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

Optimal. Leaf size=141 \[ \frac{2 \sqrt{a+b x} \sqrt{b x+c}}{x (a-c)^2}+\frac{2 b \log (x)}{(a-c)^2}+\frac{2 b (a+c) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{b x+c}}\right )}{\sqrt{a} \sqrt{c} (a-c)^2}-\frac{4 b \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{b x+c}}\right )}{(a-c)^2}-\frac{a+c}{x (a-c)^2} \]

[Out]

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

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Rubi [A]  time = 0.533415, antiderivative size = 141, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.32 \[ \frac{2 \sqrt{a+b x} \sqrt{b x+c}}{x (a-c)^2}+\frac{2 b \log (x)}{(a-c)^2}+\frac{2 b (a+c) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{b x+c}}\right )}{\sqrt{a} \sqrt{c} (a-c)^2}-\frac{4 b \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{b x+c}}\right )}{(a-c)^2}-\frac{a+c}{x (a-c)^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 44.2423, size = 124, normalized size = 0.88 \[ \frac{2 b \log{\left (x \right )}}{\left (a - c\right )^{2}} - \frac{4 b \operatorname{atanh}{\left (\frac{\sqrt{b x + c}}{\sqrt{a + b x}} \right )}}{\left (a - c\right )^{2}} - \frac{a + c}{x \left (a - c\right )^{2}} + \frac{2 \sqrt{a + b x} \sqrt{b x + c}}{x \left (a - c\right )^{2}} + \frac{2 b \left (a + c\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{a + b x}}{\sqrt{a} \sqrt{b x + c}} \right )}}{\sqrt{a} \sqrt{c} \left (a - c\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.314713, size = 153, normalized size = 1.09 \[ \frac{\frac{2 \sqrt{a+b x} \sqrt{b x+c}}{x}-\frac{b (a+c) \log (x)}{\sqrt{a} \sqrt{c}}+\frac{b (a+c) \log \left (2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{b x+c}+a b x+2 a c+b c x\right )}{\sqrt{a} \sqrt{c}}-2 b \log \left (2 \sqrt{a+b x} \sqrt{b x+c}+a+2 b x+c\right )-\frac{a+c}{x}+2 b \log (x)}{(a-c)^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [C]  time = 0.017, size = 274, normalized size = 1.9 \[ -{\frac{a}{x \left ( a-c \right ) ^{2}}}-{\frac{c}{x \left ( a-c \right ) ^{2}}}+2\,{\frac{b\ln \left ( x \right ) }{ \left ( a-c \right ) ^{2}}}+{\frac{{\it csgn} \left ( b \right ) }{x \left ( a-c \right ) ^{2}}\sqrt{bx+a}\sqrt{bx+c} \left ({\it csgn} \left ( b \right ) \ln \left ({\frac{1}{x} \left ( abx+bcx+2\,\sqrt{ac}\sqrt{{b}^{2}{x}^{2}+abx+bcx+ac}+2\,ac \right ) } \right ) xab+{\it csgn} \left ( b \right ) \ln \left ({\frac{1}{x} \left ( abx+bcx+2\,\sqrt{ac}\sqrt{{b}^{2}{x}^{2}+abx+bcx+ac}+2\,ac \right ) } \right ) xbc-2\,\ln \left ( 1/2\, \left ( 2\,{\it csgn} \left ( b \right ) \sqrt{{b}^{2}{x}^{2}+abx+bcx+ac}+2\,bx+a+c \right ){\it csgn} \left ( b \right ) \right ) xb\sqrt{ac}+2\,{\it csgn} \left ( b \right ) \sqrt{ac}\sqrt{{b}^{2}{x}^{2}+abx+bcx+ac} \right ){\frac{1}{\sqrt{ac}}}{\frac{1}{\sqrt{{b}^{2}{x}^{2}+abx+bcx+ac}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.30657, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out