Integrand size = 21, antiderivative size = 106 \[ \int \frac {\left (a+\frac {b}{x}\right )^{3/2}}{c+\frac {d}{x}} \, dx=\frac {a \sqrt {a+\frac {b}{x}} x}{c}-\frac {2 (b c-a d)^{3/2} \arctan \left (\frac {\sqrt {d} \sqrt {a+\frac {b}{x}}}{\sqrt {b c-a d}}\right )}{c^2 \sqrt {d}}+\frac {\sqrt {a} (3 b c-2 a d) \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{c^2} \] Output:
a*(a+b/x)^(1/2)*x/c-2*(-a*d+b*c)^(3/2)*arctan(d^(1/2)*(a+b/x)^(1/2)/(-a*d+ b*c)^(1/2))/c^2/d^(1/2)+a^(1/2)*(-2*a*d+3*b*c)*arctanh((a+b/x)^(1/2)/a^(1/ 2))/c^2
Time = 0.43 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.97 \[ \int \frac {\left (a+\frac {b}{x}\right )^{3/2}}{c+\frac {d}{x}} \, dx=\frac {a c \sqrt {a+\frac {b}{x}} x-\frac {2 (b c-a d)^{3/2} \arctan \left (\frac {\sqrt {d} \sqrt {a+\frac {b}{x}}}{\sqrt {b c-a d}}\right )}{\sqrt {d}}-\sqrt {a} (-3 b c+2 a d) \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{c^2} \] Input:
Integrate[(a + b/x)^(3/2)/(c + d/x),x]
Output:
(a*c*Sqrt[a + b/x]*x - (2*(b*c - a*d)^(3/2)*ArcTan[(Sqrt[d]*Sqrt[a + b/x]) /Sqrt[b*c - a*d]])/Sqrt[d] - Sqrt[a]*(-3*b*c + 2*a*d)*ArcTanh[Sqrt[a + b/x ]/Sqrt[a]])/c^2
Time = 0.48 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.08, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {899, 109, 27, 174, 73, 218, 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.
| \(\displaystyle \int \frac {\left (a+\frac {b}{x}\right )^{3/2}}{c+\frac {d}{x}} \, dx\) |
| \(\Big \downarrow \) 899 |
| \(\displaystyle -\int \frac {\left (a+\frac {b}{x}\right )^{3/2} x^2}{c+\frac {d}{x}}d\frac {1}{x}\) |
| \(\Big \downarrow \) 109 |
| \(\displaystyle \frac {\int -\frac {\left (a (3 b c-2 a d)+\frac {b (2 b c-a d)}{x}\right ) x}{2 \sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right )}d\frac {1}{x}}{c}+\frac {a x \sqrt {a+\frac {b}{x}}}{c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {a x \sqrt {a+\frac {b}{x}}}{c}-\frac {\int \frac {\left (a (3 b c-2 a d)+\frac {b (2 b c-a d)}{x}\right ) x}{\sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right )}d\frac {1}{x}}{2 c}\) |
| \(\Big \downarrow \) 174 |
| \(\displaystyle \frac {a x \sqrt {a+\frac {b}{x}}}{c}-\frac {\frac {2 (b c-a d)^2 \int \frac {1}{\sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right )}d\frac {1}{x}}{c}+\frac {a (3 b c-2 a d) \int \frac {x}{\sqrt {a+\frac {b}{x}}}d\frac {1}{x}}{c}}{2 c}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {a x \sqrt {a+\frac {b}{x}}}{c}-\frac {\frac {4 (b c-a d)^2 \int \frac {1}{c-\frac {a d}{b}+\frac {d}{b x^2}}d\sqrt {a+\frac {b}{x}}}{b c}+\frac {2 a (3 b c-2 a d) \int \frac {1}{\frac {1}{b x^2}-\frac {a}{b}}d\sqrt {a+\frac {b}{x}}}{b c}}{2 c}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {a x \sqrt {a+\frac {b}{x}}}{c}-\frac {\frac {2 a (3 b c-2 a d) \int \frac {1}{\frac {1}{b x^2}-\frac {a}{b}}d\sqrt {a+\frac {b}{x}}}{b c}+\frac {4 (b c-a d)^{3/2} \arctan \left (\frac {\sqrt {d} \sqrt {a+\frac {b}{x}}}{\sqrt {b c-a d}}\right )}{c \sqrt {d}}}{2 c}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {a x \sqrt {a+\frac {b}{x}}}{c}-\frac {\frac {4 (b c-a d)^{3/2} \arctan \left (\frac {\sqrt {d} \sqrt {a+\frac {b}{x}}}{\sqrt {b c-a d}}\right )}{c \sqrt {d}}-\frac {2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right ) (3 b c-2 a d)}{c}}{2 c}\) |
Input:
Int[(a + b/x)^(3/2)/(c + d/x),x]
Output:
(a*Sqrt[a + b/x]*x)/c - ((4*(b*c - a*d)^(3/2)*ArcTan[(Sqrt[d]*Sqrt[a + b/x ])/Sqrt[b*c - a*d]])/(c*Sqrt[d]) - (2*Sqrt[a]*(3*b*c - 2*a*d)*ArcTanh[Sqrt [a + b/x]/Sqrt[a]])/c)/(2*c)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, 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_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[(b*c - a*d)*(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*((e + f *x)^(p + 1)/(b*(b*e - a*f)*(m + 1))), x] + Simp[1/(b*(b*e - a*f)*(m + 1)) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 2)*(e + f*x)^p*Simp[a*d*(d*e*(n - 1) + c*f*(p + 1)) + b*c*(d*e*(m - n + 2) - c*f*(m + p + 2)) + d*(a*d*f*(n + p) + b*(d*e*(m + 1) - c*f*(m + n + p + 1)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 1] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p] || IntegersQ[p, m + n])
Int[(((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/(((a_.) + (b_.)*(x_))* ((c_.) + (d_.)*(x_))), x_] :> Simp[(b*g - a*h)/(b*c - a*d) Int[(e + f*x)^ p/(a + b*x), x], x] - Simp[(d*g - c*h)/(b*c - a*d) Int[(e + f*x)^p/(c + d *x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, 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[((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[((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol ] :> -Subst[Int[(a + b/x^n)^p*((c + d/x^n)^q/x^2), x], x, 1/x] /; FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && ILtQ[n, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(244\) vs. \(2(88)=176\).
Time = 0.46 (sec) , antiderivative size = 245, normalized size of antiderivative = 2.31
| method | result | size |
| risch | \(\frac {x a \sqrt {\frac {a x +b}{x}}}{c}-\frac {\left (\frac {\sqrt {a}\, \left (2 a d -3 b c \right ) \ln \left (\frac {\frac {b}{2}+a x}{\sqrt {a}}+\sqrt {a \,x^{2}+b x}\right )}{c}+\frac {2 \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \ln \left (\frac {\frac {2 \left (a d -b c \right ) d}{c^{2}}-\frac {\left (2 a d -b c \right ) \left (x +\frac {d}{c}\right )}{c}+2 \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}\, \sqrt {a \left (x +\frac {d}{c}\right )^{2}-\frac {\left (2 a d -b c \right ) \left (x +\frac {d}{c}\right )}{c}+\frac {\left (a d -b c \right ) d}{c^{2}}}}{x +\frac {d}{c}}\right )}{c^{2} \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}}\right ) \sqrt {\frac {a x +b}{x}}\, \sqrt {x \left (a x +b \right )}}{2 c \left (a x +b \right )}\) | \(245\) |
| default | \(\frac {\sqrt {\frac {a x +b}{x}}\, x \left (2 \sqrt {a \,x^{2}+b x}\, \sqrt {a}\, \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}\, b \,c^{3}+\ln \left (\frac {2 \sqrt {a \,x^{2}+b x}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}\, b^{2} c^{3}+2 a^{\frac {3}{2}} \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}\, \sqrt {x \left (a x +b \right )}\, c^{2} d -2 \sqrt {a}\, \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}\, \sqrt {x \left (a x +b \right )}\, b \,c^{3}-2 a^{\frac {5}{2}} \ln \left (\frac {2 \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}\, \sqrt {x \left (a x +b \right )}\, c -2 a d x +b c x -b d}{c x +d}\right ) d^{3}+4 a^{\frac {3}{2}} \ln \left (\frac {2 \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}\, \sqrt {x \left (a x +b \right )}\, c -2 a d x +b c x -b d}{c x +d}\right ) b c \,d^{2}-2 \sqrt {a}\, \ln \left (\frac {2 \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}\, \sqrt {x \left (a x +b \right )}\, c -2 a d x +b c x -b d}{c x +d}\right ) b^{2} c^{2} d -2 \ln \left (\frac {2 \sqrt {x \left (a x +b \right )}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}\, a^{2} c \,d^{2}+3 \ln \left (\frac {2 \sqrt {x \left (a x +b \right )}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}\, a b \,c^{2} d -\ln \left (\frac {2 \sqrt {x \left (a x +b \right )}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}\, b^{2} c^{3}\right )}{2 \sqrt {x \left (a x +b \right )}\, d \,c^{3} \sqrt {a}\, \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}}\) | \(528\) |
Input:
int((a+b/x)^(3/2)/(c+1/x*d),x,method=_RETURNVERBOSE)
Output:
1/c*x*a*((a*x+b)/x)^(1/2)-1/2/c*(a^(1/2)*(2*a*d-3*b*c)/c*ln((1/2*b+a*x)/a^ (1/2)+(a*x^2+b*x)^(1/2))+2*(a^2*d^2-2*a*b*c*d+b^2*c^2)/c^2/((a*d-b*c)*d/c^ 2)^(1/2)*ln((2*(a*d-b*c)*d/c^2-(2*a*d-b*c)/c*(x+1/c*d)+2*((a*d-b*c)*d/c^2) ^(1/2)*(a*(x+1/c*d)^2-(2*a*d-b*c)/c*(x+1/c*d)+(a*d-b*c)*d/c^2)^(1/2))/(x+1 /c*d)))*((a*x+b)/x)^(1/2)*(x*(a*x+b))^(1/2)/(a*x+b)
Time = 0.18 (sec) , antiderivative size = 529, normalized size of antiderivative = 4.99 \[ \int \frac {\left (a+\frac {b}{x}\right )^{3/2}}{c+\frac {d}{x}} \, dx=\left [\frac {2 \, a c x \sqrt {\frac {a x + b}{x}} - {\left (3 \, b c - 2 \, a d\right )} \sqrt {a} \log \left (2 \, a x - 2 \, \sqrt {a} x \sqrt {\frac {a x + b}{x}} + b\right ) - 2 \, {\left (b c - a d\right )} \sqrt {-\frac {b c - a d}{d}} \log \left (\frac {2 \, d x \sqrt {-\frac {b c - a d}{d}} \sqrt {\frac {a x + b}{x}} + b d - {\left (b c - 2 \, a d\right )} x}{c x + d}\right )}{2 \, c^{2}}, \frac {a c x \sqrt {\frac {a x + b}{x}} - {\left (3 \, b c - 2 \, a d\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {-a} x \sqrt {\frac {a x + b}{x}}}{a x + b}\right ) - {\left (b c - a d\right )} \sqrt {-\frac {b c - a d}{d}} \log \left (\frac {2 \, d x \sqrt {-\frac {b c - a d}{d}} \sqrt {\frac {a x + b}{x}} + b d - {\left (b c - 2 \, a d\right )} x}{c x + d}\right )}{c^{2}}, \frac {2 \, a c x \sqrt {\frac {a x + b}{x}} + 4 \, {\left (b c - a d\right )} \sqrt {\frac {b c - a d}{d}} \arctan \left (-\frac {d \sqrt {\frac {b c - a d}{d}} \sqrt {\frac {a x + b}{x}}}{b c - a d}\right ) - {\left (3 \, b c - 2 \, a d\right )} \sqrt {a} \log \left (2 \, a x - 2 \, \sqrt {a} x \sqrt {\frac {a x + b}{x}} + b\right )}{2 \, c^{2}}, \frac {a c x \sqrt {\frac {a x + b}{x}} - {\left (3 \, b c - 2 \, a d\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {-a} x \sqrt {\frac {a x + b}{x}}}{a x + b}\right ) + 2 \, {\left (b c - a d\right )} \sqrt {\frac {b c - a d}{d}} \arctan \left (-\frac {d \sqrt {\frac {b c - a d}{d}} \sqrt {\frac {a x + b}{x}}}{b c - a d}\right )}{c^{2}}\right ] \] Input:
integrate((a+b/x)^(3/2)/(c+d/x),x, algorithm="fricas")
Output:
[1/2*(2*a*c*x*sqrt((a*x + b)/x) - (3*b*c - 2*a*d)*sqrt(a)*log(2*a*x - 2*sq rt(a)*x*sqrt((a*x + b)/x) + b) - 2*(b*c - a*d)*sqrt(-(b*c - a*d)/d)*log((2 *d*x*sqrt(-(b*c - a*d)/d)*sqrt((a*x + b)/x) + b*d - (b*c - 2*a*d)*x)/(c*x + d)))/c^2, (a*c*x*sqrt((a*x + b)/x) - (3*b*c - 2*a*d)*sqrt(-a)*arctan(sqr t(-a)*x*sqrt((a*x + b)/x)/(a*x + b)) - (b*c - a*d)*sqrt(-(b*c - a*d)/d)*lo g((2*d*x*sqrt(-(b*c - a*d)/d)*sqrt((a*x + b)/x) + b*d - (b*c - 2*a*d)*x)/( c*x + d)))/c^2, 1/2*(2*a*c*x*sqrt((a*x + b)/x) + 4*(b*c - a*d)*sqrt((b*c - a*d)/d)*arctan(-d*sqrt((b*c - a*d)/d)*sqrt((a*x + b)/x)/(b*c - a*d)) - (3 *b*c - 2*a*d)*sqrt(a)*log(2*a*x - 2*sqrt(a)*x*sqrt((a*x + b)/x) + b))/c^2, (a*c*x*sqrt((a*x + b)/x) - (3*b*c - 2*a*d)*sqrt(-a)*arctan(sqrt(-a)*x*sqr t((a*x + b)/x)/(a*x + b)) + 2*(b*c - a*d)*sqrt((b*c - a*d)/d)*arctan(-d*sq rt((b*c - a*d)/d)*sqrt((a*x + b)/x)/(b*c - a*d)))/c^2]
\[ \int \frac {\left (a+\frac {b}{x}\right )^{3/2}}{c+\frac {d}{x}} \, dx=\int \frac {x \left (a + \frac {b}{x}\right )^{\frac {3}{2}}}{c x + d}\, dx \] Input:
integrate((a+b/x)**(3/2)/(c+d/x),x)
Output:
Integral(x*(a + b/x)**(3/2)/(c*x + d), x)
\[ \int \frac {\left (a+\frac {b}{x}\right )^{3/2}}{c+\frac {d}{x}} \, dx=\int { \frac {{\left (a + \frac {b}{x}\right )}^{\frac {3}{2}}}{c + \frac {d}{x}} \,d x } \] Input:
integrate((a+b/x)^(3/2)/(c+d/x),x, algorithm="maxima")
Output:
integrate((a + b/x)^(3/2)/(c + d/x), x)
Exception generated. \[ \int \frac {\left (a+\frac {b}{x}\right )^{3/2}}{c+\frac {d}{x}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((a+b/x)^(3/2)/(c+d/x),x, algorithm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Error: Bad Argument Type
Time = 1.00 (sec) , antiderivative size = 556, normalized size of antiderivative = 5.25 \[ \int \frac {\left (a+\frac {b}{x}\right )^{3/2}}{c+\frac {d}{x}} \, dx=\frac {a\,x\,\sqrt {a+\frac {b}{x}}}{c}-\frac {\sqrt {a}\,\mathrm {atanh}\left (\frac {58\,a^{3/2}\,b^6\,d^2\,\sqrt {a+\frac {b}{x}}}{58\,a^2\,b^6\,d^2-24\,a\,b^7\,c\,d-\frac {46\,a^3\,b^5\,d^3}{c}+\frac {12\,a^4\,b^4\,d^4}{c^2}}+\frac {46\,a^{5/2}\,b^5\,d^3\,\sqrt {a+\frac {b}{x}}}{46\,a^3\,b^5\,d^3-58\,a^2\,b^6\,c\,d^2-\frac {12\,a^4\,b^4\,d^4}{c}+24\,a\,b^7\,c^2\,d}+\frac {12\,a^{7/2}\,b^4\,d^4\,\sqrt {a+\frac {b}{x}}}{12\,a^4\,b^4\,d^4-46\,a^3\,b^5\,c\,d^3+58\,a^2\,b^6\,c^2\,d^2-24\,a\,b^7\,c^3\,d}-\frac {24\,\sqrt {a}\,b^7\,c\,d\,\sqrt {a+\frac {b}{x}}}{58\,a^2\,b^6\,d^2-24\,a\,b^7\,c\,d-\frac {46\,a^3\,b^5\,d^3}{c}+\frac {12\,a^4\,b^4\,d^4}{c^2}}\right )\,\left (2\,a\,d-3\,b\,c\right )}{c^2}+\frac {2\,\mathrm {atanh}\left (\frac {12\,a^2\,b^4\,d^2\,\sqrt {a+\frac {b}{x}}\,\sqrt {a^3\,d^4-3\,a^2\,b\,c\,d^3+3\,a\,b^2\,c^2\,d^2-b^3\,c^3\,d}}{12\,a^4\,b^4\,d^4-40\,a^3\,b^5\,c\,d^3+44\,a^2\,b^6\,c^2\,d^2-16\,a\,b^7\,c^3\,d}+\frac {16\,a\,b^5\,d\,\sqrt {a+\frac {b}{x}}\,\sqrt {a^3\,d^4-3\,a^2\,b\,c\,d^3+3\,a\,b^2\,c^2\,d^2-b^3\,c^3\,d}}{40\,a^3\,b^5\,d^3-44\,a^2\,b^6\,c\,d^2-\frac {12\,a^4\,b^4\,d^4}{c}+16\,a\,b^7\,c^2\,d}\right )\,\sqrt {d\,{\left (a\,d-b\,c\right )}^3}}{c^2\,d} \] Input:
int((a + b/x)^(3/2)/(c + d/x),x)
Output:
(a*x*(a + b/x)^(1/2))/c - (a^(1/2)*atanh((58*a^(3/2)*b^6*d^2*(a + b/x)^(1/ 2))/(58*a^2*b^6*d^2 - 24*a*b^7*c*d - (46*a^3*b^5*d^3)/c + (12*a^4*b^4*d^4) /c^2) + (46*a^(5/2)*b^5*d^3*(a + b/x)^(1/2))/(46*a^3*b^5*d^3 - 58*a^2*b^6* c*d^2 - (12*a^4*b^4*d^4)/c + 24*a*b^7*c^2*d) + (12*a^(7/2)*b^4*d^4*(a + b/ x)^(1/2))/(12*a^4*b^4*d^4 - 46*a^3*b^5*c*d^3 + 58*a^2*b^6*c^2*d^2 - 24*a*b ^7*c^3*d) - (24*a^(1/2)*b^7*c*d*(a + b/x)^(1/2))/(58*a^2*b^6*d^2 - 24*a*b^ 7*c*d - (46*a^3*b^5*d^3)/c + (12*a^4*b^4*d^4)/c^2))*(2*a*d - 3*b*c))/c^2 + (2*atanh((12*a^2*b^4*d^2*(a + b/x)^(1/2)*(a^3*d^4 - b^3*c^3*d + 3*a*b^2*c ^2*d^2 - 3*a^2*b*c*d^3)^(1/2))/(12*a^4*b^4*d^4 - 40*a^3*b^5*c*d^3 + 44*a^2 *b^6*c^2*d^2 - 16*a*b^7*c^3*d) + (16*a*b^5*d*(a + b/x)^(1/2)*(a^3*d^4 - b^ 3*c^3*d + 3*a*b^2*c^2*d^2 - 3*a^2*b*c*d^3)^(1/2))/(40*a^3*b^5*d^3 - 44*a^2 *b^6*c*d^2 - (12*a^4*b^4*d^4)/c + 16*a*b^7*c^2*d))*(d*(a*d - b*c)^3)^(1/2) )/(c^2*d)
Time = 0.27 (sec) , antiderivative size = 408, normalized size of antiderivative = 3.85 \[ \int \frac {\left (a+\frac {b}{x}\right )^{3/2}}{c+\frac {d}{x}} \, dx=\frac {\sqrt {d}\, \sqrt {a d -b c}\, \mathrm {log}\left (\sqrt {c}\, \sqrt {a x +b}-\sqrt {2 \sqrt {d}\, \sqrt {a}\, \sqrt {a d -b c}-2 a d +b c}+\sqrt {x}\, \sqrt {c}\, \sqrt {a}\right ) a d -\sqrt {d}\, \sqrt {a d -b c}\, \mathrm {log}\left (\sqrt {c}\, \sqrt {a x +b}-\sqrt {2 \sqrt {d}\, \sqrt {a}\, \sqrt {a d -b c}-2 a d +b c}+\sqrt {x}\, \sqrt {c}\, \sqrt {a}\right ) b c +\sqrt {d}\, \sqrt {a d -b c}\, \mathrm {log}\left (\sqrt {c}\, \sqrt {a x +b}+\sqrt {2 \sqrt {d}\, \sqrt {a}\, \sqrt {a d -b c}-2 a d +b c}+\sqrt {x}\, \sqrt {c}\, \sqrt {a}\right ) a d -\sqrt {d}\, \sqrt {a d -b c}\, \mathrm {log}\left (\sqrt {c}\, \sqrt {a x +b}+\sqrt {2 \sqrt {d}\, \sqrt {a}\, \sqrt {a d -b c}-2 a d +b c}+\sqrt {x}\, \sqrt {c}\, \sqrt {a}\right ) b c -\sqrt {d}\, \sqrt {a d -b c}\, \mathrm {log}\left (2 \sqrt {d}\, \sqrt {a}\, \sqrt {a d -b c}+2 \sqrt {x}\, \sqrt {a}\, \sqrt {a x +b}\, c +2 a c x +2 a d \right ) a d +\sqrt {d}\, \sqrt {a d -b c}\, \mathrm {log}\left (2 \sqrt {d}\, \sqrt {a}\, \sqrt {a d -b c}+2 \sqrt {x}\, \sqrt {a}\, \sqrt {a x +b}\, c +2 a c x +2 a d \right ) b c +\sqrt {x}\, \sqrt {a x +b}\, a c d -2 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {a x +b}+\sqrt {x}\, \sqrt {a}}{\sqrt {b}}\right ) a \,d^{2}+3 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {a x +b}+\sqrt {x}\, \sqrt {a}}{\sqrt {b}}\right ) b c d}{c^{2} d} \] Input:
int((a+b/x)^(3/2)/(c+d/x),x)
Output:
(sqrt(d)*sqrt(a*d - b*c)*log(sqrt(c)*sqrt(a*x + b) - sqrt(2*sqrt(d)*sqrt(a )*sqrt(a*d - b*c) - 2*a*d + b*c) + sqrt(x)*sqrt(c)*sqrt(a))*a*d - sqrt(d)* sqrt(a*d - b*c)*log(sqrt(c)*sqrt(a*x + b) - sqrt(2*sqrt(d)*sqrt(a)*sqrt(a* d - b*c) - 2*a*d + b*c) + sqrt(x)*sqrt(c)*sqrt(a))*b*c + sqrt(d)*sqrt(a*d - b*c)*log(sqrt(c)*sqrt(a*x + b) + sqrt(2*sqrt(d)*sqrt(a)*sqrt(a*d - b*c) - 2*a*d + b*c) + sqrt(x)*sqrt(c)*sqrt(a))*a*d - sqrt(d)*sqrt(a*d - b*c)*lo g(sqrt(c)*sqrt(a*x + b) + sqrt(2*sqrt(d)*sqrt(a)*sqrt(a*d - b*c) - 2*a*d + b*c) + sqrt(x)*sqrt(c)*sqrt(a))*b*c - sqrt(d)*sqrt(a*d - b*c)*log(2*sqrt( d)*sqrt(a)*sqrt(a*d - b*c) + 2*sqrt(x)*sqrt(a)*sqrt(a*x + b)*c + 2*a*c*x + 2*a*d)*a*d + sqrt(d)*sqrt(a*d - b*c)*log(2*sqrt(d)*sqrt(a)*sqrt(a*d - b*c ) + 2*sqrt(x)*sqrt(a)*sqrt(a*x + b)*c + 2*a*c*x + 2*a*d)*b*c + sqrt(x)*sqr t(a*x + b)*a*c*d - 2*sqrt(a)*log((sqrt(a*x + b) + sqrt(x)*sqrt(a))/sqrt(b) )*a*d**2 + 3*sqrt(a)*log((sqrt(a*x + b) + sqrt(x)*sqrt(a))/sqrt(b))*b*c*d) /(c**2*d)