3.663 \(\int \frac{(a+b x)^2}{x^{2/3}} \, dx\)

Optimal. Leaf size=34 \[ 3 a^2 \sqrt [3]{x}+\frac{3}{2} a b x^{4/3}+\frac{3}{7} b^2 x^{7/3} \]

[Out]

3*a^2*x^(1/3) + (3*a*b*x^(4/3))/2 + (3*b^2*x^(7/3))/7

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Rubi [A]  time = 0.0230266, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ 3 a^2 \sqrt [3]{x}+\frac{3}{2} a b x^{4/3}+\frac{3}{7} b^2 x^{7/3} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^2/x^(2/3),x]

[Out]

3*a^2*x^(1/3) + (3*a*b*x^(4/3))/2 + (3*b^2*x^(7/3))/7

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Rubi in Sympy [A]  time = 4.02108, size = 32, normalized size = 0.94 \[ 3 a^{2} \sqrt [3]{x} + \frac{3 a b x^{\frac{4}{3}}}{2} + \frac{3 b^{2} x^{\frac{7}{3}}}{7} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*a**2*x**(1/3) + 3*a*b*x**(4/3)/2 + 3*b**2*x**(7/3)/7

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Mathematica [A]  time = 0.00931375, size = 28, normalized size = 0.82 \[ \frac{3}{14} \sqrt [3]{x} \left (14 a^2+7 a b x+2 b^2 x^2\right ) \]

Antiderivative was successfully verified.

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

[Out]

(3*x^(1/3)*(14*a^2 + 7*a*b*x + 2*b^2*x^2))/14

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Maple [A]  time = 0.005, size = 25, normalized size = 0.7 \[{\frac{6\,{b}^{2}{x}^{2}+21\,abx+42\,{a}^{2}}{14}\sqrt [3]{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^2/x^(2/3),x)

[Out]

3/14*x^(1/3)*(2*b^2*x^2+7*a*b*x+14*a^2)

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Maxima [A]  time = 1.35263, size = 32, normalized size = 0.94 \[ \frac{3}{7} \, b^{2} x^{\frac{7}{3}} + \frac{3}{2} \, a b x^{\frac{4}{3}} + 3 \, a^{2} x^{\frac{1}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/7*b^2*x^(7/3) + 3/2*a*b*x^(4/3) + 3*a^2*x^(1/3)

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Fricas [A]  time = 0.202103, size = 32, normalized size = 0.94 \[ \frac{3}{14} \,{\left (2 \, b^{2} x^{2} + 7 \, a b x + 14 \, a^{2}\right )} x^{\frac{1}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/14*(2*b^2*x^2 + 7*a*b*x + 14*a^2)*x^(1/3)

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Sympy [A]  time = 6.65764, size = 1765, normalized size = 51.91 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((-27*a**(31/3)*(-1 + b*(a/b + x)/a)**(1/3)/(-14*a**8*b**(1/3) + 42*a**
7*b**(4/3)*(a/b + x) - 42*a**6*b**(7/3)*(a/b + x)**2 + 14*a**5*b**(10/3)*(a/b +
x)**3) + 27*a**(31/3)*exp(7*I*pi/3)/(-14*a**8*b**(1/3) + 42*a**7*b**(4/3)*(a/b +
x) - 42*a**6*b**(7/3)*(a/b + x)**2 + 14*a**5*b**(10/3)*(a/b + x)**3) + 72*a**(2
8/3)*b*(-1 + b*(a/b + x)/a)**(1/3)*(a/b + x)/(-14*a**8*b**(1/3) + 42*a**7*b**(4/
3)*(a/b + x) - 42*a**6*b**(7/3)*(a/b + x)**2 + 14*a**5*b**(10/3)*(a/b + x)**3) -
81*a**(28/3)*b*(a/b + x)*exp(7*I*pi/3)/(-14*a**8*b**(1/3) + 42*a**7*b**(4/3)*(a
/b + x) - 42*a**6*b**(7/3)*(a/b + x)**2 + 14*a**5*b**(10/3)*(a/b + x)**3) - 60*a
**(25/3)*b**2*(-1 + b*(a/b + x)/a)**(1/3)*(a/b + x)**2/(-14*a**8*b**(1/3) + 42*a
**7*b**(4/3)*(a/b + x) - 42*a**6*b**(7/3)*(a/b + x)**2 + 14*a**5*b**(10/3)*(a/b
+ x)**3) + 81*a**(25/3)*b**2*(a/b + x)**2*exp(7*I*pi/3)/(-14*a**8*b**(1/3) + 42*
a**7*b**(4/3)*(a/b + x) - 42*a**6*b**(7/3)*(a/b + x)**2 + 14*a**5*b**(10/3)*(a/b
+ x)**3) + 18*a**(22/3)*b**3*(-1 + b*(a/b + x)/a)**(1/3)*(a/b + x)**3/(-14*a**8
*b**(1/3) + 42*a**7*b**(4/3)*(a/b + x) - 42*a**6*b**(7/3)*(a/b + x)**2 + 14*a**5
*b**(10/3)*(a/b + x)**3) - 27*a**(22/3)*b**3*(a/b + x)**3*exp(7*I*pi/3)/(-14*a**
8*b**(1/3) + 42*a**7*b**(4/3)*(a/b + x) - 42*a**6*b**(7/3)*(a/b + x)**2 + 14*a**
5*b**(10/3)*(a/b + x)**3) - 9*a**(19/3)*b**4*(-1 + b*(a/b + x)/a)**(1/3)*(a/b +
x)**4/(-14*a**8*b**(1/3) + 42*a**7*b**(4/3)*(a/b + x) - 42*a**6*b**(7/3)*(a/b +
x)**2 + 14*a**5*b**(10/3)*(a/b + x)**3) + 6*a**(16/3)*b**5*(-1 + b*(a/b + x)/a)*
*(1/3)*(a/b + x)**5/(-14*a**8*b**(1/3) + 42*a**7*b**(4/3)*(a/b + x) - 42*a**6*b*
*(7/3)*(a/b + x)**2 + 14*a**5*b**(10/3)*(a/b + x)**3), Abs(b*(a/b + x)/a) > 1),
(-27*a**(31/3)*(1 - b*(a/b + x)/a)**(1/3)*exp(7*I*pi/3)/(-14*a**8*b**(1/3) + 42*
a**7*b**(4/3)*(a/b + x) - 42*a**6*b**(7/3)*(a/b + x)**2 + 14*a**5*b**(10/3)*(a/b
+ x)**3) + 27*a**(31/3)*exp(7*I*pi/3)/(-14*a**8*b**(1/3) + 42*a**7*b**(4/3)*(a/
b + x) - 42*a**6*b**(7/3)*(a/b + x)**2 + 14*a**5*b**(10/3)*(a/b + x)**3) + 72*a*
*(28/3)*b*(1 - b*(a/b + x)/a)**(1/3)*(a/b + x)*exp(7*I*pi/3)/(-14*a**8*b**(1/3)
+ 42*a**7*b**(4/3)*(a/b + x) - 42*a**6*b**(7/3)*(a/b + x)**2 + 14*a**5*b**(10/3)
*(a/b + x)**3) - 81*a**(28/3)*b*(a/b + x)*exp(7*I*pi/3)/(-14*a**8*b**(1/3) + 42*
a**7*b**(4/3)*(a/b + x) - 42*a**6*b**(7/3)*(a/b + x)**2 + 14*a**5*b**(10/3)*(a/b
+ x)**3) - 60*a**(25/3)*b**2*(1 - b*(a/b + x)/a)**(1/3)*(a/b + x)**2*exp(7*I*pi
/3)/(-14*a**8*b**(1/3) + 42*a**7*b**(4/3)*(a/b + x) - 42*a**6*b**(7/3)*(a/b + x)
**2 + 14*a**5*b**(10/3)*(a/b + x)**3) + 81*a**(25/3)*b**2*(a/b + x)**2*exp(7*I*p
i/3)/(-14*a**8*b**(1/3) + 42*a**7*b**(4/3)*(a/b + x) - 42*a**6*b**(7/3)*(a/b + x
)**2 + 14*a**5*b**(10/3)*(a/b + x)**3) + 18*a**(22/3)*b**3*(1 - b*(a/b + x)/a)**
(1/3)*(a/b + x)**3*exp(7*I*pi/3)/(-14*a**8*b**(1/3) + 42*a**7*b**(4/3)*(a/b + x)
- 42*a**6*b**(7/3)*(a/b + x)**2 + 14*a**5*b**(10/3)*(a/b + x)**3) - 27*a**(22/3
)*b**3*(a/b + x)**3*exp(7*I*pi/3)/(-14*a**8*b**(1/3) + 42*a**7*b**(4/3)*(a/b + x
) - 42*a**6*b**(7/3)*(a/b + x)**2 + 14*a**5*b**(10/3)*(a/b + x)**3) - 9*a**(19/3
)*b**4*(1 - b*(a/b + x)/a)**(1/3)*(a/b + x)**4*exp(7*I*pi/3)/(-14*a**8*b**(1/3)
+ 42*a**7*b**(4/3)*(a/b + x) - 42*a**6*b**(7/3)*(a/b + x)**2 + 14*a**5*b**(10/3)
*(a/b + x)**3) + 6*a**(16/3)*b**5*(1 - b*(a/b + x)/a)**(1/3)*(a/b + x)**5*exp(7*
I*pi/3)/(-14*a**8*b**(1/3) + 42*a**7*b**(4/3)*(a/b + x) - 42*a**6*b**(7/3)*(a/b
+ x)**2 + 14*a**5*b**(10/3)*(a/b + x)**3), True))

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GIAC/XCAS [A]  time = 0.204447, size = 32, normalized size = 0.94 \[ \frac{3}{7} \, b^{2} x^{\frac{7}{3}} + \frac{3}{2} \, a b x^{\frac{4}{3}} + 3 \, a^{2} x^{\frac{1}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/7*b^2*x^(7/3) + 3/2*a*b*x^(4/3) + 3*a^2*x^(1/3)