3.2327 \(\int (a+b \sqrt [3]{x})^{10} x \, dx\)

Optimal. Leaf size=120 \[ -\frac {3 a^5 \left (a+b \sqrt [3]{x}\right )^{11}}{11 b^6}+\frac {5 a^4 \left (a+b \sqrt [3]{x}\right )^{12}}{4 b^6}-\frac {30 a^3 \left (a+b \sqrt [3]{x}\right )^{13}}{13 b^6}+\frac {15 a^2 \left (a+b \sqrt [3]{x}\right )^{14}}{7 b^6}+\frac {3 \left (a+b \sqrt [3]{x}\right )^{16}}{16 b^6}-\frac {a \left (a+b \sqrt [3]{x}\right )^{15}}{b^6} \]

[Out]

-3/11*a^5*(a+b*x^(1/3))^11/b^6+5/4*a^4*(a+b*x^(1/3))^12/b^6-30/13*a^3*(a+b*x^(1/3))^13/b^6+15/7*a^2*(a+b*x^(1/
3))^14/b^6-a*(a+b*x^(1/3))^15/b^6+3/16*(a+b*x^(1/3))^16/b^6

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Rubi [A]  time = 0.07, antiderivative size = 120, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {266, 43} \[ \frac {15 a^2 \left (a+b \sqrt [3]{x}\right )^{14}}{7 b^6}-\frac {30 a^3 \left (a+b \sqrt [3]{x}\right )^{13}}{13 b^6}+\frac {5 a^4 \left (a+b \sqrt [3]{x}\right )^{12}}{4 b^6}-\frac {3 a^5 \left (a+b \sqrt [3]{x}\right )^{11}}{11 b^6}+\frac {3 \left (a+b \sqrt [3]{x}\right )^{16}}{16 b^6}-\frac {a \left (a+b \sqrt [3]{x}\right )^{15}}{b^6} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^(1/3))^10*x,x]

[Out]

(-3*a^5*(a + b*x^(1/3))^11)/(11*b^6) + (5*a^4*(a + b*x^(1/3))^12)/(4*b^6) - (30*a^3*(a + b*x^(1/3))^13)/(13*b^
6) + (15*a^2*(a + b*x^(1/3))^14)/(7*b^6) - (a*(a + b*x^(1/3))^15)/b^6 + (3*(a + b*x^(1/3))^16)/(16*b^6)

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 \, dx &=3 \operatorname {Subst}\left (\int x^5 (a+b x)^{10} \, dx,x,\sqrt [3]{x}\right )\\ &=3 \operatorname {Subst}\left (\int \left (-\frac {a^5 (a+b x)^{10}}{b^5}+\frac {5 a^4 (a+b x)^{11}}{b^5}-\frac {10 a^3 (a+b x)^{12}}{b^5}+\frac {10 a^2 (a+b x)^{13}}{b^5}-\frac {5 a (a+b x)^{14}}{b^5}+\frac {(a+b x)^{15}}{b^5}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac {3 a^5 \left (a+b \sqrt [3]{x}\right )^{11}}{11 b^6}+\frac {5 a^4 \left (a+b \sqrt [3]{x}\right )^{12}}{4 b^6}-\frac {30 a^3 \left (a+b \sqrt [3]{x}\right )^{13}}{13 b^6}+\frac {15 a^2 \left (a+b \sqrt [3]{x}\right )^{14}}{7 b^6}-\frac {a \left (a+b \sqrt [3]{x}\right )^{15}}{b^6}+\frac {3 \left (a+b \sqrt [3]{x}\right )^{16}}{16 b^6}\\ \end {align*}

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Mathematica [A]  time = 0.09, size = 76, normalized size = 0.63 \[ -\frac {\left (a+b \sqrt [3]{x}\right )^{11} \left (a^5-11 a^4 b \sqrt [3]{x}+66 a^3 b^2 x^{2/3}-286 a^2 b^3 x+1001 a b^4 x^{4/3}-3003 b^5 x^{5/3}\right )}{16016 b^6} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^(1/3))^10*x,x]

[Out]

-1/16016*((a + b*x^(1/3))^11*(a^5 - 11*a^4*b*x^(1/3) + 66*a^3*b^2*x^(2/3) - 286*a^2*b^3*x + 1001*a*b^4*x^(4/3)
- 3003*b^5*x^(5/3)))/b^6

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fricas [A]  time = 0.89, size = 124, normalized size = 1.03 \[ 2 \, a b^{9} x^{5} + \frac {105}{2} \, a^{4} b^{6} x^{4} + 40 \, a^{7} b^{3} x^{3} + \frac {1}{2} \, a^{10} x^{2} + \frac {27}{616} \, {\left (220 \, a^{2} b^{8} x^{4} + 1568 \, a^{5} b^{5} x^{3} + 385 \, a^{8} b^{2} x^{2}\right )} x^{\frac {2}{3}} + \frac {3}{1456} \, {\left (91 \, b^{10} x^{5} + 13440 \, a^{3} b^{7} x^{4} + 30576 \, a^{6} b^{4} x^{3} + 2080 \, a^{9} b x^{2}\right )} x^{\frac {1}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*a*b^9*x^5 + 105/2*a^4*b^6*x^4 + 40*a^7*b^3*x^3 + 1/2*a^10*x^2 + 27/616*(220*a^2*b^8*x^4 + 1568*a^5*b^5*x^3 +
385*a^8*b^2*x^2)*x^(2/3) + 3/1456*(91*b^10*x^5 + 13440*a^3*b^7*x^4 + 30576*a^6*b^4*x^3 + 2080*a^9*b*x^2)*x^(1
/3)

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giac [A]  time = 0.19, size = 112, normalized size = 0.93 \[ \frac {3}{16} \, b^{10} x^{\frac {16}{3}} + 2 \, a b^{9} x^{5} + \frac {135}{14} \, a^{2} b^{8} x^{\frac {14}{3}} + \frac {360}{13} \, a^{3} b^{7} x^{\frac {13}{3}} + \frac {105}{2} \, a^{4} b^{6} x^{4} + \frac {756}{11} \, a^{5} b^{5} x^{\frac {11}{3}} + 63 \, a^{6} b^{4} x^{\frac {10}{3}} + 40 \, a^{7} b^{3} x^{3} + \frac {135}{8} \, a^{8} b^{2} x^{\frac {8}{3}} + \frac {30}{7} \, a^{9} b x^{\frac {7}{3}} + \frac {1}{2} \, a^{10} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/16*b^10*x^(16/3) + 2*a*b^9*x^5 + 135/14*a^2*b^8*x^(14/3) + 360/13*a^3*b^7*x^(13/3) + 105/2*a^4*b^6*x^4 + 756
/11*a^5*b^5*x^(11/3) + 63*a^6*b^4*x^(10/3) + 40*a^7*b^3*x^3 + 135/8*a^8*b^2*x^(8/3) + 30/7*a^9*b*x^(7/3) + 1/2
*a^10*x^2

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maple [A]  time = 0.00, size = 113, normalized size = 0.94 \[ \frac {3 b^{10} x^{\frac {16}{3}}}{16}+2 a \,b^{9} x^{5}+\frac {135 a^{2} b^{8} x^{\frac {14}{3}}}{14}+\frac {360 a^{3} b^{7} x^{\frac {13}{3}}}{13}+\frac {105 a^{4} b^{6} x^{4}}{2}+\frac {756 a^{5} b^{5} x^{\frac {11}{3}}}{11}+63 a^{6} b^{4} x^{\frac {10}{3}}+40 a^{7} b^{3} x^{3}+\frac {135 a^{8} b^{2} x^{\frac {8}{3}}}{8}+\frac {30 a^{9} b \,x^{\frac {7}{3}}}{7}+\frac {a^{10} x^{2}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/16*b^10*x^(16/3)+2*a*b^9*x^5+135/14*a^2*b^8*x^(14/3)+360/13*a^3*b^7*x^(13/3)+105/2*a^4*b^6*x^4+756/11*a^5*b^
5*x^(11/3)+63*a^6*b^4*x^(10/3)+40*a^7*b^3*x^3+135/8*a^8*b^2*x^(8/3)+30/7*a^9*b*x^(7/3)+1/2*x^2*a^10

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maxima [A]  time = 0.99, size = 98, normalized size = 0.82 \[ \frac {3 \, {\left (b x^{\frac {1}{3}} + a\right )}^{16}}{16 \, b^{6}} - \frac {{\left (b x^{\frac {1}{3}} + a\right )}^{15} a}{b^{6}} + \frac {15 \, {\left (b x^{\frac {1}{3}} + a\right )}^{14} a^{2}}{7 \, b^{6}} - \frac {30 \, {\left (b x^{\frac {1}{3}} + a\right )}^{13} a^{3}}{13 \, b^{6}} + \frac {5 \, {\left (b x^{\frac {1}{3}} + a\right )}^{12} a^{4}}{4 \, b^{6}} - \frac {3 \, {\left (b x^{\frac {1}{3}} + a\right )}^{11} a^{5}}{11 \, b^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/16*(b*x^(1/3) + a)^16/b^6 - (b*x^(1/3) + a)^15*a/b^6 + 15/7*(b*x^(1/3) + a)^14*a^2/b^6 - 30/13*(b*x^(1/3) +
a)^13*a^3/b^6 + 5/4*(b*x^(1/3) + a)^12*a^4/b^6 - 3/11*(b*x^(1/3) + a)^11*a^5/b^6

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mupad [B]  time = 0.06, size = 112, normalized size = 0.93 \[ \frac {a^{10}\,x^2}{2}+\frac {3\,b^{10}\,x^{16/3}}{16}+2\,a\,b^9\,x^5+\frac {30\,a^9\,b\,x^{7/3}}{7}+40\,a^7\,b^3\,x^3+\frac {105\,a^4\,b^6\,x^4}{2}+\frac {135\,a^8\,b^2\,x^{8/3}}{8}+63\,a^6\,b^4\,x^{10/3}+\frac {756\,a^5\,b^5\,x^{11/3}}{11}+\frac {360\,a^3\,b^7\,x^{13/3}}{13}+\frac {135\,a^2\,b^8\,x^{14/3}}{14} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(a + b*x^(1/3))^10,x)

[Out]

(a^10*x^2)/2 + (3*b^10*x^(16/3))/16 + 2*a*b^9*x^5 + (30*a^9*b*x^(7/3))/7 + 40*a^7*b^3*x^3 + (105*a^4*b^6*x^4)/
2 + (135*a^8*b^2*x^(8/3))/8 + 63*a^6*b^4*x^(10/3) + (756*a^5*b^5*x^(11/3))/11 + (360*a^3*b^7*x^(13/3))/13 + (1
35*a^2*b^8*x^(14/3))/14

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sympy [A]  time = 3.56, size = 143, normalized size = 1.19 \[ \frac {a^{10} x^{2}}{2} + \frac {30 a^{9} b x^{\frac {7}{3}}}{7} + \frac {135 a^{8} b^{2} x^{\frac {8}{3}}}{8} + 40 a^{7} b^{3} x^{3} + 63 a^{6} b^{4} x^{\frac {10}{3}} + \frac {756 a^{5} b^{5} x^{\frac {11}{3}}}{11} + \frac {105 a^{4} b^{6} x^{4}}{2} + \frac {360 a^{3} b^{7} x^{\frac {13}{3}}}{13} + \frac {135 a^{2} b^{8} x^{\frac {14}{3}}}{14} + 2 a b^{9} x^{5} + \frac {3 b^{10} x^{\frac {16}{3}}}{16} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**10*x**2/2 + 30*a**9*b*x**(7/3)/7 + 135*a**8*b**2*x**(8/3)/8 + 40*a**7*b**3*x**3 + 63*a**6*b**4*x**(10/3) +
756*a**5*b**5*x**(11/3)/11 + 105*a**4*b**6*x**4/2 + 360*a**3*b**7*x**(13/3)/13 + 135*a**2*b**8*x**(14/3)/14 +
2*a*b**9*x**5 + 3*b**10*x**(16/3)/16

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