3.2335 \(\int \left (a+b \sqrt [3]{x}\right )^{15} x^2 \, dx\)

Optimal. Leaf size=183 \[ \frac{3 a^8 \left (a+b \sqrt [3]{x}\right )^{16}}{16 b^9}-\frac{24 a^7 \left (a+b \sqrt [3]{x}\right )^{17}}{17 b^9}+\frac{14 a^6 \left (a+b \sqrt [3]{x}\right )^{18}}{3 b^9}-\frac{168 a^5 \left (a+b \sqrt [3]{x}\right )^{19}}{19 b^9}+\frac{21 a^4 \left (a+b \sqrt [3]{x}\right )^{20}}{2 b^9}-\frac{8 a^3 \left (a+b \sqrt [3]{x}\right )^{21}}{b^9}+\frac{42 a^2 \left (a+b \sqrt [3]{x}\right )^{22}}{11 b^9}+\frac{\left (a+b \sqrt [3]{x}\right )^{24}}{8 b^9}-\frac{24 a \left (a+b \sqrt [3]{x}\right )^{23}}{23 b^9} \]

[Out]

(3*a^8*(a + b*x^(1/3))^16)/(16*b^9) - (24*a^7*(a + b*x^(1/3))^17)/(17*b^9) + (14
*a^6*(a + b*x^(1/3))^18)/(3*b^9) - (168*a^5*(a + b*x^(1/3))^19)/(19*b^9) + (21*a
^4*(a + b*x^(1/3))^20)/(2*b^9) - (8*a^3*(a + b*x^(1/3))^21)/b^9 + (42*a^2*(a + b
*x^(1/3))^22)/(11*b^9) - (24*a*(a + b*x^(1/3))^23)/(23*b^9) + (a + b*x^(1/3))^24
/(8*b^9)

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Rubi [A]  time = 0.289039, antiderivative size = 183, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{3 a^8 \left (a+b \sqrt [3]{x}\right )^{16}}{16 b^9}-\frac{24 a^7 \left (a+b \sqrt [3]{x}\right )^{17}}{17 b^9}+\frac{14 a^6 \left (a+b \sqrt [3]{x}\right )^{18}}{3 b^9}-\frac{168 a^5 \left (a+b \sqrt [3]{x}\right )^{19}}{19 b^9}+\frac{21 a^4 \left (a+b \sqrt [3]{x}\right )^{20}}{2 b^9}-\frac{8 a^3 \left (a+b \sqrt [3]{x}\right )^{21}}{b^9}+\frac{42 a^2 \left (a+b \sqrt [3]{x}\right )^{22}}{11 b^9}+\frac{\left (a+b \sqrt [3]{x}\right )^{24}}{8 b^9}-\frac{24 a \left (a+b \sqrt [3]{x}\right )^{23}}{23 b^9} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^(1/3))^15*x^2,x]

[Out]

(3*a^8*(a + b*x^(1/3))^16)/(16*b^9) - (24*a^7*(a + b*x^(1/3))^17)/(17*b^9) + (14
*a^6*(a + b*x^(1/3))^18)/(3*b^9) - (168*a^5*(a + b*x^(1/3))^19)/(19*b^9) + (21*a
^4*(a + b*x^(1/3))^20)/(2*b^9) - (8*a^3*(a + b*x^(1/3))^21)/b^9 + (42*a^2*(a + b
*x^(1/3))^22)/(11*b^9) - (24*a*(a + b*x^(1/3))^23)/(23*b^9) + (a + b*x^(1/3))^24
/(8*b^9)

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Rubi in Sympy [A]  time = 53.7429, size = 173, normalized size = 0.95 \[ \frac{3 a^{8} \left (a + b \sqrt [3]{x}\right )^{16}}{16 b^{9}} - \frac{24 a^{7} \left (a + b \sqrt [3]{x}\right )^{17}}{17 b^{9}} + \frac{14 a^{6} \left (a + b \sqrt [3]{x}\right )^{18}}{3 b^{9}} - \frac{168 a^{5} \left (a + b \sqrt [3]{x}\right )^{19}}{19 b^{9}} + \frac{21 a^{4} \left (a + b \sqrt [3]{x}\right )^{20}}{2 b^{9}} - \frac{8 a^{3} \left (a + b \sqrt [3]{x}\right )^{21}}{b^{9}} + \frac{42 a^{2} \left (a + b \sqrt [3]{x}\right )^{22}}{11 b^{9}} - \frac{24 a \left (a + b \sqrt [3]{x}\right )^{23}}{23 b^{9}} + \frac{\left (a + b \sqrt [3]{x}\right )^{24}}{8 b^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b*x**(1/3))**15*x**2,x)

[Out]

3*a**8*(a + b*x**(1/3))**16/(16*b**9) - 24*a**7*(a + b*x**(1/3))**17/(17*b**9) +
14*a**6*(a + b*x**(1/3))**18/(3*b**9) - 168*a**5*(a + b*x**(1/3))**19/(19*b**9)
+ 21*a**4*(a + b*x**(1/3))**20/(2*b**9) - 8*a**3*(a + b*x**(1/3))**21/b**9 + 42
*a**2*(a + b*x**(1/3))**22/(11*b**9) - 24*a*(a + b*x**(1/3))**23/(23*b**9) + (a
+ b*x**(1/3))**24/(8*b**9)

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Mathematica [A]  time = 0.0336197, size = 213, normalized size = 1.16 \[ \frac{a^{15} x^3}{3}+\frac{9}{2} a^{14} b x^{10/3}+\frac{315}{11} a^{13} b^2 x^{11/3}+\frac{455}{4} a^{12} b^3 x^4+315 a^{11} b^4 x^{13/3}+\frac{1287}{2} a^{10} b^5 x^{14/3}+1001 a^9 b^6 x^5+\frac{19305}{16} a^8 b^7 x^{16/3}+\frac{19305}{17} a^7 b^8 x^{17/3}+\frac{5005}{6} a^6 b^9 x^6+\frac{9009}{19} a^5 b^{10} x^{19/3}+\frac{819}{4} a^4 b^{11} x^{20/3}+65 a^3 b^{12} x^7+\frac{315}{22} a^2 b^{13} x^{22/3}+\frac{45}{23} a b^{14} x^{23/3}+\frac{b^{15} x^8}{8} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^(1/3))^15*x^2,x]

[Out]

(a^15*x^3)/3 + (9*a^14*b*x^(10/3))/2 + (315*a^13*b^2*x^(11/3))/11 + (455*a^12*b^
3*x^4)/4 + 315*a^11*b^4*x^(13/3) + (1287*a^10*b^5*x^(14/3))/2 + 1001*a^9*b^6*x^5
+ (19305*a^8*b^7*x^(16/3))/16 + (19305*a^7*b^8*x^(17/3))/17 + (5005*a^6*b^9*x^6
)/6 + (9009*a^5*b^10*x^(19/3))/19 + (819*a^4*b^11*x^(20/3))/4 + 65*a^3*b^12*x^7
+ (315*a^2*b^13*x^(22/3))/22 + (45*a*b^14*x^(23/3))/23 + (b^15*x^8)/8

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Maple [A]  time = 0.004, size = 168, normalized size = 0.9 \[{\frac{{b}^{15}{x}^{8}}{8}}+{\frac{45\,a{b}^{14}}{23}{x}^{{\frac{23}{3}}}}+{\frac{315\,{a}^{2}{b}^{13}}{22}{x}^{{\frac{22}{3}}}}+65\,{x}^{7}{a}^{3}{b}^{12}+{\frac{819\,{a}^{4}{b}^{11}}{4}{x}^{{\frac{20}{3}}}}+{\frac{9009\,{a}^{5}{b}^{10}}{19}{x}^{{\frac{19}{3}}}}+{\frac{5005\,{a}^{6}{b}^{9}{x}^{6}}{6}}+{\frac{19305\,{a}^{7}{b}^{8}}{17}{x}^{{\frac{17}{3}}}}+{\frac{19305\,{a}^{8}{b}^{7}}{16}{x}^{{\frac{16}{3}}}}+1001\,{x}^{5}{a}^{9}{b}^{6}+{\frac{1287\,{a}^{10}{b}^{5}}{2}{x}^{{\frac{14}{3}}}}+315\,{a}^{11}{b}^{4}{x}^{13/3}+{\frac{455\,{a}^{12}{b}^{3}{x}^{4}}{4}}+{\frac{315\,{a}^{13}{b}^{2}}{11}{x}^{{\frac{11}{3}}}}+{\frac{9\,{a}^{14}b}{2}{x}^{{\frac{10}{3}}}}+{\frac{{x}^{3}{a}^{15}}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b*x^(1/3))^15*x^2,x)

[Out]

1/8*b^15*x^8+45/23*a*b^14*x^(23/3)+315/22*a^2*b^13*x^(22/3)+65*x^7*a^3*b^12+819/
4*a^4*b^11*x^(20/3)+9009/19*a^5*b^10*x^(19/3)+5005/6*a^6*b^9*x^6+19305/17*a^7*b^
8*x^(17/3)+19305/16*a^8*b^7*x^(16/3)+1001*x^5*a^9*b^6+1287/2*a^10*b^5*x^(14/3)+3
15*a^11*b^4*x^(13/3)+455/4*a^12*b^3*x^4+315/11*a^13*b^2*x^(11/3)+9/2*a^14*b*x^(1
0/3)+1/3*x^3*a^15

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Maxima [A]  time = 1.45724, size = 201, normalized size = 1.1 \[ \frac{{\left (b x^{\frac{1}{3}} + a\right )}^{24}}{8 \, b^{9}} - \frac{24 \,{\left (b x^{\frac{1}{3}} + a\right )}^{23} a}{23 \, b^{9}} + \frac{42 \,{\left (b x^{\frac{1}{3}} + a\right )}^{22} a^{2}}{11 \, b^{9}} - \frac{8 \,{\left (b x^{\frac{1}{3}} + a\right )}^{21} a^{3}}{b^{9}} + \frac{21 \,{\left (b x^{\frac{1}{3}} + a\right )}^{20} a^{4}}{2 \, b^{9}} - \frac{168 \,{\left (b x^{\frac{1}{3}} + a\right )}^{19} a^{5}}{19 \, b^{9}} + \frac{14 \,{\left (b x^{\frac{1}{3}} + a\right )}^{18} a^{6}}{3 \, b^{9}} - \frac{24 \,{\left (b x^{\frac{1}{3}} + a\right )}^{17} a^{7}}{17 \, b^{9}} + \frac{3 \,{\left (b x^{\frac{1}{3}} + a\right )}^{16} a^{8}}{16 \, b^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^(1/3) + a)^15*x^2,x, algorithm="maxima")

[Out]

1/8*(b*x^(1/3) + a)^24/b^9 - 24/23*(b*x^(1/3) + a)^23*a/b^9 + 42/11*(b*x^(1/3) +
a)^22*a^2/b^9 - 8*(b*x^(1/3) + a)^21*a^3/b^9 + 21/2*(b*x^(1/3) + a)^20*a^4/b^9
- 168/19*(b*x^(1/3) + a)^19*a^5/b^9 + 14/3*(b*x^(1/3) + a)^18*a^6/b^9 - 24/17*(b
*x^(1/3) + a)^17*a^7/b^9 + 3/16*(b*x^(1/3) + a)^16*a^8/b^9

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Fricas [A]  time = 0.214222, size = 242, normalized size = 1.32 \[ \frac{1}{8} \, b^{15} x^{8} + 65 \, a^{3} b^{12} x^{7} + \frac{5005}{6} \, a^{6} b^{9} x^{6} + 1001 \, a^{9} b^{6} x^{5} + \frac{455}{4} \, a^{12} b^{3} x^{4} + \frac{1}{3} \, a^{15} x^{3} + \frac{9}{17204} \,{\left (3740 \, a b^{14} x^{7} + 391391 \, a^{4} b^{11} x^{6} + 2170740 \, a^{7} b^{8} x^{5} + 1230086 \, a^{10} b^{5} x^{4} + 54740 \, a^{13} b^{2} x^{3}\right )} x^{\frac{2}{3}} + \frac{9}{3344} \,{\left (5320 \, a^{2} b^{13} x^{7} + 176176 \, a^{5} b^{10} x^{6} + 448305 \, a^{8} b^{7} x^{5} + 117040 \, a^{11} b^{4} x^{4} + 1672 \, a^{14} b x^{3}\right )} x^{\frac{1}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^(1/3) + a)^15*x^2,x, algorithm="fricas")

[Out]

1/8*b^15*x^8 + 65*a^3*b^12*x^7 + 5005/6*a^6*b^9*x^6 + 1001*a^9*b^6*x^5 + 455/4*a
^12*b^3*x^4 + 1/3*a^15*x^3 + 9/17204*(3740*a*b^14*x^7 + 391391*a^4*b^11*x^6 + 21
70740*a^7*b^8*x^5 + 1230086*a^10*b^5*x^4 + 54740*a^13*b^2*x^3)*x^(2/3) + 9/3344*
(5320*a^2*b^13*x^7 + 176176*a^5*b^10*x^6 + 448305*a^8*b^7*x^5 + 117040*a^11*b^4*
x^4 + 1672*a^14*b*x^3)*x^(1/3)

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Sympy [A]  time = 17.3199, size = 214, normalized size = 1.17 \[ \frac{a^{15} x^{3}}{3} + \frac{9 a^{14} b x^{\frac{10}{3}}}{2} + \frac{315 a^{13} b^{2} x^{\frac{11}{3}}}{11} + \frac{455 a^{12} b^{3} x^{4}}{4} + 315 a^{11} b^{4} x^{\frac{13}{3}} + \frac{1287 a^{10} b^{5} x^{\frac{14}{3}}}{2} + 1001 a^{9} b^{6} x^{5} + \frac{19305 a^{8} b^{7} x^{\frac{16}{3}}}{16} + \frac{19305 a^{7} b^{8} x^{\frac{17}{3}}}{17} + \frac{5005 a^{6} b^{9} x^{6}}{6} + \frac{9009 a^{5} b^{10} x^{\frac{19}{3}}}{19} + \frac{819 a^{4} b^{11} x^{\frac{20}{3}}}{4} + 65 a^{3} b^{12} x^{7} + \frac{315 a^{2} b^{13} x^{\frac{22}{3}}}{22} + \frac{45 a b^{14} x^{\frac{23}{3}}}{23} + \frac{b^{15} x^{8}}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b*x**(1/3))**15*x**2,x)

[Out]

a**15*x**3/3 + 9*a**14*b*x**(10/3)/2 + 315*a**13*b**2*x**(11/3)/11 + 455*a**12*b
**3*x**4/4 + 315*a**11*b**4*x**(13/3) + 1287*a**10*b**5*x**(14/3)/2 + 1001*a**9*
b**6*x**5 + 19305*a**8*b**7*x**(16/3)/16 + 19305*a**7*b**8*x**(17/3)/17 + 5005*a
**6*b**9*x**6/6 + 9009*a**5*b**10*x**(19/3)/19 + 819*a**4*b**11*x**(20/3)/4 + 65
*a**3*b**12*x**7 + 315*a**2*b**13*x**(22/3)/22 + 45*a*b**14*x**(23/3)/23 + b**15
*x**8/8

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GIAC/XCAS [A]  time = 0.222511, size = 225, normalized size = 1.23 \[ \frac{1}{8} \, b^{15} x^{8} + \frac{45}{23} \, a b^{14} x^{\frac{23}{3}} + \frac{315}{22} \, a^{2} b^{13} x^{\frac{22}{3}} + 65 \, a^{3} b^{12} x^{7} + \frac{819}{4} \, a^{4} b^{11} x^{\frac{20}{3}} + \frac{9009}{19} \, a^{5} b^{10} x^{\frac{19}{3}} + \frac{5005}{6} \, a^{6} b^{9} x^{6} + \frac{19305}{17} \, a^{7} b^{8} x^{\frac{17}{3}} + \frac{19305}{16} \, a^{8} b^{7} x^{\frac{16}{3}} + 1001 \, a^{9} b^{6} x^{5} + \frac{1287}{2} \, a^{10} b^{5} x^{\frac{14}{3}} + 315 \, a^{11} b^{4} x^{\frac{13}{3}} + \frac{455}{4} \, a^{12} b^{3} x^{4} + \frac{315}{11} \, a^{13} b^{2} x^{\frac{11}{3}} + \frac{9}{2} \, a^{14} b x^{\frac{10}{3}} + \frac{1}{3} \, a^{15} x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^(1/3) + a)^15*x^2,x, algorithm="giac")

[Out]

1/8*b^15*x^8 + 45/23*a*b^14*x^(23/3) + 315/22*a^2*b^13*x^(22/3) + 65*a^3*b^12*x^
7 + 819/4*a^4*b^11*x^(20/3) + 9009/19*a^5*b^10*x^(19/3) + 5005/6*a^6*b^9*x^6 + 1
9305/17*a^7*b^8*x^(17/3) + 19305/16*a^8*b^7*x^(16/3) + 1001*a^9*b^6*x^5 + 1287/2
*a^10*b^5*x^(14/3) + 315*a^11*b^4*x^(13/3) + 455/4*a^12*b^3*x^4 + 315/11*a^13*b^
2*x^(11/3) + 9/2*a^14*b*x^(10/3) + 1/3*a^15*x^3