Integrand size = 19, antiderivative size = 177 \[ \int \frac {1}{x^3 \left (b \sqrt {x}+a x\right )^{3/2}} \, dx=\frac {2048 a^5 \left (b+2 a \sqrt {x}\right )}{231 b^7 \sqrt {b \sqrt {x}+a x}}-\frac {4}{11 b x^{5/2} \sqrt {b \sqrt {x}+a x}}+\frac {16 a}{33 b^2 x^2 \sqrt {b \sqrt {x}+a x}}-\frac {160 a^2}{231 b^3 x^{3/2} \sqrt {b \sqrt {x}+a x}}+\frac {256 a^3}{231 b^4 x \sqrt {b \sqrt {x}+a x}}-\frac {512 a^4}{231 b^5 \sqrt {x} \sqrt {b \sqrt {x}+a x}} \] Output:
2048/231*a^5*(b+2*a*x^(1/2))/b^7/(b*x^(1/2)+a*x)^(1/2)-4/11/b/x^(5/2)/(b*x ^(1/2)+a*x)^(1/2)+16/33*a/b^2/x^2/(b*x^(1/2)+a*x)^(1/2)-160/231*a^2/b^3/x^ (3/2)/(b*x^(1/2)+a*x)^(1/2)+256/231*a^3/b^4/x/(b*x^(1/2)+a*x)^(1/2)-512/23 1*a^4/b^5/x^(1/2)/(b*x^(1/2)+a*x)^(1/2)
Time = 0.23 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.59 \[ \int \frac {1}{x^3 \left (b \sqrt {x}+a x\right )^{3/2}} \, dx=-\frac {4 \sqrt {b \sqrt {x}+a x} \left (21 b^6-28 a b^5 \sqrt {x}+40 a^2 b^4 x-64 a^3 b^3 x^{3/2}+128 a^4 b^2 x^2-512 a^5 b x^{5/2}-1024 a^6 x^3\right )}{231 b^7 \left (b+a \sqrt {x}\right ) x^3} \] Input:
Integrate[1/(x^3*(b*Sqrt[x] + a*x)^(3/2)),x]
Output:
(-4*Sqrt[b*Sqrt[x] + a*x]*(21*b^6 - 28*a*b^5*Sqrt[x] + 40*a^2*b^4*x - 64*a ^3*b^3*x^(3/2) + 128*a^4*b^2*x^2 - 512*a^5*b*x^(5/2) - 1024*a^6*x^3))/(231 *b^7*(b + a*Sqrt[x])*x^3)
Time = 0.43 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.27, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {1921, 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.
| \(\displaystyle \int \frac {1}{x^3 \left (a x+b \sqrt {x}\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 1921 |
| \(\displaystyle \frac {12 \int \frac {1}{x^{7/2} \sqrt {\sqrt {x} b+a x}}dx}{b}+\frac {4}{b x^{5/2} \sqrt {a x+b \sqrt {x}}}\) |
| \(\Big \downarrow \) 1922 |
| \(\displaystyle \frac {12 \left (-\frac {10 a \int \frac {1}{x^3 \sqrt {\sqrt {x} b+a x}}dx}{11 b}-\frac {4 \sqrt {a x+b \sqrt {x}}}{11 b x^3}\right )}{b}+\frac {4}{b x^{5/2} \sqrt {a x+b \sqrt {x}}}\) |
| \(\Big \downarrow \) 1922 |
| \(\displaystyle \frac {12 \left (-\frac {10 a \left (-\frac {8 a \int \frac {1}{x^{5/2} \sqrt {\sqrt {x} b+a x}}dx}{9 b}-\frac {4 \sqrt {a x+b \sqrt {x}}}{9 b x^{5/2}}\right )}{11 b}-\frac {4 \sqrt {a x+b \sqrt {x}}}{11 b x^3}\right )}{b}+\frac {4}{b x^{5/2} \sqrt {a x+b \sqrt {x}}}\) |
| \(\Big \downarrow \) 1922 |
| \(\displaystyle \frac {12 \left (-\frac {10 a \left (-\frac {8 a \left (-\frac {6 a \int \frac {1}{x^2 \sqrt {\sqrt {x} b+a x}}dx}{7 b}-\frac {4 \sqrt {a x+b \sqrt {x}}}{7 b x^2}\right )}{9 b}-\frac {4 \sqrt {a x+b \sqrt {x}}}{9 b x^{5/2}}\right )}{11 b}-\frac {4 \sqrt {a x+b \sqrt {x}}}{11 b x^3}\right )}{b}+\frac {4}{b x^{5/2} \sqrt {a x+b \sqrt {x}}}\) |
| \(\Big \downarrow \) 1922 |
| \(\displaystyle \frac {12 \left (-\frac {10 a \left (-\frac {8 a \left (-\frac {6 a \left (-\frac {4 a \int \frac {1}{x^{3/2} \sqrt {\sqrt {x} b+a x}}dx}{5 b}-\frac {4 \sqrt {a x+b \sqrt {x}}}{5 b x^{3/2}}\right )}{7 b}-\frac {4 \sqrt {a x+b \sqrt {x}}}{7 b x^2}\right )}{9 b}-\frac {4 \sqrt {a x+b \sqrt {x}}}{9 b x^{5/2}}\right )}{11 b}-\frac {4 \sqrt {a x+b \sqrt {x}}}{11 b x^3}\right )}{b}+\frac {4}{b x^{5/2} \sqrt {a x+b \sqrt {x}}}\) |
| \(\Big \downarrow \) 1922 |
| \(\displaystyle \frac {12 \left (-\frac {10 a \left (-\frac {8 a \left (-\frac {6 a \left (-\frac {4 a \left (-\frac {2 a \int \frac {1}{x \sqrt {\sqrt {x} b+a x}}dx}{3 b}-\frac {4 \sqrt {a x+b \sqrt {x}}}{3 b x}\right )}{5 b}-\frac {4 \sqrt {a x+b \sqrt {x}}}{5 b x^{3/2}}\right )}{7 b}-\frac {4 \sqrt {a x+b \sqrt {x}}}{7 b x^2}\right )}{9 b}-\frac {4 \sqrt {a x+b \sqrt {x}}}{9 b x^{5/2}}\right )}{11 b}-\frac {4 \sqrt {a x+b \sqrt {x}}}{11 b x^3}\right )}{b}+\frac {4}{b x^{5/2} \sqrt {a x+b \sqrt {x}}}\) |
| \(\Big \downarrow \) 1920 |
| \(\displaystyle \frac {12 \left (-\frac {10 a \left (-\frac {8 a \left (-\frac {6 a \left (-\frac {4 a \left (\frac {8 a \sqrt {a x+b \sqrt {x}}}{3 b^2 \sqrt {x}}-\frac {4 \sqrt {a x+b \sqrt {x}}}{3 b x}\right )}{5 b}-\frac {4 \sqrt {a x+b \sqrt {x}}}{5 b x^{3/2}}\right )}{7 b}-\frac {4 \sqrt {a x+b \sqrt {x}}}{7 b x^2}\right )}{9 b}-\frac {4 \sqrt {a x+b \sqrt {x}}}{9 b x^{5/2}}\right )}{11 b}-\frac {4 \sqrt {a x+b \sqrt {x}}}{11 b x^3}\right )}{b}+\frac {4}{b x^{5/2} \sqrt {a x+b \sqrt {x}}}\) |
Input:
Int[1/(x^3*(b*Sqrt[x] + a*x)^(3/2)),x]
Output:
4/(b*x^(5/2)*Sqrt[b*Sqrt[x] + a*x]) + (12*((-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)))/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*(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 = 0.09 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.85
| method | result | size |
| derivativedivides | \(-\frac {4}{11 b \,x^{\frac {5}{2}} \sqrt {b \sqrt {x}+x a}}-\frac {24 a \left (-\frac {2}{9 b \,x^{2} \sqrt {b \sqrt {x}+x a}}-\frac {10 a \left (-\frac {2}{7 b \,x^{\frac {3}{2}} \sqrt {b \sqrt {x}+x a}}-\frac {8 a \left (-\frac {2}{5 b x \sqrt {b \sqrt {x}+x a}}-\frac {6 a \left (-\frac {2}{3 b \sqrt {x}\, \sqrt {b \sqrt {x}+x a}}+\frac {8 a \left (b +2 \sqrt {x}\, a \right )}{3 b^{3} \sqrt {b \sqrt {x}+x a}}\right )}{5 b}\right )}{7 b}\right )}{9 b}\right )}{11 b}\) | \(150\) |
| default | \(\frac {\sqrt {b \sqrt {x}+x a}\, \left (2048 \left (b \sqrt {x}+x a \right )^{\frac {3}{2}} x^{\frac {11}{2}} a^{\frac {11}{2}} b^{2}+8716 \left (b \sqrt {x}+x a \right )^{\frac {3}{2}} x^{6} a^{\frac {13}{2}} b -4620 \sqrt {b \sqrt {x}+x a}\, x^{7} a^{\frac {15}{2}} b -4620 x^{7} a^{\frac {15}{2}} \sqrt {\sqrt {x}\, \left (\sqrt {x}\, a +b \right )}\, b -512 \left (b \sqrt {x}+x a \right )^{\frac {3}{2}} x^{5} a^{\frac {9}{2}} b^{3}-2310 \sqrt {b \sqrt {x}+x a}\, x^{\frac {13}{2}} a^{\frac {13}{2}} b^{2}-2310 x^{\frac {13}{2}} a^{\frac {13}{2}} \sqrt {\sqrt {x}\, \left (\sqrt {x}\, a +b \right )}\, b^{2}-84 \left (b \sqrt {x}+x a \right )^{\frac {3}{2}} \sqrt {a}\, b^{7} x^{3}+5544 \left (b \sqrt {x}+x a \right )^{\frac {3}{2}} x^{\frac {13}{2}} a^{\frac {15}{2}}-2310 \sqrt {b \sqrt {x}+x a}\, x^{\frac {15}{2}} a^{\frac {17}{2}}-1155 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^{8} b +1155 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^{8} b -2310 x^{\frac {15}{2}} a^{\frac {17}{2}} \sqrt {\sqrt {x}\, \left (\sqrt {x}\, a +b \right )}-2310 x^{7} \ln \left (\frac {2 \sqrt {x}\, a +2 \sqrt {b \sqrt {x}+x a}\, \sqrt {a}+b}{2 \sqrt {a}}\right ) a^{7} b^{2}+2310 x^{7} \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^{2}-924 x^{\frac {13}{2}} a^{\frac {15}{2}} \left (\sqrt {x}\, \left (\sqrt {x}\, a +b \right )\right )^{\frac {3}{2}}+256 \left (b \sqrt {x}+x a \right )^{\frac {3}{2}} x^{\frac {9}{2}} a^{\frac {7}{2}} b^{4}-1155 x^{\frac {13}{2}} \ln \left (\frac {2 \sqrt {x}\, a +2 \sqrt {b \sqrt {x}+x a}\, \sqrt {a}+b}{2 \sqrt {a}}\right ) a^{6} b^{3}+1155 x^{\frac {13}{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^{6} b^{3}-160 \left (b \sqrt {x}+x a \right )^{\frac {3}{2}} x^{4} a^{\frac {5}{2}} b^{5}+112 \left (b \sqrt {x}+x a \right )^{\frac {3}{2}} x^{\frac {7}{2}} a^{\frac {3}{2}} b^{6}\right )}{231 \sqrt {\sqrt {x}\, \left (\sqrt {x}\, a +b \right )}\, b^{8} x^{\frac {13}{2}} \sqrt {a}\, \left (\sqrt {x}\, a +b \right )^{2}}\) | \(614\) |
Input:
int(1/x^3/(b*x^(1/2)+x*a)^(3/2),x,method=_RETURNVERBOSE)
Output:
-4/11/b/x^(5/2)/(b*x^(1/2)+x*a)^(1/2)-24/11*a/b*(-2/9/b/x^2/(b*x^(1/2)+x*a )^(1/2)-10/9*a/b*(-2/7/b/x^(3/2)/(b*x^(1/2)+x*a)^(1/2)-8/7*a/b*(-2/5/b/x/( b*x^(1/2)+x*a)^(1/2)-6/5*a/b*(-2/3/b/x^(1/2)/(b*x^(1/2)+x*a)^(1/2)+8/3*a*( b+2*x^(1/2)*a)/b^3/(b*x^(1/2)+x*a)^(1/2)))))
Time = 0.11 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.62 \[ \int \frac {1}{x^3 \left (b \sqrt {x}+a x\right )^{3/2}} \, dx=-\frac {4 \, {\left (512 \, a^{6} b x^{3} - 192 \, a^{4} b^{3} x^{2} - 68 \, a^{2} b^{5} x - 21 \, b^{7} - {\left (1024 \, a^{7} x^{3} - 640 \, a^{5} b^{2} x^{2} - 104 \, a^{3} b^{4} x - 49 \, a b^{6}\right )} \sqrt {x}\right )} \sqrt {a x + b \sqrt {x}}}{231 \, {\left (a^{2} b^{7} x^{4} - b^{9} x^{3}\right )}} \] Input:
integrate(1/x^3/(b*x^(1/2)+a*x)^(3/2),x, algorithm="fricas")
Output:
-4/231*(512*a^6*b*x^3 - 192*a^4*b^3*x^2 - 68*a^2*b^5*x - 21*b^7 - (1024*a^ 7*x^3 - 640*a^5*b^2*x^2 - 104*a^3*b^4*x - 49*a*b^6)*sqrt(x))*sqrt(a*x + b* sqrt(x))/(a^2*b^7*x^4 - b^9*x^3)
\[ \int \frac {1}{x^3 \left (b \sqrt {x}+a x\right )^{3/2}} \, dx=\int \frac {1}{x^{3} \left (a x + b \sqrt {x}\right )^{\frac {3}{2}}}\, dx \] Input:
integrate(1/x**3/(b*x**(1/2)+a*x)**(3/2),x)
Output:
Integral(1/(x**3*(a*x + b*sqrt(x))**(3/2)), x)
\[ \int \frac {1}{x^3 \left (b \sqrt {x}+a x\right )^{3/2}} \, dx=\int { \frac {1}{{\left (a x + b \sqrt {x}\right )}^{\frac {3}{2}} x^{3}} \,d x } \] Input:
integrate(1/x^3/(b*x^(1/2)+a*x)^(3/2),x, algorithm="maxima")
Output:
integrate(1/((a*x + b*sqrt(x))^(3/2)*x^3), x)
\[ \int \frac {1}{x^3 \left (b \sqrt {x}+a x\right )^{3/2}} \, dx=\int { \frac {1}{{\left (a x + b \sqrt {x}\right )}^{\frac {3}{2}} x^{3}} \,d x } \] Input:
integrate(1/x^3/(b*x^(1/2)+a*x)^(3/2),x, algorithm="giac")
Output:
integrate(1/((a*x + b*sqrt(x))^(3/2)*x^3), x)
Timed out. \[ \int \frac {1}{x^3 \left (b \sqrt {x}+a x\right )^{3/2}} \, dx=\int \frac {1}{x^3\,{\left (a\,x+b\,\sqrt {x}\right )}^{3/2}} \,d x \] Input:
int(1/(x^3*(a*x + b*x^(1/2))^(3/2)),x)
Output:
int(1/(x^3*(a*x + b*x^(1/2))^(3/2)), x)
Time = 0.18 (sec) , antiderivative size = 152, normalized size of antiderivative = 0.86 \[ \int \frac {1}{x^3 \left (b \sqrt {x}+a x\right )^{3/2}} \, dx=\frac {\frac {2048 \sqrt {x}\, \sqrt {\sqrt {x}\, a +b}\, a^{5} b \,x^{2}}{231}+\frac {256 \sqrt {x}\, \sqrt {\sqrt {x}\, a +b}\, a^{3} b^{3} x}{231}+\frac {16 \sqrt {x}\, \sqrt {\sqrt {x}\, a +b}\, a \,b^{5}}{33}+\frac {4096 \sqrt {\sqrt {x}\, a +b}\, a^{6} x^{3}}{231}-\frac {512 \sqrt {\sqrt {x}\, a +b}\, a^{4} b^{2} x^{2}}{231}-\frac {160 \sqrt {\sqrt {x}\, a +b}\, a^{2} b^{4} x}{231}-\frac {4 \sqrt {\sqrt {x}\, a +b}\, b^{6}}{11}-\frac {4096 x^{\frac {11}{4}} \sqrt {a}\, a^{5} b}{231}-\frac {4096 x^{\frac {13}{4}} \sqrt {a}\, a^{6}}{231}}{x^{\frac {9}{4}} b^{7} \left (\sqrt {x}\, b +a x \right )} \] Input:
int(1/x^3/(b*x^(1/2)+a*x)^(3/2),x)
Output:
(4*(512*sqrt(x)*sqrt(sqrt(x)*a + b)*a**5*b*x**2 + 64*sqrt(x)*sqrt(sqrt(x)* a + b)*a**3*b**3*x + 28*sqrt(x)*sqrt(sqrt(x)*a + b)*a*b**5 + 1024*sqrt(sqr t(x)*a + b)*a**6*x**3 - 128*sqrt(sqrt(x)*a + b)*a**4*b**2*x**2 - 40*sqrt(s qrt(x)*a + b)*a**2*b**4*x - 21*sqrt(sqrt(x)*a + b)*b**6 - 1024*x**(3/4)*sq rt(a)*a**5*b*x**2 - 1024*x**(1/4)*sqrt(a)*a**6*x**3))/(231*x**(1/4)*b**7*x **2*(sqrt(x)*b + a*x))