3.7.61 \(\int \sqrt [3]{x} (a+b x)^2 \, dx\)
Optimal. Leaf size=36 \[ \frac {3}{4} a^2 x^{4/3}+\frac {6}{7} a b x^{7/3}+\frac {3}{10} b^2 x^{10/3} \]
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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 =
{43} \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 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 \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*}
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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
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IntegrateAlgebraic [A] time = 0.01, size = 34, normalized size = 0.94 \begin {gather*} \frac {3}{140} \left (35 a^2 x^{4/3}+40 a b x^{7/3}+14 b^2 x^{10/3}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
IntegrateAlgebraic[x^(1/3)*(a + b*x)^2,x]
[Out]
(3*(35*a^2*x^(4/3) + 40*a*b*x^(7/3) + 14*b^2*x^(10/3)))/140
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fricas [A] time = 1.15, 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)
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giac [A] time = 1.04, 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="giac")
[Out]
3/10*b^2*x^(10/3) + 6/7*a*b*x^(7/3) + 3/4*a^2*x^(4/3)
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maple [A] time = 0.01, size = 25, normalized size = 0.69 \begin {gather*} \frac {3 \left (14 b^{2} x^{2}+40 a b x +35 a^{2}\right ) x^{\frac {4}{3}}}{140} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x^(1/3)*(b*x+a)^2,x)
[Out]
3/140*x^(4/3)*(14*b^2*x^2+40*a*b*x+35*a^2)
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maxima [A] time = 1.28, 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)
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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
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sympy [C] time = 2.22, 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))
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