Integrand size = 15, antiderivative size = 117 \[ \int \frac {x^2}{\left (a+\frac {b}{x}\right )^{3/2}} \, dx=\frac {35 b^3}{8 a^4 \sqrt {a+\frac {b}{x}}}+\frac {35 b^2 x}{24 a^3 \sqrt {a+\frac {b}{x}}}-\frac {7 b x^2}{12 a^2 \sqrt {a+\frac {b}{x}}}+\frac {x^3}{3 a \sqrt {a+\frac {b}{x}}}-\frac {35 b^3 \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{8 a^{9/2}} \] Output:
35/8*b^3/a^4/(a+b/x)^(1/2)+35/24*b^2*x/a^3/(a+b/x)^(1/2)-7/12*b*x^2/a^2/(a +b/x)^(1/2)+1/3*x^3/a/(a+b/x)^(1/2)-35/8*b^3*arctanh((a+b/x)^(1/2)/a^(1/2) )/a^(9/2)
Time = 0.14 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.74 \[ \int \frac {x^2}{\left (a+\frac {b}{x}\right )^{3/2}} \, dx=\frac {\sqrt {a+\frac {b}{x}} x \left (105 b^3+35 a b^2 x-14 a^2 b x^2+8 a^3 x^3\right )}{24 a^4 (b+a x)}-\frac {35 b^3 \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{8 a^{9/2}} \] Input:
Integrate[x^2/(a + b/x)^(3/2),x]
Output:
(Sqrt[a + b/x]*x*(105*b^3 + 35*a*b^2*x - 14*a^2*b*x^2 + 8*a^3*x^3))/(24*a^ 4*(b + a*x)) - (35*b^3*ArcTanh[Sqrt[a + b/x]/Sqrt[a]])/(8*a^(9/2))
Time = 0.36 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.09, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.467, Rules used = {798, 52, 52, 52, 61, 73, 221}
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.
| \(\displaystyle \int \frac {x^2}{\left (a+\frac {b}{x}\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 798 |
| \(\displaystyle -\int \frac {x^4}{\left (a+\frac {b}{x}\right )^{3/2}}d\frac {1}{x}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {7 b \int \frac {x^3}{\left (a+\frac {b}{x}\right )^{3/2}}d\frac {1}{x}}{6 a}+\frac {x^3}{3 a \sqrt {a+\frac {b}{x}}}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {7 b \left (-\frac {5 b \int \frac {x^2}{\left (a+\frac {b}{x}\right )^{3/2}}d\frac {1}{x}}{4 a}-\frac {x^2}{2 a \sqrt {a+\frac {b}{x}}}\right )}{6 a}+\frac {x^3}{3 a \sqrt {a+\frac {b}{x}}}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {7 b \left (-\frac {5 b \left (-\frac {3 b \int \frac {x}{\left (a+\frac {b}{x}\right )^{3/2}}d\frac {1}{x}}{2 a}-\frac {x}{a \sqrt {a+\frac {b}{x}}}\right )}{4 a}-\frac {x^2}{2 a \sqrt {a+\frac {b}{x}}}\right )}{6 a}+\frac {x^3}{3 a \sqrt {a+\frac {b}{x}}}\) |
| \(\Big \downarrow \) 61 |
| \(\displaystyle \frac {7 b \left (-\frac {5 b \left (-\frac {3 b \left (\frac {\int \frac {x}{\sqrt {a+\frac {b}{x}}}d\frac {1}{x}}{a}+\frac {2}{a \sqrt {a+\frac {b}{x}}}\right )}{2 a}-\frac {x}{a \sqrt {a+\frac {b}{x}}}\right )}{4 a}-\frac {x^2}{2 a \sqrt {a+\frac {b}{x}}}\right )}{6 a}+\frac {x^3}{3 a \sqrt {a+\frac {b}{x}}}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {7 b \left (-\frac {5 b \left (-\frac {3 b \left (\frac {2 \int \frac {1}{\frac {1}{b x^2}-\frac {a}{b}}d\sqrt {a+\frac {b}{x}}}{a b}+\frac {2}{a \sqrt {a+\frac {b}{x}}}\right )}{2 a}-\frac {x}{a \sqrt {a+\frac {b}{x}}}\right )}{4 a}-\frac {x^2}{2 a \sqrt {a+\frac {b}{x}}}\right )}{6 a}+\frac {x^3}{3 a \sqrt {a+\frac {b}{x}}}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {7 b \left (-\frac {5 b \left (-\frac {3 b \left (\frac {2}{a \sqrt {a+\frac {b}{x}}}-\frac {2 \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{a^{3/2}}\right )}{2 a}-\frac {x}{a \sqrt {a+\frac {b}{x}}}\right )}{4 a}-\frac {x^2}{2 a \sqrt {a+\frac {b}{x}}}\right )}{6 a}+\frac {x^3}{3 a \sqrt {a+\frac {b}{x}}}\) |
Input:
Int[x^2/(a + b/x)^(3/2),x]
Output:
x^3/(3*a*Sqrt[a + b/x]) + (7*b*(-1/2*x^2/(a*Sqrt[a + b/x]) - (5*b*(-(x/(a* Sqrt[a + b/x])) - (3*b*(2/(a*Sqrt[a + b/x]) - (2*ArcTanh[Sqrt[a + b/x]/Sqr t[a]])/a^(3/2)))/(2*a)))/(4*a)))/(6*a)
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( m + n + 2)/((b*c - a*d)*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && ILtQ[m, -1] && FractionQ[n] && LtQ[n, 0]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( m + n + 2)/((b*c - a*d)*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && LtQ[m, -1] && !(LtQ[n, -1] && (EqQ[a, 0 ] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d , m, n, x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[1/n Subst [Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]
Time = 0.29 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.20
| method | result | size |
| risch | \(\frac {\left (8 a^{2} x^{2}-22 a b x +57 b^{2}\right ) \left (a x +b \right )}{24 a^{4} \sqrt {\frac {a x +b}{x}}}+\frac {\left (-\frac {35 b^{3} \ln \left (\frac {\frac {b}{2}+a x}{\sqrt {a}}+\sqrt {a \,x^{2}+b x}\right )}{16 a^{\frac {9}{2}}}+\frac {2 b^{3} \sqrt {a \left (x +\frac {b}{a}\right )^{2}-b \left (x +\frac {b}{a}\right )}}{a^{5} \left (x +\frac {b}{a}\right )}\right ) \sqrt {x \left (a x +b \right )}}{x \sqrt {\frac {a x +b}{x}}}\) | \(140\) |
| default | \(\frac {\sqrt {\frac {a x +b}{x}}\, x \left (16 \left (a \,x^{2}+b x \right )^{\frac {3}{2}} a^{\frac {9}{2}} x^{2}-60 \sqrt {a \,x^{2}+b x}\, a^{\frac {9}{2}} b \,x^{3}+32 \left (a \,x^{2}+b x \right )^{\frac {3}{2}} a^{\frac {7}{2}} b x -150 \sqrt {a \,x^{2}+b x}\, a^{\frac {7}{2}} b^{2} x^{2}+240 a^{\frac {7}{2}} \sqrt {x \left (a x +b \right )}\, b^{2} x^{2}-120 a^{3} \ln \left (\frac {2 \sqrt {x \left (a x +b \right )}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) b^{3} x^{2}+16 \left (a \,x^{2}+b x \right )^{\frac {3}{2}} a^{\frac {5}{2}} b^{2}-120 \sqrt {a \,x^{2}+b x}\, a^{\frac {5}{2}} b^{3} x -96 a^{\frac {5}{2}} \left (x \left (a x +b \right )\right )^{\frac {3}{2}} b^{2}+480 a^{\frac {5}{2}} \sqrt {x \left (a x +b \right )}\, b^{3} x -240 a^{2} \ln \left (\frac {2 \sqrt {x \left (a x +b \right )}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) b^{4} x +15 \ln \left (\frac {2 \sqrt {a \,x^{2}+b x}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a^{3} b^{3} x^{2}-30 \sqrt {a \,x^{2}+b x}\, a^{\frac {3}{2}} b^{4}+240 a^{\frac {3}{2}} \sqrt {x \left (a x +b \right )}\, b^{4}-120 a \ln \left (\frac {2 \sqrt {x \left (a x +b \right )}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) b^{5}+30 \ln \left (\frac {2 \sqrt {a \,x^{2}+b x}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a^{2} b^{4} x +15 \ln \left (\frac {2 \sqrt {a \,x^{2}+b x}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a \,b^{5}\right )}{48 a^{\frac {11}{2}} \sqrt {x \left (a x +b \right )}\, \left (a x +b \right )^{2}}\) | \(458\) |
Input:
int(x^2/(a+b/x)^(3/2),x,method=_RETURNVERBOSE)
Output:
1/24*(8*a^2*x^2-22*a*b*x+57*b^2)*(a*x+b)/a^4/((a*x+b)/x)^(1/2)+(-35/16*b^3 /a^(9/2)*ln((1/2*b+a*x)/a^(1/2)+(a*x^2+b*x)^(1/2))+2*b^3/a^5/(x+b/a)*(a*(x +b/a)^2-b*(x+b/a))^(1/2))/x/((a*x+b)/x)^(1/2)*(x*(a*x+b))^(1/2)
Time = 0.08 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.81 \[ \int \frac {x^2}{\left (a+\frac {b}{x}\right )^{3/2}} \, dx=\left [\frac {105 \, {\left (a b^{3} x + b^{4}\right )} \sqrt {a} \log \left (2 \, a x - 2 \, \sqrt {a} x \sqrt {\frac {a x + b}{x}} + b\right ) + 2 \, {\left (8 \, a^{4} x^{4} - 14 \, a^{3} b x^{3} + 35 \, a^{2} b^{2} x^{2} + 105 \, a b^{3} x\right )} \sqrt {\frac {a x + b}{x}}}{48 \, {\left (a^{6} x + a^{5} b\right )}}, \frac {105 \, {\left (a b^{3} x + b^{4}\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {-a} x \sqrt {\frac {a x + b}{x}}}{a x + b}\right ) + {\left (8 \, a^{4} x^{4} - 14 \, a^{3} b x^{3} + 35 \, a^{2} b^{2} x^{2} + 105 \, a b^{3} x\right )} \sqrt {\frac {a x + b}{x}}}{24 \, {\left (a^{6} x + a^{5} b\right )}}\right ] \] Input:
integrate(x^2/(a+b/x)^(3/2),x, algorithm="fricas")
Output:
[1/48*(105*(a*b^3*x + b^4)*sqrt(a)*log(2*a*x - 2*sqrt(a)*x*sqrt((a*x + b)/ x) + b) + 2*(8*a^4*x^4 - 14*a^3*b*x^3 + 35*a^2*b^2*x^2 + 105*a*b^3*x)*sqrt ((a*x + b)/x))/(a^6*x + a^5*b), 1/24*(105*(a*b^3*x + b^4)*sqrt(-a)*arctan( sqrt(-a)*x*sqrt((a*x + b)/x)/(a*x + b)) + (8*a^4*x^4 - 14*a^3*b*x^3 + 35*a ^2*b^2*x^2 + 105*a*b^3*x)*sqrt((a*x + b)/x))/(a^6*x + a^5*b)]
Time = 16.51 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.14 \[ \int \frac {x^2}{\left (a+\frac {b}{x}\right )^{3/2}} \, dx=\frac {x^{\frac {7}{2}}}{3 a \sqrt {b} \sqrt {\frac {a x}{b} + 1}} - \frac {7 \sqrt {b} x^{\frac {5}{2}}}{12 a^{2} \sqrt {\frac {a x}{b} + 1}} + \frac {35 b^{\frac {3}{2}} x^{\frac {3}{2}}}{24 a^{3} \sqrt {\frac {a x}{b} + 1}} + \frac {35 b^{\frac {5}{2}} \sqrt {x}}{8 a^{4} \sqrt {\frac {a x}{b} + 1}} - \frac {35 b^{3} \operatorname {asinh}{\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}} \right )}}{8 a^{\frac {9}{2}}} \] Input:
integrate(x**2/(a+b/x)**(3/2),x)
Output:
x**(7/2)/(3*a*sqrt(b)*sqrt(a*x/b + 1)) - 7*sqrt(b)*x**(5/2)/(12*a**2*sqrt( a*x/b + 1)) + 35*b**(3/2)*x**(3/2)/(24*a**3*sqrt(a*x/b + 1)) + 35*b**(5/2) *sqrt(x)/(8*a**4*sqrt(a*x/b + 1)) - 35*b**3*asinh(sqrt(a)*sqrt(x)/sqrt(b)) /(8*a**(9/2))
Time = 0.11 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.32 \[ \int \frac {x^2}{\left (a+\frac {b}{x}\right )^{3/2}} \, dx=\frac {105 \, {\left (a + \frac {b}{x}\right )}^{3} b^{3} - 280 \, {\left (a + \frac {b}{x}\right )}^{2} a b^{3} + 231 \, {\left (a + \frac {b}{x}\right )} a^{2} b^{3} - 48 \, a^{3} b^{3}}{24 \, {\left ({\left (a + \frac {b}{x}\right )}^{\frac {7}{2}} a^{4} - 3 \, {\left (a + \frac {b}{x}\right )}^{\frac {5}{2}} a^{5} + 3 \, {\left (a + \frac {b}{x}\right )}^{\frac {3}{2}} a^{6} - \sqrt {a + \frac {b}{x}} a^{7}\right )}} + \frac {35 \, b^{3} \log \left (\frac {\sqrt {a + \frac {b}{x}} - \sqrt {a}}{\sqrt {a + \frac {b}{x}} + \sqrt {a}}\right )}{16 \, a^{\frac {9}{2}}} \] Input:
integrate(x^2/(a+b/x)^(3/2),x, algorithm="maxima")
Output:
1/24*(105*(a + b/x)^3*b^3 - 280*(a + b/x)^2*a*b^3 + 231*(a + b/x)*a^2*b^3 - 48*a^3*b^3)/((a + b/x)^(7/2)*a^4 - 3*(a + b/x)^(5/2)*a^5 + 3*(a + b/x)^( 3/2)*a^6 - sqrt(a + b/x)*a^7) + 35/16*b^3*log((sqrt(a + b/x) - sqrt(a))/(s qrt(a + b/x) + sqrt(a)))/a^(9/2)
Time = 0.15 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.29 \[ \int \frac {x^2}{\left (a+\frac {b}{x}\right )^{3/2}} \, dx=\frac {1}{24} \, \sqrt {a x^{2} + b x} {\left (2 \, x {\left (\frac {4 \, x}{a^{2} \mathrm {sgn}\left (x\right )} - \frac {11 \, b}{a^{3} \mathrm {sgn}\left (x\right )}\right )} + \frac {57 \, b^{2}}{a^{4} \mathrm {sgn}\left (x\right )}\right )} + \frac {35 \, b^{3} \log \left ({\left | 2 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )} \sqrt {a} + b \right |}\right )}{16 \, a^{\frac {9}{2}} \mathrm {sgn}\left (x\right )} + \frac {2 \, b^{4}}{{\left ({\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )} \sqrt {a} + b\right )} a^{\frac {9}{2}} \mathrm {sgn}\left (x\right )} - \frac {{\left (35 \, b^{3} \log \left ({\left | b \right |}\right ) + 32 \, b^{3}\right )} \mathrm {sgn}\left (x\right )}{16 \, a^{\frac {9}{2}}} \] Input:
integrate(x^2/(a+b/x)^(3/2),x, algorithm="giac")
Output:
1/24*sqrt(a*x^2 + b*x)*(2*x*(4*x/(a^2*sgn(x)) - 11*b/(a^3*sgn(x))) + 57*b^ 2/(a^4*sgn(x))) + 35/16*b^3*log(abs(2*(sqrt(a)*x - sqrt(a*x^2 + b*x))*sqrt (a) + b))/(a^(9/2)*sgn(x)) + 2*b^4/(((sqrt(a)*x - sqrt(a*x^2 + b*x))*sqrt( a) + b)*a^(9/2)*sgn(x)) - 1/16*(35*b^3*log(abs(b)) + 32*b^3)*sgn(x)/a^(9/2 )
Time = 0.82 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.79 \[ \int \frac {x^2}{\left (a+\frac {b}{x}\right )^{3/2}} \, dx=\frac {35\,b^3}{8\,a^4\,\sqrt {a+\frac {b}{x}}}-\frac {35\,b^3\,\mathrm {atanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{8\,a^{9/2}}+\frac {x^3}{3\,a\,\sqrt {a+\frac {b}{x}}}-\frac {7\,b\,x^2}{12\,a^2\,\sqrt {a+\frac {b}{x}}}+\frac {35\,b^2\,x}{24\,a^3\,\sqrt {a+\frac {b}{x}}} \] Input:
int(x^2/(a + b/x)^(3/2),x)
Output:
(35*b^3)/(8*a^4*(a + b/x)^(1/2)) - (35*b^3*atanh((a + b/x)^(1/2)/a^(1/2))) /(8*a^(9/2)) + x^3/(3*a*(a + b/x)^(1/2)) - (7*b*x^2)/(12*a^2*(a + b/x)^(1/ 2)) + (35*b^2*x)/(24*a^3*(a + b/x)^(1/2))
Time = 0.26 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.84 \[ \int \frac {x^2}{\left (a+\frac {b}{x}\right )^{3/2}} \, dx=\frac {-840 \sqrt {a}\, \sqrt {a x +b}\, \mathrm {log}\left (\frac {\sqrt {a x +b}+\sqrt {x}\, \sqrt {a}}{\sqrt {b}}\right ) b^{3}+525 \sqrt {a}\, \sqrt {a x +b}\, b^{3}+64 \sqrt {x}\, a^{4} x^{3}-112 \sqrt {x}\, a^{3} b \,x^{2}+280 \sqrt {x}\, a^{2} b^{2} x +840 \sqrt {x}\, a \,b^{3}}{192 \sqrt {a x +b}\, a^{5}} \] Input:
int(x^2/(a+b/x)^(3/2),x)
Output:
( - 840*sqrt(a)*sqrt(a*x + b)*log((sqrt(a*x + b) + sqrt(x)*sqrt(a))/sqrt(b ))*b**3 + 525*sqrt(a)*sqrt(a*x + b)*b**3 + 64*sqrt(x)*a**4*x**3 - 112*sqrt (x)*a**3*b*x**2 + 280*sqrt(x)*a**2*b**2*x + 840*sqrt(x)*a*b**3)/(192*sqrt( a*x + b)*a**5)