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

Optimal. Leaf size=157 \[ -\frac{4 a \sqrt{a+b x}}{x (b-c)^3}+\frac{4 a \sqrt{a+c x}}{x (b-c)^3}+\frac{2 (b+3 c) \sqrt{a+b x}}{(b-c)^3}-\frac{2 (3 b+c) \sqrt{a+c x}}{(b-c)^3}-\frac{6 \sqrt{a} (b+c) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{(b-c)^3}+\frac{6 \sqrt{a} (b+c) \tanh ^{-1}\left (\frac{\sqrt{a+c x}}{\sqrt{a}}\right )}{(b-c)^3} \]

[Out]

(2*(b + 3*c)*Sqrt[a + b*x])/(b - c)^3 - (4*a*Sqrt[a + b*x])/((b - c)^3*x) - (2*(
3*b + c)*Sqrt[a + c*x])/(b - c)^3 + (4*a*Sqrt[a + c*x])/((b - c)^3*x) - (6*Sqrt[
a]*(b + c)*ArcTanh[Sqrt[a + b*x]/Sqrt[a]])/(b - c)^3 + (6*Sqrt[a]*(b + c)*ArcTan
h[Sqrt[a + c*x]/Sqrt[a]])/(b - c)^3

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Rubi [A]  time = 0.451189, antiderivative size = 223, normalized size of antiderivative = 1.42, number of steps used = 14, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217 \[ -\frac{4 a \sqrt{a+b x}}{x (b-c)^3}+\frac{4 a \sqrt{a+c x}}{x (b-c)^3}+\frac{2 (b+3 c) \sqrt{a+b x}}{(b-c)^3}-\frac{2 (3 b+c) \sqrt{a+c x}}{(b-c)^3}-\frac{2 \sqrt{a} (b+3 c) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{(b-c)^3}-\frac{4 \sqrt{a} b \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{(b-c)^3}+\frac{4 \sqrt{a} c \tanh ^{-1}\left (\frac{\sqrt{a+c x}}{\sqrt{a}}\right )}{(b-c)^3}+\frac{2 \sqrt{a} (3 b+c) \tanh ^{-1}\left (\frac{\sqrt{a+c x}}{\sqrt{a}}\right )}{(b-c)^3} \]

Antiderivative was successfully verified.

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

[Out]

(2*(b + 3*c)*Sqrt[a + b*x])/(b - c)^3 - (4*a*Sqrt[a + b*x])/((b - c)^3*x) - (2*(
3*b + c)*Sqrt[a + c*x])/(b - c)^3 + (4*a*Sqrt[a + c*x])/((b - c)^3*x) - (4*Sqrt[
a]*b*ArcTanh[Sqrt[a + b*x]/Sqrt[a]])/(b - c)^3 - (2*Sqrt[a]*(b + 3*c)*ArcTanh[Sq
rt[a + b*x]/Sqrt[a]])/(b - c)^3 + (4*Sqrt[a]*c*ArcTanh[Sqrt[a + c*x]/Sqrt[a]])/(
b - c)^3 + (2*Sqrt[a]*(3*b + c)*ArcTanh[Sqrt[a + c*x]/Sqrt[a]])/(b - c)^3

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Rubi in Sympy [A]  time = 38.5623, size = 199, normalized size = 1.27 \[ - \frac{4 \sqrt{a} b \operatorname{atanh}{\left (\frac{\sqrt{a + b x}}{\sqrt{a}} \right )}}{\left (b - c\right )^{3}} + \frac{4 \sqrt{a} c \operatorname{atanh}{\left (\frac{\sqrt{a + c x}}{\sqrt{a}} \right )}}{\left (b - c\right )^{3}} - \frac{2 \sqrt{a} \left (b + 3 c\right ) \operatorname{atanh}{\left (\frac{\sqrt{a + b x}}{\sqrt{a}} \right )}}{\left (b - c\right )^{3}} + \frac{2 \sqrt{a} \left (3 b + c\right ) \operatorname{atanh}{\left (\frac{\sqrt{a + c x}}{\sqrt{a}} \right )}}{\left (b - c\right )^{3}} - \frac{4 a \sqrt{a + b x}}{x \left (b - c\right )^{3}} + \frac{4 a \sqrt{a + c x}}{x \left (b - c\right )^{3}} + \frac{2 \sqrt{a + b x} \left (b + 3 c\right )}{\left (b - c\right )^{3}} - \frac{2 \sqrt{a + c x} \left (3 b + c\right )}{\left (b - c\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-4*sqrt(a)*b*atanh(sqrt(a + b*x)/sqrt(a))/(b - c)**3 + 4*sqrt(a)*c*atanh(sqrt(a
+ c*x)/sqrt(a))/(b - c)**3 - 2*sqrt(a)*(b + 3*c)*atanh(sqrt(a + b*x)/sqrt(a))/(b
- c)**3 + 2*sqrt(a)*(3*b + c)*atanh(sqrt(a + c*x)/sqrt(a))/(b - c)**3 - 4*a*sqr
t(a + b*x)/(x*(b - c)**3) + 4*a*sqrt(a + c*x)/(x*(b - c)**3) + 2*sqrt(a + b*x)*(
b + 3*c)/(b - c)**3 - 2*sqrt(a + c*x)*(3*b + c)/(b - c)**3

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Mathematica [A]  time = 0.23465, size = 143, normalized size = 0.91 \[ \sqrt{a+b x} \left (\frac{2 (b+3 c)}{(b-c)^3}-\frac{4 a}{x (b-c)^3}\right )+\sqrt{a+c x} \left (\frac{4 a}{x (b-c)^3}-\frac{2 (3 b+c)}{(b-c)^3}\right )-\frac{6 \sqrt{a} (b+c) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{(b-c)^3}+\frac{6 \sqrt{a} (b+c) \tanh ^{-1}\left (\frac{\sqrt{a+c x}}{\sqrt{a}}\right )}{(b-c)^3} \]

Antiderivative was successfully verified.

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

[Out]

((2*(b + 3*c))/(b - c)^3 - (4*a)/((b - c)^3*x))*Sqrt[a + b*x] + ((-2*(3*b + c))/
(b - c)^3 + (4*a)/((b - c)^3*x))*Sqrt[a + c*x] - (6*Sqrt[a]*(b + c)*ArcTanh[Sqrt
[a + b*x]/Sqrt[a]])/(b - c)^3 + (6*Sqrt[a]*(b + c)*ArcTanh[Sqrt[a + c*x]/Sqrt[a]
])/(b - c)^3

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Maple [A]  time = 0.005, size = 237, normalized size = 1.5 \[{\frac{b}{ \left ( b-c \right ) ^{3}} \left ( 2\,\sqrt{bx+a}-2\,\sqrt{a}{\it Artanh} \left ({\frac{\sqrt{bx+a}}{\sqrt{a}}} \right ) \right ) }+8\,{\frac{ab}{ \left ( b-c \right ) ^{3}} \left ( -1/2\,{\frac{\sqrt{bx+a}}{bx}}-1/2\,{\frac{1}{\sqrt{a}}{\it Artanh} \left ({\frac{\sqrt{bx+a}}{\sqrt{a}}} \right ) } \right ) }-8\,{\frac{ac}{ \left ( b-c \right ) ^{3}} \left ( -1/2\,{\frac{\sqrt{cx+a}}{cx}}-1/2\,{\frac{1}{\sqrt{a}}{\it Artanh} \left ({\frac{\sqrt{cx+a}}{\sqrt{a}}} \right ) } \right ) }+3\,{\frac{c}{ \left ( b-c \right ) ^{3}} \left ( 2\,\sqrt{bx+a}-2\,\sqrt{a}{\it Artanh} \left ({\frac{\sqrt{bx+a}}{\sqrt{a}}} \right ) \right ) }-3\,{\frac{b}{ \left ( b-c \right ) ^{3}} \left ( 2\,\sqrt{cx+a}-2\,\sqrt{a}{\it Artanh} \left ({\frac{\sqrt{cx+a}}{\sqrt{a}}} \right ) \right ) }-{\frac{c}{ \left ( b-c \right ) ^{3}} \left ( 2\,\sqrt{cx+a}-2\,\sqrt{a}{\it Artanh} \left ({\frac{\sqrt{cx+a}}{\sqrt{a}}} \right ) \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/(b-c)^3*b*(2*(b*x+a)^(1/2)-2*a^(1/2)*arctanh((b*x+a)^(1/2)/a^(1/2)))+8/(b-c)^3
*a*b*(-1/2*(b*x+a)^(1/2)/x/b-1/2*arctanh((b*x+a)^(1/2)/a^(1/2))/a^(1/2))-8/(b-c)
^3*a*c*(-1/2*(c*x+a)^(1/2)/c/x-1/2/a^(1/2)*arctanh((c*x+a)^(1/2)/a^(1/2)))+3/(b-
c)^3*c*(2*(b*x+a)^(1/2)-2*a^(1/2)*arctanh((b*x+a)^(1/2)/a^(1/2)))-3/(b-c)^3*b*(2
*(c*x+a)^(1/2)-2*a^(1/2)*arctanh((c*x+a)^(1/2)/a^(1/2)))-1/(b-c)^3*c*(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. \[ \int \frac{x}{{\left (\sqrt{b x + a} + \sqrt{c x + a}\right )}^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.292785, size = 1, normalized size = 0.01 \[ \left [-\frac{3 \, \sqrt{a}{\left (b + c\right )} x \log \left (\frac{b x + 2 \, \sqrt{b x + a} \sqrt{a} + 2 \, a}{x}\right ) + 3 \, \sqrt{a}{\left (b + c\right )} x \log \left (\frac{c x - 2 \, \sqrt{c x + a} \sqrt{a} + 2 \, a}{x}\right ) - 2 \,{\left ({\left (b + 3 \, c\right )} x - 2 \, a\right )} \sqrt{b x + a} + 2 \,{\left ({\left (3 \, b + c\right )} x - 2 \, a\right )} \sqrt{c x + a}}{{\left (b^{3} - 3 \, b^{2} c + 3 \, b c^{2} - c^{3}\right )} x}, -\frac{2 \,{\left (3 \, \sqrt{-a}{\left (b + c\right )} x \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right ) - 3 \, \sqrt{-a}{\left (b + c\right )} x \arctan \left (\frac{\sqrt{c x + a}}{\sqrt{-a}}\right ) -{\left ({\left (b + 3 \, c\right )} x - 2 \, a\right )} \sqrt{b x + a} +{\left ({\left (3 \, b + c\right )} x - 2 \, a\right )} \sqrt{c x + a}\right )}}{{\left (b^{3} - 3 \, b^{2} c + 3 \, b c^{2} - c^{3}\right )} x}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-(3*sqrt(a)*(b + c)*x*log((b*x + 2*sqrt(b*x + a)*sqrt(a) + 2*a)/x) + 3*sqrt(a)*
(b + c)*x*log((c*x - 2*sqrt(c*x + a)*sqrt(a) + 2*a)/x) - 2*((b + 3*c)*x - 2*a)*s
qrt(b*x + a) + 2*((3*b + c)*x - 2*a)*sqrt(c*x + a))/((b^3 - 3*b^2*c + 3*b*c^2 -
c^3)*x), -2*(3*sqrt(-a)*(b + c)*x*arctan(sqrt(b*x + a)/sqrt(-a)) - 3*sqrt(-a)*(b
+ c)*x*arctan(sqrt(c*x + a)/sqrt(-a)) - ((b + 3*c)*x - 2*a)*sqrt(b*x + a) + ((3
*b + c)*x - 2*a)*sqrt(c*x + a))/((b^3 - 3*b^2*c + 3*b*c^2 - c^3)*x)]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(x/(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/(sqrt(b*x + a) + sqrt(c*x + a))^3,x, algorithm="giac")

[Out]

Timed out