Integrand size = 21, antiderivative size = 138 \[ \int \left (a+\frac {b}{x}\right )^{5/2} \left (c+\frac {d}{x}\right )^2 \, dx=-4 a c (b c+a d) \sqrt {a+\frac {b}{x}}-\frac {2}{3} c (b c+2 a d) \left (a+\frac {b}{x}\right )^{3/2}-\frac {4}{5} c d \left (a+\frac {b}{x}\right )^{5/2}-\frac {2 d^2 \left (a+\frac {b}{x}\right )^{7/2}}{7 b}+a^2 c^2 \sqrt {a+\frac {b}{x}} x+a^{3/2} c (5 b c+4 a d) \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right ) \] Output:
-4*a*c*(a*d+b*c)*(a+b/x)^(1/2)-2/3*c*(2*a*d+b*c)*(a+b/x)^(3/2)-4/5*c*d*(a+ b/x)^(5/2)-2/7*d^2*(a+b/x)^(7/2)/b+a^2*c^2*(a+b/x)^(1/2)*x+a^(3/2)*c*(4*a* d+5*b*c)*arctanh((a+b/x)^(1/2)/a^(1/2))
Time = 0.43 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.05 \[ \int \left (a+\frac {b}{x}\right )^{5/2} \left (c+\frac {d}{x}\right )^2 \, dx=\frac {\sqrt {a+\frac {b}{x}} \left (-30 a^3 d^2 x^3-2 b^3 \left (15 d^2+42 c d x+35 c^2 x^2\right )+a^2 b x^2 \left (-90 d^2-644 c d x+105 c^2 x^2\right )-2 a b^2 x \left (45 d^2+154 c d x+245 c^2 x^2\right )\right )}{105 b x^3}+a^{3/2} c (5 b c+4 a d) \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right ) \] Input:
Integrate[(a + b/x)^(5/2)*(c + d/x)^2,x]
Output:
(Sqrt[a + b/x]*(-30*a^3*d^2*x^3 - 2*b^3*(15*d^2 + 42*c*d*x + 35*c^2*x^2) + a^2*b*x^2*(-90*d^2 - 644*c*d*x + 105*c^2*x^2) - 2*a*b^2*x*(45*d^2 + 154*c *d*x + 245*c^2*x^2)))/(105*b*x^3) + a^(3/2)*c*(5*b*c + 4*a*d)*ArcTanh[Sqrt [a + b/x]/Sqrt[a]]
Time = 0.43 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.99, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {899, 100, 27, 90, 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.
| \(\displaystyle \int \left (a+\frac {b}{x}\right )^{5/2} \left (c+\frac {d}{x}\right )^2 \, dx\) |
| \(\Big \downarrow \) 899 |
| \(\displaystyle -\int \left (a+\frac {b}{x}\right )^{5/2} \left (c+\frac {d}{x}\right )^2 x^2d\frac {1}{x}\) |
| \(\Big \downarrow \) 100 |
| \(\displaystyle \frac {c^2 x \left (a+\frac {b}{x}\right )^{7/2}}{a}-\frac {\int \frac {1}{2} \left (a+\frac {b}{x}\right )^{5/2} \left (\frac {2 a d^2}{x}+c (5 b c+4 a d)\right ) xd\frac {1}{x}}{a}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {c^2 x \left (a+\frac {b}{x}\right )^{7/2}}{a}-\frac {\int \left (a+\frac {b}{x}\right )^{5/2} \left (\frac {2 a d^2}{x}+c (5 b c+4 a d)\right ) xd\frac {1}{x}}{2 a}\) |
| \(\Big \downarrow \) 90 |
| \(\displaystyle \frac {c^2 x \left (a+\frac {b}{x}\right )^{7/2}}{a}-\frac {c (4 a d+5 b c) \int \left (a+\frac {b}{x}\right )^{5/2} xd\frac {1}{x}+\frac {4 a d^2 \left (a+\frac {b}{x}\right )^{7/2}}{7 b}}{2 a}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {c^2 x \left (a+\frac {b}{x}\right )^{7/2}}{a}-\frac {c (4 a d+5 b c) \left (a \int \left (a+\frac {b}{x}\right )^{3/2} xd\frac {1}{x}+\frac {2}{5} \left (a+\frac {b}{x}\right )^{5/2}\right )+\frac {4 a d^2 \left (a+\frac {b}{x}\right )^{7/2}}{7 b}}{2 a}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {c^2 x \left (a+\frac {b}{x}\right )^{7/2}}{a}-\frac {c (4 a d+5 b c) \left (a \left (a \int \sqrt {a+\frac {b}{x}} xd\frac {1}{x}+\frac {2}{3} \left (a+\frac {b}{x}\right )^{3/2}\right )+\frac {2}{5} \left (a+\frac {b}{x}\right )^{5/2}\right )+\frac {4 a d^2 \left (a+\frac {b}{x}\right )^{7/2}}{7 b}}{2 a}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {c^2 x \left (a+\frac {b}{x}\right )^{7/2}}{a}-\frac {c (4 a d+5 b c) \left (a \left (a \left (a \int \frac {x}{\sqrt {a+\frac {b}{x}}}d\frac {1}{x}+2 \sqrt {a+\frac {b}{x}}\right )+\frac {2}{3} \left (a+\frac {b}{x}\right )^{3/2}\right )+\frac {2}{5} \left (a+\frac {b}{x}\right )^{5/2}\right )+\frac {4 a d^2 \left (a+\frac {b}{x}\right )^{7/2}}{7 b}}{2 a}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {c^2 x \left (a+\frac {b}{x}\right )^{7/2}}{a}-\frac {c (4 a d+5 b c) \left (a \left (a \left (\frac {2 a \int \frac {1}{\frac {1}{b x^2}-\frac {a}{b}}d\sqrt {a+\frac {b}{x}}}{b}+2 \sqrt {a+\frac {b}{x}}\right )+\frac {2}{3} \left (a+\frac {b}{x}\right )^{3/2}\right )+\frac {2}{5} \left (a+\frac {b}{x}\right )^{5/2}\right )+\frac {4 a d^2 \left (a+\frac {b}{x}\right )^{7/2}}{7 b}}{2 a}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {c^2 x \left (a+\frac {b}{x}\right )^{7/2}}{a}-\frac {c \left (a \left (a \left (2 \sqrt {a+\frac {b}{x}}-2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )\right )+\frac {2}{3} \left (a+\frac {b}{x}\right )^{3/2}\right )+\frac {2}{5} \left (a+\frac {b}{x}\right )^{5/2}\right ) (4 a d+5 b c)+\frac {4 a d^2 \left (a+\frac {b}{x}\right )^{7/2}}{7 b}}{2 a}\) |
Input:
Int[(a + b/x)^(5/2)*(c + d/x)^2,x]
Output:
(c^2*(a + b/x)^(7/2)*x)/a - ((4*a*d^2*(a + b/x)^(7/2))/(7*b) + c*(5*b*c + 4*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_)*(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] :> 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*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), x] + Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(d*f*(n + p + 2)) Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0]
Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^( p_), x_] :> Simp[(b*c - a*d)^2*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d^2*(d *e - c*f)*(n + 1))), x] - Simp[1/(d^2*(d*e - c*f)*(n + 1)) Int[(c + d*x)^ (n + 1)*(e + f*x)^p*Simp[a^2*d^2*f*(n + p + 2) + b^2*c*(d*e*(n + 1) + c*f*( p + 1)) - 2*a*b*d*(d*e*(n + 1) + c*f*(p + 1)) - b^2*d*(d*e - c*f)*(n + 1)*x , x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && (LtQ[n, -1] || (EqQ[n + p + 3, 0] && NeQ[n, -1] && (SumSimplerQ[n, 1] || !SumSimplerQ[p, 1])))
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 = 0.22 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.41
| method | result | size |
| risch | \(-\frac {\left (-105 a^{2} b \,c^{2} x^{4}+30 a^{3} d^{2} x^{3}+644 a^{2} b c d \,x^{3}+490 a \,b^{2} c^{2} x^{3}+90 x^{2} a^{2} b \,d^{2}+308 x^{2} a \,b^{2} c d +70 x^{2} b^{3} c^{2}+90 a \,b^{2} d^{2} x +84 b^{3} c d x +30 b^{3} d^{2}\right ) \sqrt {\frac {a x +b}{x}}}{105 x^{3} b}+\frac {\left (4 a d +5 b c \right ) a^{\frac {3}{2}} c \ln \left (\frac {\frac {b}{2}+a x}{\sqrt {a}}+\sqrt {a \,x^{2}+b x}\right ) \sqrt {\frac {a x +b}{x}}\, \sqrt {x \left (a x +b \right )}}{2 a x +2 b}\) | \(194\) |
| default | \(-\frac {\sqrt {\frac {a x +b}{x}}\, \left (-840 \sqrt {a \,x^{2}+b x}\, a^{\frac {7}{2}} c d \,x^{5}-1050 \sqrt {a \,x^{2}+b x}\, a^{\frac {5}{2}} b \,c^{2} x^{5}-420 \ln \left (\frac {2 \sqrt {a \,x^{2}+b x}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a^{3} b c d \,x^{5}-525 \ln \left (\frac {2 \sqrt {a \,x^{2}+b x}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a^{2} b^{2} c^{2} x^{5}+840 \left (a \,x^{2}+b x \right )^{\frac {3}{2}} a^{\frac {5}{2}} c d \,x^{3}+840 \left (a \,x^{2}+b x \right )^{\frac {3}{2}} a^{\frac {3}{2}} b \,c^{2} x^{3}+60 \left (a \,x^{2}+b x \right )^{\frac {3}{2}} a^{\frac {5}{2}} d^{2} x^{2}+448 \left (a \,x^{2}+b x \right )^{\frac {3}{2}} a^{\frac {3}{2}} b c d \,x^{2}+140 \left (a \,x^{2}+b x \right )^{\frac {3}{2}} \sqrt {a}\, b^{2} c^{2} x^{2}+120 \left (a \,x^{2}+b x \right )^{\frac {3}{2}} a^{\frac {3}{2}} b \,d^{2} x +168 \left (a \,x^{2}+b x \right )^{\frac {3}{2}} \sqrt {a}\, b^{2} c d x +60 \left (a \,x^{2}+b x \right )^{\frac {3}{2}} \sqrt {a}\, b^{2} d^{2}\right )}{210 x^{4} b \sqrt {x \left (a x +b \right )}\, \sqrt {a}}\) | \(336\) |
Input:
int((a+b/x)^(5/2)*(c+1/x*d)^2,x,method=_RETURNVERBOSE)
Output:
-1/105*(-105*a^2*b*c^2*x^4+30*a^3*d^2*x^3+644*a^2*b*c*d*x^3+490*a*b^2*c^2* x^3+90*a^2*b*d^2*x^2+308*a*b^2*c*d*x^2+70*b^3*c^2*x^2+90*a*b^2*d^2*x+84*b^ 3*c*d*x+30*b^3*d^2)/x^3/b*((a*x+b)/x)^(1/2)+1/2*(4*a*d+5*b*c)*a^(3/2)*c*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 = 0.13 (sec) , antiderivative size = 355, normalized size of antiderivative = 2.57 \[ \int \left (a+\frac {b}{x}\right )^{5/2} \left (c+\frac {d}{x}\right )^2 \, dx=\left [\frac {105 \, {\left (5 \, a b^{2} c^{2} + 4 \, a^{2} b c d\right )} \sqrt {a} x^{3} \log \left (2 \, a x + 2 \, \sqrt {a} x \sqrt {\frac {a x + b}{x}} + b\right ) + 2 \, {\left (105 \, a^{2} b c^{2} x^{4} - 30 \, b^{3} d^{2} - 2 \, {\left (245 \, a b^{2} c^{2} + 322 \, a^{2} b c d + 15 \, a^{3} d^{2}\right )} x^{3} - 2 \, {\left (35 \, b^{3} c^{2} + 154 \, a b^{2} c d + 45 \, a^{2} b d^{2}\right )} x^{2} - 6 \, {\left (14 \, b^{3} c d + 15 \, a b^{2} d^{2}\right )} x\right )} \sqrt {\frac {a x + b}{x}}}{210 \, b x^{3}}, -\frac {105 \, {\left (5 \, a b^{2} c^{2} + 4 \, a^{2} b c d\right )} \sqrt {-a} x^{3} \arctan \left (\frac {\sqrt {-a} x \sqrt {\frac {a x + b}{x}}}{a x + b}\right ) - {\left (105 \, a^{2} b c^{2} x^{4} - 30 \, b^{3} d^{2} - 2 \, {\left (245 \, a b^{2} c^{2} + 322 \, a^{2} b c d + 15 \, a^{3} d^{2}\right )} x^{3} - 2 \, {\left (35 \, b^{3} c^{2} + 154 \, a b^{2} c d + 45 \, a^{2} b d^{2}\right )} x^{2} - 6 \, {\left (14 \, b^{3} c d + 15 \, a b^{2} d^{2}\right )} x\right )} \sqrt {\frac {a x + b}{x}}}{105 \, b x^{3}}\right ] \] Input:
integrate((a+b/x)^(5/2)*(c+d/x)^2,x, algorithm="fricas")
Output:
[1/210*(105*(5*a*b^2*c^2 + 4*a^2*b*c*d)*sqrt(a)*x^3*log(2*a*x + 2*sqrt(a)* x*sqrt((a*x + b)/x) + b) + 2*(105*a^2*b*c^2*x^4 - 30*b^3*d^2 - 2*(245*a*b^ 2*c^2 + 322*a^2*b*c*d + 15*a^3*d^2)*x^3 - 2*(35*b^3*c^2 + 154*a*b^2*c*d + 45*a^2*b*d^2)*x^2 - 6*(14*b^3*c*d + 15*a*b^2*d^2)*x)*sqrt((a*x + b)/x))/(b *x^3), -1/105*(105*(5*a*b^2*c^2 + 4*a^2*b*c*d)*sqrt(-a)*x^3*arctan(sqrt(-a )*x*sqrt((a*x + b)/x)/(a*x + b)) - (105*a^2*b*c^2*x^4 - 30*b^3*d^2 - 2*(24 5*a*b^2*c^2 + 322*a^2*b*c*d + 15*a^3*d^2)*x^3 - 2*(35*b^3*c^2 + 154*a*b^2* c*d + 45*a^2*b*d^2)*x^2 - 6*(14*b^3*c*d + 15*a*b^2*d^2)*x)*sqrt((a*x + b)/ x))/(b*x^3)]
Time = 31.88 (sec) , antiderivative size = 1853, normalized size of antiderivative = 13.43 \[ \int \left (a+\frac {b}{x}\right )^{5/2} \left (c+\frac {d}{x}\right )^2 \, dx=\text {Too large to display} \] Input:
integrate((a+b/x)**(5/2)*(c+d/x)**2,x)
Output:
-16*a**(19/2)*b**(13/2)*d**2*x**6*sqrt(a*x/b + 1)/(105*a**(13/2)*b**7*x**( 13/2) + 315*a**(11/2)*b**8*x**(11/2) + 315*a**(9/2)*b**9*x**(9/2) + 105*a* *(7/2)*b**10*x**(7/2)) - 40*a**(17/2)*b**(15/2)*d**2*x**5*sqrt(a*x/b + 1)/ (105*a**(13/2)*b**7*x**(13/2) + 315*a**(11/2)*b**8*x**(11/2) + 315*a**(9/2 )*b**9*x**(9/2) + 105*a**(7/2)*b**10*x**(7/2)) - 30*a**(15/2)*b**(17/2)*d* *2*x**4*sqrt(a*x/b + 1)/(105*a**(13/2)*b**7*x**(13/2) + 315*a**(11/2)*b**8 *x**(11/2) + 315*a**(9/2)*b**9*x**(9/2) + 105*a**(7/2)*b**10*x**(7/2)) - 4 0*a**(13/2)*b**(19/2)*d**2*x**3*sqrt(a*x/b + 1)/(105*a**(13/2)*b**7*x**(13 /2) + 315*a**(11/2)*b**8*x**(11/2) + 315*a**(9/2)*b**9*x**(9/2) + 105*a**( 7/2)*b**10*x**(7/2)) + 8*a**(13/2)*b**(5/2)*d**2*x**3*sqrt(a*x/b + 1)/(15* a**(7/2)*b**3*x**(7/2) + 15*a**(5/2)*b**4*x**(5/2)) - 100*a**(11/2)*b**(21 /2)*d**2*x**2*sqrt(a*x/b + 1)/(105*a**(13/2)*b**7*x**(13/2) + 315*a**(11/2 )*b**8*x**(11/2) + 315*a**(9/2)*b**9*x**(9/2) + 105*a**(7/2)*b**10*x**(7/2 )) + 8*a**(11/2)*b**(7/2)*c*d*x**3*sqrt(a*x/b + 1)/(15*a**(7/2)*b**3*x**(7 /2) + 15*a**(5/2)*b**4*x**(5/2)) + 4*a**(11/2)*b**(7/2)*d**2*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)) - 96*a**(9 /2)*b**(23/2)*d**2*x*sqrt(a*x/b + 1)/(105*a**(13/2)*b**7*x**(13/2) + 315*a **(11/2)*b**8*x**(11/2) + 315*a**(9/2)*b**9*x**(9/2) + 105*a**(7/2)*b**10* x**(7/2)) + 4*a**(9/2)*b**(9/2)*c*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)) - 16*a**(9/2)*b**(9/2)*d**2*x*sq...
Time = 0.11 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.31 \[ \int \left (a+\frac {b}{x}\right )^{5/2} \left (c+\frac {d}{x}\right )^2 \, dx=-\frac {2 \, {\left (a + \frac {b}{x}\right )}^{\frac {7}{2}} d^{2}}{7 \, b} + \frac {1}{6} \, {\left (6 \, \sqrt {a + \frac {b}{x}} a^{2} x - 15 \, a^{\frac {3}{2}} b \log \left (\frac {\sqrt {a + \frac {b}{x}} - \sqrt {a}}{\sqrt {a + \frac {b}{x}} + \sqrt {a}}\right ) - 4 \, {\left (a + \frac {b}{x}\right )}^{\frac {3}{2}} b - 24 \, \sqrt {a + \frac {b}{x}} a b\right )} c^{2} - \frac {2}{15} \, {\left (15 \, a^{\frac {5}{2}} \log \left (\frac {\sqrt {a + \frac {b}{x}} - \sqrt {a}}{\sqrt {a + \frac {b}{x}} + \sqrt {a}}\right ) + 6 \, {\left (a + \frac {b}{x}\right )}^{\frac {5}{2}} + 10 \, {\left (a + \frac {b}{x}\right )}^{\frac {3}{2}} a + 30 \, \sqrt {a + \frac {b}{x}} a^{2}\right )} c d \] Input:
integrate((a+b/x)^(5/2)*(c+d/x)^2,x, algorithm="maxima")
Output:
-2/7*(a + b/x)^(7/2)*d^2/b + 1/6*(6*sqrt(a + b/x)*a^2*x - 15*a^(3/2)*b*log ((sqrt(a + b/x) - sqrt(a))/(sqrt(a + b/x) + sqrt(a))) - 4*(a + b/x)^(3/2)* b - 24*sqrt(a + b/x)*a*b)*c^2 - 2/15*(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)*c*d
Exception generated. \[ \int \left (a+\frac {b}{x}\right )^{5/2} \left (c+\frac {d}{x}\right )^2 \, dx=\text {Exception raised: TypeError} \] Input:
integrate((a+b/x)^(5/2)*(c+d/x)^2,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 = 3.45 (sec) , antiderivative size = 271, normalized size of antiderivative = 1.96 \[ \int \left (a+\frac {b}{x}\right )^{5/2} \left (c+\frac {d}{x}\right )^2 \, dx={\left (a+\frac {b}{x}\right )}^{3/2}\,\left (\frac {2\,a\,\left (\frac {4\,a\,d^2-4\,b\,c\,d}{b}-\frac {4\,a\,d^2}{b}\right )}{3}-\frac {2\,{\left (a\,d-b\,c\right )}^2}{3\,b}+\frac {2\,a^2\,d^2}{3\,b}\right )+\left (\frac {4\,a\,d^2-4\,b\,c\,d}{5\,b}-\frac {4\,a\,d^2}{5\,b}\right )\,{\left (a+\frac {b}{x}\right )}^{5/2}-\sqrt {a+\frac {b}{x}}\,\left (a^2\,\left (\frac {4\,a\,d^2-4\,b\,c\,d}{b}-\frac {4\,a\,d^2}{b}\right )-2\,a\,\left (2\,a\,\left (\frac {4\,a\,d^2-4\,b\,c\,d}{b}-\frac {4\,a\,d^2}{b}\right )-\frac {2\,{\left (a\,d-b\,c\right )}^2}{b}+\frac {2\,a^2\,d^2}{b}\right )\right )-\frac {2\,d^2\,{\left (a+\frac {b}{x}\right )}^{7/2}}{7\,b}+a^2\,c^2\,x\,\sqrt {a+\frac {b}{x}}-a^{3/2}\,c\,\mathrm {atan}\left (\frac {\sqrt {a+\frac {b}{x}}\,1{}\mathrm {i}}{\sqrt {a}}\right )\,\left (4\,a\,d+5\,b\,c\right )\,1{}\mathrm {i} \] Input:
int((a + b/x)^(5/2)*(c + d/x)^2,x)
Output:
(a + b/x)^(3/2)*((2*a*((4*a*d^2 - 4*b*c*d)/b - (4*a*d^2)/b))/3 - (2*(a*d - b*c)^2)/(3*b) + (2*a^2*d^2)/(3*b)) + ((4*a*d^2 - 4*b*c*d)/(5*b) - (4*a*d^ 2)/(5*b))*(a + b/x)^(5/2) - (a + b/x)^(1/2)*(a^2*((4*a*d^2 - 4*b*c*d)/b - (4*a*d^2)/b) - 2*a*(2*a*((4*a*d^2 - 4*b*c*d)/b - (4*a*d^2)/b) - (2*(a*d - b*c)^2)/b + (2*a^2*d^2)/b)) - (2*d^2*(a + b/x)^(7/2))/(7*b) + a^2*c^2*x*(a + b/x)^(1/2) - a^(3/2)*c*atan(((a + b/x)^(1/2)*1i)/a^(1/2))*(4*a*d + 5*b* c)*1i
Time = 0.23 (sec) , antiderivative size = 298, normalized size of antiderivative = 2.16 \[ \int \left (a+\frac {b}{x}\right )^{5/2} \left (c+\frac {d}{x}\right )^2 \, dx=\frac {-30 \sqrt {x}\, \sqrt {a x +b}\, a^{3} d^{2} x^{3}+105 \sqrt {x}\, \sqrt {a x +b}\, a^{2} b \,c^{2} x^{4}-644 \sqrt {x}\, \sqrt {a x +b}\, a^{2} b c d \,x^{3}-90 \sqrt {x}\, \sqrt {a x +b}\, a^{2} b \,d^{2} x^{2}-490 \sqrt {x}\, \sqrt {a x +b}\, a \,b^{2} c^{2} x^{3}-308 \sqrt {x}\, \sqrt {a x +b}\, a \,b^{2} c d \,x^{2}-90 \sqrt {x}\, \sqrt {a x +b}\, a \,b^{2} d^{2} x -70 \sqrt {x}\, \sqrt {a x +b}\, b^{3} c^{2} x^{2}-84 \sqrt {x}\, \sqrt {a x +b}\, b^{3} c d x -30 \sqrt {x}\, \sqrt {a x +b}\, b^{3} d^{2}+420 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {a x +b}+\sqrt {x}\, \sqrt {a}}{\sqrt {b}}\right ) a^{2} b c d \,x^{4}+525 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {a x +b}+\sqrt {x}\, \sqrt {a}}{\sqrt {b}}\right ) a \,b^{2} c^{2} x^{4}-30 \sqrt {a}\, a^{3} d^{2} x^{4}+284 \sqrt {a}\, a^{2} b c d \,x^{4}+385 \sqrt {a}\, a \,b^{2} c^{2} x^{4}}{105 b \,x^{4}} \] Input:
int((a+b/x)^(5/2)*(c+d/x)^2,x)
Output:
( - 30*sqrt(x)*sqrt(a*x + b)*a**3*d**2*x**3 + 105*sqrt(x)*sqrt(a*x + b)*a* *2*b*c**2*x**4 - 644*sqrt(x)*sqrt(a*x + b)*a**2*b*c*d*x**3 - 90*sqrt(x)*sq rt(a*x + b)*a**2*b*d**2*x**2 - 490*sqrt(x)*sqrt(a*x + b)*a*b**2*c**2*x**3 - 308*sqrt(x)*sqrt(a*x + b)*a*b**2*c*d*x**2 - 90*sqrt(x)*sqrt(a*x + b)*a*b **2*d**2*x - 70*sqrt(x)*sqrt(a*x + b)*b**3*c**2*x**2 - 84*sqrt(x)*sqrt(a*x + b)*b**3*c*d*x - 30*sqrt(x)*sqrt(a*x + b)*b**3*d**2 + 420*sqrt(a)*log((s qrt(a*x + b) + sqrt(x)*sqrt(a))/sqrt(b))*a**2*b*c*d*x**4 + 525*sqrt(a)*log ((sqrt(a*x + b) + sqrt(x)*sqrt(a))/sqrt(b))*a*b**2*c**2*x**4 - 30*sqrt(a)* a**3*d**2*x**4 + 284*sqrt(a)*a**2*b*c*d*x**4 + 385*sqrt(a)*a*b**2*c**2*x** 4)/(105*b*x**4)