\(\int \frac {1}{x (a+b x)^{3/2} (c+d x)^{3/2}} \, dx\) [389]

Optimal result

Integrand size = 22, antiderivative size = 121 \[ \int \frac {1}{x (a+b x)^{3/2} (c+d x)^{3/2}} \, dx=\frac {2 b}{a (b c-a d) \sqrt {a+b x} \sqrt {c+d x}}+\frac {2 d (b c+a d) \sqrt {a+b x}}{a c (b c-a d)^2 \sqrt {c+d x}}-\frac {2 \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{a^{3/2} c^{3/2}} \] Output:

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

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.84 \[ \int \frac {1}{x (a+b x)^{3/2} (c+d x)^{3/2}} \, dx=\frac {2 \sqrt {a+b x} \left (a d^2+\frac {b^2 c (c+d x)}{a+b x}\right )}{a c (-b c+a d)^2 \sqrt {c+d x}}-\frac {2 \text {arctanh}\left (\frac {\sqrt {a} \sqrt {c+d x}}{\sqrt {c} \sqrt {a+b x}}\right )}{a^{3/2} c^{3/2}} \] Input:

Integrate[1/(x*(a + b*x)^(3/2)*(c + d*x)^(3/2)),x]
 

Output:

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

Rubi [A] (verified)

Time = 0.24 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.17, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {115, 27, 169, 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.

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

\(\Big \downarrow \) 115

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 169

\(\displaystyle \frac {\frac {2 d \sqrt {a+b x} (a d+b c)}{c \sqrt {c+d x} (b c-a d)}-\frac {2 \int -\frac {(b c-a d)^2}{2 x \sqrt {a+b x} \sqrt {c+d x}}dx}{c (b c-a d)}}{a (b c-a d)}+\frac {2 b}{a \sqrt {a+b x} \sqrt {c+d x} (b c-a d)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {(b c-a d) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}}dx}{c}+\frac {2 d \sqrt {a+b x} (a d+b c)}{c \sqrt {c+d x} (b c-a d)}}{a (b c-a d)}+\frac {2 b}{a \sqrt {a+b x} \sqrt {c+d x} (b c-a d)}\)

\(\Big \downarrow \) 104

\(\displaystyle \frac {\frac {2 (b c-a d) \int \frac {1}{\frac {c (a+b x)}{c+d x}-a}d\frac {\sqrt {a+b x}}{\sqrt {c+d x}}}{c}+\frac {2 d \sqrt {a+b x} (a d+b c)}{c \sqrt {c+d x} (b c-a d)}}{a (b c-a d)}+\frac {2 b}{a \sqrt {a+b x} \sqrt {c+d x} (b c-a d)}\)

\(\Big \downarrow \) 221

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

Input:

Int[1/(x*(a + b*x)^(3/2)*(c + d*x)^(3/2)),x]
 

Output:

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

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 104

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]
 

rule 115

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[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*(m + 1) 
 - b*(d*e*(m + n + 2) + c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], 
 x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && LtQ[m, -1] && IntegersQ[2*m, 2 
*n, 2*p]
 

rule 169

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] && LtQ[m, -1] && IntegersQ[ 
2*m, 2*n, 2*p]
 

rule 221

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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(637\) vs. \(2(101)=202\).

Time = 0.29 (sec) , antiderivative size = 638, normalized size of antiderivative = 5.27

method	result	size
default	\(\frac {-\ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (x d +c \right )}+2 a c}{x}\right ) a^{2} b \,d^{3} x^{2}+2 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (x d +c \right )}+2 a c}{x}\right ) a \,b^{2} c \,d^{2} x^{2}-\ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (x d +c \right )}+2 a c}{x}\right ) b^{3} c^{2} d \,x^{2}-\ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (x d +c \right )}+2 a c}{x}\right ) a^{3} d^{3} x +\ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (x d +c \right )}+2 a c}{x}\right ) a^{2} b c \,d^{2} x +\ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (x d +c \right )}+2 a c}{x}\right ) a \,b^{2} c^{2} d x -\ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (x d +c \right )}+2 a c}{x}\right ) b^{3} c^{3} x -\ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (x d +c \right )}+2 a c}{x}\right ) a^{3} c \,d^{2}+2 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (x d +c \right )}+2 a c}{x}\right ) a^{2} b \,c^{2} d -\ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (x d +c \right )}+2 a c}{x}\right ) a \,b^{2} c^{3}+2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, a b \,d^{2} x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, b^{2} c d x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, a^{2} d^{2}+2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, b^{2} c^{2}}{\sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {a c}\, \left (a d -b c \right )^{2} \sqrt {b x +a}\, \sqrt {x d +c}\, a c}\)	\(638\)

Input:

int(1/x/(b*x+a)^(3/2)/(d*x+c)^(3/2),x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 349 vs. \(2 (101) = 202\).

Time = 0.23 (sec) , antiderivative size = 719, normalized size of antiderivative = 5.94 \[ \int \frac {1}{x (a+b x)^{3/2} (c+d x)^{3/2}} \, dx=\left [\frac {{\left (a b^{2} c^{3} - 2 \, a^{2} b c^{2} d + a^{3} c d^{2} + {\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x^{2} + {\left (b^{3} c^{3} - a b^{2} c^{2} d - a^{2} b c d^{2} + a^{3} d^{3}\right )} x\right )} \sqrt {a c} \log \left (\frac {8 \, a^{2} c^{2} + {\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} - 4 \, {\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {a c} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (a b c^{2} + a^{2} c d\right )} x}{x^{2}}\right ) + 4 \, {\left (a b^{2} c^{3} + a^{3} c d^{2} + {\left (a b^{2} c^{2} d + a^{2} b c d^{2}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (a^{3} b^{2} c^{5} - 2 \, a^{4} b c^{4} d + a^{5} c^{3} d^{2} + {\left (a^{2} b^{3} c^{4} d - 2 \, a^{3} b^{2} c^{3} d^{2} + a^{4} b c^{2} d^{3}\right )} x^{2} + {\left (a^{2} b^{3} c^{5} - a^{3} b^{2} c^{4} d - a^{4} b c^{3} d^{2} + a^{5} c^{2} d^{3}\right )} x\right )}}, \frac {{\left (a b^{2} c^{3} - 2 \, a^{2} b c^{2} d + a^{3} c d^{2} + {\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x^{2} + {\left (b^{3} c^{3} - a b^{2} c^{2} d - a^{2} b c d^{2} + a^{3} d^{3}\right )} x\right )} \sqrt {-a c} \arctan \left (\frac {{\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {-a c} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (a b c d x^{2} + a^{2} c^{2} + {\left (a b c^{2} + a^{2} c d\right )} x\right )}}\right ) + 2 \, {\left (a b^{2} c^{3} + a^{3} c d^{2} + {\left (a b^{2} c^{2} d + a^{2} b c d^{2}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{a^{3} b^{2} c^{5} - 2 \, a^{4} b c^{4} d + a^{5} c^{3} d^{2} + {\left (a^{2} b^{3} c^{4} d - 2 \, a^{3} b^{2} c^{3} d^{2} + a^{4} b c^{2} d^{3}\right )} x^{2} + {\left (a^{2} b^{3} c^{5} - a^{3} b^{2} c^{4} d - a^{4} b c^{3} d^{2} + a^{5} c^{2} d^{3}\right )} x}\right ] \] Input:

integrate(1/x/(b*x+a)^(3/2)/(d*x+c)^(3/2),x, algorithm="fricas")
 

Output:

[1/2*((a*b^2*c^3 - 2*a^2*b*c^2*d + a^3*c*d^2 + (b^3*c^2*d - 2*a*b^2*c*d^2 
+ a^2*b*d^3)*x^2 + (b^3*c^3 - a*b^2*c^2*d - a^2*b*c*d^2 + a^3*d^3)*x)*sqrt 
(a*c)*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*(a*b^2*c^3 + a^3*c*d^2 + (a*b^2*c^2*d + a^2*b*c*d^2)*x)*sqrt( 
b*x + a)*sqrt(d*x + c))/(a^3*b^2*c^5 - 2*a^4*b*c^4*d + a^5*c^3*d^2 + (a^2* 
b^3*c^4*d - 2*a^3*b^2*c^3*d^2 + a^4*b*c^2*d^3)*x^2 + (a^2*b^3*c^5 - a^3*b^ 
2*c^4*d - a^4*b*c^3*d^2 + a^5*c^2*d^3)*x), ((a*b^2*c^3 - 2*a^2*b*c^2*d + a 
^3*c*d^2 + (b^3*c^2*d - 2*a*b^2*c*d^2 + a^2*b*d^3)*x^2 + (b^3*c^3 - a*b^2* 
c^2*d - a^2*b*c*d^2 + a^3*d^3)*x)*sqrt(-a*c)*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*(a*b^2*c^3 + a^3*c*d^2 + (a*b^2*c^2*d + a^2*b*c*d^ 
2)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(a^3*b^2*c^5 - 2*a^4*b*c^4*d + a^5*c^3* 
d^2 + (a^2*b^3*c^4*d - 2*a^3*b^2*c^3*d^2 + a^4*b*c^2*d^3)*x^2 + (a^2*b^3*c 
^5 - a^3*b^2*c^4*d - a^4*b*c^3*d^2 + a^5*c^2*d^3)*x)]
 

Sympy [F]

\[ \int \frac {1}{x (a+b x)^{3/2} (c+d x)^{3/2}} \, dx=\int \frac {1}{x \left (a + b x\right )^{\frac {3}{2}} \left (c + d x\right )^{\frac {3}{2}}}\, dx \] Input:

integrate(1/x/(b*x+a)**(3/2)/(d*x+c)**(3/2),x)
 

Output:

Integral(1/(x*(a + b*x)**(3/2)*(c + d*x)**(3/2)), x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {1}{x (a+b x)^{3/2} (c+d x)^{3/2}} \, dx=\text {Exception raised: ValueError} \] Input:

integrate(1/x/(b*x+a)^(3/2)/(d*x+c)^(3/2),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. 241 vs. \(2 (101) = 202\).

Time = 0.39 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.99 \[ \int \frac {1}{x (a+b x)^{3/2} (c+d x)^{3/2}} \, dx=\frac {2 \, \sqrt {b x + a} b^{2} d^{2}}{{\left (b^{2} c^{3} {\left | b \right |} - 2 \, a b c^{2} d {\left | b \right |} + a^{2} c d^{2} {\left | b \right |}\right )} \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}} + \frac {4 \, \sqrt {b d} b^{3}}{{\left (a b c {\left | b \right |} - a^{2} d {\left | b \right |}\right )} {\left (b^{2} c - a b d - {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2}\right )}} - \frac {2 \, \sqrt {b d} b \arctan \left (-\frac {b^{2} c + a b d - {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2}}{2 \, \sqrt {-a b c d} b}\right )}{\sqrt {-a b c d} a c {\left | b \right |}} \] Input:

integrate(1/x/(b*x+a)^(3/2)/(d*x+c)^(3/2),x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{x (a+b x)^{3/2} (c+d x)^{3/2}} \, dx=\int \frac {1}{x\,{\left (a+b\,x\right )}^{3/2}\,{\left (c+d\,x\right )}^{3/2}} \,d x \] Input:

int(1/(x*(a + b*x)^(3/2)*(c + d*x)^(3/2)),x)
 

Output:

int(1/(x*(a + b*x)^(3/2)*(c + d*x)^(3/2)), x)
 

Reduce [B] (verification not implemented)

Time = 1.11 (sec) , antiderivative size = 1216, normalized size of antiderivative = 10.05 \[ \int \frac {1}{x (a+b x)^{3/2} (c+d x)^{3/2}} \, dx =\text {Too large to display} \] Input:

int(1/x/(b*x+a)^(3/2)/(d*x+c)^(3/2),x)
 

Output:

(sqrt(c)*sqrt(a)*sqrt(a + b*x)*log( - sqrt(2*sqrt(d)*sqrt(c)*sqrt(b)*sqrt( 
a) + a*d + b*c) + sqrt(d)*sqrt(a + b*x) + sqrt(b)*sqrt(c + d*x))*a**2*c*d* 
*2 + sqrt(c)*sqrt(a)*sqrt(a + b*x)*log( - sqrt(2*sqrt(d)*sqrt(c)*sqrt(b)*s 
qrt(a) + a*d + b*c) + sqrt(d)*sqrt(a + b*x) + sqrt(b)*sqrt(c + d*x))*a**2* 
d**3*x - 2*sqrt(c)*sqrt(a)*sqrt(a + b*x)*log( - sqrt(2*sqrt(d)*sqrt(c)*sqr 
t(b)*sqrt(a) + a*d + b*c) + sqrt(d)*sqrt(a + b*x) + sqrt(b)*sqrt(c + d*x)) 
*a*b*c**2*d - 2*sqrt(c)*sqrt(a)*sqrt(a + b*x)*log( - sqrt(2*sqrt(d)*sqrt(c 
)*sqrt(b)*sqrt(a) + a*d + b*c) + sqrt(d)*sqrt(a + b*x) + sqrt(b)*sqrt(c + 
d*x))*a*b*c*d**2*x + sqrt(c)*sqrt(a)*sqrt(a + b*x)*log( - sqrt(2*sqrt(d)*s 
qrt(c)*sqrt(b)*sqrt(a) + a*d + b*c) + sqrt(d)*sqrt(a + b*x) + sqrt(b)*sqrt 
(c + d*x))*b**2*c**3 + sqrt(c)*sqrt(a)*sqrt(a + b*x)*log( - sqrt(2*sqrt(d) 
*sqrt(c)*sqrt(b)*sqrt(a) + a*d + b*c) + sqrt(d)*sqrt(a + b*x) + sqrt(b)*sq 
rt(c + d*x))*b**2*c**2*d*x + sqrt(c)*sqrt(a)*sqrt(a + b*x)*log(sqrt(2*sqrt 
(d)*sqrt(c)*sqrt(b)*sqrt(a) + a*d + b*c) + sqrt(d)*sqrt(a + b*x) + sqrt(b) 
*sqrt(c + d*x))*a**2*c*d**2 + sqrt(c)*sqrt(a)*sqrt(a + b*x)*log(sqrt(2*sqr 
t(d)*sqrt(c)*sqrt(b)*sqrt(a) + a*d + b*c) + sqrt(d)*sqrt(a + b*x) + sqrt(b 
)*sqrt(c + d*x))*a**2*d**3*x - 2*sqrt(c)*sqrt(a)*sqrt(a + b*x)*log(sqrt(2* 
sqrt(d)*sqrt(c)*sqrt(b)*sqrt(a) + a*d + b*c) + sqrt(d)*sqrt(a + b*x) + sqr 
t(b)*sqrt(c + d*x))*a*b*c**2*d - 2*sqrt(c)*sqrt(a)*sqrt(a + b*x)*log(sqrt( 
2*sqrt(d)*sqrt(c)*sqrt(b)*sqrt(a) + a*d + b*c) + sqrt(d)*sqrt(a + b*x) ...