Integrand size = 19, antiderivative size = 111 \[ \int \left (a+\frac {b}{x}\right )^{5/2} \left (c+\frac {d}{x}\right ) \, dx=-2 a (2 b c+a d) \sqrt {a+\frac {b}{x}}-\frac {2}{3} (b c+a d) \left (a+\frac {b}{x}\right )^{3/2}-\frac {2}{5} d \left (a+\frac {b}{x}\right )^{5/2}+a^2 c \sqrt {a+\frac {b}{x}} x+a^{3/2} (5 b c+2 a d) \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right ) \] Output:
-2*a*(a*d+2*b*c)*(a+b/x)^(1/2)-2/3*(a*d+b*c)*(a+b/x)^(3/2)-2/5*d*(a+b/x)^( 5/2)+a^2*c*(a+b/x)^(1/2)*x+a^(3/2)*(2*a*d+5*b*c)*arctanh((a+b/x)^(1/2)/a^( 1/2))
Time = 0.31 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.85 \[ \int \left (a+\frac {b}{x}\right )^{5/2} \left (c+\frac {d}{x}\right ) \, dx=\frac {\sqrt {a+\frac {b}{x}} \left (-2 b^2 (3 d+5 c x)+a^2 x^2 (-46 d+15 c x)-2 a b x (11 d+35 c x)\right )}{15 x^2}+a^{3/2} (5 b c+2 a d) \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right ) \] Input:
Integrate[(a + b/x)^(5/2)*(c + d/x),x]
Output:
(Sqrt[a + b/x]*(-2*b^2*(3*d + 5*c*x) + a^2*x^2*(-46*d + 15*c*x) - 2*a*b*x* (11*d + 35*c*x)))/(15*x^2) + a^(3/2)*(5*b*c + 2*a*d)*ArcTanh[Sqrt[a + b/x] /Sqrt[a]]
Time = 0.37 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.98, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {899, 87, 60, 60, 60, 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.
| \(\displaystyle \int \left (a+\frac {b}{x}\right )^{5/2} \left (c+\frac {d}{x}\right ) \, dx\) |
| \(\Big \downarrow \) 899 |
| \(\displaystyle -\int \left (a+\frac {b}{x}\right )^{5/2} \left (c+\frac {d}{x}\right ) x^2d\frac {1}{x}\) |
| \(\Big \downarrow \) 87 |
| \(\displaystyle \frac {c x \left (a+\frac {b}{x}\right )^{7/2}}{a}-\frac {(2 a d+5 b c) \int \left (a+\frac {b}{x}\right )^{5/2} xd\frac {1}{x}}{2 a}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {c x \left (a+\frac {b}{x}\right )^{7/2}}{a}-\frac {(2 a d+5 b c) \left (a \int \left (a+\frac {b}{x}\right )^{3/2} xd\frac {1}{x}+\frac {2}{5} \left (a+\frac {b}{x}\right )^{5/2}\right )}{2 a}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {c x \left (a+\frac {b}{x}\right )^{7/2}}{a}-\frac {(2 a d+5 b c) \left (a \left (a \int \sqrt {a+\frac {b}{x}} xd\frac {1}{x}+\frac {2}{3} \left (a+\frac {b}{x}\right )^{3/2}\right )+\frac {2}{5} \left (a+\frac {b}{x}\right )^{5/2}\right )}{2 a}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {c x \left (a+\frac {b}{x}\right )^{7/2}}{a}-\frac {(2 a d+5 b c) \left (a \left (a \left (a \int \frac {x}{\sqrt {a+\frac {b}{x}}}d\frac {1}{x}+2 \sqrt {a+\frac {b}{x}}\right )+\frac {2}{3} \left (a+\frac {b}{x}\right )^{3/2}\right )+\frac {2}{5} \left (a+\frac {b}{x}\right )^{5/2}\right )}{2 a}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {c x \left (a+\frac {b}{x}\right )^{7/2}}{a}-\frac {(2 a d+5 b c) \left (a \left (a \left (\frac {2 a \int \frac {1}{\frac {1}{b x^2}-\frac {a}{b}}d\sqrt {a+\frac {b}{x}}}{b}+2 \sqrt {a+\frac {b}{x}}\right )+\frac {2}{3} \left (a+\frac {b}{x}\right )^{3/2}\right )+\frac {2}{5} \left (a+\frac {b}{x}\right )^{5/2}\right )}{2 a}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {c x \left (a+\frac {b}{x}\right )^{7/2}}{a}-\frac {\left (a \left (a \left (2 \sqrt {a+\frac {b}{x}}-2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )\right )+\frac {2}{3} \left (a+\frac {b}{x}\right )^{3/2}\right )+\frac {2}{5} \left (a+\frac {b}{x}\right )^{5/2}\right ) (2 a d+5 b c)}{2 a}\) |
Input:
Int[(a + b/x)^(5/2)*(c + d/x),x]
Output:
(c*(a + b/x)^(7/2)*x)/a - ((5*b*c + 2*a*d)*((2*(a + b/x)^(5/2))/5 + a*((2* (a + b/x)^(3/2))/3 + a*(2*Sqrt[a + b/x] - 2*Sqrt[a]*ArcTanh[Sqrt[a + b/x]/ Sqrt[a]]))))/(2*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_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)) Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || Intege rQ[p] || !(IntegerQ[n] || !(EqQ[e, 0] || !(EqQ[c, 0] || LtQ[p, n]))))
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]
Time = 0.19 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.16
| method | result | size |
| risch | \(\frac {\left (15 a^{2} c \,x^{3}-46 a^{2} d \,x^{2}-70 a b c \,x^{2}-22 a x d b -10 b^{2} c x -6 d \,b^{2}\right ) \sqrt {\frac {a x +b}{x}}}{15 x^{2}}+\frac {\left (2 a d +5 b c \right ) a^{\frac {3}{2}} \ln \left (\frac {\frac {b}{2}+a x}{\sqrt {a}}+\sqrt {a \,x^{2}+b x}\right ) \sqrt {\frac {a x +b}{x}}\, \sqrt {x \left (a x +b \right )}}{2 a x +2 b}\) | \(129\) |
| default | \(-\frac {\sqrt {\frac {a x +b}{x}}\, \left (-60 \sqrt {a \,x^{2}+b x}\, a^{\frac {7}{2}} d \,x^{4}-150 \sqrt {a \,x^{2}+b x}\, a^{\frac {5}{2}} b c \,x^{4}-30 \ln \left (\frac {2 \sqrt {a \,x^{2}+b x}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a^{3} b d \,x^{4}-75 \ln \left (\frac {2 \sqrt {a \,x^{2}+b x}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a^{2} b^{2} c \,x^{4}+60 \left (a \,x^{2}+b x \right )^{\frac {3}{2}} a^{\frac {5}{2}} d \,x^{2}+120 \left (a \,x^{2}+b x \right )^{\frac {3}{2}} a^{\frac {3}{2}} b c \,x^{2}+32 \left (a \,x^{2}+b x \right )^{\frac {3}{2}} a^{\frac {3}{2}} b d x +20 \left (a \,x^{2}+b x \right )^{\frac {3}{2}} \sqrt {a}\, b^{2} c x +12 \left (a \,x^{2}+b x \right )^{\frac {3}{2}} \sqrt {a}\, b^{2} d \right )}{30 x^{3} b \sqrt {x \left (a x +b \right )}\, \sqrt {a}}\) | \(253\) |
Input:
int((a+b/x)^(5/2)*(c+1/x*d),x,method=_RETURNVERBOSE)
Output:
1/15*(15*a^2*c*x^3-46*a^2*d*x^2-70*a*b*c*x^2-22*a*b*d*x-10*b^2*c*x-6*b^2*d )/x^2*((a*x+b)/x)^(1/2)+1/2*(2*a*d+5*b*c)*a^(3/2)*ln((1/2*b+a*x)/a^(1/2)+( a*x^2+b*x)^(1/2))*((a*x+b)/x)^(1/2)*(x*(a*x+b))^(1/2)/(a*x+b)
Time = 0.12 (sec) , antiderivative size = 227, normalized size of antiderivative = 2.05 \[ \int \left (a+\frac {b}{x}\right )^{5/2} \left (c+\frac {d}{x}\right ) \, dx=\left [\frac {15 \, {\left (5 \, a b c + 2 \, a^{2} d\right )} \sqrt {a} x^{2} \log \left (2 \, a x + 2 \, \sqrt {a} x \sqrt {\frac {a x + b}{x}} + b\right ) + 2 \, {\left (15 \, a^{2} c x^{3} - 6 \, b^{2} d - 2 \, {\left (35 \, a b c + 23 \, a^{2} d\right )} x^{2} - 2 \, {\left (5 \, b^{2} c + 11 \, a b d\right )} x\right )} \sqrt {\frac {a x + b}{x}}}{30 \, x^{2}}, -\frac {15 \, {\left (5 \, a b c + 2 \, a^{2} d\right )} \sqrt {-a} x^{2} \arctan \left (\frac {\sqrt {-a} x \sqrt {\frac {a x + b}{x}}}{a x + b}\right ) - {\left (15 \, a^{2} c x^{3} - 6 \, b^{2} d - 2 \, {\left (35 \, a b c + 23 \, a^{2} d\right )} x^{2} - 2 \, {\left (5 \, b^{2} c + 11 \, a b d\right )} x\right )} \sqrt {\frac {a x + b}{x}}}{15 \, x^{2}}\right ] \] Input:
integrate((a+b/x)^(5/2)*(c+d/x),x, algorithm="fricas")
Output:
[1/30*(15*(5*a*b*c + 2*a^2*d)*sqrt(a)*x^2*log(2*a*x + 2*sqrt(a)*x*sqrt((a* x + b)/x) + b) + 2*(15*a^2*c*x^3 - 6*b^2*d - 2*(35*a*b*c + 23*a^2*d)*x^2 - 2*(5*b^2*c + 11*a*b*d)*x)*sqrt((a*x + b)/x))/x^2, -1/15*(15*(5*a*b*c + 2* a^2*d)*sqrt(-a)*x^2*arctan(sqrt(-a)*x*sqrt((a*x + b)/x)/(a*x + b)) - (15*a ^2*c*x^3 - 6*b^2*d - 2*(35*a*b*c + 23*a^2*d)*x^2 - 2*(5*b^2*c + 11*a*b*d)* x)*sqrt((a*x + b)/x))/x^2]
Time = 21.31 (sec) , antiderivative size = 534, normalized size of antiderivative = 4.81 \[ \int \left (a+\frac {b}{x}\right )^{5/2} \left (c+\frac {d}{x}\right ) \, dx=\frac {4 a^{\frac {11}{2}} b^{\frac {7}{2}} d x^{3} \sqrt {\frac {a x}{b} + 1}}{15 a^{\frac {7}{2}} b^{3} x^{\frac {7}{2}} + 15 a^{\frac {5}{2}} b^{4} x^{\frac {5}{2}}} + \frac {2 a^{\frac {9}{2}} b^{\frac {9}{2}} d x^{2} \sqrt {\frac {a x}{b} + 1}}{15 a^{\frac {7}{2}} b^{3} x^{\frac {7}{2}} + 15 a^{\frac {5}{2}} b^{4} x^{\frac {5}{2}}} - \frac {8 a^{\frac {7}{2}} b^{\frac {11}{2}} d x \sqrt {\frac {a x}{b} + 1}}{15 a^{\frac {7}{2}} b^{3} x^{\frac {7}{2}} + 15 a^{\frac {5}{2}} b^{4} x^{\frac {5}{2}}} - \frac {6 a^{\frac {5}{2}} b^{\frac {13}{2}} d \sqrt {\frac {a x}{b} + 1}}{15 a^{\frac {7}{2}} b^{3} x^{\frac {7}{2}} + 15 a^{\frac {5}{2}} b^{4} x^{\frac {5}{2}}} + a^{\frac {3}{2}} b c \operatorname {asinh}{\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}} \right )} - \frac {4 a^{6} b^{3} d x^{\frac {7}{2}}}{15 a^{\frac {7}{2}} b^{3} x^{\frac {7}{2}} + 15 a^{\frac {5}{2}} b^{4} x^{\frac {5}{2}}} - \frac {4 a^{5} b^{4} d x^{\frac {5}{2}}}{15 a^{\frac {7}{2}} b^{3} x^{\frac {7}{2}} + 15 a^{\frac {5}{2}} b^{4} x^{\frac {5}{2}}} + a^{2} \sqrt {b} c \sqrt {x} \sqrt {\frac {a x}{b} + 1} - a^{2} d \left (\begin {cases} \frac {2 a \operatorname {atan}{\left (\frac {\sqrt {a + \frac {b}{x}}}{\sqrt {- a}} \right )}}{\sqrt {- a}} + 2 \sqrt {a + \frac {b}{x}} & \text {for}\: b \neq 0 \\- \sqrt {a} \log {\left (x \right )} & \text {otherwise} \end {cases}\right ) - 2 a b c \left (\begin {cases} \frac {2 a \operatorname {atan}{\left (\frac {\sqrt {a + \frac {b}{x}}}{\sqrt {- a}} \right )}}{\sqrt {- a}} + 2 \sqrt {a + \frac {b}{x}} & \text {for}\: b \neq 0 \\- \sqrt {a} \log {\left (x \right )} & \text {otherwise} \end {cases}\right ) + 2 a b d \left (\begin {cases} - \frac {\sqrt {a}}{x} & \text {for}\: b = 0 \\- \frac {2 \left (a + \frac {b}{x}\right )^{\frac {3}{2}}}{3 b} & \text {otherwise} \end {cases}\right ) + b^{2} c \left (\begin {cases} - \frac {\sqrt {a}}{x} & \text {for}\: b = 0 \\- \frac {2 \left (a + \frac {b}{x}\right )^{\frac {3}{2}}}{3 b} & \text {otherwise} \end {cases}\right ) \] Input:
integrate((a+b/x)**(5/2)*(c+d/x),x)
Output:
4*a**(11/2)*b**(7/2)*d*x**3*sqrt(a*x/b + 1)/(15*a**(7/2)*b**3*x**(7/2) + 1 5*a**(5/2)*b**4*x**(5/2)) + 2*a**(9/2)*b**(9/2)*d*x**2*sqrt(a*x/b + 1)/(15 *a**(7/2)*b**3*x**(7/2) + 15*a**(5/2)*b**4*x**(5/2)) - 8*a**(7/2)*b**(11/2 )*d*x*sqrt(a*x/b + 1)/(15*a**(7/2)*b**3*x**(7/2) + 15*a**(5/2)*b**4*x**(5/ 2)) - 6*a**(5/2)*b**(13/2)*d*sqrt(a*x/b + 1)/(15*a**(7/2)*b**3*x**(7/2) + 15*a**(5/2)*b**4*x**(5/2)) + a**(3/2)*b*c*asinh(sqrt(a)*sqrt(x)/sqrt(b)) - 4*a**6*b**3*d*x**(7/2)/(15*a**(7/2)*b**3*x**(7/2) + 15*a**(5/2)*b**4*x**( 5/2)) - 4*a**5*b**4*d*x**(5/2)/(15*a**(7/2)*b**3*x**(7/2) + 15*a**(5/2)*b* *4*x**(5/2)) + a**2*sqrt(b)*c*sqrt(x)*sqrt(a*x/b + 1) - a**2*d*Piecewise(( 2*a*atan(sqrt(a + b/x)/sqrt(-a))/sqrt(-a) + 2*sqrt(a + b/x), Ne(b, 0)), (- sqrt(a)*log(x), True)) - 2*a*b*c*Piecewise((2*a*atan(sqrt(a + b/x)/sqrt(-a ))/sqrt(-a) + 2*sqrt(a + b/x), Ne(b, 0)), (-sqrt(a)*log(x), True)) + 2*a*b *d*Piecewise((-sqrt(a)/x, Eq(b, 0)), (-2*(a + b/x)**(3/2)/(3*b), True)) + b**2*c*Piecewise((-sqrt(a)/x, Eq(b, 0)), (-2*(a + b/x)**(3/2)/(3*b), True) )
Time = 0.11 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.45 \[ \int \left (a+\frac {b}{x}\right )^{5/2} \left (c+\frac {d}{x}\right ) \, dx=\frac {1}{6} \, {\left (6 \, \sqrt {a + \frac {b}{x}} a^{2} x - 15 \, a^{\frac {3}{2}} b \log \left (\frac {\sqrt {a + \frac {b}{x}} - \sqrt {a}}{\sqrt {a + \frac {b}{x}} + \sqrt {a}}\right ) - 4 \, {\left (a + \frac {b}{x}\right )}^{\frac {3}{2}} b - 24 \, \sqrt {a + \frac {b}{x}} a b\right )} c - \frac {1}{15} \, {\left (15 \, a^{\frac {5}{2}} \log \left (\frac {\sqrt {a + \frac {b}{x}} - \sqrt {a}}{\sqrt {a + \frac {b}{x}} + \sqrt {a}}\right ) + 6 \, {\left (a + \frac {b}{x}\right )}^{\frac {5}{2}} + 10 \, {\left (a + \frac {b}{x}\right )}^{\frac {3}{2}} a + 30 \, \sqrt {a + \frac {b}{x}} a^{2}\right )} d \] Input:
integrate((a+b/x)^(5/2)*(c+d/x),x, algorithm="maxima")
Output:
1/6*(6*sqrt(a + b/x)*a^2*x - 15*a^(3/2)*b*log((sqrt(a + b/x) - sqrt(a))/(s qrt(a + b/x) + sqrt(a))) - 4*(a + b/x)^(3/2)*b - 24*sqrt(a + b/x)*a*b)*c - 1/15*(15*a^(5/2)*log((sqrt(a + b/x) - sqrt(a))/(sqrt(a + b/x) + sqrt(a))) + 6*(a + b/x)^(5/2) + 10*(a + b/x)^(3/2)*a + 30*sqrt(a + b/x)*a^2)*d
Exception generated. \[ \int \left (a+\frac {b}{x}\right )^{5/2} \left (c+\frac {d}{x}\right ) \, dx=\text {Exception raised: TypeError} \] Input:
integrate((a+b/x)^(5/2)*(c+d/x),x, algorithm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Limit: Max order reached or unable to make series expansion Error: Bad Argument Value
Time = 3.01 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.89 \[ \int \left (a+\frac {b}{x}\right )^{5/2} \left (c+\frac {d}{x}\right ) \, dx=-\frac {2\,d\,{\left (a+\frac {b}{x}\right )}^{5/2}}{5}-2\,a^2\,d\,\sqrt {a+\frac {b}{x}}-\frac {2\,a\,d\,{\left (a+\frac {b}{x}\right )}^{3/2}}{3}-\frac {2\,c\,x\,{\left (a+\frac {b}{x}\right )}^{5/2}\,{{}}_2{\mathrm {F}}_1\left (-\frac {5}{2},-\frac {3}{2};\ -\frac {1}{2};\ -\frac {a\,x}{b}\right )}{3\,{\left (\frac {a\,x}{b}+1\right )}^{5/2}}-a^{5/2}\,d\,\mathrm {atan}\left (\frac {\sqrt {a+\frac {b}{x}}\,1{}\mathrm {i}}{\sqrt {a}}\right )\,2{}\mathrm {i} \] Input:
int((a + b/x)^(5/2)*(c + d/x),x)
Output:
- (2*d*(a + b/x)^(5/2))/5 - 2*a^2*d*(a + b/x)^(1/2) - a^(5/2)*d*atan(((a + b/x)^(1/2)*1i)/a^(1/2))*2i - (2*a*d*(a + b/x)^(3/2))/3 - (2*c*x*(a + b/x) ^(5/2)*hypergeom([-5/2, -3/2], -1/2, -(a*x)/b))/(3*((a*x)/b + 1)^(5/2))
Time = 0.27 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.59 \[ \int \left (a+\frac {b}{x}\right )^{5/2} \left (c+\frac {d}{x}\right ) \, dx=\frac {60 \sqrt {x}\, \sqrt {a x +b}\, a^{2} c \,x^{3}-184 \sqrt {x}\, \sqrt {a x +b}\, a^{2} d \,x^{2}-280 \sqrt {x}\, \sqrt {a x +b}\, a b c \,x^{2}-88 \sqrt {x}\, \sqrt {a x +b}\, a b d x -40 \sqrt {x}\, \sqrt {a x +b}\, b^{2} c x -24 \sqrt {x}\, \sqrt {a x +b}\, b^{2} d +120 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {a x +b}+\sqrt {x}\, \sqrt {a}}{\sqrt {b}}\right ) a^{2} d \,x^{3}+300 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {a x +b}+\sqrt {x}\, \sqrt {a}}{\sqrt {b}}\right ) a b c \,x^{3}+40 \sqrt {a}\, a^{2} d \,x^{3}+163 \sqrt {a}\, a b c \,x^{3}}{60 x^{3}} \] Input:
int((a+b/x)^(5/2)*(c+d/x),x)
Output:
(60*sqrt(x)*sqrt(a*x + b)*a**2*c*x**3 - 184*sqrt(x)*sqrt(a*x + b)*a**2*d*x **2 - 280*sqrt(x)*sqrt(a*x + b)*a*b*c*x**2 - 88*sqrt(x)*sqrt(a*x + b)*a*b* d*x - 40*sqrt(x)*sqrt(a*x + b)*b**2*c*x - 24*sqrt(x)*sqrt(a*x + b)*b**2*d + 120*sqrt(a)*log((sqrt(a*x + b) + sqrt(x)*sqrt(a))/sqrt(b))*a**2*d*x**3 + 300*sqrt(a)*log((sqrt(a*x + b) + sqrt(x)*sqrt(a))/sqrt(b))*a*b*c*x**3 + 4 0*sqrt(a)*a**2*d*x**3 + 163*sqrt(a)*a*b*c*x**3)/(60*x**3)