3.2328 \(\int (a+b \sqrt [3]{x})^{10} \, dx\)

Optimal. Leaf size=59 \[ \frac{3 a^2 \left (a+b \sqrt [3]{x}\right )^{11}}{11 b^3}+\frac{3 \left (a+b \sqrt [3]{x}\right )^{13}}{13 b^3}-\frac{a \left (a+b \sqrt [3]{x}\right )^{12}}{2 b^3} \]

[Out]

(3*a^2*(a + b*x^(1/3))^11)/(11*b^3) - (a*(a + b*x^(1/3))^12)/(2*b^3) + (3*(a + b*x^(1/3))^13)/(13*b^3)

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Rubi [A]  time = 0.0376548, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {190, 43} \[ \frac{3 a^2 \left (a+b \sqrt [3]{x}\right )^{11}}{11 b^3}+\frac{3 \left (a+b \sqrt [3]{x}\right )^{13}}{13 b^3}-\frac{a \left (a+b \sqrt [3]{x}\right )^{12}}{2 b^3} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^(1/3))^10,x]

[Out]

(3*a^2*(a + b*x^(1/3))^11)/(11*b^3) - (a*(a + b*x^(1/3))^12)/(2*b^3) + (3*(a + b*x^(1/3))^13)/(13*b^3)

Rule 190

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

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 \left (a+b \sqrt [3]{x}\right )^{10} \, dx &=3 \operatorname{Subst}\left (\int x^2 (a+b x)^{10} \, dx,x,\sqrt [3]{x}\right )\\ &=3 \operatorname{Subst}\left (\int \left (\frac{a^2 (a+b x)^{10}}{b^2}-\frac{2 a (a+b x)^{11}}{b^2}+\frac{(a+b x)^{12}}{b^2}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=\frac{3 a^2 \left (a+b \sqrt [3]{x}\right )^{11}}{11 b^3}-\frac{a \left (a+b \sqrt [3]{x}\right )^{12}}{2 b^3}+\frac{3 \left (a+b \sqrt [3]{x}\right )^{13}}{13 b^3}\\ \end{align*}

Mathematica [A]  time = 0.0319234, size = 41, normalized size = 0.69 \[ \frac{\left (a+b \sqrt [3]{x}\right )^{11} \left (a^2-11 a b \sqrt [3]{x}+66 b^2 x^{2/3}\right )}{286 b^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^(1/3))^10,x]

[Out]

((a + b*x^(1/3))^11*(a^2 - 11*a*b*x^(1/3) + 66*b^2*x^(2/3)))/(286*b^3)

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Maple [B]  time = 0.002, size = 110, normalized size = 1.9 \begin{align*} x{a}^{10}+{\frac{3\,{b}^{10}}{13}{x}^{{\frac{13}{3}}}}+{\frac{5\,a{b}^{9}{x}^{4}}{2}}+{\frac{135\,{a}^{2}{b}^{8}}{11}{x}^{{\frac{11}{3}}}}+36\,{a}^{3}{b}^{7}{x}^{10/3}+70\,{a}^{4}{b}^{6}{x}^{3}+{\frac{189\,{a}^{5}{b}^{5}}{2}{x}^{{\frac{8}{3}}}}+90\,{a}^{6}{b}^{4}{x}^{7/3}+60\,{a}^{7}{b}^{3}{x}^{2}+27\,{a}^{8}{b}^{2}{x}^{5/3}+{\frac{15\,{a}^{9}b}{2}{x}^{{\frac{4}{3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*x^(1/3))^10,x)

[Out]

x*a^10+3/13*b^10*x^(13/3)+5/2*a*b^9*x^4+135/11*a^2*b^8*x^(11/3)+36*a^3*b^7*x^(10/3)+70*a^4*b^6*x^3+189/2*a^5*b
^5*x^(8/3)+90*a^6*b^4*x^(7/3)+60*a^7*b^3*x^2+27*a^8*b^2*x^(5/3)+15/2*a^9*b*x^(4/3)

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Maxima [A]  time = 0.946849, size = 63, normalized size = 1.07 \begin{align*} \frac{3 \,{\left (b x^{\frac{1}{3}} + a\right )}^{13}}{13 \, b^{3}} - \frac{{\left (b x^{\frac{1}{3}} + a\right )}^{12} a}{2 \, b^{3}} + \frac{3 \,{\left (b x^{\frac{1}{3}} + a\right )}^{11} a^{2}}{11 \, b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/3))^10,x, algorithm="maxima")

[Out]

3/13*(b*x^(1/3) + a)^13/b^3 - 1/2*(b*x^(1/3) + a)^12*a/b^3 + 3/11*(b*x^(1/3) + a)^11*a^2/b^3

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Fricas [B]  time = 1.45207, size = 270, normalized size = 4.58 \begin{align*} \frac{5}{2} \, a b^{9} x^{4} + 70 \, a^{4} b^{6} x^{3} + 60 \, a^{7} b^{3} x^{2} + a^{10} x + \frac{27}{22} \,{\left (10 \, a^{2} b^{8} x^{3} + 77 \, a^{5} b^{5} x^{2} + 22 \, a^{8} b^{2} x\right )} x^{\frac{2}{3}} + \frac{3}{26} \,{\left (2 \, b^{10} x^{4} + 312 \, a^{3} b^{7} x^{3} + 780 \, a^{6} b^{4} x^{2} + 65 \, a^{9} b x\right )} x^{\frac{1}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/3))^10,x, algorithm="fricas")

[Out]

5/2*a*b^9*x^4 + 70*a^4*b^6*x^3 + 60*a^7*b^3*x^2 + a^10*x + 27/22*(10*a^2*b^8*x^3 + 77*a^5*b^5*x^2 + 22*a^8*b^2
*x)*x^(2/3) + 3/26*(2*b^10*x^4 + 312*a^3*b^7*x^3 + 780*a^6*b^4*x^2 + 65*a^9*b*x)*x^(1/3)

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Sympy [B]  time = 2.63423, size = 136, normalized size = 2.31 \begin{align*} a^{10} x + \frac{15 a^{9} b x^{\frac{4}{3}}}{2} + 27 a^{8} b^{2} x^{\frac{5}{3}} + 60 a^{7} b^{3} x^{2} + 90 a^{6} b^{4} x^{\frac{7}{3}} + \frac{189 a^{5} b^{5} x^{\frac{8}{3}}}{2} + 70 a^{4} b^{6} x^{3} + 36 a^{3} b^{7} x^{\frac{10}{3}} + \frac{135 a^{2} b^{8} x^{\frac{11}{3}}}{11} + \frac{5 a b^{9} x^{4}}{2} + \frac{3 b^{10} x^{\frac{13}{3}}}{13} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x**(1/3))**10,x)

[Out]

a**10*x + 15*a**9*b*x**(4/3)/2 + 27*a**8*b**2*x**(5/3) + 60*a**7*b**3*x**2 + 90*a**6*b**4*x**(7/3) + 189*a**5*
b**5*x**(8/3)/2 + 70*a**4*b**6*x**3 + 36*a**3*b**7*x**(10/3) + 135*a**2*b**8*x**(11/3)/11 + 5*a*b**9*x**4/2 +
3*b**10*x**(13/3)/13

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Giac [B]  time = 1.16008, size = 147, normalized size = 2.49 \begin{align*} \frac{3}{13} \, b^{10} x^{\frac{13}{3}} + \frac{5}{2} \, a b^{9} x^{4} + \frac{135}{11} \, a^{2} b^{8} x^{\frac{11}{3}} + 36 \, a^{3} b^{7} x^{\frac{10}{3}} + 70 \, a^{4} b^{6} x^{3} + \frac{189}{2} \, a^{5} b^{5} x^{\frac{8}{3}} + 90 \, a^{6} b^{4} x^{\frac{7}{3}} + 60 \, a^{7} b^{3} x^{2} + 27 \, a^{8} b^{2} x^{\frac{5}{3}} + \frac{15}{2} \, a^{9} b x^{\frac{4}{3}} + a^{10} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/3))^10,x, algorithm="giac")

[Out]

3/13*b^10*x^(13/3) + 5/2*a*b^9*x^4 + 135/11*a^2*b^8*x^(11/3) + 36*a^3*b^7*x^(10/3) + 70*a^4*b^6*x^3 + 189/2*a^
5*b^5*x^(8/3) + 90*a^6*b^4*x^(7/3) + 60*a^7*b^3*x^2 + 27*a^8*b^2*x^(5/3) + 15/2*a^9*b*x^(4/3) + a^10*x