3.665 \(\int \frac{(a+b x)^2}{x^{5/3}} \, dx\)
Optimal. Leaf size=34 \[ -\frac{3 a^2}{2 x^{2/3}}+6 a b \sqrt [3]{x}+\frac{3}{4} b^2 x^{4/3} \]
[Out]
(-3*a^2)/(2*x^(2/3)) + 6*a*b*x^(1/3) + (3*b^2*x^(4/3))/4
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Rubi [A] time = 0.0071943, 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, Rules used =
{43} \[ -\frac{3 a^2}{2 x^{2/3}}+6 a b \sqrt [3]{x}+\frac{3}{4} b^2 x^{4/3} \]
Antiderivative was successfully verified.
[In]
Int[(a + b*x)^2/x^(5/3),x]
[Out]
(-3*a^2)/(2*x^(2/3)) + 6*a*b*x^(1/3) + (3*b^2*x^(4/3))/4
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}{x^{5/3}} \, dx &=\int \left (\frac{a^2}{x^{5/3}}+\frac{2 a b}{x^{2/3}}+b^2 \sqrt [3]{x}\right ) \, dx\\ &=-\frac{3 a^2}{2 x^{2/3}}+6 a b \sqrt [3]{x}+\frac{3}{4} b^2 x^{4/3}\\ \end{align*}
Mathematica [A] time = 0.0074894, size = 27, normalized size = 0.79 \[ \frac{3 \left (-2 a^2+8 a b x+b^2 x^2\right )}{4 x^{2/3}} \]
Antiderivative was successfully verified.
[In]
Integrate[(a + b*x)^2/x^(5/3),x]
[Out]
(3*(-2*a^2 + 8*a*b*x + b^2*x^2))/(4*x^(2/3))
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Maple [A] time = 0.004, size = 25, normalized size = 0.7 \begin{align*} -{\frac{-3\,{b}^{2}{x}^{2}-24\,abx+6\,{a}^{2}}{4}{x}^{-{\frac{2}{3}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((b*x+a)^2/x^(5/3),x)
[Out]
-3/4*(-b^2*x^2-8*a*b*x+2*a^2)/x^(2/3)
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Maxima [A] time = 1.5608, size = 32, normalized size = 0.94 \begin{align*} \frac{3}{4} \, b^{2} x^{\frac{4}{3}} + 6 \, a b x^{\frac{1}{3}} - \frac{3 \, a^{2}}{2 \, x^{\frac{2}{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a)^2/x^(5/3),x, algorithm="maxima")
[Out]
3/4*b^2*x^(4/3) + 6*a*b*x^(1/3) - 3/2*a^2/x^(2/3)
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Fricas [A] time = 1.45979, size = 55, normalized size = 1.62 \begin{align*} \frac{3 \,{\left (b^{2} x^{2} + 8 \, a b x - 2 \, a^{2}\right )}}{4 \, x^{\frac{2}{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a)^2/x^(5/3),x, algorithm="fricas")
[Out]
3/4*(b^2*x^2 + 8*a*b*x - 2*a^2)/x^(2/3)
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Sympy [C] time = 2.18521, size = 1958, normalized size = 57.59 \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**(5/3),x)
[Out]
Piecewise((-27*a**(28/3)*b**(2/3)*(-1 + b*(a/b + x)/a)**(1/3)*exp(2*I*pi/3)/(-4*a**8*exp(2*I*pi/3) + 12*a**7*b
*(a/b + x)*exp(2*I*pi/3) - 12*a**6*b**2*(a/b + x)**2*exp(2*I*pi/3) + 4*a**5*b**3*(a/b + x)**3*exp(2*I*pi/3)) -
27*a**(28/3)*b**(2/3)/(-4*a**8*exp(2*I*pi/3) + 12*a**7*b*(a/b + x)*exp(2*I*pi/3) - 12*a**6*b**2*(a/b + x)**2*
exp(2*I*pi/3) + 4*a**5*b**3*(a/b + x)**3*exp(2*I*pi/3)) + 72*a**(25/3)*b**(5/3)*(-1 + b*(a/b + x)/a)**(1/3)*(a
/b + x)*exp(2*I*pi/3)/(-4*a**8*exp(2*I*pi/3) + 12*a**7*b*(a/b + x)*exp(2*I*pi/3) - 12*a**6*b**2*(a/b + x)**2*e
xp(2*I*pi/3) + 4*a**5*b**3*(a/b + x)**3*exp(2*I*pi/3)) + 81*a**(25/3)*b**(5/3)*(a/b + x)/(-4*a**8*exp(2*I*pi/3
) + 12*a**7*b*(a/b + x)*exp(2*I*pi/3) - 12*a**6*b**2*(a/b + x)**2*exp(2*I*pi/3) + 4*a**5*b**3*(a/b + x)**3*exp
(2*I*pi/3)) - 60*a**(22/3)*b**(8/3)*(-1 + b*(a/b + x)/a)**(1/3)*(a/b + x)**2*exp(2*I*pi/3)/(-4*a**8*exp(2*I*pi
/3) + 12*a**7*b*(a/b + x)*exp(2*I*pi/3) - 12*a**6*b**2*(a/b + x)**2*exp(2*I*pi/3) + 4*a**5*b**3*(a/b + x)**3*e
xp(2*I*pi/3)) - 81*a**(22/3)*b**(8/3)*(a/b + x)**2/(-4*a**8*exp(2*I*pi/3) + 12*a**7*b*(a/b + x)*exp(2*I*pi/3)
- 12*a**6*b**2*(a/b + x)**2*exp(2*I*pi/3) + 4*a**5*b**3*(a/b + x)**3*exp(2*I*pi/3)) + 12*a**(19/3)*b**(11/3)*(
-1 + b*(a/b + x)/a)**(1/3)*(a/b + x)**3*exp(2*I*pi/3)/(-4*a**8*exp(2*I*pi/3) + 12*a**7*b*(a/b + x)*exp(2*I*pi/
3) - 12*a**6*b**2*(a/b + x)**2*exp(2*I*pi/3) + 4*a**5*b**3*(a/b + x)**3*exp(2*I*pi/3)) + 27*a**(19/3)*b**(11/3
)*(a/b + x)**3/(-4*a**8*exp(2*I*pi/3) + 12*a**7*b*(a/b + x)*exp(2*I*pi/3) - 12*a**6*b**2*(a/b + x)**2*exp(2*I*
pi/3) + 4*a**5*b**3*(a/b + x)**3*exp(2*I*pi/3)) + 3*a**(16/3)*b**(14/3)*(-1 + b*(a/b + x)/a)**(1/3)*(a/b + x)*
*4*exp(2*I*pi/3)/(-4*a**8*exp(2*I*pi/3) + 12*a**7*b*(a/b + x)*exp(2*I*pi/3) - 12*a**6*b**2*(a/b + x)**2*exp(2*
I*pi/3) + 4*a**5*b**3*(a/b + x)**3*exp(2*I*pi/3)), Abs(b*(a/b + x))/Abs(a) > 1), (27*a**(28/3)*b**(2/3)*(1 - b
*(a/b + x)/a)**(1/3)/(-4*a**8*exp(2*I*pi/3) + 12*a**7*b*(a/b + x)*exp(2*I*pi/3) - 12*a**6*b**2*(a/b + x)**2*ex
p(2*I*pi/3) + 4*a**5*b**3*(a/b + x)**3*exp(2*I*pi/3)) - 27*a**(28/3)*b**(2/3)/(-4*a**8*exp(2*I*pi/3) + 12*a**7
*b*(a/b + x)*exp(2*I*pi/3) - 12*a**6*b**2*(a/b + x)**2*exp(2*I*pi/3) + 4*a**5*b**3*(a/b + x)**3*exp(2*I*pi/3))
- 72*a**(25/3)*b**(5/3)*(1 - b*(a/b + x)/a)**(1/3)*(a/b + x)/(-4*a**8*exp(2*I*pi/3) + 12*a**7*b*(a/b + x)*exp
(2*I*pi/3) - 12*a**6*b**2*(a/b + x)**2*exp(2*I*pi/3) + 4*a**5*b**3*(a/b + x)**3*exp(2*I*pi/3)) + 81*a**(25/3)*
b**(5/3)*(a/b + x)/(-4*a**8*exp(2*I*pi/3) + 12*a**7*b*(a/b + x)*exp(2*I*pi/3) - 12*a**6*b**2*(a/b + x)**2*exp(
2*I*pi/3) + 4*a**5*b**3*(a/b + x)**3*exp(2*I*pi/3)) + 60*a**(22/3)*b**(8/3)*(1 - b*(a/b + x)/a)**(1/3)*(a/b +
x)**2/(-4*a**8*exp(2*I*pi/3) + 12*a**7*b*(a/b + x)*exp(2*I*pi/3) - 12*a**6*b**2*(a/b + x)**2*exp(2*I*pi/3) + 4
*a**5*b**3*(a/b + x)**3*exp(2*I*pi/3)) - 81*a**(22/3)*b**(8/3)*(a/b + x)**2/(-4*a**8*exp(2*I*pi/3) + 12*a**7*b
*(a/b + x)*exp(2*I*pi/3) - 12*a**6*b**2*(a/b + x)**2*exp(2*I*pi/3) + 4*a**5*b**3*(a/b + x)**3*exp(2*I*pi/3)) -
12*a**(19/3)*b**(11/3)*(1 - b*(a/b + x)/a)**(1/3)*(a/b + x)**3/(-4*a**8*exp(2*I*pi/3) + 12*a**7*b*(a/b + x)*e
xp(2*I*pi/3) - 12*a**6*b**2*(a/b + x)**2*exp(2*I*pi/3) + 4*a**5*b**3*(a/b + x)**3*exp(2*I*pi/3)) + 27*a**(19/3
)*b**(11/3)*(a/b + x)**3/(-4*a**8*exp(2*I*pi/3) + 12*a**7*b*(a/b + x)*exp(2*I*pi/3) - 12*a**6*b**2*(a/b + x)**
2*exp(2*I*pi/3) + 4*a**5*b**3*(a/b + x)**3*exp(2*I*pi/3)) - 3*a**(16/3)*b**(14/3)*(1 - b*(a/b + x)/a)**(1/3)*(
a/b + x)**4/(-4*a**8*exp(2*I*pi/3) + 12*a**7*b*(a/b + x)*exp(2*I*pi/3) - 12*a**6*b**2*(a/b + x)**2*exp(2*I*pi/
3) + 4*a**5*b**3*(a/b + x)**3*exp(2*I*pi/3)), True))
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Giac [A] time = 1.05725, size = 32, normalized size = 0.94 \begin{align*} \frac{3}{4} \, b^{2} x^{\frac{4}{3}} + 6 \, a b x^{\frac{1}{3}} - \frac{3 \, a^{2}}{2 \, x^{\frac{2}{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a)^2/x^(5/3),x, algorithm="giac")
[Out]
3/4*b^2*x^(4/3) + 6*a*b*x^(1/3) - 3/2*a^2/x^(2/3)