\(\int \frac {x}{(a+b x)^{5/2} (c+d x)^{5/2}} \, dx\) [416]

Optimal result

Integrand size = 20, antiderivative size = 160 \[ \int \frac {x}{(a+b x)^{5/2} (c+d x)^{5/2}} \, dx=\frac {2 a}{3 b (b c-a d) (a+b x)^{3/2} (c+d x)^{3/2}}-\frac {2 (b c+a d)}{b (b c-a d)^2 \sqrt {a+b x} (c+d x)^{3/2}}-\frac {8 d (b c+a d) \sqrt {a+b x}}{3 b (b c-a d)^3 (c+d x)^{3/2}}-\frac {16 d (b c+a d) \sqrt {a+b x}}{3 (b c-a d)^4 \sqrt {c+d x}} \] Output:

2/3*a/b/(-a*d+b*c)/(b*x+a)^(3/2)/(d*x+c)^(3/2)-2*(a*d+b*c)/b/(-a*d+b*c)^2/ 
(b*x+a)^(1/2)/(d*x+c)^(3/2)-8/3*d*(a*d+b*c)*(b*x+a)^(1/2)/b/(-a*d+b*c)^3/( 
d*x+c)^(3/2)-16/3*d*(a*d+b*c)*(b*x+a)^(1/2)/(-a*d+b*c)^4/(d*x+c)^(1/2)
 

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.83 \[ \int \frac {x}{(a+b x)^{5/2} (c+d x)^{5/2}} \, dx=-\frac {2 (a+b x)^{3/2} \left (-c d^2+\frac {6 b c d (c+d x)}{a+b x}+\frac {3 a d^2 (c+d x)}{a+b x}+\frac {3 b^2 c (c+d x)^2}{(a+b x)^2}+\frac {6 a b d (c+d x)^2}{(a+b x)^2}-\frac {a b^2 (c+d x)^3}{(a+b x)^3}\right )}{3 (b c-a d)^4 (c+d x)^{3/2}} \] Input:

Integrate[x/((a + b*x)^(5/2)*(c + d*x)^(5/2)),x]
 

Output:

(-2*(a + b*x)^(3/2)*(-(c*d^2) + (6*b*c*d*(c + d*x))/(a + b*x) + (3*a*d^2*( 
c + d*x))/(a + b*x) + (3*b^2*c*(c + d*x)^2)/(a + b*x)^2 + (6*a*b*d*(c + d* 
x)^2)/(a + b*x)^2 - (a*b^2*(c + d*x)^3)/(a + b*x)^3))/(3*(b*c - a*d)^4*(c 
+ d*x)^(3/2))
 

Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.06, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {87, 55, 55, 48}

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.

\(\displaystyle \int \frac {x}{(a+b x)^{5/2} (c+d x)^{5/2}} \, dx\)

\(\Big \downarrow \) 87

\(\displaystyle -\frac {(a d+b c) \int \frac {1}{(a+b x)^{5/2} (c+d x)^{3/2}}dx}{d (b c-a d)}-\frac {2 c}{3 d (a+b x)^{3/2} (c+d x)^{3/2} (b c-a d)}\)

\(\Big \downarrow \) 55

\(\displaystyle -\frac {(a d+b c) \left (-\frac {4 d \int \frac {1}{(a+b x)^{3/2} (c+d x)^{3/2}}dx}{3 (b c-a d)}-\frac {2}{3 (a+b x)^{3/2} \sqrt {c+d x} (b c-a d)}\right )}{d (b c-a d)}-\frac {2 c}{3 d (a+b x)^{3/2} (c+d x)^{3/2} (b c-a d)}\)

\(\Big \downarrow \) 55

\(\displaystyle -\frac {(a d+b c) \left (-\frac {4 d \left (-\frac {2 d \int \frac {1}{\sqrt {a+b x} (c+d x)^{3/2}}dx}{b c-a d}-\frac {2}{\sqrt {a+b x} \sqrt {c+d x} (b c-a d)}\right )}{3 (b c-a d)}-\frac {2}{3 (a+b x)^{3/2} \sqrt {c+d x} (b c-a d)}\right )}{d (b c-a d)}-\frac {2 c}{3 d (a+b x)^{3/2} (c+d x)^{3/2} (b c-a d)}\)

\(\Big \downarrow \) 48

\(\displaystyle -\frac {2 c}{3 d (a+b x)^{3/2} (c+d x)^{3/2} (b c-a d)}-\frac {(a d+b c) \left (-\frac {4 d \left (-\frac {4 d \sqrt {a+b x}}{\sqrt {c+d x} (b c-a d)^2}-\frac {2}{\sqrt {a+b x} \sqrt {c+d x} (b c-a d)}\right )}{3 (b c-a d)}-\frac {2}{3 (a+b x)^{3/2} \sqrt {c+d x} (b c-a d)}\right )}{d (b c-a d)}\)

Input:

Int[x/((a + b*x)^(5/2)*(c + d*x)^(5/2)),x]
 

Output:

(-2*c)/(3*d*(b*c - a*d)*(a + b*x)^(3/2)*(c + d*x)^(3/2)) - ((b*c + a*d)*(- 
2/(3*(b*c - a*d)*(a + b*x)^(3/2)*Sqrt[c + d*x]) - (4*d*(-2/((b*c - a*d)*Sq 
rt[a + b*x]*Sqrt[c + d*x]) - (4*d*Sqrt[a + b*x])/((b*c - a*d)^2*Sqrt[c + d 
*x])))/(3*(b*c - a*d))))/(d*(b*c - a*d))
 

Defintions of rubi rules used

rule 48

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp 
[(a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{ 
a, b, c, d, m, n}, x] && EqQ[m + n + 2, 0] && NeQ[m, -1]
 

rule 55

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(S 
implify[m + n + 2]/((b*c - a*d)*(m + 1)))   Int[(a + b*x)^Simplify[m + 1]*( 
c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && ILtQ[Simplify[m + n + 
 2], 0] && NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[ 
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (SumSimplerQ[m, 1] ||  !SumSimp 
lerQ[n, 1])
 

rule 87

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p 
_.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p 
+ 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p 
+ 1)))/(f*(p + 1)*(c*f - d*e))   Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] 
/; FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || Intege 
rQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || LtQ[p, n]))))
 

Maple [A] (verified)

Time = 0.29 (sec) , antiderivative size = 157, normalized size of antiderivative = 0.98

method	result	size
default	\(-\frac {2 \left (8 x^{3} a \,b^{2} d^{3}+8 x^{3} b^{3} c \,d^{2}+12 x^{2} a^{2} b \,d^{3}+24 x^{2} a \,b^{2} c \,d^{2}+12 x^{2} b^{3} c^{2} d +3 a^{3} d^{3} x +21 a^{2} b c \,d^{2} x +21 a \,b^{2} c^{2} d x +3 b^{3} c^{3} x +2 a^{3} c \,d^{2}+12 a^{2} b \,c^{2} d +2 b^{2} c^{3} a \right )}{3 \left (a d -b c \right )^{4} \left (b x +a \right )^{\frac {3}{2}} \left (x d +c \right )^{\frac {3}{2}}}\)	\(157\)
gosper	\(-\frac {2 \left (8 x^{3} a \,b^{2} d^{3}+8 x^{3} b^{3} c \,d^{2}+12 x^{2} a^{2} b \,d^{3}+24 x^{2} a \,b^{2} c \,d^{2}+12 x^{2} b^{3} c^{2} d +3 a^{3} d^{3} x +21 a^{2} b c \,d^{2} x +21 a \,b^{2} c^{2} d x +3 b^{3} c^{3} x +2 a^{3} c \,d^{2}+12 a^{2} b \,c^{2} d +2 b^{2} c^{3} a \right )}{3 \left (b x +a \right )^{\frac {3}{2}} \left (x d +c \right )^{\frac {3}{2}} \left (d^{4} a^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +c^{4} b^{4}\right )}\)	\(198\)
orering	\(-\frac {2 \left (8 x^{3} a \,b^{2} d^{3}+8 x^{3} b^{3} c \,d^{2}+12 x^{2} a^{2} b \,d^{3}+24 x^{2} a \,b^{2} c \,d^{2}+12 x^{2} b^{3} c^{2} d +3 a^{3} d^{3} x +21 a^{2} b c \,d^{2} x +21 a \,b^{2} c^{2} d x +3 b^{3} c^{3} x +2 a^{3} c \,d^{2}+12 a^{2} b \,c^{2} d +2 b^{2} c^{3} a \right )}{3 \left (b x +a \right )^{\frac {3}{2}} \left (x d +c \right )^{\frac {3}{2}} \left (d^{4} a^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +c^{4} b^{4}\right )}\)	\(198\)

Input:

int(x/(b*x+a)^(5/2)/(d*x+c)^(5/2),x,method=_RETURNVERBOSE)
 

Output:

-2/3*(8*a*b^2*d^3*x^3+8*b^3*c*d^2*x^3+12*a^2*b*d^3*x^2+24*a*b^2*c*d^2*x^2+ 
12*b^3*c^2*d*x^2+3*a^3*d^3*x+21*a^2*b*c*d^2*x+21*a*b^2*c^2*d*x+3*b^3*c^3*x 
+2*a^3*c*d^2+12*a^2*b*c^2*d+2*a*b^2*c^3)/(a*d-b*c)^4/(b*x+a)^(3/2)/(d*x+c) 
^(3/2)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 468 vs. \(2 (138) = 276\).

Time = 0.48 (sec) , antiderivative size = 468, normalized size of antiderivative = 2.92 \[ \int \frac {x}{(a+b x)^{5/2} (c+d x)^{5/2}} \, dx=-\frac {2 \, {\left (2 \, a b^{2} c^{3} + 12 \, a^{2} b c^{2} d + 2 \, a^{3} c d^{2} + 8 \, {\left (b^{3} c d^{2} + a b^{2} d^{3}\right )} x^{3} + 12 \, {\left (b^{3} c^{2} d + 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x^{2} + 3 \, {\left (b^{3} c^{3} + 7 \, a b^{2} c^{2} d + 7 \, a^{2} b c d^{2} + a^{3} d^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{3 \, {\left (a^{2} b^{4} c^{6} - 4 \, a^{3} b^{3} c^{5} d + 6 \, a^{4} b^{2} c^{4} d^{2} - 4 \, a^{5} b c^{3} d^{3} + a^{6} c^{2} d^{4} + {\left (b^{6} c^{4} d^{2} - 4 \, a b^{5} c^{3} d^{3} + 6 \, a^{2} b^{4} c^{2} d^{4} - 4 \, a^{3} b^{3} c d^{5} + a^{4} b^{2} d^{6}\right )} x^{4} + 2 \, {\left (b^{6} c^{5} d - 3 \, a b^{5} c^{4} d^{2} + 2 \, a^{2} b^{4} c^{3} d^{3} + 2 \, a^{3} b^{3} c^{2} d^{4} - 3 \, a^{4} b^{2} c d^{5} + a^{5} b d^{6}\right )} x^{3} + {\left (b^{6} c^{6} - 9 \, a^{2} b^{4} c^{4} d^{2} + 16 \, a^{3} b^{3} c^{3} d^{3} - 9 \, a^{4} b^{2} c^{2} d^{4} + a^{6} d^{6}\right )} x^{2} + 2 \, {\left (a b^{5} c^{6} - 3 \, a^{2} b^{4} c^{5} d + 2 \, a^{3} b^{3} c^{4} d^{2} + 2 \, a^{4} b^{2} c^{3} d^{3} - 3 \, a^{5} b c^{2} d^{4} + a^{6} c d^{5}\right )} x\right )}} \] Input:

integrate(x/(b*x+a)^(5/2)/(d*x+c)^(5/2),x, algorithm="fricas")
 

Output:

-2/3*(2*a*b^2*c^3 + 12*a^2*b*c^2*d + 2*a^3*c*d^2 + 8*(b^3*c*d^2 + a*b^2*d^ 
3)*x^3 + 12*(b^3*c^2*d + 2*a*b^2*c*d^2 + a^2*b*d^3)*x^2 + 3*(b^3*c^3 + 7*a 
*b^2*c^2*d + 7*a^2*b*c*d^2 + a^3*d^3)*x)*sqrt(b*x + a)*sqrt(d*x + c)/(a^2* 
b^4*c^6 - 4*a^3*b^3*c^5*d + 6*a^4*b^2*c^4*d^2 - 4*a^5*b*c^3*d^3 + a^6*c^2* 
d^4 + (b^6*c^4*d^2 - 4*a*b^5*c^3*d^3 + 6*a^2*b^4*c^2*d^4 - 4*a^3*b^3*c*d^5 
 + a^4*b^2*d^6)*x^4 + 2*(b^6*c^5*d - 3*a*b^5*c^4*d^2 + 2*a^2*b^4*c^3*d^3 + 
 2*a^3*b^3*c^2*d^4 - 3*a^4*b^2*c*d^5 + a^5*b*d^6)*x^3 + (b^6*c^6 - 9*a^2*b 
^4*c^4*d^2 + 16*a^3*b^3*c^3*d^3 - 9*a^4*b^2*c^2*d^4 + a^6*d^6)*x^2 + 2*(a* 
b^5*c^6 - 3*a^2*b^4*c^5*d + 2*a^3*b^3*c^4*d^2 + 2*a^4*b^2*c^3*d^3 - 3*a^5* 
b*c^2*d^4 + a^6*c*d^5)*x)
 

Sympy [F]

\[ \int \frac {x}{(a+b x)^{5/2} (c+d x)^{5/2}} \, dx=\int \frac {x}{\left (a + b x\right )^{\frac {5}{2}} \left (c + d x\right )^{\frac {5}{2}}}\, dx \] Input:

integrate(x/(b*x+a)**(5/2)/(d*x+c)**(5/2),x)
 

Output:

Integral(x/((a + b*x)**(5/2)*(c + d*x)**(5/2)), x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {x}{(a+b x)^{5/2} (c+d x)^{5/2}} \, dx=\text {Exception raised: ValueError} \] Input:

integrate(x/(b*x+a)^(5/2)/(d*x+c)^(5/2),x, algorithm="maxima")
 

Output:

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
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 829 vs. \(2 (138) = 276\).

Time = 0.36 (sec) , antiderivative size = 829, normalized size of antiderivative = 5.18 \[ \int \frac {x}{(a+b x)^{5/2} (c+d x)^{5/2}} \, dx =\text {Too large to display} \] Input:

integrate(x/(b*x+a)^(5/2)/(d*x+c)^(5/2),x, algorithm="giac")
 

Output:

-2/3*(sqrt(b*x + a)*((5*b^8*c^4*d^3*abs(b) - 12*a*b^7*c^3*d^4*abs(b) + 6*a 
^2*b^6*c^2*d^5*abs(b) + 4*a^3*b^5*c*d^6*abs(b) - 3*a^4*b^4*d^7*abs(b))*(b* 
x + a)/(b^9*c^7*d - 7*a*b^8*c^6*d^2 + 21*a^2*b^7*c^5*d^3 - 35*a^3*b^6*c^4* 
d^4 + 35*a^4*b^5*c^3*d^5 - 21*a^5*b^4*c^2*d^6 + 7*a^6*b^3*c*d^7 - a^7*b^2* 
d^8) + 3*(2*b^9*c^5*d^2*abs(b) - 7*a*b^8*c^4*d^3*abs(b) + 8*a^2*b^7*c^3*d^ 
4*abs(b) - 2*a^3*b^6*c^2*d^5*abs(b) - 2*a^4*b^5*c*d^6*abs(b) + a^5*b^4*d^7 
*abs(b))/(b^9*c^7*d - 7*a*b^8*c^6*d^2 + 21*a^2*b^7*c^5*d^3 - 35*a^3*b^6*c^ 
4*d^4 + 35*a^4*b^5*c^3*d^5 - 21*a^5*b^4*c^2*d^6 + 7*a^6*b^3*c*d^7 - a^7*b^ 
2*d^8))/(b^2*c + (b*x + a)*b*d - a*b*d)^(3/2) + 2*(3*sqrt(b*d)*b^8*c^3 - s 
qrt(b*d)*a*b^7*c^2*d - 7*sqrt(b*d)*a^2*b^6*c*d^2 + 5*sqrt(b*d)*a^3*b^5*d^3 
 - 6*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b 
*d))^2*b^6*c^2 - 6*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^5*c*d + 12*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - s 
qrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^2*b^4*d^2 + 3*sqrt(b*d)*(sqrt(b*d) 
*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*b^4*c + 3*sqrt(b*d 
)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*a*b^3* 
d)/((b^3*c^3*abs(b) - 3*a*b^2*c^2*d*abs(b) + 3*a^2*b*c*d^2*abs(b) - a^3*d^ 
3*abs(b))*(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)^3))/b
 

Mupad [B] (verification not implemented)

Time = 1.53 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.49 \[ \int \frac {x}{(a+b x)^{5/2} (c+d x)^{5/2}} \, dx=-\frac {\sqrt {c+d\,x}\,\left (\frac {8\,x^2\,{\left (a\,d+b\,c\right )}^2}{d\,{\left (a\,d-b\,c\right )}^4}+\frac {16\,b\,x^3\,\left (a\,d+b\,c\right )}{3\,{\left (a\,d-b\,c\right )}^4}+\frac {x\,\left (6\,a^3\,d^3+42\,a^2\,b\,c\,d^2+42\,a\,b^2\,c^2\,d+6\,b^3\,c^3\right )}{3\,b\,d^2\,{\left (a\,d-b\,c\right )}^4}+\frac {4\,a\,c\,\left (a^2\,d^2+6\,a\,b\,c\,d+b^2\,c^2\right )}{3\,b\,d^2\,{\left (a\,d-b\,c\right )}^4}\right )}{x^3\,\sqrt {a+b\,x}+\frac {a\,c^2\,\sqrt {a+b\,x}}{b\,d^2}+\frac {x^2\,\left (a\,d+2\,b\,c\right )\,\sqrt {a+b\,x}}{b\,d}+\frac {c\,x\,\left (2\,a\,d+b\,c\right )\,\sqrt {a+b\,x}}{b\,d^2}} \] Input:

int(x/((a + b*x)^(5/2)*(c + d*x)^(5/2)),x)
 

Output:

-((c + d*x)^(1/2)*((8*x^2*(a*d + b*c)^2)/(d*(a*d - b*c)^4) + (16*b*x^3*(a* 
d + b*c))/(3*(a*d - b*c)^4) + (x*(6*a^3*d^3 + 6*b^3*c^3 + 42*a*b^2*c^2*d + 
 42*a^2*b*c*d^2))/(3*b*d^2*(a*d - b*c)^4) + (4*a*c*(a^2*d^2 + b^2*c^2 + 6* 
a*b*c*d))/(3*b*d^2*(a*d - b*c)^4)))/(x^3*(a + b*x)^(1/2) + (a*c^2*(a + b*x 
)^(1/2))/(b*d^2) + (x^2*(a*d + 2*b*c)*(a + b*x)^(1/2))/(b*d) + (c*x*(2*a*d 
 + b*c)*(a + b*x)^(1/2))/(b*d^2))
 

Reduce [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 698, normalized size of antiderivative = 4.36 \[ \int \frac {x}{(a+b x)^{5/2} (c+d x)^{5/2}} \, dx=\frac {\frac {16 \sqrt {d}\, \sqrt {b}\, \sqrt {b x +a}\, a^{2} c^{2} d}{3}+\frac {32 \sqrt {d}\, \sqrt {b}\, \sqrt {b x +a}\, a^{2} c \,d^{2} x}{3}+\frac {16 \sqrt {d}\, \sqrt {b}\, \sqrt {b x +a}\, a^{2} d^{3} x^{2}}{3}+\frac {16 \sqrt {d}\, \sqrt {b}\, \sqrt {b x +a}\, a b \,c^{3}}{3}+16 \sqrt {d}\, \sqrt {b}\, \sqrt {b x +a}\, a b \,c^{2} d x +16 \sqrt {d}\, \sqrt {b}\, \sqrt {b x +a}\, a b c \,d^{2} x^{2}+\frac {16 \sqrt {d}\, \sqrt {b}\, \sqrt {b x +a}\, a b \,d^{3} x^{3}}{3}+\frac {16 \sqrt {d}\, \sqrt {b}\, \sqrt {b x +a}\, b^{2} c^{3} x}{3}+\frac {32 \sqrt {d}\, \sqrt {b}\, \sqrt {b x +a}\, b^{2} c^{2} d \,x^{2}}{3}+\frac {16 \sqrt {d}\, \sqrt {b}\, \sqrt {b x +a}\, b^{2} c \,d^{2} x^{3}}{3}-\frac {4 \sqrt {d x +c}\, a^{3} c \,d^{2}}{3}-2 \sqrt {d x +c}\, a^{3} d^{3} x -8 \sqrt {d x +c}\, a^{2} b \,c^{2} d -14 \sqrt {d x +c}\, a^{2} b c \,d^{2} x -8 \sqrt {d x +c}\, a^{2} b \,d^{3} x^{2}-\frac {4 \sqrt {d x +c}\, a \,b^{2} c^{3}}{3}-14 \sqrt {d x +c}\, a \,b^{2} c^{2} d x -16 \sqrt {d x +c}\, a \,b^{2} c \,d^{2} x^{2}-\frac {16 \sqrt {d x +c}\, a \,b^{2} d^{3} x^{3}}{3}-2 \sqrt {d x +c}\, b^{3} c^{3} x -8 \sqrt {d x +c}\, b^{3} c^{2} d \,x^{2}-\frac {16 \sqrt {d x +c}\, b^{3} c \,d^{2} x^{3}}{3}}{\sqrt {b x +a}\, \left (a^{4} b \,d^{6} x^{3}-4 a^{3} b^{2} c \,d^{5} x^{3}+6 a^{2} b^{3} c^{2} d^{4} x^{3}-4 a \,b^{4} c^{3} d^{3} x^{3}+b^{5} c^{4} d^{2} x^{3}+a^{5} d^{6} x^{2}-2 a^{4} b c \,d^{5} x^{2}-2 a^{3} b^{2} c^{2} d^{4} x^{2}+8 a^{2} b^{3} c^{3} d^{3} x^{2}-7 a \,b^{4} c^{4} d^{2} x^{2}+2 b^{5} c^{5} d \,x^{2}+2 a^{5} c \,d^{5} x -7 a^{4} b \,c^{2} d^{4} x +8 a^{3} b^{2} c^{3} d^{3} x -2 a^{2} b^{3} c^{4} d^{2} x -2 a \,b^{4} c^{5} d x +b^{5} c^{6} x +a^{5} c^{2} d^{4}-4 a^{4} b \,c^{3} d^{3}+6 a^{3} b^{2} c^{4} d^{2}-4 a^{2} b^{3} c^{5} d +a \,b^{4} c^{6}\right )} \] Input:

int(x/(b*x+a)^(5/2)/(d*x+c)^(5/2),x)
 

Output:

(2*(8*sqrt(d)*sqrt(b)*sqrt(a + b*x)*a**2*c**2*d + 16*sqrt(d)*sqrt(b)*sqrt( 
a + b*x)*a**2*c*d**2*x + 8*sqrt(d)*sqrt(b)*sqrt(a + b*x)*a**2*d**3*x**2 + 
8*sqrt(d)*sqrt(b)*sqrt(a + b*x)*a*b*c**3 + 24*sqrt(d)*sqrt(b)*sqrt(a + b*x 
)*a*b*c**2*d*x + 24*sqrt(d)*sqrt(b)*sqrt(a + b*x)*a*b*c*d**2*x**2 + 8*sqrt 
(d)*sqrt(b)*sqrt(a + b*x)*a*b*d**3*x**3 + 8*sqrt(d)*sqrt(b)*sqrt(a + b*x)* 
b**2*c**3*x + 16*sqrt(d)*sqrt(b)*sqrt(a + b*x)*b**2*c**2*d*x**2 + 8*sqrt(d 
)*sqrt(b)*sqrt(a + b*x)*b**2*c*d**2*x**3 - 2*sqrt(c + d*x)*a**3*c*d**2 - 3 
*sqrt(c + d*x)*a**3*d**3*x - 12*sqrt(c + d*x)*a**2*b*c**2*d - 21*sqrt(c + 
d*x)*a**2*b*c*d**2*x - 12*sqrt(c + d*x)*a**2*b*d**3*x**2 - 2*sqrt(c + d*x) 
*a*b**2*c**3 - 21*sqrt(c + d*x)*a*b**2*c**2*d*x - 24*sqrt(c + d*x)*a*b**2* 
c*d**2*x**2 - 8*sqrt(c + d*x)*a*b**2*d**3*x**3 - 3*sqrt(c + d*x)*b**3*c**3 
*x - 12*sqrt(c + d*x)*b**3*c**2*d*x**2 - 8*sqrt(c + d*x)*b**3*c*d**2*x**3) 
)/(3*sqrt(a + b*x)*(a**5*c**2*d**4 + 2*a**5*c*d**5*x + a**5*d**6*x**2 - 4* 
a**4*b*c**3*d**3 - 7*a**4*b*c**2*d**4*x - 2*a**4*b*c*d**5*x**2 + a**4*b*d* 
*6*x**3 + 6*a**3*b**2*c**4*d**2 + 8*a**3*b**2*c**3*d**3*x - 2*a**3*b**2*c* 
*2*d**4*x**2 - 4*a**3*b**2*c*d**5*x**3 - 4*a**2*b**3*c**5*d - 2*a**2*b**3* 
c**4*d**2*x + 8*a**2*b**3*c**3*d**3*x**2 + 6*a**2*b**3*c**2*d**4*x**3 + a* 
b**4*c**6 - 2*a*b**4*c**5*d*x - 7*a*b**4*c**4*d**2*x**2 - 4*a*b**4*c**3*d* 
*3*x**3 + b**5*c**6*x + 2*b**5*c**5*d*x**2 + b**5*c**4*d**2*x**3))