Integrand size = 21, antiderivative size = 209 \[ \int \frac {\left (a+\frac {b}{x}\right )^{3/2}}{\left (c+\frac {d}{x}\right )^3} \, dx=-\frac {(b c-3 a d) \sqrt {a+\frac {b}{x}}}{2 c^2 \left (c+\frac {d}{x}\right )^2}-\frac {3 (b c-4 a d) \sqrt {a+\frac {b}{x}}}{4 c^3 \left (c+\frac {d}{x}\right )}+\frac {a \sqrt {a+\frac {b}{x}} x}{c \left (c+\frac {d}{x}\right )^2}-\frac {3 \left (b^2 c^2-8 a b c d+8 a^2 d^2\right ) \arctan \left (\frac {\sqrt {d} \sqrt {a+\frac {b}{x}}}{\sqrt {b c-a d}}\right )}{4 c^4 \sqrt {d} \sqrt {b c-a d}}+\frac {3 \sqrt {a} (b c-2 a d) \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{c^4} \]
3*(-2*a*d+b*c)*arctanh((a+b/x)^(1/2)/a^(1/2))*a^(1/2)/c^4-3/4*(8*a^2*d^2-8 *a*b*c*d+b^2*c^2)*arctan(d^(1/2)*(a+b/x)^(1/2)/(-a*d+b*c)^(1/2))/c^4/d^(1/ 2)/(-a*d+b*c)^(1/2)-1/2*(-3*a*d+b*c)*(a+b/x)^(1/2)/c^2/(c+d/x)^2-3/4*(-4*a *d+b*c)*(a+b/x)^(1/2)/c^3/(c+d/x)+a*x*(a+b/x)^(1/2)/c/(c+d/x)^2
Time = 0.66 (sec) , antiderivative size = 169, normalized size of antiderivative = 0.81 \[ \int \frac {\left (a+\frac {b}{x}\right )^{3/2}}{\left (c+\frac {d}{x}\right )^3} \, dx=\frac {\frac {c \sqrt {a+\frac {b}{x}} x \left (-b c (3 d+5 c x)+2 a \left (6 d^2+9 c d x+2 c^2 x^2\right )\right )}{(d+c x)^2}-\frac {3 \left (b^2 c^2-8 a b c d+8 a^2 d^2\right ) \arctan \left (\frac {\sqrt {d} \sqrt {a+\frac {b}{x}}}{\sqrt {b c-a d}}\right )}{\sqrt {d} \sqrt {b c-a d}}-12 \sqrt {a} (-b c+2 a d) \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{4 c^4} \]
((c*Sqrt[a + b/x]*x*(-(b*c*(3*d + 5*c*x)) + 2*a*(6*d^2 + 9*c*d*x + 2*c^2*x ^2)))/(d + c*x)^2 - (3*(b^2*c^2 - 8*a*b*c*d + 8*a^2*d^2)*ArcTan[(Sqrt[d]*S qrt[a + b/x])/Sqrt[b*c - a*d]])/(Sqrt[d]*Sqrt[b*c - a*d]) - 12*Sqrt[a]*(-( b*c) + 2*a*d)*ArcTanh[Sqrt[a + b/x]/Sqrt[a]])/(4*c^4)
Time = 0.41 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.08, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.524, Rules used = {899, 109, 27, 168, 27, 168, 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.
| \(\displaystyle \int \frac {\left (a+\frac {b}{x}\right )^{3/2}}{\left (c+\frac {d}{x}\right )^3} \, dx\) |
| \(\Big \downarrow \) 899 |
| \(\displaystyle -\int \frac {\left (a+\frac {b}{x}\right )^{3/2} x^2}{\left (c+\frac {d}{x}\right )^3}d\frac {1}{x}\) |
| \(\Big \downarrow \) 109 |
| \(\displaystyle \frac {\int -\frac {\left (\frac {b (2 b c-5 a d)}{x}+3 a (b c-2 a d)\right ) x}{2 \sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right )^3}d\frac {1}{x}}{c}+\frac {a x \sqrt {a+\frac {b}{x}}}{c \left (c+\frac {d}{x}\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {a x \sqrt {a+\frac {b}{x}}}{c \left (c+\frac {d}{x}\right )^2}-\frac {\int \frac {\left (\frac {b (2 b c-5 a d)}{x}+3 a (b c-2 a d)\right ) x}{\sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right )^3}d\frac {1}{x}}{2 c}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {a x \sqrt {a+\frac {b}{x}}}{c \left (c+\frac {d}{x}\right )^2}-\frac {\frac {\sqrt {a+\frac {b}{x}} (b c-3 a d)}{c \left (c+\frac {d}{x}\right )^2}-\frac {\int -\frac {3 (b c-a d) \left (\frac {b (b c-3 a d)}{x}+2 a (b c-2 a d)\right ) x}{\sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right )^2}d\frac {1}{x}}{2 c (b c-a d)}}{2 c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {a x \sqrt {a+\frac {b}{x}}}{c \left (c+\frac {d}{x}\right )^2}-\frac {\frac {3 \int \frac {\left (\frac {b (b c-3 a d)}{x}+2 a (b c-2 a d)\right ) x}{\sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right )^2}d\frac {1}{x}}{2 c}+\frac {\sqrt {a+\frac {b}{x}} (b c-3 a d)}{c \left (c+\frac {d}{x}\right )^2}}{2 c}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {a x \sqrt {a+\frac {b}{x}}}{c \left (c+\frac {d}{x}\right )^2}-\frac {\frac {3 \left (\frac {\sqrt {a+\frac {b}{x}} (b c-4 a d)}{c \left (c+\frac {d}{x}\right )}-\frac {\int -\frac {(b c-a d) \left (\frac {b (b c-4 a d)}{x}+4 a (b c-2 a d)\right ) x}{2 \sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right )}d\frac {1}{x}}{c (b c-a d)}\right )}{2 c}+\frac {\sqrt {a+\frac {b}{x}} (b c-3 a d)}{c \left (c+\frac {d}{x}\right )^2}}{2 c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {a x \sqrt {a+\frac {b}{x}}}{c \left (c+\frac {d}{x}\right )^2}-\frac {\frac {3 \left (\frac {\int \frac {\left (\frac {b (b c-4 a d)}{x}+4 a (b c-2 a d)\right ) x}{\sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right )}d\frac {1}{x}}{2 c}+\frac {\sqrt {a+\frac {b}{x}} (b c-4 a d)}{c \left (c+\frac {d}{x}\right )}\right )}{2 c}+\frac {\sqrt {a+\frac {b}{x}} (b c-3 a d)}{c \left (c+\frac {d}{x}\right )^2}}{2 c}\) |
| \(\Big \downarrow \) 174 |
| \(\displaystyle \frac {a x \sqrt {a+\frac {b}{x}}}{c \left (c+\frac {d}{x}\right )^2}-\frac {\frac {3 \left (\frac {\frac {\left (8 a^2 d^2-8 a b c d+b^2 c^2\right ) \int \frac {1}{\sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right )}d\frac {1}{x}}{c}+\frac {4 a (b c-2 a d) \int \frac {x}{\sqrt {a+\frac {b}{x}}}d\frac {1}{x}}{c}}{2 c}+\frac {\sqrt {a+\frac {b}{x}} (b c-4 a d)}{c \left (c+\frac {d}{x}\right )}\right )}{2 c}+\frac {\sqrt {a+\frac {b}{x}} (b c-3 a d)}{c \left (c+\frac {d}{x}\right )^2}}{2 c}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {a x \sqrt {a+\frac {b}{x}}}{c \left (c+\frac {d}{x}\right )^2}-\frac {\frac {3 \left (\frac {\frac {2 \left (8 a^2 d^2-8 a b c d+b^2 c^2\right ) \int \frac {1}{c-\frac {a d}{b}+\frac {d}{b x^2}}d\sqrt {a+\frac {b}{x}}}{b c}+\frac {8 a (b c-2 a d) \int \frac {1}{\frac {1}{b x^2}-\frac {a}{b}}d\sqrt {a+\frac {b}{x}}}{b c}}{2 c}+\frac {\sqrt {a+\frac {b}{x}} (b c-4 a d)}{c \left (c+\frac {d}{x}\right )}\right )}{2 c}+\frac {\sqrt {a+\frac {b}{x}} (b c-3 a d)}{c \left (c+\frac {d}{x}\right )^2}}{2 c}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {a x \sqrt {a+\frac {b}{x}}}{c \left (c+\frac {d}{x}\right )^2}-\frac {\frac {3 \left (\frac {\frac {8 a (b c-2 a d) \int \frac {1}{\frac {1}{b x^2}-\frac {a}{b}}d\sqrt {a+\frac {b}{x}}}{b c}+\frac {2 \left (8 a^2 d^2-8 a b c d+b^2 c^2\right ) \arctan \left (\frac {\sqrt {d} \sqrt {a+\frac {b}{x}}}{\sqrt {b c-a d}}\right )}{c \sqrt {d} \sqrt {b c-a d}}}{2 c}+\frac {\sqrt {a+\frac {b}{x}} (b c-4 a d)}{c \left (c+\frac {d}{x}\right )}\right )}{2 c}+\frac {\sqrt {a+\frac {b}{x}} (b c-3 a d)}{c \left (c+\frac {d}{x}\right )^2}}{2 c}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {a x \sqrt {a+\frac {b}{x}}}{c \left (c+\frac {d}{x}\right )^2}-\frac {\frac {3 \left (\frac {\frac {2 \left (8 a^2 d^2-8 a b c d+b^2 c^2\right ) \arctan \left (\frac {\sqrt {d} \sqrt {a+\frac {b}{x}}}{\sqrt {b c-a d}}\right )}{c \sqrt {d} \sqrt {b c-a d}}-\frac {8 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right ) (b c-2 a d)}{c}}{2 c}+\frac {\sqrt {a+\frac {b}{x}} (b c-4 a d)}{c \left (c+\frac {d}{x}\right )}\right )}{2 c}+\frac {\sqrt {a+\frac {b}{x}} (b c-3 a d)}{c \left (c+\frac {d}{x}\right )^2}}{2 c}\) |
(a*Sqrt[a + b/x]*x)/(c*(c + d/x)^2) - (((b*c - 3*a*d)*Sqrt[a + b/x])/(c*(c + d/x)^2) + (3*(((b*c - 4*a*d)*Sqrt[a + b/x])/(c*(c + d/x)) + ((2*(b^2*c^ 2 - 8*a*b*c*d + 8*a^2*d^2)*ArcTan[(Sqrt[d]*Sqrt[a + b/x])/Sqrt[b*c - a*d]] )/(c*Sqrt[d]*Sqrt[b*c - a*d]) - (8*Sqrt[a]*(b*c - 2*a*d)*ArcTanh[Sqrt[a + b/x]/Sqrt[a]])/c)/(2*c)))/(2*c))/(2*c)
3.3.37.3.1 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*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[((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] && ILtQ[m, -1]
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. \(1007\) vs. \(2(181)=362\).
Time = 0.27 (sec) , antiderivative size = 1008, normalized size of antiderivative = 4.82
| method | result | size |
| risch | \(\text {Expression too large to display}\) | \(1008\) |
| default | \(\text {Expression too large to display}\) | \(1817\) |
1/c^3*x*a*((a*x+b)/x)^(1/2)-1/2/c^3*(3*a^(1/2)*(2*a*d-b*c)/c*ln((1/2*b+a*x )/a^(1/2)+(a*x^2+b*x)^(1/2))-(-12*a^2*d^2+12*a*b*c*d-2*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+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^3*d*(2*a^2*d^2-3*a*b*c*d+b^2*c^2)*(-1/(a*d-b*c)/d*c^2/(x+d/c)* (a*(x+d/c)^2-(2*a*d-b*c)/c*(x+d/c)+(a*d-b*c)*d/c^2)^(1/2)-1/2*(2*a*d-b*c)* c/(a*d-b*c)/d/((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)))-2*d^2*(a^2*d^2-2*a*b*c*d+b^2*c^2)/c^4*(-1/2/( a*d-b*c)/d*c^2/(x+d/c)^2*(a*(x+d/c)^2-(2*a*d-b*c)/c*(x+d/c)+(a*d-b*c)*d/c^ 2)^(1/2)+3/4*(2*a*d-b*c)*c/(a*d-b*c)/d*(-1/(a*d-b*c)/d*c^2/(x+d/c)*(a*(x+d /c)^2-(2*a*d-b*c)/c*(x+d/c)+(a*d-b*c)*d/c^2)^(1/2)-1/2*(2*a*d-b*c)*c/(a*d- b*c)/d/((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)))+1/2*a/(a*d-b*c)/d*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))))*((a*x+b)/x)^(1 /2)*(x*(a*x+b))^(1/2)/(a*x+b)
Leaf count of result is larger than twice the leaf count of optimal. 433 vs. \(2 (181) = 362\).
Time = 0.34 (sec) , antiderivative size = 1765, normalized size of antiderivative = 8.44 \[ \int \frac {\left (a+\frac {b}{x}\right )^{3/2}}{\left (c+\frac {d}{x}\right )^3} \, dx=\text {Too large to display} \]
[-1/8*(12*(b^2*c^2*d^3 - 3*a*b*c*d^4 + 2*a^2*d^5 + (b^2*c^4*d - 3*a*b*c^3* d^2 + 2*a^2*c^2*d^3)*x^2 + 2*(b^2*c^3*d^2 - 3*a*b*c^2*d^3 + 2*a^2*c*d^4)*x )*sqrt(a)*log(2*a*x - 2*sqrt(a)*x*sqrt((a*x + b)/x) + b) + 3*(b^2*c^2*d^2 - 8*a*b*c*d^3 + 8*a^2*d^4 + (b^2*c^4 - 8*a*b*c^3*d + 8*a^2*c^2*d^2)*x^2 + 2*(b^2*c^3*d - 8*a*b*c^2*d^2 + 8*a^2*c*d^3)*x)*sqrt(-b*c*d + a*d^2)*log((b *d - (b*c - 2*a*d)*x + 2*sqrt(-b*c*d + a*d^2)*x*sqrt((a*x + b)/x))/(c*x + d)) - 2*(4*(a*b*c^4*d - a^2*c^3*d^2)*x^3 - (5*b^2*c^4*d - 23*a*b*c^3*d^2 + 18*a^2*c^2*d^3)*x^2 - 3*(b^2*c^3*d^2 - 5*a*b*c^2*d^3 + 4*a^2*c*d^4)*x)*sq rt((a*x + b)/x))/(b*c^5*d^3 - a*c^4*d^4 + (b*c^7*d - a*c^6*d^2)*x^2 + 2*(b *c^6*d^2 - a*c^5*d^3)*x), -1/8*(24*(b^2*c^2*d^3 - 3*a*b*c*d^4 + 2*a^2*d^5 + (b^2*c^4*d - 3*a*b*c^3*d^2 + 2*a^2*c^2*d^3)*x^2 + 2*(b^2*c^3*d^2 - 3*a*b *c^2*d^3 + 2*a^2*c*d^4)*x)*sqrt(-a)*arctan(sqrt(-a)*sqrt((a*x + b)/x)/a) + 3*(b^2*c^2*d^2 - 8*a*b*c*d^3 + 8*a^2*d^4 + (b^2*c^4 - 8*a*b*c^3*d + 8*a^2 *c^2*d^2)*x^2 + 2*(b^2*c^3*d - 8*a*b*c^2*d^2 + 8*a^2*c*d^3)*x)*sqrt(-b*c*d + a*d^2)*log((b*d - (b*c - 2*a*d)*x + 2*sqrt(-b*c*d + a*d^2)*x*sqrt((a*x + b)/x))/(c*x + d)) - 2*(4*(a*b*c^4*d - a^2*c^3*d^2)*x^3 - (5*b^2*c^4*d - 23*a*b*c^3*d^2 + 18*a^2*c^2*d^3)*x^2 - 3*(b^2*c^3*d^2 - 5*a*b*c^2*d^3 + 4* a^2*c*d^4)*x)*sqrt((a*x + b)/x))/(b*c^5*d^3 - a*c^4*d^4 + (b*c^7*d - a*c^6 *d^2)*x^2 + 2*(b*c^6*d^2 - a*c^5*d^3)*x), 1/4*(3*(b^2*c^2*d^2 - 8*a*b*c*d^ 3 + 8*a^2*d^4 + (b^2*c^4 - 8*a*b*c^3*d + 8*a^2*c^2*d^2)*x^2 + 2*(b^2*c^...
Timed out. \[ \int \frac {\left (a+\frac {b}{x}\right )^{3/2}}{\left (c+\frac {d}{x}\right )^3} \, dx=\text {Timed out} \]
\[ \int \frac {\left (a+\frac {b}{x}\right )^{3/2}}{\left (c+\frac {d}{x}\right )^3} \, dx=\int { \frac {{\left (a + \frac {b}{x}\right )}^{\frac {3}{2}}}{{\left (c + \frac {d}{x}\right )}^{3}} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 720 vs. \(2 (181) = 362\).
Time = 0.38 (sec) , antiderivative size = 720, normalized size of antiderivative = 3.44 \[ \int \frac {\left (a+\frac {b}{x}\right )^{3/2}}{\left (c+\frac {d}{x}\right )^3} \, dx=\frac {\sqrt {a x^{2} + b x} a \mathrm {sgn}\left (x\right )}{c^{3}} + \frac {3 \, {\left (b^{2} c^{2} \mathrm {sgn}\left (x\right ) - 8 \, a b c d \mathrm {sgn}\left (x\right ) + 8 \, a^{2} d^{2} \mathrm {sgn}\left (x\right )\right )} \arctan \left (-\frac {{\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )} c + \sqrt {a} d}{\sqrt {b c d - a d^{2}}}\right )}{4 \, \sqrt {b c d - a d^{2}} c^{4}} - \frac {3 \, {\left (a b c \mathrm {sgn}\left (x\right ) - 2 \, a^{2} d \mathrm {sgn}\left (x\right )\right )} \log \left ({\left | -2 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )} \sqrt {a} - b \right |}\right )}{2 \, \sqrt {a} c^{4}} + \frac {{\left (3 \, \sqrt {a} b^{2} c^{2} \arctan \left (\frac {\sqrt {a} d}{\sqrt {b c d - a d^{2}}}\right ) - 24 \, a^{\frac {3}{2}} b c d \arctan \left (\frac {\sqrt {a} d}{\sqrt {b c d - a d^{2}}}\right ) + 24 \, a^{\frac {5}{2}} d^{2} \arctan \left (\frac {\sqrt {a} d}{\sqrt {b c d - a d^{2}}}\right ) + 6 \, \sqrt {b c d - a d^{2}} a b c \log \left ({\left | b \right |}\right ) - 12 \, \sqrt {b c d - a d^{2}} a^{2} d \log \left ({\left | b \right |}\right ) + 5 \, \sqrt {b c d - a d^{2}} a b c - 10 \, \sqrt {b c d - a d^{2}} a^{2} d\right )} \mathrm {sgn}\left (x\right )}{4 \, \sqrt {b c d - a d^{2}} \sqrt {a} c^{4}} + \frac {5 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{3} b^{2} c^{3} \mathrm {sgn}\left (x\right ) - 24 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{3} a b c^{2} d \mathrm {sgn}\left (x\right ) + 24 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{3} a^{2} c d^{2} \mathrm {sgn}\left (x\right ) - {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{2} \sqrt {a} b^{2} c^{2} d \mathrm {sgn}\left (x\right ) - 24 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{2} a^{\frac {3}{2}} b c d^{2} \mathrm {sgn}\left (x\right ) + 40 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{2} a^{\frac {5}{2}} d^{3} \mathrm {sgn}\left (x\right ) + 3 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )} b^{3} c^{2} d \mathrm {sgn}\left (x\right ) - 28 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )} a b^{2} c d^{2} \mathrm {sgn}\left (x\right ) + 40 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )} a^{2} b d^{3} \mathrm {sgn}\left (x\right ) - 5 \, \sqrt {a} b^{3} c d^{2} \mathrm {sgn}\left (x\right ) + 10 \, a^{\frac {3}{2}} b^{2} d^{3} \mathrm {sgn}\left (x\right )}{4 \, {\left ({\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{2} c + 2 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )} \sqrt {a} d + b d\right )}^{2} c^{4}} \]
sqrt(a*x^2 + b*x)*a*sgn(x)/c^3 + 3/4*(b^2*c^2*sgn(x) - 8*a*b*c*d*sgn(x) + 8*a^2*d^2*sgn(x))*arctan(-((sqrt(a)*x - sqrt(a*x^2 + b*x))*c + sqrt(a)*d)/ sqrt(b*c*d - a*d^2))/(sqrt(b*c*d - a*d^2)*c^4) - 3/2*(a*b*c*sgn(x) - 2*a^2 *d*sgn(x))*log(abs(-2*(sqrt(a)*x - sqrt(a*x^2 + b*x))*sqrt(a) - b))/(sqrt( a)*c^4) + 1/4*(3*sqrt(a)*b^2*c^2*arctan(sqrt(a)*d/sqrt(b*c*d - a*d^2)) - 2 4*a^(3/2)*b*c*d*arctan(sqrt(a)*d/sqrt(b*c*d - a*d^2)) + 24*a^(5/2)*d^2*arc tan(sqrt(a)*d/sqrt(b*c*d - a*d^2)) + 6*sqrt(b*c*d - a*d^2)*a*b*c*log(abs(b )) - 12*sqrt(b*c*d - a*d^2)*a^2*d*log(abs(b)) + 5*sqrt(b*c*d - a*d^2)*a*b* c - 10*sqrt(b*c*d - a*d^2)*a^2*d)*sgn(x)/(sqrt(b*c*d - a*d^2)*sqrt(a)*c^4) + 1/4*(5*(sqrt(a)*x - sqrt(a*x^2 + b*x))^3*b^2*c^3*sgn(x) - 24*(sqrt(a)*x - sqrt(a*x^2 + b*x))^3*a*b*c^2*d*sgn(x) + 24*(sqrt(a)*x - sqrt(a*x^2 + b* x))^3*a^2*c*d^2*sgn(x) - (sqrt(a)*x - sqrt(a*x^2 + b*x))^2*sqrt(a)*b^2*c^2 *d*sgn(x) - 24*(sqrt(a)*x - sqrt(a*x^2 + b*x))^2*a^(3/2)*b*c*d^2*sgn(x) + 40*(sqrt(a)*x - sqrt(a*x^2 + b*x))^2*a^(5/2)*d^3*sgn(x) + 3*(sqrt(a)*x - s qrt(a*x^2 + b*x))*b^3*c^2*d*sgn(x) - 28*(sqrt(a)*x - sqrt(a*x^2 + b*x))*a* b^2*c*d^2*sgn(x) + 40*(sqrt(a)*x - sqrt(a*x^2 + b*x))*a^2*b*d^3*sgn(x) - 5 *sqrt(a)*b^3*c*d^2*sgn(x) + 10*a^(3/2)*b^2*d^3*sgn(x))/(((sqrt(a)*x - sqrt (a*x^2 + b*x))^2*c + 2*(sqrt(a)*x - sqrt(a*x^2 + b*x))*sqrt(a)*d + b*d)^2* c^4)
Time = 7.51 (sec) , antiderivative size = 1664, normalized size of antiderivative = 7.96 \[ \int \frac {\left (a+\frac {b}{x}\right )^{3/2}}{\left (c+\frac {d}{x}\right )^3} \, dx=\text {Too large to display} \]
- ((3*(a + b/x)^(1/2)*(3*a*b^3*c^2 + 4*a^3*b*d^2 - 7*a^2*b^2*c*d))/(4*c^3) - ((a + b/x)^(3/2)*(5*b^3*c^2 + 24*a^2*b*d^2 - 24*a*b^2*c*d))/(4*c^3) + ( 3*b*(a + b/x)^(5/2)*(4*a*d^2 - b*c*d))/(4*c^3))/((a + b/x)^2*(3*a*d^2 - 2* b*c*d) - (a + b/x)*(3*a^2*d^2 + b^2*c^2 - 4*a*b*c*d) - d^2*(a + b/x)^3 + a ^3*d^2 + a*b^2*c^2 - 2*a^2*b*c*d) - (3*a^(1/2)*atanh((27*a^(1/2)*b^7*d*(a + b/x)^(1/2))/(8*((27*a*b^7*d)/8 - (27*a^2*b^6*d^2)/(4*c))) + (27*a^(3/2)* b^6*d^2*(a + b/x)^(1/2))/(4*((27*a^2*b^6*d^2)/4 - (27*a*b^7*c*d)/8)))*(2*a *d - b*c))/c^4 - (atan((((((a + b/x)^(1/2)*(9*b^6*c^4*d + 1152*a^4*b^2*d^5 - 144*a*b^5*c^3*d^2 - 1728*a^3*b^3*c*d^4 + 864*a^2*b^4*c^2*d^3))/(8*c^6) - (3*(d*(a*d - b*c))^(1/2)*((9*a*b^4*c^9*d^2 - 12*a^2*b^3*c^8*d^3)/c^9 - ( 3*(64*b^3*c^9*d^2 - 128*a*b^2*c^8*d^3)*(a + b/x)^(1/2)*(d*(a*d - b*c))^(1/ 2)*(8*a^2*d^2 + b^2*c^2 - 8*a*b*c*d))/(64*c^6*(a*c^4*d^2 - b*c^5*d)))*(8*a ^2*d^2 + b^2*c^2 - 8*a*b*c*d))/(8*(a*c^4*d^2 - b*c^5*d)))*(d*(a*d - b*c))^ (1/2)*(8*a^2*d^2 + b^2*c^2 - 8*a*b*c*d)*3i)/(8*(a*c^4*d^2 - b*c^5*d)) + (( ((a + b/x)^(1/2)*(9*b^6*c^4*d + 1152*a^4*b^2*d^5 - 144*a*b^5*c^3*d^2 - 172 8*a^3*b^3*c*d^4 + 864*a^2*b^4*c^2*d^3))/(8*c^6) + (3*(d*(a*d - b*c))^(1/2) *((9*a*b^4*c^9*d^2 - 12*a^2*b^3*c^8*d^3)/c^9 + (3*(64*b^3*c^9*d^2 - 128*a* b^2*c^8*d^3)*(a + b/x)^(1/2)*(d*(a*d - b*c))^(1/2)*(8*a^2*d^2 + b^2*c^2 - 8*a*b*c*d))/(64*c^6*(a*c^4*d^2 - b*c^5*d)))*(8*a^2*d^2 + b^2*c^2 - 8*a*b*c *d))/(8*(a*c^4*d^2 - b*c^5*d)))*(d*(a*d - b*c))^(1/2)*(8*a^2*d^2 + b^2*...