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

Optimal. Leaf size=165 \[ -\frac{2 (a+b x)^{3/2} (b x+c)^{3/2}}{3 b^2 (a-c)^2}+\frac{(a+c) (a+b x)^{3/2} \sqrt{b x+c}}{2 b^2 (a-c)^2}-\frac{(a+c) \sqrt{a+b x} \sqrt{b x+c}}{4 b^2 (a-c)}-\frac{(a+c) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{b x+c}}\right )}{4 b^2}+\frac{2 b x^3}{3 (a-c)^2}+\frac{x^2 (a+c)}{2 (a-c)^2} \]

[Out]

((a + c)*x^2)/(2*(a - c)^2) + (2*b*x^3)/(3*(a - c)^2) - ((a + c)*Sqrt[a + b*x]*S
qrt[c + b*x])/(4*b^2*(a - c)) + ((a + c)*(a + b*x)^(3/2)*Sqrt[c + b*x])/(2*b^2*(
a - c)^2) - (2*(a + b*x)^(3/2)*(c + b*x)^(3/2))/(3*b^2*(a - c)^2) - ((a + c)*Arc
Tanh[Sqrt[a + b*x]/Sqrt[c + b*x]])/(4*b^2)

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Rubi [A]  time = 0.44974, antiderivative size = 165, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.261 \[ -\frac{2 (a+b x)^{3/2} (b x+c)^{3/2}}{3 b^2 (a-c)^2}+\frac{(a+c) (a+b x)^{3/2} \sqrt{b x+c}}{2 b^2 (a-c)^2}-\frac{(a+c) \sqrt{a+b x} \sqrt{b x+c}}{4 b^2 (a-c)}-\frac{(a+c) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{b x+c}}\right )}{4 b^2}+\frac{2 b x^3}{3 (a-c)^2}+\frac{x^2 (a+c)}{2 (a-c)^2} \]

Antiderivative was successfully verified.

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

[Out]

((a + c)*x^2)/(2*(a - c)^2) + (2*b*x^3)/(3*(a - c)^2) - ((a + c)*Sqrt[a + b*x]*S
qrt[c + b*x])/(4*b^2*(a - c)) + ((a + c)*(a + b*x)^(3/2)*Sqrt[c + b*x])/(2*b^2*(
a - c)^2) - (2*(a + b*x)^(3/2)*(c + b*x)^(3/2))/(3*b^2*(a - c)^2) - ((a + c)*Arc
Tanh[Sqrt[a + b*x]/Sqrt[c + b*x]])/(4*b^2)

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{2 b x^{3}}{3 \left (a - c\right )^{2}} + \frac{\left (a + c\right ) \int x\, dx}{\left (a - c\right )^{2}} - \frac{\left (a + c\right ) \operatorname{atanh}{\left (\frac{\sqrt{b x + c}}{\sqrt{a + b x}} \right )}}{4 b^{2}} + \frac{\left (a + c\right ) \sqrt{a + b x} \sqrt{b x + c}}{4 b^{2} \left (a - c\right )} + \frac{\left (a + c\right ) \sqrt{a + b x} \left (b x + c\right )^{\frac{3}{2}}}{2 b^{2} \left (a - c\right )^{2}} - \frac{2 \left (a + b x\right )^{\frac{3}{2}} \left (b x + c\right )^{\frac{3}{2}}}{3 b^{2} \left (a - c\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.28808, size = 124, normalized size = 0.75 \[ \frac{2 \sqrt{a+b x} \sqrt{b x+c} \left (3 a^2-2 b x (a+c)-2 a c-8 b^2 x^2+3 c^2\right )+12 b^2 x^2 (a+c)-3 (a-c)^2 (a+c) \log \left (2 \sqrt{a+b x} \sqrt{b x+c}+a+2 b x+c\right )+16 b^3 x^3}{24 b^2 (a-c)^2} \]

Antiderivative was successfully verified.

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

[Out]

(12*b^2*(a + c)*x^2 + 16*b^3*x^3 + 2*Sqrt[a + b*x]*Sqrt[c + b*x]*(3*a^2 - 2*a*c
+ 3*c^2 - 2*b*(a + c)*x - 8*b^2*x^2) - 3*(a - c)^2*(a + c)*Log[a + c + 2*b*x + 2
*Sqrt[a + b*x]*Sqrt[c + b*x]])/(24*b^2*(a - c)^2)

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Maple [C]  time = 0.015, size = 431, normalized size = 2.6 \[{\frac{a{x}^{2}}{2\, \left ( a-c \right ) ^{2}}}+{\frac{c{x}^{2}}{2\, \left ( a-c \right ) ^{2}}}+{\frac{2\,b{x}^{3}}{3\, \left ( a-c \right ) ^{2}}}-{\frac{{\it csgn} \left ( b \right ) }{24\, \left ( a-c \right ) ^{2}{b}^{2}}\sqrt{bx+a}\sqrt{bx+c} \left ( 16\,{\it csgn} \left ( b \right ){x}^{2}{b}^{2}\sqrt{{b}^{2}{x}^{2}+abx+bcx+ac}+4\,{\it csgn} \left ( b \right ) \sqrt{{b}^{2}{x}^{2}+abx+bcx+ac}xab+4\,{\it csgn} \left ( b \right ) \sqrt{{b}^{2}{x}^{2}+abx+bcx+ac}xbc-6\,{\it csgn} \left ( b \right ) \sqrt{{b}^{2}{x}^{2}+abx+bcx+ac}{a}^{2}+4\,{\it csgn} \left ( b \right ) \sqrt{{b}^{2}{x}^{2}+abx+bcx+ac}ac-6\,{\it csgn} \left ( b \right ) \sqrt{{b}^{2}{x}^{2}+abx+bcx+ac}{c}^{2}+3\,\ln \left ( 1/2\, \left ( 2\,{\it csgn} \left ( b \right ) \sqrt{{b}^{2}{x}^{2}+abx+bcx+ac}+2\,bx+a+c \right ){\it csgn} \left ( b \right ) \right ){a}^{3}-3\,\ln \left ( 1/2\, \left ( 2\,{\it csgn} \left ( b \right ) \sqrt{{b}^{2}{x}^{2}+abx+bcx+ac}+2\,bx+a+c \right ){\it csgn} \left ( b \right ) \right ){a}^{2}c-3\,\ln \left ( 1/2\, \left ( 2\,{\it csgn} \left ( b \right ) \sqrt{{b}^{2}{x}^{2}+abx+bcx+ac}+2\,bx+a+c \right ){\it csgn} \left ( b \right ) \right ) a{c}^{2}+3\,\ln \left ( 1/2\, \left ( 2\,{\it csgn} \left ( b \right ) \sqrt{{b}^{2}{x}^{2}+abx+bcx+ac}+2\,bx+a+c \right ){\it csgn} \left ( b \right ) \right ){c}^{3} \right ){\frac{1}{\sqrt{{b}^{2}{x}^{2}+abx+bcx+ac}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*x^2/(a-c)^2*a+1/2*x^2/(a-c)^2*c+2/3*b*x^3/(a-c)^2-1/24/(a-c)^2*(b*x+a)^(1/2)
*(b*x+c)^(1/2)*(16*csgn(b)*x^2*b^2*(b^2*x^2+a*b*x+b*c*x+a*c)^(1/2)+4*csgn(b)*(b^
2*x^2+a*b*x+b*c*x+a*c)^(1/2)*x*a*b+4*csgn(b)*(b^2*x^2+a*b*x+b*c*x+a*c)^(1/2)*x*b
*c-6*csgn(b)*(b^2*x^2+a*b*x+b*c*x+a*c)^(1/2)*a^2+4*csgn(b)*(b^2*x^2+a*b*x+b*c*x+
a*c)^(1/2)*a*c-6*csgn(b)*(b^2*x^2+a*b*x+b*c*x+a*c)^(1/2)*c^2+3*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))*a^3-3*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))*a^2*c-3*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))*a*c^2+3*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))*c^3)*csgn(b)/b^2/(b^2*x^2+a*b*x+b*c
*x+a*c)^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.312584, size = 1385, normalized size = 8.39 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/96*(4096*b^6*x^6 + 5*a^6 - 66*a^5*c - 261*a^4*c^2 + 132*a^3*c^3 - 261*a^2*c^4
- 66*a*c^5 + 5*c^6 + 9216*(a*b^5 + b^5*c)*x^5 + 6144*(a^2*b^4 + 3*a*b^4*c + b^4*
c^2)*x^4 + 32*(17*a^3*b^3 + 327*a^2*b^3*c + 327*a*b^3*c^2 + 17*b^3*c^3)*x^3 - 48
*(11*a^4*b^2 - 8*a^3*b^2*c - 102*a^2*b^2*c^2 - 8*a*b^2*c^3 + 11*b^2*c^4)*x^2 - 2
*(2048*b^5*x^5 + 3*a^5 - 113*a^4*c - 18*a^3*c^2 - 18*a^2*c^3 - 113*a*c^4 + 3*c^5
+ 3584*(a*b^4 + b^4*c)*x^4 + 512*(3*a^2*b^3 + 10*a*b^3*c + 3*b^3*c^2)*x^3 - 176
*(a^3*b^2 - 9*a^2*b^2*c - 9*a*b^2*c^2 + b^2*c^3)*x^2 - 64*(2*a^4*b + 5*a^3*b*c -
6*a^2*b*c^2 + 5*a*b*c^3 + 2*b*c^4)*x)*sqrt(b*x + a)*sqrt(b*x + c) - 6*(9*a^5*b
+ 141*a^4*b*c - 22*a^3*b*c^2 - 22*a^2*b*c^3 + 141*a*b*c^4 + 9*b*c^5)*x + 12*(a^6
+ 14*a^5*c - a^4*c^2 - 28*a^3*c^3 - a^2*c^4 + 14*a*c^5 + c^6 + 32*(a^3*b^3 - a^
2*b^3*c - a*b^3*c^2 + b^3*c^3)*x^3 + 48*(a^4*b^2 - 2*a^2*b^2*c^2 + b^2*c^4)*x^2
- 2*(3*a^5 + 7*a^4*c - 10*a^3*c^2 - 10*a^2*c^3 + 7*a*c^4 + 3*c^5 + 16*(a^3*b^2 -
a^2*b^2*c - a*b^2*c^2 + b^2*c^3)*x^2 + 16*(a^4*b - 2*a^2*b*c^2 + b*c^4)*x)*sqrt
(b*x + a)*sqrt(b*x + c) + 6*(3*a^5*b + 7*a^4*b*c - 10*a^3*b*c^2 - 10*a^2*b*c^3 +
7*a*b*c^4 + 3*b*c^5)*x)*log(-2*b*x + 2*sqrt(b*x + a)*sqrt(b*x + c) - a - c))/(a
^5*b^2 + 13*a^4*b^2*c - 14*a^3*b^2*c^2 - 14*a^2*b^2*c^3 + 13*a*b^2*c^4 + b^2*c^5
+ 32*(a^2*b^5 - 2*a*b^5*c + b^5*c^2)*x^3 + 48*(a^3*b^4 - a^2*b^4*c - a*b^4*c^2
+ b^4*c^3)*x^2 - 2*(3*a^4*b^2 + 4*a^3*b^2*c - 14*a^2*b^2*c^2 + 4*a*b^2*c^3 + 3*b
^2*c^4 + 16*(a^2*b^4 - 2*a*b^4*c + b^4*c^2)*x^2 + 16*(a^3*b^3 - a^2*b^3*c - a*b^
3*c^2 + b^3*c^3)*x)*sqrt(b*x + a)*sqrt(b*x + c) + 6*(3*a^4*b^3 + 4*a^3*b^3*c - 1
4*a^2*b^3*c^2 + 4*a*b^3*c^3 + 3*b^3*c^4)*x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out