3.250 \(\int \frac{x^2}{\left (\sqrt{a+b x}+\sqrt{c+b x}\right )^3} \, dx\)

Optimal. Leaf size=375 \[ -\frac{8 a^3 (a+b x)^{3/2}}{3 b^3 (a-c)^3}+\frac{24 a^2 (a+b x)^{5/2}}{5 b^3 (a-c)^3}+\frac{2 a^2 (a+3 c) (a+b x)^{3/2}}{3 b^3 (a-c)^3}+\frac{8 c^3 (b x+c)^{3/2}}{3 b^3 (a-c)^3}-\frac{24 c^2 (b x+c)^{5/2}}{5 b^3 (a-c)^3}-\frac{2 c^2 (3 a+c) (b x+c)^{3/2}}{3 b^3 (a-c)^3}+\frac{8 (a+b x)^{9/2}}{9 b^3 (a-c)^3}+\frac{2 (a+3 c) (a+b x)^{7/2}}{7 b^3 (a-c)^3}-\frac{24 a (a+b x)^{7/2}}{7 b^3 (a-c)^3}-\frac{4 a (a+3 c) (a+b x)^{5/2}}{5 b^3 (a-c)^3}-\frac{8 (b x+c)^{9/2}}{9 b^3 (a-c)^3}+\frac{24 c (b x+c)^{7/2}}{7 b^3 (a-c)^3}-\frac{2 (3 a+c) (b x+c)^{7/2}}{7 b^3 (a-c)^3}+\frac{4 c (3 a+c) (b x+c)^{5/2}}{5 b^3 (a-c)^3} \]

[Out]

(-8*a^3*(a + b*x)^(3/2))/(3*b^3*(a - c)^3) + (2*a^2*(a + 3*c)*(a + b*x)^(3/2))/(
3*b^3*(a - c)^3) + (24*a^2*(a + b*x)^(5/2))/(5*b^3*(a - c)^3) - (4*a*(a + 3*c)*(
a + b*x)^(5/2))/(5*b^3*(a - c)^3) - (24*a*(a + b*x)^(7/2))/(7*b^3*(a - c)^3) + (
2*(a + 3*c)*(a + b*x)^(7/2))/(7*b^3*(a - c)^3) + (8*(a + b*x)^(9/2))/(9*b^3*(a -
c)^3) + (8*c^3*(c + b*x)^(3/2))/(3*b^3*(a - c)^3) - (2*c^2*(3*a + c)*(c + b*x)^
(3/2))/(3*b^3*(a - c)^3) - (24*c^2*(c + b*x)^(5/2))/(5*b^3*(a - c)^3) + (4*c*(3*
a + c)*(c + b*x)^(5/2))/(5*b^3*(a - c)^3) + (24*c*(c + b*x)^(7/2))/(7*b^3*(a - c
)^3) - (2*(3*a + c)*(c + b*x)^(7/2))/(7*b^3*(a - c)^3) - (8*(c + b*x)^(9/2))/(9*
b^3*(a - c)^3)

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Rubi [A]  time = 0.7333, antiderivative size = 375, 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 \[ -\frac{8 a^3 (a+b x)^{3/2}}{3 b^3 (a-c)^3}+\frac{24 a^2 (a+b x)^{5/2}}{5 b^3 (a-c)^3}+\frac{2 a^2 (a+3 c) (a+b x)^{3/2}}{3 b^3 (a-c)^3}+\frac{8 c^3 (b x+c)^{3/2}}{3 b^3 (a-c)^3}-\frac{24 c^2 (b x+c)^{5/2}}{5 b^3 (a-c)^3}-\frac{2 c^2 (3 a+c) (b x+c)^{3/2}}{3 b^3 (a-c)^3}+\frac{8 (a+b x)^{9/2}}{9 b^3 (a-c)^3}+\frac{2 (a+3 c) (a+b x)^{7/2}}{7 b^3 (a-c)^3}-\frac{24 a (a+b x)^{7/2}}{7 b^3 (a-c)^3}-\frac{4 a (a+3 c) (a+b x)^{5/2}}{5 b^3 (a-c)^3}-\frac{8 (b x+c)^{9/2}}{9 b^3 (a-c)^3}+\frac{24 c (b x+c)^{7/2}}{7 b^3 (a-c)^3}-\frac{2 (3 a+c) (b x+c)^{7/2}}{7 b^3 (a-c)^3}+\frac{4 c (3 a+c) (b x+c)^{5/2}}{5 b^3 (a-c)^3} \]

Antiderivative was successfully verified.

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

[Out]

(-8*a^3*(a + b*x)^(3/2))/(3*b^3*(a - c)^3) + (2*a^2*(a + 3*c)*(a + b*x)^(3/2))/(
3*b^3*(a - c)^3) + (24*a^2*(a + b*x)^(5/2))/(5*b^3*(a - c)^3) - (4*a*(a + 3*c)*(
a + b*x)^(5/2))/(5*b^3*(a - c)^3) - (24*a*(a + b*x)^(7/2))/(7*b^3*(a - c)^3) + (
2*(a + 3*c)*(a + b*x)^(7/2))/(7*b^3*(a - c)^3) + (8*(a + b*x)^(9/2))/(9*b^3*(a -
c)^3) + (8*c^3*(c + b*x)^(3/2))/(3*b^3*(a - c)^3) - (2*c^2*(3*a + c)*(c + b*x)^
(3/2))/(3*b^3*(a - c)^3) - (24*c^2*(c + b*x)^(5/2))/(5*b^3*(a - c)^3) + (4*c*(3*
a + c)*(c + b*x)^(5/2))/(5*b^3*(a - c)^3) + (24*c*(c + b*x)^(7/2))/(7*b^3*(a - c
)^3) - (2*(3*a + c)*(c + b*x)^(7/2))/(7*b^3*(a - c)^3) - (8*(c + b*x)^(9/2))/(9*
b^3*(a - c)^3)

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Rubi in Sympy [A]  time = 78.8331, size = 342, normalized size = 0.91 \[ - \frac{8 a^{3} \left (a + b x\right )^{\frac{3}{2}}}{3 b^{3} \left (a - c\right )^{3}} + \frac{2 a^{2} \left (a + 3 c\right ) \left (a + b x\right )^{\frac{3}{2}}}{3 b^{3} \left (a - c\right )^{3}} + \frac{24 a^{2} \left (a + b x\right )^{\frac{5}{2}}}{5 b^{3} \left (a - c\right )^{3}} - \frac{4 a \left (a + 3 c\right ) \left (a + b x\right )^{\frac{5}{2}}}{5 b^{3} \left (a - c\right )^{3}} - \frac{24 a \left (a + b x\right )^{\frac{7}{2}}}{7 b^{3} \left (a - c\right )^{3}} + \frac{8 c^{3} \left (b x + c\right )^{\frac{3}{2}}}{3 b^{3} \left (a - c\right )^{3}} - \frac{2 c^{2} \left (3 a + c\right ) \left (b x + c\right )^{\frac{3}{2}}}{3 b^{3} \left (a - c\right )^{3}} - \frac{24 c^{2} \left (b x + c\right )^{\frac{5}{2}}}{5 b^{3} \left (a - c\right )^{3}} + \frac{4 c \left (3 a + c\right ) \left (b x + c\right )^{\frac{5}{2}}}{5 b^{3} \left (a - c\right )^{3}} + \frac{24 c \left (b x + c\right )^{\frac{7}{2}}}{7 b^{3} \left (a - c\right )^{3}} + \frac{2 \left (a + 3 c\right ) \left (a + b x\right )^{\frac{7}{2}}}{7 b^{3} \left (a - c\right )^{3}} + \frac{8 \left (a + b x\right )^{\frac{9}{2}}}{9 b^{3} \left (a - c\right )^{3}} - \frac{2 \left (3 a + c\right ) \left (b x + c\right )^{\frac{7}{2}}}{7 b^{3} \left (a - c\right )^{3}} - \frac{8 \left (b x + c\right )^{\frac{9}{2}}}{9 b^{3} \left (a - c\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-8*a**3*(a + b*x)**(3/2)/(3*b**3*(a - c)**3) + 2*a**2*(a + 3*c)*(a + b*x)**(3/2)
/(3*b**3*(a - c)**3) + 24*a**2*(a + b*x)**(5/2)/(5*b**3*(a - c)**3) - 4*a*(a + 3
*c)*(a + b*x)**(5/2)/(5*b**3*(a - c)**3) - 24*a*(a + b*x)**(7/2)/(7*b**3*(a - c)
**3) + 8*c**3*(b*x + c)**(3/2)/(3*b**3*(a - c)**3) - 2*c**2*(3*a + c)*(b*x + c)*
*(3/2)/(3*b**3*(a - c)**3) - 24*c**2*(b*x + c)**(5/2)/(5*b**3*(a - c)**3) + 4*c*
(3*a + c)*(b*x + c)**(5/2)/(5*b**3*(a - c)**3) + 24*c*(b*x + c)**(7/2)/(7*b**3*(
a - c)**3) + 2*(a + 3*c)*(a + b*x)**(7/2)/(7*b**3*(a - c)**3) + 8*(a + b*x)**(9/
2)/(9*b**3*(a - c)**3) - 2*(3*a + c)*(b*x + c)**(7/2)/(7*b**3*(a - c)**3) - 8*(b
*x + c)**(9/2)/(9*b**3*(a - c)**3)

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Mathematica [A]  time = 0.536062, size = 138, normalized size = 0.37 \[ -\frac{2 \left ((a+b x)^{3/2} \left (40 a^3-12 a^2 (5 b x+6 c)+3 a b x (25 b x+36 c)-5 b^2 x^2 (28 b x+27 c)\right )+(b x+c)^{3/2} \left (9 a \left (15 b^2 x^2-12 b c x+8 c^2\right )+5 \left (28 b^3 x^3-15 b^2 c x^2+12 b c^2 x-8 c^3\right )\right )\right )}{315 b^3 (a-c)^3} \]

Antiderivative was successfully verified.

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

[Out]

(-2*((a + b*x)^(3/2)*(40*a^3 - 12*a^2*(6*c + 5*b*x) + 3*a*b*x*(36*c + 25*b*x) -
5*b^2*x^2*(27*c + 28*b*x)) + (c + b*x)^(3/2)*(9*a*(8*c^2 - 12*b*c*x + 15*b^2*x^2
) + 5*(-8*c^3 + 12*b*c^2*x - 15*b^2*c*x^2 + 28*b^3*x^3))))/(315*b^3*(a - c)^3)

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Maple [A]  time = 0.007, size = 294, normalized size = 0.8 \[ 2\,{\frac{a \left ( 1/7\, \left ( bx+a \right ) ^{7/2}-2/5\, \left ( bx+a \right ) ^{5/2}a+1/3\,{a}^{2} \left ( bx+a \right ) ^{3/2} \right ) }{ \left ( a-c \right ) ^{3}{b}^{3}}}+6\,{\frac{c \left ( 1/7\, \left ( bx+a \right ) ^{7/2}-2/5\, \left ( bx+a \right ) ^{5/2}a+1/3\,{a}^{2} \left ( bx+a \right ) ^{3/2} \right ) }{ \left ( a-c \right ) ^{3}{b}^{3}}}-6\,{\frac{a \left ( 1/7\, \left ( bx+c \right ) ^{7/2}-2/5\, \left ( bx+c \right ) ^{5/2}c+1/3\,{c}^{2} \left ( bx+c \right ) ^{3/2} \right ) }{ \left ( a-c \right ) ^{3}{b}^{3}}}-2\,{\frac{c \left ( 1/7\, \left ( bx+c \right ) ^{7/2}-2/5\, \left ( bx+c \right ) ^{5/2}c+1/3\,{c}^{2} \left ( bx+c \right ) ^{3/2} \right ) }{ \left ( a-c \right ) ^{3}{b}^{3}}}+8\,{\frac{1/9\, \left ( bx+a \right ) ^{9/2}-3/7\,a \left ( bx+a \right ) ^{7/2}+3/5\,{a}^{2} \left ( bx+a \right ) ^{5/2}-1/3\,{a}^{3} \left ( bx+a \right ) ^{3/2}}{ \left ( a-c \right ) ^{3}{b}^{3}}}-8\,{\frac{1/9\, \left ( bx+c \right ) ^{9/2}-3/7\, \left ( bx+c \right ) ^{7/2}c+3/5\, \left ( bx+c \right ) ^{5/2}{c}^{2}-1/3\,{c}^{3} \left ( bx+c \right ) ^{3/2}}{ \left ( a-c \right ) ^{3}{b}^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/(a-c)^3*a/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/
(a-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/(a
-c)^3*a/b^3*(1/7*(b*x+c)^(7/2)-2/5*(b*x+c)^(5/2)*c+1/3*c^2*(b*x+c)^(3/2))-2/(a-c
)^3*c/b^3*(1/7*(b*x+c)^(7/2)-2/5*(b*x+c)^(5/2)*c+1/3*c^2*(b*x+c)^(3/2))+8/(a-c)^
3/b^3*(1/9*(b*x+a)^(9/2)-3/7*a*(b*x+a)^(7/2)+3/5*a^2*(b*x+a)^(5/2)-1/3*a^3*(b*x+
a)^(3/2))-8/(a-c)^3/b^3*(1/9*(b*x+c)^(9/2)-3/7*(b*x+c)^(7/2)*c+3/5*(b*x+c)^(5/2)
*c^2-1/3*c^3*(b*x+c)^(3/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2}}{{\left (\sqrt{b x + a} + \sqrt{b x + c}\right )}^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/(sqrt(b*x + a) + sqrt(b*x + c))^3,x, algorithm="maxima")

[Out]

integrate(x^2/(sqrt(b*x + a) + sqrt(b*x + c))^3, x)

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Fricas [A]  time = 0.27866, size = 281, normalized size = 0.75 \[ \frac{2 \,{\left ({\left (140 \, b^{4} x^{4} - 40 \, a^{4} + 72 \, a^{3} c + 5 \,{\left (13 \, a b^{3} + 27 \, b^{3} c\right )} x^{3} - 3 \,{\left (5 \, a^{2} b^{2} - 9 \, a b^{2} c\right )} x^{2} + 4 \,{\left (5 \, a^{3} b - 9 \, a^{2} b c\right )} x\right )} \sqrt{b x + a} -{\left (140 \, b^{4} x^{4} + 72 \, a c^{3} - 40 \, c^{4} + 5 \,{\left (27 \, a b^{3} + 13 \, b^{3} c\right )} x^{3} + 3 \,{\left (9 \, a b^{2} c - 5 \, b^{2} c^{2}\right )} x^{2} - 4 \,{\left (9 \, a b c^{2} - 5 \, b c^{3}\right )} x\right )} \sqrt{b x + c}\right )}}{315 \,{\left (a^{3} b^{3} - 3 \, a^{2} b^{3} c + 3 \, a b^{3} c^{2} - b^{3} c^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/(sqrt(b*x + a) + sqrt(b*x + c))^3,x, algorithm="fricas")

[Out]

2/315*((140*b^4*x^4 - 40*a^4 + 72*a^3*c + 5*(13*a*b^3 + 27*b^3*c)*x^3 - 3*(5*a^2
*b^2 - 9*a*b^2*c)*x^2 + 4*(5*a^3*b - 9*a^2*b*c)*x)*sqrt(b*x + a) - (140*b^4*x^4
+ 72*a*c^3 - 40*c^4 + 5*(27*a*b^3 + 13*b^3*c)*x^3 + 3*(9*a*b^2*c - 5*b^2*c^2)*x^
2 - 4*(9*a*b*c^2 - 5*b*c^3)*x)*sqrt(b*x + c))/(a^3*b^3 - 3*a^2*b^3*c + 3*a*b^3*c
^2 - b^3*c^3)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(x**2/(sqrt(a + b*x) + sqrt(b*x + c))**3, 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^2/(sqrt(b*x + a) + sqrt(b*x + c))^3,x, algorithm="giac")

[Out]

Timed out