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

Optimal. Leaf size=147 \[ \frac{2 a^2 (a+b x)^{3/2}}{3 b^3 (b-c)}-\frac{2 a^2 (a+c x)^{3/2}}{3 c^3 (b-c)}+\frac{2 (a+b x)^{7/2}}{7 b^3 (b-c)}-\frac{4 a (a+b x)^{5/2}}{5 b^3 (b-c)}-\frac{2 (a+c x)^{7/2}}{7 c^3 (b-c)}+\frac{4 a (a+c x)^{5/2}}{5 c^3 (b-c)} \]

[Out]

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

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Rubi [A]  time = 0.233471, antiderivative size = 147, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08 \[ \frac{2 a^2 (a+b x)^{3/2}}{3 b^3 (b-c)}-\frac{2 a^2 (a+c x)^{3/2}}{3 c^3 (b-c)}+\frac{2 (a+b x)^{7/2}}{7 b^3 (b-c)}-\frac{4 a (a+b x)^{5/2}}{5 b^3 (b-c)}-\frac{2 (a+c x)^{7/2}}{7 c^3 (b-c)}+\frac{4 a (a+c x)^{5/2}}{5 c^3 (b-c)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 27.4787, size = 121, normalized size = 0.82 \[ - \frac{2 a^{2} \left (a + c x\right )^{\frac{3}{2}}}{3 c^{3} \left (b - c\right )} + \frac{2 a^{2} \left (a + b x\right )^{\frac{3}{2}}}{3 b^{3} \left (b - c\right )} + \frac{4 a \left (a + c x\right )^{\frac{5}{2}}}{5 c^{3} \left (b - c\right )} - \frac{4 a \left (a + b x\right )^{\frac{5}{2}}}{5 b^{3} \left (b - c\right )} - \frac{2 \left (a + c x\right )^{\frac{7}{2}}}{7 c^{3} \left (b - c\right )} + \frac{2 \left (a + b x\right )^{\frac{7}{2}}}{7 b^{3} \left (b - c\right )} \]

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)),x)

[Out]

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

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Mathematica [A]  time = 0.205245, size = 157, normalized size = 1.07 \[ \sqrt{a+b x} \left (\frac{16 a^3}{105 b^3 (b-c)}-\frac{8 a^2 x}{105 b^2 (b-c)}+\frac{2 a x^2}{35 b (b-c)}+\frac{2 x^3}{7 (b-c)}\right )+\sqrt{a+c x} \left (-\frac{16 a^3}{105 c^3 (b-c)}+\frac{8 a^2 x}{105 c^2 (b-c)}-\frac{2 a x^2}{35 c (b-c)}-\frac{2 x^3}{7 (b-c)}\right ) \]

Antiderivative was successfully verified.

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

[Out]

Sqrt[a + c*x]*((-16*a^3)/(105*(b - c)*c^3) + (8*a^2*x)/(105*(b - c)*c^2) - (2*a*
x^2)/(35*(b - c)*c) - (2*x^3)/(7*(b - c))) + Sqrt[a + b*x]*((16*a^3)/(105*b^3*(b
- c)) - (8*a^2*x)/(105*b^2*(b - c)) + (2*a*x^2)/(35*b*(b - c)) + (2*x^3)/(7*(b
- c)))

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Maple [A]  time = 0.005, size = 90, normalized size = 0.6 \[ 2\,{\frac{1/7\, \left ( bx+a \right ) ^{7/2}-2/5\, \left ( bx+a \right ) ^{5/2}a+1/3\,{a}^{2} \left ( bx+a \right ) ^{3/2}}{ \left ( b-c \right ){b}^{3}}}-2\,{\frac{1/7\, \left ( cx+a \right ) ^{7/2}-2/5\, \left ( cx+a \right ) ^{5/2}a+1/3\,{a}^{2} \left ( cx+a \right ) ^{3/2}}{ \left ( b-c \right ){c}^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/(b-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))-2/(b-c
)/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))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.288328, size = 165, normalized size = 1.12 \[ \frac{2 \,{\left ({\left (15 \, b^{3} c^{3} x^{3} + 3 \, a b^{2} c^{3} x^{2} - 4 \, a^{2} b c^{3} x + 8 \, a^{3} c^{3}\right )} \sqrt{b x + a} -{\left (15 \, b^{3} c^{3} x^{3} + 3 \, a b^{3} c^{2} x^{2} - 4 \, a^{2} b^{3} c x + 8 \, a^{3} b^{3}\right )} \sqrt{c x + a}\right )}}{105 \,{\left (b^{4} c^{3} - b^{3} c^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/105*((15*b^3*c^3*x^3 + 3*a*b^2*c^3*x^2 - 4*a^2*b*c^3*x + 8*a^3*c^3)*sqrt(b*x +
a) - (15*b^3*c^3*x^3 + 3*a*b^3*c^2*x^2 - 4*a^2*b^3*c*x + 8*a^3*b^3)*sqrt(c*x +
a))/(b^4*c^3 - b^3*c^4)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out