}{13 b x^{7/2}}\) |
| \(\Big \downarrow \) 1922 |
| \(\displaystyle -\frac {12 a \left (-\frac {10 a \left (-\frac {8 a \left (-\frac {6 a \left (-\frac {4 a \left (-\frac {2 a \int \frac {1}{x \sqrt {\sqrt {x} b+a x}}dx}{3 b}-\frac {4 \sqrt {a x+b \sqrt {x}}}{3 b x}\right )}{5 b}-\frac {4 \sqrt {a x+b \sqrt {x}}}{5 b x^{3/2}}\right )}{7 b}-\frac {4 \sqrt {a x+b \sqrt {x}}}{7 b x^2}\right )}{9 b}-\frac {4 \sqrt {a x+b \sqrt {x}}}{9 b x^{5/2}}\right )}{11 b}-\frac {4 \sqrt {a x+b \sqrt {x}}}{11 b x^3}\right )}{13 b}-\frac {4 \sqrt {a x+b \sqrt {x}}}{13 b x^{7/2}}\) |
| \(\Big \downarrow \) 1920 |
| \(\displaystyle -\frac {12 a \left (-\frac {10 a \left (-\frac {8 a \left (-\frac {6 a \left (-\frac {4 a \left (\frac {8 a \sqrt {a x+b \sqrt {x}}}{3 b^2 \sqrt {x}}-\frac {4 \sqrt {a x+b \sqrt {x}}}{3 b x}\right )}{5 b}-\frac {4 \sqrt {a x+b \sqrt {x}}}{5 b x^{3/2}}\right )}{7 b}-\frac {4 \sqrt {a x+b \sqrt {x}}}{7 b x^2}\right )}{9 b}-\frac {4 \sqrt {a x+b \sqrt {x}}}{9 b x^{5/2}}\right )}{11 b}-\frac {4 \sqrt {a x+b \sqrt {x}}}{11 b x^3}\right )}{13 b}-\frac {4 \sqrt {a x+b \sqrt {x}}}{13 b x^{7/2}}\) |
Input:
Int[1/(x^4*Sqrt[b*Sqrt[x] + a*x]),x]
Output:
(-4*Sqrt[b*Sqrt[x] + a*x])/(13*b*x^(7/2)) - (12*a*((-4*Sqrt[b*Sqrt[x] + a* x])/(11*b*x^3) - (10*a*((-4*Sqrt[b*Sqrt[x] + a*x])/(9*b*x^(5/2)) - (8*a*(( -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)))/(9*b)))/(11*b)))/(13*b)
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 = 0.09 (sec) , antiderivative size = 171, normalized size of antiderivative = 0.86
| method | result | size |
| derivativedivides | \(-\frac {4 \sqrt {b \sqrt {x}+x a}}{13 b \,x^{\frac {7}{2}}}-\frac {24 a \left (-\frac {2 \sqrt {b \sqrt {x}+x a}}{11 b \,x^{3}}-\frac {10 a \left (-\frac {2 \sqrt {b \sqrt {x}+x a}}{9 b \,x^{\frac {5}{2}}}-\frac {8 a \left (-\frac {2 \sqrt {b \sqrt {x}+x a}}{7 b \,x^{2}}-\frac {6 a \left (-\frac {2 \sqrt {b \sqrt {x}+x a}}{5 b \,x^{\frac {3}{2}}}-\frac {4 a \left (-\frac {2 \sqrt {b \sqrt {x}+x a}}{3 b x}+\frac {4 a \sqrt {b \sqrt {x}+x a}}{3 b^{2} \sqrt {x}}\right )}{5 b}\right )}{7 b}\right )}{9 b}\right )}{11 b}\right )}{13 b}\) | \(171\) |
| default | \(-\frac {\sqrt {b \sqrt {x}+x a}\, \left (12012 \left (b \sqrt {x}+x a \right )^{\frac {3}{2}} x^{\frac {13}{2}} a^{\frac {13}{2}}-6006 \sqrt {b \sqrt {x}+x a}\, x^{\frac {15}{2}} a^{\frac {15}{2}}-3003 x^{\frac {15}{2}} \ln \left (\frac {2 \sqrt {x}\, a +2 \sqrt {b \sqrt {x}+x a}\, \sqrt {a}+b}{2 \sqrt {a}}\right ) a^{7} b -6006 x^{\frac {15}{2}} a^{\frac {15}{2}} \sqrt {\sqrt {x}\, \left (\sqrt {x}\, a +b \right )}+3003 x^{\frac {15}{2}} \ln \left (\frac {2 \sqrt {x}\, a +2 \sqrt {\sqrt {x}\, \left (\sqrt {x}\, a +b \right )}\, \sqrt {a}+b}{2 \sqrt {a}}\right ) a^{7} b +5868 \left (b \sqrt {x}+x a \right )^{\frac {3}{2}} x^{\frac {11}{2}} a^{\frac {9}{2}} b^{2}+3052 \left (b \sqrt {x}+x a \right )^{\frac {3}{2}} x^{\frac {9}{2}} a^{\frac {5}{2}} b^{4}-7916 a^{\frac {11}{2}} \left (b \sqrt {x}+x a \right )^{\frac {3}{2}} b \,x^{6}+924 \left (b \sqrt {x}+x a \right )^{\frac {3}{2}} x^{\frac {7}{2}} \sqrt {a}\, b^{6}-4332 \left (b \sqrt {x}+x a \right )^{\frac {3}{2}} x^{5} a^{\frac {7}{2}} b^{3}-1932 \left (b \sqrt {x}+x a \right )^{\frac {3}{2}} a^{\frac {3}{2}} b^{5} x^{4}\right )}{3003 \sqrt {\sqrt {x}\, \left (\sqrt {x}\, a +b \right )}\, b^{8} x^{\frac {15}{2}} \sqrt {a}}\) | \(306\) |
Input:
int(1/x^4/(b*x^(1/2)+x*a)^(1/2),x,method=_RETURNVERBOSE)
Output:
-4/13*(b*x^(1/2)+x*a)^(1/2)/b/x^(7/2)-24/13*a/b*(-2/11*(b*x^(1/2)+x*a)^(1/ 2)/b/x^3-10/11*a/b*(-2/9*(b*x^(1/2)+x*a)^(1/2)/b/x^(5/2)-8/9*a/b*(-2/7*(b* x^(1/2)+x*a)^(1/2)/b/x^2-6/7*a/b*(-2/5*(b*x^(1/2)+x*a)^(1/2)/b/x^(3/2)-4/5 *a/b*(-2/3*(b*x^(1/2)+x*a)^(1/2)/b/x+4/3*a*(b*x^(1/2)+x*a)^(1/2)/b^2/x^(1/ 2))))))
Time = 0.10 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.43 \[ \int \frac {1}{x^4 \sqrt {b \sqrt {x}+a x}} \, dx=\frac {4 \, {\left (512 \, a^{5} b x^{3} + 320 \, a^{3} b^{3} x^{2} + 252 \, a b^{5} x - {\left (1024 \, a^{6} x^{3} + 384 \, a^{4} b^{2} x^{2} + 280 \, a^{2} b^{4} x + 231 \, b^{6}\right )} \sqrt {x}\right )} \sqrt {a x + b \sqrt {x}}}{3003 \, b^{7} x^{4}} \] Input:
integrate(1/x^4/(b*x^(1/2)+a*x)^(1/2),x, algorithm="fricas")
Output:
4/3003*(512*a^5*b*x^3 + 320*a^3*b^3*x^2 + 252*a*b^5*x - (1024*a^6*x^3 + 38 4*a^4*b^2*x^2 + 280*a^2*b^4*x + 231*b^6)*sqrt(x))*sqrt(a*x + b*sqrt(x))/(b ^7*x^4)
\[ \int \frac {1}{x^4 \sqrt {b \sqrt {x}+a x}} \, dx=\int \frac {1}{x^{4} \sqrt {a x + b \sqrt {x}}}\, dx \] Input:
integrate(1/x**4/(b*x**(1/2)+a*x)**(1/2),x)
Output:
Integral(1/(x**4*sqrt(a*x + b*sqrt(x))), x)
\[ \int \frac {1}{x^4 \sqrt {b \sqrt {x}+a x}} \, dx=\int { \frac {1}{\sqrt {a x + b \sqrt {x}} x^{4}} \,d x } \] Input:
integrate(1/x^4/(b*x^(1/2)+a*x)^(1/2),x, algorithm="maxima")
Output:
integrate(1/(sqrt(a*x + b*sqrt(x))*x^4), x)
Time = 0.12 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.04 \[ \int \frac {1}{x^4 \sqrt {b \sqrt {x}+a x}} \, dx=\frac {4 \, {\left (27456 \, a^{3} {\left (\sqrt {a} \sqrt {x} - \sqrt {a x + b \sqrt {x}}\right )}^{6} + 72072 \, a^{\frac {5}{2}} b {\left (\sqrt {a} \sqrt {x} - \sqrt {a x + b \sqrt {x}}\right )}^{5} + 80080 \, a^{2} b^{2} {\left (\sqrt {a} \sqrt {x} - \sqrt {a x + b \sqrt {x}}\right )}^{4} + 48048 \, a^{\frac {3}{2}} b^{3} {\left (\sqrt {a} \sqrt {x} - \sqrt {a x + b \sqrt {x}}\right )}^{3} + 16380 \, a b^{4} {\left (\sqrt {a} \sqrt {x} - \sqrt {a x + b \sqrt {x}}\right )}^{2} + 3003 \, \sqrt {a} b^{5} {\left (\sqrt {a} \sqrt {x} - \sqrt {a x + b \sqrt {x}}\right )} + 231 \, b^{6}\right )}}{3003 \, {\left (\sqrt {a} \sqrt {x} - \sqrt {a x + b \sqrt {x}}\right )}^{13}} \] Input:
integrate(1/x^4/(b*x^(1/2)+a*x)^(1/2),x, algorithm="giac")
Output:
4/3003*(27456*a^3*(sqrt(a)*sqrt(x) - sqrt(a*x + b*sqrt(x)))^6 + 72072*a^(5 /2)*b*(sqrt(a)*sqrt(x) - sqrt(a*x + b*sqrt(x)))^5 + 80080*a^2*b^2*(sqrt(a) *sqrt(x) - sqrt(a*x + b*sqrt(x)))^4 + 48048*a^(3/2)*b^3*(sqrt(a)*sqrt(x) - sqrt(a*x + b*sqrt(x)))^3 + 16380*a*b^4*(sqrt(a)*sqrt(x) - sqrt(a*x + b*sq rt(x)))^2 + 3003*sqrt(a)*b^5*(sqrt(a)*sqrt(x) - sqrt(a*x + b*sqrt(x))) + 2 31*b^6)/(sqrt(a)*sqrt(x) - sqrt(a*x + b*sqrt(x)))^13
Timed out. \[ \int \frac {1}{x^4 \sqrt {b \sqrt {x}+a x}} \, dx=\int \frac {1}{x^4\,\sqrt {a\,x+b\,\sqrt {x}}} \,d x \] Input:
int(1/(x^4*(a*x + b*x^(1/2))^(1/2)),x)
Output:
int(1/(x^4*(a*x + b*x^(1/2))^(1/2)), x)
Time = 0.17 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.70 \[ \int \frac {1}{x^4 \sqrt {b \sqrt {x}+a x}} \, dx=\frac {\frac {2048 x^{\frac {11}{4}} \sqrt {\sqrt {x}\, a +b}\, a^{5} b}{3003}+\frac {1280 x^{\frac {7}{4}} \sqrt {\sqrt {x}\, a +b}\, a^{3} b^{3}}{3003}+\frac {48 x^{\frac {3}{4}} \sqrt {\sqrt {x}\, a +b}\, a \,b^{5}}{143}-\frac {4096 x^{\frac {13}{4}} \sqrt {\sqrt {x}\, a +b}\, a^{6}}{3003}-\frac {512 x^{\frac {9}{4}} \sqrt {\sqrt {x}\, a +b}\, a^{4} b^{2}}{1001}-\frac {160 x^{\frac {5}{4}} \sqrt {\sqrt {x}\, a +b}\, a^{2} b^{4}}{429}-\frac {4 x^{\frac {1}{4}} \sqrt {\sqrt {x}\, a +b}\, b^{6}}{13}+\frac {4096 \sqrt {x}\, \sqrt {a}\, a^{6} x^{3}}{3003}}{\sqrt {x}\, b^{7} x^{3}} \] Input:
int(1/x^4/(b*x^(1/2)+a*x)^(1/2),x)
Output:
(4*(512*x**(3/4)*sqrt(sqrt(x)*a + b)*a**5*b*x**2 + 320*x**(3/4)*sqrt(sqrt( x)*a + b)*a**3*b**3*x + 252*x**(3/4)*sqrt(sqrt(x)*a + b)*a*b**5 - 1024*x** (1/4)*sqrt(sqrt(x)*a + b)*a**6*x**3 - 384*x**(1/4)*sqrt(sqrt(x)*a + b)*a** 4*b**2*x**2 - 280*x**(1/4)*sqrt(sqrt(x)*a + b)*a**2*b**4*x - 231*x**(1/4)* sqrt(sqrt(x)*a + b)*b**6 + 1024*sqrt(x)*sqrt(a)*a**6*x**3))/(3003*sqrt(x)* b**7*x**3)