\(\int \frac {x^2}{(\sqrt {a+b x}+\sqrt {a+c x})^3} \, dx\) [40]

Optimal result

Integrand size = 25, antiderivative size = 155 \[ \int \frac {x^2}{\left (\sqrt {a+b x}+\sqrt {a+c x}\right )^3} \, dx=\frac {8 a \sqrt {a+b x}}{(b-c)^3}+\frac {2 (b+3 c) (a+b x)^{3/2}}{3 b (b-c)^3}-\frac {8 a \sqrt {a+c x}}{(b-c)^3}-\frac {2 (3 b+c) (a+c x)^{3/2}}{3 (b-c)^3 c}-\frac {8 a^{3/2} \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{(b-c)^3}+\frac {8 a^{3/2} \text {arctanh}\left (\frac {\sqrt {a+c x}}{\sqrt {a}}\right )}{(b-c)^3} \] Output:

8*a*(b*x+a)^(1/2)/(b-c)^3+2/3*(b+3*c)*(b*x+a)^(3/2)/b/(b-c)^3-8*a*(c*x+a)^ 
(1/2)/(b-c)^3-2/3*(3*b+c)*(c*x+a)^(3/2)/(b-c)^3/c-8*a^(3/2)*arctanh((b*x+a 
)^(1/2)/a^(1/2))/(b-c)^3+8*a^(3/2)*arctanh((c*x+a)^(1/2)/a^(1/2))/(b-c)^3
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(1004\) vs. \(2(155)=310\).

Time = 10.12 (sec) , antiderivative size = 1004, normalized size of antiderivative = 6.48 \[ \int \frac {x^2}{\left (\sqrt {a+b x}+\sqrt {a+c x}\right )^3} \, dx=\frac {-2 a \sqrt {b-c} \left (b \sqrt {a-\frac {a b}{c}} c^3 x^2 \left (b^2 x+3 b c x-3 b \sqrt {a+b x} \sqrt {a+c x}-c \sqrt {a+b x} \sqrt {a+c x}\right )+a^3 \left (b c^2 \left (12 \sqrt {a-\frac {a b}{c}}-3 \sqrt {a+b x}-53 \sqrt {a+c x}\right )+b^2 c \left (-15 \sqrt {a-\frac {a b}{c}}+24 \sqrt {a+b x}-2 \sqrt {a+c x}\right )+b^3 \left (\sqrt {a-\frac {a b}{c}}-3 \sqrt {a+b x}+3 \sqrt {a+c x}\right )+2 c^3 \left (9 \sqrt {a-\frac {a b}{c}}-9 \sqrt {a+b x}+26 \sqrt {a+c x}\right )\right )+a c^2 x \left (-4 \sqrt {a-\frac {a b}{c}} c^2 \sqrt {a+b x} \sqrt {a+c x}+b^3 x \left (-3 \sqrt {a-\frac {a b}{c}}+3 \sqrt {a+b x}-9 \sqrt {a+c x}\right )+b \left (-22 \sqrt {a-\frac {a b}{c}} c \sqrt {a+b x} \sqrt {a+c x}+3 c^2 x \left (6 \sqrt {a-\frac {a b}{c}}-3 \sqrt {a+b x}+\sqrt {a+c x}\right )\right )+b^2 \left (6 \sqrt {a-\frac {a b}{c}} \sqrt {a+b x} \sqrt {a+c x}+c x \left (9 \sqrt {a-\frac {a b}{c}}+6 \sqrt {a+b x}+6 \sqrt {a+c x}\right )\right )\right )+a^2 \left (-3 b^3 c x \left (\sqrt {a-\frac {a b}{c}}+2 \sqrt {a+c x}\right )+b^2 \left (9 \sqrt {a-\frac {a b}{c}} c \sqrt {a+b x} \sqrt {a+c x}+c^2 x \left (-9 \sqrt {a-\frac {a b}{c}}+30 \sqrt {a+b x}-44 \sqrt {a+c x}\right )\right )+2 c^3 \left (-26 \sqrt {a-\frac {a b}{c}} \sqrt {a+b x} \sqrt {a+c x}+c x \left (9 \sqrt {a-\frac {a b}{c}}-9 \sqrt {a+b x}+2 \sqrt {a+c x}\right )\right )+b \left (27 \sqrt {a-\frac {a b}{c}} c^2 \sqrt {a+b x} \sqrt {a+c x}+c^3 x \left (30 \sqrt {a-\frac {a b}{c}}-12 \sqrt {a+b x}+46 \sqrt {a+c x}\right )\right )\right )\right )-48 a^3 c^{3/2} (-b+c) \left (b c x \left (3 \sqrt {a-\frac {a b}{c}}-\sqrt {a+b x}\right )+a \left (-b \sqrt {a-\frac {a b}{c}}+4 \sqrt {a-\frac {a b}{c}} c+3 b \sqrt {a+b x}-4 c \sqrt {a+b x}\right )\right ) \arctan \left (\frac {\sqrt {b-c} \sqrt {a+c x}}{\sqrt {c} \left (-\sqrt {a-\frac {a b}{c}}+\sqrt {a+b x}+\sqrt {a+c x}\right )}\right )}{3 (b-c)^{5/2} c \left (a (b-c)+\sqrt {a-\frac {a b}{c}} c \sqrt {a+b x}\right )^3} \] Input:

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

Output:

(-2*a*Sqrt[b - c]*(b*Sqrt[a - (a*b)/c]*c^3*x^2*(b^2*x + 3*b*c*x - 3*b*Sqrt 
[a + b*x]*Sqrt[a + c*x] - c*Sqrt[a + b*x]*Sqrt[a + c*x]) + a^3*(b*c^2*(12* 
Sqrt[a - (a*b)/c] - 3*Sqrt[a + b*x] - 53*Sqrt[a + c*x]) + b^2*c*(-15*Sqrt[ 
a - (a*b)/c] + 24*Sqrt[a + b*x] - 2*Sqrt[a + c*x]) + b^3*(Sqrt[a - (a*b)/c 
] - 3*Sqrt[a + b*x] + 3*Sqrt[a + c*x]) + 2*c^3*(9*Sqrt[a - (a*b)/c] - 9*Sq 
rt[a + b*x] + 26*Sqrt[a + c*x])) + a*c^2*x*(-4*Sqrt[a - (a*b)/c]*c^2*Sqrt[ 
a + b*x]*Sqrt[a + c*x] + b^3*x*(-3*Sqrt[a - (a*b)/c] + 3*Sqrt[a + b*x] - 9 
*Sqrt[a + c*x]) + b*(-22*Sqrt[a - (a*b)/c]*c*Sqrt[a + b*x]*Sqrt[a + c*x] + 
 3*c^2*x*(6*Sqrt[a - (a*b)/c] - 3*Sqrt[a + b*x] + Sqrt[a + c*x])) + b^2*(6 
*Sqrt[a - (a*b)/c]*Sqrt[a + b*x]*Sqrt[a + c*x] + c*x*(9*Sqrt[a - (a*b)/c] 
+ 6*Sqrt[a + b*x] + 6*Sqrt[a + c*x]))) + a^2*(-3*b^3*c*x*(Sqrt[a - (a*b)/c 
] + 2*Sqrt[a + c*x]) + b^2*(9*Sqrt[a - (a*b)/c]*c*Sqrt[a + b*x]*Sqrt[a + c 
*x] + c^2*x*(-9*Sqrt[a - (a*b)/c] + 30*Sqrt[a + b*x] - 44*Sqrt[a + c*x])) 
+ 2*c^3*(-26*Sqrt[a - (a*b)/c]*Sqrt[a + b*x]*Sqrt[a + c*x] + c*x*(9*Sqrt[a 
 - (a*b)/c] - 9*Sqrt[a + b*x] + 2*Sqrt[a + c*x])) + b*(27*Sqrt[a - (a*b)/c 
]*c^2*Sqrt[a + b*x]*Sqrt[a + c*x] + c^3*x*(30*Sqrt[a - (a*b)/c] - 12*Sqrt[ 
a + b*x] + 46*Sqrt[a + c*x])))) - 48*a^3*c^(3/2)*(-b + c)*(b*c*x*(3*Sqrt[a 
 - (a*b)/c] - Sqrt[a + b*x]) + a*(-(b*Sqrt[a - (a*b)/c]) + 4*Sqrt[a - (a*b 
)/c]*c + 3*b*Sqrt[a + b*x] - 4*c*Sqrt[a + b*x]))*ArcTan[(Sqrt[b - c]*Sqrt[ 
a + c*x])/(Sqrt[c]*(-Sqrt[a - (a*b)/c] + Sqrt[a + b*x] + Sqrt[a + c*x])...
 

Rubi [A] (verified)

Time = 0.67 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.78, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {7241, 2009}

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.

\(\displaystyle \int \frac {x^2}{\left (\sqrt {a+b x}+\sqrt {a+c x}\right )^3} \, dx\)

\(\Big \downarrow \) 7241

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {-8 a^{3/2} \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )+8 a^{3/2} \text {arctanh}\left (\frac {\sqrt {a+c x}}{\sqrt {a}}\right )+\frac {2 (b+3 c) (a+b x)^{3/2}}{3 b}-\frac {2 (3 b+c) (a+c x)^{3/2}}{3 c}+8 a \sqrt {a+b x}-8 a \sqrt {a+c x}}{(b-c)^3}\)

Input:

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

Output:

(8*a*Sqrt[a + b*x] + (2*(b + 3*c)*(a + b*x)^(3/2))/(3*b) - 8*a*Sqrt[a + c* 
x] - (2*(3*b + c)*(a + c*x)^(3/2))/(3*c) - 8*a^(3/2)*ArcTanh[Sqrt[a + b*x] 
/Sqrt[a]] + 8*a^(3/2)*ArcTanh[Sqrt[a + c*x]/Sqrt[a]])/(b - c)^3
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 7241

Int[(u_.)*((e_.)*Sqrt[(a_.) + (b_.)*(x_)^(n_.)] + (f_.)*Sqrt[(c_.) + (d_.)* 
(x_)^(n_.)])^(m_), x_Symbol] :> Simp[(b*e^2 - d*f^2)^m   Int[ExpandIntegran 
d[(u*x^(m*n))/(e*Sqrt[a + b*x^n] - f*Sqrt[c + d*x^n])^m, x], x], x] /; Free 
Q[{a, b, c, d, e, f, n}, x] && ILtQ[m, 0] && EqQ[a*e^2 - c*f^2, 0]
 

Maple [A] (verified)

Time = 0.01 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.95

method	result	size
default	\(\frac {2 \left (b x +a \right )^{\frac {3}{2}}}{3 \left (b -c \right )^{3}}+\frac {2 c \left (b x +a \right )^{\frac {3}{2}}}{\left (b -c \right )^{3} b}-\frac {2 b \left (c x +a \right )^{\frac {3}{2}}}{\left (b -c \right )^{3} c}-\frac {2 \left (c x +a \right )^{\frac {3}{2}}}{3 \left (b -c \right )^{3}}+\frac {4 a \left (2 \sqrt {b x +a}-2 \sqrt {a}\, \operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )\right )}{\left (b -c \right )^{3}}-\frac {4 a \left (2 \sqrt {c x +a}-2 \sqrt {a}\, \operatorname {arctanh}\left (\frac {\sqrt {c x +a}}{\sqrt {a}}\right )\right )}{\left (b -c \right )^{3}}\)	\(148\)

Input:

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

Output:

2/3/(b-c)^3*(b*x+a)^(3/2)+2/(b-c)^3*c*(b*x+a)^(3/2)/b-2/(b-c)^3*b*(c*x+a)^ 
(3/2)/c-2/3/(b-c)^3*(c*x+a)^(3/2)+4*a/(b-c)^3*(2*(b*x+a)^(1/2)-2*a^(1/2)*a 
rctanh((b*x+a)^(1/2)/a^(1/2)))-4*a/(b-c)^3*(2*(c*x+a)^(1/2)-2*a^(1/2)*arct 
anh((c*x+a)^(1/2)/a^(1/2)))
 

Fricas [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 315, normalized size of antiderivative = 2.03 \[ \int \frac {x^2}{\left (\sqrt {a+b x}+\sqrt {a+c x}\right )^3} \, dx=\left [-\frac {2 \, {\left (6 \, a^{\frac {3}{2}} b c \log \left (\frac {b x + 2 \, \sqrt {b x + a} \sqrt {a} + 2 \, a}{x}\right ) + 6 \, a^{\frac {3}{2}} b c \log \left (\frac {c x - 2 \, \sqrt {c x + a} \sqrt {a} + 2 \, a}{x}\right ) - {\left (13 \, a b c + 3 \, a c^{2} + {\left (b^{2} c + 3 \, b c^{2}\right )} x\right )} \sqrt {b x + a} + {\left (3 \, a b^{2} + 13 \, a b c + {\left (3 \, b^{2} c + b c^{2}\right )} x\right )} \sqrt {c x + a}\right )}}{3 \, {\left (b^{4} c - 3 \, b^{3} c^{2} + 3 \, b^{2} c^{3} - b c^{4}\right )}}, \frac {2 \, {\left (12 \, \sqrt {-a} a b c \arctan \left (\frac {\sqrt {-a}}{\sqrt {b x + a}}\right ) - 12 \, \sqrt {-a} a b c \arctan \left (\frac {\sqrt {-a}}{\sqrt {c x + a}}\right ) + {\left (13 \, a b c + 3 \, a c^{2} + {\left (b^{2} c + 3 \, b c^{2}\right )} x\right )} \sqrt {b x + a} - {\left (3 \, a b^{2} + 13 \, a b c + {\left (3 \, b^{2} c + b c^{2}\right )} x\right )} \sqrt {c x + a}\right )}}{3 \, {\left (b^{4} c - 3 \, b^{3} c^{2} + 3 \, b^{2} c^{3} - b c^{4}\right )}}\right ] \] Input:

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

Output:

[-2/3*(6*a^(3/2)*b*c*log((b*x + 2*sqrt(b*x + a)*sqrt(a) + 2*a)/x) + 6*a^(3 
/2)*b*c*log((c*x - 2*sqrt(c*x + a)*sqrt(a) + 2*a)/x) - (13*a*b*c + 3*a*c^2 
 + (b^2*c + 3*b*c^2)*x)*sqrt(b*x + a) + (3*a*b^2 + 13*a*b*c + (3*b^2*c + b 
*c^2)*x)*sqrt(c*x + a))/(b^4*c - 3*b^3*c^2 + 3*b^2*c^3 - b*c^4), 2/3*(12*s 
qrt(-a)*a*b*c*arctan(sqrt(-a)/sqrt(b*x + a)) - 12*sqrt(-a)*a*b*c*arctan(sq 
rt(-a)/sqrt(c*x + a)) + (13*a*b*c + 3*a*c^2 + (b^2*c + 3*b*c^2)*x)*sqrt(b* 
x + a) - (3*a*b^2 + 13*a*b*c + (3*b^2*c + b*c^2)*x)*sqrt(c*x + a))/(b^4*c 
- 3*b^3*c^2 + 3*b^2*c^3 - b*c^4)]
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2366 vs. \(2 (131) = 262\).

Time = 1.30 (sec) , antiderivative size = 2366, normalized size of antiderivative = 15.26 \[ \int \frac {x^2}{\left (\sqrt {a+b x}+\sqrt {a+c x}\right )^3} \, dx=\text {Too large to display} \] Input:

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

Output:

2/3*(12*a^2*b^2*arctan(sqrt(b*x + a)/sqrt(-a))/((b^3 - 3*b^2*c + 3*b*c^2 - 
 c^3)*sqrt(-a)) - sqrt(a*b^2 + (b*x + a)*b*c - a*b*c)*((3*b^5*c*abs(b) - 8 
*b^4*c^2*abs(b) + 6*b^3*c^3*abs(b) - b*c^5*abs(b))*(b*x + a)/(b^8*c - 6*b^ 
7*c^2 + 15*b^6*c^3 - 20*b^5*c^4 + 15*b^4*c^5 - 6*b^3*c^6 + b^2*c^7) + (3*a 
*b^6*abs(b) + a*b^5*c*abs(b) - 22*a*b^4*c^2*abs(b) + 30*a*b^3*c^3*abs(b) - 
 13*a*b^2*c^4*abs(b) + a*b*c^5*abs(b))/(b^8*c - 6*b^7*c^2 + 15*b^6*c^3 - 2 
0*b^5*c^4 + 15*b^4*c^5 - 6*b^3*c^6 + b^2*c^7)) + ((b*x + a)^(3/2)*b^8 + 12 
*sqrt(b*x + a)*a*b^8 - 3*(b*x + a)^(3/2)*b^7*c - 72*sqrt(b*x + a)*a*b^7*c 
- 3*(b*x + a)^(3/2)*b^6*c^2 + 180*sqrt(b*x + a)*a*b^6*c^2 + 25*(b*x + a)^( 
3/2)*b^5*c^3 - 240*sqrt(b*x + a)*a*b^5*c^3 - 45*(b*x + a)^(3/2)*b^4*c^4 + 
180*sqrt(b*x + a)*a*b^4*c^4 + 39*(b*x + a)^(3/2)*b^3*c^5 - 72*sqrt(b*x + a 
)*a*b^3*c^5 - 17*(b*x + a)^(3/2)*b^2*c^6 + 12*sqrt(b*x + a)*a*b^2*c^6 + 3* 
(b*x + a)^(3/2)*b*c^7)/(b^9 - 9*b^8*c + 36*b^7*c^2 - 84*b^6*c^3 + 126*b^5* 
c^4 - 126*b^4*c^5 + 84*b^3*c^6 - 36*b^2*c^7 + 9*b*c^8 - c^9) - 12*(2*(a*b^ 
4 - 3*a*b^3*c + 3*a*b^2*c^2 - a*b*c^3)^2*(a*b^3*c - a*b^2*c^2)*sqrt(-a)*ab 
s(b)*sgn(b^3 - 3*b^2*c + 3*b*c^2 - c^3) + 2*(a*b^4 - 3*a*b^3*c + 3*a*b^2*c 
^2 - a*b*c^3)^2*(a*b^3 - a*b^2*c)*sqrt(-a*b*c)*abs(b) + (a^2*b^7 - 5*a^2*b 
^6*c + 10*a^2*b^5*c^2 - 10*a^2*b^4*c^3 + 5*a^2*b^3*c^4 - a^2*b^2*c^5)*sqrt 
(-a*b*c)*abs(-a*b^4 + 3*a*b^3*c - 3*a*b^2*c^2 + a*b*c^3)*abs(b)*sgn(b^3 - 
3*b^2*c + 3*b*c^2 - c^3) + (a^2*b^8 - 5*a^2*b^7*c + 10*a^2*b^6*c^2 - 10...
 

Mupad [B] (verification not implemented)

Time = 26.37 (sec) , antiderivative size = 762, normalized size of antiderivative = 4.92 \[ \int \frac {x^2}{\left (\sqrt {a+b x}+\sqrt {a+c x}\right )^3} \, dx =\text {Too large to display} \] Input:

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

Output:

(4*a^(3/2)*b^4 - (4*a^(3/2)*c^4*((4*((a + b*x)^(1/2) - a^(1/2))^3)/((a + c 
*x)^(1/2) - a^(1/2))^3 - (15*((a + b*x)^(1/2) - a^(1/2))^4)/((a + c*x)^(1/ 
2) - a^(1/2))^4 + (24*((a + b*x)^(1/2) - a^(1/2))^5)/((a + c*x)^(1/2) - a^ 
(1/2))^5 + (6*log(((a + b*x)^(1/2) - a^(1/2))/((a + c*x)^(1/2) - a^(1/2))) 
*((a + b*x)^(1/2) - a^(1/2))^6)/((a + c*x)^(1/2) - a^(1/2))^6))/3 - (4*a^( 
3/2)*b^2*c^2*((24*((a + b*x)^(1/2) - a^(1/2)))/((a + c*x)^(1/2) - a^(1/2)) 
 + (12*((a + b*x)^(1/2) - a^(1/2))^2)/((a + c*x)^(1/2) - a^(1/2))^2 + (12* 
((a + b*x)^(1/2) - a^(1/2))^3)/((a + c*x)^(1/2) - a^(1/2))^3 - (15*((a + b 
*x)^(1/2) - a^(1/2))^4)/((a + c*x)^(1/2) - a^(1/2))^4 + (18*log(((a + b*x) 
^(1/2) - a^(1/2))/((a + c*x)^(1/2) - a^(1/2)))*((a + b*x)^(1/2) - a^(1/2)) 
^2)/((a + c*x)^(1/2) - a^(1/2))^2 - 3))/3 + (4*a^(3/2)*b*c^3*((6*((a + b*x 
)^(1/2) - a^(1/2))^2)/((a + c*x)^(1/2) - a^(1/2))^2 - (12*((a + b*x)^(1/2) 
 - a^(1/2))^3)/((a + c*x)^(1/2) - a^(1/2))^3 + (66*((a + b*x)^(1/2) - a^(1 
/2))^4)/((a + c*x)^(1/2) - a^(1/2))^4 - (24*((a + b*x)^(1/2) - a^(1/2))^5) 
/((a + c*x)^(1/2) - a^(1/2))^5 + (18*log(((a + b*x)^(1/2) - a^(1/2))/((a + 
 c*x)^(1/2) - a^(1/2)))*((a + b*x)^(1/2) - a^(1/2))^4)/((a + c*x)^(1/2) - 
a^(1/2))^4))/3 + (4*a^(3/2)*b^3*c*(6*log(((a + b*x)^(1/2) - a^(1/2))/((a + 
 c*x)^(1/2) - a^(1/2))) - (24*((a + b*x)^(1/2) - a^(1/2)))/((a + c*x)^(1/2 
) - a^(1/2)) + (6*((a + b*x)^(1/2) - a^(1/2))^2)/((a + c*x)^(1/2) - a^(1/2 
))^2 - (4*((a + b*x)^(1/2) - a^(1/2))^3)/((a + c*x)^(1/2) - a^(1/2))^3 ...
 

Reduce [B] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 322, normalized size of antiderivative = 2.08 \[ \int \frac {x^2}{\left (\sqrt {a+b x}+\sqrt {a+c x}\right )^3} \, dx=\frac {16 \sqrt {b}\, \sqrt {a}\, \sqrt {2 \sqrt {c}\, \sqrt {b}-b -c}\, \mathit {atan} \left (\frac {\sqrt {c}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {c x +a}}{\sqrt {a}\, \sqrt {2 \sqrt {c}\, \sqrt {b}-b -c}}\right ) a b c +16 \sqrt {c}\, \sqrt {a}\, \sqrt {2 \sqrt {c}\, \sqrt {b}-b -c}\, \mathit {atan} \left (\frac {\sqrt {c}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {c x +a}}{\sqrt {a}\, \sqrt {2 \sqrt {c}\, \sqrt {b}-b -c}}\right ) a b c +\frac {26 \sqrt {b x +a}\, a \,b^{2} c}{3}-\frac {20 \sqrt {b x +a}\, a b \,c^{2}}{3}-2 \sqrt {b x +a}\, a \,c^{3}+\frac {2 \sqrt {b x +a}\, b^{3} c x}{3}+\frac {4 \sqrt {b x +a}\, b^{2} c^{2} x}{3}-2 \sqrt {b x +a}\, b \,c^{3} x -2 \sqrt {c x +a}\, a \,b^{3}-\frac {20 \sqrt {c x +a}\, a \,b^{2} c}{3}+\frac {26 \sqrt {c x +a}\, a b \,c^{2}}{3}-2 \sqrt {c x +a}\, b^{3} c x +\frac {4 \sqrt {c x +a}\, b^{2} c^{2} x}{3}+\frac {2 \sqrt {c x +a}\, b \,c^{3} x}{3}}{b c \left (b^{4}-4 b^{3} c +6 b^{2} c^{2}-4 b \,c^{3}+c^{4}\right )} \] Input:

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

Output:

(2*(24*sqrt(b)*sqrt(a)*sqrt(2*sqrt(c)*sqrt(b) - b - c)*atan((sqrt(c)*sqrt( 
a + b*x) + sqrt(b)*sqrt(a + c*x))/(sqrt(a)*sqrt(2*sqrt(c)*sqrt(b) - b - c) 
))*a*b*c + 24*sqrt(c)*sqrt(a)*sqrt(2*sqrt(c)*sqrt(b) - b - c)*atan((sqrt(c 
)*sqrt(a + b*x) + sqrt(b)*sqrt(a + c*x))/(sqrt(a)*sqrt(2*sqrt(c)*sqrt(b) - 
 b - c)))*a*b*c + 13*sqrt(a + b*x)*a*b**2*c - 10*sqrt(a + b*x)*a*b*c**2 - 
3*sqrt(a + b*x)*a*c**3 + sqrt(a + b*x)*b**3*c*x + 2*sqrt(a + b*x)*b**2*c** 
2*x - 3*sqrt(a + b*x)*b*c**3*x - 3*sqrt(a + c*x)*a*b**3 - 10*sqrt(a + c*x) 
*a*b**2*c + 13*sqrt(a + c*x)*a*b*c**2 - 3*sqrt(a + c*x)*b**3*c*x + 2*sqrt( 
a + c*x)*b**2*c**2*x + sqrt(a + c*x)*b*c**3*x))/(3*b*c*(b**4 - 4*b**3*c + 
6*b**2*c**2 - 4*b*c**3 + c**4))