Integrand size = 22, antiderivative size = 279 \[ \int \frac {\sqrt {a+b x}}{x^5 \sqrt {c+d x}} \, dx=-\frac {\sqrt {a+b x} \sqrt {c+d x}}{4 c x^4}-\frac {(b c-7 a d) \sqrt {a+b x} \sqrt {c+d x}}{24 a c^2 x^3}+\frac {\left (5 b^2 c^2+6 a b c d-35 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{96 a^2 c^3 x^2}-\frac {\left (15 b^3 c^3+17 a b^2 c^2 d+25 a^2 b c d^2-105 a^3 d^3\right ) \sqrt {a+b x} \sqrt {c+d x}}{192 a^3 c^4 x}+\frac {(b c-a d) \left (5 b^3 c^3+9 a b^2 c^2 d+15 a^2 b c d^2+35 a^3 d^3\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{64 a^{7/2} c^{9/2}} \]
1/64*(-a*d+b*c)*(35*a^3*d^3+15*a^2*b*c*d^2+9*a*b^2*c^2*d+5*b^3*c^3)*arctan h(c^(1/2)*(b*x+a)^(1/2)/a^(1/2)/(d*x+c)^(1/2))/a^(7/2)/c^(9/2)-1/4*(b*x+a) ^(1/2)*(d*x+c)^(1/2)/c/x^4-1/24*(-7*a*d+b*c)*(b*x+a)^(1/2)*(d*x+c)^(1/2)/a /c^2/x^3+1/96*(-35*a^2*d^2+6*a*b*c*d+5*b^2*c^2)*(b*x+a)^(1/2)*(d*x+c)^(1/2 )/a^2/c^3/x^2-1/192*(-105*a^3*d^3+25*a^2*b*c*d^2+17*a*b^2*c^2*d+15*b^3*c^3 )*(b*x+a)^(1/2)*(d*x+c)^(1/2)/a^3/c^4/x
Time = 10.19 (sec) , antiderivative size = 244, normalized size of antiderivative = 0.87 \[ \int \frac {\sqrt {a+b x}}{x^5 \sqrt {c+d x}} \, dx=\frac {-48 (a+b x)^{3/2} \sqrt {c+d x}+\frac {8 (5 b c+7 a d) x (a+b x)^{3/2} \sqrt {c+d x}}{a c}-\frac {2 \left (15 b^2 c^2+22 a b c d+35 a^2 d^2\right ) x^2 (a+b x)^{3/2} \sqrt {c+d x}}{a^2 c^2}+\frac {3 \left (5 b^3 c^3+9 a b^2 c^2 d+15 a^2 b c d^2+35 a^3 d^3\right ) x^3 \left (\sqrt {a} \sqrt {c} \sqrt {a+b x} \sqrt {c+d x}+(b c-a d) x \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )\right )}{a^{5/2} c^{7/2}}}{192 a c x^4} \]
(-48*(a + b*x)^(3/2)*Sqrt[c + d*x] + (8*(5*b*c + 7*a*d)*x*(a + b*x)^(3/2)* Sqrt[c + d*x])/(a*c) - (2*(15*b^2*c^2 + 22*a*b*c*d + 35*a^2*d^2)*x^2*(a + b*x)^(3/2)*Sqrt[c + d*x])/(a^2*c^2) + (3*(5*b^3*c^3 + 9*a*b^2*c^2*d + 15*a ^2*b*c*d^2 + 35*a^3*d^3)*x^3*(Sqrt[a]*Sqrt[c]*Sqrt[a + b*x]*Sqrt[c + d*x] + (b*c - a*d)*x*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])) /(a^(5/2)*c^(7/2)))/(192*a*c*x^4)
Time = 0.40 (sec) , antiderivative size = 299, normalized size of antiderivative = 1.07, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.455, Rules used = {110, 27, 168, 27, 168, 27, 168, 27, 104, 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.
| \(\displaystyle \int \frac {\sqrt {a+b x}}{x^5 \sqrt {c+d x}} \, dx\) |
| \(\Big \downarrow \) 110 |
| \(\displaystyle \frac {\int \frac {b c-7 a d-6 b d x}{2 x^4 \sqrt {a+b x} \sqrt {c+d x}}dx}{4 c}-\frac {\sqrt {a+b x} \sqrt {c+d x}}{4 c x^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {b c-7 a d-6 b d x}{x^4 \sqrt {a+b x} \sqrt {c+d x}}dx}{8 c}-\frac {\sqrt {a+b x} \sqrt {c+d x}}{4 c x^4}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {-\frac {\int \frac {5 b^2 c^2+6 a b d c-35 a^2 d^2+4 b d (b c-7 a d) x}{2 x^3 \sqrt {a+b x} \sqrt {c+d x}}dx}{3 a c}-\frac {\sqrt {a+b x} \sqrt {c+d x} (b c-7 a d)}{3 a c x^3}}{8 c}-\frac {\sqrt {a+b x} \sqrt {c+d x}}{4 c x^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {\int \frac {5 b^2 c^2+6 a b d c-35 a^2 d^2+4 b d (b c-7 a d) x}{x^3 \sqrt {a+b x} \sqrt {c+d x}}dx}{6 a c}-\frac {\sqrt {a+b x} \sqrt {c+d x} (b c-7 a d)}{3 a c x^3}}{8 c}-\frac {\sqrt {a+b x} \sqrt {c+d x}}{4 c x^4}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {-\frac {-\frac {\int \frac {15 b^3 c^3+17 a b^2 d c^2+25 a^2 b d^2 c-105 a^3 d^3+2 b d \left (5 b^2 c^2+6 a b d c-35 a^2 d^2\right ) x}{2 x^2 \sqrt {a+b x} \sqrt {c+d x}}dx}{2 a c}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (\frac {5 b^2 c}{a}-\frac {35 a d^2}{c}+6 b d\right )}{2 x^2}}{6 a c}-\frac {\sqrt {a+b x} \sqrt {c+d x} (b c-7 a d)}{3 a c x^3}}{8 c}-\frac {\sqrt {a+b x} \sqrt {c+d x}}{4 c x^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {-\frac {\int \frac {15 b^3 c^3+17 a b^2 d c^2+25 a^2 b d^2 c-105 a^3 d^3+2 b d \left (5 b^2 c^2+6 a b d c-35 a^2 d^2\right ) x}{x^2 \sqrt {a+b x} \sqrt {c+d x}}dx}{4 a c}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (\frac {5 b^2 c}{a}-\frac {35 a d^2}{c}+6 b d\right )}{2 x^2}}{6 a c}-\frac {\sqrt {a+b x} \sqrt {c+d x} (b c-7 a d)}{3 a c x^3}}{8 c}-\frac {\sqrt {a+b x} \sqrt {c+d x}}{4 c x^4}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {-\frac {-\frac {-\frac {\int \frac {3 (b c-a d) \left (5 b^3 c^3+9 a b^2 d c^2+15 a^2 b d^2 c+35 a^3 d^3\right )}{2 x \sqrt {a+b x} \sqrt {c+d x}}dx}{a c}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (-105 a^3 d^3+25 a^2 b c d^2+17 a b^2 c^2 d+15 b^3 c^3\right )}{a c x}}{4 a c}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (\frac {5 b^2 c}{a}-\frac {35 a d^2}{c}+6 b d\right )}{2 x^2}}{6 a c}-\frac {\sqrt {a+b x} \sqrt {c+d x} (b c-7 a d)}{3 a c x^3}}{8 c}-\frac {\sqrt {a+b x} \sqrt {c+d x}}{4 c x^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {-\frac {-\frac {3 (b c-a d) \left (35 a^3 d^3+15 a^2 b c d^2+9 a b^2 c^2 d+5 b^3 c^3\right ) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}}dx}{2 a c}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (-105 a^3 d^3+25 a^2 b c d^2+17 a b^2 c^2 d+15 b^3 c^3\right )}{a c x}}{4 a c}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (\frac {5 b^2 c}{a}-\frac {35 a d^2}{c}+6 b d\right )}{2 x^2}}{6 a c}-\frac {\sqrt {a+b x} \sqrt {c+d x} (b c-7 a d)}{3 a c x^3}}{8 c}-\frac {\sqrt {a+b x} \sqrt {c+d x}}{4 c x^4}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle \frac {-\frac {-\frac {-\frac {3 (b c-a d) \left (35 a^3 d^3+15 a^2 b c d^2+9 a b^2 c^2 d+5 b^3 c^3\right ) \int \frac {1}{\frac {c (a+b x)}{c+d x}-a}d\frac {\sqrt {a+b x}}{\sqrt {c+d x}}}{a c}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (-105 a^3 d^3+25 a^2 b c d^2+17 a b^2 c^2 d+15 b^3 c^3\right )}{a c x}}{4 a c}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (\frac {5 b^2 c}{a}-\frac {35 a d^2}{c}+6 b d\right )}{2 x^2}}{6 a c}-\frac {\sqrt {a+b x} \sqrt {c+d x} (b c-7 a d)}{3 a c x^3}}{8 c}-\frac {\sqrt {a+b x} \sqrt {c+d x}}{4 c x^4}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {-\frac {-\frac {\frac {3 (b c-a d) \left (35 a^3 d^3+15 a^2 b c d^2+9 a b^2 c^2 d+5 b^3 c^3\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{a^{3/2} c^{3/2}}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (-105 a^3 d^3+25 a^2 b c d^2+17 a b^2 c^2 d+15 b^3 c^3\right )}{a c x}}{4 a c}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (\frac {5 b^2 c}{a}-\frac {35 a d^2}{c}+6 b d\right )}{2 x^2}}{6 a c}-\frac {\sqrt {a+b x} \sqrt {c+d x} (b c-7 a d)}{3 a c x^3}}{8 c}-\frac {\sqrt {a+b x} \sqrt {c+d x}}{4 c x^4}\) |
-1/4*(Sqrt[a + b*x]*Sqrt[c + d*x])/(c*x^4) + (-1/3*((b*c - 7*a*d)*Sqrt[a + b*x]*Sqrt[c + d*x])/(a*c*x^3) - (-1/2*(((5*b^2*c)/a + 6*b*d - (35*a*d^2)/ c)*Sqrt[a + b*x]*Sqrt[c + d*x])/x^2 - (-(((15*b^3*c^3 + 17*a*b^2*c^2*d + 2 5*a^2*b*c*d^2 - 105*a^3*d^3)*Sqrt[a + b*x]*Sqrt[c + d*x])/(a*c*x)) + (3*(b *c - a*d)*(5*b^3*c^3 + 9*a*b^2*c^2*d + 15*a^2*b*c*d^2 + 35*a^3*d^3)*ArcTan h[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(a^(3/2)*c^(3/2)))/(4* a*c))/(6*a*c))/(8*c)
3.6.84.3.1 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_))/((e_.) + (f_.)*(x _)), x_] :> With[{q = Denominator[m]}, Simp[q Subst[Int[x^(q*(m + 1) - 1) /(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^(1/q)], x] ] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && L tQ[-1, m, 0] && SimplerQ[a + b*x, c + d*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[((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]
Leaf count of result is larger than twice the leaf count of optimal. \(592\) vs. \(2(241)=482\).
Time = 1.52 (sec) , antiderivative size = 593, normalized size of antiderivative = 2.13
| method | result | size |
| default | \(-\frac {\sqrt {b x +a}\, \sqrt {d x +c}\, \left (105 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a^{4} d^{4} x^{4}-60 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a^{3} b c \,d^{3} x^{4}-18 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a^{2} b^{2} c^{2} d^{2} x^{4}-12 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a \,b^{3} c^{3} d \,x^{4}-15 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) b^{4} c^{4} x^{4}-210 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a^{3} d^{3} x^{3}+50 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a^{2} b c \,d^{2} x^{3}+34 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a \,b^{2} c^{2} d \,x^{3}+30 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, b^{3} c^{3} x^{3}+140 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, a^{3} c \,d^{2} x^{2}-24 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, a^{2} b \,c^{2} d \,x^{2}-20 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, a \,b^{2} c^{3} x^{2}-112 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, a^{3} c^{2} d x +16 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, a^{2} b \,c^{3} x +96 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a^{3} c^{3} \sqrt {a c}\right )}{384 a^{3} c^{4} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, x^{4} \sqrt {a c}}\) | \(593\) |
-1/384*(b*x+a)^(1/2)*(d*x+c)^(1/2)/a^3/c^4*(105*ln((a*d*x+b*c*x+2*(a*c)^(1 /2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*a^4*d^4*x^4-60*ln((a*d*x+b*c*x+2*(a* c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*a^3*b*c*d^3*x^4-18*ln((a*d*x+b* c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*a^2*b^2*c^2*d^2*x^4-12 *ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*a*b^3*c^3 *d*x^4-15*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)* b^4*c^4*x^4-210*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a^3*d^3*x^3+50*(a*c)^( 1/2)*((b*x+a)*(d*x+c))^(1/2)*a^2*b*c*d^2*x^3+34*(a*c)^(1/2)*((b*x+a)*(d*x+ c))^(1/2)*a*b^2*c^2*d*x^3+30*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*b^3*c^3*x ^3+140*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)*a^3*c*d^2*x^2-24*((b*x+a)*(d*x+ c))^(1/2)*(a*c)^(1/2)*a^2*b*c^2*d*x^2-20*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/ 2)*a*b^2*c^3*x^2-112*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)*a^3*c^2*d*x+16*(( b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)*a^2*b*c^3*x+96*((b*x+a)*(d*x+c))^(1/2)*a ^3*c^3*(a*c)^(1/2))/((b*x+a)*(d*x+c))^(1/2)/x^4/(a*c)^(1/2)
Time = 1.53 (sec) , antiderivative size = 570, normalized size of antiderivative = 2.04 \[ \int \frac {\sqrt {a+b x}}{x^5 \sqrt {c+d x}} \, dx=\left [-\frac {3 \, {\left (5 \, b^{4} c^{4} + 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} + 20 \, a^{3} b c d^{3} - 35 \, a^{4} d^{4}\right )} \sqrt {a c} x^{4} \log \left (\frac {8 \, a^{2} c^{2} + {\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} - 4 \, {\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {a c} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (a b c^{2} + a^{2} c d\right )} x}{x^{2}}\right ) + 4 \, {\left (48 \, a^{4} c^{4} + {\left (15 \, a b^{3} c^{4} + 17 \, a^{2} b^{2} c^{3} d + 25 \, a^{3} b c^{2} d^{2} - 105 \, a^{4} c d^{3}\right )} x^{3} - 2 \, {\left (5 \, a^{2} b^{2} c^{4} + 6 \, a^{3} b c^{3} d - 35 \, a^{4} c^{2} d^{2}\right )} x^{2} + 8 \, {\left (a^{3} b c^{4} - 7 \, a^{4} c^{3} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{768 \, a^{4} c^{5} x^{4}}, -\frac {3 \, {\left (5 \, b^{4} c^{4} + 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} + 20 \, a^{3} b c d^{3} - 35 \, a^{4} d^{4}\right )} \sqrt {-a c} x^{4} \arctan \left (\frac {{\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {-a c} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (a b c d x^{2} + a^{2} c^{2} + {\left (a b c^{2} + a^{2} c d\right )} x\right )}}\right ) + 2 \, {\left (48 \, a^{4} c^{4} + {\left (15 \, a b^{3} c^{4} + 17 \, a^{2} b^{2} c^{3} d + 25 \, a^{3} b c^{2} d^{2} - 105 \, a^{4} c d^{3}\right )} x^{3} - 2 \, {\left (5 \, a^{2} b^{2} c^{4} + 6 \, a^{3} b c^{3} d - 35 \, a^{4} c^{2} d^{2}\right )} x^{2} + 8 \, {\left (a^{3} b c^{4} - 7 \, a^{4} c^{3} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{384 \, a^{4} c^{5} x^{4}}\right ] \]
[-1/768*(3*(5*b^4*c^4 + 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 + 20*a^3*b*c*d^3 - 35*a^4*d^4)*sqrt(a*c)*x^4*log((8*a^2*c^2 + (b^2*c^2 + 6*a*b*c*d + a^2*d ^2)*x^2 - 4*(2*a*c + (b*c + a*d)*x)*sqrt(a*c)*sqrt(b*x + a)*sqrt(d*x + c) + 8*(a*b*c^2 + a^2*c*d)*x)/x^2) + 4*(48*a^4*c^4 + (15*a*b^3*c^4 + 17*a^2*b ^2*c^3*d + 25*a^3*b*c^2*d^2 - 105*a^4*c*d^3)*x^3 - 2*(5*a^2*b^2*c^4 + 6*a^ 3*b*c^3*d - 35*a^4*c^2*d^2)*x^2 + 8*(a^3*b*c^4 - 7*a^4*c^3*d)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(a^4*c^5*x^4), -1/384*(3*(5*b^4*c^4 + 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 + 20*a^3*b*c*d^3 - 35*a^4*d^4)*sqrt(-a*c)*x^4*arctan(1/2 *(2*a*c + (b*c + a*d)*x)*sqrt(-a*c)*sqrt(b*x + a)*sqrt(d*x + c)/(a*b*c*d*x ^2 + a^2*c^2 + (a*b*c^2 + a^2*c*d)*x)) + 2*(48*a^4*c^4 + (15*a*b^3*c^4 + 1 7*a^2*b^2*c^3*d + 25*a^3*b*c^2*d^2 - 105*a^4*c*d^3)*x^3 - 2*(5*a^2*b^2*c^4 + 6*a^3*b*c^3*d - 35*a^4*c^2*d^2)*x^2 + 8*(a^3*b*c^4 - 7*a^4*c^3*d)*x)*sq rt(b*x + a)*sqrt(d*x + c))/(a^4*c^5*x^4)]
\[ \int \frac {\sqrt {a+b x}}{x^5 \sqrt {c+d x}} \, dx=\int \frac {\sqrt {a + b x}}{x^{5} \sqrt {c + d x}}\, dx \]
Exception generated. \[ \int \frac {\sqrt {a+b x}}{x^5 \sqrt {c+d x}} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(a*d-b*c>0)', see `assume?` for m ore detail
Leaf count of result is larger than twice the leaf count of optimal. 3690 vs. \(2 (241) = 482\).
Time = 1.42 (sec) , antiderivative size = 3690, normalized size of antiderivative = 13.23 \[ \int \frac {\sqrt {a+b x}}{x^5 \sqrt {c+d x}} \, dx=\text {Too large to display} \]
1/192*b*(3*(5*sqrt(b*d)*b^5*c^4 + 4*sqrt(b*d)*a*b^4*c^3*d + 6*sqrt(b*d)*a^ 2*b^3*c^2*d^2 + 20*sqrt(b*d)*a^3*b^2*c*d^3 - 35*sqrt(b*d)*a^4*b*d^4)*arcta n(-1/2*(b^2*c + a*b*d - (sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)* b*d - a*b*d))^2)/(sqrt(-a*b*c*d)*b))/(sqrt(-a*b*c*d)*a^3*b*c^4) - 2*(15*sq rt(b*d)*b^19*c^11 - 103*sqrt(b*d)*a*b^18*c^10*d + 309*sqrt(b*d)*a^2*b^17*c ^9*d^2 - 669*sqrt(b*d)*a^3*b^16*c^8*d^3 + 1638*sqrt(b*d)*a^4*b^15*c^7*d^4 - 3990*sqrt(b*d)*a^5*b^14*c^6*d^5 + 7098*sqrt(b*d)*a^6*b^13*c^5*d^6 - 8394 *sqrt(b*d)*a^7*b^12*c^4*d^7 + 6459*sqrt(b*d)*a^8*b^11*c^3*d^8 - 3123*sqrt( b*d)*a^9*b^10*c^2*d^9 + 865*sqrt(b*d)*a^10*b^9*c*d^10 - 105*sqrt(b*d)*a^11 *b^8*d^11 - 105*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a )*b*d - a*b*d))^2*b^17*c^10 + 386*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqr t(b^2*c + (b*x + a)*b*d - a*b*d))^2*a*b^16*c^9*d - 429*sqrt(b*d)*(sqrt(b*d )*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^2*b^15*c^8*d^2 + 1048*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a *b*d))^2*a^3*b^14*c^7*d^3 - 4930*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt (b^2*c + (b*x + a)*b*d - a*b*d))^2*a^4*b^13*c^6*d^4 + 9420*sqrt(b*d)*(sqrt (b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^5*b^12*c^5* d^5 - 6386*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^6*b^11*c^4*d^6 - 2536*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^7*b^10*c^3*d^7 + 6507*sqrt(b*d...
Timed out. \[ \int \frac {\sqrt {a+b x}}{x^5 \sqrt {c+d x}} \, dx=\text {Hanged} \]