3.662 \(\int \frac{(a+b x)^2}{\sqrt [3]{x}} \, dx\)
Optimal. Leaf size=36 \[ \frac{3}{2} a^2 x^{2/3}+\frac{6}{5} a b x^{5/3}+\frac{3}{8} b^2 x^{8/3} \]
[Out]
(3*a^2*x^(2/3))/2 + (6*a*b*x^(5/3))/5 + (3*b^2*x^(8/3))/8
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Rubi [A] time = 0.0076617, antiderivative size = 36, 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, Rules used =
{43} \[ \frac{3}{2} a^2 x^{2/3}+\frac{6}{5} a b x^{5/3}+\frac{3}{8} b^2 x^{8/3} \]
Antiderivative was successfully verified.
[In]
Int[(a + b*x)^2/x^(1/3),x]
[Out]
(3*a^2*x^(2/3))/2 + (6*a*b*x^(5/3))/5 + (3*b^2*x^(8/3))/8
Rule 43
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])
Rubi steps
\begin{align*} \int \frac{(a+b x)^2}{\sqrt [3]{x}} \, dx &=\int \left (\frac{a^2}{\sqrt [3]{x}}+2 a b x^{2/3}+b^2 x^{5/3}\right ) \, dx\\ &=\frac{3}{2} a^2 x^{2/3}+\frac{6}{5} a b x^{5/3}+\frac{3}{8} b^2 x^{8/3}\\ \end{align*}
Mathematica [A] time = 0.0073862, size = 28, normalized size = 0.78 \[ \frac{3}{40} x^{2/3} \left (20 a^2+16 a b x+5 b^2 x^2\right ) \]
Antiderivative was successfully verified.
[In]
Integrate[(a + b*x)^2/x^(1/3),x]
[Out]
(3*x^(2/3)*(20*a^2 + 16*a*b*x + 5*b^2*x^2))/40
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Maple [A] time = 0.003, size = 25, normalized size = 0.7 \begin{align*}{\frac{15\,{b}^{2}{x}^{2}+48\,abx+60\,{a}^{2}}{40}{x}^{{\frac{2}{3}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((b*x+a)^2/x^(1/3),x)
[Out]
3/40*x^(2/3)*(5*b^2*x^2+16*a*b*x+20*a^2)
________________________________________________________________________________________
Maxima [A] time = 1.02346, size = 32, normalized size = 0.89 \begin{align*} \frac{3}{8} \, b^{2} x^{\frac{8}{3}} + \frac{6}{5} \, a b x^{\frac{5}{3}} + \frac{3}{2} \, a^{2} x^{\frac{2}{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a)^2/x^(1/3),x, algorithm="maxima")
[Out]
3/8*b^2*x^(8/3) + 6/5*a*b*x^(5/3) + 3/2*a^2*x^(2/3)
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Fricas [A] time = 1.50496, size = 62, normalized size = 1.72 \begin{align*} \frac{3}{40} \,{\left (5 \, b^{2} x^{2} + 16 \, a b x + 20 \, a^{2}\right )} x^{\frac{2}{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a)^2/x^(1/3),x, algorithm="fricas")
[Out]
3/40*(5*b^2*x^2 + 16*a*b*x + 20*a^2)*x^(2/3)
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Sympy [C] time = 2.16043, size = 1766, normalized size = 49.06 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a)**2/x**(1/3),x)
[Out]
Piecewise((-27*a**(32/3)*(-1 + b*(a/b + x)/a)**(2/3)/(-40*a**8*b**(2/3) + 120*a**7*b**(5/3)*(a/b + x) - 120*a*
*6*b**(8/3)*(a/b + x)**2 + 40*a**5*b**(11/3)*(a/b + x)**3) + 27*a**(32/3)*exp(2*I*pi/3)/(-40*a**8*b**(2/3) + 1
20*a**7*b**(5/3)*(a/b + x) - 120*a**6*b**(8/3)*(a/b + x)**2 + 40*a**5*b**(11/3)*(a/b + x)**3) + 63*a**(29/3)*b
*(-1 + b*(a/b + x)/a)**(2/3)*(a/b + x)/(-40*a**8*b**(2/3) + 120*a**7*b**(5/3)*(a/b + x) - 120*a**6*b**(8/3)*(a
/b + x)**2 + 40*a**5*b**(11/3)*(a/b + x)**3) - 81*a**(29/3)*b*(a/b + x)*exp(2*I*pi/3)/(-40*a**8*b**(2/3) + 120
*a**7*b**(5/3)*(a/b + x) - 120*a**6*b**(8/3)*(a/b + x)**2 + 40*a**5*b**(11/3)*(a/b + x)**3) - 42*a**(26/3)*b**
2*(-1 + b*(a/b + x)/a)**(2/3)*(a/b + x)**2/(-40*a**8*b**(2/3) + 120*a**7*b**(5/3)*(a/b + x) - 120*a**6*b**(8/3
)*(a/b + x)**2 + 40*a**5*b**(11/3)*(a/b + x)**3) + 81*a**(26/3)*b**2*(a/b + x)**2*exp(2*I*pi/3)/(-40*a**8*b**(
2/3) + 120*a**7*b**(5/3)*(a/b + x) - 120*a**6*b**(8/3)*(a/b + x)**2 + 40*a**5*b**(11/3)*(a/b + x)**3) + 18*a**
(23/3)*b**3*(-1 + b*(a/b + x)/a)**(2/3)*(a/b + x)**3/(-40*a**8*b**(2/3) + 120*a**7*b**(5/3)*(a/b + x) - 120*a*
*6*b**(8/3)*(a/b + x)**2 + 40*a**5*b**(11/3)*(a/b + x)**3) - 27*a**(23/3)*b**3*(a/b + x)**3*exp(2*I*pi/3)/(-40
*a**8*b**(2/3) + 120*a**7*b**(5/3)*(a/b + x) - 120*a**6*b**(8/3)*(a/b + x)**2 + 40*a**5*b**(11/3)*(a/b + x)**3
) - 27*a**(20/3)*b**4*(-1 + b*(a/b + x)/a)**(2/3)*(a/b + x)**4/(-40*a**8*b**(2/3) + 120*a**7*b**(5/3)*(a/b + x
) - 120*a**6*b**(8/3)*(a/b + x)**2 + 40*a**5*b**(11/3)*(a/b + x)**3) + 15*a**(17/3)*b**5*(-1 + b*(a/b + x)/a)*
*(2/3)*(a/b + x)**5/(-40*a**8*b**(2/3) + 120*a**7*b**(5/3)*(a/b + x) - 120*a**6*b**(8/3)*(a/b + x)**2 + 40*a**
5*b**(11/3)*(a/b + x)**3), Abs(b*(a/b + x))/Abs(a) > 1), (-27*a**(32/3)*(1 - b*(a/b + x)/a)**(2/3)*exp(2*I*pi/
3)/(-40*a**8*b**(2/3) + 120*a**7*b**(5/3)*(a/b + x) - 120*a**6*b**(8/3)*(a/b + x)**2 + 40*a**5*b**(11/3)*(a/b
+ x)**3) + 27*a**(32/3)*exp(2*I*pi/3)/(-40*a**8*b**(2/3) + 120*a**7*b**(5/3)*(a/b + x) - 120*a**6*b**(8/3)*(a/
b + x)**2 + 40*a**5*b**(11/3)*(a/b + x)**3) + 63*a**(29/3)*b*(1 - b*(a/b + x)/a)**(2/3)*(a/b + x)*exp(2*I*pi/3
)/(-40*a**8*b**(2/3) + 120*a**7*b**(5/3)*(a/b + x) - 120*a**6*b**(8/3)*(a/b + x)**2 + 40*a**5*b**(11/3)*(a/b +
x)**3) - 81*a**(29/3)*b*(a/b + x)*exp(2*I*pi/3)/(-40*a**8*b**(2/3) + 120*a**7*b**(5/3)*(a/b + x) - 120*a**6*b
**(8/3)*(a/b + x)**2 + 40*a**5*b**(11/3)*(a/b + x)**3) - 42*a**(26/3)*b**2*(1 - b*(a/b + x)/a)**(2/3)*(a/b + x
)**2*exp(2*I*pi/3)/(-40*a**8*b**(2/3) + 120*a**7*b**(5/3)*(a/b + x) - 120*a**6*b**(8/3)*(a/b + x)**2 + 40*a**5
*b**(11/3)*(a/b + x)**3) + 81*a**(26/3)*b**2*(a/b + x)**2*exp(2*I*pi/3)/(-40*a**8*b**(2/3) + 120*a**7*b**(5/3)
*(a/b + x) - 120*a**6*b**(8/3)*(a/b + x)**2 + 40*a**5*b**(11/3)*(a/b + x)**3) + 18*a**(23/3)*b**3*(1 - b*(a/b
+ x)/a)**(2/3)*(a/b + x)**3*exp(2*I*pi/3)/(-40*a**8*b**(2/3) + 120*a**7*b**(5/3)*(a/b + x) - 120*a**6*b**(8/3)
*(a/b + x)**2 + 40*a**5*b**(11/3)*(a/b + x)**3) - 27*a**(23/3)*b**3*(a/b + x)**3*exp(2*I*pi/3)/(-40*a**8*b**(2
/3) + 120*a**7*b**(5/3)*(a/b + x) - 120*a**6*b**(8/3)*(a/b + x)**2 + 40*a**5*b**(11/3)*(a/b + x)**3) - 27*a**(
20/3)*b**4*(1 - b*(a/b + x)/a)**(2/3)*(a/b + x)**4*exp(2*I*pi/3)/(-40*a**8*b**(2/3) + 120*a**7*b**(5/3)*(a/b +
x) - 120*a**6*b**(8/3)*(a/b + x)**2 + 40*a**5*b**(11/3)*(a/b + x)**3) + 15*a**(17/3)*b**5*(1 - b*(a/b + x)/a)
**(2/3)*(a/b + x)**5*exp(2*I*pi/3)/(-40*a**8*b**(2/3) + 120*a**7*b**(5/3)*(a/b + x) - 120*a**6*b**(8/3)*(a/b +
x)**2 + 40*a**5*b**(11/3)*(a/b + x)**3), True))
________________________________________________________________________________________
Giac [A] time = 1.0473, size = 32, normalized size = 0.89 \begin{align*} \frac{3}{8} \, b^{2} x^{\frac{8}{3}} + \frac{6}{5} \, a b x^{\frac{5}{3}} + \frac{3}{2} \, a^{2} x^{\frac{2}{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a)^2/x^(1/3),x, algorithm="giac")
[Out]
3/8*b^2*x^(8/3) + 6/5*a*b*x^(5/3) + 3/2*a^2*x^(2/3)