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

Optimal. Leaf size=142 \[ \frac{a^2 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{b} \sqrt{a+c x}}\right )}{2 b^{3/2} c^{3/2}}+\frac{2 a x}{(b-c)^2}-\frac{a \sqrt{a+b x} \sqrt{a+c x}}{2 b c (b-c)}-\frac{(a+b x)^{3/2} \sqrt{a+c x}}{b (b-c)^2}+\frac{x^2 (b+c)}{2 (b-c)^2} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.447945, antiderivative size = 142, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16 \[ \frac{a^2 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{b} \sqrt{a+c x}}\right )}{2 b^{3/2} c^{3/2}}+\frac{2 a x}{(b-c)^2}-\frac{a \sqrt{a+b x} \sqrt{a+c x}}{2 b c (b-c)}-\frac{(a+b x)^{3/2} \sqrt{a+c x}}{b (b-c)^2}+\frac{x^2 (b+c)}{2 (b-c)^2} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{a^{2} \operatorname{atanh}{\left (\frac{\sqrt{b} \sqrt{a + c x}}{\sqrt{c} \sqrt{a + b x}} \right )}}{2 b^{\frac{3}{2}} c^{\frac{3}{2}}} + \frac{2 a x}{\left (b - c\right )^{2}} - \frac{a \sqrt{a + b x} \sqrt{a + c x}}{2 b c \left (b - c\right )} + \frac{\left (b + c\right ) \int x\, dx}{\left (b - c\right )^{2}} - \frac{\left (a + b x\right )^{\frac{3}{2}} \sqrt{a + c x}}{b \left (b - c\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**2/((b*x+a)**(1/2)+(c*x+a)**(1/2))**2,x)

[Out]

a**2*atanh(sqrt(b)*sqrt(a + c*x)/(sqrt(c)*sqrt(a + b*x)))/(2*b**(3/2)*c**(3/2))
+ 2*a*x/(b - c)**2 - a*sqrt(a + b*x)*sqrt(a + c*x)/(2*b*c*(b - c)) + (b + c)*Int
egral(x, x)/(b - c)**2 - (a + b*x)**(3/2)*sqrt(a + c*x)/(b*(b - c)**2)

_______________________________________________________________________________________

Mathematica [A]  time = 0.275629, size = 132, normalized size = 0.93 \[ \frac{1}{4} \left (\frac{a^2 \log \left (2 \sqrt{b} \sqrt{c} \sqrt{a+b x} \sqrt{a+c x}+a b+a c+2 b c x\right )}{b^{3/2} c^{3/2}}+\frac{8 a x}{(b-c)^2}-\frac{2 \sqrt{a+b x} \sqrt{a+c x} (a (b+c)+2 b c x)}{b c (b-c)^2}+\frac{2 x^2 (b+c)}{(b-c)^2}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [B]  time = 0.009, size = 385, normalized size = 2.7 \[{\frac{b{x}^{2}}{2\, \left ( b-c \right ) ^{2}}}+{\frac{c{x}^{2}}{2\, \left ( b-c \right ) ^{2}}}+2\,{\frac{ax}{ \left ( b-c \right ) ^{2}}}-{\frac{1}{ \left ( b-c \right ) ^{2}c}\sqrt{bx+a} \left ( cx+a \right ) ^{{\frac{3}{2}}}}+{\frac{a}{2\, \left ( b-c \right ) ^{2}c}\sqrt{cx+a}\sqrt{bx+a}}-{\frac{a}{2\, \left ( b-c \right ) ^{2}b}\sqrt{cx+a}\sqrt{bx+a}}+{\frac{{a}^{2}b}{4\, \left ( b-c \right ) ^{2}c}\sqrt{ \left ( cx+a \right ) \left ( bx+a \right ) }\ln \left ({1 \left ({\frac{ab}{2}}+{\frac{ac}{2}}+bcx \right ){\frac{1}{\sqrt{bc}}}}+\sqrt{bc{x}^{2}+ \left ( ab+ac \right ) x+{a}^{2}} \right ){\frac{1}{\sqrt{cx+a}}}{\frac{1}{\sqrt{bx+a}}}{\frac{1}{\sqrt{bc}}}}-{\frac{{a}^{2}}{2\, \left ( b-c \right ) ^{2}}\sqrt{ \left ( cx+a \right ) \left ( bx+a \right ) }\ln \left ({1 \left ({\frac{ab}{2}}+{\frac{ac}{2}}+bcx \right ){\frac{1}{\sqrt{bc}}}}+\sqrt{bc{x}^{2}+ \left ( ab+ac \right ) x+{a}^{2}} \right ){\frac{1}{\sqrt{cx+a}}}{\frac{1}{\sqrt{bx+a}}}{\frac{1}{\sqrt{bc}}}}+{\frac{{a}^{2}c}{4\, \left ( b-c \right ) ^{2}b}\sqrt{ \left ( cx+a \right ) \left ( bx+a \right ) }\ln \left ({1 \left ({\frac{ab}{2}}+{\frac{ac}{2}}+bcx \right ){\frac{1}{\sqrt{bc}}}}+\sqrt{bc{x}^{2}+ \left ( ab+ac \right ) x+{a}^{2}} \right ){\frac{1}{\sqrt{cx+a}}}{\frac{1}{\sqrt{bx+a}}}{\frac{1}{\sqrt{bc}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2/((b*x+a)^(1/2)+(c*x+a)^(1/2))^2,x)

[Out]

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

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2}}{{\left (\sqrt{b x + a} + \sqrt{c x + a}\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/(sqrt(b*x + a) + sqrt(c*x + a))^2,x, algorithm="maxima")

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.294556, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/(sqrt(b*x + a) + sqrt(c*x + a))^2,x, algorithm="fricas")

[Out]

[1/4*(2*(2*(3*b^3*c + 10*b^2*c^2 + 3*b*c^3)*x^3 + (a*b^3 + 47*a*b^2*c + 47*a*b*c
^2 + a*c^3)*x^2 + 4*(a^2*b^2 + 14*a^2*b*c + a^2*c^2)*x)*sqrt(b*c)*sqrt(b*x + a)*
sqrt(c*x + a) - (8*a^4*b^2 - 16*a^4*b*c + 8*a^4*c^2 + (a^2*b^4 + 4*a^2*b^3*c - 1
0*a^2*b^2*c^2 + 4*a^2*b*c^3 + a^2*c^4)*x^2 - 4*(2*a^3*b^2 - 4*a^3*b*c + 2*a^3*c^
2 + (a^2*b^3 - a^2*b^2*c - a^2*b*c^2 + a^2*c^3)*x)*sqrt(b*x + a)*sqrt(c*x + a) +
8*(a^3*b^3 - a^3*b^2*c - a^3*b*c^2 + a^3*c^3)*x)*log(-(2*a*b*c*x - 2*(b*c*x - s
qrt(b*c)*a)*sqrt(b*x + a)*sqrt(c*x + a) - (2*b*c*x^2 + 2*a^2 + (a*b + a*c)*x)*sq
rt(b*c))/((b + c)*x - 2*sqrt(b*x + a)*sqrt(c*x + a) + 2*a)) - 2*((b^4*c + 15*b^3
*c^2 + 15*b^2*c^3 + b*c^4)*x^4 + 8*(3*a*b^3*c + 10*a*b^2*c^2 + 3*a*b*c^3)*x^3 +
(3*a^2*b^3 + 77*a^2*b^2*c + 77*a^2*b*c^2 + 3*a^2*c^3)*x^2 + 4*(a^3*b^2 + 14*a^3*
b*c + a^3*c^2)*x)*sqrt(b*c))/(4*(2*a*b^3*c - 4*a*b^2*c^2 + 2*a*b*c^3 + (b^4*c -
b^3*c^2 - b^2*c^3 + b*c^4)*x)*sqrt(b*c)*sqrt(b*x + a)*sqrt(c*x + a) - (8*a^2*b^3
*c - 16*a^2*b^2*c^2 + 8*a^2*b*c^3 + (b^5*c + 4*b^4*c^2 - 10*b^3*c^3 + 4*b^2*c^4
+ b*c^5)*x^2 + 8*(a*b^4*c - a*b^3*c^2 - a*b^2*c^3 + a*b*c^4)*x)*sqrt(b*c)), 1/2*
((2*(3*b^3*c + 10*b^2*c^2 + 3*b*c^3)*x^3 + (a*b^3 + 47*a*b^2*c + 47*a*b*c^2 + a*
c^3)*x^2 + 4*(a^2*b^2 + 14*a^2*b*c + a^2*c^2)*x)*sqrt(-b*c)*sqrt(b*x + a)*sqrt(c
*x + a) - (8*a^4*b^2 - 16*a^4*b*c + 8*a^4*c^2 + (a^2*b^4 + 4*a^2*b^3*c - 10*a^2*
b^2*c^2 + 4*a^2*b*c^3 + a^2*c^4)*x^2 - 4*(2*a^3*b^2 - 4*a^3*b*c + 2*a^3*c^2 + (a
^2*b^3 - a^2*b^2*c - a^2*b*c^2 + a^2*c^3)*x)*sqrt(b*x + a)*sqrt(c*x + a) + 8*(a^
3*b^3 - a^3*b^2*c - a^3*b*c^2 + a^3*c^3)*x)*arctan((sqrt(-b*c)*sqrt(b*x + a)*sqr
t(c*x + a) - sqrt(-b*c)*a)/(b*c*x)) - ((b^4*c + 15*b^3*c^2 + 15*b^2*c^3 + b*c^4)
*x^4 + 8*(3*a*b^3*c + 10*a*b^2*c^2 + 3*a*b*c^3)*x^3 + (3*a^2*b^3 + 77*a^2*b^2*c
+ 77*a^2*b*c^2 + 3*a^2*c^3)*x^2 + 4*(a^3*b^2 + 14*a^3*b*c + a^3*c^2)*x)*sqrt(-b*
c))/(4*(2*a*b^3*c - 4*a*b^2*c^2 + 2*a*b*c^3 + (b^4*c - b^3*c^2 - b^2*c^3 + b*c^4
)*x)*sqrt(-b*c)*sqrt(b*x + a)*sqrt(c*x + a) - (8*a^2*b^3*c - 16*a^2*b^2*c^2 + 8*
a^2*b*c^3 + (b^5*c + 4*b^4*c^2 - 10*b^3*c^3 + 4*b^2*c^4 + b*c^5)*x^2 + 8*(a*b^4*
c - a*b^3*c^2 - a*b^2*c^3 + a*b*c^4)*x)*sqrt(-b*c))]

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2}}{\left (\sqrt{a + b x} + \sqrt{a + c x}\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2/((b*x+a)**(1/2)+(c*x+a)**(1/2))**2,x)

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/(sqrt(b*x + a) + sqrt(c*x + a))^2,x, algorithm="giac")

[Out]

Timed out