Integrand size = 22, antiderivative size = 233 \[ \int \frac {\sqrt {a+b x} (c+d x)^{3/2}}{x^5} \, dx=-\frac {(b c-a d)^2 (5 b c+3 a d) \sqrt {a+b x} \sqrt {c+d x}}{64 a^3 c^2 x}+\frac {(b c-a d) (5 b c+3 a d) \sqrt {a+b x} (c+d x)^{3/2}}{96 a^2 c^2 x^2}+\frac {(5 b c+3 a d) \sqrt {a+b x} (c+d x)^{5/2}}{24 a c^2 x^3}-\frac {(a+b x)^{3/2} (c+d x)^{5/2}}{4 a c x^4}+\frac {(b c-a d)^3 (5 b c+3 a d) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{64 a^{7/2} c^{5/2}} \]
-1/4*(b*x+a)^(3/2)*(d*x+c)^(5/2)/a/c/x^4+1/64*(-a*d+b*c)^3*(3*a*d+5*b*c)*a rctanh(c^(1/2)*(b*x+a)^(1/2)/a^(1/2)/(d*x+c)^(1/2))/a^(7/2)/c^(5/2)+1/96*( -a*d+b*c)*(3*a*d+5*b*c)*(d*x+c)^(3/2)*(b*x+a)^(1/2)/a^2/c^2/x^2+1/24*(3*a* d+5*b*c)*(d*x+c)^(5/2)*(b*x+a)^(1/2)/a/c^2/x^3-1/64*(-a*d+b*c)^2*(3*a*d+5* b*c)*(b*x+a)^(1/2)*(d*x+c)^(1/2)/a^3/c^2/x
Time = 0.41 (sec) , antiderivative size = 200, normalized size of antiderivative = 0.86 \[ \int \frac {\sqrt {a+b x} (c+d x)^{3/2}}{x^5} \, dx=\frac {(-b c+a d)^3 \left (\frac {\sqrt {a} \sqrt {c} \sqrt {a+b x} \sqrt {c+d x} \left (15 b^3 c^3 x^3-a b^2 c^2 x^2 (10 c+31 d x)+a^2 b c x \left (8 c^2+20 c d x+9 d^2 x^2\right )+a^3 \left (48 c^3+72 c^2 d x+6 c d^2 x^2-9 d^3 x^3\right )\right )}{(b c-a d)^3 x^4}-3 (5 b c+3 a d) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )\right )}{192 a^{7/2} c^{5/2}} \]
((-(b*c) + a*d)^3*((Sqrt[a]*Sqrt[c]*Sqrt[a + b*x]*Sqrt[c + d*x]*(15*b^3*c^ 3*x^3 - a*b^2*c^2*x^2*(10*c + 31*d*x) + a^2*b*c*x*(8*c^2 + 20*c*d*x + 9*d^ 2*x^2) + a^3*(48*c^3 + 72*c^2*d*x + 6*c*d^2*x^2 - 9*d^3*x^3)))/((b*c - a*d )^3*x^4) - 3*(5*b*c + 3*a*d)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt [c + d*x])]))/(192*a^(7/2)*c^(5/2))
Time = 0.28 (sec) , antiderivative size = 215, normalized size of antiderivative = 0.92, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {107, 105, 105, 105, 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.
| \(\displaystyle \int \frac {\sqrt {a+b x} (c+d x)^{3/2}}{x^5} \, dx\) |
| \(\Big \downarrow \) 107 |
| \(\displaystyle -\frac {(3 a d+5 b c) \int \frac {\sqrt {a+b x} (c+d x)^{3/2}}{x^4}dx}{8 a c}-\frac {(a+b x)^{3/2} (c+d x)^{5/2}}{4 a c x^4}\) |
| \(\Big \downarrow \) 105 |
| \(\displaystyle -\frac {(3 a d+5 b c) \left (\frac {(b c-a d) \int \frac {(c+d x)^{3/2}}{x^3 \sqrt {a+b x}}dx}{6 c}-\frac {\sqrt {a+b x} (c+d x)^{5/2}}{3 c x^3}\right )}{8 a c}-\frac {(a+b x)^{3/2} (c+d x)^{5/2}}{4 a c x^4}\) |
| \(\Big \downarrow \) 105 |
| \(\displaystyle -\frac {(3 a d+5 b c) \left (\frac {(b c-a d) \left (-\frac {3 (b c-a d) \int \frac {\sqrt {c+d x}}{x^2 \sqrt {a+b x}}dx}{4 a}-\frac {\sqrt {a+b x} (c+d x)^{3/2}}{2 a x^2}\right )}{6 c}-\frac {\sqrt {a+b x} (c+d x)^{5/2}}{3 c x^3}\right )}{8 a c}-\frac {(a+b x)^{3/2} (c+d x)^{5/2}}{4 a c x^4}\) |
| \(\Big \downarrow \) 105 |
| \(\displaystyle -\frac {(3 a d+5 b c) \left (\frac {(b c-a d) \left (-\frac {3 (b c-a d) \left (-\frac {(b c-a d) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}}dx}{2 a}-\frac {\sqrt {a+b x} \sqrt {c+d x}}{a x}\right )}{4 a}-\frac {\sqrt {a+b x} (c+d x)^{3/2}}{2 a x^2}\right )}{6 c}-\frac {\sqrt {a+b x} (c+d x)^{5/2}}{3 c x^3}\right )}{8 a c}-\frac {(a+b x)^{3/2} (c+d x)^{5/2}}{4 a c x^4}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle -\frac {(3 a d+5 b c) \left (\frac {(b c-a d) \left (-\frac {3 (b c-a d) \left (-\frac {(b c-a d) \int \frac {1}{\frac {c (a+b x)}{c+d x}-a}d\frac {\sqrt {a+b x}}{\sqrt {c+d x}}}{a}-\frac {\sqrt {a+b x} \sqrt {c+d x}}{a x}\right )}{4 a}-\frac {\sqrt {a+b x} (c+d x)^{3/2}}{2 a x^2}\right )}{6 c}-\frac {\sqrt {a+b x} (c+d x)^{5/2}}{3 c x^3}\right )}{8 a c}-\frac {(a+b x)^{3/2} (c+d x)^{5/2}}{4 a c x^4}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {(3 a d+5 b c) \left (\frac {(b c-a d) \left (-\frac {3 (b c-a d) \left (\frac {(b c-a d) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{a^{3/2} \sqrt {c}}-\frac {\sqrt {a+b x} \sqrt {c+d x}}{a x}\right )}{4 a}-\frac {\sqrt {a+b x} (c+d x)^{3/2}}{2 a x^2}\right )}{6 c}-\frac {\sqrt {a+b x} (c+d x)^{5/2}}{3 c x^3}\right )}{8 a c}-\frac {(a+b x)^{3/2} (c+d x)^{5/2}}{4 a c x^4}\) |
-1/4*((a + b*x)^(3/2)*(c + d*x)^(5/2))/(a*c*x^4) - ((5*b*c + 3*a*d)*(-1/3* (Sqrt[a + b*x]*(c + d*x)^(5/2))/(c*x^3) + ((b*c - a*d)*(-1/2*(Sqrt[a + b*x ]*(c + d*x)^(3/2))/(a*x^2) - (3*(b*c - a*d)*(-((Sqrt[a + b*x]*Sqrt[c + d*x ])/(a*x)) + ((b*c - a*d)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(a^(3/2)*Sqrt[c])))/(4*a)))/(6*c)))/(8*a*c)
3.6.64.3.1 Defintions of rubi rules used
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[n*((d*e - c*f)/((m + 1)*(b*e - a*f))) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[m + n + p + 2, 0] && GtQ[n, 0] && (SumSimplerQ[m, 1] || !SumSimplerQ[p, 1]) && NeQ[m, -1]
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[(a*d*f*(m + 1) + b*c*f*(n + 1) + b*d*e*(p + 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, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x ] && EqQ[Simplify[m + n + p + 3], 0] && (LtQ[m, -1] || SumSimplerQ[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. \(2(195)=390\).
Time = 0.53 (sec) , antiderivative size = 593, normalized size of antiderivative = 2.55
| method | result | size |
| default | \(-\frac {\sqrt {b x +a}\, \sqrt {d x +c}\, \left (9 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a^{4} d^{4} x^{4}-12 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a^{3} b c \,d^{3} x^{4}-18 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a^{2} b^{2} c^{2} d^{2} x^{4}+36 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a \,b^{3} c^{3} d \,x^{4}-15 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) b^{4} c^{4} x^{4}-18 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a^{3} d^{3} x^{3}+18 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a^{2} b c \,d^{2} x^{3}-62 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a \,b^{2} c^{2} d \,x^{3}+30 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, b^{3} c^{3} x^{3}+12 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, a^{3} c \,d^{2} x^{2}+40 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, a^{2} b \,c^{2} d \,x^{2}-20 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, a \,b^{2} c^{3} x^{2}+144 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, a^{3} c^{2} d x +16 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, a^{2} b \,c^{3} x +96 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a^{3} c^{3} \sqrt {a c}\right )}{384 a^{3} c^{2} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, x^{4} \sqrt {a c}}\) | \(593\) |
-1/384*(b*x+a)^(1/2)*(d*x+c)^(1/2)/a^3/c^2*(9*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^4-12*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^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+36*l n((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^4-18*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a^3*d^3*x^3+18*(a*c)^(1/2 )*((b*x+a)*(d*x+c))^(1/2)*a^2*b*c*d^2*x^3-62*(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+ 12*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)*a^3*c*d^2*x^2+40*((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+144*((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 = 1.21 (sec) , antiderivative size = 570, normalized size of antiderivative = 2.45 \[ \int \frac {\sqrt {a+b x} (c+d x)^{3/2}}{x^5} \, dx=\left [-\frac {3 \, {\left (5 \, b^{4} c^{4} - 12 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} + 4 \, a^{3} b c d^{3} - 3 \, a^{4} d^{4}\right )} \sqrt {a c} x^{4} \log \left (\frac {8 \, a^{2} c^{2} + {\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} - 4 \, {\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {a c} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (a b c^{2} + a^{2} c d\right )} x}{x^{2}}\right ) + 4 \, {\left (48 \, a^{4} c^{4} + {\left (15 \, a b^{3} c^{4} - 31 \, a^{2} b^{2} c^{3} d + 9 \, a^{3} b c^{2} d^{2} - 9 \, a^{4} c d^{3}\right )} x^{3} - 2 \, {\left (5 \, a^{2} b^{2} c^{4} - 10 \, a^{3} b c^{3} d - 3 \, a^{4} c^{2} d^{2}\right )} x^{2} + 8 \, {\left (a^{3} b c^{4} + 9 \, a^{4} c^{3} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{768 \, a^{4} c^{3} x^{4}}, -\frac {3 \, {\left (5 \, b^{4} c^{4} - 12 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} + 4 \, a^{3} b c d^{3} - 3 \, a^{4} d^{4}\right )} \sqrt {-a c} x^{4} \arctan \left (\frac {{\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {-a c} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (a b c d x^{2} + a^{2} c^{2} + {\left (a b c^{2} + a^{2} c d\right )} x\right )}}\right ) + 2 \, {\left (48 \, a^{4} c^{4} + {\left (15 \, a b^{3} c^{4} - 31 \, a^{2} b^{2} c^{3} d + 9 \, a^{3} b c^{2} d^{2} - 9 \, a^{4} c d^{3}\right )} x^{3} - 2 \, {\left (5 \, a^{2} b^{2} c^{4} - 10 \, a^{3} b c^{3} d - 3 \, a^{4} c^{2} d^{2}\right )} x^{2} + 8 \, {\left (a^{3} b c^{4} + 9 \, a^{4} c^{3} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{384 \, a^{4} c^{3} x^{4}}\right ] \]
[-1/768*(3*(5*b^4*c^4 - 12*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 + 4*a^3*b*c*d^3 - 3*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 - 31*a^2*b^ 2*c^3*d + 9*a^3*b*c^2*d^2 - 9*a^4*c*d^3)*x^3 - 2*(5*a^2*b^2*c^4 - 10*a^3*b *c^3*d - 3*a^4*c^2*d^2)*x^2 + 8*(a^3*b*c^4 + 9*a^4*c^3*d)*x)*sqrt(b*x + a) *sqrt(d*x + c))/(a^4*c^3*x^4), -1/384*(3*(5*b^4*c^4 - 12*a*b^3*c^3*d + 6*a ^2*b^2*c^2*d^2 + 4*a^3*b*c*d^3 - 3*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 - 31*a^2 *b^2*c^3*d + 9*a^3*b*c^2*d^2 - 9*a^4*c*d^3)*x^3 - 2*(5*a^2*b^2*c^4 - 10*a^ 3*b*c^3*d - 3*a^4*c^2*d^2)*x^2 + 8*(a^3*b*c^4 + 9*a^4*c^3*d)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(a^4*c^3*x^4)]
\[ \int \frac {\sqrt {a+b x} (c+d x)^{3/2}}{x^5} \, dx=\int \frac {\sqrt {a + b x} \left (c + d x\right )^{\frac {3}{2}}}{x^{5}}\, dx \]
Exception generated. \[ \int \frac {\sqrt {a+b x} (c+d x)^{3/2}}{x^5} \, dx=\text {Exception raised: ValueError} \]
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. 3834 vs. \(2 (195) = 390\).
Time = 1.54 (sec) , antiderivative size = 3834, normalized size of antiderivative = 16.45 \[ \int \frac {\sqrt {a+b x} (c+d x)^{3/2}}{x^5} \, dx=\text {Too large to display} \]
1/192*(3*(5*sqrt(b*d)*b^5*c^4*abs(b) - 12*sqrt(b*d)*a*b^4*c^3*d*abs(b) + 6 *sqrt(b*d)*a^2*b^3*c^2*d^2*abs(b) + 4*sqrt(b*d)*a^3*b^2*c*d^3*abs(b) - 3*s qrt(b*d)*a^4*b*d^4*abs(b))*arctan(-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^2) - 2*(15*sqrt(b*d)*b^19*c^11*abs(b) - 151*sqrt(b*d)*a *b^18*c^10*d*abs(b) + 677*sqrt(b*d)*a^2*b^17*c^9*d^2*abs(b) - 1789*sqrt(b* d)*a^3*b^16*c^8*d^3*abs(b) + 3110*sqrt(b*d)*a^4*b^15*c^7*d^4*abs(b) - 3766 *sqrt(b*d)*a^5*b^14*c^6*d^5*abs(b) + 3290*sqrt(b*d)*a^6*b^13*c^5*d^6*abs(b ) - 2122*sqrt(b*d)*a^7*b^12*c^4*d^7*abs(b) + 1019*sqrt(b*d)*a^8*b^11*c^3*d ^8*abs(b) - 355*sqrt(b*d)*a^9*b^10*c^2*d^9*abs(b) + 81*sqrt(b*d)*a^10*b^9* c*d^10*abs(b) - 9*sqrt(b*d)*a^11*b^8*d^11*abs(b) - 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*abs(b) + 722*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a* b*d))^2*a*b^16*c^9*d*abs(b) - 1933*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sq rt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^2*b^15*c^8*d^2*abs(b) + 2328*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*abs(b) - 514*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*abs(b) - 2068*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*abs(b) + 2958*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*...
Timed out. \[ \int \frac {\sqrt {a+b x} (c+d x)^{3/2}}{x^5} \, dx=\int \frac {\sqrt {a+b\,x}\,{\left (c+d\,x\right )}^{3/2}}{x^5} \,d x \]