3.276 \(\int \frac{1}{x^2 \left (\sqrt{a+b x}+\sqrt{a+c x}\right )^2} \, dx\)

Optimal. Leaf size=174 \[ \frac{2 (a+b x)^{3/2} (a+c x)^{3/2}}{3 a^2 x^3 (b-c)^2}-\frac{(b+c) \sqrt{a+b x} (a+c x)^{3/2}}{2 a^2 x^2 (b-c)^2}-\frac{(b+c) \sqrt{a+b x} \sqrt{a+c x}}{4 a^2 x (b-c)}+\frac{(b+c) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a+c x}}\right )}{4 a^2}-\frac{2 a}{3 x^3 (b-c)^2}-\frac{b+c}{2 x^2 (b-c)^2} \]

[Out]

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

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Rubi [A]  time = 0.537159, antiderivative size = 174, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{2 (a+b x)^{3/2} (a+c x)^{3/2}}{3 a^2 x^3 (b-c)^2}-\frac{(b+c) \sqrt{a+b x} (a+c x)^{3/2}}{2 a^2 x^2 (b-c)^2}-\frac{(b+c) \sqrt{a+b x} \sqrt{a+c x}}{4 a^2 x (b-c)}+\frac{(b+c) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a+c x}}\right )}{4 a^2}-\frac{2 a}{3 x^3 (b-c)^2}-\frac{b+c}{2 x^2 (b-c)^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 36.1827, size = 150, normalized size = 0.86 \[ - \frac{2 a}{3 x^{3} \left (b - c\right )^{2}} - \frac{b + c}{2 x^{2} \left (b - c\right )^{2}} + \frac{\left (b + c\right ) \operatorname{atanh}{\left (\frac{\sqrt{a + b x}}{\sqrt{a + c x}} \right )}}{4 a^{2}} + \frac{\sqrt{a + b x} \sqrt{a + c x} \left (b + c\right )}{4 a^{2} x \left (b - c\right )} - \frac{\left (a + b x\right )^{\frac{3}{2}} \sqrt{a + c x} \left (b + c\right )}{2 a^{2} x^{2} \left (b - c\right )^{2}} + \frac{2 \left (a + b x\right )^{\frac{3}{2}} \left (a + c x\right )^{\frac{3}{2}}}{3 a^{2} x^{3} \left (b - c\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.43688, size = 164, normalized size = 0.94 \[ \frac{\frac{2 \left (-8 a^3+a^2 \left (8 \sqrt{a+b x} \sqrt{a+c x}-6 b x-6 c x\right )+x^2 \left (-3 b^2+2 b c-3 c^2\right ) \sqrt{a+b x} \sqrt{a+c x}+2 a x (b+c) \sqrt{a+b x} \sqrt{a+c x}\right )}{x^3 (b-c)^2}+3 (b+c) \log \left (2 \sqrt{a+b x} \sqrt{a+c x}+2 a+b x+c x\right )-3 (b+c) \log (x)}{24 a^2} \]

Antiderivative was successfully verified.

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

[Out]

((2*(-8*a^3 + 2*a*(b + c)*x*Sqrt[a + b*x]*Sqrt[a + c*x] + (-3*b^2 + 2*b*c - 3*c^
2)*x^2*Sqrt[a + b*x]*Sqrt[a + c*x] + a^2*(-6*b*x - 6*c*x + 8*Sqrt[a + b*x]*Sqrt[
a + c*x])))/((b - c)^2*x^3) - 3*(b + c)*Log[x] + 3*(b + c)*Log[2*a + b*x + c*x +
2*Sqrt[a + b*x]*Sqrt[a + c*x]])/(24*a^2)

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Maple [C]  time = 0.019, size = 457, normalized size = 2.6 \[ -{\frac{b}{2\,{x}^{2} \left ( b-c \right ) ^{2}}}-{\frac{c}{2\,{x}^{2} \left ( b-c \right ) ^{2}}}-{\frac{2\,a}{3\, \left ( b-c \right ) ^{2}{x}^{3}}}-{\frac{{\it csgn} \left ( a \right ) }{24\, \left ( b-c \right ) ^{2}{a}^{2}{x}^{3}}\sqrt{bx+a}\sqrt{cx+a} \left ( -3\,\ln \left ({\frac{a \left ( 2\,{\it csgn} \left ( a \right ) \sqrt{bc{x}^{2}+abx+acx+{a}^{2}}+bx+cx+2\,a \right ) }{x}} \right ){x}^{3}{b}^{3}+3\,\ln \left ({\frac{a \left ( 2\,{\it csgn} \left ( a \right ) \sqrt{bc{x}^{2}+abx+acx+{a}^{2}}+bx+cx+2\,a \right ) }{x}} \right ){x}^{3}{b}^{2}c+3\,\ln \left ({\frac{a \left ( 2\,{\it csgn} \left ( a \right ) \sqrt{bc{x}^{2}+abx+acx+{a}^{2}}+bx+cx+2\,a \right ) }{x}} \right ){x}^{3}b{c}^{2}-3\,\ln \left ({\frac{a \left ( 2\,{\it csgn} \left ( a \right ) \sqrt{bc{x}^{2}+abx+acx+{a}^{2}}+bx+cx+2\,a \right ) }{x}} \right ){x}^{3}{c}^{3}+6\,{\it csgn} \left ( a \right ) \sqrt{bc{x}^{2}+abx+acx+{a}^{2}}{x}^{2}{b}^{2}-4\,{\it csgn} \left ( a \right ) \sqrt{bc{x}^{2}+abx+acx+{a}^{2}}{x}^{2}bc+6\,{\it csgn} \left ( a \right ) \sqrt{bc{x}^{2}+abx+acx+{a}^{2}}{x}^{2}{c}^{2}-4\,{\it csgn} \left ( a \right ) \sqrt{bc{x}^{2}+abx+acx+{a}^{2}}xab-4\,{\it csgn} \left ( a \right ) \sqrt{bc{x}^{2}+abx+acx+{a}^{2}}xac-16\,{\it csgn} \left ( a \right ) \sqrt{bc{x}^{2}+abx+acx+{a}^{2}}{a}^{2} \right ){\frac{1}{\sqrt{bc{x}^{2}+abx+acx+{a}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/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-2/3*a/(b-c)^2/x^3-1/24/(b-c)^2*(b*x+a)^(1/2
)*(c*x+a)^(1/2)/a^2*(-3*ln(a*(2*csgn(a)*(b*c*x^2+a*b*x+a*c*x+a^2)^(1/2)+b*x+c*x+
2*a)/x)*x^3*b^3+3*ln(a*(2*csgn(a)*(b*c*x^2+a*b*x+a*c*x+a^2)^(1/2)+b*x+c*x+2*a)/x
)*x^3*b^2*c+3*ln(a*(2*csgn(a)*(b*c*x^2+a*b*x+a*c*x+a^2)^(1/2)+b*x+c*x+2*a)/x)*x^
3*b*c^2-3*ln(a*(2*csgn(a)*(b*c*x^2+a*b*x+a*c*x+a^2)^(1/2)+b*x+c*x+2*a)/x)*x^3*c^
3+6*csgn(a)*(b*c*x^2+a*b*x+a*c*x+a^2)^(1/2)*x^2*b^2-4*csgn(a)*(b*c*x^2+a*b*x+a*c
*x+a^2)^(1/2)*x^2*b*c+6*csgn(a)*(b*c*x^2+a*b*x+a*c*x+a^2)^(1/2)*x^2*c^2-4*csgn(a
)*(b*c*x^2+a*b*x+a*c*x+a^2)^(1/2)*x*a*b-4*csgn(a)*(b*c*x^2+a*b*x+a*c*x+a^2)^(1/2
)*x*a*c-16*csgn(a)*(b*c*x^2+a*b*x+a*c*x+a^2)^(1/2)*a^2)*csgn(a)/(b*c*x^2+a*b*x+a
*c*x+a^2)^(1/2)/x^3

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x^{2}{\left (\sqrt{b x + a} + \sqrt{c x + a}\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(x^2*(sqrt(b*x + a) + sqrt(c*x + a))^2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.285803, size = 1432, normalized size = 8.23 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(x^2*(sqrt(b*x + a) + sqrt(c*x + a))^2),x, algorithm="fricas")

[Out]

-1/96*((5*b^6 - 66*b^5*c - 261*b^4*c^2 + 132*b^3*c^3 - 261*b^2*c^4 - 66*b*c^5 +
5*c^6)*x^6 + 4096*a^6 - 6*(9*a*b^5 + 141*a*b^4*c - 22*a*b^3*c^2 - 22*a*b^2*c^3 +
141*a*b*c^4 + 9*a*c^5)*x^5 - 48*(11*a^2*b^4 - 8*a^2*b^3*c - 102*a^2*b^2*c^2 - 8
*a^2*b*c^3 + 11*a^2*c^4)*x^4 + 32*(17*a^3*b^3 + 327*a^3*b^2*c + 327*a^3*b*c^2 +
17*a^3*c^3)*x^3 + 6144*(a^4*b^2 + 3*a^4*b*c + a^4*c^2)*x^2 - 2*((3*b^5 - 113*b^4
*c - 18*b^3*c^2 - 18*b^2*c^3 - 113*b*c^4 + 3*c^5)*x^5 + 2048*a^5 - 64*(2*a*b^4 +
5*a*b^3*c - 6*a*b^2*c^2 + 5*a*b*c^3 + 2*a*c^4)*x^4 - 176*(a^2*b^3 - 9*a^2*b^2*c
- 9*a^2*b*c^2 + a^2*c^3)*x^3 + 512*(3*a^3*b^2 + 10*a^3*b*c + 3*a^3*c^2)*x^2 + 3
584*(a^4*b + a^4*c)*x)*sqrt(b*x + a)*sqrt(c*x + a) + 9216*(a^5*b + a^5*c)*x + 12
*((b^6 + 14*b^5*c - b^4*c^2 - 28*b^3*c^3 - b^2*c^4 + 14*b*c^5 + c^6)*x^6 + 6*(3*
a*b^5 + 7*a*b^4*c - 10*a*b^3*c^2 - 10*a*b^2*c^3 + 7*a*b*c^4 + 3*a*c^5)*x^5 + 48*
(a^2*b^4 - 2*a^2*b^2*c^2 + a^2*c^4)*x^4 + 32*(a^3*b^3 - a^3*b^2*c - a^3*b*c^2 +
a^3*c^3)*x^3 - 2*((3*b^5 + 7*b^4*c - 10*b^3*c^2 - 10*b^2*c^3 + 7*b*c^4 + 3*c^5)*
x^5 + 16*(a*b^4 - 2*a*b^2*c^2 + a*c^4)*x^4 + 16*(a^2*b^3 - a^2*b^2*c - a^2*b*c^2
+ a^2*c^3)*x^3)*sqrt(b*x + a)*sqrt(c*x + a))*log(-((b + c)*x - 2*sqrt(b*x + a)*
sqrt(c*x + a) + 2*a)/x))/((a^2*b^5 + 13*a^2*b^4*c - 14*a^2*b^3*c^2 - 14*a^2*b^2*
c^3 + 13*a^2*b*c^4 + a^2*c^5)*x^6 + 6*(3*a^3*b^4 + 4*a^3*b^3*c - 14*a^3*b^2*c^2
+ 4*a^3*b*c^3 + 3*a^3*c^4)*x^5 + 48*(a^4*b^3 - a^4*b^2*c - a^4*b*c^2 + a^4*c^3)*
x^4 + 32*(a^5*b^2 - 2*a^5*b*c + a^5*c^2)*x^3 - 2*((3*a^2*b^4 + 4*a^2*b^3*c - 14*
a^2*b^2*c^2 + 4*a^2*b*c^3 + 3*a^2*c^4)*x^5 + 16*(a^3*b^3 - a^3*b^2*c - a^3*b*c^2
+ a^3*c^3)*x^4 + 16*(a^4*b^2 - 2*a^4*b*c + a^4*c^2)*x^3)*sqrt(b*x + a)*sqrt(c*x
+ a))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x^{2} \left (\sqrt{a + b x} + \sqrt{a + c x}\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out