3.4.99 \(\int \frac {x^3}{\sqrt [3]{-a+b x}} \, dx\) [399]
Optimal. Leaf size=80 \[ \frac {3 a^3 (-a+b x)^{2/3}}{2 b^4}+\frac {9 a^2 (-a+b x)^{5/3}}{5 b^4}+\frac {9 a (-a+b x)^{8/3}}{8 b^4}+\frac {3 (-a+b x)^{11/3}}{11 b^4} \]
[Out]
3/2*a^3*(b*x-a)^(2/3)/b^4+9/5*a^2*(b*x-a)^(5/3)/b^4+9/8*a*(b*x-a)^(8/3)/b^4+3/11*(b*x-a)^(11/3)/b^4
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Rubi [A]
time = 0.01, antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {45}
\begin {gather*} \frac {3 a^3 (b x-a)^{2/3}}{2 b^4}+\frac {9 a^2 (b x-a)^{5/3}}{5 b^4}+\frac {3 (b x-a)^{11/3}}{11 b^4}+\frac {9 a (b x-a)^{8/3}}{8 b^4} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[x^3/(-a + b*x)^(1/3),x]
[Out]
(3*a^3*(-a + b*x)^(2/3))/(2*b^4) + (9*a^2*(-a + b*x)^(5/3))/(5*b^4) + (9*a*(-a + b*x)^(8/3))/(8*b^4) + (3*(-a
+ b*x)^(11/3))/(11*b^4)
Rule 45
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}{\sqrt [3]{-a+b x}} \, dx &=\int \left (\frac {a^3}{b^3 \sqrt [3]{-a+b x}}+\frac {3 a^2 (-a+b x)^{2/3}}{b^3}+\frac {3 a (-a+b x)^{5/3}}{b^3}+\frac {(-a+b x)^{8/3}}{b^3}\right ) \, dx\\ &=\frac {3 a^3 (-a+b x)^{2/3}}{2 b^4}+\frac {9 a^2 (-a+b x)^{5/3}}{5 b^4}+\frac {9 a (-a+b x)^{8/3}}{8 b^4}+\frac {3 (-a+b x)^{11/3}}{11 b^4}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 48, normalized size = 0.60 \begin {gather*} \frac {3 (-a+b x)^{2/3} \left (81 a^3+54 a^2 b x+45 a b^2 x^2+40 b^3 x^3\right )}{440 b^4} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[x^3/(-a + b*x)^(1/3),x]
[Out]
(3*(-a + b*x)^(2/3)*(81*a^3 + 54*a^2*b*x + 45*a*b^2*x^2 + 40*b^3*x^3))/(440*b^4)
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Maple [A]
time = 0.11, size = 58, normalized size = 0.72
| | |
| method |
result |
size |
| | |
| gosper |
\(\frac {3 \left (40 b^{3} x^{3}+45 a \,b^{2} x^{2}+54 a^{2} b x +81 a^{3}\right ) \left (b x -a \right )^{\frac {2}{3}}}{440 b^{4}}\) |
\(45\) |
| trager |
\(\frac {3 \left (40 b^{3} x^{3}+45 a \,b^{2} x^{2}+54 a^{2} b x +81 a^{3}\right ) \left (b x -a \right )^{\frac {2}{3}}}{440 b^{4}}\) |
\(45\) |
| risch |
\(-\frac {3 \left (-b x +a \right ) \left (40 b^{3} x^{3}+45 a \,b^{2} x^{2}+54 a^{2} b x +81 a^{3}\right )}{440 b^{4} \left (b x -a \right )^{\frac {1}{3}}}\) |
\(51\) |
| derivativedivides |
\(\frac {\frac {3 \left (b x -a \right )^{\frac {11}{3}}}{11}+\frac {9 a \left (b x -a \right )^{\frac {8}{3}}}{8}+\frac {9 a^{2} \left (b x -a \right )^{\frac {5}{3}}}{5}+\frac {3 a^{3} \left (b x -a \right )^{\frac {2}{3}}}{2}}{b^{4}}\) |
\(58\) |
| default |
\(\frac {\frac {3 \left (b x -a \right )^{\frac {11}{3}}}{11}+\frac {9 a \left (b x -a \right )^{\frac {8}{3}}}{8}+\frac {9 a^{2} \left (b x -a \right )^{\frac {5}{3}}}{5}+\frac {3 a^{3} \left (b x -a \right )^{\frac {2}{3}}}{2}}{b^{4}}\) |
\(58\) |
| | |
|
|
|
|
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x^3/(b*x-a)^(1/3),x,method=_RETURNVERBOSE)
[Out]
3/b^4*(1/11*(b*x-a)^(11/3)+3/8*a*(b*x-a)^(8/3)+3/5*a^2*(b*x-a)^(5/3)+1/2*a^3*(b*x-a)^(2/3))
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Maxima [A]
time = 0.27, size = 64, normalized size = 0.80 \begin {gather*} \frac {3 \, {\left (b x - a\right )}^{\frac {11}{3}}}{11 \, b^{4}} + \frac {9 \, {\left (b x - a\right )}^{\frac {8}{3}} a}{8 \, b^{4}} + \frac {9 \, {\left (b x - a\right )}^{\frac {5}{3}} a^{2}}{5 \, b^{4}} + \frac {3 \, {\left (b x - a\right )}^{\frac {2}{3}} a^{3}}{2 \, b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^3/(b*x-a)^(1/3),x, algorithm="maxima")
[Out]
3/11*(b*x - a)^(11/3)/b^4 + 9/8*(b*x - a)^(8/3)*a/b^4 + 9/5*(b*x - a)^(5/3)*a^2/b^4 + 3/2*(b*x - a)^(2/3)*a^3/
b^4
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Fricas [A]
time = 0.70, size = 44, normalized size = 0.55 \begin {gather*} \frac {3 \, {\left (40 \, b^{3} x^{3} + 45 \, a b^{2} x^{2} + 54 \, a^{2} b x + 81 \, a^{3}\right )} {\left (b x - a\right )}^{\frac {2}{3}}}{440 \, b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^3/(b*x-a)^(1/3),x, algorithm="fricas")
[Out]
3/440*(40*b^3*x^3 + 45*a*b^2*x^2 + 54*a^2*b*x + 81*a^3)*(b*x - a)^(2/3)/b^4
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Sympy [C] Result contains complex when optimal does not.
time = 1.32, size = 4974, normalized size = 62.18 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x**3/(b*x-a)**(1/3),x)
[Out]
Piecewise((243*a**(71/3)*(-1 + b*x/a)**(2/3)*exp(I*pi/3)/(440*a**20*b**4*exp(I*pi/3) - 2640*a**19*b**5*x*exp(I
*pi/3) + 6600*a**18*b**6*x**2*exp(I*pi/3) - 8800*a**17*b**7*x**3*exp(I*pi/3) + 6600*a**16*b**8*x**4*exp(I*pi/3
) - 2640*a**15*b**9*x**5*exp(I*pi/3) + 440*a**14*b**10*x**6*exp(I*pi/3)) + 243*a**(71/3)/(440*a**20*b**4*exp(I
*pi/3) - 2640*a**19*b**5*x*exp(I*pi/3) + 6600*a**18*b**6*x**2*exp(I*pi/3) - 8800*a**17*b**7*x**3*exp(I*pi/3) +
6600*a**16*b**8*x**4*exp(I*pi/3) - 2640*a**15*b**9*x**5*exp(I*pi/3) + 440*a**14*b**10*x**6*exp(I*pi/3)) - 129
6*a**(68/3)*b*x*(-1 + b*x/a)**(2/3)*exp(I*pi/3)/(440*a**20*b**4*exp(I*pi/3) - 2640*a**19*b**5*x*exp(I*pi/3) +
6600*a**18*b**6*x**2*exp(I*pi/3) - 8800*a**17*b**7*x**3*exp(I*pi/3) + 6600*a**16*b**8*x**4*exp(I*pi/3) - 2640*
a**15*b**9*x**5*exp(I*pi/3) + 440*a**14*b**10*x**6*exp(I*pi/3)) - 1458*a**(68/3)*b*x/(440*a**20*b**4*exp(I*pi/
3) - 2640*a**19*b**5*x*exp(I*pi/3) + 6600*a**18*b**6*x**2*exp(I*pi/3) - 8800*a**17*b**7*x**3*exp(I*pi/3) + 660
0*a**16*b**8*x**4*exp(I*pi/3) - 2640*a**15*b**9*x**5*exp(I*pi/3) + 440*a**14*b**10*x**6*exp(I*pi/3)) + 2808*a*
*(65/3)*b**2*x**2*(-1 + b*x/a)**(2/3)*exp(I*pi/3)/(440*a**20*b**4*exp(I*pi/3) - 2640*a**19*b**5*x*exp(I*pi/3)
+ 6600*a**18*b**6*x**2*exp(I*pi/3) - 8800*a**17*b**7*x**3*exp(I*pi/3) + 6600*a**16*b**8*x**4*exp(I*pi/3) - 264
0*a**15*b**9*x**5*exp(I*pi/3) + 440*a**14*b**10*x**6*exp(I*pi/3)) + 3645*a**(65/3)*b**2*x**2/(440*a**20*b**4*e
xp(I*pi/3) - 2640*a**19*b**5*x*exp(I*pi/3) + 6600*a**18*b**6*x**2*exp(I*pi/3) - 8800*a**17*b**7*x**3*exp(I*pi/
3) + 6600*a**16*b**8*x**4*exp(I*pi/3) - 2640*a**15*b**9*x**5*exp(I*pi/3) + 440*a**14*b**10*x**6*exp(I*pi/3)) -
3120*a**(62/3)*b**3*x**3*(-1 + b*x/a)**(2/3)*exp(I*pi/3)/(440*a**20*b**4*exp(I*pi/3) - 2640*a**19*b**5*x*exp(
I*pi/3) + 6600*a**18*b**6*x**2*exp(I*pi/3) - 8800*a**17*b**7*x**3*exp(I*pi/3) + 6600*a**16*b**8*x**4*exp(I*pi/
3) - 2640*a**15*b**9*x**5*exp(I*pi/3) + 440*a**14*b**10*x**6*exp(I*pi/3)) - 4860*a**(62/3)*b**3*x**3/(440*a**2
0*b**4*exp(I*pi/3) - 2640*a**19*b**5*x*exp(I*pi/3) + 6600*a**18*b**6*x**2*exp(I*pi/3) - 8800*a**17*b**7*x**3*e
xp(I*pi/3) + 6600*a**16*b**8*x**4*exp(I*pi/3) - 2640*a**15*b**9*x**5*exp(I*pi/3) + 440*a**14*b**10*x**6*exp(I*
pi/3)) + 1710*a**(59/3)*b**4*x**4*(-1 + b*x/a)**(2/3)*exp(I*pi/3)/(440*a**20*b**4*exp(I*pi/3) - 2640*a**19*b**
5*x*exp(I*pi/3) + 6600*a**18*b**6*x**2*exp(I*pi/3) - 8800*a**17*b**7*x**3*exp(I*pi/3) + 6600*a**16*b**8*x**4*e
xp(I*pi/3) - 2640*a**15*b**9*x**5*exp(I*pi/3) + 440*a**14*b**10*x**6*exp(I*pi/3)) + 3645*a**(59/3)*b**4*x**4/(
440*a**20*b**4*exp(I*pi/3) - 2640*a**19*b**5*x*exp(I*pi/3) + 6600*a**18*b**6*x**2*exp(I*pi/3) - 8800*a**17*b**
7*x**3*exp(I*pi/3) + 6600*a**16*b**8*x**4*exp(I*pi/3) - 2640*a**15*b**9*x**5*exp(I*pi/3) + 440*a**14*b**10*x**
6*exp(I*pi/3)) + 72*a**(56/3)*b**5*x**5*(-1 + b*x/a)**(2/3)*exp(I*pi/3)/(440*a**20*b**4*exp(I*pi/3) - 2640*a**
19*b**5*x*exp(I*pi/3) + 6600*a**18*b**6*x**2*exp(I*pi/3) - 8800*a**17*b**7*x**3*exp(I*pi/3) + 6600*a**16*b**8*
x**4*exp(I*pi/3) - 2640*a**15*b**9*x**5*exp(I*pi/3) + 440*a**14*b**10*x**6*exp(I*pi/3)) - 1458*a**(56/3)*b**5*
x**5/(440*a**20*b**4*exp(I*pi/3) - 2640*a**19*b**5*x*exp(I*pi/3) + 6600*a**18*b**6*x**2*exp(I*pi/3) - 8800*a**
17*b**7*x**3*exp(I*pi/3) + 6600*a**16*b**8*x**4*exp(I*pi/3) - 2640*a**15*b**9*x**5*exp(I*pi/3) + 440*a**14*b**
10*x**6*exp(I*pi/3)) - 1104*a**(53/3)*b**6*x**6*(-1 + b*x/a)**(2/3)*exp(I*pi/3)/(440*a**20*b**4*exp(I*pi/3) -
2640*a**19*b**5*x*exp(I*pi/3) + 6600*a**18*b**6*x**2*exp(I*pi/3) - 8800*a**17*b**7*x**3*exp(I*pi/3) + 6600*a**
16*b**8*x**4*exp(I*pi/3) - 2640*a**15*b**9*x**5*exp(I*pi/3) + 440*a**14*b**10*x**6*exp(I*pi/3)) + 243*a**(53/3
)*b**6*x**6/(440*a**20*b**4*exp(I*pi/3) - 2640*a**19*b**5*x*exp(I*pi/3) + 6600*a**18*b**6*x**2*exp(I*pi/3) - 8
800*a**17*b**7*x**3*exp(I*pi/3) + 6600*a**16*b**8*x**4*exp(I*pi/3) - 2640*a**15*b**9*x**5*exp(I*pi/3) + 440*a*
*14*b**10*x**6*exp(I*pi/3)) + 1152*a**(50/3)*b**7*x**7*(-1 + b*x/a)**(2/3)*exp(I*pi/3)/(440*a**20*b**4*exp(I*p
i/3) - 2640*a**19*b**5*x*exp(I*pi/3) + 6600*a**18*b**6*x**2*exp(I*pi/3) - 8800*a**17*b**7*x**3*exp(I*pi/3) + 6
600*a**16*b**8*x**4*exp(I*pi/3) - 2640*a**15*b**9*x**5*exp(I*pi/3) + 440*a**14*b**10*x**6*exp(I*pi/3)) - 585*a
**(47/3)*b**8*x**8*(-1 + b*x/a)**(2/3)*exp(I*pi/3)/(440*a**20*b**4*exp(I*pi/3) - 2640*a**19*b**5*x*exp(I*pi/3)
+ 6600*a**18*b**6*x**2*exp(I*pi/3) - 8800*a**17*b**7*x**3*exp(I*pi/3) + 6600*a**16*b**8*x**4*exp(I*pi/3) - 26
40*a**15*b**9*x**5*exp(I*pi/3) + 440*a**14*b**10*x**6*exp(I*pi/3)) + 120*a**(44/3)*b**9*x**9*(-1 + b*x/a)**(2/
3)*exp(I*pi/3)/(440*a**20*b**4*exp(I*pi/3) - 2640*a**19*b**5*x*exp(I*pi/3) + 6600*a**18*b**6*x**2*exp(I*pi/3)
- 8800*a**17*b**7*x**3*exp(I*pi/3) + 6600*a**16*b**8*x**4*exp(I*pi/3) - 2640*a**15*b**9*x**5*exp(I*pi/3) + 440
*a**14*b**10*x**6*exp(I*pi/3)), Abs(b*x/a) > 1), (-243*a**(71/3)*(1 - b*x/a)**(2/3)/(440*a**20*b**4*exp(I*pi/3
) - 2640*a**19*b**5*x*exp(I*pi/3) + 6600*a**18*b**6*x**2*exp(I*pi/3) - 8800*a**17*b**7*x**3*exp(I*pi/3) + 6600
*a**16*b**8*x**4*exp(I*pi/3) - 2640*a**15*b**9*...
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Giac [A]
time = 1.11, size = 57, normalized size = 0.71 \begin {gather*} \frac {3 \, {\left (40 \, {\left (b x - a\right )}^{\frac {11}{3}} + 165 \, {\left (b x - a\right )}^{\frac {8}{3}} a + 264 \, {\left (b x - a\right )}^{\frac {5}{3}} a^{2} + 220 \, {\left (b x - a\right )}^{\frac {2}{3}} a^{3}\right )}}{440 \, b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^3/(b*x-a)^(1/3),x, algorithm="giac")
[Out]
3/440*(40*(b*x - a)^(11/3) + 165*(b*x - a)^(8/3)*a + 264*(b*x - a)^(5/3)*a^2 + 220*(b*x - a)^(2/3)*a^3)/b^4
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Mupad [B]
time = 0.05, size = 64, normalized size = 0.80 \begin {gather*} \frac {3\,{\left (b\,x-a\right )}^{11/3}}{11\,b^4}+\frac {9\,a\,{\left (b\,x-a\right )}^{8/3}}{8\,b^4}+\frac {3\,a^3\,{\left (b\,x-a\right )}^{2/3}}{2\,b^4}+\frac {9\,a^2\,{\left (b\,x-a\right )}^{5/3}}{5\,b^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x^3/(b*x - a)^(1/3),x)
[Out]
(3*(b*x - a)^(11/3))/(11*b^4) + (9*a*(b*x - a)^(8/3))/(8*b^4) + (3*a^3*(b*x - a)^(2/3))/(2*b^4) + (9*a^2*(b*x
- a)^(5/3))/(5*b^4)
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