3.2404 \(\int (a+\frac {b}{\sqrt [3]{x}})^2 x^3 \, dx\)

Optimal. Leaf size=34 \[ \frac {a^2 x^4}{4}+\frac {6}{11} a b x^{11/3}+\frac {3}{10} b^2 x^{10/3} \]

[Out]

3/10*b^2*x^(10/3)+6/11*a*b*x^(11/3)+1/4*a^2*x^4

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Rubi [A]  time = 0.03, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {263, 266, 43} \[ \frac {a^2 x^4}{4}+\frac {6}{11} a b x^{11/3}+\frac {3}{10} b^2 x^{10/3} \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x^(1/3))^2*x^3,x]

[Out]

(3*b^2*x^(10/3))/10 + (6*a*b*x^(11/3))/11 + (a^2*x^4)/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])

Rule 263

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[x^(m + n*p)*(b + a/x^n)^p, x] /; FreeQ[{a, b, m
, n}, x] && IntegerQ[p] && NegQ[n]

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rubi steps

\begin {align*} \int \left (a+\frac {b}{\sqrt [3]{x}}\right )^2 x^3 \, dx &=\int \left (b+a \sqrt [3]{x}\right )^2 x^{7/3} \, dx\\ &=3 \operatorname {Subst}\left (\int x^9 (b+a x)^2 \, dx,x,\sqrt [3]{x}\right )\\ &=3 \operatorname {Subst}\left (\int \left (b^2 x^9+2 a b x^{10}+a^2 x^{11}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=\frac {3}{10} b^2 x^{10/3}+\frac {6}{11} a b x^{11/3}+\frac {a^2 x^4}{4}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 34, normalized size = 1.00 \[ \frac {a^2 x^4}{4}+\frac {6}{11} a b x^{11/3}+\frac {3}{10} b^2 x^{10/3} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x^(1/3))^2*x^3,x]

[Out]

(3*b^2*x^(10/3))/10 + (6*a*b*x^(11/3))/11 + (a^2*x^4)/4

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fricas [A]  time = 0.77, size = 24, normalized size = 0.71 \[ \frac {1}{4} \, a^{2} x^{4} + \frac {6}{11} \, a b x^{\frac {11}{3}} + \frac {3}{10} \, b^{2} x^{\frac {10}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x^(1/3))^2*x^3,x, algorithm="fricas")

[Out]

1/4*a^2*x^4 + 6/11*a*b*x^(11/3) + 3/10*b^2*x^(10/3)

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giac [A]  time = 0.15, size = 24, normalized size = 0.71 \[ \frac {1}{4} \, a^{2} x^{4} + \frac {6}{11} \, a b x^{\frac {11}{3}} + \frac {3}{10} \, b^{2} x^{\frac {10}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x^(1/3))^2*x^3,x, algorithm="giac")

[Out]

1/4*a^2*x^4 + 6/11*a*b*x^(11/3) + 3/10*b^2*x^(10/3)

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maple [A]  time = 0.00, size = 25, normalized size = 0.74 \[ \frac {a^{2} x^{4}}{4}+\frac {6 a b \,x^{\frac {11}{3}}}{11}+\frac {3 b^{2} x^{\frac {10}{3}}}{10} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b/x^(1/3))^2*x^3,x)

[Out]

3/10*b^2*x^(10/3)+6/11*a*b*x^(11/3)+1/4*a^2*x^4

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maxima [A]  time = 0.59, size = 26, normalized size = 0.76 \[ \frac {1}{220} \, {\left (55 \, a^{2} + \frac {120 \, a b}{x^{\frac {1}{3}}} + \frac {66 \, b^{2}}{x^{\frac {2}{3}}}\right )} x^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x^(1/3))^2*x^3,x, algorithm="maxima")

[Out]

1/220*(55*a^2 + 120*a*b/x^(1/3) + 66*b^2/x^(2/3))*x^4

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mupad [B]  time = 1.14, size = 24, normalized size = 0.71 \[ \frac {a^2\,x^4}{4}+\frac {3\,b^2\,x^{10/3}}{10}+\frac {6\,a\,b\,x^{11/3}}{11} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(a + b/x^(1/3))^2,x)

[Out]

(a^2*x^4)/4 + (3*b^2*x^(10/3))/10 + (6*a*b*x^(11/3))/11

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sympy [A]  time = 1.79, size = 31, normalized size = 0.91 \[ \frac {a^{2} x^{4}}{4} + \frac {6 a b x^{\frac {11}{3}}}{11} + \frac {3 b^{2} x^{\frac {10}{3}}}{10} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x**(1/3))**2*x**3,x)

[Out]

a**2*x**4/4 + 6*a*b*x**(11/3)/11 + 3*b**2*x**(10/3)/10

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