3.2831 \(\int \left (c (a+b x)^{2/3}\right )^{3/2} \, dx\)

Optimal. Leaf size=32 \[ \frac{c (a+b x)^{5/3} \sqrt{c (a+b x)^{2/3}}}{2 b} \]

[Out]

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

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Rubi [A]  time = 0.0262344, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{c (a+b x)^{5/3} \sqrt{c (a+b x)^{2/3}}}{2 b} \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

c*sqrt(c*(a + b*x)**(2/3))*Integral(x, (x, a + b*x))/(b*(a + b*x)**(1/3))

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Mathematica [A]  time = 0.0268066, size = 34, normalized size = 1.06 \[ \frac{x (2 a+b x) \left (c (a+b x)^{2/3}\right )^{3/2}}{2 (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.006, size = 29, normalized size = 0.9 \[{\frac{x \left ( bx+2\,a \right ) }{2\,bx+2\,a} \left ( c \left ( bx+a \right ) ^{{\frac{2}{3}}} \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*(b*x+a)^(2/3))^(3/2),x)

[Out]

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

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Maxima [A]  time = 1.33913, size = 20, normalized size = 0.62 \[ \frac{1}{2} \,{\left (b x^{2} + 2 \, a x\right )} c^{\frac{3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(b*x^2 + 2*a*x)*c^(3/2)

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Fricas [A]  time = 0.210873, size = 23, normalized size = 0.72 \[ \frac{1}{2} \,{\left (b c x^{2} + 2 \, a c x\right )} \sqrt{c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(b*c*x^2 + 2*a*c*x)*sqrt(c)

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Sympy [A]  time = 20.1222, size = 65, normalized size = 2.03 \[ \begin{cases} \frac{2 a^{2} c^{\frac{3}{2}} x}{2 a + 2 b x} + \frac{3 a b c^{\frac{3}{2}} x^{2}}{2 a + 2 b x} + \frac{b^{2} c^{\frac{3}{2}} x^{3}}{2 a + 2 b x} & \text{for}\: a \neq 0 \vee b \neq 0 \\0 & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((2*a**2*c**(3/2)*x/(2*a + 2*b*x) + 3*a*b*c**(3/2)*x**2/(2*a + 2*b*x) +
b**2*c**(3/2)*x**3/(2*a + 2*b*x), Ne(a, 0) | Ne(b, 0)), (0, True))

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GIAC/XCAS [A]  time = 0.211809, size = 20, normalized size = 0.62 \[ \frac{1}{2} \,{\left (b x^{2} + 2 \, a x\right )} c^{\frac{3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(b*x^2 + 2*a*x)*c^(3/2)