\(\int \frac {1}{(a+b x)^{5/2} (c+d x)^{5/2}} \, dx\) [417]

Optimal result

Integrand size = 19, antiderivative size = 135 \[ \int \frac {1}{(a+b x)^{5/2} (c+d x)^{5/2}} \, dx=-\frac {2}{3 (b c-a d) (a+b x)^{3/2} (c+d x)^{3/2}}+\frac {4 d}{(b c-a d)^2 \sqrt {a+b x} (c+d x)^{3/2}}+\frac {16 d^2 \sqrt {a+b x}}{3 (b c-a d)^3 (c+d x)^{3/2}}+\frac {32 b d^2 \sqrt {a+b x}}{3 (b c-a d)^4 \sqrt {c+d x}} \] Output:

-2/3/(-a*d+b*c)/(b*x+a)^(3/2)/(d*x+c)^(3/2)+4*d/(-a*d+b*c)^2/(b*x+a)^(1/2) 
/(d*x+c)^(3/2)+16/3*d^2*(b*x+a)^(1/2)/(-a*d+b*c)^3/(d*x+c)^(3/2)+32/3*b*d^ 
2*(b*x+a)^(1/2)/(-a*d+b*c)^4/(d*x+c)^(1/2)
 

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.87 \[ \int \frac {1}{(a+b x)^{5/2} (c+d x)^{5/2}} \, dx=\frac {-2 a^3 d^3+6 a^2 b d^2 (3 c+2 d x)+6 a b^2 d \left (3 c^2+12 c d x+8 d^2 x^2\right )+b^3 \left (-2 c^3+12 c^2 d x+48 c d^2 x^2+32 d^3 x^3\right )}{3 (b c-a d)^4 (a+b x)^{3/2} (c+d x)^{3/2}} \] Input:

Integrate[1/((a + b*x)^(5/2)*(c + d*x)^(5/2)),x]
 

Output:

(-2*a^3*d^3 + 6*a^2*b*d^2*(3*c + 2*d*x) + 6*a*b^2*d*(3*c^2 + 12*c*d*x + 8* 
d^2*x^2) + b^3*(-2*c^3 + 12*c^2*d*x + 48*c*d^2*x^2 + 32*d^3*x^3))/(3*(b*c 
- a*d)^4*(a + b*x)^(3/2)*(c + d*x)^(3/2))
 

Rubi [A] (verified)

Time = 0.21 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.16, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {55, 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.

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

\(\Big \downarrow \) 55

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

\(\Big \downarrow \) 55

\(\displaystyle -\frac {2 d \left (-\frac {4 d \int \frac {1}{\sqrt {a+b x} (c+d x)^{5/2}}dx}{b c-a d}-\frac {2}{\sqrt {a+b x} (c+d x)^{3/2} (b c-a d)}\right )}{b c-a d}-\frac {2}{3 (a+b x)^{3/2} (c+d x)^{3/2} (b c-a d)}\)

\(\Big \downarrow \) 55

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

\(\Big \downarrow \) 48

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

Input:

Int[1/((a + b*x)^(5/2)*(c + d*x)^(5/2)),x]
 

Output:

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

Maple [A] (verified)

Time = 0.24 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.00

method	result	size
default	\(-\frac {2}{3 \left (-a d +b c \right ) \left (b x +a \right )^{\frac {3}{2}} \left (x d +c \right )^{\frac {3}{2}}}-\frac {2 d \left (-\frac {2}{\left (-a d +b c \right ) \sqrt {b x +a}\, \left (x d +c \right )^{\frac {3}{2}}}-\frac {4 d \left (-\frac {2 \sqrt {b x +a}}{3 \left (a d -b c \right ) \left (x d +c \right )^{\frac {3}{2}}}+\frac {4 b \sqrt {b x +a}}{3 \left (a d -b c \right )^{2} \sqrt {x d +c}}\right )}{-a d +b c}\right )}{-a d +b c}\)	\(135\)
gosper	\(-\frac {2 \left (-16 d^{3} x^{3} b^{3}-24 x^{2} a \,b^{2} d^{3}-24 x^{2} b^{3} c \,d^{2}-6 x \,a^{2} b \,d^{3}-36 x a \,b^{2} c \,d^{2}-6 x \,b^{3} c^{2} d +a^{3} d^{3}-9 a^{2} b c \,d^{2}-9 a \,b^{2} c^{2} d +b^{3} c^{3}\right )}{3 \left (b x +a \right )^{\frac {3}{2}} \left (x d +c \right )^{\frac {3}{2}} \left (d^{4} a^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +c^{4} b^{4}\right )}\)	\(169\)
orering	\(-\frac {2 \left (-16 d^{3} x^{3} b^{3}-24 x^{2} a \,b^{2} d^{3}-24 x^{2} b^{3} c \,d^{2}-6 x \,a^{2} b \,d^{3}-36 x a \,b^{2} c \,d^{2}-6 x \,b^{3} c^{2} d +a^{3} d^{3}-9 a^{2} b c \,d^{2}-9 a \,b^{2} c^{2} d +b^{3} c^{3}\right )}{3 \left (b x +a \right )^{\frac {3}{2}} \left (x d +c \right )^{\frac {3}{2}} \left (d^{4} a^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +c^{4} b^{4}\right )}\)	\(169\)

Input:

int(1/(b*x+a)^(5/2)/(d*x+c)^(5/2),x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 447 vs. \(2 (113) = 226\).

Time = 0.43 (sec) , antiderivative size = 447, normalized size of antiderivative = 3.31 \[ \int \frac {1}{(a+b x)^{5/2} (c+d x)^{5/2}} \, dx=\frac {2 \, {\left (16 \, b^{3} d^{3} x^{3} - b^{3} c^{3} + 9 \, a b^{2} c^{2} d + 9 \, a^{2} b c d^{2} - a^{3} d^{3} + 24 \, {\left (b^{3} c d^{2} + a b^{2} d^{3}\right )} x^{2} + 6 \, {\left (b^{3} c^{2} d + 6 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{3 \, {\left (a^{2} b^{4} c^{6} - 4 \, a^{3} b^{3} c^{5} d + 6 \, a^{4} b^{2} c^{4} d^{2} - 4 \, a^{5} b c^{3} d^{3} + a^{6} c^{2} d^{4} + {\left (b^{6} c^{4} d^{2} - 4 \, a b^{5} c^{3} d^{3} + 6 \, a^{2} b^{4} c^{2} d^{4} - 4 \, a^{3} b^{3} c d^{5} + a^{4} b^{2} d^{6}\right )} x^{4} + 2 \, {\left (b^{6} c^{5} d - 3 \, a b^{5} c^{4} d^{2} + 2 \, a^{2} b^{4} c^{3} d^{3} + 2 \, a^{3} b^{3} c^{2} d^{4} - 3 \, a^{4} b^{2} c d^{5} + a^{5} b d^{6}\right )} x^{3} + {\left (b^{6} c^{6} - 9 \, a^{2} b^{4} c^{4} d^{2} + 16 \, a^{3} b^{3} c^{3} d^{3} - 9 \, a^{4} b^{2} c^{2} d^{4} + a^{6} d^{6}\right )} x^{2} + 2 \, {\left (a b^{5} c^{6} - 3 \, a^{2} b^{4} c^{5} d + 2 \, a^{3} b^{3} c^{4} d^{2} + 2 \, a^{4} b^{2} c^{3} d^{3} - 3 \, a^{5} b c^{2} d^{4} + a^{6} c d^{5}\right )} x\right )}} \] Input:

integrate(1/(b*x+a)^(5/2)/(d*x+c)^(5/2),x, algorithm="fricas")
 

Output:

2/3*(16*b^3*d^3*x^3 - b^3*c^3 + 9*a*b^2*c^2*d + 9*a^2*b*c*d^2 - a^3*d^3 + 
24*(b^3*c*d^2 + a*b^2*d^3)*x^2 + 6*(b^3*c^2*d + 6*a*b^2*c*d^2 + a^2*b*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]

\[ \int \frac {1}{(a+b x)^{5/2} (c+d x)^{5/2}} \, dx=\int \frac {1}{\left (a + b x\right )^{\frac {5}{2}} \left (c + d x\right )^{\frac {5}{2}}}\, dx \] Input:

integrate(1/(b*x+a)**(5/2)/(d*x+c)**(5/2),x)
                                                                                    
                                                                                    
 

Output:

Integral(1/((a + b*x)**(5/2)*(c + d*x)**(5/2)), x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {1}{(a+b x)^{5/2} (c+d x)^{5/2}} \, dx=\text {Exception raised: ValueError} \] Input:

integrate(1/(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. 670 vs. \(2 (113) = 226\).

Time = 0.25 (sec) , antiderivative size = 670, normalized size of antiderivative = 4.96 \[ \int \frac {1}{(a+b x)^{5/2} (c+d x)^{5/2}} \, dx=\frac {2 \, \sqrt {b x + a} {\left (\frac {8 \, {\left (b^{7} c^{3} d^{4} {\left | b \right |} - 3 \, a b^{6} c^{2} d^{5} {\left | b \right |} + 3 \, a^{2} b^{5} c d^{6} {\left | b \right |} - a^{3} b^{4} d^{7} {\left | b \right |}\right )} {\left (b x + a\right )}}{b^{9} c^{7} d - 7 \, a b^{8} c^{6} d^{2} + 21 \, a^{2} b^{7} c^{5} d^{3} - 35 \, a^{3} b^{6} c^{4} d^{4} + 35 \, a^{4} b^{5} c^{3} d^{5} - 21 \, a^{5} b^{4} c^{2} d^{6} + 7 \, a^{6} b^{3} c d^{7} - a^{7} b^{2} d^{8}} + \frac {9 \, {\left (b^{8} c^{4} d^{3} {\left | b \right |} - 4 \, a b^{7} c^{3} d^{4} {\left | b \right |} + 6 \, a^{2} b^{6} c^{2} d^{5} {\left | b \right |} - 4 \, a^{3} b^{5} c d^{6} {\left | b \right |} + a^{4} b^{4} d^{7} {\left | b \right |}\right )}}{b^{9} c^{7} d - 7 \, a b^{8} c^{6} d^{2} + 21 \, a^{2} b^{7} c^{5} d^{3} - 35 \, a^{3} b^{6} c^{4} d^{4} + 35 \, a^{4} b^{5} c^{3} d^{5} - 21 \, a^{5} b^{4} c^{2} d^{6} + 7 \, a^{6} b^{3} c d^{7} - a^{7} b^{2} d^{8}}\right )}}{3 \, {\left (b^{2} c + {\left (b x + a\right )} b d - a b d\right )}^{\frac {3}{2}}} + \frac {8 \, {\left (4 \, \sqrt {b d} b^{7} c^{2} d - 8 \, \sqrt {b d} a b^{6} c d^{2} + 4 \, \sqrt {b d} a^{2} b^{5} d^{3} - 9 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} b^{5} c d + 9 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a b^{4} d^{2} + 3 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4} b^{3} d\right )}}{3 \, {\left (b^{3} c^{3} {\left | b \right |} - 3 \, a b^{2} c^{2} d {\left | b \right |} + 3 \, a^{2} b c d^{2} {\left | b \right |} - a^{3} d^{3} {\left | b \right |}\right )} {\left (b^{2} c - a b d - {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2}\right )}^{3}} \] Input:

integrate(1/(b*x+a)^(5/2)/(d*x+c)^(5/2),x, algorithm="giac")
 

Output:

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

Mupad [B] (verification not implemented)

Time = 1.52 (sec) , antiderivative size = 224, normalized size of antiderivative = 1.66 \[ \int \frac {1}{(a+b x)^{5/2} (c+d x)^{5/2}} \, dx=\frac {\sqrt {c+d\,x}\,\left (\frac {16\,b\,x^2\,\left (a\,d+b\,c\right )}{{\left (a\,d-b\,c\right )}^4}-\frac {2\,a^3\,d^3-18\,a^2\,b\,c\,d^2-18\,a\,b^2\,c^2\,d+2\,b^3\,c^3}{3\,b\,d^2\,{\left (a\,d-b\,c\right )}^4}+\frac {32\,b^2\,d\,x^3}{3\,{\left (a\,d-b\,c\right )}^4}+\frac {4\,x\,\left (a^2\,d^2+6\,a\,b\,c\,d+b^2\,c^2\right )}{d\,{\left (a\,d-b\,c\right )}^4}\right )}{x^3\,\sqrt {a+b\,x}+\frac {a\,c^2\,\sqrt {a+b\,x}}{b\,d^2}+\frac {x^2\,\left (a\,d+2\,b\,c\right )\,\sqrt {a+b\,x}}{b\,d}+\frac {c\,x\,\left (2\,a\,d+b\,c\right )\,\sqrt {a+b\,x}}{b\,d^2}} \] Input:

int(1/((a + b*x)^(5/2)*(c + d*x)^(5/2)),x)
 

Output:

((c + d*x)^(1/2)*((16*b*x^2*(a*d + b*c))/(a*d - b*c)^4 - (2*a^3*d^3 + 2*b^ 
3*c^3 - 18*a*b^2*c^2*d - 18*a^2*b*c*d^2)/(3*b*d^2*(a*d - b*c)^4) + (32*b^2 
*d*x^3)/(3*(a*d - b*c)^4) + (4*x*(a^2*d^2 + b^2*c^2 + 6*a*b*c*d))/(d*(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 = 0.21 (sec) , antiderivative size = 579, normalized size of antiderivative = 4.29 \[ \int \frac {1}{(a+b x)^{5/2} (c+d x)^{5/2}} \, dx=\frac {-\frac {32 \sqrt {d}\, \sqrt {b}\, \sqrt {b x +a}\, a b \,c^{2} d}{3}-\frac {64 \sqrt {d}\, \sqrt {b}\, \sqrt {b x +a}\, a b c \,d^{2} x}{3}-\frac {32 \sqrt {d}\, \sqrt {b}\, \sqrt {b x +a}\, a b \,d^{3} x^{2}}{3}-\frac {32 \sqrt {d}\, \sqrt {b}\, \sqrt {b x +a}\, b^{2} c^{2} d x}{3}-\frac {64 \sqrt {d}\, \sqrt {b}\, \sqrt {b x +a}\, b^{2} c \,d^{2} x^{2}}{3}-\frac {32 \sqrt {d}\, \sqrt {b}\, \sqrt {b x +a}\, b^{2} d^{3} x^{3}}{3}-\frac {2 \sqrt {d x +c}\, a^{3} d^{3}}{3}+6 \sqrt {d x +c}\, a^{2} b c \,d^{2}+4 \sqrt {d x +c}\, a^{2} b \,d^{3} x +6 \sqrt {d x +c}\, a \,b^{2} c^{2} d +24 \sqrt {d x +c}\, a \,b^{2} c \,d^{2} x +16 \sqrt {d x +c}\, a \,b^{2} d^{3} x^{2}-\frac {2 \sqrt {d x +c}\, b^{3} c^{3}}{3}+4 \sqrt {d x +c}\, b^{3} c^{2} d x +16 \sqrt {d x +c}\, b^{3} c \,d^{2} x^{2}+\frac {32 \sqrt {d x +c}\, b^{3} d^{3} x^{3}}{3}}{\sqrt {b x +a}\, \left (a^{4} b \,d^{6} x^{3}-4 a^{3} b^{2} c \,d^{5} x^{3}+6 a^{2} b^{3} c^{2} d^{4} x^{3}-4 a \,b^{4} c^{3} d^{3} x^{3}+b^{5} c^{4} d^{2} x^{3}+a^{5} d^{6} x^{2}-2 a^{4} b c \,d^{5} x^{2}-2 a^{3} b^{2} c^{2} d^{4} x^{2}+8 a^{2} b^{3} c^{3} d^{3} x^{2}-7 a \,b^{4} c^{4} d^{2} x^{2}+2 b^{5} c^{5} d \,x^{2}+2 a^{5} c \,d^{5} x -7 a^{4} b \,c^{2} d^{4} x +8 a^{3} b^{2} c^{3} d^{3} x -2 a^{2} b^{3} c^{4} d^{2} x -2 a \,b^{4} c^{5} d x +b^{5} c^{6} x +a^{5} c^{2} d^{4}-4 a^{4} b \,c^{3} d^{3}+6 a^{3} b^{2} c^{4} d^{2}-4 a^{2} b^{3} c^{5} d +a \,b^{4} c^{6}\right )} \] Input:

int(1/(b*x+a)^(5/2)/(d*x+c)^(5/2),x)
 

Output:

(2*( - 16*sqrt(d)*sqrt(b)*sqrt(a + b*x)*a*b*c**2*d - 32*sqrt(d)*sqrt(b)*sq 
rt(a + b*x)*a*b*c*d**2*x - 16*sqrt(d)*sqrt(b)*sqrt(a + b*x)*a*b*d**3*x**2 
- 16*sqrt(d)*sqrt(b)*sqrt(a + b*x)*b**2*c**2*d*x - 32*sqrt(d)*sqrt(b)*sqrt 
(a + b*x)*b**2*c*d**2*x**2 - 16*sqrt(d)*sqrt(b)*sqrt(a + b*x)*b**2*d**3*x* 
*3 - sqrt(c + d*x)*a**3*d**3 + 9*sqrt(c + d*x)*a**2*b*c*d**2 + 6*sqrt(c + 
d*x)*a**2*b*d**3*x + 9*sqrt(c + d*x)*a*b**2*c**2*d + 36*sqrt(c + d*x)*a*b* 
*2*c*d**2*x + 24*sqrt(c + d*x)*a*b**2*d**3*x**2 - sqrt(c + d*x)*b**3*c**3 
+ 6*sqrt(c + d*x)*b**3*c**2*d*x + 24*sqrt(c + d*x)*b**3*c*d**2*x**2 + 16*s 
qrt(c + d*x)*b**3*d**3*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))