3.2320 \(\int \left (a+b \sqrt [3]{x}\right )^{10} x \, dx\)

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*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)

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Rubi [A]  time = 0.165851, antiderivative size = 120, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ -\frac{3 a^5 \left (a+b \sqrt [3]{x}\right )^{11}}{11 b^6}+\frac{5 a^4 \left (a+b \sqrt [3]{x}\right )^{12}}{4 b^6}-\frac{30 a^3 \left (a+b \sqrt [3]{x}\right )^{13}}{13 b^6}+\frac{15 a^2 \left (a+b \sqrt [3]{x}\right )^{14}}{7 b^6}+\frac{3 \left (a+b \sqrt [3]{x}\right )^{16}}{16 b^6}-\frac{a \left (a+b \sqrt [3]{x}\right )^{15}}{b^6} \]

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)

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Rubi in Sympy [A]  time = 30.0399, size = 112, normalized size = 0.93 \[ - \frac{3 a^{5} \left (a + b \sqrt [3]{x}\right )^{11}}{11 b^{6}} + \frac{5 a^{4} \left (a + b \sqrt [3]{x}\right )^{12}}{4 b^{6}} - \frac{30 a^{3} \left (a + b \sqrt [3]{x}\right )^{13}}{13 b^{6}} + \frac{15 a^{2} \left (a + b \sqrt [3]{x}\right )^{14}}{7 b^{6}} - \frac{a \left (a + b \sqrt [3]{x}\right )^{15}}{b^{6}} + \frac{3 \left (a + b \sqrt [3]{x}\right )^{16}}{16 b^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((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)

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Mathematica [A]  time = 0.022489, size = 142, normalized size = 1.18 \[ \frac{a^{10} x^2}{2}+\frac{30}{7} a^9 b x^{7/3}+\frac{135}{8} a^8 b^2 x^{8/3}+40 a^7 b^3 x^3+63 a^6 b^4 x^{10/3}+\frac{756}{11} a^5 b^5 x^{11/3}+\frac{105}{2} a^4 b^6 x^4+\frac{360}{13} a^3 b^7 x^{13/3}+\frac{135}{14} a^2 b^8 x^{14/3}+2 a b^9 x^5+\frac{3}{16} b^{10} x^{16/3} \]

Antiderivative was successfully verified.

[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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Maple [A]  time = 0.003, size = 113, normalized size = 0.9 \[{\frac{3\,{b}^{10}}{16}{x}^{{\frac{16}{3}}}}+2\,a{b}^{9}{x}^{5}+{\frac{135\,{a}^{2}{b}^{8}}{14}{x}^{{\frac{14}{3}}}}+{\frac{360\,{a}^{3}{b}^{7}}{13}{x}^{{\frac{13}{3}}}}+{\frac{105\,{x}^{4}{a}^{4}{b}^{6}}{2}}+{\frac{756\,{a}^{5}{b}^{5}}{11}{x}^{{\frac{11}{3}}}}+63\,{a}^{6}{b}^{4}{x}^{10/3}+40\,{a}^{7}{b}^{3}{x}^{3}+{\frac{135\,{a}^{8}{b}^{2}}{8}{x}^{{\frac{8}{3}}}}+{\frac{30\,{a}^{9}b}{7}{x}^{{\frac{7}{3}}}}+{\frac{{x}^{2}{a}^{10}}{2}} \]

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)+1
05/2*x^4*a^4*b^6+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 = 1.44269, size = 132, normalized size = 1.1 \[ \frac{3 \,{\left (b x^{\frac{1}{3}} + a\right )}^{16}}{16 \, b^{6}} - \frac{{\left (b x^{\frac{1}{3}} + a\right )}^{15} a}{b^{6}} + \frac{15 \,{\left (b x^{\frac{1}{3}} + a\right )}^{14} a^{2}}{7 \, b^{6}} - \frac{30 \,{\left (b x^{\frac{1}{3}} + a\right )}^{13} a^{3}}{13 \, b^{6}} + \frac{5 \,{\left (b x^{\frac{1}{3}} + a\right )}^{12} a^{4}}{4 \, b^{6}} - \frac{3 \,{\left (b x^{\frac{1}{3}} + a\right )}^{11} a^{5}}{11 \, b^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^(1/3) + a)^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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Fricas [A]  time = 0.236912, size = 167, normalized size = 1.39 \[ 2 \, a b^{9} x^{5} + \frac{105}{2} \, a^{4} b^{6} x^{4} + 40 \, a^{7} b^{3} x^{3} + \frac{1}{2} \, a^{10} x^{2} + \frac{27}{616} \,{\left (220 \, a^{2} b^{8} x^{4} + 1568 \, a^{5} b^{5} x^{3} + 385 \, a^{8} b^{2} x^{2}\right )} x^{\frac{2}{3}} + \frac{3}{1456} \,{\left (91 \, b^{10} x^{5} + 13440 \, a^{3} b^{7} x^{4} + 30576 \, a^{6} b^{4} x^{3} + 2080 \, a^{9} b x^{2}\right )} x^{\frac{1}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^(1/3) + a)^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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Sympy [A]  time = 3.53187, 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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GIAC/XCAS [A]  time = 0.247145, size = 151, normalized size = 1.26 \[ \frac{3}{16} \, b^{10} x^{\frac{16}{3}} + 2 \, a b^{9} x^{5} + \frac{135}{14} \, a^{2} b^{8} x^{\frac{14}{3}} + \frac{360}{13} \, a^{3} b^{7} x^{\frac{13}{3}} + \frac{105}{2} \, a^{4} b^{6} x^{4} + \frac{756}{11} \, a^{5} b^{5} x^{\frac{11}{3}} + 63 \, a^{6} b^{4} x^{\frac{10}{3}} + 40 \, a^{7} b^{3} x^{3} + \frac{135}{8} \, a^{8} b^{2} x^{\frac{8}{3}} + \frac{30}{7} \, a^{9} b x^{\frac{7}{3}} + \frac{1}{2} \, a^{10} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^(1/3) + a)^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^(1
3/3) + 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