3.2325 \(\int (a+b \sqrt [3]{x})^{10} x^3 \, dx\)

Optimal. Leaf size=144 \[ \frac {a^{10} x^4}{4}+\frac {30}{13} a^9 b x^{13/3}+\frac {135}{14} a^8 b^2 x^{14/3}+24 a^7 b^3 x^5+\frac {315}{8} a^6 b^4 x^{16/3}+\frac {756}{17} a^5 b^5 x^{17/3}+35 a^4 b^6 x^6+\frac {360}{19} a^3 b^7 x^{19/3}+\frac {27}{4} a^2 b^8 x^{20/3}+\frac {10}{7} a b^9 x^7+\frac {3}{22} b^{10} x^{22/3} \]

[Out]

1/4*a^10*x^4+30/13*a^9*b*x^(13/3)+135/14*a^8*b^2*x^(14/3)+24*a^7*b^3*x^5+315/8*a^6*b^4*x^(16/3)+756/17*a^5*b^5
*x^(17/3)+35*a^4*b^6*x^6+360/19*a^3*b^7*x^(19/3)+27/4*a^2*b^8*x^(20/3)+10/7*a*b^9*x^7+3/22*b^10*x^(22/3)

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Rubi [A]  time = 0.09, antiderivative size = 144, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {266, 43} \[ \frac {135}{14} a^8 b^2 x^{14/3}+24 a^7 b^3 x^5+\frac {315}{8} a^6 b^4 x^{16/3}+\frac {756}{17} a^5 b^5 x^{17/3}+35 a^4 b^6 x^6+\frac {360}{19} a^3 b^7 x^{19/3}+\frac {27}{4} a^2 b^8 x^{20/3}+\frac {30}{13} a^9 b x^{13/3}+\frac {a^{10} x^4}{4}+\frac {10}{7} a b^9 x^7+\frac {3}{22} b^{10} x^{22/3} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^(1/3))^10*x^3,x]

[Out]

(a^10*x^4)/4 + (30*a^9*b*x^(13/3))/13 + (135*a^8*b^2*x^(14/3))/14 + 24*a^7*b^3*x^5 + (315*a^6*b^4*x^(16/3))/8
+ (756*a^5*b^5*x^(17/3))/17 + 35*a^4*b^6*x^6 + (360*a^3*b^7*x^(19/3))/19 + (27*a^2*b^8*x^(20/3))/4 + (10*a*b^9
*x^7)/7 + (3*b^10*x^(22/3))/22

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 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+b \sqrt [3]{x}\right )^{10} x^3 \, dx &=3 \operatorname {Subst}\left (\int x^{11} (a+b x)^{10} \, dx,x,\sqrt [3]{x}\right )\\ &=3 \operatorname {Subst}\left (\int \left (a^{10} x^{11}+10 a^9 b x^{12}+45 a^8 b^2 x^{13}+120 a^7 b^3 x^{14}+210 a^6 b^4 x^{15}+252 a^5 b^5 x^{16}+210 a^4 b^6 x^{17}+120 a^3 b^7 x^{18}+45 a^2 b^8 x^{19}+10 a b^9 x^{20}+b^{10} x^{21}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=\frac {a^{10} x^4}{4}+\frac {30}{13} a^9 b x^{13/3}+\frac {135}{14} a^8 b^2 x^{14/3}+24 a^7 b^3 x^5+\frac {315}{8} a^6 b^4 x^{16/3}+\frac {756}{17} a^5 b^5 x^{17/3}+35 a^4 b^6 x^6+\frac {360}{19} a^3 b^7 x^{19/3}+\frac {27}{4} a^2 b^8 x^{20/3}+\frac {10}{7} a b^9 x^7+\frac {3}{22} b^{10} x^{22/3}\\ \end {align*}

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Mathematica [A]  time = 0.08, size = 144, normalized size = 1.00 \[ \frac {a^{10} x^4}{4}+\frac {30}{13} a^9 b x^{13/3}+\frac {135}{14} a^8 b^2 x^{14/3}+24 a^7 b^3 x^5+\frac {315}{8} a^6 b^4 x^{16/3}+\frac {756}{17} a^5 b^5 x^{17/3}+35 a^4 b^6 x^6+\frac {360}{19} a^3 b^7 x^{19/3}+\frac {27}{4} a^2 b^8 x^{20/3}+\frac {10}{7} a b^9 x^7+\frac {3}{22} b^{10} x^{22/3} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^(1/3))^10*x^3,x]

[Out]

(a^10*x^4)/4 + (30*a^9*b*x^(13/3))/13 + (135*a^8*b^2*x^(14/3))/14 + 24*a^7*b^3*x^5 + (315*a^6*b^4*x^(16/3))/8
+ (756*a^5*b^5*x^(17/3))/17 + 35*a^4*b^6*x^6 + (360*a^3*b^7*x^(19/3))/19 + (27*a^2*b^8*x^(20/3))/4 + (10*a*b^9
*x^7)/7 + (3*b^10*x^(22/3))/22

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fricas [A]  time = 0.66, size = 124, normalized size = 0.86 \[ \frac {10}{7} \, a b^{9} x^{7} + 35 \, a^{4} b^{6} x^{6} + 24 \, a^{7} b^{3} x^{5} + \frac {1}{4} \, a^{10} x^{4} + \frac {27}{476} \, {\left (119 \, a^{2} b^{8} x^{6} + 784 \, a^{5} b^{5} x^{5} + 170 \, a^{8} b^{2} x^{4}\right )} x^{\frac {2}{3}} + \frac {3}{21736} \, {\left (988 \, b^{10} x^{7} + 137280 \, a^{3} b^{7} x^{6} + 285285 \, a^{6} b^{4} x^{5} + 16720 \, a^{9} b x^{4}\right )} x^{\frac {1}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

10/7*a*b^9*x^7 + 35*a^4*b^6*x^6 + 24*a^7*b^3*x^5 + 1/4*a^10*x^4 + 27/476*(119*a^2*b^8*x^6 + 784*a^5*b^5*x^5 +
170*a^8*b^2*x^4)*x^(2/3) + 3/21736*(988*b^10*x^7 + 137280*a^3*b^7*x^6 + 285285*a^6*b^4*x^5 + 16720*a^9*b*x^4)*
x^(1/3)

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giac [A]  time = 0.16, size = 112, normalized size = 0.78 \[ \frac {3}{22} \, b^{10} x^{\frac {22}{3}} + \frac {10}{7} \, a b^{9} x^{7} + \frac {27}{4} \, a^{2} b^{8} x^{\frac {20}{3}} + \frac {360}{19} \, a^{3} b^{7} x^{\frac {19}{3}} + 35 \, a^{4} b^{6} x^{6} + \frac {756}{17} \, a^{5} b^{5} x^{\frac {17}{3}} + \frac {315}{8} \, a^{6} b^{4} x^{\frac {16}{3}} + 24 \, a^{7} b^{3} x^{5} + \frac {135}{14} \, a^{8} b^{2} x^{\frac {14}{3}} + \frac {30}{13} \, a^{9} b x^{\frac {13}{3}} + \frac {1}{4} \, a^{10} x^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/22*b^10*x^(22/3) + 10/7*a*b^9*x^7 + 27/4*a^2*b^8*x^(20/3) + 360/19*a^3*b^7*x^(19/3) + 35*a^4*b^6*x^6 + 756/1
7*a^5*b^5*x^(17/3) + 315/8*a^6*b^4*x^(16/3) + 24*a^7*b^3*x^5 + 135/14*a^8*b^2*x^(14/3) + 30/13*a^9*b*x^(13/3)
+ 1/4*a^10*x^4

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maple [A]  time = 0.00, size = 113, normalized size = 0.78 \[ \frac {3 b^{10} x^{\frac {22}{3}}}{22}+\frac {10 a \,b^{9} x^{7}}{7}+\frac {27 a^{2} b^{8} x^{\frac {20}{3}}}{4}+\frac {360 a^{3} b^{7} x^{\frac {19}{3}}}{19}+35 a^{4} b^{6} x^{6}+\frac {756 a^{5} b^{5} x^{\frac {17}{3}}}{17}+\frac {315 a^{6} b^{4} x^{\frac {16}{3}}}{8}+24 a^{7} b^{3} x^{5}+\frac {135 a^{8} b^{2} x^{\frac {14}{3}}}{14}+\frac {30 a^{9} b \,x^{\frac {13}{3}}}{13}+\frac {a^{10} x^{4}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*a^10*x^4+30/13*a^9*b*x^(13/3)+135/14*a^8*b^2*x^(14/3)+24*a^7*b^3*x^5+315/8*a^6*b^4*x^(16/3)+756/17*a^5*b^5
*x^(17/3)+35*a^4*b^6*x^6+360/19*a^3*b^7*x^(19/3)+27/4*a^2*b^8*x^(20/3)+10/7*a*b^9*x^7+3/22*b^10*x^(22/3)

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maxima [A]  time = 0.90, size = 200, normalized size = 1.39 \[ \frac {3 \, {\left (b x^{\frac {1}{3}} + a\right )}^{22}}{22 \, b^{12}} - \frac {11 \, {\left (b x^{\frac {1}{3}} + a\right )}^{21} a}{7 \, b^{12}} + \frac {33 \, {\left (b x^{\frac {1}{3}} + a\right )}^{20} a^{2}}{4 \, b^{12}} - \frac {495 \, {\left (b x^{\frac {1}{3}} + a\right )}^{19} a^{3}}{19 \, b^{12}} + \frac {55 \, {\left (b x^{\frac {1}{3}} + a\right )}^{18} a^{4}}{b^{12}} - \frac {1386 \, {\left (b x^{\frac {1}{3}} + a\right )}^{17} a^{5}}{17 \, b^{12}} + \frac {693 \, {\left (b x^{\frac {1}{3}} + a\right )}^{16} a^{6}}{8 \, b^{12}} - \frac {66 \, {\left (b x^{\frac {1}{3}} + a\right )}^{15} a^{7}}{b^{12}} + \frac {495 \, {\left (b x^{\frac {1}{3}} + a\right )}^{14} a^{8}}{14 \, b^{12}} - \frac {165 \, {\left (b x^{\frac {1}{3}} + a\right )}^{13} a^{9}}{13 \, b^{12}} + \frac {11 \, {\left (b x^{\frac {1}{3}} + a\right )}^{12} a^{10}}{4 \, b^{12}} - \frac {3 \, {\left (b x^{\frac {1}{3}} + a\right )}^{11} a^{11}}{11 \, b^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/22*(b*x^(1/3) + a)^22/b^12 - 11/7*(b*x^(1/3) + a)^21*a/b^12 + 33/4*(b*x^(1/3) + a)^20*a^2/b^12 - 495/19*(b*x
^(1/3) + a)^19*a^3/b^12 + 55*(b*x^(1/3) + a)^18*a^4/b^12 - 1386/17*(b*x^(1/3) + a)^17*a^5/b^12 + 693/8*(b*x^(1
/3) + a)^16*a^6/b^12 - 66*(b*x^(1/3) + a)^15*a^7/b^12 + 495/14*(b*x^(1/3) + a)^14*a^8/b^12 - 165/13*(b*x^(1/3)
+ a)^13*a^9/b^12 + 11/4*(b*x^(1/3) + a)^12*a^10/b^12 - 3/11*(b*x^(1/3) + a)^11*a^11/b^12

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mupad [B]  time = 0.06, size = 112, normalized size = 0.78 \[ \frac {a^{10}\,x^4}{4}+\frac {3\,b^{10}\,x^{22/3}}{22}+\frac {10\,a\,b^9\,x^7}{7}+\frac {30\,a^9\,b\,x^{13/3}}{13}+24\,a^7\,b^3\,x^5+35\,a^4\,b^6\,x^6+\frac {135\,a^8\,b^2\,x^{14/3}}{14}+\frac {315\,a^6\,b^4\,x^{16/3}}{8}+\frac {756\,a^5\,b^5\,x^{17/3}}{17}+\frac {360\,a^3\,b^7\,x^{19/3}}{19}+\frac {27\,a^2\,b^8\,x^{20/3}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(a + b*x^(1/3))^10,x)

[Out]

(a^10*x^4)/4 + (3*b^10*x^(22/3))/22 + (10*a*b^9*x^7)/7 + (30*a^9*b*x^(13/3))/13 + 24*a^7*b^3*x^5 + 35*a^4*b^6*
x^6 + (135*a^8*b^2*x^(14/3))/14 + (315*a^6*b^4*x^(16/3))/8 + (756*a^5*b^5*x^(17/3))/17 + (360*a^3*b^7*x^(19/3)
)/19 + (27*a^2*b^8*x^(20/3))/4

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sympy [A]  time = 4.20, size = 144, normalized size = 1.00 \[ \frac {a^{10} x^{4}}{4} + \frac {30 a^{9} b x^{\frac {13}{3}}}{13} + \frac {135 a^{8} b^{2} x^{\frac {14}{3}}}{14} + 24 a^{7} b^{3} x^{5} + \frac {315 a^{6} b^{4} x^{\frac {16}{3}}}{8} + \frac {756 a^{5} b^{5} x^{\frac {17}{3}}}{17} + 35 a^{4} b^{6} x^{6} + \frac {360 a^{3} b^{7} x^{\frac {19}{3}}}{19} + \frac {27 a^{2} b^{8} x^{\frac {20}{3}}}{4} + \frac {10 a b^{9} x^{7}}{7} + \frac {3 b^{10} x^{\frac {22}{3}}}{22} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**10*x**4/4 + 30*a**9*b*x**(13/3)/13 + 135*a**8*b**2*x**(14/3)/14 + 24*a**7*b**3*x**5 + 315*a**6*b**4*x**(16/
3)/8 + 756*a**5*b**5*x**(17/3)/17 + 35*a**4*b**6*x**6 + 360*a**3*b**7*x**(19/3)/19 + 27*a**2*b**8*x**(20/3)/4
+ 10*a*b**9*x**7/7 + 3*b**10*x**(22/3)/22

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