Integrand size = 22, antiderivative size = 229 \[ \int \frac {(a+b x)^{5/2}}{x^5 \sqrt {c+d x}} \, dx=\frac {5 (b c-a d)^2 (b c+7 a d) \sqrt {a+b x} \sqrt {c+d x}}{64 a c^4 x}+\frac {5 (b c-a d) (b c+7 a d) (a+b x)^{3/2} \sqrt {c+d x}}{96 a c^3 x^2}+\frac {(b c+7 a d) (a+b x)^{5/2} \sqrt {c+d x}}{24 a c^2 x^3}-\frac {(a+b x)^{7/2} \sqrt {c+d x}}{4 a c x^4}+\frac {5 (b c-a d)^3 (b c+7 a d) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{64 a^{3/2} c^{9/2}} \]
5/64*(-a*d+b*c)^3*(7*a*d+b*c)*arctanh(c^(1/2)*(b*x+a)^(1/2)/a^(1/2)/(d*x+c )^(1/2))/a^(3/2)/c^(9/2)+5/96*(-a*d+b*c)*(7*a*d+b*c)*(b*x+a)^(3/2)*(d*x+c) ^(1/2)/a/c^3/x^2+1/24*(7*a*d+b*c)*(b*x+a)^(5/2)*(d*x+c)^(1/2)/a/c^2/x^3-1/ 4*(b*x+a)^(7/2)*(d*x+c)^(1/2)/a/c/x^4+5/64*(-a*d+b*c)^2*(7*a*d+b*c)*(b*x+a )^(1/2)*(d*x+c)^(1/2)/a/c^4/x
Time = 0.43 (sec) , antiderivative size = 199, normalized size of antiderivative = 0.87 \[ \int \frac {(a+b x)^{5/2}}{x^5 \sqrt {c+d x}} \, dx=\frac {(-b c+a d)^3 \left (-\frac {\sqrt {a} \sqrt {c} \sqrt {a+b x} \sqrt {c+d x} \left (-15 b^3 c^3 x^3+a b^2 c^2 x^2 (-118 c+191 d x)+a^2 b c x \left (-136 c^2+172 c d x-265 d^2 x^2\right )+a^3 \left (-48 c^3+56 c^2 d x-70 c d^2 x^2+105 d^3 x^3\right )\right )}{(b c-a d)^3 x^4}-15 (b c+7 a d) \text {arctanh}\left (\frac {\sqrt {a} \sqrt {c+d x}}{\sqrt {c} \sqrt {a+b x}}\right )\right )}{192 a^{3/2} c^{9/2}} \]
((-(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*(-118*c + 191*d*x) + a^2*b*c*x*(-136*c^2 + 172*c* d*x - 265*d^2*x^2) + a^3*(-48*c^3 + 56*c^2*d*x - 70*c*d^2*x^2 + 105*d^3*x^ 3)))/((b*c - a*d)^3*x^4)) - 15*(b*c + 7*a*d)*ArcTanh[(Sqrt[a]*Sqrt[c + d*x ])/(Sqrt[c]*Sqrt[a + b*x])]))/(192*a^(3/2)*c^(9/2))
Time = 0.28 (sec) , antiderivative size = 215, normalized size of antiderivative = 0.94, 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.
| \(\displaystyle \int \frac {(a+b x)^{5/2}}{x^5 \sqrt {c+d x}} \, dx\) |
| \(\Big \downarrow \) 107 |
| \(\displaystyle -\frac {(7 a d+b c) \int \frac {(a+b x)^{5/2}}{x^4 \sqrt {c+d x}}dx}{8 a c}-\frac {(a+b x)^{7/2} \sqrt {c+d x}}{4 a c x^4}\) |
| \(\Big \downarrow \) 105 |
| \(\displaystyle -\frac {(7 a d+b c) \left (\frac {5 (b c-a d) \int \frac {(a+b x)^{3/2}}{x^3 \sqrt {c+d x}}dx}{6 c}-\frac {(a+b x)^{5/2} \sqrt {c+d x}}{3 c x^3}\right )}{8 a c}-\frac {(a+b x)^{7/2} \sqrt {c+d x}}{4 a c x^4}\) |
| \(\Big \downarrow \) 105 |
| \(\displaystyle -\frac {(7 a d+b c) \left (\frac {5 (b c-a d) \left (\frac {3 (b c-a d) \int \frac {\sqrt {a+b x}}{x^2 \sqrt {c+d x}}dx}{4 c}-\frac {(a+b x)^{3/2} \sqrt {c+d x}}{2 c x^2}\right )}{6 c}-\frac {(a+b x)^{5/2} \sqrt {c+d x}}{3 c x^3}\right )}{8 a c}-\frac {(a+b x)^{7/2} \sqrt {c+d x}}{4 a c x^4}\) |
| \(\Big \downarrow \) 105 |
| \(\displaystyle -\frac {(7 a d+b c) \left (\frac {5 (b c-a d) \left (\frac {3 (b c-a d) \left (\frac {(b c-a d) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}}dx}{2 c}-\frac {\sqrt {a+b x} \sqrt {c+d x}}{c x}\right )}{4 c}-\frac {(a+b x)^{3/2} \sqrt {c+d x}}{2 c x^2}\right )}{6 c}-\frac {(a+b x)^{5/2} \sqrt {c+d x}}{3 c x^3}\right )}{8 a c}-\frac {(a+b x)^{7/2} \sqrt {c+d x}}{4 a c x^4}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle -\frac {(7 a d+b c) \left (\frac {5 (b c-a d) \left (\frac {3 (b c-a d) \left (\frac {(b c-a d) \int \frac {1}{\frac {c (a+b x)}{c+d x}-a}d\frac {\sqrt {a+b x}}{\sqrt {c+d x}}}{c}-\frac {\sqrt {a+b x} \sqrt {c+d x}}{c x}\right )}{4 c}-\frac {(a+b x)^{3/2} \sqrt {c+d x}}{2 c x^2}\right )}{6 c}-\frac {(a+b x)^{5/2} \sqrt {c+d x}}{3 c x^3}\right )}{8 a c}-\frac {(a+b x)^{7/2} \sqrt {c+d x}}{4 a c x^4}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {(7 a d+b c) \left (\frac {5 (b c-a d) \left (\frac {3 (b c-a d) \left (-\frac {(b c-a d) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{\sqrt {a} c^{3/2}}-\frac {\sqrt {a+b x} \sqrt {c+d x}}{c x}\right )}{4 c}-\frac {(a+b x)^{3/2} \sqrt {c+d x}}{2 c x^2}\right )}{6 c}-\frac {(a+b x)^{5/2} \sqrt {c+d x}}{3 c x^3}\right )}{8 a c}-\frac {(a+b x)^{7/2} \sqrt {c+d x}}{4 a c x^4}\) |
-1/4*((a + b*x)^(7/2)*Sqrt[c + d*x])/(a*c*x^4) - ((b*c + 7*a*d)*(-1/3*((a + b*x)^(5/2)*Sqrt[c + d*x])/(c*x^3) + (5*(b*c - a*d)*(-1/2*((a + b*x)^(3/2 )*Sqrt[c + d*x])/(c*x^2) + (3*(b*c - a*d)*(-((Sqrt[a + b*x]*Sqrt[c + d*x]) /(c*x)) - ((b*c - a*d)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d *x])])/(Sqrt[a]*c^(3/2))))/(4*c)))/(6*c)))/(8*a*c)
3.7.81.3.1 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. \(2(191)=382\).
Time = 0.56 (sec) , antiderivative size = 593, normalized size of antiderivative = 2.59
| method | result | size |
| default | \(-\frac {\sqrt {b x +a}\, \sqrt {d x +c}\, \left (105 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a^{4} d^{4} x^{4}-300 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a^{3} b c \,d^{3} x^{4}+270 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a^{2} b^{2} c^{2} d^{2} x^{4}-60 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a \,b^{3} c^{3} d \,x^{4}-15 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) b^{4} c^{4} x^{4}-210 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a^{3} d^{3} x^{3}+530 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a^{2} b c \,d^{2} x^{3}-382 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a \,b^{2} c^{2} d \,x^{3}+30 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, b^{3} c^{3} x^{3}+140 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, a^{3} c \,d^{2} x^{2}-344 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, a^{2} b \,c^{2} d \,x^{2}+236 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, a \,b^{2} c^{3} x^{2}-112 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, a^{3} c^{2} d x +272 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, a^{2} b \,c^{3} x +96 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a^{3} c^{3} \sqrt {a c}\right )}{384 a \,c^{4} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, x^{4} \sqrt {a c}}\) | \(593\) |
-1/384*(b*x+a)^(1/2)*(d*x+c)^(1/2)/a/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^4-300*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+270*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^4-60 *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*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+530*(a*c)^ (1/2)*((b*x+a)*(d*x+c))^(1/2)*a^2*b*c*d^2*x^3-382*(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^2-344*((b*x+a)*(d *x+c))^(1/2)*(a*c)^(1/2)*a^2*b*c^2*d*x^2+236*((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+2 72*((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 = 1.73 (sec) , antiderivative size = 570, normalized size of antiderivative = 2.49 \[ \int \frac {(a+b x)^{5/2}}{x^5 \sqrt {c+d x}} \, dx=\left [-\frac {15 \, {\left (b^{4} c^{4} + 4 \, a b^{3} c^{3} d - 18 \, a^{2} b^{2} c^{2} d^{2} + 20 \, a^{3} b c d^{3} - 7 \, a^{4} d^{4}\right )} \sqrt {a c} x^{4} \log \left (\frac {8 \, a^{2} c^{2} + {\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} - 4 \, {\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {a c} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (a b c^{2} + a^{2} c d\right )} x}{x^{2}}\right ) + 4 \, {\left (48 \, a^{4} c^{4} + {\left (15 \, a b^{3} c^{4} - 191 \, a^{2} b^{2} c^{3} d + 265 \, a^{3} b c^{2} d^{2} - 105 \, a^{4} c d^{3}\right )} x^{3} + 2 \, {\left (59 \, a^{2} b^{2} c^{4} - 86 \, a^{3} b c^{3} d + 35 \, a^{4} c^{2} d^{2}\right )} x^{2} + 8 \, {\left (17 \, a^{3} b c^{4} - 7 \, a^{4} c^{3} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{768 \, a^{2} c^{5} x^{4}}, -\frac {15 \, {\left (b^{4} c^{4} + 4 \, a b^{3} c^{3} d - 18 \, a^{2} b^{2} c^{2} d^{2} + 20 \, a^{3} b c d^{3} - 7 \, a^{4} d^{4}\right )} \sqrt {-a c} x^{4} \arctan \left (\frac {{\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {-a c} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (a b c d x^{2} + a^{2} c^{2} + {\left (a b c^{2} + a^{2} c d\right )} x\right )}}\right ) + 2 \, {\left (48 \, a^{4} c^{4} + {\left (15 \, a b^{3} c^{4} - 191 \, a^{2} b^{2} c^{3} d + 265 \, a^{3} b c^{2} d^{2} - 105 \, a^{4} c d^{3}\right )} x^{3} + 2 \, {\left (59 \, a^{2} b^{2} c^{4} - 86 \, a^{3} b c^{3} d + 35 \, a^{4} c^{2} d^{2}\right )} x^{2} + 8 \, {\left (17 \, a^{3} b c^{4} - 7 \, a^{4} c^{3} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{384 \, a^{2} c^{5} x^{4}}\right ] \]
[-1/768*(15*(b^4*c^4 + 4*a*b^3*c^3*d - 18*a^2*b^2*c^2*d^2 + 20*a^3*b*c*d^3 - 7*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 - 191*a^2*b ^2*c^3*d + 265*a^3*b*c^2*d^2 - 105*a^4*c*d^3)*x^3 + 2*(59*a^2*b^2*c^4 - 86 *a^3*b*c^3*d + 35*a^4*c^2*d^2)*x^2 + 8*(17*a^3*b*c^4 - 7*a^4*c^3*d)*x)*sqr t(b*x + a)*sqrt(d*x + c))/(a^2*c^5*x^4), -1/384*(15*(b^4*c^4 + 4*a*b^3*c^3 *d - 18*a^2*b^2*c^2*d^2 + 20*a^3*b*c*d^3 - 7*a^4*d^4)*sqrt(-a*c)*x^4*arcta n(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 - 191*a^2*b^2*c^3*d + 265*a^3*b*c^2*d^2 - 105*a^4*c*d^3)*x^3 + 2*(59*a^2 *b^2*c^4 - 86*a^3*b*c^3*d + 35*a^4*c^2*d^2)*x^2 + 8*(17*a^3*b*c^4 - 7*a^4* c^3*d)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(a^2*c^5*x^4)]
\[ \int \frac {(a+b x)^{5/2}}{x^5 \sqrt {c+d x}} \, dx=\int \frac {\left (a + b x\right )^{\frac {5}{2}}}{x^{5} \sqrt {c + d x}}\, dx \]
Exception generated. \[ \int \frac {(a+b x)^{5/2}}{x^5 \sqrt {c+d x}} \, dx=\text {Exception raised: ValueError} \]
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. 3689 vs. \(2 (191) = 382\).
Time = 1.53 (sec) , antiderivative size = 3689, normalized size of antiderivative = 16.11 \[ \int \frac {(a+b x)^{5/2}}{x^5 \sqrt {c+d x}} \, dx=\text {Too large to display} \]
1/192*b*(15*(sqrt(b*d)*b^5*c^4 + 4*sqrt(b*d)*a*b^4*c^3*d - 18*sqrt(b*d)*a^ 2*b^3*c^2*d^2 + 20*sqrt(b*d)*a^3*b^2*c*d^3 - 7*sqrt(b*d)*a^4*b*d^4)*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*b*c^4) - 2*(15*sqrt( b*d)*b^19*c^11 - 311*sqrt(b*d)*a*b^18*c^10*d + 2213*sqrt(b*d)*a^2*b^17*c^9 *d^2 - 8413*sqrt(b*d)*a^3*b^16*c^8*d^3 + 20006*sqrt(b*d)*a^4*b^15*c^7*d^4 - 31990*sqrt(b*d)*a^5*b^14*c^6*d^5 + 35546*sqrt(b*d)*a^6*b^13*c^5*d^6 - 27 658*sqrt(b*d)*a^7*b^12*c^4*d^7 + 14843*sqrt(b*d)*a^8*b^11*c^3*d^8 - 5251*s qrt(b*d)*a^9*b^10*c^2*d^9 + 1105*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 + 2098*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 - 11245*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 + 28312*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 - 37250*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 + 20780*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 + 8782*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 - 22760*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 + 1...
Timed out. \[ \int \frac {(a+b x)^{5/2}}{x^5 \sqrt {c+d x}} \, dx=\int \frac {{\left (a+b\,x\right )}^{5/2}}{x^5\,\sqrt {c+d\,x}} \,d x \]