Integrand size = 15, antiderivative size = 96 \[ \int \frac {x^2}{\sqrt {a+\frac {b}{x}}} \, dx=\frac {5 b^2 \sqrt {a+\frac {b}{x}} x}{8 a^3}-\frac {5 b \sqrt {a+\frac {b}{x}} x^2}{12 a^2}+\frac {\sqrt {a+\frac {b}{x}} x^3}{3 a}-\frac {5 b^3 \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{8 a^{7/2}} \] Output:
5/8*b^2*(a+b/x)^(1/2)*x/a^3-5/12*b*(a+b/x)^(1/2)*x^2/a^2+1/3*(a+b/x)^(1/2) *x^3/a-5/8*b^3*arctanh((a+b/x)^(1/2)/a^(1/2))/a^(7/2)
Time = 0.09 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.72 \[ \int \frac {x^2}{\sqrt {a+\frac {b}{x}}} \, dx=\frac {\sqrt {a+\frac {b}{x}} x \left (15 b^2-10 a b x+8 a^2 x^2\right )}{24 a^3}-\frac {5 b^3 \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{8 a^{7/2}} \] Input:
Integrate[x^2/Sqrt[a + b/x],x]
Output:
(Sqrt[a + b/x]*x*(15*b^2 - 10*a*b*x + 8*a^2*x^2))/(24*a^3) - (5*b^3*ArcTan h[Sqrt[a + b/x]/Sqrt[a]])/(8*a^(7/2))
Time = 0.32 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.07, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {798, 52, 52, 52, 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.
\(\displaystyle \int \frac {x^2}{\sqrt {a+\frac {b}{x}}} \, dx\) |
\(\Big \downarrow \) 798 |
\(\displaystyle -\int \frac {x^4}{\sqrt {a+\frac {b}{x}}}d\frac {1}{x}\) |
\(\Big \downarrow \) 52 |
\(\displaystyle \frac {5 b \int \frac {x^3}{\sqrt {a+\frac {b}{x}}}d\frac {1}{x}}{6 a}+\frac {x^3 \sqrt {a+\frac {b}{x}}}{3 a}\) |
\(\Big \downarrow \) 52 |
\(\displaystyle \frac {5 b \left (-\frac {3 b \int \frac {x^2}{\sqrt {a+\frac {b}{x}}}d\frac {1}{x}}{4 a}-\frac {x^2 \sqrt {a+\frac {b}{x}}}{2 a}\right )}{6 a}+\frac {x^3 \sqrt {a+\frac {b}{x}}}{3 a}\) |
\(\Big \downarrow \) 52 |
\(\displaystyle \frac {5 b \left (-\frac {3 b \left (-\frac {b \int \frac {x}{\sqrt {a+\frac {b}{x}}}d\frac {1}{x}}{2 a}-\frac {x \sqrt {a+\frac {b}{x}}}{a}\right )}{4 a}-\frac {x^2 \sqrt {a+\frac {b}{x}}}{2 a}\right )}{6 a}+\frac {x^3 \sqrt {a+\frac {b}{x}}}{3 a}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {5 b \left (-\frac {3 b \left (-\frac {\int \frac {1}{\frac {1}{b x^2}-\frac {a}{b}}d\sqrt {a+\frac {b}{x}}}{a}-\frac {x \sqrt {a+\frac {b}{x}}}{a}\right )}{4 a}-\frac {x^2 \sqrt {a+\frac {b}{x}}}{2 a}\right )}{6 a}+\frac {x^3 \sqrt {a+\frac {b}{x}}}{3 a}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {5 b \left (-\frac {3 b \left (\frac {b \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{a^{3/2}}-\frac {x \sqrt {a+\frac {b}{x}}}{a}\right )}{4 a}-\frac {x^2 \sqrt {a+\frac {b}{x}}}{2 a}\right )}{6 a}+\frac {x^3 \sqrt {a+\frac {b}{x}}}{3 a}\) |
Input:
Int[x^2/Sqrt[a + b/x],x]
Output:
(Sqrt[a + b/x]*x^3)/(3*a) + (5*b*(-1/2*(Sqrt[a + b/x]*x^2)/a - (3*b*(-((Sq rt[a + b/x]*x)/a) + (b*ArcTanh[Sqrt[a + b/x]/Sqrt[a]])/a^(3/2)))/(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] :> 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 = 0.25 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.01
method | result | size |
risch | \(\frac {\left (8 a^{2} x^{2}-10 a b x +15 b^{2}\right ) \left (a x +b \right )}{24 a^{3} \sqrt {\frac {a x +b}{x}}}-\frac {5 b^{3} \ln \left (\frac {\frac {b}{2}+a x}{\sqrt {a}}+\sqrt {a \,x^{2}+b x}\right ) \sqrt {x \left (a x +b \right )}}{16 a^{\frac {7}{2}} x \sqrt {\frac {a x +b}{x}}}\) | \(97\) |
default | \(\frac {\sqrt {\frac {a x +b}{x}}\, x \left (16 \left (a \,x^{2}+b x \right )^{\frac {3}{2}} a^{\frac {5}{2}}-36 \sqrt {a \,x^{2}+b x}\, a^{\frac {5}{2}} b x +48 \sqrt {x \left (a x +b \right )}\, a^{\frac {3}{2}} b^{2}-24 \ln \left (\frac {2 \sqrt {x \left (a x +b \right )}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a \,b^{3}-18 \sqrt {a \,x^{2}+b x}\, a^{\frac {3}{2}} b^{2}+9 \ln \left (\frac {2 \sqrt {a \,x^{2}+b x}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a \,b^{3}\right )}{48 a^{\frac {9}{2}} \sqrt {x \left (a x +b \right )}}\) | \(164\) |
Input:
int(x^2/(a+b/x)^(1/2),x,method=_RETURNVERBOSE)
Output:
1/24*(8*a^2*x^2-10*a*b*x+15*b^2)*(a*x+b)/a^3/((a*x+b)/x)^(1/2)-5/16*b^3/a^ (7/2)*ln((1/2*b+a*x)/a^(1/2)+(a*x^2+b*x)^(1/2))/x/((a*x+b)/x)^(1/2)*(x*(a* x+b))^(1/2)
Time = 0.10 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.62 \[ \int \frac {x^2}{\sqrt {a+\frac {b}{x}}} \, dx=\left [\frac {15 \, \sqrt {a} b^{3} \log \left (2 \, a x - 2 \, \sqrt {a} x \sqrt {\frac {a x + b}{x}} + b\right ) + 2 \, {\left (8 \, a^{3} x^{3} - 10 \, a^{2} b x^{2} + 15 \, a b^{2} x\right )} \sqrt {\frac {a x + b}{x}}}{48 \, a^{4}}, \frac {15 \, \sqrt {-a} b^{3} \arctan \left (\frac {\sqrt {-a} x \sqrt {\frac {a x + b}{x}}}{a x + b}\right ) + {\left (8 \, a^{3} x^{3} - 10 \, a^{2} b x^{2} + 15 \, a b^{2} x\right )} \sqrt {\frac {a x + b}{x}}}{24 \, a^{4}}\right ] \] Input:
integrate(x^2/(a+b/x)^(1/2),x, algorithm="fricas")
Output:
[1/48*(15*sqrt(a)*b^3*log(2*a*x - 2*sqrt(a)*x*sqrt((a*x + b)/x) + b) + 2*( 8*a^3*x^3 - 10*a^2*b*x^2 + 15*a*b^2*x)*sqrt((a*x + b)/x))/a^4, 1/24*(15*sq rt(-a)*b^3*arctan(sqrt(-a)*x*sqrt((a*x + b)/x)/(a*x + b)) + (8*a^3*x^3 - 1 0*a^2*b*x^2 + 15*a*b^2*x)*sqrt((a*x + b)/x))/a^4]
Time = 9.26 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.33 \[ \int \frac {x^2}{\sqrt {a+\frac {b}{x}}} \, dx=\frac {x^{\frac {7}{2}}}{3 \sqrt {b} \sqrt {\frac {a x}{b} + 1}} - \frac {\sqrt {b} x^{\frac {5}{2}}}{12 a \sqrt {\frac {a x}{b} + 1}} + \frac {5 b^{\frac {3}{2}} x^{\frac {3}{2}}}{24 a^{2} \sqrt {\frac {a x}{b} + 1}} + \frac {5 b^{\frac {5}{2}} \sqrt {x}}{8 a^{3} \sqrt {\frac {a x}{b} + 1}} - \frac {5 b^{3} \operatorname {asinh}{\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}} \right )}}{8 a^{\frac {7}{2}}} \] Input:
integrate(x**2/(a+b/x)**(1/2),x)
Output:
x**(7/2)/(3*sqrt(b)*sqrt(a*x/b + 1)) - sqrt(b)*x**(5/2)/(12*a*sqrt(a*x/b + 1)) + 5*b**(3/2)*x**(3/2)/(24*a**2*sqrt(a*x/b + 1)) + 5*b**(5/2)*sqrt(x)/ (8*a**3*sqrt(a*x/b + 1)) - 5*b**3*asinh(sqrt(a)*sqrt(x)/sqrt(b))/(8*a**(7/ 2))
Time = 0.11 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.43 \[ \int \frac {x^2}{\sqrt {a+\frac {b}{x}}} \, dx=\frac {5 \, b^{3} \log \left (\frac {\sqrt {a + \frac {b}{x}} - \sqrt {a}}{\sqrt {a + \frac {b}{x}} + \sqrt {a}}\right )}{16 \, a^{\frac {7}{2}}} + \frac {15 \, {\left (a + \frac {b}{x}\right )}^{\frac {5}{2}} b^{3} - 40 \, {\left (a + \frac {b}{x}\right )}^{\frac {3}{2}} a b^{3} + 33 \, \sqrt {a + \frac {b}{x}} a^{2} b^{3}}{24 \, {\left ({\left (a + \frac {b}{x}\right )}^{3} a^{3} - 3 \, {\left (a + \frac {b}{x}\right )}^{2} a^{4} + 3 \, {\left (a + \frac {b}{x}\right )} a^{5} - a^{6}\right )}} \] Input:
integrate(x^2/(a+b/x)^(1/2),x, algorithm="maxima")
Output:
5/16*b^3*log((sqrt(a + b/x) - sqrt(a))/(sqrt(a + b/x) + sqrt(a)))/a^(7/2) + 1/24*(15*(a + b/x)^(5/2)*b^3 - 40*(a + b/x)^(3/2)*a*b^3 + 33*sqrt(a + b/ x)*a^2*b^3)/((a + b/x)^3*a^3 - 3*(a + b/x)^2*a^4 + 3*(a + b/x)*a^5 - a^6)
Time = 0.14 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.08 \[ \int \frac {x^2}{\sqrt {a+\frac {b}{x}}} \, dx=\frac {1}{24} \, \sqrt {a x^{2} + b x} {\left (2 \, x {\left (\frac {4 \, x}{a \mathrm {sgn}\left (x\right )} - \frac {5 \, b}{a^{2} \mathrm {sgn}\left (x\right )}\right )} + \frac {15 \, b^{2}}{a^{3} \mathrm {sgn}\left (x\right )}\right )} - \frac {5 \, b^{3} \log \left ({\left | b \right |}\right ) \mathrm {sgn}\left (x\right )}{16 \, a^{\frac {7}{2}}} + \frac {5 \, b^{3} \log \left ({\left | 2 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )} \sqrt {a} + b \right |}\right )}{16 \, a^{\frac {7}{2}} \mathrm {sgn}\left (x\right )} \] Input:
integrate(x^2/(a+b/x)^(1/2),x, algorithm="giac")
Output:
1/24*sqrt(a*x^2 + b*x)*(2*x*(4*x/(a*sgn(x)) - 5*b/(a^2*sgn(x))) + 15*b^2/( a^3*sgn(x))) - 5/16*b^3*log(abs(b))*sgn(x)/a^(7/2) + 5/16*b^3*log(abs(2*(s qrt(a)*x - sqrt(a*x^2 + b*x))*sqrt(a) + b))/(a^(7/2)*sgn(x))
Time = 0.63 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.80 \[ \int \frac {x^2}{\sqrt {a+\frac {b}{x}}} \, dx=\frac {11\,x^3\,\sqrt {a+\frac {b}{x}}}{8\,a}-\frac {5\,x^3\,{\left (a+\frac {b}{x}\right )}^{3/2}}{3\,a^2}+\frac {5\,x^3\,{\left (a+\frac {b}{x}\right )}^{5/2}}{8\,a^3}+\frac {b^3\,\mathrm {atan}\left (\frac {\sqrt {a+\frac {b}{x}}\,1{}\mathrm {i}}{\sqrt {a}}\right )\,5{}\mathrm {i}}{8\,a^{7/2}} \] Input:
int(x^2/(a + b/x)^(1/2),x)
Output:
(b^3*atan(((a + b/x)^(1/2)*1i)/a^(1/2))*5i)/(8*a^(7/2)) + (11*x^3*(a + b/x )^(1/2))/(8*a) - (5*x^3*(a + b/x)^(3/2))/(3*a^2) + (5*x^3*(a + b/x)^(5/2)) /(8*a^3)
Time = 0.23 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.79 \[ \int \frac {x^2}{\sqrt {a+\frac {b}{x}}} \, dx=\frac {8 \sqrt {x}\, \sqrt {a x +b}\, a^{3} x^{2}-10 \sqrt {x}\, \sqrt {a x +b}\, a^{2} b x +15 \sqrt {x}\, \sqrt {a x +b}\, a \,b^{2}-15 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {a x +b}+\sqrt {x}\, \sqrt {a}}{\sqrt {b}}\right ) b^{3}}{24 a^{4}} \] Input:
int(x^2/(a+b/x)^(1/2),x)
Output:
(8*sqrt(x)*sqrt(a*x + b)*a**3*x**2 - 10*sqrt(x)*sqrt(a*x + b)*a**2*b*x + 1 5*sqrt(x)*sqrt(a*x + b)*a*b**2 - 15*sqrt(a)*log((sqrt(a*x + b) + sqrt(x)*s qrt(a))/sqrt(b))*b**3)/(24*a**4)