3.2305 \(\int \left (a+b \sqrt [3]{x}\right )^5 x^4 \, dx\)

Optimal. Leaf size=77 \[ \frac{a^5 x^5}{5}+\frac{15}{16} a^4 b x^{16/3}+\frac{30}{17} a^3 b^2 x^{17/3}+\frac{5}{3} a^2 b^3 x^6+\frac{15}{19} a b^4 x^{19/3}+\frac{3}{20} b^5 x^{20/3} \]

[Out]

(a^5*x^5)/5 + (15*a^4*b*x^(16/3))/16 + (30*a^3*b^2*x^(17/3))/17 + (5*a^2*b^3*x^6
)/3 + (15*a*b^4*x^(19/3))/19 + (3*b^5*x^(20/3))/20

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Rubi [A]  time = 0.120376, antiderivative size = 77, 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 \[ \frac{a^5 x^5}{5}+\frac{15}{16} a^4 b x^{16/3}+\frac{30}{17} a^3 b^2 x^{17/3}+\frac{5}{3} a^2 b^3 x^6+\frac{15}{19} a b^4 x^{19/3}+\frac{3}{20} b^5 x^{20/3} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^(1/3))^5*x^4,x]

[Out]

(a^5*x^5)/5 + (15*a^4*b*x^(16/3))/16 + (30*a^3*b^2*x^(17/3))/17 + (5*a^2*b^3*x^6
)/3 + (15*a*b^4*x^(19/3))/19 + (3*b^5*x^(20/3))/20

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Rubi in Sympy [A]  time = 21.3437, size = 75, normalized size = 0.97 \[ \frac{a^{5} x^{5}}{5} + \frac{15 a^{4} b x^{\frac{16}{3}}}{16} + \frac{30 a^{3} b^{2} x^{\frac{17}{3}}}{17} + \frac{5 a^{2} b^{3} x^{6}}{3} + \frac{15 a b^{4} x^{\frac{19}{3}}}{19} + \frac{3 b^{5} x^{\frac{20}{3}}}{20} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b*x**(1/3))**5*x**4,x)

[Out]

a**5*x**5/5 + 15*a**4*b*x**(16/3)/16 + 30*a**3*b**2*x**(17/3)/17 + 5*a**2*b**3*x
**6/3 + 15*a*b**4*x**(19/3)/19 + 3*b**5*x**(20/3)/20

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Mathematica [A]  time = 0.0161966, size = 77, normalized size = 1. \[ \frac{a^5 x^5}{5}+\frac{15}{16} a^4 b x^{16/3}+\frac{30}{17} a^3 b^2 x^{17/3}+\frac{5}{3} a^2 b^3 x^6+\frac{15}{19} a b^4 x^{19/3}+\frac{3}{20} b^5 x^{20/3} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^(1/3))^5*x^4,x]

[Out]

(a^5*x^5)/5 + (15*a^4*b*x^(16/3))/16 + (30*a^3*b^2*x^(17/3))/17 + (5*a^2*b^3*x^6
)/3 + (15*a*b^4*x^(19/3))/19 + (3*b^5*x^(20/3))/20

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Maple [A]  time = 0.003, size = 58, normalized size = 0.8 \[{\frac{{a}^{5}{x}^{5}}{5}}+{\frac{15\,{a}^{4}b}{16}{x}^{{\frac{16}{3}}}}+{\frac{30\,{a}^{3}{b}^{2}}{17}{x}^{{\frac{17}{3}}}}+{\frac{5\,{a}^{2}{b}^{3}{x}^{6}}{3}}+{\frac{15\,a{b}^{4}}{19}{x}^{{\frac{19}{3}}}}+{\frac{3\,{b}^{5}}{20}{x}^{{\frac{20}{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b*x^(1/3))^5*x^4,x)

[Out]

1/5*a^5*x^5+15/16*a^4*b*x^(16/3)+30/17*a^3*b^2*x^(17/3)+5/3*a^2*b^3*x^6+15/19*a*
b^4*x^(19/3)+3/20*b^5*x^(20/3)

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Maxima [A]  time = 1.43507, size = 339, normalized size = 4.4 \[ \frac{3 \,{\left (b x^{\frac{1}{3}} + a\right )}^{20}}{20 \, b^{15}} - \frac{42 \,{\left (b x^{\frac{1}{3}} + a\right )}^{19} a}{19 \, b^{15}} + \frac{91 \,{\left (b x^{\frac{1}{3}} + a\right )}^{18} a^{2}}{6 \, b^{15}} - \frac{1092 \,{\left (b x^{\frac{1}{3}} + a\right )}^{17} a^{3}}{17 \, b^{15}} + \frac{3003 \,{\left (b x^{\frac{1}{3}} + a\right )}^{16} a^{4}}{16 \, b^{15}} - \frac{2002 \,{\left (b x^{\frac{1}{3}} + a\right )}^{15} a^{5}}{5 \, b^{15}} + \frac{1287 \,{\left (b x^{\frac{1}{3}} + a\right )}^{14} a^{6}}{2 \, b^{15}} - \frac{792 \,{\left (b x^{\frac{1}{3}} + a\right )}^{13} a^{7}}{b^{15}} + \frac{3003 \,{\left (b x^{\frac{1}{3}} + a\right )}^{12} a^{8}}{4 \, b^{15}} - \frac{546 \,{\left (b x^{\frac{1}{3}} + a\right )}^{11} a^{9}}{b^{15}} + \frac{3003 \,{\left (b x^{\frac{1}{3}} + a\right )}^{10} a^{10}}{10 \, b^{15}} - \frac{364 \,{\left (b x^{\frac{1}{3}} + a\right )}^{9} a^{11}}{3 \, b^{15}} + \frac{273 \,{\left (b x^{\frac{1}{3}} + a\right )}^{8} a^{12}}{8 \, b^{15}} - \frac{6 \,{\left (b x^{\frac{1}{3}} + a\right )}^{7} a^{13}}{b^{15}} + \frac{{\left (b x^{\frac{1}{3}} + a\right )}^{6} a^{14}}{2 \, b^{15}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^(1/3) + a)^5*x^4,x, algorithm="maxima")

[Out]

3/20*(b*x^(1/3) + a)^20/b^15 - 42/19*(b*x^(1/3) + a)^19*a/b^15 + 91/6*(b*x^(1/3)
+ a)^18*a^2/b^15 - 1092/17*(b*x^(1/3) + a)^17*a^3/b^15 + 3003/16*(b*x^(1/3) + a
)^16*a^4/b^15 - 2002/5*(b*x^(1/3) + a)^15*a^5/b^15 + 1287/2*(b*x^(1/3) + a)^14*a
^6/b^15 - 792*(b*x^(1/3) + a)^13*a^7/b^15 + 3003/4*(b*x^(1/3) + a)^12*a^8/b^15 -
546*(b*x^(1/3) + a)^11*a^9/b^15 + 3003/10*(b*x^(1/3) + a)^10*a^10/b^15 - 364/3*
(b*x^(1/3) + a)^9*a^11/b^15 + 273/8*(b*x^(1/3) + a)^8*a^12/b^15 - 6*(b*x^(1/3) +
a)^7*a^13/b^15 + 1/2*(b*x^(1/3) + a)^6*a^14/b^15

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Fricas [A]  time = 0.212505, size = 93, normalized size = 1.21 \[ \frac{5}{3} \, a^{2} b^{3} x^{6} + \frac{1}{5} \, a^{5} x^{5} + \frac{3}{340} \,{\left (17 \, b^{5} x^{6} + 200 \, a^{3} b^{2} x^{5}\right )} x^{\frac{2}{3}} + \frac{15}{304} \,{\left (16 \, a b^{4} x^{6} + 19 \, a^{4} b x^{5}\right )} x^{\frac{1}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^(1/3) + a)^5*x^4,x, algorithm="fricas")

[Out]

5/3*a^2*b^3*x^6 + 1/5*a^5*x^5 + 3/340*(17*b^5*x^6 + 200*a^3*b^2*x^5)*x^(2/3) + 1
5/304*(16*a*b^4*x^6 + 19*a^4*b*x^5)*x^(1/3)

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Sympy [A]  time = 9.91422, size = 75, normalized size = 0.97 \[ \frac{a^{5} x^{5}}{5} + \frac{15 a^{4} b x^{\frac{16}{3}}}{16} + \frac{30 a^{3} b^{2} x^{\frac{17}{3}}}{17} + \frac{5 a^{2} b^{3} x^{6}}{3} + \frac{15 a b^{4} x^{\frac{19}{3}}}{19} + \frac{3 b^{5} x^{\frac{20}{3}}}{20} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b*x**(1/3))**5*x**4,x)

[Out]

a**5*x**5/5 + 15*a**4*b*x**(16/3)/16 + 30*a**3*b**2*x**(17/3)/17 + 5*a**2*b**3*x
**6/3 + 15*a*b**4*x**(19/3)/19 + 3*b**5*x**(20/3)/20

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GIAC/XCAS [A]  time = 0.240798, size = 77, normalized size = 1. \[ \frac{3}{20} \, b^{5} x^{\frac{20}{3}} + \frac{15}{19} \, a b^{4} x^{\frac{19}{3}} + \frac{5}{3} \, a^{2} b^{3} x^{6} + \frac{30}{17} \, a^{3} b^{2} x^{\frac{17}{3}} + \frac{15}{16} \, a^{4} b x^{\frac{16}{3}} + \frac{1}{5} \, a^{5} x^{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^(1/3) + a)^5*x^4,x, algorithm="giac")

[Out]

3/20*b^5*x^(20/3) + 15/19*a*b^4*x^(19/3) + 5/3*a^2*b^3*x^6 + 30/17*a^3*b^2*x^(17
/3) + 15/16*a^4*b*x^(16/3) + 1/5*a^5*x^5