\(\int \frac {x^2}{(a+b x)^{3/2} (c+d x)^{3/2}} \, dx\) [386]

Optimal result

Integrand size = 22, antiderivative size = 130 \[ \int \frac {x^2}{(a+b x)^{3/2} (c+d x)^{3/2}} \, dx=-\frac {2 a^2}{b^2 (b c-a d) \sqrt {a+b x} \sqrt {c+d x}}-\frac {2 \left (b^2 c^2+a^2 d^2\right ) \sqrt {a+b x}}{b^2 d (b c-a d)^2 \sqrt {c+d x}}+\frac {2 \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{b^{3/2} d^{3/2}} \] Output:

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

Mathematica [A] (verified)

Time = 0.16 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.78 \[ \int \frac {x^2}{(a+b x)^{3/2} (c+d x)^{3/2}} \, dx=-\frac {2 \sqrt {a+b x} \left (b c^2+\frac {a^2 d (c+d x)}{a+b x}\right )}{b d (b c-a d)^2 \sqrt {c+d x}}+\frac {2 \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {a+b x}}\right )}{b^{3/2} d^{3/2}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.24 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.16, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {100, 27, 87, 66, 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.

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

\(\Big \downarrow \) 100

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 87

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

\(\Big \downarrow \) 66

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

(-2*a^2)/(b^2*(b*c - a*d)*Sqrt[a + b*x]*Sqrt[c + d*x]) - ((2*(b^2*c^2 + a^ 
2*d^2)*Sqrt[a + b*x])/(d*(b*c - a*d)*Sqrt[c + d*x]) - (2*Sqrt[b]*(b*c - a* 
d)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/d^(3/2))/(b^2 
*(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 66

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[ 
2   Subst[Int[1/(b - d*x^2), x], x, Sqrt[a + b*x]/Sqrt[c + d*x]], x] /; Fre 
eQ[{a, b, c, d}, x] &&  !GtQ[c - a*(d/b), 0]
 

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]))))
 

rule 100

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^( 
p_), x_] :> Simp[(b*c - a*d)^2*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d^2*(d 
*e - c*f)*(n + 1))), x] - Simp[1/(d^2*(d*e - c*f)*(n + 1))   Int[(c + d*x)^ 
(n + 1)*(e + f*x)^p*Simp[a^2*d^2*f*(n + p + 2) + b^2*c*(d*e*(n + 1) + c*f*( 
p + 1)) - 2*a*b*d*(d*e*(n + 1) + c*f*(p + 1)) - b^2*d*(d*e - c*f)*(n + 1)*x 
, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && (LtQ[n, -1] || (EqQ[n 
 + p + 3, 0] && NeQ[n, -1] && (SumSimplerQ[n, 1] ||  !SumSimplerQ[p, 1])))
 

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. \(653\) vs. \(2(110)=220\).

Time = 0.28 (sec) , antiderivative size = 654, normalized size of antiderivative = 5.03

method	result	size
default	\(\frac {\ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}+a d +b c}{2 \sqrt {d b}}\right ) a^{2} b \,d^{3} x^{2}-2 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}+a d +b c}{2 \sqrt {d b}}\right ) a \,b^{2} c \,d^{2} x^{2}+\ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}+a d +b c}{2 \sqrt {d b}}\right ) b^{3} c^{2} d \,x^{2}+\ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}+a d +b c}{2 \sqrt {d b}}\right ) a^{3} d^{3} x -\ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}+a d +b c}{2 \sqrt {d b}}\right ) a^{2} b c \,d^{2} x -\ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}+a d +b c}{2 \sqrt {d b}}\right ) a \,b^{2} c^{2} d x +\ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}+a d +b c}{2 \sqrt {d b}}\right ) b^{3} c^{3} x +\ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}+a d +b c}{2 \sqrt {d b}}\right ) a^{3} c \,d^{2}-2 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}+a d +b c}{2 \sqrt {d b}}\right ) a^{2} b \,c^{2} d +\ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}+a d +b c}{2 \sqrt {d b}}\right ) a \,b^{2} c^{3}-2 a^{2} d^{2} x \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}-2 b^{2} c^{2} x \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}-2 a^{2} c d \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}-2 a b \,c^{2} \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}}{\sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \left (a d -b c \right )^{2} \sqrt {d b}\, \sqrt {b x +a}\, \sqrt {x d +c}\, d b}\)	\(654\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 348 vs. \(2 (110) = 220\).

Time = 0.14 (sec) , antiderivative size = 710, normalized size of antiderivative = 5.46 \[ \int \frac {x^2}{(a+b x)^{3/2} (c+d x)^{3/2}} \, dx=\left [\frac {{\left (a b^{2} c^{3} - 2 \, a^{2} b c^{2} d + a^{3} c d^{2} + {\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x^{2} + {\left (b^{3} c^{3} - a b^{2} c^{2} d - a^{2} b c d^{2} + a^{3} d^{3}\right )} x\right )} \sqrt {b d} \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 4 \, {\left (2 \, b d x + b c + a d\right )} \sqrt {b d} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x\right ) - 4 \, {\left (a b^{2} c^{2} d + a^{2} b c d^{2} + {\left (b^{3} c^{2} d + a^{2} b d^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (a b^{4} c^{3} d^{2} - 2 \, a^{2} b^{3} c^{2} d^{3} + a^{3} b^{2} c d^{4} + {\left (b^{5} c^{2} d^{3} - 2 \, a b^{4} c d^{4} + a^{2} b^{3} d^{5}\right )} x^{2} + {\left (b^{5} c^{3} d^{2} - a b^{4} c^{2} d^{3} - a^{2} b^{3} c d^{4} + a^{3} b^{2} d^{5}\right )} x\right )}}, -\frac {{\left (a b^{2} c^{3} - 2 \, a^{2} b c^{2} d + a^{3} c d^{2} + {\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x^{2} + {\left (b^{3} c^{3} - a b^{2} c^{2} d - a^{2} b c d^{2} + a^{3} d^{3}\right )} x\right )} \sqrt {-b d} \arctan \left (\frac {{\left (2 \, b d x + b c + a d\right )} \sqrt {-b d} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (b^{2} d^{2} x^{2} + a b c d + {\left (b^{2} c d + a b d^{2}\right )} x\right )}}\right ) + 2 \, {\left (a b^{2} c^{2} d + a^{2} b c d^{2} + {\left (b^{3} c^{2} d + a^{2} b d^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{a b^{4} c^{3} d^{2} - 2 \, a^{2} b^{3} c^{2} d^{3} + a^{3} b^{2} c d^{4} + {\left (b^{5} c^{2} d^{3} - 2 \, a b^{4} c d^{4} + a^{2} b^{3} d^{5}\right )} x^{2} + {\left (b^{5} c^{3} d^{2} - a b^{4} c^{2} d^{3} - a^{2} b^{3} c d^{4} + a^{3} b^{2} d^{5}\right )} x}\right ] \] Input:

integrate(x^2/(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 
(b*d)*log(8*b^2*d^2*x^2 + b^2*c^2 + 6*a*b*c*d + a^2*d^2 + 4*(2*b*d*x + b*c 
 + a*d)*sqrt(b*d)*sqrt(b*x + a)*sqrt(d*x + c) + 8*(b^2*c*d + a*b*d^2)*x) - 
 4*(a*b^2*c^2*d + a^2*b*c*d^2 + (b^3*c^2*d + a^2*b*d^3)*x)*sqrt(b*x + a)*s 
qrt(d*x + c))/(a*b^4*c^3*d^2 - 2*a^2*b^3*c^2*d^3 + a^3*b^2*c*d^4 + (b^5*c^ 
2*d^3 - 2*a*b^4*c*d^4 + a^2*b^3*d^5)*x^2 + (b^5*c^3*d^2 - a*b^4*c^2*d^3 - 
a^2*b^3*c*d^4 + a^3*b^2*d^5)*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(-b*d)*arctan(1/2*(2*b*d*x + b*c + a*d)*sqrt( 
-b*d)*sqrt(b*x + a)*sqrt(d*x + c)/(b^2*d^2*x^2 + a*b*c*d + (b^2*c*d + a*b* 
d^2)*x)) + 2*(a*b^2*c^2*d + a^2*b*c*d^2 + (b^3*c^2*d + a^2*b*d^3)*x)*sqrt( 
b*x + a)*sqrt(d*x + c))/(a*b^4*c^3*d^2 - 2*a^2*b^3*c^2*d^3 + a^3*b^2*c*d^4 
 + (b^5*c^2*d^3 - 2*a*b^4*c*d^4 + a^2*b^3*d^5)*x^2 + (b^5*c^3*d^2 - a*b^4* 
c^2*d^3 - a^2*b^3*c*d^4 + a^3*b^2*d^5)*x)]
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

integrate(x^2/(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 [A] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 210, normalized size of antiderivative = 1.62 \[ \int \frac {x^2}{(a+b x)^{3/2} (c+d x)^{3/2}} \, dx=-\frac {\frac {2 \, \sqrt {b x + a} b^{4} c^{2} {\left | b \right |}}{{\left (b^{4} c^{2} d - 2 \, a b^{3} c d^{2} + a^{2} b^{2} d^{3}\right )} \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}} + \frac {4 \, \sqrt {b d} a^{2} b^{2}}{{\left (b^{2} c - a b d - {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2}\right )} {\left (b c {\left | b \right |} - a d {\left | b \right |}\right )}} + \frac {\sqrt {b d} b \log \left ({\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2}\right )}{d^{2} {\left | b \right |}}}{b^{2}} \] Input:

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 510, normalized size of antiderivative = 3.92 \[ \int \frac {x^2}{(a+b x)^{3/2} (c+d x)^{3/2}} \, dx=\frac {2 \sqrt {d}\, \sqrt {b}\, \sqrt {b x +a}\, \mathrm {log}\left (\frac {\sqrt {d}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {d x +c}}{\sqrt {a d -b c}}\right ) a^{2} c \,d^{2}+2 \sqrt {d}\, \sqrt {b}\, \sqrt {b x +a}\, \mathrm {log}\left (\frac {\sqrt {d}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {d x +c}}{\sqrt {a d -b c}}\right ) a^{2} d^{3} x -4 \sqrt {d}\, \sqrt {b}\, \sqrt {b x +a}\, \mathrm {log}\left (\frac {\sqrt {d}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {d x +c}}{\sqrt {a d -b c}}\right ) a b \,c^{2} d -4 \sqrt {d}\, \sqrt {b}\, \sqrt {b x +a}\, \mathrm {log}\left (\frac {\sqrt {d}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {d x +c}}{\sqrt {a d -b c}}\right ) a b c \,d^{2} x +2 \sqrt {d}\, \sqrt {b}\, \sqrt {b x +a}\, \mathrm {log}\left (\frac {\sqrt {d}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {d x +c}}{\sqrt {a d -b c}}\right ) b^{2} c^{3}+2 \sqrt {d}\, \sqrt {b}\, \sqrt {b x +a}\, \mathrm {log}\left (\frac {\sqrt {d}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {d x +c}}{\sqrt {a d -b c}}\right ) b^{2} c^{2} d x -2 \sqrt {d}\, \sqrt {b}\, \sqrt {b x +a}\, a^{2} c \,d^{2}-2 \sqrt {d}\, \sqrt {b}\, \sqrt {b x +a}\, a^{2} d^{3} x -2 \sqrt {d}\, \sqrt {b}\, \sqrt {b x +a}\, b^{2} c^{3}-2 \sqrt {d}\, \sqrt {b}\, \sqrt {b x +a}\, b^{2} c^{2} d x -2 \sqrt {d x +c}\, a^{2} b c \,d^{2}-2 \sqrt {d x +c}\, a^{2} b \,d^{3} x -2 \sqrt {d x +c}\, a \,b^{2} c^{2} d -2 \sqrt {d x +c}\, b^{3} c^{2} d x}{\sqrt {b x +a}\, b^{2} d^{2} \left (a^{2} d^{3} x -2 a b c \,d^{2} x +b^{2} c^{2} d x +a^{2} c \,d^{2}-2 a b \,c^{2} d +b^{2} c^{3}\right )} \] Input:

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

Output:

(2*(sqrt(d)*sqrt(b)*sqrt(a + b*x)*log((sqrt(d)*sqrt(a + b*x) + sqrt(b)*sqr 
t(c + d*x))/sqrt(a*d - b*c))*a**2*c*d**2 + sqrt(d)*sqrt(b)*sqrt(a + b*x)*l 
og((sqrt(d)*sqrt(a + b*x) + sqrt(b)*sqrt(c + d*x))/sqrt(a*d - b*c))*a**2*d 
**3*x - 2*sqrt(d)*sqrt(b)*sqrt(a + b*x)*log((sqrt(d)*sqrt(a + b*x) + sqrt( 
b)*sqrt(c + d*x))/sqrt(a*d - b*c))*a*b*c**2*d - 2*sqrt(d)*sqrt(b)*sqrt(a + 
 b*x)*log((sqrt(d)*sqrt(a + b*x) + sqrt(b)*sqrt(c + d*x))/sqrt(a*d - b*c)) 
*a*b*c*d**2*x + sqrt(d)*sqrt(b)*sqrt(a + b*x)*log((sqrt(d)*sqrt(a + b*x) + 
 sqrt(b)*sqrt(c + d*x))/sqrt(a*d - b*c))*b**2*c**3 + sqrt(d)*sqrt(b)*sqrt( 
a + b*x)*log((sqrt(d)*sqrt(a + b*x) + sqrt(b)*sqrt(c + d*x))/sqrt(a*d - b* 
c))*b**2*c**2*d*x - sqrt(d)*sqrt(b)*sqrt(a + b*x)*a**2*c*d**2 - sqrt(d)*sq 
rt(b)*sqrt(a + b*x)*a**2*d**3*x - sqrt(d)*sqrt(b)*sqrt(a + b*x)*b**2*c**3 
- sqrt(d)*sqrt(b)*sqrt(a + b*x)*b**2*c**2*d*x - sqrt(c + d*x)*a**2*b*c*d** 
2 - sqrt(c + d*x)*a**2*b*d**3*x - sqrt(c + d*x)*a*b**2*c**2*d - sqrt(c + d 
*x)*b**3*c**2*d*x))/(sqrt(a + b*x)*b**2*d**2*(a**2*c*d**2 + a**2*d**3*x - 
2*a*b*c**2*d - 2*a*b*c*d**2*x + b**2*c**3 + b**2*c**2*d*x))