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

Optimal. Leaf size=155 \[ -\frac{8 a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{(b-c)^3}+\frac{8 a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a+c x}}{\sqrt{a}}\right )}{(b-c)^3}+\frac{8 a \sqrt{a+b x}}{(b-c)^3}-\frac{8 a \sqrt{a+c x}}{(b-c)^3}+\frac{2 (b+3 c) (a+b x)^{3/2}}{3 b (b-c)^3}-\frac{2 (3 b+c) (a+c x)^{3/2}}{3 c (b-c)^3} \]

[Out]

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

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Rubi [A]  time = 0.396067, antiderivative size = 155, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16 \[ -\frac{8 a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{(b-c)^3}+\frac{8 a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a+c x}}{\sqrt{a}}\right )}{(b-c)^3}+\frac{8 a \sqrt{a+b x}}{(b-c)^3}-\frac{8 a \sqrt{a+c x}}{(b-c)^3}+\frac{2 (b+3 c) (a+b x)^{3/2}}{3 b (b-c)^3}-\frac{2 (3 b+c) (a+c x)^{3/2}}{3 c (b-c)^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 32.8726, size = 131, normalized size = 0.85 \[ - \frac{8 a^{\frac{3}{2}} \operatorname{atanh}{\left (\frac{\sqrt{a + b x}}{\sqrt{a}} \right )}}{\left (b - c\right )^{3}} + \frac{8 a^{\frac{3}{2}} \operatorname{atanh}{\left (\frac{\sqrt{a + c x}}{\sqrt{a}} \right )}}{\left (b - c\right )^{3}} + \frac{8 a \sqrt{a + b x}}{\left (b - c\right )^{3}} - \frac{8 a \sqrt{a + c x}}{\left (b - c\right )^{3}} - \frac{2 \left (a + c x\right )^{\frac{3}{2}} \left (b + \frac{c}{3}\right )}{c \left (b - c\right )^{3}} + \frac{2 \left (a + b x\right )^{\frac{3}{2}} \left (\frac{b}{3} + c\right )}{b \left (b - c\right )^{3}} \]

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))**3,x)

[Out]

-8*a**(3/2)*atanh(sqrt(a + b*x)/sqrt(a))/(b - c)**3 + 8*a**(3/2)*atanh(sqrt(a +
c*x)/sqrt(a))/(b - c)**3 + 8*a*sqrt(a + b*x)/(b - c)**3 - 8*a*sqrt(a + c*x)/(b -
c)**3 - 2*(a + c*x)**(3/2)*(b + c/3)/(c*(b - c)**3) + 2*(a + b*x)**(3/2)*(b/3 +
c)/(b*(b - c)**3)

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Mathematica [A]  time = 0.286157, size = 127, normalized size = 0.82 \[ -\frac{2 \left (12 a^{3/2} b c \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )-12 a^{3/2} b c \tanh ^{-1}\left (\frac{\sqrt{a+c x}}{\sqrt{a}}\right )+b \sqrt{a+c x} (a (3 b+13 c)+c x (3 b+c))-c \sqrt{a+b x} (a (13 b+3 c)+b x (b+3 c))\right )}{3 b c (b-c)^3} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(b*Sqrt[a + c*x]*(a*(3*b + 13*c) + c*(3*b + c)*x) - c*Sqrt[a + b*x]*(a*(13*b
+ 3*c) + b*(b + 3*c)*x) + 12*a^(3/2)*b*c*ArcTanh[Sqrt[a + b*x]/Sqrt[a]] - 12*a^
(3/2)*b*c*ArcTanh[Sqrt[a + c*x]/Sqrt[a]]))/(3*b*(b - c)^3*c)

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Maple [A]  time = 0.005, size = 148, normalized size = 1. \[{\frac{2}{3\, \left ( b-c \right ) ^{3}} \left ( bx+a \right ) ^{{\frac{3}{2}}}}+2\,{\frac{c \left ( bx+a \right ) ^{3/2}}{ \left ( b-c \right ) ^{3}b}}-2\,{\frac{b \left ( cx+a \right ) ^{3/2}}{ \left ( b-c \right ) ^{3}c}}-{\frac{2}{3\, \left ( b-c \right ) ^{3}} \left ( cx+a \right ) ^{{\frac{3}{2}}}}+4\,{\frac{a}{ \left ( b-c \right ) ^{3}} \left ( 2\,\sqrt{bx+a}-2\,\sqrt{a}{\it Artanh} \left ({\frac{\sqrt{bx+a}}{\sqrt{a}}} \right ) \right ) }-4\,{\frac{a}{ \left ( b-c \right ) ^{3}} \left ( 2\,\sqrt{cx+a}-2\,\sqrt{a}{\it Artanh} \left ({\frac{\sqrt{cx+a}}{\sqrt{a}}} \right ) \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2}}{{\left (\sqrt{b x + a} + \sqrt{c x + a}\right )}^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/(sqrt(b*x + a) + sqrt(c*x + a))^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.280438, size = 1, normalized size = 0.01 \[ \left [-\frac{2 \,{\left (6 \, a^{\frac{3}{2}} b c \log \left (\frac{b x + 2 \, \sqrt{b x + a} \sqrt{a} + 2 \, a}{x}\right ) + 6 \, a^{\frac{3}{2}} b c \log \left (\frac{c x - 2 \, \sqrt{c x + a} \sqrt{a} + 2 \, a}{x}\right ) -{\left (13 \, a b c + 3 \, a c^{2} +{\left (b^{2} c + 3 \, b c^{2}\right )} x\right )} \sqrt{b x + a} +{\left (3 \, a b^{2} + 13 \, a b c +{\left (3 \, b^{2} c + b c^{2}\right )} x\right )} \sqrt{c x + a}\right )}}{3 \,{\left (b^{4} c - 3 \, b^{3} c^{2} + 3 \, b^{2} c^{3} - b c^{4}\right )}}, -\frac{2 \,{\left (12 \, \sqrt{-a} a b c \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right ) - 12 \, \sqrt{-a} a b c \arctan \left (\frac{\sqrt{c x + a}}{\sqrt{-a}}\right ) -{\left (13 \, a b c + 3 \, a c^{2} +{\left (b^{2} c + 3 \, b c^{2}\right )} x\right )} \sqrt{b x + a} +{\left (3 \, a b^{2} + 13 \, a b c +{\left (3 \, b^{2} c + b c^{2}\right )} x\right )} \sqrt{c x + a}\right )}}{3 \,{\left (b^{4} c - 3 \, b^{3} c^{2} + 3 \, b^{2} c^{3} - b c^{4}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/(sqrt(b*x + a) + sqrt(c*x + a))^3,x, algorithm="fricas")

[Out]

[-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*sqrt(-a)*a*b*c*arctan(sq
rt(b*x + a)/sqrt(-a)) - 12*sqrt(-a)*a*b*c*arctan(sqrt(c*x + a)/sqrt(-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)]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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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))^3,x, algorithm="giac")

[Out]

Timed out