Integrand size = 21, antiderivative size = 213 \[ \int \frac {\sqrt {a+\frac {b}{x}}}{\left (c+\frac {d}{x}\right )^3} \, dx=\frac {3 d \sqrt {a+\frac {b}{x}}}{2 c^2 \left (c+\frac {d}{x}\right )^2}+\frac {d (11 b c-12 a d) \sqrt {a+\frac {b}{x}}}{4 c^3 (b c-a d) \left (c+\frac {d}{x}\right )}+\frac {\sqrt {a+\frac {b}{x}} x}{c \left (c+\frac {d}{x}\right )^2}+\frac {\sqrt {d} \left (15 b^2 c^2-40 a b c d+24 a^2 d^2\right ) \arctan \left (\frac {\sqrt {d} \sqrt {a+\frac {b}{x}}}{\sqrt {b c-a d}}\right )}{4 c^4 (b c-a d)^{3/2}}+\frac {(b c-6 a d) \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{\sqrt {a} c^4} \]
(-6*a*d+b*c)*arctanh((a+b/x)^(1/2)/a^(1/2))/c^4/a^(1/2)+1/4*(24*a^2*d^2-40 *a*b*c*d+15*b^2*c^2)*arctan(d^(1/2)*(a+b/x)^(1/2)/(-a*d+b*c)^(1/2))*d^(1/2 )/c^4/(-a*d+b*c)^(3/2)+3/2*d*(a+b/x)^(1/2)/c^2/(c+d/x)^2+1/4*d*(-12*a*d+11 *b*c)*(a+b/x)^(1/2)/c^3/(-a*d+b*c)/(c+d/x)+x*(a+b/x)^(1/2)/c/(c+d/x)^2
Time = 1.43 (sec) , antiderivative size = 189, normalized size of antiderivative = 0.89 \[ \int \frac {\sqrt {a+\frac {b}{x}}}{\left (c+\frac {d}{x}\right )^3} \, dx=\frac {\frac {c \sqrt {a+\frac {b}{x}} x \left (-2 a d \left (6 d^2+9 c d x+2 c^2 x^2\right )+b c \left (11 d^2+17 c d x+4 c^2 x^2\right )\right )}{(b c-a d) (d+c x)^2}+\frac {\sqrt {d} \left (15 b^2 c^2-40 a b c d+24 a^2 d^2\right ) \arctan \left (\frac {\sqrt {d} \sqrt {a+\frac {b}{x}}}{\sqrt {b c-a d}}\right )}{(b c-a d)^{3/2}}+\frac {4 (b c-6 a d) \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{\sqrt {a}}}{4 c^4} \]
((c*Sqrt[a + b/x]*x*(-2*a*d*(6*d^2 + 9*c*d*x + 2*c^2*x^2) + b*c*(11*d^2 + 17*c*d*x + 4*c^2*x^2)))/((b*c - a*d)*(d + c*x)^2) + (Sqrt[d]*(15*b^2*c^2 - 40*a*b*c*d + 24*a^2*d^2)*ArcTan[(Sqrt[d]*Sqrt[a + b/x])/Sqrt[b*c - a*d]]) /(b*c - a*d)^(3/2) + (4*(b*c - 6*a*d)*ArcTanh[Sqrt[a + b/x]/Sqrt[a]])/Sqrt [a])/(4*c^4)
Time = 0.43 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.17, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {899, 110, 27, 168, 25, 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.
| \(\displaystyle \int \frac {\sqrt {a+\frac {b}{x}}}{\left (c+\frac {d}{x}\right )^3} \, dx\) |
| \(\Big \downarrow \) 899 |
| \(\displaystyle -\int \frac {\sqrt {a+\frac {b}{x}} x^2}{\left (c+\frac {d}{x}\right )^3}d\frac {1}{x}\) |
| \(\Big \downarrow \) 110 |
| \(\displaystyle \frac {x \sqrt {a+\frac {b}{x}}}{c \left (c+\frac {d}{x}\right )^2}-\frac {\int \frac {\left (b c-6 a d-\frac {5 b d}{x}\right ) x}{2 \sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right )^3}d\frac {1}{x}}{c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {x \sqrt {a+\frac {b}{x}}}{c \left (c+\frac {d}{x}\right )^2}-\frac {\int \frac {\left (b c-6 a d-\frac {5 b d}{x}\right ) x}{\sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right )^3}d\frac {1}{x}}{2 c}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {x \sqrt {a+\frac {b}{x}}}{c \left (c+\frac {d}{x}\right )^2}-\frac {-\frac {\int -\frac {(b c-a d) \left (2 (b c-6 a d)-\frac {9 b d}{x}\right ) x}{\sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right )^2}d\frac {1}{x}}{2 c (b c-a d)}-\frac {3 d \sqrt {a+\frac {b}{x}}}{c \left (c+\frac {d}{x}\right )^2}}{2 c}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {x \sqrt {a+\frac {b}{x}}}{c \left (c+\frac {d}{x}\right )^2}-\frac {\frac {\int \frac {(b c-a d) \left (2 (b c-6 a d)-\frac {9 b d}{x}\right ) x}{\sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right )^2}d\frac {1}{x}}{2 c (b c-a d)}-\frac {3 d \sqrt {a+\frac {b}{x}}}{c \left (c+\frac {d}{x}\right )^2}}{2 c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {x \sqrt {a+\frac {b}{x}}}{c \left (c+\frac {d}{x}\right )^2}-\frac {\frac {\int \frac {\left (2 (b c-6 a d)-\frac {9 b d}{x}\right ) x}{\sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right )^2}d\frac {1}{x}}{2 c}-\frac {3 d \sqrt {a+\frac {b}{x}}}{c \left (c+\frac {d}{x}\right )^2}}{2 c}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {x \sqrt {a+\frac {b}{x}}}{c \left (c+\frac {d}{x}\right )^2}-\frac {\frac {-\frac {\int -\frac {\left (4 (b c-6 a d) (b c-a d)-\frac {b d (11 b c-12 a d)}{x}\right ) x}{2 \sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right )}d\frac {1}{x}}{c (b c-a d)}-\frac {d \sqrt {a+\frac {b}{x}} (11 b c-12 a d)}{c \left (c+\frac {d}{x}\right ) (b c-a d)}}{2 c}-\frac {3 d \sqrt {a+\frac {b}{x}}}{c \left (c+\frac {d}{x}\right )^2}}{2 c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {x \sqrt {a+\frac {b}{x}}}{c \left (c+\frac {d}{x}\right )^2}-\frac {\frac {\frac {\int \frac {\left (4 (b c-6 a d) (b c-a d)-\frac {b d (11 b c-12 a d)}{x}\right ) x}{\sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right )}d\frac {1}{x}}{2 c (b c-a d)}-\frac {d \sqrt {a+\frac {b}{x}} (11 b c-12 a d)}{c \left (c+\frac {d}{x}\right ) (b c-a d)}}{2 c}-\frac {3 d \sqrt {a+\frac {b}{x}}}{c \left (c+\frac {d}{x}\right )^2}}{2 c}\) |
| \(\Big \downarrow \) 174 |
| \(\displaystyle \frac {x \sqrt {a+\frac {b}{x}}}{c \left (c+\frac {d}{x}\right )^2}-\frac {\frac {\frac {\frac {4 (b c-6 a d) (b c-a d) \int \frac {x}{\sqrt {a+\frac {b}{x}}}d\frac {1}{x}}{c}-\frac {d \left (24 a^2 d^2-40 a b c d+15 b^2 c^2\right ) \int \frac {1}{\sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right )}d\frac {1}{x}}{c}}{2 c (b c-a d)}-\frac {d \sqrt {a+\frac {b}{x}} (11 b c-12 a d)}{c \left (c+\frac {d}{x}\right ) (b c-a d)}}{2 c}-\frac {3 d \sqrt {a+\frac {b}{x}}}{c \left (c+\frac {d}{x}\right )^2}}{2 c}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {x \sqrt {a+\frac {b}{x}}}{c \left (c+\frac {d}{x}\right )^2}-\frac {\frac {\frac {\frac {8 (b c-6 a d) (b c-a d) \int \frac {1}{\frac {1}{b x^2}-\frac {a}{b}}d\sqrt {a+\frac {b}{x}}}{b c}-\frac {2 d \left (24 a^2 d^2-40 a b c d+15 b^2 c^2\right ) \int \frac {1}{c-\frac {a d}{b}+\frac {d}{b x^2}}d\sqrt {a+\frac {b}{x}}}{b c}}{2 c (b c-a d)}-\frac {d \sqrt {a+\frac {b}{x}} (11 b c-12 a d)}{c \left (c+\frac {d}{x}\right ) (b c-a d)}}{2 c}-\frac {3 d \sqrt {a+\frac {b}{x}}}{c \left (c+\frac {d}{x}\right )^2}}{2 c}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {x \sqrt {a+\frac {b}{x}}}{c \left (c+\frac {d}{x}\right )^2}-\frac {\frac {\frac {\frac {8 (b c-6 a d) (b c-a d) \int \frac {1}{\frac {1}{b x^2}-\frac {a}{b}}d\sqrt {a+\frac {b}{x}}}{b c}-\frac {2 \sqrt {d} \left (24 a^2 d^2-40 a b c d+15 b^2 c^2\right ) \arctan \left (\frac {\sqrt {d} \sqrt {a+\frac {b}{x}}}{\sqrt {b c-a d}}\right )}{c \sqrt {b c-a d}}}{2 c (b c-a d)}-\frac {d \sqrt {a+\frac {b}{x}} (11 b c-12 a d)}{c \left (c+\frac {d}{x}\right ) (b c-a d)}}{2 c}-\frac {3 d \sqrt {a+\frac {b}{x}}}{c \left (c+\frac {d}{x}\right )^2}}{2 c}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {x \sqrt {a+\frac {b}{x}}}{c \left (c+\frac {d}{x}\right )^2}-\frac {\frac {\frac {-\frac {2 \sqrt {d} \left (24 a^2 d^2-40 a b c d+15 b^2 c^2\right ) \arctan \left (\frac {\sqrt {d} \sqrt {a+\frac {b}{x}}}{\sqrt {b c-a d}}\right )}{c \sqrt {b c-a d}}-\frac {8 \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right ) (b c-6 a d) (b c-a d)}{\sqrt {a} c}}{2 c (b c-a d)}-\frac {d \sqrt {a+\frac {b}{x}} (11 b c-12 a d)}{c \left (c+\frac {d}{x}\right ) (b c-a d)}}{2 c}-\frac {3 d \sqrt {a+\frac {b}{x}}}{c \left (c+\frac {d}{x}\right )^2}}{2 c}\) |
(Sqrt[a + b/x]*x)/(c*(c + d/x)^2) - ((-3*d*Sqrt[a + b/x])/(c*(c + d/x)^2) + (-((d*(11*b*c - 12*a*d)*Sqrt[a + b/x])/(c*(b*c - a*d)*(c + d/x))) + ((-2 *Sqrt[d]*(15*b^2*c^2 - 40*a*b*c*d + 24*a^2*d^2)*ArcTan[(Sqrt[d]*Sqrt[a + b /x])/Sqrt[b*c - a*d]])/(c*Sqrt[b*c - a*d]) - (8*(b*c - 6*a*d)*(b*c - a*d)* ArcTanh[Sqrt[a + b/x]/Sqrt[a]])/(Sqrt[a]*c))/(2*c*(b*c - a*d)))/(2*c))/(2* c)
3.3.30.3.1 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[(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/((m + 1)*(b*e - a*f))), x] - Simp[1/((m + 1)*(b*e - a*f)) Int[(a + b*x)^(m + 1) *(c + d*x)^(n - 1)*(e + f*x)^p*Simp[d*e*n + c*f*(m + p + 2) + d*f*(m + n + p + 2)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && Gt Q[n, 0] && (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. \(968\) vs. \(2(185)=370\).
Time = 0.26 (sec) , antiderivative size = 969, normalized size of antiderivative = 4.55
| method | result | size |
| risch | \(\frac {x \sqrt {\frac {a x +b}{x}}}{c^{3}}-\frac {\left (\frac {\left (6 a d -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 d^{2} \left (4 a d -3 b c \right ) \left (-\frac {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}}}}{\left (a d -b c \right ) d \left (x +\frac {d}{c}\right )}-\frac {\left (2 a d -b c \right ) c \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 )}{2 \left (a d -b c \right ) d \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}}\right )}{c^{3}}-\frac {2 d^{3} \left (a d -b c \right ) \left (-\frac {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}}}}{2 \left (a d -b c \right ) d \left (x +\frac {d}{c}\right )^{2}}+\frac {3 \left (2 a d -b c \right ) c \left (-\frac {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}}}}{\left (a d -b c \right ) d \left (x +\frac {d}{c}\right )}-\frac {\left (2 a d -b c \right ) c \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 )}{2 \left (a d -b c \right ) d \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}}\right )}{4 \left (a d -b c \right ) d}+\frac {a \,c^{2} \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 )}{2 \left (a d -b c \right ) d \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}}\right )}{c^{4}}+\frac {6 d \left (2 a d -b c \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^{3} \left (a x +b \right )}\) | \(969\) |
| default | \(\text {Expression too large to display}\) | \(1972\) |
1/c^3*x*((a*x+b)/x)^(1/2)-1/2/c^3*((6*a*d-b*c)/c*ln((1/2*b+a*x)/a^(1/2)+(a *x^2+b*x)^(1/2))/a^(1/2)+2/c^3*d^2*(4*a*d-3*b*c)*(-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^3*(a*d-b*c)/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)))+6/c^2*d*(2*a*d-b*c)/((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. 428 vs. \(2 (185) = 370\).
Time = 0.41 (sec) , antiderivative size = 1749, normalized size of antiderivative = 8.21 \[ \int \frac {\sqrt {a+\frac {b}{x}}}{\left (c+\frac {d}{x}\right )^3} \, dx=\text {Too large to display} \]
[-1/8*(4*(b^2*c^2*d^2 - 7*a*b*c*d^3 + 6*a^2*d^4 + (b^2*c^4 - 7*a*b*c^3*d + 6*a^2*c^2*d^2)*x^2 + 2*(b^2*c^3*d - 7*a*b*c^2*d^2 + 6*a^2*c*d^3)*x)*sqrt( a)*log(2*a*x - 2*sqrt(a)*x*sqrt((a*x + b)/x) + b) + (15*a*b^2*c^2*d^2 - 40 *a^2*b*c*d^3 + 24*a^3*d^4 + (15*a*b^2*c^4 - 40*a^2*b*c^3*d + 24*a^3*c^2*d^ 2)*x^2 + 2*(15*a*b^2*c^3*d - 40*a^2*b*c^2*d^2 + 24*a^3*c*d^3)*x)*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)) - 2*(4*(a*b*c^4 - a^2*c^3*d)*x^3 + (17* a*b*c^3*d - 18*a^2*c^2*d^2)*x^2 + (11*a*b*c^2*d^2 - 12*a^2*c*d^3)*x)*sqrt( (a*x + b)/x))/(a*b*c^5*d^2 - a^2*c^4*d^3 + (a*b*c^7 - a^2*c^6*d)*x^2 + 2*( a*b*c^6*d - a^2*c^5*d^2)*x), 1/4*((15*a*b^2*c^2*d^2 - 40*a^2*b*c*d^3 + 24* a^3*d^4 + (15*a*b^2*c^4 - 40*a^2*b*c^3*d + 24*a^3*c^2*d^2)*x^2 + 2*(15*a*b ^2*c^3*d - 40*a^2*b*c^2*d^2 + 24*a^3*c*d^3)*x)*sqrt(d/(b*c - a*d))*arctan( -(b*c - a*d)*x*sqrt(d/(b*c - a*d))*sqrt((a*x + b)/x)/(a*d*x + b*d)) - 2*(b ^2*c^2*d^2 - 7*a*b*c*d^3 + 6*a^2*d^4 + (b^2*c^4 - 7*a*b*c^3*d + 6*a^2*c^2* d^2)*x^2 + 2*(b^2*c^3*d - 7*a*b*c^2*d^2 + 6*a^2*c*d^3)*x)*sqrt(a)*log(2*a* x - 2*sqrt(a)*x*sqrt((a*x + b)/x) + b) + (4*(a*b*c^4 - a^2*c^3*d)*x^3 + (1 7*a*b*c^3*d - 18*a^2*c^2*d^2)*x^2 + (11*a*b*c^2*d^2 - 12*a^2*c*d^3)*x)*sqr t((a*x + b)/x))/(a*b*c^5*d^2 - a^2*c^4*d^3 + (a*b*c^7 - a^2*c^6*d)*x^2 + 2 *(a*b*c^6*d - a^2*c^5*d^2)*x), -1/8*(8*(b^2*c^2*d^2 - 7*a*b*c*d^3 + 6*a^2* d^4 + (b^2*c^4 - 7*a*b*c^3*d + 6*a^2*c^2*d^2)*x^2 + 2*(b^2*c^3*d - 7*a*...
\[ \int \frac {\sqrt {a+\frac {b}{x}}}{\left (c+\frac {d}{x}\right )^3} \, dx=\int \frac {x^{3} \sqrt {a + \frac {b}{x}}}{\left (c x + d\right )^{3}}\, dx \]
\[ \int \frac {\sqrt {a+\frac {b}{x}}}{\left (c+\frac {d}{x}\right )^3} \, dx=\int { \frac {\sqrt {a + \frac {b}{x}}}{{\left (c + \frac {d}{x}\right )}^{3}} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 809 vs. \(2 (185) = 370\).
Time = 0.34 (sec) , antiderivative size = 809, normalized size of antiderivative = 3.80 \[ \int \frac {\sqrt {a+\frac {b}{x}}}{\left (c+\frac {d}{x}\right )^3} \, dx=-\frac {{\left (15 \, \sqrt {a} b^{2} c^{2} d \arctan \left (\frac {\sqrt {a} d}{\sqrt {b c d - a d^{2}}}\right ) - 40 \, a^{\frac {3}{2}} b c d^{2} \arctan \left (\frac {\sqrt {a} d}{\sqrt {b c d - a d^{2}}}\right ) + 24 \, a^{\frac {5}{2}} d^{3} \arctan \left (\frac {\sqrt {a} d}{\sqrt {b c d - a d^{2}}}\right ) - 2 \, \sqrt {b c d - a d^{2}} b^{2} c^{2} \log \left ({\left | b \right |}\right ) + 14 \, \sqrt {b c d - a d^{2}} a b c d \log \left ({\left | b \right |}\right ) - 12 \, \sqrt {b c d - a d^{2}} a^{2} d^{2} \log \left ({\left | b \right |}\right ) + 9 \, \sqrt {b c d - a d^{2}} a b c d - 10 \, \sqrt {b c d - a d^{2}} a^{2} d^{2}\right )} \mathrm {sgn}\left (x\right )}{4 \, {\left (\sqrt {b c d - a d^{2}} \sqrt {a} b c^{5} - \sqrt {b c d - a d^{2}} a^{\frac {3}{2}} c^{4} d\right )}} - \frac {{\left (15 \, b^{2} c^{2} d \mathrm {sgn}\left (x\right ) - 40 \, a b c d^{2} \mathrm {sgn}\left (x\right ) + 24 \, a^{2} d^{3} \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 \, {\left (b c^{5} - a c^{4} d\right )} \sqrt {b c d - a d^{2}}} + \frac {\sqrt {a x^{2} + b x} \mathrm {sgn}\left (x\right )}{c^{3}} - \frac {9 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{3} b^{2} c^{3} d \mathrm {sgn}\left (x\right ) - 32 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{3} a b c^{2} d^{2} \mathrm {sgn}\left (x\right ) + 24 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{3} a^{2} c d^{3} \mathrm {sgn}\left (x\right ) + 3 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{2} \sqrt {a} b^{2} c^{2} d^{2} \mathrm {sgn}\left (x\right ) - 40 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{2} a^{\frac {3}{2}} b c d^{3} \mathrm {sgn}\left (x\right ) + 40 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{2} a^{\frac {5}{2}} d^{4} \mathrm {sgn}\left (x\right ) + 7 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )} b^{3} c^{2} d^{2} \mathrm {sgn}\left (x\right ) - 44 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )} a b^{2} c d^{3} \mathrm {sgn}\left (x\right ) + 40 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )} a^{2} b d^{4} \mathrm {sgn}\left (x\right ) - 9 \, \sqrt {a} b^{3} c d^{3} \mathrm {sgn}\left (x\right ) + 10 \, a^{\frac {3}{2}} b^{2} d^{4} \mathrm {sgn}\left (x\right )}{4 \, {\left (b c^{5} - a c^{4} d\right )} {\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}} - \frac {{\left (b c \mathrm {sgn}\left (x\right ) - 6 \, a 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}} \]
-1/4*(15*sqrt(a)*b^2*c^2*d*arctan(sqrt(a)*d/sqrt(b*c*d - a*d^2)) - 40*a^(3 /2)*b*c*d^2*arctan(sqrt(a)*d/sqrt(b*c*d - a*d^2)) + 24*a^(5/2)*d^3*arctan( sqrt(a)*d/sqrt(b*c*d - a*d^2)) - 2*sqrt(b*c*d - a*d^2)*b^2*c^2*log(abs(b)) + 14*sqrt(b*c*d - a*d^2)*a*b*c*d*log(abs(b)) - 12*sqrt(b*c*d - a*d^2)*a^2 *d^2*log(abs(b)) + 9*sqrt(b*c*d - a*d^2)*a*b*c*d - 10*sqrt(b*c*d - a*d^2)* a^2*d^2)*sgn(x)/(sqrt(b*c*d - a*d^2)*sqrt(a)*b*c^5 - sqrt(b*c*d - a*d^2)*a ^(3/2)*c^4*d) - 1/4*(15*b^2*c^2*d*sgn(x) - 40*a*b*c*d^2*sgn(x) + 24*a^2*d^ 3*sgn(x))*arctan(-((sqrt(a)*x - sqrt(a*x^2 + b*x))*c + sqrt(a)*d)/sqrt(b*c *d - a*d^2))/((b*c^5 - a*c^4*d)*sqrt(b*c*d - a*d^2)) + sqrt(a*x^2 + b*x)*s gn(x)/c^3 - 1/4*(9*(sqrt(a)*x - sqrt(a*x^2 + b*x))^3*b^2*c^3*d*sgn(x) - 32 *(sqrt(a)*x - sqrt(a*x^2 + b*x))^3*a*b*c^2*d^2*sgn(x) + 24*(sqrt(a)*x - sq rt(a*x^2 + b*x))^3*a^2*c*d^3*sgn(x) + 3*(sqrt(a)*x - sqrt(a*x^2 + b*x))^2* sqrt(a)*b^2*c^2*d^2*sgn(x) - 40*(sqrt(a)*x - sqrt(a*x^2 + b*x))^2*a^(3/2)* b*c*d^3*sgn(x) + 40*(sqrt(a)*x - sqrt(a*x^2 + b*x))^2*a^(5/2)*d^4*sgn(x) + 7*(sqrt(a)*x - sqrt(a*x^2 + b*x))*b^3*c^2*d^2*sgn(x) - 44*(sqrt(a)*x - sq rt(a*x^2 + b*x))*a*b^2*c*d^3*sgn(x) + 40*(sqrt(a)*x - sqrt(a*x^2 + b*x))*a ^2*b*d^4*sgn(x) - 9*sqrt(a)*b^3*c*d^3*sgn(x) + 10*a^(3/2)*b^2*d^4*sgn(x))/ ((b*c^5 - a*c^4*d)*((sqrt(a)*x - sqrt(a*x^2 + b*x))^2*c + 2*(sqrt(a)*x - s qrt(a*x^2 + b*x))*sqrt(a)*d + b*d)^2) - 1/2*(b*c*sgn(x) - 6*a*d*sgn(x))*lo g(abs(2*(sqrt(a)*x - sqrt(a*x^2 + b*x))*sqrt(a) + b))/(sqrt(a)*c^4)
Time = 7.94 (sec) , antiderivative size = 1895, normalized size of antiderivative = 8.90 \[ \int \frac {\sqrt {a+\frac {b}{x}}}{\left (c+\frac {d}{x}\right )^3} \, dx=\text {Too large to display} \]
(log((a + b/x)^(1/2)*(d*(a*d - b*c)^3)^(1/2) - a^2*d^2 - b^2*c^2 + 2*a*b*c *d)*(d*(a*d - b*c)^3)^(1/2)*(3*a^2*d^2 + (15*b^2*c^2)/8 - 5*a*b*c*d))/(b^3 *c^7 - a^3*c^4*d^3 + 3*a^2*b*c^5*d^2 - 3*a*b^2*c^6*d) - ((b*(a + b/x)^(1/2 )*(12*a^2*d^2 + 4*b^2*c^2 - 17*a*b*c*d))/(4*c^3) + (b*(a + b/x)^(5/2)*(12* a*d^3 - 11*b*c*d^2))/(4*c^3*(a*d - b*c)) - (d*(a + b/x)^(3/2)*(17*b^3*c^2 + 24*a^2*b*d^2 - 40*a*b^2*c*d))/(4*c^3*(a*d - b*c)))/((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) - (log((a + b/x)^(1/2)*(d*(a*d - b* c)^3)^(1/2) + a^2*d^2 + b^2*c^2 - 2*a*b*c*d)*(d*(a*d - b*c)^3)^(1/2)*(24*a ^2*d^2 + 15*b^2*c^2 - 40*a*b*c*d))/(8*(b^3*c^7 - a^3*c^4*d^3 + 3*a^2*b*c^5 *d^2 - 3*a*b^2*c^6*d)) - (atan((((((a + b/x)^(1/2)*(1152*a^4*b^2*d^7 + 241 *b^6*c^4*d^3 - 1424*a*b^5*c^3*d^4 - 3264*a^3*b^3*c*d^6 + 3296*a^2*b^4*c^2* d^5))/(8*(b^2*c^8 + a^2*c^6*d^2 - 2*a*b*c^7*d)) - ((6*a*d - b*c)*((4*b^6*c ^11*d^2 - 21*a*b^5*c^10*d^3 + 29*a^2*b^4*c^9*d^4 - 12*a^3*b^3*c^8*d^5)/(b^ 2*c^11 + a^2*c^9*d^2 - 2*a*b*c^10*d) - ((a + b/x)^(1/2)*(6*a*d - b*c)*(64* b^5*c^11*d^2 - 256*a*b^4*c^10*d^3 + 320*a^2*b^3*c^9*d^4 - 128*a^3*b^2*c^8* d^5))/(16*a^(1/2)*c^4*(b^2*c^8 + a^2*c^6*d^2 - 2*a*b*c^7*d))))/(2*a^(1/2)* c^4))*(6*a*d - b*c)*1i)/(2*a^(1/2)*c^4) + ((((a + b/x)^(1/2)*(1152*a^4*b^2 *d^7 + 241*b^6*c^4*d^3 - 1424*a*b^5*c^3*d^4 - 3264*a^3*b^3*c*d^6 + 3296*a^ 2*b^4*c^2*d^5))/(8*(b^2*c^8 + a^2*c^6*d^2 - 2*a*b*c^7*d)) + ((6*a*d - b...