Integrand size = 15, antiderivative size = 95 \[ \int \frac {\sqrt {x}}{\left (a+\frac {b}{x}\right )^3} \, dx=-\frac {35 b \sqrt {x}}{4 a^4}+\frac {35 x^{3/2}}{12 a^3}-\frac {x^{7/2}}{2 a (b+a x)^2}-\frac {7 x^{5/2}}{4 a^2 (b+a x)}+\frac {35 b^{3/2} \arctan \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )}{4 a^{9/2}} \]
35/12*x^(3/2)/a^3-1/2*x^(7/2)/a/(a*x+b)^2-7/4*x^(5/2)/a^2/(a*x+b)+35/4*b^( 3/2)*arctan(a^(1/2)*x^(1/2)/b^(1/2))/a^(9/2)-35/4*b*x^(1/2)/a^4
Time = 0.19 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.85 \[ \int \frac {\sqrt {x}}{\left (a+\frac {b}{x}\right )^3} \, dx=\frac {\sqrt {x} \left (-105 b^3-175 a b^2 x-56 a^2 b x^2+8 a^3 x^3\right )}{12 a^4 (b+a x)^2}+\frac {35 b^{3/2} \arctan \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )}{4 a^{9/2}} \]
(Sqrt[x]*(-105*b^3 - 175*a*b^2*x - 56*a^2*b*x^2 + 8*a^3*x^3))/(12*a^4*(b + a*x)^2) + (35*b^(3/2)*ArcTan[(Sqrt[a]*Sqrt[x])/Sqrt[b]])/(4*a^(9/2))
Time = 0.20 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.17, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.467, Rules used = {795, 51, 51, 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.
\(\displaystyle \int \frac {\sqrt {x}}{\left (a+\frac {b}{x}\right )^3} \, dx\) |
\(\Big \downarrow \) 795 |
\(\displaystyle \int \frac {x^{7/2}}{(a x+b)^3}dx\) |
\(\Big \downarrow \) 51 |
\(\displaystyle \frac {7 \int \frac {x^{5/2}}{(b+a x)^2}dx}{4 a}-\frac {x^{7/2}}{2 a (a x+b)^2}\) |
\(\Big \downarrow \) 51 |
\(\displaystyle \frac {7 \left (\frac {5 \int \frac {x^{3/2}}{b+a x}dx}{2 a}-\frac {x^{5/2}}{a (a x+b)}\right )}{4 a}-\frac {x^{7/2}}{2 a (a x+b)^2}\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {7 \left (\frac {5 \left (\frac {2 x^{3/2}}{3 a}-\frac {b \int \frac {\sqrt {x}}{b+a x}dx}{a}\right )}{2 a}-\frac {x^{5/2}}{a (a x+b)}\right )}{4 a}-\frac {x^{7/2}}{2 a (a x+b)^2}\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {7 \left (\frac {5 \left (\frac {2 x^{3/2}}{3 a}-\frac {b \left (\frac {2 \sqrt {x}}{a}-\frac {b \int \frac {1}{\sqrt {x} (b+a x)}dx}{a}\right )}{a}\right )}{2 a}-\frac {x^{5/2}}{a (a x+b)}\right )}{4 a}-\frac {x^{7/2}}{2 a (a x+b)^2}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {7 \left (\frac {5 \left (\frac {2 x^{3/2}}{3 a}-\frac {b \left (\frac {2 \sqrt {x}}{a}-\frac {2 b \int \frac {1}{b+a x}d\sqrt {x}}{a}\right )}{a}\right )}{2 a}-\frac {x^{5/2}}{a (a x+b)}\right )}{4 a}-\frac {x^{7/2}}{2 a (a x+b)^2}\) |
\(\Big \downarrow \) 218 |
\(\displaystyle \frac {7 \left (\frac {5 \left (\frac {2 x^{3/2}}{3 a}-\frac {b \left (\frac {2 \sqrt {x}}{a}-\frac {2 \sqrt {b} \arctan \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )}{a^{3/2}}\right )}{a}\right )}{2 a}-\frac {x^{5/2}}{a (a x+b)}\right )}{4 a}-\frac {x^{7/2}}{2 a (a x+b)^2}\) |
-1/2*x^(7/2)/(a*(b + a*x)^2) + (7*(-(x^(5/2)/(a*(b + a*x))) + (5*((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))/(2*a)))/(4*a)
3.17.83.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + 1))), x] - Simp[d*(n/(b*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d, n}, x ] && ILtQ[m, -1] && FractionQ[n] && GtQ[n, 0]
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 = 0.08 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.68
method | result | size |
risch | \(\frac {2 \left (a x -9 b \right ) \sqrt {x}}{3 a^{4}}+\frac {b^{2} \left (\frac {-\frac {13 a \,x^{\frac {3}{2}}}{4}-\frac {11 b \sqrt {x}}{4}}{\left (a x +b \right )^{2}}+\frac {35 \arctan \left (\frac {a \sqrt {x}}{\sqrt {a b}}\right )}{4 \sqrt {a b}}\right )}{a^{4}}\) | \(65\) |
derivativedivides | \(\frac {\frac {2 a \,x^{\frac {3}{2}}}{3}-6 b \sqrt {x}}{a^{4}}+\frac {2 b^{2} \left (\frac {-\frac {13 a \,x^{\frac {3}{2}}}{8}-\frac {11 b \sqrt {x}}{8}}{\left (a x +b \right )^{2}}+\frac {35 \arctan \left (\frac {a \sqrt {x}}{\sqrt {a b}}\right )}{8 \sqrt {a b}}\right )}{a^{4}}\) | \(68\) |
default | \(\frac {\frac {2 a \,x^{\frac {3}{2}}}{3}-6 b \sqrt {x}}{a^{4}}+\frac {2 b^{2} \left (\frac {-\frac {13 a \,x^{\frac {3}{2}}}{8}-\frac {11 b \sqrt {x}}{8}}{\left (a x +b \right )^{2}}+\frac {35 \arctan \left (\frac {a \sqrt {x}}{\sqrt {a b}}\right )}{8 \sqrt {a b}}\right )}{a^{4}}\) | \(68\) |
2/3*(a*x-9*b)*x^(1/2)/a^4+b^2/a^4*(2*(-13/8*a*x^(3/2)-11/8*b*x^(1/2))/(a*x +b)^2+35/4/(a*b)^(1/2)*arctan(a*x^(1/2)/(a*b)^(1/2)))
Time = 0.27 (sec) , antiderivative size = 227, normalized size of antiderivative = 2.39 \[ \int \frac {\sqrt {x}}{\left (a+\frac {b}{x}\right )^3} \, dx=\left [\frac {105 \, {\left (a^{2} b x^{2} + 2 \, a b^{2} x + b^{3}\right )} \sqrt {-\frac {b}{a}} \log \left (\frac {a x + 2 \, a \sqrt {x} \sqrt {-\frac {b}{a}} - b}{a x + b}\right ) + 2 \, {\left (8 \, a^{3} x^{3} - 56 \, a^{2} b x^{2} - 175 \, a b^{2} x - 105 \, b^{3}\right )} \sqrt {x}}{24 \, {\left (a^{6} x^{2} + 2 \, a^{5} b x + a^{4} b^{2}\right )}}, \frac {105 \, {\left (a^{2} b x^{2} + 2 \, a b^{2} x + b^{3}\right )} \sqrt {\frac {b}{a}} \arctan \left (\frac {a \sqrt {x} \sqrt {\frac {b}{a}}}{b}\right ) + {\left (8 \, a^{3} x^{3} - 56 \, a^{2} b x^{2} - 175 \, a b^{2} x - 105 \, b^{3}\right )} \sqrt {x}}{12 \, {\left (a^{6} x^{2} + 2 \, a^{5} b x + a^{4} b^{2}\right )}}\right ] \]
[1/24*(105*(a^2*b*x^2 + 2*a*b^2*x + b^3)*sqrt(-b/a)*log((a*x + 2*a*sqrt(x) *sqrt(-b/a) - b)/(a*x + b)) + 2*(8*a^3*x^3 - 56*a^2*b*x^2 - 175*a*b^2*x - 105*b^3)*sqrt(x))/(a^6*x^2 + 2*a^5*b*x + a^4*b^2), 1/12*(105*(a^2*b*x^2 + 2*a*b^2*x + b^3)*sqrt(b/a)*arctan(a*sqrt(x)*sqrt(b/a)/b) + (8*a^3*x^3 - 56 *a^2*b*x^2 - 175*a*b^2*x - 105*b^3)*sqrt(x))/(a^6*x^2 + 2*a^5*b*x + a^4*b^ 2)]
Leaf count of result is larger than twice the leaf count of optimal. 762 vs. \(2 (87) = 174\).
Time = 7.29 (sec) , antiderivative size = 762, normalized size of antiderivative = 8.02 \[ \int \frac {\sqrt {x}}{\left (a+\frac {b}{x}\right )^3} \, dx=\begin {cases} \tilde {\infty } x^{\frac {9}{2}} & \text {for}\: a = 0 \wedge b = 0 \\\frac {2 x^{\frac {9}{2}}}{9 b^{3}} & \text {for}\: a = 0 \\\frac {2 x^{\frac {3}{2}}}{3 a^{3}} & \text {for}\: b = 0 \\\frac {16 a^{4} x^{\frac {7}{2}} \sqrt {- \frac {b}{a}}}{24 a^{7} x^{2} \sqrt {- \frac {b}{a}} + 48 a^{6} b x \sqrt {- \frac {b}{a}} + 24 a^{5} b^{2} \sqrt {- \frac {b}{a}}} - \frac {112 a^{3} b x^{\frac {5}{2}} \sqrt {- \frac {b}{a}}}{24 a^{7} x^{2} \sqrt {- \frac {b}{a}} + 48 a^{6} b x \sqrt {- \frac {b}{a}} + 24 a^{5} b^{2} \sqrt {- \frac {b}{a}}} - \frac {350 a^{2} b^{2} x^{\frac {3}{2}} \sqrt {- \frac {b}{a}}}{24 a^{7} x^{2} \sqrt {- \frac {b}{a}} + 48 a^{6} b x \sqrt {- \frac {b}{a}} + 24 a^{5} b^{2} \sqrt {- \frac {b}{a}}} + \frac {105 a^{2} b^{2} x^{2} \log {\left (\sqrt {x} - \sqrt {- \frac {b}{a}} \right )}}{24 a^{7} x^{2} \sqrt {- \frac {b}{a}} + 48 a^{6} b x \sqrt {- \frac {b}{a}} + 24 a^{5} b^{2} \sqrt {- \frac {b}{a}}} - \frac {105 a^{2} b^{2} x^{2} \log {\left (\sqrt {x} + \sqrt {- \frac {b}{a}} \right )}}{24 a^{7} x^{2} \sqrt {- \frac {b}{a}} + 48 a^{6} b x \sqrt {- \frac {b}{a}} + 24 a^{5} b^{2} \sqrt {- \frac {b}{a}}} - \frac {210 a b^{3} \sqrt {x} \sqrt {- \frac {b}{a}}}{24 a^{7} x^{2} \sqrt {- \frac {b}{a}} + 48 a^{6} b x \sqrt {- \frac {b}{a}} + 24 a^{5} b^{2} \sqrt {- \frac {b}{a}}} + \frac {210 a b^{3} x \log {\left (\sqrt {x} - \sqrt {- \frac {b}{a}} \right )}}{24 a^{7} x^{2} \sqrt {- \frac {b}{a}} + 48 a^{6} b x \sqrt {- \frac {b}{a}} + 24 a^{5} b^{2} \sqrt {- \frac {b}{a}}} - \frac {210 a b^{3} x \log {\left (\sqrt {x} + \sqrt {- \frac {b}{a}} \right )}}{24 a^{7} x^{2} \sqrt {- \frac {b}{a}} + 48 a^{6} b x \sqrt {- \frac {b}{a}} + 24 a^{5} b^{2} \sqrt {- \frac {b}{a}}} + \frac {105 b^{4} \log {\left (\sqrt {x} - \sqrt {- \frac {b}{a}} \right )}}{24 a^{7} x^{2} \sqrt {- \frac {b}{a}} + 48 a^{6} b x \sqrt {- \frac {b}{a}} + 24 a^{5} b^{2} \sqrt {- \frac {b}{a}}} - \frac {105 b^{4} \log {\left (\sqrt {x} + \sqrt {- \frac {b}{a}} \right )}}{24 a^{7} x^{2} \sqrt {- \frac {b}{a}} + 48 a^{6} b x \sqrt {- \frac {b}{a}} + 24 a^{5} b^{2} \sqrt {- \frac {b}{a}}} & \text {otherwise} \end {cases} \]
Piecewise((zoo*x**(9/2), Eq(a, 0) & Eq(b, 0)), (2*x**(9/2)/(9*b**3), Eq(a, 0)), (2*x**(3/2)/(3*a**3), Eq(b, 0)), (16*a**4*x**(7/2)*sqrt(-b/a)/(24*a* *7*x**2*sqrt(-b/a) + 48*a**6*b*x*sqrt(-b/a) + 24*a**5*b**2*sqrt(-b/a)) - 1 12*a**3*b*x**(5/2)*sqrt(-b/a)/(24*a**7*x**2*sqrt(-b/a) + 48*a**6*b*x*sqrt( -b/a) + 24*a**5*b**2*sqrt(-b/a)) - 350*a**2*b**2*x**(3/2)*sqrt(-b/a)/(24*a **7*x**2*sqrt(-b/a) + 48*a**6*b*x*sqrt(-b/a) + 24*a**5*b**2*sqrt(-b/a)) + 105*a**2*b**2*x**2*log(sqrt(x) - sqrt(-b/a))/(24*a**7*x**2*sqrt(-b/a) + 48 *a**6*b*x*sqrt(-b/a) + 24*a**5*b**2*sqrt(-b/a)) - 105*a**2*b**2*x**2*log(s qrt(x) + sqrt(-b/a))/(24*a**7*x**2*sqrt(-b/a) + 48*a**6*b*x*sqrt(-b/a) + 2 4*a**5*b**2*sqrt(-b/a)) - 210*a*b**3*sqrt(x)*sqrt(-b/a)/(24*a**7*x**2*sqrt (-b/a) + 48*a**6*b*x*sqrt(-b/a) + 24*a**5*b**2*sqrt(-b/a)) + 210*a*b**3*x* log(sqrt(x) - sqrt(-b/a))/(24*a**7*x**2*sqrt(-b/a) + 48*a**6*b*x*sqrt(-b/a ) + 24*a**5*b**2*sqrt(-b/a)) - 210*a*b**3*x*log(sqrt(x) + sqrt(-b/a))/(24* a**7*x**2*sqrt(-b/a) + 48*a**6*b*x*sqrt(-b/a) + 24*a**5*b**2*sqrt(-b/a)) + 105*b**4*log(sqrt(x) - sqrt(-b/a))/(24*a**7*x**2*sqrt(-b/a) + 48*a**6*b*x *sqrt(-b/a) + 24*a**5*b**2*sqrt(-b/a)) - 105*b**4*log(sqrt(x) + sqrt(-b/a) )/(24*a**7*x**2*sqrt(-b/a) + 48*a**6*b*x*sqrt(-b/a) + 24*a**5*b**2*sqrt(-b /a)), True))
Time = 0.30 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.93 \[ \int \frac {\sqrt {x}}{\left (a+\frac {b}{x}\right )^3} \, dx=\frac {8 \, a^{3} - \frac {56 \, a^{2} b}{x} - \frac {175 \, a b^{2}}{x^{2}} - \frac {105 \, b^{3}}{x^{3}}}{12 \, {\left (\frac {a^{6}}{x^{\frac {3}{2}}} + \frac {2 \, a^{5} b}{x^{\frac {5}{2}}} + \frac {a^{4} b^{2}}{x^{\frac {7}{2}}}\right )}} - \frac {35 \, b^{2} \arctan \left (\frac {b}{\sqrt {a b} \sqrt {x}}\right )}{4 \, \sqrt {a b} a^{4}} \]
1/12*(8*a^3 - 56*a^2*b/x - 175*a*b^2/x^2 - 105*b^3/x^3)/(a^6/x^(3/2) + 2*a ^5*b/x^(5/2) + a^4*b^2/x^(7/2)) - 35/4*b^2*arctan(b/(sqrt(a*b)*sqrt(x)))/( sqrt(a*b)*a^4)
Time = 0.28 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.81 \[ \int \frac {\sqrt {x}}{\left (a+\frac {b}{x}\right )^3} \, dx=\frac {35 \, b^{2} \arctan \left (\frac {a \sqrt {x}}{\sqrt {a b}}\right )}{4 \, \sqrt {a b} a^{4}} - \frac {13 \, a b^{2} x^{\frac {3}{2}} + 11 \, b^{3} \sqrt {x}}{4 \, {\left (a x + b\right )}^{2} a^{4}} + \frac {2 \, {\left (a^{6} x^{\frac {3}{2}} - 9 \, a^{5} b \sqrt {x}\right )}}{3 \, a^{9}} \]
35/4*b^2*arctan(a*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*a^4) - 1/4*(13*a*b^2*x^(3/ 2) + 11*b^3*sqrt(x))/((a*x + b)^2*a^4) + 2/3*(a^6*x^(3/2) - 9*a^5*b*sqrt(x ))/a^9
Time = 0.07 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.85 \[ \int \frac {\sqrt {x}}{\left (a+\frac {b}{x}\right )^3} \, dx=\frac {2\,x^{3/2}}{3\,a^3}-\frac {\frac {11\,b^3\,\sqrt {x}}{4}+\frac {13\,a\,b^2\,x^{3/2}}{4}}{a^6\,x^2+2\,a^5\,b\,x+a^4\,b^2}-\frac {6\,b\,\sqrt {x}}{a^4}+\frac {35\,b^{3/2}\,\mathrm {atan}\left (\frac {\sqrt {a}\,\sqrt {x}}{\sqrt {b}}\right )}{4\,a^{9/2}} \]