3.2808 \(\int \sqrt{c (a+b x)^3} \, dx\)

Optimal. Leaf size=25 \[ \frac{2 (a+b x) \sqrt{c (a+b x)^3}}{5 b} \]

[Out]

(2*(a + b*x)*Sqrt[c*(a + b*x)^3])/(5*b)

_______________________________________________________________________________________

Rubi [A]  time = 0.029352, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \frac{2 (a+b x) \sqrt{c (a+b x)^3}}{5 b} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[c*(a + b*x)^3],x]

[Out]

(2*(a + b*x)*Sqrt[c*(a + b*x)^3])/(5*b)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 9.29164, size = 51, normalized size = 2.04 \[ \frac{2 \left (3 a + 3 b x\right ) \sqrt{a^{3} c + 3 a^{2} b c x + 3 a b^{2} c x^{2} + b^{3} c x^{3}}}{15 b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*(3*a + 3*b*x)*sqrt(a**3*c + 3*a**2*b*c*x + 3*a*b**2*c*x**2 + b**3*c*x**3)/(15*
b)

_______________________________________________________________________________________

Mathematica [A]  time = 0.0114887, size = 25, normalized size = 1. \[ \frac{2 (a+b x) \sqrt{c (a+b x)^3}}{5 b} \]

Antiderivative was successfully verified.

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

[Out]

(2*(a + b*x)*Sqrt[c*(a + b*x)^3])/(5*b)

_______________________________________________________________________________________

Maple [A]  time = 0.004, size = 22, normalized size = 0.9 \[{\frac{2\,bx+2\,a}{5\,b}\sqrt{c \left ( bx+a \right ) ^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*(b*x+a)^3)^(1/2),x)

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 1.39501, size = 32, normalized size = 1.28 \[ \frac{2 \,{\left (b \sqrt{c} x + a \sqrt{c}\right )}{\left (b x + a\right )}^{\frac{3}{2}}}{5 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.214091, size = 59, normalized size = 2.36 \[ \frac{2 \, \sqrt{b^{3} c x^{3} + 3 \, a b^{2} c x^{2} + 3 \, a^{2} b c x + a^{3} c}{\left (b x + a\right )}}{5 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(sqrt(c*(a + b*x)**3), x)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.219801, size = 89, normalized size = 3.56 \[ \frac{2 \,{\left (5 \,{\left (b c x + a c\right )}^{\frac{3}{2}} a{\rm sign}\left (b x + a\right ) - \frac{{\left (5 \,{\left (b c x + a c\right )}^{\frac{3}{2}} a c - 3 \,{\left (b c x + a c\right )}^{\frac{5}{2}}\right )}{\rm sign}\left (b x + a\right )}{c}\right )}}{15 \, b c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/15*(5*(b*c*x + a*c)^(3/2)*a*sign(b*x + a) - (5*(b*c*x + a*c)^(3/2)*a*c - 3*(b*
c*x + a*c)^(5/2))*sign(b*x + a)/c)/(b*c)