3.277 \(\int \frac{x^4}{\left (\sqrt{a+b x}+\sqrt{a+c x}\right )^3} \, dx\)

Optimal. Leaf size=277 \[ \frac{2 a^2 (b+3 c) (a+b x)^{3/2}}{3 b^3 (b-c)^3}-\frac{8 a^2 (a+b x)^{3/2}}{3 b^2 (b-c)^3}-\frac{2 a^2 (3 b+c) (a+c x)^{3/2}}{3 c^3 (b-c)^3}+\frac{8 a^2 (a+c x)^{3/2}}{3 c^2 (b-c)^3}+\frac{2 (b+3 c) (a+b x)^{7/2}}{7 b^3 (b-c)^3}-\frac{4 a (b+3 c) (a+b x)^{5/2}}{5 b^3 (b-c)^3}+\frac{8 a (a+b x)^{5/2}}{5 b^2 (b-c)^3}-\frac{2 (3 b+c) (a+c x)^{7/2}}{7 c^3 (b-c)^3}+\frac{4 a (3 b+c) (a+c x)^{5/2}}{5 c^3 (b-c)^3}-\frac{8 a (a+c x)^{5/2}}{5 c^2 (b-c)^3} \]

[Out]

(-8*a^2*(a + b*x)^(3/2))/(3*b^2*(b - c)^3) + (2*a^2*(b + 3*c)*(a + b*x)^(3/2))/(
3*b^3*(b - c)^3) + (8*a*(a + b*x)^(5/2))/(5*b^2*(b - c)^3) - (4*a*(b + 3*c)*(a +
b*x)^(5/2))/(5*b^3*(b - c)^3) + (2*(b + 3*c)*(a + b*x)^(7/2))/(7*b^3*(b - c)^3)
+ (8*a^2*(a + c*x)^(3/2))/(3*(b - c)^3*c^2) - (2*a^2*(3*b + c)*(a + c*x)^(3/2))
/(3*(b - c)^3*c^3) - (8*a*(a + c*x)^(5/2))/(5*(b - c)^3*c^2) + (4*a*(3*b + c)*(a
+ c*x)^(5/2))/(5*(b - c)^3*c^3) - (2*(3*b + c)*(a + c*x)^(7/2))/(7*(b - c)^3*c^
3)

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Rubi [A]  time = 0.609445, antiderivative size = 277, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08 \[ \frac{2 a^2 (b+3 c) (a+b x)^{3/2}}{3 b^3 (b-c)^3}-\frac{8 a^2 (a+b x)^{3/2}}{3 b^2 (b-c)^3}-\frac{2 a^2 (3 b+c) (a+c x)^{3/2}}{3 c^3 (b-c)^3}+\frac{8 a^2 (a+c x)^{3/2}}{3 c^2 (b-c)^3}+\frac{2 (b+3 c) (a+b x)^{7/2}}{7 b^3 (b-c)^3}-\frac{4 a (b+3 c) (a+b x)^{5/2}}{5 b^3 (b-c)^3}+\frac{8 a (a+b x)^{5/2}}{5 b^2 (b-c)^3}-\frac{2 (3 b+c) (a+c x)^{7/2}}{7 c^3 (b-c)^3}+\frac{4 a (3 b+c) (a+c x)^{5/2}}{5 c^3 (b-c)^3}-\frac{8 a (a+c x)^{5/2}}{5 c^2 (b-c)^3} \]

Antiderivative was successfully verified.

[In]  Int[x^4/(Sqrt[a + b*x] + Sqrt[a + c*x])^3,x]

[Out]

(-8*a^2*(a + b*x)^(3/2))/(3*b^2*(b - c)^3) + (2*a^2*(b + 3*c)*(a + b*x)^(3/2))/(
3*b^3*(b - c)^3) + (8*a*(a + b*x)^(5/2))/(5*b^2*(b - c)^3) - (4*a*(b + 3*c)*(a +
b*x)^(5/2))/(5*b^3*(b - c)^3) + (2*(b + 3*c)*(a + b*x)^(7/2))/(7*b^3*(b - c)^3)
+ (8*a^2*(a + c*x)^(3/2))/(3*(b - c)^3*c^2) - (2*a^2*(3*b + c)*(a + c*x)^(3/2))
/(3*(b - c)^3*c^3) - (8*a*(a + c*x)^(5/2))/(5*(b - c)^3*c^2) + (4*a*(3*b + c)*(a
+ c*x)^(5/2))/(5*(b - c)^3*c^3) - (2*(3*b + c)*(a + c*x)^(7/2))/(7*(b - c)^3*c^
3)

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Rubi in Sympy [A]  time = 58.3058, size = 253, normalized size = 0.91 \[ \frac{8 a^{2} \left (a + c x\right )^{\frac{3}{2}}}{3 c^{2} \left (b - c\right )^{3}} - \frac{2 a^{2} \left (a + c x\right )^{\frac{3}{2}} \left (3 b + c\right )}{3 c^{3} \left (b - c\right )^{3}} - \frac{8 a^{2} \left (a + b x\right )^{\frac{3}{2}}}{3 b^{2} \left (b - c\right )^{3}} + \frac{2 a^{2} \left (a + b x\right )^{\frac{3}{2}} \left (b + 3 c\right )}{3 b^{3} \left (b - c\right )^{3}} - \frac{8 a \left (a + c x\right )^{\frac{5}{2}}}{5 c^{2} \left (b - c\right )^{3}} + \frac{4 a \left (a + c x\right )^{\frac{5}{2}} \left (3 b + c\right )}{5 c^{3} \left (b - c\right )^{3}} + \frac{8 a \left (a + b x\right )^{\frac{5}{2}}}{5 b^{2} \left (b - c\right )^{3}} - \frac{4 a \left (a + b x\right )^{\frac{5}{2}} \left (b + 3 c\right )}{5 b^{3} \left (b - c\right )^{3}} - \frac{2 \left (a + c x\right )^{\frac{7}{2}} \left (3 b + c\right )}{7 c^{3} \left (b - c\right )^{3}} + \frac{2 \left (a + b x\right )^{\frac{7}{2}} \left (b + 3 c\right )}{7 b^{3} \left (b - c\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**4/((b*x+a)**(1/2)+(c*x+a)**(1/2))**3,x)

[Out]

8*a**2*(a + c*x)**(3/2)/(3*c**2*(b - c)**3) - 2*a**2*(a + c*x)**(3/2)*(3*b + c)/
(3*c**3*(b - c)**3) - 8*a**2*(a + b*x)**(3/2)/(3*b**2*(b - c)**3) + 2*a**2*(a +
b*x)**(3/2)*(b + 3*c)/(3*b**3*(b - c)**3) - 8*a*(a + c*x)**(5/2)/(5*c**2*(b - c)
**3) + 4*a*(a + c*x)**(5/2)*(3*b + c)/(5*c**3*(b - c)**3) + 8*a*(a + b*x)**(5/2)
/(5*b**2*(b - c)**3) - 4*a*(a + b*x)**(5/2)*(b + 3*c)/(5*b**3*(b - c)**3) - 2*(a
+ c*x)**(7/2)*(3*b + c)/(7*c**3*(b - c)**3) + 2*(a + b*x)**(7/2)*(b + 3*c)/(7*b
**3*(b - c)**3)

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Mathematica [A]  time = 0.51002, size = 114, normalized size = 0.41 \[ -\frac{2 \left (b^3 (a+c x)^{3/2} \left (8 a^2 (b-2 c)-12 a c x (b-2 c)+5 c^2 x^2 (3 b+c)\right )+c^3 (a+b x)^{3/2} \left (8 a^2 (2 b-c)+12 a b x (c-2 b)-5 b^2 x^2 (b+3 c)\right )\right )}{35 b^3 c^3 (b-c)^3} \]

Antiderivative was successfully verified.

[In]  Integrate[x^4/(Sqrt[a + b*x] + Sqrt[a + c*x])^3,x]

[Out]

(-2*(b^3*(a + c*x)^(3/2)*(8*a^2*(b - 2*c) - 12*a*(b - 2*c)*c*x + 5*c^2*(3*b + c)
*x^2) + c^3*(a + b*x)^(3/2)*(8*a^2*(2*b - c) + 12*a*b*(-2*b + c)*x - 5*b^2*(b +
3*c)*x^2)))/(35*b^3*(b - c)^3*c^3)

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Maple [A]  time = 0.005, size = 246, normalized size = 0.9 \[ 2\,{\frac{1/7\, \left ( bx+a \right ) ^{7/2}-2/5\, \left ( bx+a \right ) ^{5/2}a+1/3\,{a}^{2} \left ( bx+a \right ) ^{3/2}}{ \left ( b-c \right ) ^{3}{b}^{2}}}+8\,{\frac{a \left ( 1/5\, \left ( bx+a \right ) ^{5/2}-1/3\, \left ( bx+a \right ) ^{3/2}a \right ) }{ \left ( b-c \right ) ^{3}{b}^{2}}}-8\,{\frac{a \left ( 1/5\, \left ( cx+a \right ) ^{5/2}-1/3\, \left ( cx+a \right ) ^{3/2}a \right ) }{ \left ( b-c \right ) ^{3}{c}^{2}}}+6\,{\frac{c \left ( 1/7\, \left ( bx+a \right ) ^{7/2}-2/5\, \left ( bx+a \right ) ^{5/2}a+1/3\,{a}^{2} \left ( bx+a \right ) ^{3/2} \right ) }{ \left ( b-c \right ) ^{3}{b}^{3}}}-6\,{\frac{b \left ( 1/7\, \left ( cx+a \right ) ^{7/2}-2/5\, \left ( cx+a \right ) ^{5/2}a+1/3\,{a}^{2} \left ( cx+a \right ) ^{3/2} \right ) }{ \left ( b-c \right ) ^{3}{c}^{3}}}-2\,{\frac{1/7\, \left ( cx+a \right ) ^{7/2}-2/5\, \left ( cx+a \right ) ^{5/2}a+1/3\,{a}^{2} \left ( cx+a \right ) ^{3/2}}{ \left ( b-c \right ) ^{3}{c}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4/((b*x+a)^(1/2)+(c*x+a)^(1/2))^3,x)

[Out]

2/(b-c)^3/b^2*(1/7*(b*x+a)^(7/2)-2/5*(b*x+a)^(5/2)*a+1/3*a^2*(b*x+a)^(3/2))+8/(b
-c)^3*a/b^2*(1/5*(b*x+a)^(5/2)-1/3*(b*x+a)^(3/2)*a)-8/(b-c)^3*a/c^2*(1/5*(c*x+a)
^(5/2)-1/3*(c*x+a)^(3/2)*a)+6/(b-c)^3*c/b^3*(1/7*(b*x+a)^(7/2)-2/5*(b*x+a)^(5/2)
*a+1/3*a^2*(b*x+a)^(3/2))-6/(b-c)^3*b/c^3*(1/7*(c*x+a)^(7/2)-2/5*(c*x+a)^(5/2)*a
+1/3*a^2*(c*x+a)^(3/2))-2/(b-c)^3/c^2*(1/7*(c*x+a)^(7/2)-2/5*(c*x+a)^(5/2)*a+1/3
*a^2*(c*x+a)^(3/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{4}}{{\left (\sqrt{b x + a} + \sqrt{c x + a}\right )}^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/(sqrt(b*x + a) + sqrt(c*x + a))^3,x, algorithm="maxima")

[Out]

integrate(x^4/(sqrt(b*x + a) + sqrt(c*x + a))^3, x)

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Fricas [A]  time = 0.267037, size = 304, normalized size = 1.1 \[ -\frac{2 \,{\left ({\left (16 \, a^{3} b c^{3} - 8 \, a^{3} c^{4} - 5 \,{\left (b^{4} c^{3} + 3 \, b^{3} c^{4}\right )} x^{3} -{\left (29 \, a b^{3} c^{3} + 3 \, a b^{2} c^{4}\right )} x^{2} - 4 \,{\left (2 \, a^{2} b^{2} c^{3} - a^{2} b c^{4}\right )} x\right )} \sqrt{b x + a} +{\left (8 \, a^{3} b^{4} - 16 \, a^{3} b^{3} c + 5 \,{\left (3 \, b^{4} c^{3} + b^{3} c^{4}\right )} x^{3} +{\left (3 \, a b^{4} c^{2} + 29 \, a b^{3} c^{3}\right )} x^{2} - 4 \,{\left (a^{2} b^{4} c - 2 \, a^{2} b^{3} c^{2}\right )} x\right )} \sqrt{c x + a}\right )}}{35 \,{\left (b^{6} c^{3} - 3 \, b^{5} c^{4} + 3 \, b^{4} c^{5} - b^{3} c^{6}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/(sqrt(b*x + a) + sqrt(c*x + a))^3,x, algorithm="fricas")

[Out]

-2/35*((16*a^3*b*c^3 - 8*a^3*c^4 - 5*(b^4*c^3 + 3*b^3*c^4)*x^3 - (29*a*b^3*c^3 +
3*a*b^2*c^4)*x^2 - 4*(2*a^2*b^2*c^3 - a^2*b*c^4)*x)*sqrt(b*x + a) + (8*a^3*b^4
- 16*a^3*b^3*c + 5*(3*b^4*c^3 + b^3*c^4)*x^3 + (3*a*b^4*c^2 + 29*a*b^3*c^3)*x^2
- 4*(a^2*b^4*c - 2*a^2*b^3*c^2)*x)*sqrt(c*x + a))/(b^6*c^3 - 3*b^5*c^4 + 3*b^4*c
^5 - b^3*c^6)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**4/((b*x+a)**(1/2)+(c*x+a)**(1/2))**3,x)

[Out]

Integral(x**4/(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^4/(sqrt(b*x + a) + sqrt(c*x + a))^3,x, algorithm="giac")

[Out]

Exception raised: NotImplementedError