Integrand size = 17, antiderivative size = 99 \[ \int \left (a+\frac {b}{x}\right )^{5/2} x^{3/2} \, dx=\frac {46}{15} b^2 \sqrt {a+\frac {b}{x}} \sqrt {x}+\frac {22}{15} a b \sqrt {a+\frac {b}{x}} x^{3/2}+\frac {2}{5} a^2 \sqrt {a+\frac {b}{x}} x^{5/2}-2 b^{5/2} \text {arctanh}\left (\frac {\sqrt {b}}{\sqrt {a+\frac {b}{x}} \sqrt {x}}\right ) \] Output:
46/15*b^2*(a+b/x)^(1/2)*x^(1/2)+22/15*a*b*(a+b/x)^(1/2)*x^(3/2)+2/5*a^2*(a +b/x)^(1/2)*x^(5/2)-2*b^(5/2)*arctanh(b^(1/2)/(a+b/x)^(1/2)/x^(1/2))
Time = 5.47 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.83 \[ \int \left (a+\frac {b}{x}\right )^{5/2} x^{3/2} \, dx=\frac {\sqrt {a+\frac {b}{x}} \sqrt {x} \left (\frac {2}{15} \sqrt {b+a x} \left (23 b^2+11 a b x+3 a^2 x^2\right )-2 b^{5/2} \text {arctanh}\left (\frac {\sqrt {b+a x}}{\sqrt {b}}\right )\right )}{\sqrt {b+a x}} \] Input:
Integrate[(a + b/x)^(5/2)*x^(3/2),x]
Output:
(Sqrt[a + b/x]*Sqrt[x]*((2*Sqrt[b + a*x]*(23*b^2 + 11*a*b*x + 3*a^2*x^2))/ 15 - 2*b^(5/2)*ArcTanh[Sqrt[b + a*x]/Sqrt[b]]))/Sqrt[b + a*x]
Time = 0.33 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.97, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.353, Rules used = {860, 247, 247, 247, 224, 219}
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.
\(\displaystyle \int x^{3/2} \left (a+\frac {b}{x}\right )^{5/2} \, dx\) |
\(\Big \downarrow \) 860 |
\(\displaystyle -2 \int \left (a+\frac {b}{x}\right )^{5/2} x^3d\frac {1}{\sqrt {x}}\) |
\(\Big \downarrow \) 247 |
\(\displaystyle -2 \left (b \int \left (a+\frac {b}{x}\right )^{3/2} x^2d\frac {1}{\sqrt {x}}-\frac {1}{5} x^{5/2} \left (a+\frac {b}{x}\right )^{5/2}\right )\) |
\(\Big \downarrow \) 247 |
\(\displaystyle -2 \left (b \left (b \int \sqrt {a+\frac {b}{x}} xd\frac {1}{\sqrt {x}}-\frac {1}{3} x^{3/2} \left (a+\frac {b}{x}\right )^{3/2}\right )-\frac {1}{5} x^{5/2} \left (a+\frac {b}{x}\right )^{5/2}\right )\) |
\(\Big \downarrow \) 247 |
\(\displaystyle -2 \left (b \left (b \left (b \int \frac {1}{\sqrt {a+\frac {b}{x}}}d\frac {1}{\sqrt {x}}-\sqrt {x} \sqrt {a+\frac {b}{x}}\right )-\frac {1}{3} x^{3/2} \left (a+\frac {b}{x}\right )^{3/2}\right )-\frac {1}{5} x^{5/2} \left (a+\frac {b}{x}\right )^{5/2}\right )\) |
\(\Big \downarrow \) 224 |
\(\displaystyle -2 \left (b \left (b \left (b \int \frac {1}{1-\frac {b}{x}}d\frac {1}{\sqrt {a+\frac {b}{x}} \sqrt {x}}-\sqrt {x} \sqrt {a+\frac {b}{x}}\right )-\frac {1}{3} x^{3/2} \left (a+\frac {b}{x}\right )^{3/2}\right )-\frac {1}{5} x^{5/2} \left (a+\frac {b}{x}\right )^{5/2}\right )\) |
\(\Big \downarrow \) 219 |
\(\displaystyle -2 \left (b \left (b \left (\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b}}{\sqrt {x} \sqrt {a+\frac {b}{x}}}\right )-\sqrt {x} \sqrt {a+\frac {b}{x}}\right )-\frac {1}{3} x^{3/2} \left (a+\frac {b}{x}\right )^{3/2}\right )-\frac {1}{5} x^{5/2} \left (a+\frac {b}{x}\right )^{5/2}\right )\) |
Input:
Int[(a + b/x)^(5/2)*x^(3/2),x]
Output:
-2*(-1/5*((a + b/x)^(5/2)*x^(5/2)) + b*(-1/3*((a + b/x)^(3/2)*x^(3/2)) + b *(-(Sqrt[a + b/x]*Sqrt[x]) + Sqrt[b]*ArcTanh[Sqrt[b]/(Sqrt[a + b/x]*Sqrt[x ])])))
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] && !GtQ[a, 0]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^ (m + 1)*((a + b*x^2)^p/(c*(m + 1))), x] - Simp[2*b*(p/(c^2*(m + 1))) Int[ (c*x)^(m + 2)*(a + b*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && GtQ[p, 0] && LtQ[m, -1] && !ILtQ[(m + 2*p + 3)/2, 0] && IntBinomialQ[a, b, c, 2, m, p, x]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Simp[-k/c Subst[Int[(a + b/(c^n*x^(k*n)))^p/x^(k*(m + 1 ) + 1), x], x, 1/(c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && ILtQ[n, 0] && FractionQ[m]
Time = 0.24 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.82
method | result | size |
default | \(-\frac {2 \sqrt {\frac {a x +b}{x}}\, \sqrt {x}\, \left (-3 a^{2} x^{2} \sqrt {a x +b}+15 b^{\frac {5}{2}} \operatorname {arctanh}\left (\frac {\sqrt {a x +b}}{\sqrt {b}}\right )-11 a b x \sqrt {a x +b}-23 b^{2} \sqrt {a x +b}\right )}{15 \sqrt {a x +b}}\) | \(81\) |
Input:
int((a+b/x)^(5/2)*x^(3/2),x,method=_RETURNVERBOSE)
Output:
-2/15*((a*x+b)/x)^(1/2)*x^(1/2)*(-3*a^2*x^2*(a*x+b)^(1/2)+15*b^(5/2)*arcta nh((a*x+b)^(1/2)/b^(1/2))-11*a*b*x*(a*x+b)^(1/2)-23*b^2*(a*x+b)^(1/2))/(a* x+b)^(1/2)
Time = 0.08 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.47 \[ \int \left (a+\frac {b}{x}\right )^{5/2} x^{3/2} \, dx=\left [b^{\frac {5}{2}} \log \left (\frac {a x - 2 \, \sqrt {b} \sqrt {x} \sqrt {\frac {a x + b}{x}} + 2 \, b}{x}\right ) + \frac {2}{15} \, {\left (3 \, a^{2} x^{2} + 11 \, a b x + 23 \, b^{2}\right )} \sqrt {x} \sqrt {\frac {a x + b}{x}}, 2 \, \sqrt {-b} b^{2} \arctan \left (\frac {\sqrt {-b} \sqrt {x} \sqrt {\frac {a x + b}{x}}}{a x + b}\right ) + \frac {2}{15} \, {\left (3 \, a^{2} x^{2} + 11 \, a b x + 23 \, b^{2}\right )} \sqrt {x} \sqrt {\frac {a x + b}{x}}\right ] \] Input:
integrate((a+b/x)^(5/2)*x^(3/2),x, algorithm="fricas")
Output:
[b^(5/2)*log((a*x - 2*sqrt(b)*sqrt(x)*sqrt((a*x + b)/x) + 2*b)/x) + 2/15*( 3*a^2*x^2 + 11*a*b*x + 23*b^2)*sqrt(x)*sqrt((a*x + b)/x), 2*sqrt(-b)*b^2*a rctan(sqrt(-b)*sqrt(x)*sqrt((a*x + b)/x)/(a*x + b)) + 2/15*(3*a^2*x^2 + 11 *a*b*x + 23*b^2)*sqrt(x)*sqrt((a*x + b)/x)]
Time = 12.99 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.98 \[ \int \left (a+\frac {b}{x}\right )^{5/2} x^{3/2} \, dx=\frac {2 a^{2} \sqrt {b} x^{2} \sqrt {\frac {a x}{b} + 1}}{5} + \frac {22 a b^{\frac {3}{2}} x \sqrt {\frac {a x}{b} + 1}}{15} + \frac {46 b^{\frac {5}{2}} \sqrt {\frac {a x}{b} + 1}}{15} + b^{\frac {5}{2}} \log {\left (\frac {a x}{b} \right )} - 2 b^{\frac {5}{2}} \log {\left (\sqrt {\frac {a x}{b} + 1} + 1 \right )} \] Input:
integrate((a+b/x)**(5/2)*x**(3/2),x)
Output:
2*a**2*sqrt(b)*x**2*sqrt(a*x/b + 1)/5 + 22*a*b**(3/2)*x*sqrt(a*x/b + 1)/15 + 46*b**(5/2)*sqrt(a*x/b + 1)/15 + b**(5/2)*log(a*x/b) - 2*b**(5/2)*log(s qrt(a*x/b + 1) + 1)
Time = 0.12 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.92 \[ \int \left (a+\frac {b}{x}\right )^{5/2} x^{3/2} \, dx=\frac {2}{5} \, {\left (a + \frac {b}{x}\right )}^{\frac {5}{2}} x^{\frac {5}{2}} + \frac {2}{3} \, {\left (a + \frac {b}{x}\right )}^{\frac {3}{2}} b x^{\frac {3}{2}} + b^{\frac {5}{2}} \log \left (\frac {\sqrt {a + \frac {b}{x}} \sqrt {x} - \sqrt {b}}{\sqrt {a + \frac {b}{x}} \sqrt {x} + \sqrt {b}}\right ) + 2 \, \sqrt {a + \frac {b}{x}} b^{2} \sqrt {x} \] Input:
integrate((a+b/x)^(5/2)*x^(3/2),x, algorithm="maxima")
Output:
2/5*(a + b/x)^(5/2)*x^(5/2) + 2/3*(a + b/x)^(3/2)*b*x^(3/2) + b^(5/2)*log( (sqrt(a + b/x)*sqrt(x) - sqrt(b))/(sqrt(a + b/x)*sqrt(x) + sqrt(b))) + 2*s qrt(a + b/x)*b^2*sqrt(x)
Time = 0.13 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.00 \[ \int \left (a+\frac {b}{x}\right )^{5/2} x^{3/2} \, dx=\frac {2 \, b^{3} \arctan \left (\frac {\sqrt {a x + b}}{\sqrt {-b}}\right ) \mathrm {sgn}\left (x\right )}{\sqrt {-b}} + \frac {2}{5} \, {\left (a x + b\right )}^{\frac {5}{2}} \mathrm {sgn}\left (x\right ) + \frac {2}{3} \, {\left (a x + b\right )}^{\frac {3}{2}} b \mathrm {sgn}\left (x\right ) + 2 \, \sqrt {a x + b} b^{2} \mathrm {sgn}\left (x\right ) - \frac {2 \, {\left (15 \, b^{3} \arctan \left (\frac {\sqrt {b}}{\sqrt {-b}}\right ) + 23 \, \sqrt {-b} b^{\frac {5}{2}}\right )} \mathrm {sgn}\left (x\right )}{15 \, \sqrt {-b}} \] Input:
integrate((a+b/x)^(5/2)*x^(3/2),x, algorithm="giac")
Output:
2*b^3*arctan(sqrt(a*x + b)/sqrt(-b))*sgn(x)/sqrt(-b) + 2/5*(a*x + b)^(5/2) *sgn(x) + 2/3*(a*x + b)^(3/2)*b*sgn(x) + 2*sqrt(a*x + b)*b^2*sgn(x) - 2/15 *(15*b^3*arctan(sqrt(b)/sqrt(-b)) + 23*sqrt(-b)*b^(5/2))*sgn(x)/sqrt(-b)
Timed out. \[ \int \left (a+\frac {b}{x}\right )^{5/2} x^{3/2} \, dx=\int x^{3/2}\,{\left (a+\frac {b}{x}\right )}^{5/2} \,d x \] Input:
int(x^(3/2)*(a + b/x)^(5/2),x)
Output:
int(x^(3/2)*(a + b/x)^(5/2), x)
Time = 0.21 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.73 \[ \int \left (a+\frac {b}{x}\right )^{5/2} x^{3/2} \, dx=\frac {2 \sqrt {a x +b}\, a^{2} x^{2}}{5}+\frac {22 \sqrt {a x +b}\, a b x}{15}+\frac {46 \sqrt {a x +b}\, b^{2}}{15}+\sqrt {b}\, \mathrm {log}\left (\sqrt {a x +b}-\sqrt {b}\right ) b^{2}-\sqrt {b}\, \mathrm {log}\left (\sqrt {a x +b}+\sqrt {b}\right ) b^{2} \] Input:
int((a+b/x)^(5/2)*x^(3/2),x)
Output:
(6*sqrt(a*x + b)*a**2*x**2 + 22*sqrt(a*x + b)*a*b*x + 46*sqrt(a*x + b)*b** 2 + 15*sqrt(b)*log(sqrt(a*x + b) - sqrt(b))*b**2 - 15*sqrt(b)*log(sqrt(a*x + b) + sqrt(b))*b**2)/15