3.7.61 \(\int \sqrt [3]{x} (a+b x)^2 \, dx\) [661]
Optimal. Leaf size=36 \[ \frac {3}{4} a^2 x^{4/3}+\frac {6}{7} a b x^{7/3}+\frac {3}{10} b^2 x^{10/3} \]
[Out]
3/4*a^2*x^(4/3)+6/7*a*b*x^(7/3)+3/10*b^2*x^(10/3)
________________________________________________________________________________________
Rubi [A]
time = 0.01, antiderivative size = 36, normalized size of antiderivative = 1.00, 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 = {45}
\begin {gather*} \frac {3}{4} a^2 x^{4/3}+\frac {6}{7} a b x^{7/3}+\frac {3}{10} b^2 x^{10/3} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[x^(1/3)*(a + b*x)^2,x]
[Out]
(3*a^2*x^(4/3))/4 + (6*a*b*x^(7/3))/7 + (3*b^2*x^(10/3))/10
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 \sqrt [3]{x} (a+b x)^2 \, dx &=\int \left (a^2 \sqrt [3]{x}+2 a b x^{4/3}+b^2 x^{7/3}\right ) \, dx\\ &=\frac {3}{4} a^2 x^{4/3}+\frac {6}{7} a b x^{7/3}+\frac {3}{10} b^2 x^{10/3}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.01, size = 28, normalized size = 0.78 \begin {gather*} \frac {3}{140} x^{4/3} \left (35 a^2+40 a b x+14 b^2 x^2\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[x^(1/3)*(a + b*x)^2,x]
[Out]
(3*x^(4/3)*(35*a^2 + 40*a*b*x + 14*b^2*x^2))/140
________________________________________________________________________________________
Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 2 in
optimal.
time = 13.43, size = 1235, normalized size = 34.31
result too large to display
Warning: Unable to verify antiderivative.
[In]
mathics('Integrate[x^(1/3)*(a + b*x)^2,x]')
[Out]
Piecewise[{{3 a ^ (1 / 3) (9 -1 ^ (1 / 3) a ^ 3 + 35 a ^ 2 b x (b x / a) ^ (1 / 3) + 40 a b ^ 2 x ^ 2 (b x / a
) ^ (1 / 3) + 14 b ^ 3 x ^ 3 (b x / a) ^ (1 / 3)) / (140 b ^ (4 / 3)), Abs[(a + b x) / a] > 1}}, -27 a ^ (34 /
3) / (140 E ^ (I 2 Pi / 3) a ^ 8 b ^ (4 / 3) - 420 E ^ (I 2 Pi / 3) a ^ 7 b ^ (7 / 3) (a / b + x) + 420 E ^ (
I 2 Pi / 3) a ^ 6 b ^ (10 / 3) (a / b + x) ^ 2 - 140 E ^ (I 2 Pi / 3) a ^ 5 b ^ (13 / 3) (a / b + x) ^ 3) + 27
a ^ (34 / 3) (1 - b (a / b + x) / a) ^ (1 / 3) / (140 E ^ (I 2 Pi / 3) a ^ 8 b ^ (4 / 3) - 420 E ^ (I 2 Pi /
3) a ^ 7 b ^ (7 / 3) (a / b + x) + 420 E ^ (I 2 Pi / 3) a ^ 6 b ^ (10 / 3) (a / b + x) ^ 2 - 140 E ^ (I 2 Pi /
3) a ^ 5 b ^ (13 / 3) (a / b + x) ^ 3) - 72 a ^ (31 / 3) b (a / b + x) (1 - b (a / b + x) / a) ^ (1 / 3) / (1
40 E ^ (I 2 Pi / 3) a ^ 8 b ^ (4 / 3) - 420 E ^ (I 2 Pi / 3) a ^ 7 b ^ (7 / 3) (a / b + x) + 420 E ^ (I 2 Pi /
3) a ^ 6 b ^ (10 / 3) (a / b + x) ^ 2 - 140 E ^ (I 2 Pi / 3) a ^ 5 b ^ (13 / 3) (a / b + x) ^ 3) + 81 a ^ (31
/ 3) b (a / b + x) / (140 E ^ (I 2 Pi / 3) a ^ 8 b ^ (4 / 3) - 420 E ^ (I 2 Pi / 3) a ^ 7 b ^ (7 / 3) (a / b
+ x) + 420 E ^ (I 2 Pi / 3) a ^ 6 b ^ (10 / 3) (a / b + x) ^ 2 - 140 E ^ (I 2 Pi / 3) a ^ 5 b ^ (13 / 3) (a /
b + x) ^ 3) - 81 a ^ (28 / 3) b ^ 2 (a / b + x) ^ 2 / (140 E ^ (I 2 Pi / 3) a ^ 8 b ^ (4 / 3) - 420 E ^ (I 2 P
i / 3) a ^ 7 b ^ (7 / 3) (a / b + x) + 420 E ^ (I 2 Pi / 3) a ^ 6 b ^ (10 / 3) (a / b + x) ^ 2 - 140 E ^ (I 2
Pi / 3) a ^ 5 b ^ (13 / 3) (a / b + x) ^ 3) + 60 a ^ (28 / 3) b ^ 2 (1 - b (a / b + x) / a) ^ (1 / 3) (a / b +
x) ^ 2 / (140 E ^ (I 2 Pi / 3) a ^ 8 b ^ (4 / 3) - 420 E ^ (I 2 Pi / 3) a ^ 7 b ^ (7 / 3) (a / b + x) + 420 E
^ (I 2 Pi / 3) a ^ 6 b ^ (10 / 3) (a / b + x) ^ 2 - 140 E ^ (I 2 Pi / 3) a ^ 5 b ^ (13 / 3) (a / b + x) ^ 3)
- 60 a ^ (25 / 3) b ^ 3 (1 - b (a / b + x) / a) ^ (1 / 3) (a / b + x) ^ 3 / (140 E ^ (I 2 Pi / 3) a ^ 8 b ^ (4
/ 3) - 420 E ^ (I 2 Pi / 3) a ^ 7 b ^ (7 / 3) (a / b + x) + 420 E ^ (I 2 Pi / 3) a ^ 6 b ^ (10 / 3) (a / b +
x) ^ 2 - 140 E ^ (I 2 Pi / 3) a ^ 5 b ^ (13 / 3) (a / b + x) ^ 3) + 27 a ^ (25 / 3) b ^ 3 (a / b + x) ^ 3 / (1
40 E ^ (I 2 Pi / 3) a ^ 8 b ^ (4 / 3) - 420 E ^ (I 2 Pi / 3) a ^ 7 b ^ (7 / 3) (a / b + x) + 420 E ^ (I 2 Pi /
3) a ^ 6 b ^ (10 / 3) (a / b + x) ^ 2 - 140 E ^ (I 2 Pi / 3) a ^ 5 b ^ (13 / 3) (a / b + x) ^ 3) + 135 a ^ (2
2 / 3) b ^ 4 (1 - b (a / b + x) / a) ^ (1 / 3) (a / b + x) ^ 4 / (140 E ^ (I 2 Pi / 3) a ^ 8 b ^ (4 / 3) - 420
E ^ (I 2 Pi / 3) a ^ 7 b ^ (7 / 3) (a / b + x) + 420 E ^ (I 2 Pi / 3) a ^ 6 b ^ (10 / 3) (a / b + x) ^ 2 - 14
0 E ^ (I 2 Pi / 3) a ^ 5 b ^ (13 / 3) (a / b + x) ^ 3) - 132 a ^ (19 / 3) b ^ 5 (1 - b (a / b + x) / a) ^ (1 /
3) (a / b + x) ^ 5 / (140 E ^ (I 2 Pi / 3) a ^ 8 b ^ (4 / 3) - 420 E ^ (I 2 Pi / 3) a ^ 7 b ^ (7 / 3) (a / b
+ x) + 420 E ^ (I 2 Pi / 3) a ^ 6 b ^ (10 / 3) (a / b + x) ^ 2 - 140 E ^ (I 2 Pi / 3) a ^ 5 b ^ (13 / 3) (a /
b + x) ^ 3) + 42 a ^ (16 / 3) b ^ 6 (1 - b (a / b + x) / a) ^ (1 / 3) (a / b + x) ^ 6 / (140 E ^ (I 2 Pi / 3)
a ^ 8 b ^ (4 / 3) - 420 E ^ (I 2 Pi / 3) a ^ 7 b ^ (7 / 3) (a / b + x) + 420 E ^ (I 2 Pi / 3) a ^ 6 b ^ (10 /
3) (a / b + x) ^ 2 - 140 E ^ (I 2 Pi / 3) a ^ 5 b ^ (13 / 3) (a / b + x) ^ 3)]
________________________________________________________________________________________
Maple [A]
time = 0.10, size = 25, normalized size = 0.69
| | |
| method |
result |
size |
| | |
| gosper |
\(\frac {3 x^{\frac {4}{3}} \left (14 x^{2} b^{2}+40 a b x +35 a^{2}\right )}{140}\) |
\(25\) |
| derivativedivides |
\(\frac {3 a^{2} x^{\frac {4}{3}}}{4}+\frac {6 a b \,x^{\frac {7}{3}}}{7}+\frac {3 b^{2} x^{\frac {10}{3}}}{10}\) |
\(25\) |
| default |
\(\frac {3 a^{2} x^{\frac {4}{3}}}{4}+\frac {6 a b \,x^{\frac {7}{3}}}{7}+\frac {3 b^{2} x^{\frac {10}{3}}}{10}\) |
\(25\) |
| trager |
\(\frac {3 x^{\frac {4}{3}} \left (14 x^{2} b^{2}+40 a b x +35 a^{2}\right )}{140}\) |
\(25\) |
| risch |
\(\frac {3 x^{\frac {4}{3}} \left (14 x^{2} b^{2}+40 a b x +35 a^{2}\right )}{140}\) |
\(25\) |
| | |
|
|
|
|
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x^(1/3)*(b*x+a)^2,x,method=_RETURNVERBOSE)
[Out]
3/4*a^2*x^(4/3)+6/7*a*b*x^(7/3)+3/10*b^2*x^(10/3)
________________________________________________________________________________________
Maxima [A]
time = 0.27, size = 24, normalized size = 0.67 \begin {gather*} \frac {3}{10} \, b^{2} x^{\frac {10}{3}} + \frac {6}{7} \, a b x^{\frac {7}{3}} + \frac {3}{4} \, a^{2} x^{\frac {4}{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^(1/3)*(b*x+a)^2,x, algorithm="maxima")
[Out]
3/10*b^2*x^(10/3) + 6/7*a*b*x^(7/3) + 3/4*a^2*x^(4/3)
________________________________________________________________________________________
Fricas [A]
time = 0.30, size = 27, normalized size = 0.75 \begin {gather*} \frac {3}{140} \, {\left (14 \, b^{2} x^{3} + 40 \, a b x^{2} + 35 \, a^{2} x\right )} x^{\frac {1}{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^(1/3)*(b*x+a)^2,x, algorithm="fricas")
[Out]
3/140*(14*b^2*x^3 + 40*a*b*x^2 + 35*a^2*x)*x^(1/3)
________________________________________________________________________________________
Sympy [C] Result contains complex when optimal does not.
time = 1.12, size = 2633, normalized size = 73.14
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x**(1/3)*(b*x+a)**2,x)
[Out]
Piecewise((27*a**(34/3)*(-1 + b*(a/b + x)/a)**(1/3)*exp(2*I*pi/3)/(-140*a**8*b**(4/3)*exp(2*I*pi/3) + 420*a**7
*b**(7/3)*(a/b + x)*exp(2*I*pi/3) - 420*a**6*b**(10/3)*(a/b + x)**2*exp(2*I*pi/3) + 140*a**5*b**(13/3)*(a/b +
x)**3*exp(2*I*pi/3)) + 27*a**(34/3)/(-140*a**8*b**(4/3)*exp(2*I*pi/3) + 420*a**7*b**(7/3)*(a/b + x)*exp(2*I*pi
/3) - 420*a**6*b**(10/3)*(a/b + x)**2*exp(2*I*pi/3) + 140*a**5*b**(13/3)*(a/b + x)**3*exp(2*I*pi/3)) - 72*a**(
31/3)*b*(-1 + b*(a/b + x)/a)**(1/3)*(a/b + x)*exp(2*I*pi/3)/(-140*a**8*b**(4/3)*exp(2*I*pi/3) + 420*a**7*b**(7
/3)*(a/b + x)*exp(2*I*pi/3) - 420*a**6*b**(10/3)*(a/b + x)**2*exp(2*I*pi/3) + 140*a**5*b**(13/3)*(a/b + x)**3*
exp(2*I*pi/3)) - 81*a**(31/3)*b*(a/b + x)/(-140*a**8*b**(4/3)*exp(2*I*pi/3) + 420*a**7*b**(7/3)*(a/b + x)*exp(
2*I*pi/3) - 420*a**6*b**(10/3)*(a/b + x)**2*exp(2*I*pi/3) + 140*a**5*b**(13/3)*(a/b + x)**3*exp(2*I*pi/3)) + 6
0*a**(28/3)*b**2*(-1 + b*(a/b + x)/a)**(1/3)*(a/b + x)**2*exp(2*I*pi/3)/(-140*a**8*b**(4/3)*exp(2*I*pi/3) + 42
0*a**7*b**(7/3)*(a/b + x)*exp(2*I*pi/3) - 420*a**6*b**(10/3)*(a/b + x)**2*exp(2*I*pi/3) + 140*a**5*b**(13/3)*(
a/b + x)**3*exp(2*I*pi/3)) + 81*a**(28/3)*b**2*(a/b + x)**2/(-140*a**8*b**(4/3)*exp(2*I*pi/3) + 420*a**7*b**(7
/3)*(a/b + x)*exp(2*I*pi/3) - 420*a**6*b**(10/3)*(a/b + x)**2*exp(2*I*pi/3) + 140*a**5*b**(13/3)*(a/b + x)**3*
exp(2*I*pi/3)) - 60*a**(25/3)*b**3*(-1 + b*(a/b + x)/a)**(1/3)*(a/b + x)**3*exp(2*I*pi/3)/(-140*a**8*b**(4/3)*
exp(2*I*pi/3) + 420*a**7*b**(7/3)*(a/b + x)*exp(2*I*pi/3) - 420*a**6*b**(10/3)*(a/b + x)**2*exp(2*I*pi/3) + 14
0*a**5*b**(13/3)*(a/b + x)**3*exp(2*I*pi/3)) - 27*a**(25/3)*b**3*(a/b + x)**3/(-140*a**8*b**(4/3)*exp(2*I*pi/3
) + 420*a**7*b**(7/3)*(a/b + x)*exp(2*I*pi/3) - 420*a**6*b**(10/3)*(a/b + x)**2*exp(2*I*pi/3) + 140*a**5*b**(1
3/3)*(a/b + x)**3*exp(2*I*pi/3)) + 135*a**(22/3)*b**4*(-1 + b*(a/b + x)/a)**(1/3)*(a/b + x)**4*exp(2*I*pi/3)/(
-140*a**8*b**(4/3)*exp(2*I*pi/3) + 420*a**7*b**(7/3)*(a/b + x)*exp(2*I*pi/3) - 420*a**6*b**(10/3)*(a/b + x)**2
*exp(2*I*pi/3) + 140*a**5*b**(13/3)*(a/b + x)**3*exp(2*I*pi/3)) - 132*a**(19/3)*b**5*(-1 + b*(a/b + x)/a)**(1/
3)*(a/b + x)**5*exp(2*I*pi/3)/(-140*a**8*b**(4/3)*exp(2*I*pi/3) + 420*a**7*b**(7/3)*(a/b + x)*exp(2*I*pi/3) -
420*a**6*b**(10/3)*(a/b + x)**2*exp(2*I*pi/3) + 140*a**5*b**(13/3)*(a/b + x)**3*exp(2*I*pi/3)) + 42*a**(16/3)*
b**6*(-1 + b*(a/b + x)/a)**(1/3)*(a/b + x)**6*exp(2*I*pi/3)/(-140*a**8*b**(4/3)*exp(2*I*pi/3) + 420*a**7*b**(7
/3)*(a/b + x)*exp(2*I*pi/3) - 420*a**6*b**(10/3)*(a/b + x)**2*exp(2*I*pi/3) + 140*a**5*b**(13/3)*(a/b + x)**3*
exp(2*I*pi/3)), Abs(b*(a/b + x)/a) > 1), (-27*a**(34/3)*(1 - b*(a/b + x)/a)**(1/3)/(-140*a**8*b**(4/3)*exp(2*I
*pi/3) + 420*a**7*b**(7/3)*(a/b + x)*exp(2*I*pi/3) - 420*a**6*b**(10/3)*(a/b + x)**2*exp(2*I*pi/3) + 140*a**5*
b**(13/3)*(a/b + x)**3*exp(2*I*pi/3)) + 27*a**(34/3)/(-140*a**8*b**(4/3)*exp(2*I*pi/3) + 420*a**7*b**(7/3)*(a/
b + x)*exp(2*I*pi/3) - 420*a**6*b**(10/3)*(a/b + x)**2*exp(2*I*pi/3) + 140*a**5*b**(13/3)*(a/b + x)**3*exp(2*I
*pi/3)) + 72*a**(31/3)*b*(1 - b*(a/b + x)/a)**(1/3)*(a/b + x)/(-140*a**8*b**(4/3)*exp(2*I*pi/3) + 420*a**7*b**
(7/3)*(a/b + x)*exp(2*I*pi/3) - 420*a**6*b**(10/3)*(a/b + x)**2*exp(2*I*pi/3) + 140*a**5*b**(13/3)*(a/b + x)**
3*exp(2*I*pi/3)) - 81*a**(31/3)*b*(a/b + x)/(-140*a**8*b**(4/3)*exp(2*I*pi/3) + 420*a**7*b**(7/3)*(a/b + x)*ex
p(2*I*pi/3) - 420*a**6*b**(10/3)*(a/b + x)**2*exp(2*I*pi/3) + 140*a**5*b**(13/3)*(a/b + x)**3*exp(2*I*pi/3)) -
60*a**(28/3)*b**2*(1 - b*(a/b + x)/a)**(1/3)*(a/b + x)**2/(-140*a**8*b**(4/3)*exp(2*I*pi/3) + 420*a**7*b**(7/
3)*(a/b + x)*exp(2*I*pi/3) - 420*a**6*b**(10/3)*(a/b + x)**2*exp(2*I*pi/3) + 140*a**5*b**(13/3)*(a/b + x)**3*e
xp(2*I*pi/3)) + 81*a**(28/3)*b**2*(a/b + x)**2/(-140*a**8*b**(4/3)*exp(2*I*pi/3) + 420*a**7*b**(7/3)*(a/b + x)
*exp(2*I*pi/3) - 420*a**6*b**(10/3)*(a/b + x)**2*exp(2*I*pi/3) + 140*a**5*b**(13/3)*(a/b + x)**3*exp(2*I*pi/3)
) + 60*a**(25/3)*b**3*(1 - b*(a/b + x)/a)**(1/3)*(a/b + x)**3/(-140*a**8*b**(4/3)*exp(2*I*pi/3) + 420*a**7*b**
(7/3)*(a/b + x)*exp(2*I*pi/3) - 420*a**6*b**(10/3)*(a/b + x)**2*exp(2*I*pi/3) + 140*a**5*b**(13/3)*(a/b + x)**
3*exp(2*I*pi/3)) - 27*a**(25/3)*b**3*(a/b + x)**3/(-140*a**8*b**(4/3)*exp(2*I*pi/3) + 420*a**7*b**(7/3)*(a/b +
x)*exp(2*I*pi/3) - 420*a**6*b**(10/3)*(a/b + x)**2*exp(2*I*pi/3) + 140*a**5*b**(13/3)*(a/b + x)**3*exp(2*I*pi
/3)) - 135*a**(22/3)*b**4*(1 - b*(a/b + x)/a)**(1/3)*(a/b + x)**4/(-140*a**8*b**(4/3)*exp(2*I*pi/3) + 420*a**7
*b**(7/3)*(a/b + x)*exp(2*I*pi/3) - 420*a**6*b**(10/3)*(a/b + x)**2*exp(2*I*pi/3) + 140*a**5*b**(13/3)*(a/b +
x)**3*exp(2*I*pi/3)) + 132*a**(19/3)*b**5*(1 - b*(a/b + x)/a)**(1/3)*(a/b + x)**5/(-140*a**8*b**(4/3)*exp(2*I*
pi/3) + 420*a**7*b**(7/3)*(a/b + x)*exp(2*I*pi/3) - 420*a**6*b**(10/3)*(a/b + x)**2*exp(2*I*pi/3) + 140*a**5*b
**(13/3)*(a/b + x)**3*exp(2*I*pi/3)) - 42*a**(16/3)*b**6*(1 - b*(a/b + x)/a)**(1/3)*(a/b + x)**6/(-140*a**8*b*
*(4/3)*exp(2*I*pi/3) + 420*a**7*b**(7/3)*(a/b + x)*exp(2*I*pi/3) - 420*a**6*b**(10/3)*(a/b + x)**2*exp(2*I*pi/
3) + 140*a**5*b**(13/3)*(a/b + x)**3*exp(2*I*pi/3)), True))
________________________________________________________________________________________
Giac [A]
time = 0.00, size = 43, normalized size = 1.19 \begin {gather*} \frac {3}{10} b^{2} x^{\frac {1}{3}} x^{3}+\frac {6}{7} a b x^{\frac {1}{3}} x^{2}+\frac {3}{4} a^{2} x^{\frac {1}{3}} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^(1/3)*(b*x+a)^2,x)
[Out]
3/10*b^2*x^(10/3) + 6/7*a*b*x^(7/3) + 3/4*a^2*x^(4/3)
________________________________________________________________________________________
Mupad [B]
time = 0.04, size = 24, normalized size = 0.67 \begin {gather*} \frac {3\,x^{4/3}\,\left (35\,a^2+40\,a\,b\,x+14\,b^2\,x^2\right )}{140} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x^(1/3)*(a + b*x)^2,x)
[Out]
(3*x^(4/3)*(35*a^2 + 14*b^2*x^2 + 40*a*b*x))/140
________________________________________________________________________________________