3.378 \(\int x^3 (a+b x)^{2/3} \, dx\)
Optimal. Leaf size=72 \[ \frac{9 a^2 (a+b x)^{8/3}}{8 b^4}-\frac{3 a^3 (a+b x)^{5/3}}{5 b^4}+\frac{3 (a+b x)^{14/3}}{14 b^4}-\frac{9 a (a+b x)^{11/3}}{11 b^4} \]
[Out]
(-3*a^3*(a + b*x)^(5/3))/(5*b^4) + (9*a^2*(a + b*x)^(8/3))/(8*b^4) - (9*a*(a + b*x)^(11/3))/(11*b^4) + (3*(a +
b*x)^(14/3))/(14*b^4)
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Rubi [A] time = 0.018402, antiderivative size = 72, 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} \[ \frac{9 a^2 (a+b x)^{8/3}}{8 b^4}-\frac{3 a^3 (a+b x)^{5/3}}{5 b^4}+\frac{3 (a+b x)^{14/3}}{14 b^4}-\frac{9 a (a+b x)^{11/3}}{11 b^4} \]
Antiderivative was successfully verified.
[In]
Int[x^3*(a + b*x)^(2/3),x]
[Out]
(-3*a^3*(a + b*x)^(5/3))/(5*b^4) + (9*a^2*(a + b*x)^(8/3))/(8*b^4) - (9*a*(a + b*x)^(11/3))/(11*b^4) + (3*(a +
b*x)^(14/3))/(14*b^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 x^3 (a+b x)^{2/3} \, dx &=\int \left (-\frac{a^3 (a+b x)^{2/3}}{b^3}+\frac{3 a^2 (a+b x)^{5/3}}{b^3}-\frac{3 a (a+b x)^{8/3}}{b^3}+\frac{(a+b x)^{11/3}}{b^3}\right ) \, dx\\ &=-\frac{3 a^3 (a+b x)^{5/3}}{5 b^4}+\frac{9 a^2 (a+b x)^{8/3}}{8 b^4}-\frac{9 a (a+b x)^{11/3}}{11 b^4}+\frac{3 (a+b x)^{14/3}}{14 b^4}\\ \end{align*}
Mathematica [A] time = 0.0534829, size = 46, normalized size = 0.64 \[ \frac{3 (a+b x)^{5/3} \left (135 a^2 b x-81 a^3-180 a b^2 x^2+220 b^3 x^3\right )}{3080 b^4} \]
Antiderivative was successfully verified.
[In]
Integrate[x^3*(a + b*x)^(2/3),x]
[Out]
(3*(a + b*x)^(5/3)*(-81*a^3 + 135*a^2*b*x - 180*a*b^2*x^2 + 220*b^3*x^3))/(3080*b^4)
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Maple [A] time = 0.005, size = 43, normalized size = 0.6 \begin{align*} -{\frac{-660\,{b}^{3}{x}^{3}+540\,a{b}^{2}{x}^{2}-405\,{a}^{2}bx+243\,{a}^{3}}{3080\,{b}^{4}} \left ( bx+a \right ) ^{{\frac{5}{3}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x^3*(b*x+a)^(2/3),x)
[Out]
-3/3080*(b*x+a)^(5/3)*(-220*b^3*x^3+180*a*b^2*x^2-135*a^2*b*x+81*a^3)/b^4
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Maxima [A] time = 1.09555, size = 76, normalized size = 1.06 \begin{align*} \frac{3 \,{\left (b x + a\right )}^{\frac{14}{3}}}{14 \, b^{4}} - \frac{9 \,{\left (b x + a\right )}^{\frac{11}{3}} a}{11 \, b^{4}} + \frac{9 \,{\left (b x + a\right )}^{\frac{8}{3}} a^{2}}{8 \, b^{4}} - \frac{3 \,{\left (b x + a\right )}^{\frac{5}{3}} a^{3}}{5 \, b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^3*(b*x+a)^(2/3),x, algorithm="maxima")
[Out]
3/14*(b*x + a)^(14/3)/b^4 - 9/11*(b*x + a)^(11/3)*a/b^4 + 9/8*(b*x + a)^(8/3)*a^2/b^4 - 3/5*(b*x + a)^(5/3)*a^
3/b^4
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Fricas [A] time = 1.85305, size = 130, normalized size = 1.81 \begin{align*} \frac{3 \,{\left (220 \, b^{4} x^{4} + 40 \, a b^{3} x^{3} - 45 \, a^{2} b^{2} x^{2} + 54 \, a^{3} b x - 81 \, a^{4}\right )}{\left (b x + a\right )}^{\frac{2}{3}}}{3080 \, b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^3*(b*x+a)^(2/3),x, algorithm="fricas")
[Out]
3/3080*(220*b^4*x^4 + 40*a*b^3*x^3 - 45*a^2*b^2*x^2 + 54*a^3*b*x - 81*a^4)*(b*x + a)^(2/3)/b^4
________________________________________________________________________________________
Sympy [B] time = 3.7386, size = 1742, normalized size = 24.19 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x**3*(b*x+a)**(2/3),x)
[Out]
-243*a**(74/3)*(1 + b*x/a)**(2/3)/(3080*a**20*b**4 + 18480*a**19*b**5*x + 46200*a**18*b**6*x**2 + 61600*a**17*
b**7*x**3 + 46200*a**16*b**8*x**4 + 18480*a**15*b**9*x**5 + 3080*a**14*b**10*x**6) + 243*a**(74/3)/(3080*a**20
*b**4 + 18480*a**19*b**5*x + 46200*a**18*b**6*x**2 + 61600*a**17*b**7*x**3 + 46200*a**16*b**8*x**4 + 18480*a**
15*b**9*x**5 + 3080*a**14*b**10*x**6) - 1296*a**(71/3)*b*x*(1 + b*x/a)**(2/3)/(3080*a**20*b**4 + 18480*a**19*b
**5*x + 46200*a**18*b**6*x**2 + 61600*a**17*b**7*x**3 + 46200*a**16*b**8*x**4 + 18480*a**15*b**9*x**5 + 3080*a
**14*b**10*x**6) + 1458*a**(71/3)*b*x/(3080*a**20*b**4 + 18480*a**19*b**5*x + 46200*a**18*b**6*x**2 + 61600*a*
*17*b**7*x**3 + 46200*a**16*b**8*x**4 + 18480*a**15*b**9*x**5 + 3080*a**14*b**10*x**6) - 2808*a**(68/3)*b**2*x
**2*(1 + b*x/a)**(2/3)/(3080*a**20*b**4 + 18480*a**19*b**5*x + 46200*a**18*b**6*x**2 + 61600*a**17*b**7*x**3 +
46200*a**16*b**8*x**4 + 18480*a**15*b**9*x**5 + 3080*a**14*b**10*x**6) + 3645*a**(68/3)*b**2*x**2/(3080*a**20
*b**4 + 18480*a**19*b**5*x + 46200*a**18*b**6*x**2 + 61600*a**17*b**7*x**3 + 46200*a**16*b**8*x**4 + 18480*a**
15*b**9*x**5 + 3080*a**14*b**10*x**6) - 3120*a**(65/3)*b**3*x**3*(1 + b*x/a)**(2/3)/(3080*a**20*b**4 + 18480*a
**19*b**5*x + 46200*a**18*b**6*x**2 + 61600*a**17*b**7*x**3 + 46200*a**16*b**8*x**4 + 18480*a**15*b**9*x**5 +
3080*a**14*b**10*x**6) + 4860*a**(65/3)*b**3*x**3/(3080*a**20*b**4 + 18480*a**19*b**5*x + 46200*a**18*b**6*x**
2 + 61600*a**17*b**7*x**3 + 46200*a**16*b**8*x**4 + 18480*a**15*b**9*x**5 + 3080*a**14*b**10*x**6) - 1050*a**(
62/3)*b**4*x**4*(1 + b*x/a)**(2/3)/(3080*a**20*b**4 + 18480*a**19*b**5*x + 46200*a**18*b**6*x**2 + 61600*a**17
*b**7*x**3 + 46200*a**16*b**8*x**4 + 18480*a**15*b**9*x**5 + 3080*a**14*b**10*x**6) + 3645*a**(62/3)*b**4*x**4
/(3080*a**20*b**4 + 18480*a**19*b**5*x + 46200*a**18*b**6*x**2 + 61600*a**17*b**7*x**3 + 46200*a**16*b**8*x**4
+ 18480*a**15*b**9*x**5 + 3080*a**14*b**10*x**6) + 4032*a**(59/3)*b**5*x**5*(1 + b*x/a)**(2/3)/(3080*a**20*b*
*4 + 18480*a**19*b**5*x + 46200*a**18*b**6*x**2 + 61600*a**17*b**7*x**3 + 46200*a**16*b**8*x**4 + 18480*a**15*
b**9*x**5 + 3080*a**14*b**10*x**6) + 1458*a**(59/3)*b**5*x**5/(3080*a**20*b**4 + 18480*a**19*b**5*x + 46200*a*
*18*b**6*x**2 + 61600*a**17*b**7*x**3 + 46200*a**16*b**8*x**4 + 18480*a**15*b**9*x**5 + 3080*a**14*b**10*x**6)
+ 11004*a**(56/3)*b**6*x**6*(1 + b*x/a)**(2/3)/(3080*a**20*b**4 + 18480*a**19*b**5*x + 46200*a**18*b**6*x**2
+ 61600*a**17*b**7*x**3 + 46200*a**16*b**8*x**4 + 18480*a**15*b**9*x**5 + 3080*a**14*b**10*x**6) + 243*a**(56/
3)*b**6*x**6/(3080*a**20*b**4 + 18480*a**19*b**5*x + 46200*a**18*b**6*x**2 + 61600*a**17*b**7*x**3 + 46200*a**
16*b**8*x**4 + 18480*a**15*b**9*x**5 + 3080*a**14*b**10*x**6) + 14352*a**(53/3)*b**7*x**7*(1 + b*x/a)**(2/3)/(
3080*a**20*b**4 + 18480*a**19*b**5*x + 46200*a**18*b**6*x**2 + 61600*a**17*b**7*x**3 + 46200*a**16*b**8*x**4 +
18480*a**15*b**9*x**5 + 3080*a**14*b**10*x**6) + 10485*a**(50/3)*b**8*x**8*(1 + b*x/a)**(2/3)/(3080*a**20*b**
4 + 18480*a**19*b**5*x + 46200*a**18*b**6*x**2 + 61600*a**17*b**7*x**3 + 46200*a**16*b**8*x**4 + 18480*a**15*b
**9*x**5 + 3080*a**14*b**10*x**6) + 4080*a**(47/3)*b**9*x**9*(1 + b*x/a)**(2/3)/(3080*a**20*b**4 + 18480*a**19
*b**5*x + 46200*a**18*b**6*x**2 + 61600*a**17*b**7*x**3 + 46200*a**16*b**8*x**4 + 18480*a**15*b**9*x**5 + 3080
*a**14*b**10*x**6) + 660*a**(44/3)*b**10*x**10*(1 + b*x/a)**(2/3)/(3080*a**20*b**4 + 18480*a**19*b**5*x + 4620
0*a**18*b**6*x**2 + 61600*a**17*b**7*x**3 + 46200*a**16*b**8*x**4 + 18480*a**15*b**9*x**5 + 3080*a**14*b**10*x
**6)
________________________________________________________________________________________
Giac [A] time = 1.23861, size = 66, normalized size = 0.92 \begin{align*} \frac{3 \,{\left (220 \,{\left (b x + a\right )}^{\frac{14}{3}} - 840 \,{\left (b x + a\right )}^{\frac{11}{3}} a + 1155 \,{\left (b x + a\right )}^{\frac{8}{3}} a^{2} - 616 \,{\left (b x + a\right )}^{\frac{5}{3}} a^{3}\right )}}{3080 \, b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^3*(b*x+a)^(2/3),x, algorithm="giac")
[Out]
3/3080*(220*(b*x + a)^(14/3) - 840*(b*x + a)^(11/3)*a + 1155*(b*x + a)^(8/3)*a^2 - 616*(b*x + a)^(5/3)*a^3)/b^
4