3.413 \(\int \frac{x^3}{(a+b x)^{4/3}} \, dx\)
Optimal. Leaf size=70 \[ \frac{3 a^3}{b^4 \sqrt [3]{a+b x}}+\frac{9 a^2 (a+b x)^{2/3}}{2 b^4}-\frac{9 a (a+b x)^{5/3}}{5 b^4}+\frac{3 (a+b x)^{8/3}}{8 b^4} \]
[Out]
(3*a^3)/(b^4*(a + b*x)^(1/3)) + (9*a^2*(a + b*x)^(2/3))/(2*b^4) - (9*a*(a + b*x)^(5/3))/(5*b^4) + (3*(a + b*x)
^(8/3))/(8*b^4)
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Rubi [A] time = 0.0178824, antiderivative size = 70, 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} \[ \frac{3 a^3}{b^4 \sqrt [3]{a+b x}}+\frac{9 a^2 (a+b x)^{2/3}}{2 b^4}-\frac{9 a (a+b x)^{5/3}}{5 b^4}+\frac{3 (a+b x)^{8/3}}{8 b^4} \]
Antiderivative was successfully verified.
[In]
Int[x^3/(a + b*x)^(4/3),x]
[Out]
(3*a^3)/(b^4*(a + b*x)^(1/3)) + (9*a^2*(a + b*x)^(2/3))/(2*b^4) - (9*a*(a + b*x)^(5/3))/(5*b^4) + (3*(a + b*x)
^(8/3))/(8*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 \frac{x^3}{(a+b x)^{4/3}} \, dx &=\int \left (-\frac{a^3}{b^3 (a+b x)^{4/3}}+\frac{3 a^2}{b^3 \sqrt [3]{a+b x}}-\frac{3 a (a+b x)^{2/3}}{b^3}+\frac{(a+b x)^{5/3}}{b^3}\right ) \, dx\\ &=\frac{3 a^3}{b^4 \sqrt [3]{a+b x}}+\frac{9 a^2 (a+b x)^{2/3}}{2 b^4}-\frac{9 a (a+b x)^{5/3}}{5 b^4}+\frac{3 (a+b x)^{8/3}}{8 b^4}\\ \end{align*}
Mathematica [A] time = 0.0301732, size = 46, normalized size = 0.66 \[ \frac{3 \left (27 a^2 b x+81 a^3-9 a b^2 x^2+5 b^3 x^3\right )}{40 b^4 \sqrt [3]{a+b x}} \]
Antiderivative was successfully verified.
[In]
Integrate[x^3/(a + b*x)^(4/3),x]
[Out]
(3*(81*a^3 + 27*a^2*b*x - 9*a*b^2*x^2 + 5*b^3*x^3))/(40*b^4*(a + b*x)^(1/3))
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Maple [A] time = 0.004, size = 43, normalized size = 0.6 \begin{align*}{\frac{15\,{b}^{3}{x}^{3}-27\,a{b}^{2}{x}^{2}+81\,{a}^{2}bx+243\,{a}^{3}}{40\,{b}^{4}}{\frac{1}{\sqrt [3]{bx+a}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x^3/(b*x+a)^(4/3),x)
[Out]
3/40/(b*x+a)^(1/3)*(5*b^3*x^3-9*a*b^2*x^2+27*a^2*b*x+81*a^3)/b^4
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Maxima [A] time = 1.11649, size = 76, normalized size = 1.09 \begin{align*} \frac{3 \,{\left (b x + a\right )}^{\frac{8}{3}}}{8 \, b^{4}} - \frac{9 \,{\left (b x + a\right )}^{\frac{5}{3}} a}{5 \, b^{4}} + \frac{9 \,{\left (b x + a\right )}^{\frac{2}{3}} a^{2}}{2 \, b^{4}} + \frac{3 \, a^{3}}{{\left (b x + a\right )}^{\frac{1}{3}} b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^3/(b*x+a)^(4/3),x, algorithm="maxima")
[Out]
3/8*(b*x + a)^(8/3)/b^4 - 9/5*(b*x + a)^(5/3)*a/b^4 + 9/2*(b*x + a)^(2/3)*a^2/b^4 + 3*a^3/((b*x + a)^(1/3)*b^4
)
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Fricas [A] time = 1.512, size = 116, normalized size = 1.66 \begin{align*} \frac{3 \,{\left (5 \, b^{3} x^{3} - 9 \, a b^{2} x^{2} + 27 \, a^{2} b x + 81 \, a^{3}\right )}{\left (b x + a\right )}^{\frac{2}{3}}}{40 \,{\left (b^{5} x + a b^{4}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^3/(b*x+a)^(4/3),x, algorithm="fricas")
[Out]
3/40*(5*b^3*x^3 - 9*a*b^2*x^2 + 27*a^2*b*x + 81*a^3)*(b*x + a)^(2/3)/(b^5*x + a*b^4)
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Sympy [B] time = 3.92976, size = 1538, normalized size = 21.97 \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)**(4/3),x)
[Out]
243*a**(68/3)*(1 + b*x/a)**(2/3)/(40*a**20*b**4 + 240*a**19*b**5*x + 600*a**18*b**6*x**2 + 800*a**17*b**7*x**3
+ 600*a**16*b**8*x**4 + 240*a**15*b**9*x**5 + 40*a**14*b**10*x**6) - 243*a**(68/3)/(40*a**20*b**4 + 240*a**19
*b**5*x + 600*a**18*b**6*x**2 + 800*a**17*b**7*x**3 + 600*a**16*b**8*x**4 + 240*a**15*b**9*x**5 + 40*a**14*b**
10*x**6) + 1296*a**(65/3)*b*x*(1 + b*x/a)**(2/3)/(40*a**20*b**4 + 240*a**19*b**5*x + 600*a**18*b**6*x**2 + 800
*a**17*b**7*x**3 + 600*a**16*b**8*x**4 + 240*a**15*b**9*x**5 + 40*a**14*b**10*x**6) - 1458*a**(65/3)*b*x/(40*a
**20*b**4 + 240*a**19*b**5*x + 600*a**18*b**6*x**2 + 800*a**17*b**7*x**3 + 600*a**16*b**8*x**4 + 240*a**15*b**
9*x**5 + 40*a**14*b**10*x**6) + 2808*a**(62/3)*b**2*x**2*(1 + b*x/a)**(2/3)/(40*a**20*b**4 + 240*a**19*b**5*x
+ 600*a**18*b**6*x**2 + 800*a**17*b**7*x**3 + 600*a**16*b**8*x**4 + 240*a**15*b**9*x**5 + 40*a**14*b**10*x**6)
- 3645*a**(62/3)*b**2*x**2/(40*a**20*b**4 + 240*a**19*b**5*x + 600*a**18*b**6*x**2 + 800*a**17*b**7*x**3 + 60
0*a**16*b**8*x**4 + 240*a**15*b**9*x**5 + 40*a**14*b**10*x**6) + 3120*a**(59/3)*b**3*x**3*(1 + b*x/a)**(2/3)/(
40*a**20*b**4 + 240*a**19*b**5*x + 600*a**18*b**6*x**2 + 800*a**17*b**7*x**3 + 600*a**16*b**8*x**4 + 240*a**15
*b**9*x**5 + 40*a**14*b**10*x**6) - 4860*a**(59/3)*b**3*x**3/(40*a**20*b**4 + 240*a**19*b**5*x + 600*a**18*b**
6*x**2 + 800*a**17*b**7*x**3 + 600*a**16*b**8*x**4 + 240*a**15*b**9*x**5 + 40*a**14*b**10*x**6) + 1830*a**(56/
3)*b**4*x**4*(1 + b*x/a)**(2/3)/(40*a**20*b**4 + 240*a**19*b**5*x + 600*a**18*b**6*x**2 + 800*a**17*b**7*x**3
+ 600*a**16*b**8*x**4 + 240*a**15*b**9*x**5 + 40*a**14*b**10*x**6) - 3645*a**(56/3)*b**4*x**4/(40*a**20*b**4 +
240*a**19*b**5*x + 600*a**18*b**6*x**2 + 800*a**17*b**7*x**3 + 600*a**16*b**8*x**4 + 240*a**15*b**9*x**5 + 40
*a**14*b**10*x**6) + 528*a**(53/3)*b**5*x**5*(1 + b*x/a)**(2/3)/(40*a**20*b**4 + 240*a**19*b**5*x + 600*a**18*
b**6*x**2 + 800*a**17*b**7*x**3 + 600*a**16*b**8*x**4 + 240*a**15*b**9*x**5 + 40*a**14*b**10*x**6) - 1458*a**(
53/3)*b**5*x**5/(40*a**20*b**4 + 240*a**19*b**5*x + 600*a**18*b**6*x**2 + 800*a**17*b**7*x**3 + 600*a**16*b**8
*x**4 + 240*a**15*b**9*x**5 + 40*a**14*b**10*x**6) + 96*a**(50/3)*b**6*x**6*(1 + b*x/a)**(2/3)/(40*a**20*b**4
+ 240*a**19*b**5*x + 600*a**18*b**6*x**2 + 800*a**17*b**7*x**3 + 600*a**16*b**8*x**4 + 240*a**15*b**9*x**5 + 4
0*a**14*b**10*x**6) - 243*a**(50/3)*b**6*x**6/(40*a**20*b**4 + 240*a**19*b**5*x + 600*a**18*b**6*x**2 + 800*a*
*17*b**7*x**3 + 600*a**16*b**8*x**4 + 240*a**15*b**9*x**5 + 40*a**14*b**10*x**6) + 48*a**(47/3)*b**7*x**7*(1 +
b*x/a)**(2/3)/(40*a**20*b**4 + 240*a**19*b**5*x + 600*a**18*b**6*x**2 + 800*a**17*b**7*x**3 + 600*a**16*b**8*
x**4 + 240*a**15*b**9*x**5 + 40*a**14*b**10*x**6) + 15*a**(44/3)*b**8*x**8*(1 + b*x/a)**(2/3)/(40*a**20*b**4 +
240*a**19*b**5*x + 600*a**18*b**6*x**2 + 800*a**17*b**7*x**3 + 600*a**16*b**8*x**4 + 240*a**15*b**9*x**5 + 40
*a**14*b**10*x**6)
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Giac [A] time = 1.20517, size = 84, normalized size = 1.2 \begin{align*} \frac{3 \, a^{3}}{{\left (b x + a\right )}^{\frac{1}{3}} b^{4}} + \frac{3 \,{\left (5 \,{\left (b x + a\right )}^{\frac{8}{3}} b^{28} - 24 \,{\left (b x + a\right )}^{\frac{5}{3}} a b^{28} + 60 \,{\left (b x + a\right )}^{\frac{2}{3}} a^{2} b^{28}\right )}}{40 \, b^{32}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^3/(b*x+a)^(4/3),x, algorithm="giac")
[Out]
3*a^3/((b*x + a)^(1/3)*b^4) + 3/40*(5*(b*x + a)^(8/3)*b^28 - 24*(b*x + a)^(5/3)*a*b^28 + 60*(b*x + a)^(2/3)*a^
2*b^28)/b^32