3.252 \(\int \frac{1}{\left (\sqrt{a+b x}+\sqrt{c+b x}\right )^3} \, dx\)

Optimal. Leaf size=64 \[ \frac{(a-c)^2}{10 b \left (\sqrt{a+b x}+\sqrt{b x+c}\right )^5}-\frac{1}{2 b \left (\sqrt{a+b x}+\sqrt{b x+c}\right )} \]

[Out]

(a - c)^2/(10*b*(Sqrt[a + b*x] + Sqrt[c + b*x])^5) - 1/(2*b*(Sqrt[a + b*x] + Sqr
t[c + b*x]))

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Rubi [B]  time = 0.213021, antiderivative size = 151, normalized size of antiderivative = 2.36, number of steps used = 6, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095 \[ \frac{8 (a+b x)^{5/2}}{5 b (a-c)^3}+\frac{2 (a+3 c) (a+b x)^{3/2}}{3 b (a-c)^3}-\frac{8 a (a+b x)^{3/2}}{3 b (a-c)^3}-\frac{8 (b x+c)^{5/2}}{5 b (a-c)^3}+\frac{8 c (b x+c)^{3/2}}{3 b (a-c)^3}-\frac{2 (3 a+c) (b x+c)^{3/2}}{3 b (a-c)^3} \]

Antiderivative was successfully verified.

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

[Out]

(-8*a*(a + b*x)^(3/2))/(3*b*(a - c)^3) + (2*(a + 3*c)*(a + b*x)^(3/2))/(3*b*(a -
c)^3) + (8*(a + b*x)^(5/2))/(5*b*(a - c)^3) + (8*c*(c + b*x)^(3/2))/(3*b*(a - c
)^3) - (2*(3*a + c)*(c + b*x)^(3/2))/(3*b*(a - c)^3) - (8*(c + b*x)^(5/2))/(5*b*
(a - c)^3)

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Rubi in Sympy [A]  time = 25.8229, size = 124, normalized size = 1.94 \[ - \frac{8 a \left (a + b x\right )^{\frac{3}{2}}}{3 b \left (a - c\right )^{3}} + \frac{8 c \left (b x + c\right )^{\frac{3}{2}}}{3 b \left (a - c\right )^{3}} + \frac{2 \left (a + 3 c\right ) \left (a + b x\right )^{\frac{3}{2}}}{3 b \left (a - c\right )^{3}} + \frac{8 \left (a + b x\right )^{\frac{5}{2}}}{5 b \left (a - c\right )^{3}} - \frac{2 \left (3 a + c\right ) \left (b x + c\right )^{\frac{3}{2}}}{3 b \left (a - c\right )^{3}} - \frac{8 \left (b x + c\right )^{\frac{5}{2}}}{5 b \left (a - c\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-8*a*(a + b*x)**(3/2)/(3*b*(a - c)**3) + 8*c*(b*x + c)**(3/2)/(3*b*(a - c)**3) +
2*(a + 3*c)*(a + b*x)**(3/2)/(3*b*(a - c)**3) + 8*(a + b*x)**(5/2)/(5*b*(a - c)
**3) - 2*(3*a + c)*(b*x + c)**(3/2)/(3*b*(a - c)**3) - 8*(b*x + c)**(5/2)/(5*b*(
a - c)**3)

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Mathematica [A]  time = 0.252068, size = 55, normalized size = 0.86 \[ -\frac{2 \left ((a+b x)^{3/2} (a-4 b x-5 c)+(b x+c)^{3/2} (5 a+4 b x-c)\right )}{5 b (a-c)^3} \]

Antiderivative was successfully verified.

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

[Out]

(-2*((a - 5*c - 4*b*x)*(a + b*x)^(3/2) + (c + b*x)^(3/2)*(5*a - c + 4*b*x)))/(5*
b*(a - c)^3)

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Maple [B]  time = 0.004, size = 146, normalized size = 2.3 \[{\frac{2\,a}{3\,b \left ( a-c \right ) ^{3}} \left ( bx+a \right ) ^{{\frac{3}{2}}}}+2\,{\frac{c \left ( bx+a \right ) ^{3/2}}{b \left ( a-c \right ) ^{3}}}-2\,{\frac{a \left ( bx+c \right ) ^{3/2}}{b \left ( a-c \right ) ^{3}}}-{\frac{2\,c}{3\,b \left ( a-c \right ) ^{3}} \left ( bx+c \right ) ^{{\frac{3}{2}}}}+8\,{\frac{1/5\, \left ( bx+a \right ) ^{5/2}-1/3\, \left ( bx+a \right ) ^{3/2}a}{b \left ( a-c \right ) ^{3}}}-8\,{\frac{1/5\, \left ( bx+c \right ) ^{5/2}-1/3\, \left ( bx+c \right ) ^{3/2}c}{b \left ( a-c \right ) ^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*a*(b*x+a)^(3/2)/b/(a-c)^3+2/(a-c)^3*c*(b*x+a)^(3/2)/b-2/(a-c)^3*a*(b*x+c)^(3
/2)/b-2/3*c*(b*x+c)^(3/2)/b/(a-c)^3+8/(a-c)^3/b*(1/5*(b*x+a)^(5/2)-1/3*(b*x+a)^(
3/2)*a)-8/(a-c)^3/b*(1/5*(b*x+c)^(5/2)-1/3*(b*x+c)^(3/2)*c)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (\sqrt{b x + a} + \sqrt{b x + c}\right )}^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.262813, size = 143, normalized size = 2.23 \[ \frac{2 \,{\left ({\left (4 \, b^{2} x^{2} - a^{2} + 5 \, a c +{\left (3 \, a b + 5 \, b c\right )} x\right )} \sqrt{b x + a} -{\left (4 \, b^{2} x^{2} + 5 \, a c - c^{2} +{\left (5 \, a b + 3 \, b c\right )} x\right )} \sqrt{b x + c}\right )}}{5 \,{\left (a^{3} b - 3 \, a^{2} b c + 3 \, a b c^{2} - b c^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/5*((4*b^2*x^2 - a^2 + 5*a*c + (3*a*b + 5*b*c)*x)*sqrt(b*x + a) - (4*b^2*x^2 +
5*a*c - c^2 + (5*a*b + 3*b*c)*x)*sqrt(b*x + c))/(a^3*b - 3*a^2*b*c + 3*a*b*c^2 -
b*c^3)

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Sympy [A]  time = 6.79849, size = 384, normalized size = 6. \[ \begin{cases} - \frac{2 a}{5 a b \sqrt{a + b x} + 15 a b \sqrt{b x + c} + 20 b^{2} x \sqrt{a + b x} + 20 b^{2} x \sqrt{b x + c} + 15 b c \sqrt{a + b x} + 5 b c \sqrt{b x + c}} - \frac{4 b x}{5 a b \sqrt{a + b x} + 15 a b \sqrt{b x + c} + 20 b^{2} x \sqrt{a + b x} + 20 b^{2} x \sqrt{b x + c} + 15 b c \sqrt{a + b x} + 5 b c \sqrt{b x + c}} - \frac{2 c}{5 a b \sqrt{a + b x} + 15 a b \sqrt{b x + c} + 20 b^{2} x \sqrt{a + b x} + 20 b^{2} x \sqrt{b x + c} + 15 b c \sqrt{a + b x} + 5 b c \sqrt{b x + c}} - \frac{6 \sqrt{a + b x} \sqrt{b x + c}}{5 a b \sqrt{a + b x} + 15 a b \sqrt{b x + c} + 20 b^{2} x \sqrt{a + b x} + 20 b^{2} x \sqrt{b x + c} + 15 b c \sqrt{a + b x} + 5 b c \sqrt{b x + c}} & \text{for}\: b \neq 0 \\\frac{x}{\left (\sqrt{a} + \sqrt{c}\right )^{3}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((-2*a/(5*a*b*sqrt(a + b*x) + 15*a*b*sqrt(b*x + c) + 20*b**2*x*sqrt(a +
b*x) + 20*b**2*x*sqrt(b*x + c) + 15*b*c*sqrt(a + b*x) + 5*b*c*sqrt(b*x + c)) -
4*b*x/(5*a*b*sqrt(a + b*x) + 15*a*b*sqrt(b*x + c) + 20*b**2*x*sqrt(a + b*x) + 20
*b**2*x*sqrt(b*x + c) + 15*b*c*sqrt(a + b*x) + 5*b*c*sqrt(b*x + c)) - 2*c/(5*a*b
*sqrt(a + b*x) + 15*a*b*sqrt(b*x + c) + 20*b**2*x*sqrt(a + b*x) + 20*b**2*x*sqrt
(b*x + c) + 15*b*c*sqrt(a + b*x) + 5*b*c*sqrt(b*x + c)) - 6*sqrt(a + b*x)*sqrt(b
*x + c)/(5*a*b*sqrt(a + b*x) + 15*a*b*sqrt(b*x + c) + 20*b**2*x*sqrt(a + b*x) +
20*b**2*x*sqrt(b*x + c) + 15*b*c*sqrt(a + b*x) + 5*b*c*sqrt(b*x + c)), Ne(b, 0))
, (x/(sqrt(a) + sqrt(c))**3, True))

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

[Out]

Timed out