Integrand size = 21, antiderivative size = 205 \[ \int \frac {1}{x^{7/2} \left (b \sqrt {x}+a x\right )^{3/2}} \, dx=-\frac {4096 a^6 \left (b+2 a \sqrt {x}\right )}{429 b^8 \sqrt {b \sqrt {x}+a x}}-\frac {4}{13 b x^3 \sqrt {b \sqrt {x}+a x}}+\frac {56 a}{143 b^2 x^{5/2} \sqrt {b \sqrt {x}+a x}}-\frac {224 a^2}{429 b^3 x^2 \sqrt {b \sqrt {x}+a x}}+\frac {320 a^3}{429 b^4 x^{3/2} \sqrt {b \sqrt {x}+a x}}-\frac {512 a^4}{429 b^5 x \sqrt {b \sqrt {x}+a x}}+\frac {1024 a^5}{429 b^6 \sqrt {x} \sqrt {b \sqrt {x}+a x}} \] Output:
-4096/429*a^6*(b+2*a*x^(1/2))/b^8/(b*x^(1/2)+a*x)^(1/2)-4/13/b/x^3/(b*x^(1 /2)+a*x)^(1/2)+56/143*a/b^2/x^(5/2)/(b*x^(1/2)+a*x)^(1/2)-224/429*a^2/b^3/ x^2/(b*x^(1/2)+a*x)^(1/2)+320/429*a^3/b^4/x^(3/2)/(b*x^(1/2)+a*x)^(1/2)-51 2/429*a^4/b^5/x/(b*x^(1/2)+a*x)^(1/2)+1024/429*a^5/b^6/x^(1/2)/(b*x^(1/2)+ a*x)^(1/2)
Time = 0.26 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.59 \[ \int \frac {1}{x^{7/2} \left (b \sqrt {x}+a x\right )^{3/2}} \, dx=-\frac {4 \sqrt {b \sqrt {x}+a x} \left (33 b^7-42 a b^6 \sqrt {x}+56 a^2 b^5 x-80 a^3 b^4 x^{3/2}+128 a^4 b^3 x^2-256 a^5 b^2 x^{5/2}+1024 a^6 b x^3+2048 a^7 x^{7/2}\right )}{429 b^8 \left (b+a \sqrt {x}\right ) x^{7/2}} \] Input:
Integrate[1/(x^(7/2)*(b*Sqrt[x] + a*x)^(3/2)),x]
Output:
(-4*Sqrt[b*Sqrt[x] + a*x]*(33*b^7 - 42*a*b^6*Sqrt[x] + 56*a^2*b^5*x - 80*a ^3*b^4*x^(3/2) + 128*a^4*b^3*x^2 - 256*a^5*b^2*x^(5/2) + 1024*a^6*b*x^3 + 2048*a^7*x^(7/2)))/(429*b^8*(b + a*Sqrt[x])*x^(7/2))
Time = 0.49 (sec) , antiderivative size = 259, normalized size of antiderivative = 1.26, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {1921, 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.
| \(\displaystyle \int \frac {1}{x^{7/2} \left (a x+b \sqrt {x}\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 1921 |
| \(\displaystyle \frac {14 \int \frac {1}{x^4 \sqrt {\sqrt {x} b+a x}}dx}{b}+\frac {4}{b x^3 \sqrt {a x+b \sqrt {x}}}\) |
| \(\Big \downarrow \) 1922 |
| \(\displaystyle \frac {14 \left (-\frac {12 a \int \frac {1}{x^{7/2} \sqrt {\sqrt {x} b+a x}}dx}{13 b}-\frac {4 \sqrt {a x+b \sqrt {x}}}{13 b x^{7/2}}\right )}{b}+\frac {4}{b x^3 \sqrt {a x+b \sqrt {x}}}\) |
| \(\Big \downarrow \) 1922 |
| \(\displaystyle \frac {14 \left (-\frac {12 a \left (-\frac {10 a \int \frac {1}{x^3 \sqrt {\sqrt {x} b+a x}}dx}{11 b}-\frac {4 \sqrt {a x+b \sqrt {x}}}{11 b x^3}\right )}{13 b}-\frac {4 \sqrt {a x+b \sqrt {x}}}{13 b x^{7/2}}\right )}{b}+\frac {4}{b x^3 \sqrt {a x+b \sqrt {x}}}\) |
| \(\Big \downarrow \) 1922 |
| \(\displaystyle \frac {14 \left (-\frac {12 a \left (-\frac {10 a \left (-\frac {8 a \int \frac {1}{x^{5/2} \sqrt {\sqrt {x} b+a x}}dx}{9 b}-\frac {4 \sqrt {a x+b \sqrt {x}}}{9 b x^{5/2}}\right )}{11 b}-\frac {4 \sqrt {a x+b \sqrt {x}}}{11 b x^3}\right )}{13 b}-\frac {4 \sqrt {a x+b \sqrt {x}}}{13 b x^{7/2}}\right )}{b}+\frac {4}{b x^3 \sqrt {a x+b \sqrt {x}}}\) |
| \(\Big \downarrow \) 1922 |
| \(\displaystyle \frac {14 \left (-\frac {12 a \left (-\frac {10 a \left (-\frac {8 a \left (-\frac {6 a \int \frac {1}{x^2 \sqrt {\sqrt {x} b+a x}}dx}{7 b}-\frac {4 \sqrt {a x+b \sqrt {x}}}{7 b x^2}\right )}{9 b}-\frac {4 \sqrt {a x+b \sqrt {x}}}{9 b x^{5/2}}\right )}{11 b}-\frac {4 \sqrt {a x+b \sqrt {x}}}{11 b x^3}\right )}{13 b}-\frac {4 \sqrt {a x+b \sqrt {x}}}{13 b x^{7/2}}\right )}{b}+\frac {4}{b x^3 \sqrt {a x+b \sqrt {x}}}\) |
| \(\Big \downarrow \) 1922 |
| \(\displaystyle \frac {14 \left (-\frac {12 a \left (-\frac {10 a \left (-\frac {8 a \left (-\frac {6 a \left (-\frac {4 a \int \frac {1}{x^{3/2} \sqrt {\sqrt {x} b+a x}}dx}{5 b}-\frac {4 \sqrt {a x+b \sqrt {x}}}{5 b x^{3/2}}\right )}{7 b}-\frac {4 \sqrt {a x+b \sqrt {x}}}{7 b x^2}\right )}{9 b}-\frac {4 \sqrt {a x+b \sqrt {x}}}{9 b x^{5/2}}\right )}{11 b}-\frac {4 \sqrt {a x+b \sqrt {x}}}{11 b x^3}\right )}{13 b}-\frac {4 \sqrt {a x+b \sqrt {x}}}{13 b x^{7/2}}\right )}{b}+\frac {4}{b x^3 \sqrt {a x+b \sqrt {x}}}\) |
| \(\Big \downarrow \) 1922 |
| \(\displaystyle \frac {14 \left (-\frac {12 a \left (-\frac {10 a \left (-\frac {8 a \left (-\frac {6 a \left (-\frac {4 a \left (-\frac {2 a \int \frac {1}{x \sqrt {\sqrt {x} b+a x}}dx}{3 b}-\frac {4 \sqrt {a x+b \sqrt {x}}}{3 b x}\right )}{5 b}-\frac {4 \sqrt {a x+b \sqrt {x}}}{5 b x^{3/2}}\right )}{7 b}-\frac {4 \sqrt {a x+b \sqrt {x}}}{7 b x^2}\right )}{9 b}-\frac {4 \sqrt {a x+b \sqrt {x}}}{9 b x^{5/2}}\right )}{11 b}-\frac {4 \sqrt {a x+b \sqrt {x}}}{11 b x^3}\right )}{13 b}-\frac {4 \sqrt {a x+b \sqrt {x}}}{13 b x^{7/2}}\right )}{b}+\frac {4}{b x^3 \sqrt {a x+b \sqrt {x}}}\) |
| \(\Big \downarrow \) 1920 |
| \(\displaystyle \frac {14 \left (-\frac {12 a \left (-\frac {10 a \left (-\frac {8 a \left (-\frac {6 a \left (-\frac {4 a \left (\frac {8 a \sqrt {a x+b \sqrt {x}}}{3 b^2 \sqrt {x}}-\frac {4 \sqrt {a x+b \sqrt {x}}}{3 b x}\right )}{5 b}-\frac {4 \sqrt {a x+b \sqrt {x}}}{5 b x^{3/2}}\right )}{7 b}-\frac {4 \sqrt {a x+b \sqrt {x}}}{7 b x^2}\right )}{9 b}-\frac {4 \sqrt {a x+b \sqrt {x}}}{9 b x^{5/2}}\right )}{11 b}-\frac {4 \sqrt {a x+b \sqrt {x}}}{11 b x^3}\right )}{13 b}-\frac {4 \sqrt {a x+b \sqrt {x}}}{13 b x^{7/2}}\right )}{b}+\frac {4}{b x^3 \sqrt {a x+b \sqrt {x}}}\) |
Input:
Int[1/(x^(7/2)*(b*Sqrt[x] + a*x)^(3/2)),x]
Output:
4/(b*x^3*Sqrt[b*Sqrt[x] + a*x]) + (14*((-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)))/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 = 176, normalized size of antiderivative = 0.86
| method | result | size |
| derivativedivides | \(-\frac {4}{13 b \,x^{3} \sqrt {b \sqrt {x}+x a}}-\frac {28 a \left (-\frac {2}{11 b \,x^{\frac {5}{2}} \sqrt {b \sqrt {x}+x a}}-\frac {12 a \left (-\frac {2}{9 b \,x^{2} \sqrt {b \sqrt {x}+x a}}-\frac {10 a \left (-\frac {2}{7 b \,x^{\frac {3}{2}} \sqrt {b \sqrt {x}+x a}}-\frac {8 a \left (-\frac {2}{5 b x \sqrt {b \sqrt {x}+x a}}-\frac {6 a \left (-\frac {2}{3 b \sqrt {x}\, \sqrt {b \sqrt {x}+x a}}+\frac {8 a \left (b +2 \sqrt {x}\, a \right )}{3 b^{3} \sqrt {b \sqrt {x}+x a}}\right )}{5 b}\right )}{7 b}\right )}{9 b}\right )}{11 b}\right )}{13 b}\) | \(176\) |
| default | \(\frac {2 \sqrt {b \sqrt {x}+x a}\, \left (2574 x^{\frac {17}{2}} \sqrt {b \sqrt {x}+x a}\, a^{\frac {19}{2}}+2574 x^{\frac {17}{2}} a^{\frac {19}{2}} \sqrt {\sqrt {x}\, \left (\sqrt {x}\, a +b \right )}-6006 x^{\frac {15}{2}} \left (b \sqrt {x}+x a \right )^{\frac {3}{2}} a^{\frac {17}{2}}+858 x^{\frac {15}{2}} a^{\frac {17}{2}} \left (\sqrt {x}\, \left (\sqrt {x}\, a +b \right )\right )^{\frac {3}{2}}+2574 x^{\frac {15}{2}} \sqrt {b \sqrt {x}+x a}\, a^{\frac {15}{2}} b^{2}+2574 x^{\frac {15}{2}} a^{\frac {15}{2}} \sqrt {\sqrt {x}\, \left (\sqrt {x}\, a +b \right )}\, b^{2}-2048 x^{\frac {13}{2}} \left (b \sqrt {x}+x a \right )^{\frac {3}{2}} a^{\frac {13}{2}} b^{2}-1287 x^{\frac {17}{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^{9} b +1287 x^{\frac {17}{2}} \ln \left (\frac {2 \sqrt {x}\, a +2 \sqrt {b \sqrt {x}+x a}\, \sqrt {a}+b}{2 \sqrt {a}}\right ) a^{9} b -256 x^{\frac {11}{2}} \left (b \sqrt {x}+x a \right )^{\frac {3}{2}} a^{\frac {9}{2}} b^{4}+5148 x^{8} \sqrt {b \sqrt {x}+x a}\, a^{\frac {17}{2}} b +5148 x^{8} a^{\frac {17}{2}} \sqrt {\sqrt {x}\, \left (\sqrt {x}\, a +b \right )}\, b -1287 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^{3}+1287 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^{3}-9244 x^{7} \left (b \sqrt {x}+x a \right )^{\frac {3}{2}} a^{\frac {15}{2}} b -112 x^{\frac {9}{2}} \left (b \sqrt {x}+x a \right )^{\frac {3}{2}} a^{\frac {5}{2}} b^{6}+512 x^{6} \left (b \sqrt {x}+x a \right )^{\frac {3}{2}} a^{\frac {11}{2}} b^{3}+160 x^{5} \left (b \sqrt {x}+x a \right )^{\frac {3}{2}} a^{\frac {7}{2}} b^{5}-66 x^{\frac {7}{2}} \left (b \sqrt {x}+x a \right )^{\frac {3}{2}} \sqrt {a}\, b^{8}-2574 x^{8} \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^{2}+2574 x^{8} \ln \left (\frac {2 \sqrt {x}\, a +2 \sqrt {b \sqrt {x}+x a}\, \sqrt {a}+b}{2 \sqrt {a}}\right ) a^{8} b^{2}+84 x^{4} \left (b \sqrt {x}+x a \right )^{\frac {3}{2}} a^{\frac {3}{2}} b^{7}\right )}{429 \sqrt {\sqrt {x}\, \left (\sqrt {x}\, a +b \right )}\, b^{9} x^{\frac {15}{2}} \left (\sqrt {x}\, a +b \right )^{2} \sqrt {a}}\) | \(636\) |
Input:
int(1/x^(7/2)/(b*x^(1/2)+x*a)^(3/2),x,method=_RETURNVERBOSE)
Output:
-4/13/b/x^3/(b*x^(1/2)+x*a)^(1/2)-28/13*a/b*(-2/11/b/x^(5/2)/(b*x^(1/2)+x* a)^(1/2)-12/11*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 = 0.14 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.60 \[ \int \frac {1}{x^{7/2} \left (b \sqrt {x}+a x\right )^{3/2}} \, dx=\frac {4 \, {\left (1024 \, a^{7} b x^{4} - 384 \, a^{5} b^{3} x^{3} - 136 \, a^{3} b^{5} x^{2} - 75 \, a b^{7} x - {\left (2048 \, a^{8} x^{4} - 1280 \, a^{6} b^{2} x^{3} - 208 \, a^{4} b^{4} x^{2} - 98 \, a^{2} b^{6} x - 33 \, b^{8}\right )} \sqrt {x}\right )} \sqrt {a x + b \sqrt {x}}}{429 \, {\left (a^{2} b^{8} x^{5} - b^{10} x^{4}\right )}} \] Input:
integrate(1/x^(7/2)/(b*x^(1/2)+a*x)^(3/2),x, algorithm="fricas")
Output:
4/429*(1024*a^7*b*x^4 - 384*a^5*b^3*x^3 - 136*a^3*b^5*x^2 - 75*a*b^7*x - ( 2048*a^8*x^4 - 1280*a^6*b^2*x^3 - 208*a^4*b^4*x^2 - 98*a^2*b^6*x - 33*b^8) *sqrt(x))*sqrt(a*x + b*sqrt(x))/(a^2*b^8*x^5 - b^10*x^4)
\[ \int \frac {1}{x^{7/2} \left (b \sqrt {x}+a x\right )^{3/2}} \, dx=\int \frac {1}{x^{\frac {7}{2}} \left (a x + b \sqrt {x}\right )^{\frac {3}{2}}}\, dx \] Input:
integrate(1/x**(7/2)/(b*x**(1/2)+a*x)**(3/2),x)
Output:
Integral(1/(x**(7/2)*(a*x + b*sqrt(x))**(3/2)), x)
\[ \int \frac {1}{x^{7/2} \left (b \sqrt {x}+a x\right )^{3/2}} \, dx=\int { \frac {1}{{\left (a x + b \sqrt {x}\right )}^{\frac {3}{2}} x^{\frac {7}{2}}} \,d x } \] Input:
integrate(1/x^(7/2)/(b*x^(1/2)+a*x)^(3/2),x, algorithm="maxima")
Output:
integrate(1/((a*x + b*sqrt(x))^(3/2)*x^(7/2)), x)
\[ \int \frac {1}{x^{7/2} \left (b \sqrt {x}+a x\right )^{3/2}} \, dx=\int { \frac {1}{{\left (a x + b \sqrt {x}\right )}^{\frac {3}{2}} x^{\frac {7}{2}}} \,d x } \] Input:
integrate(1/x^(7/2)/(b*x^(1/2)+a*x)^(3/2),x, algorithm="giac")
Output:
integrate(1/((a*x + b*sqrt(x))^(3/2)*x^(7/2)), x)
Timed out. \[ \int \frac {1}{x^{7/2} \left (b \sqrt {x}+a x\right )^{3/2}} \, dx=\int \frac {1}{x^{7/2}\,{\left (a\,x+b\,\sqrt {x}\right )}^{3/2}} \,d x \] Input:
int(1/(x^(7/2)*(a*x + b*x^(1/2))^(3/2)),x)
Output:
int(1/(x^(7/2)*(a*x + b*x^(1/2))^(3/2)), x)
Time = 0.37 (sec) , antiderivative size = 170, normalized size of antiderivative = 0.83 \[ \int \frac {1}{x^{7/2} \left (b \sqrt {x}+a x\right )^{3/2}} \, dx=\frac {-\frac {8192 \sqrt {x}\, \sqrt {\sqrt {x}\, a +b}\, a^{7} x^{3}}{429}+\frac {1024 \sqrt {x}\, \sqrt {\sqrt {x}\, a +b}\, a^{5} b^{2} x^{2}}{429}+\frac {320 \sqrt {x}\, \sqrt {\sqrt {x}\, a +b}\, a^{3} b^{4} x}{429}+\frac {56 \sqrt {x}\, \sqrt {\sqrt {x}\, a +b}\, a \,b^{6}}{143}-\frac {4096 \sqrt {\sqrt {x}\, a +b}\, a^{6} b \,x^{3}}{429}-\frac {512 \sqrt {\sqrt {x}\, a +b}\, a^{4} b^{3} x^{2}}{429}-\frac {224 \sqrt {\sqrt {x}\, a +b}\, a^{2} b^{5} x}{429}-\frac {4 \sqrt {\sqrt {x}\, a +b}\, b^{7}}{13}+\frac {8192 x^{\frac {15}{4}} \sqrt {a}\, a^{7}}{429}+\frac {8192 x^{\frac {13}{4}} \sqrt {a}\, a^{6} b}{429}}{x^{\frac {13}{4}} b^{8} \left (\sqrt {x}\, a +b \right )} \] Input:
int(1/x^(7/2)/(b*x^(1/2)+a*x)^(3/2),x)
Output:
(4*( - 2048*sqrt(x)*sqrt(sqrt(x)*a + b)*a**7*x**3 + 256*sqrt(x)*sqrt(sqrt( x)*a + b)*a**5*b**2*x**2 + 80*sqrt(x)*sqrt(sqrt(x)*a + b)*a**3*b**4*x + 42 *sqrt(x)*sqrt(sqrt(x)*a + b)*a*b**6 - 1024*sqrt(sqrt(x)*a + b)*a**6*b*x**3 - 128*sqrt(sqrt(x)*a + b)*a**4*b**3*x**2 - 56*sqrt(sqrt(x)*a + b)*a**2*b* *5*x - 33*sqrt(sqrt(x)*a + b)*b**7 + 2048*x**(3/4)*sqrt(a)*a**7*x**3 + 204 8*x**(1/4)*sqrt(a)*a**6*b*x**3))/(429*x**(1/4)*b**8*x**3*(sqrt(x)*a + b))