Integrand size = 15, antiderivative size = 68 \[ \int \frac {x^{3/2}}{a+\frac {b}{x}} \, dx=\frac {2 b^2 \sqrt {x}}{a^3}-\frac {2 b x^{3/2}}{3 a^2}+\frac {2 x^{5/2}}{5 a}-\frac {2 b^{5/2} \arctan \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )}{a^{7/2}} \] Output:
2*b^2*x^(1/2)/a^3-2/3*b*x^(3/2)/a^2+2/5*x^(5/2)/a-2*b^(5/2)*arctan(a^(1/2) *x^(1/2)/b^(1/2))/a^(7/2)
Time = 0.05 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.90 \[ \int \frac {x^{3/2}}{a+\frac {b}{x}} \, dx=\frac {2 \sqrt {x} \left (15 b^2-5 a b x+3 a^2 x^2\right )}{15 a^3}-\frac {2 b^{5/2} \arctan \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )}{a^{7/2}} \] Input:
Integrate[x^(3/2)/(a + b/x),x]
Output:
(2*Sqrt[x]*(15*b^2 - 5*a*b*x + 3*a^2*x^2))/(15*a^3) - (2*b^(5/2)*ArcTan[(S qrt[a]*Sqrt[x])/Sqrt[b]])/a^(7/2)
Time = 0.28 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.15, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {795, 60, 60, 60, 73, 218}
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.
\(\displaystyle \int \frac {x^{3/2}}{a+\frac {b}{x}} \, dx\) |
\(\Big \downarrow \) 795 |
\(\displaystyle \int \frac {x^{5/2}}{a x+b}dx\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {2 x^{5/2}}{5 a}-\frac {b \int \frac {x^{3/2}}{b+a x}dx}{a}\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {2 x^{5/2}}{5 a}-\frac {b \left (\frac {2 x^{3/2}}{3 a}-\frac {b \int \frac {\sqrt {x}}{b+a x}dx}{a}\right )}{a}\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {2 x^{5/2}}{5 a}-\frac {b \left (\frac {2 x^{3/2}}{3 a}-\frac {b \left (\frac {2 \sqrt {x}}{a}-\frac {b \int \frac {1}{\sqrt {x} (b+a x)}dx}{a}\right )}{a}\right )}{a}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {2 x^{5/2}}{5 a}-\frac {b \left (\frac {2 x^{3/2}}{3 a}-\frac {b \left (\frac {2 \sqrt {x}}{a}-\frac {2 b \int \frac {1}{b+a x}d\sqrt {x}}{a}\right )}{a}\right )}{a}\) |
\(\Big \downarrow \) 218 |
\(\displaystyle \frac {2 x^{5/2}}{5 a}-\frac {b \left (\frac {2 x^{3/2}}{3 a}-\frac {b \left (\frac {2 \sqrt {x}}{a}-\frac {2 \sqrt {b} \arctan \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )}{a^{3/2}}\right )}{a}\right )}{a}\) |
Input:
Int[x^(3/2)/(a + b/x),x]
Output:
(2*x^(5/2))/(5*a) - (b*((2*x^(3/2))/(3*a) - (b*((2*Sqrt[x])/a - (2*Sqrt[b] *ArcTan[(Sqrt[a]*Sqrt[x])/Sqrt[b]])/a^(3/2)))/a))/a
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( b*(m + n + 1))) Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !Integer Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && IntLinear Q[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)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[x^(m + n*p)* (b + a/x^n)^p, x] /; FreeQ[{a, b, m, n}, x] && IntegerQ[p] && NegQ[n]
Time = 0.16 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.78
method | result | size |
risch | \(\frac {2 \left (3 a^{2} x^{2}-5 a b x +15 b^{2}\right ) \sqrt {x}}{15 a^{3}}-\frac {2 b^{3} \arctan \left (\frac {a \sqrt {x}}{\sqrt {a b}}\right )}{a^{3} \sqrt {a b}}\) | \(53\) |
derivativedivides | \(\frac {\frac {2 a^{2} x^{\frac {5}{2}}}{5}-\frac {2 a b \,x^{\frac {3}{2}}}{3}+2 b^{2} \sqrt {x}}{a^{3}}-\frac {2 b^{3} \arctan \left (\frac {a \sqrt {x}}{\sqrt {a b}}\right )}{a^{3} \sqrt {a b}}\) | \(54\) |
default | \(\frac {\frac {2 a^{2} x^{\frac {5}{2}}}{5}-\frac {2 a b \,x^{\frac {3}{2}}}{3}+2 b^{2} \sqrt {x}}{a^{3}}-\frac {2 b^{3} \arctan \left (\frac {a \sqrt {x}}{\sqrt {a b}}\right )}{a^{3} \sqrt {a b}}\) | \(54\) |
Input:
int(x^(3/2)/(a+b/x),x,method=_RETURNVERBOSE)
Output:
2/15*(3*a^2*x^2-5*a*b*x+15*b^2)*x^(1/2)/a^3-2*b^3/a^3/(a*b)^(1/2)*arctan(a *x^(1/2)/(a*b)^(1/2))
Time = 0.08 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.94 \[ \int \frac {x^{3/2}}{a+\frac {b}{x}} \, dx=\left [\frac {15 \, b^{2} \sqrt {-\frac {b}{a}} \log \left (\frac {a x - 2 \, a \sqrt {x} \sqrt {-\frac {b}{a}} - b}{a x + b}\right ) + 2 \, {\left (3 \, a^{2} x^{2} - 5 \, a b x + 15 \, b^{2}\right )} \sqrt {x}}{15 \, a^{3}}, -\frac {2 \, {\left (15 \, b^{2} \sqrt {\frac {b}{a}} \arctan \left (\frac {a \sqrt {x} \sqrt {\frac {b}{a}}}{b}\right ) - {\left (3 \, a^{2} x^{2} - 5 \, a b x + 15 \, b^{2}\right )} \sqrt {x}\right )}}{15 \, a^{3}}\right ] \] Input:
integrate(x^(3/2)/(a+b/x),x, algorithm="fricas")
Output:
[1/15*(15*b^2*sqrt(-b/a)*log((a*x - 2*a*sqrt(x)*sqrt(-b/a) - b)/(a*x + b)) + 2*(3*a^2*x^2 - 5*a*b*x + 15*b^2)*sqrt(x))/a^3, -2/15*(15*b^2*sqrt(b/a)* arctan(a*sqrt(x)*sqrt(b/a)/b) - (3*a^2*x^2 - 5*a*b*x + 15*b^2)*sqrt(x))/a^ 3]
Time = 1.17 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.79 \[ \int \frac {x^{3/2}}{a+\frac {b}{x}} \, dx=\begin {cases} \tilde {\infty } x^{\frac {7}{2}} & \text {for}\: a = 0 \wedge b = 0 \\\frac {2 x^{\frac {7}{2}}}{7 b} & \text {for}\: a = 0 \\\frac {2 x^{\frac {5}{2}}}{5 a} & \text {for}\: b = 0 \\\frac {2 x^{\frac {5}{2}}}{5 a} - \frac {2 b x^{\frac {3}{2}}}{3 a^{2}} + \frac {2 b^{2} \sqrt {x}}{a^{3}} - \frac {b^{3} \log {\left (\sqrt {x} - \sqrt {- \frac {b}{a}} \right )}}{a^{4} \sqrt {- \frac {b}{a}}} + \frac {b^{3} \log {\left (\sqrt {x} + \sqrt {- \frac {b}{a}} \right )}}{a^{4} \sqrt {- \frac {b}{a}}} & \text {otherwise} \end {cases} \] Input:
integrate(x**(3/2)/(a+b/x),x)
Output:
Piecewise((zoo*x**(7/2), Eq(a, 0) & Eq(b, 0)), (2*x**(7/2)/(7*b), Eq(a, 0) ), (2*x**(5/2)/(5*a), Eq(b, 0)), (2*x**(5/2)/(5*a) - 2*b*x**(3/2)/(3*a**2) + 2*b**2*sqrt(x)/a**3 - b**3*log(sqrt(x) - sqrt(-b/a))/(a**4*sqrt(-b/a)) + b**3*log(sqrt(x) + sqrt(-b/a))/(a**4*sqrt(-b/a)), True))
Time = 0.11 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.79 \[ \int \frac {x^{3/2}}{a+\frac {b}{x}} \, dx=\frac {2 \, {\left (3 \, a^{2} - \frac {5 \, a b}{x} + \frac {15 \, b^{2}}{x^{2}}\right )} x^{\frac {5}{2}}}{15 \, a^{3}} + \frac {2 \, b^{3} \arctan \left (\frac {b}{\sqrt {a b} \sqrt {x}}\right )}{\sqrt {a b} a^{3}} \] Input:
integrate(x^(3/2)/(a+b/x),x, algorithm="maxima")
Output:
2/15*(3*a^2 - 5*a*b/x + 15*b^2/x^2)*x^(5/2)/a^3 + 2*b^3*arctan(b/(sqrt(a*b )*sqrt(x)))/(sqrt(a*b)*a^3)
Time = 0.13 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.87 \[ \int \frac {x^{3/2}}{a+\frac {b}{x}} \, dx=-\frac {2 \, b^{3} \arctan \left (\frac {a \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b} a^{3}} + \frac {2 \, {\left (3 \, a^{4} x^{\frac {5}{2}} - 5 \, a^{3} b x^{\frac {3}{2}} + 15 \, a^{2} b^{2} \sqrt {x}\right )}}{15 \, a^{5}} \] Input:
integrate(x^(3/2)/(a+b/x),x, algorithm="giac")
Output:
-2*b^3*arctan(a*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*a^3) + 2/15*(3*a^4*x^(5/2) - 5*a^3*b*x^(3/2) + 15*a^2*b^2*sqrt(x))/a^5
Time = 0.06 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.71 \[ \int \frac {x^{3/2}}{a+\frac {b}{x}} \, dx=\frac {2\,x^{5/2}}{5\,a}-\frac {2\,b\,x^{3/2}}{3\,a^2}+\frac {2\,b^2\,\sqrt {x}}{a^3}-\frac {2\,b^{5/2}\,\mathrm {atan}\left (\frac {\sqrt {a}\,\sqrt {x}}{\sqrt {b}}\right )}{a^{7/2}} \] Input:
int(x^(3/2)/(a + b/x),x)
Output:
(2*x^(5/2))/(5*a) - (2*b*x^(3/2))/(3*a^2) + (2*b^2*x^(1/2))/a^3 - (2*b^(5/ 2)*atan((a^(1/2)*x^(1/2))/b^(1/2)))/a^(7/2)
Time = 0.20 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.81 \[ \int \frac {x^{3/2}}{a+\frac {b}{x}} \, dx=\frac {-2 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {\sqrt {x}\, a}{\sqrt {b}\, \sqrt {a}}\right ) b^{2}+\frac {2 \sqrt {x}\, a^{3} x^{2}}{5}-\frac {2 \sqrt {x}\, a^{2} b x}{3}+2 \sqrt {x}\, a \,b^{2}}{a^{4}} \] Input:
int(x^(3/2)/(a+b/x),x)
Output:
(2*( - 15*sqrt(b)*sqrt(a)*atan((sqrt(x)*a)/(sqrt(b)*sqrt(a)))*b**2 + 3*sqr t(x)*a**3*x**2 - 5*sqrt(x)*a**2*b*x + 15*sqrt(x)*a*b**2))/(15*a**4)