Integrand size = 21, antiderivative size = 201 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{5/2} \left (c+\frac {d}{x}\right )} \, dx=\frac {b (5 b c-3 a d)}{3 a^2 c (b c-a d) \left (a+\frac {b}{x}\right )^{3/2}}+\frac {b \left (5 b^2 c^2-8 a b c d+a^2 d^2\right )}{a^3 c (b c-a d)^2 \sqrt {a+\frac {b}{x}}}+\frac {x}{a c \left (a+\frac {b}{x}\right )^{3/2}}-\frac {2 d^{7/2} \arctan \left (\frac {\sqrt {d} \sqrt {a+\frac {b}{x}}}{\sqrt {b c-a d}}\right )}{c^2 (b c-a d)^{5/2}}-\frac {(5 b c+2 a d) \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{a^{7/2} c^2} \]
1/3*b*(-3*a*d+5*b*c)/a^2/c/(-a*d+b*c)/(a+b/x)^(3/2)+x/a/c/(a+b/x)^(3/2)-2* d^(7/2)*arctan(d^(1/2)*(a+b/x)^(1/2)/(-a*d+b*c)^(1/2))/c^2/(-a*d+b*c)^(5/2 )-(2*a*d+5*b*c)*arctanh((a+b/x)^(1/2)/a^(1/2))/a^(7/2)/c^2+b*(a^2*d^2-8*a* b*c*d+5*b^2*c^2)/a^3/c/(-a*d+b*c)^2/(a+b/x)^(1/2)
Time = 0.77 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{5/2} \left (c+\frac {d}{x}\right )} \, dx=\frac {\frac {c \sqrt {a+\frac {b}{x}} x \left (15 b^4 c^2+3 a^4 d^2 x^2+6 a^3 b d x (d-c x)+4 a b^3 c (-6 d+5 c x)+a^2 b^2 \left (3 d^2-32 c d x+3 c^2 x^2\right )\right )}{a^3 (b c-a d)^2 (b+a x)^2}-\frac {6 d^{7/2} \arctan \left (\frac {\sqrt {d} \sqrt {a+\frac {b}{x}}}{\sqrt {b c-a d}}\right )}{(b c-a d)^{5/2}}-\frac {3 (5 b c+2 a d) \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{a^{7/2}}}{3 c^2} \]
((c*Sqrt[a + b/x]*x*(15*b^4*c^2 + 3*a^4*d^2*x^2 + 6*a^3*b*d*x*(d - c*x) + 4*a*b^3*c*(-6*d + 5*c*x) + a^2*b^2*(3*d^2 - 32*c*d*x + 3*c^2*x^2)))/(a^3*( b*c - a*d)^2*(b + a*x)^2) - (6*d^(7/2)*ArcTan[(Sqrt[d]*Sqrt[a + b/x])/Sqrt [b*c - a*d]])/(b*c - a*d)^(5/2) - (3*(5*b*c + 2*a*d)*ArcTanh[Sqrt[a + b/x] /Sqrt[a]])/a^(7/2))/(3*c^2)
Time = 0.45 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.24, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.524, Rules used = {899, 114, 27, 169, 27, 169, 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.
| \(\displaystyle \int \frac {1}{\left (a+\frac {b}{x}\right )^{5/2} \left (c+\frac {d}{x}\right )} \, dx\) |
| \(\Big \downarrow \) 899 |
| \(\displaystyle -\int \frac {x^2}{\left (a+\frac {b}{x}\right )^{5/2} \left (c+\frac {d}{x}\right )}d\frac {1}{x}\) |
| \(\Big \downarrow \) 114 |
| \(\displaystyle \frac {\int \frac {\left (5 b c+2 a d+\frac {5 b d}{x}\right ) x}{2 \left (a+\frac {b}{x}\right )^{5/2} \left (c+\frac {d}{x}\right )}d\frac {1}{x}}{a c}+\frac {x}{a c \left (a+\frac {b}{x}\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\left (5 b c+2 a d+\frac {5 b d}{x}\right ) x}{\left (a+\frac {b}{x}\right )^{5/2} \left (c+\frac {d}{x}\right )}d\frac {1}{x}}{2 a c}+\frac {x}{a c \left (a+\frac {b}{x}\right )^{3/2}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {\frac {2 \int \frac {3 \left (\frac {b d (5 b c-3 a d)}{x}+(b c-a d) (5 b c+2 a d)\right ) x}{2 \left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right )}d\frac {1}{x}}{3 a (b c-a d)}+\frac {2 b (5 b c-3 a d)}{3 a \left (a+\frac {b}{x}\right )^{3/2} (b c-a d)}}{2 a c}+\frac {x}{a c \left (a+\frac {b}{x}\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {\left (\frac {b d (5 b c-3 a d)}{x}+(b c-a d) (5 b c+2 a d)\right ) x}{\left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right )}d\frac {1}{x}}{a (b c-a d)}+\frac {2 b (5 b c-3 a d)}{3 a \left (a+\frac {b}{x}\right )^{3/2} (b c-a d)}}{2 a c}+\frac {x}{a c \left (a+\frac {b}{x}\right )^{3/2}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {\frac {\frac {2 \int \frac {\left ((5 b c+2 a d) (b c-a d)^2+\frac {b d \left (5 b^2 c^2-8 a b d c+a^2 d^2\right )}{x}\right ) x}{2 \sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right )}d\frac {1}{x}}{a (b c-a d)}+\frac {2 b \left (a^2 d^2-8 a b c d+5 b^2 c^2\right )}{a \sqrt {a+\frac {b}{x}} (b c-a d)}}{a (b c-a d)}+\frac {2 b (5 b c-3 a d)}{3 a \left (a+\frac {b}{x}\right )^{3/2} (b c-a d)}}{2 a c}+\frac {x}{a c \left (a+\frac {b}{x}\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {\left ((5 b c+2 a d) (b c-a d)^2+\frac {b d \left (5 b^2 c^2-8 a b d c+a^2 d^2\right )}{x}\right ) x}{\sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right )}d\frac {1}{x}}{a (b c-a d)}+\frac {2 b \left (a^2 d^2-8 a b c d+5 b^2 c^2\right )}{a \sqrt {a+\frac {b}{x}} (b c-a d)}}{a (b c-a d)}+\frac {2 b (5 b c-3 a d)}{3 a \left (a+\frac {b}{x}\right )^{3/2} (b c-a d)}}{2 a c}+\frac {x}{a c \left (a+\frac {b}{x}\right )^{3/2}}\) |
| \(\Big \downarrow \) 174 |
| \(\displaystyle \frac {\frac {\frac {\frac {(b c-a d)^2 (2 a d+5 b c) \int \frac {x}{\sqrt {a+\frac {b}{x}}}d\frac {1}{x}}{c}-\frac {2 a^3 d^4 \int \frac {1}{\sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right )}d\frac {1}{x}}{c}}{a (b c-a d)}+\frac {2 b \left (a^2 d^2-8 a b c d+5 b^2 c^2\right )}{a \sqrt {a+\frac {b}{x}} (b c-a d)}}{a (b c-a d)}+\frac {2 b (5 b c-3 a d)}{3 a \left (a+\frac {b}{x}\right )^{3/2} (b c-a d)}}{2 a c}+\frac {x}{a c \left (a+\frac {b}{x}\right )^{3/2}}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {\frac {\frac {\frac {2 (b c-a d)^2 (2 a d+5 b c) \int \frac {1}{\frac {1}{b x^2}-\frac {a}{b}}d\sqrt {a+\frac {b}{x}}}{b c}-\frac {4 a^3 d^4 \int \frac {1}{c-\frac {a d}{b}+\frac {d}{b x^2}}d\sqrt {a+\frac {b}{x}}}{b c}}{a (b c-a d)}+\frac {2 b \left (a^2 d^2-8 a b c d+5 b^2 c^2\right )}{a \sqrt {a+\frac {b}{x}} (b c-a d)}}{a (b c-a d)}+\frac {2 b (5 b c-3 a d)}{3 a \left (a+\frac {b}{x}\right )^{3/2} (b c-a d)}}{2 a c}+\frac {x}{a c \left (a+\frac {b}{x}\right )^{3/2}}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {\frac {\frac {\frac {2 (b c-a d)^2 (2 a d+5 b c) \int \frac {1}{\frac {1}{b x^2}-\frac {a}{b}}d\sqrt {a+\frac {b}{x}}}{b c}-\frac {4 a^3 d^{7/2} \arctan \left (\frac {\sqrt {d} \sqrt {a+\frac {b}{x}}}{\sqrt {b c-a d}}\right )}{c \sqrt {b c-a d}}}{a (b c-a d)}+\frac {2 b \left (a^2 d^2-8 a b c d+5 b^2 c^2\right )}{a \sqrt {a+\frac {b}{x}} (b c-a d)}}{a (b c-a d)}+\frac {2 b (5 b c-3 a d)}{3 a \left (a+\frac {b}{x}\right )^{3/2} (b c-a d)}}{2 a c}+\frac {x}{a c \left (a+\frac {b}{x}\right )^{3/2}}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\frac {\frac {-\frac {4 a^3 d^{7/2} \arctan \left (\frac {\sqrt {d} \sqrt {a+\frac {b}{x}}}{\sqrt {b c-a d}}\right )}{c \sqrt {b c-a d}}-\frac {2 \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right ) (b c-a d)^2 (2 a d+5 b c)}{\sqrt {a} c}}{a (b c-a d)}+\frac {2 b \left (a^2 d^2-8 a b c d+5 b^2 c^2\right )}{a \sqrt {a+\frac {b}{x}} (b c-a d)}}{a (b c-a d)}+\frac {2 b (5 b c-3 a d)}{3 a \left (a+\frac {b}{x}\right )^{3/2} (b c-a d)}}{2 a c}+\frac {x}{a c \left (a+\frac {b}{x}\right )^{3/2}}\) |
x/(a*c*(a + b/x)^(3/2)) + ((2*b*(5*b*c - 3*a*d))/(3*a*(b*c - a*d)*(a + b/x )^(3/2)) + ((2*b*(5*b^2*c^2 - 8*a*b*c*d + a^2*d^2))/(a*(b*c - a*d)*Sqrt[a + b/x]) + ((-4*a^3*d^(7/2)*ArcTan[(Sqrt[d]*Sqrt[a + b/x])/Sqrt[b*c - a*d]] )/(c*Sqrt[b*c - a*d]) - (2*(b*c - a*d)^2*(5*b*c + 2*a*d)*ArcTanh[Sqrt[a + b/x]/Sqrt[a]])/(Sqrt[a]*c))/(a*(b*c - a*d)))/(a*(b*c - a*d)))/(2*a*c)
3.3.63.3.1 Defintions of rubi rules used
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*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1 )/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Simp[1/((m + 1)*(b*c - a*d)*(b*e - a*f)) Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) - b*(d*e*(m + n + 2) + c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && ILtQ[m, -1] && (IntegerQ[n] || IntegersQ[2*n, 2*p] || ILtQ[m + n + p + 3, 0])
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + S imp[1/((m + 1)*(b*c - a*d)*(b*e - a*f)) Int[(a + b*x)^(m + 1)*(c + d*x)^n *(e + f*x)^p*Simp[(a*d*f*g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a* h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && IntegersQ[ 2*m, 2*n, 2*p]
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. \(402\) vs. \(2(177)=354\).
Time = 0.33 (sec) , antiderivative size = 403, normalized size of antiderivative = 2.00
| method | result | size |
| risch | \(\frac {a x +b}{a^{3} c \sqrt {\frac {a x +b}{x}}}-\frac {\left (\frac {\left (2 a d +5 b c \right ) \ln \left (\frac {\frac {b}{2}+a x}{\sqrt {a}}+\sqrt {a \,x^{2}+b x}\right )}{c \sqrt {a}}-\frac {2 c \,b^{4} \left (\frac {2 \sqrt {a \left (x +\frac {b}{a}\right )^{2}-b \left (x +\frac {b}{a}\right )}}{3 b \left (x +\frac {b}{a}\right )^{2}}+\frac {4 a \sqrt {a \left (x +\frac {b}{a}\right )^{2}-b \left (x +\frac {b}{a}\right )}}{3 b^{2} \left (x +\frac {b}{a}\right )}\right )}{\left (a d -b c \right ) a^{2}}+\frac {2 a^{3} d^{4} \ln \left (\frac {\frac {2 \left (a d -b c \right ) d}{c^{2}}-\frac {\left (2 a d -b c \right ) \left (x +\frac {d}{c}\right )}{c}+2 \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}\, \sqrt {a \left (x +\frac {d}{c}\right )^{2}-\frac {\left (2 a d -b c \right ) \left (x +\frac {d}{c}\right )}{c}+\frac {\left (a d -b c \right ) d}{c^{2}}}}{x +\frac {d}{c}}\right )}{c^{2} \left (a d -b c \right )^{2} \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}}+\frac {4 c \,b^{2} \left (4 a d -3 b c \right ) \sqrt {a \left (x +\frac {b}{a}\right )^{2}-b \left (x +\frac {b}{a}\right )}}{\left (a d -b c \right )^{2} a \left (x +\frac {b}{a}\right )}\right ) \sqrt {x \left (a x +b \right )}}{2 a^{3} c x \sqrt {\frac {a x +b}{x}}}\) | \(403\) |
| default | \(\text {Expression too large to display}\) | \(1767\) |
1/a^3/c*(a*x+b)/((a*x+b)/x)^(1/2)-1/2/a^3/c*((2*a*d+5*b*c)/c*ln((1/2*b+a*x )/a^(1/2)+(a*x^2+b*x)^(1/2))/a^(1/2)-2*c*b^4/(a*d-b*c)/a^2*(2/3/b/(x+b/a)^ 2*(a*(x+b/a)^2-b*(x+b/a))^(1/2)+4/3*a/b^2/(x+b/a)*(a*(x+b/a)^2-b*(x+b/a))^ (1/2))+2/c^2*a^3*d^4/(a*d-b*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+d/c)+2*((a*d-b*c)*d/c^2)^(1/2)*(a*(x+d/c)^2-(2*a*d-b *c)/c*(x+d/c)+(a*d-b*c)*d/c^2)^(1/2))/(x+d/c))+4*c*b^2*(4*a*d-3*b*c)/(a*d- b*c)^2/a/(x+b/a)*(a*(x+b/a)^2-b*(x+b/a))^(1/2))/x/((a*x+b)/x)^(1/2)*(x*(a* x+b))^(1/2)
Leaf count of result is larger than twice the leaf count of optimal. 489 vs. \(2 (177) = 354\).
Time = 1.52 (sec) , antiderivative size = 1990, normalized size of antiderivative = 9.90 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{5/2} \left (c+\frac {d}{x}\right )} \, dx=\text {Too large to display} \]
[1/6*(3*(5*b^5*c^3 - 8*a*b^4*c^2*d + a^2*b^3*c*d^2 + 2*a^3*b^2*d^3 + (5*a^ 2*b^3*c^3 - 8*a^3*b^2*c^2*d + a^4*b*c*d^2 + 2*a^5*d^3)*x^2 + 2*(5*a*b^4*c^ 3 - 8*a^2*b^3*c^2*d + a^3*b^2*c*d^2 + 2*a^4*b*d^3)*x)*sqrt(a)*log(2*a*x - 2*sqrt(a)*x*sqrt((a*x + b)/x) + b) + 6*(a^6*d^3*x^2 + 2*a^5*b*d^3*x + a^4* b^2*d^3)*sqrt(-d/(b*c - a*d))*log(-(2*(b*c - a*d)*x*sqrt(-d/(b*c - a*d))*s qrt((a*x + b)/x) - b*d + (b*c - 2*a*d)*x)/(c*x + d)) + 2*(3*(a^3*b^2*c^3 - 2*a^4*b*c^2*d + a^5*c*d^2)*x^3 + 2*(10*a^2*b^3*c^3 - 16*a^3*b^2*c^2*d + 3 *a^4*b*c*d^2)*x^2 + 3*(5*a*b^4*c^3 - 8*a^2*b^3*c^2*d + a^3*b^2*c*d^2)*x)*s qrt((a*x + b)/x))/(a^4*b^4*c^4 - 2*a^5*b^3*c^3*d + a^6*b^2*c^2*d^2 + (a^6* b^2*c^4 - 2*a^7*b*c^3*d + a^8*c^2*d^2)*x^2 + 2*(a^5*b^3*c^4 - 2*a^6*b^2*c^ 3*d + a^7*b*c^2*d^2)*x), 1/3*(3*(5*b^5*c^3 - 8*a*b^4*c^2*d + a^2*b^3*c*d^2 + 2*a^3*b^2*d^3 + (5*a^2*b^3*c^3 - 8*a^3*b^2*c^2*d + a^4*b*c*d^2 + 2*a^5* d^3)*x^2 + 2*(5*a*b^4*c^3 - 8*a^2*b^3*c^2*d + a^3*b^2*c*d^2 + 2*a^4*b*d^3) *x)*sqrt(-a)*arctan(sqrt(-a)*sqrt((a*x + b)/x)/a) + 3*(a^6*d^3*x^2 + 2*a^5 *b*d^3*x + a^4*b^2*d^3)*sqrt(-d/(b*c - a*d))*log(-(2*(b*c - a*d)*x*sqrt(-d /(b*c - a*d))*sqrt((a*x + b)/x) - b*d + (b*c - 2*a*d)*x)/(c*x + d)) + (3*( a^3*b^2*c^3 - 2*a^4*b*c^2*d + a^5*c*d^2)*x^3 + 2*(10*a^2*b^3*c^3 - 16*a^3* b^2*c^2*d + 3*a^4*b*c*d^2)*x^2 + 3*(5*a*b^4*c^3 - 8*a^2*b^3*c^2*d + a^3*b^ 2*c*d^2)*x)*sqrt((a*x + b)/x))/(a^4*b^4*c^4 - 2*a^5*b^3*c^3*d + a^6*b^2*c^ 2*d^2 + (a^6*b^2*c^4 - 2*a^7*b*c^3*d + a^8*c^2*d^2)*x^2 + 2*(a^5*b^3*c^...
\[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{5/2} \left (c+\frac {d}{x}\right )} \, dx=\int \frac {x}{\left (a + \frac {b}{x}\right )^{\frac {5}{2}} \left (c x + d\right )}\, dx \]
\[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{5/2} \left (c+\frac {d}{x}\right )} \, dx=\int { \frac {1}{{\left (a + \frac {b}{x}\right )}^{\frac {5}{2}} {\left (c + \frac {d}{x}\right )}} \,d x } \]
Exception generated. \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{5/2} \left (c+\frac {d}{x}\right )} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Error: Bad Argument Type
Time = 8.88 (sec) , antiderivative size = 5387, normalized size of antiderivative = 26.80 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{5/2} \left (c+\frac {d}{x}\right )} \, dx=\text {Too large to display} \]
- ((2*b^2)/(3*(a^2*d - a*b*c)) + (2*b^2*(a + b/x)*(8*a*d - 5*b*c))/(3*(a^2 *d - a*b*c)^2) + (b*(a + b/x)^2*(a^2*d^2 + 5*b^2*c^2 - 8*a*b*c*d))/(a^2*c* (a^2*d - a*b*c)*(a*d - b*c)))/(a*(a + b/x)^(3/2) - (a + b/x)^(5/2)) - (ata n(((((a + b/x)^(1/2)*(50*a^9*b^14*c^15*d^3 - 460*a^10*b^13*c^14*d^4 + 1858 *a^11*b^12*c^13*d^5 - 4280*a^12*b^11*c^12*d^6 + 6060*a^13*b^10*c^11*d^7 - 5160*a^14*b^9*c^10*d^8 + 2108*a^15*b^8*c^9*d^9 + 336*a^16*b^7*c^8*d^10 - 7 50*a^17*b^6*c^7*d^11 + 180*a^18*b^5*c^6*d^12 + 130*a^19*b^4*c^5*d^13 - 88* a^20*b^3*c^4*d^14 + 16*a^21*b^2*c^3*d^15) - ((2*a*d + 5*b*c)*(20*a^12*b^14 *c^17*d^2 - 212*a^13*b^13*c^16*d^3 + 1012*a^14*b^12*c^15*d^4 - 2860*a^15*b ^11*c^14*d^5 + 5288*a^16*b^10*c^13*d^6 - 6664*a^17*b^9*c^12*d^7 + 5768*a^1 8*b^8*c^11*d^8 - 3352*a^19*b^7*c^10*d^9 + 1220*a^20*b^6*c^9*d^10 - 228*a^2 1*b^5*c^8*d^11 + 4*a^22*b^4*c^7*d^12 + 4*a^23*b^3*c^6*d^13 - ((a + b/x)^(1 /2)*(2*a*d + 5*b*c)*(8*a^15*b^13*c^18*d^2 - 96*a^16*b^12*c^17*d^3 + 520*a^ 17*b^11*c^16*d^4 - 1680*a^18*b^10*c^15*d^5 + 3600*a^19*b^9*c^14*d^6 - 5376 *a^20*b^8*c^13*d^7 + 5712*a^21*b^7*c^12*d^8 - 4320*a^22*b^6*c^11*d^9 + 228 0*a^23*b^5*c^10*d^10 - 800*a^24*b^4*c^9*d^11 + 168*a^25*b^3*c^8*d^12 - 16* a^26*b^2*c^7*d^13))/(2*c^2*(a^7)^(1/2))))/(2*c^2*(a^7)^(1/2)))*(2*a*d + 5* b*c)*1i)/(2*c^2*(a^7)^(1/2)) + (((a + b/x)^(1/2)*(50*a^9*b^14*c^15*d^3 - 4 60*a^10*b^13*c^14*d^4 + 1858*a^11*b^12*c^13*d^5 - 4280*a^12*b^11*c^12*d^6 + 6060*a^13*b^10*c^11*d^7 - 5160*a^14*b^9*c^10*d^8 + 2108*a^15*b^8*c^9*...