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

Optimal. Leaf size=163 \[ \frac{2 (b+3 c) (a+b x)^{5/2}}{5 b^2 (b-c)^3}-\frac{2 a (b+3 c) (a+b x)^{3/2}}{3 b^2 (b-c)^3}-\frac{2 (3 b+c) (a+c x)^{5/2}}{5 c^2 (b-c)^3}+\frac{2 a (3 b+c) (a+c x)^{3/2}}{3 c^2 (b-c)^3}+\frac{8 a (a+b x)^{3/2}}{3 b (b-c)^3}-\frac{8 a (a+c x)^{3/2}}{3 c (b-c)^3} \]

[Out]

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

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Rubi [A]  time = 0.437136, antiderivative size = 163, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08 \[ \frac{2 (b+3 c) (a+b x)^{5/2}}{5 b^2 (b-c)^3}-\frac{2 a (b+3 c) (a+b x)^{3/2}}{3 b^2 (b-c)^3}-\frac{2 (3 b+c) (a+c x)^{5/2}}{5 c^2 (b-c)^3}+\frac{2 a (3 b+c) (a+c x)^{3/2}}{3 c^2 (b-c)^3}+\frac{8 a (a+b x)^{3/2}}{3 b (b-c)^3}-\frac{8 a (a+c x)^{3/2}}{3 c (b-c)^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 35.1659, size = 144, normalized size = 0.88 \[ - \frac{8 a \left (a + c x\right )^{\frac{3}{2}}}{3 c \left (b - c\right )^{3}} + \frac{2 a \left (a + c x\right )^{\frac{3}{2}} \left (3 b + c\right )}{3 c^{2} \left (b - c\right )^{3}} + \frac{8 a \left (a + b x\right )^{\frac{3}{2}}}{3 b \left (b - c\right )^{3}} - \frac{2 a \left (a + b x\right )^{\frac{3}{2}} \left (b + 3 c\right )}{3 b^{2} \left (b - c\right )^{3}} - \frac{2 \left (a + c x\right )^{\frac{5}{2}} \left (3 b + c\right )}{5 c^{2} \left (b - c\right )^{3}} + \frac{2 \left (a + b x\right )^{\frac{5}{2}} \left (b + 3 c\right )}{5 b^{2} \left (b - c\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.340223, size = 80, normalized size = 0.49 \[ \frac{2 \left (c^2 (a+b x)^{3/2} (a (6 b-2 c)+b x (b+3 c))-b^2 (a+c x)^{3/2} (c x (3 b+c)-2 a (b-3 c))\right )}{5 b^2 c^2 (b-c)^3} \]

Antiderivative was successfully verified.

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

[Out]

(2*(-(b^2*(a + c*x)^(3/2)*(-2*a*(b - 3*c) + c*(3*b + c)*x)) + c^2*(a + b*x)^(3/2
)*(a*(6*b - 2*c) + b*(b + 3*c)*x)))/(5*b^2*(b - c)^3*c^2)

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Maple [A]  time = 0.005, size = 172, normalized size = 1.1 \[ 2\,{\frac{1/5\, \left ( bx+a \right ) ^{5/2}-1/3\, \left ( bx+a \right ) ^{3/2}a}{ \left ( b-c \right ) ^{3}b}}+{\frac{8\,a}{3\, \left ( b-c \right ) ^{3}b} \left ( bx+a \right ) ^{{\frac{3}{2}}}}-{\frac{8\,a}{3\, \left ( b-c \right ) ^{3}c} \left ( cx+a \right ) ^{{\frac{3}{2}}}}+6\,{\frac{c \left ( 1/5\, \left ( bx+a \right ) ^{5/2}-1/3\, \left ( bx+a \right ) ^{3/2}a \right ) }{ \left ( b-c \right ) ^{3}{b}^{2}}}-6\,{\frac{b \left ( 1/5\, \left ( cx+a \right ) ^{5/2}-1/3\, \left ( cx+a \right ) ^{3/2}a \right ) }{ \left ( b-c \right ) ^{3}{c}^{2}}}-2\,{\frac{1/5\, \left ( cx+a \right ) ^{5/2}-1/3\, \left ( cx+a \right ) ^{3/2}a}{ \left ( b-c \right ) ^{3}c}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/(b-c)^3/b*(1/5*(b*x+a)^(5/2)-1/3*(b*x+a)^(3/2)*a)+8/3*a*(b*x+a)^(3/2)/b/(b-c)^
3-8/3*a*(c*x+a)^(3/2)/(b-c)^3/c+6/(b-c)^3*c/b^2*(1/5*(b*x+a)^(5/2)-1/3*(b*x+a)^(
3/2)*a)-6/(b-c)^3*b/c^2*(1/5*(c*x+a)^(5/2)-1/3*(c*x+a)^(3/2)*a)-2/(b-c)^3/c*(1/5
*(c*x+a)^(5/2)-1/3*(c*x+a)^(3/2)*a)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.273866, size = 225, normalized size = 1.38 \[ \frac{2 \,{\left ({\left (6 \, a^{2} b c^{2} - 2 \, a^{2} c^{3} +{\left (b^{3} c^{2} + 3 \, b^{2} c^{3}\right )} x^{2} +{\left (7 \, a b^{2} c^{2} + a b c^{3}\right )} x\right )} \sqrt{b x + a} +{\left (2 \, a^{2} b^{3} - 6 \, a^{2} b^{2} c -{\left (3 \, b^{3} c^{2} + b^{2} c^{3}\right )} x^{2} -{\left (a b^{3} c + 7 \, a b^{2} c^{2}\right )} x\right )} \sqrt{c x + a}\right )}}{5 \,{\left (b^{5} c^{2} - 3 \, b^{4} c^{3} + 3 \, b^{3} c^{4} - b^{2} c^{5}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/5*((6*a^2*b*c^2 - 2*a^2*c^3 + (b^3*c^2 + 3*b^2*c^3)*x^2 + (7*a*b^2*c^2 + a*b*c
^3)*x)*sqrt(b*x + a) + (2*a^2*b^3 - 6*a^2*b^2*c - (3*b^3*c^2 + b^2*c^3)*x^2 - (a
*b^3*c + 7*a*b^2*c^2)*x)*sqrt(c*x + a))/(b^5*c^2 - 3*b^4*c^3 + 3*b^3*c^4 - b^2*c
^5)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: NotImplementedError