3.2830 \(\int \left (c (a+b x)^{3/2}\right )^{2/3} \, dx\)

Optimal. Leaf size=27 \[ \frac{(a+b x) \left (c (a+b x)^{3/2}\right )^{2/3}}{2 b} \]

[Out]

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

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Rubi [A]  time = 0.0252825, antiderivative size = 27, 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 \[ \frac{(a+b x) \left (c (a+b x)^{3/2}\right )^{2/3}}{2 b} \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0269301, size = 34, normalized size = 1.26 \[ \frac{x (2 a+b x) \left (c (a+b x)^{3/2}\right )^{2/3}}{2 (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.003, size = 29, normalized size = 1.1 \[{\frac{x \left ( bx+2\,a \right ) }{2\,bx+2\,a} \left ( c \left ( bx+a \right ) ^{{\frac{3}{2}}} \right ) ^{{\frac{2}{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*(b*x+a)^(3/2))^(2/3),x)

[Out]

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

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Maxima [A]  time = 1.33921, size = 28, normalized size = 1.04 \[ \frac{\left ({\left (b x + a\right )}^{\frac{3}{2}} c\right )^{\frac{2}{3}}{\left (b x + a\right )}}{2 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.321922, size = 50, normalized size = 1.85 \[ \frac{{\left (b x^{2} + 2 \, a x\right )} \left ({\left (b c x + a c\right )} \sqrt{b x + a}\right )^{\frac{2}{3}}}{2 \,{\left (b x + a\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.220161, size = 20, normalized size = 0.74 \[ \frac{{\left (b x + a\right )}^{2} c^{\frac{2}{3}}}{2 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(b*x + a)^2*c^(2/3)/b