3.2306 \(\int \left (a+b \sqrt [3]{x}\right )^5 x^3 \, dx\)

Optimal. Leaf size=75 \[ \frac{a^5 x^4}{4}+\frac{15}{13} a^4 b x^{13/3}+\frac{15}{7} a^3 b^2 x^{14/3}+2 a^2 b^3 x^5+\frac{15}{16} a b^4 x^{16/3}+\frac{3}{17} b^5 x^{17/3} \]

[Out]

(a^5*x^4)/4 + (15*a^4*b*x^(13/3))/13 + (15*a^3*b^2*x^(14/3))/7 + 2*a^2*b^3*x^5 +
(15*a*b^4*x^(16/3))/16 + (3*b^5*x^(17/3))/17

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Rubi [A]  time = 0.111378, antiderivative size = 75, 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 \[ \frac{a^5 x^4}{4}+\frac{15}{13} a^4 b x^{13/3}+\frac{15}{7} a^3 b^2 x^{14/3}+2 a^2 b^3 x^5+\frac{15}{16} a b^4 x^{16/3}+\frac{3}{17} b^5 x^{17/3} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^(1/3))^5*x^3,x]

[Out]

(a^5*x^4)/4 + (15*a^4*b*x^(13/3))/13 + (15*a^3*b^2*x^(14/3))/7 + 2*a^2*b^3*x^5 +
(15*a*b^4*x^(16/3))/16 + (3*b^5*x^(17/3))/17

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Rubi in Sympy [A]  time = 18.7879, size = 73, normalized size = 0.97 \[ \frac{a^{5} x^{4}}{4} + \frac{15 a^{4} b x^{\frac{13}{3}}}{13} + \frac{15 a^{3} b^{2} x^{\frac{14}{3}}}{7} + 2 a^{2} b^{3} x^{5} + \frac{15 a b^{4} x^{\frac{16}{3}}}{16} + \frac{3 b^{5} x^{\frac{17}{3}}}{17} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**5*x**4/4 + 15*a**4*b*x**(13/3)/13 + 15*a**3*b**2*x**(14/3)/7 + 2*a**2*b**3*x*
*5 + 15*a*b**4*x**(16/3)/16 + 3*b**5*x**(17/3)/17

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Mathematica [A]  time = 0.0141429, size = 75, normalized size = 1. \[ \frac{a^5 x^4}{4}+\frac{15}{13} a^4 b x^{13/3}+\frac{15}{7} a^3 b^2 x^{14/3}+2 a^2 b^3 x^5+\frac{15}{16} a b^4 x^{16/3}+\frac{3}{17} b^5 x^{17/3} \]

Antiderivative was successfully verified.

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

[Out]

(a^5*x^4)/4 + (15*a^4*b*x^(13/3))/13 + (15*a^3*b^2*x^(14/3))/7 + 2*a^2*b^3*x^5 +
(15*a*b^4*x^(16/3))/16 + (3*b^5*x^(17/3))/17

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Maple [A]  time = 0.003, size = 58, normalized size = 0.8 \[{\frac{{a}^{5}{x}^{4}}{4}}+{\frac{15\,{a}^{4}b}{13}{x}^{{\frac{13}{3}}}}+{\frac{15\,{a}^{3}{b}^{2}}{7}{x}^{{\frac{14}{3}}}}+2\,{a}^{2}{b}^{3}{x}^{5}+{\frac{15\,a{b}^{4}}{16}{x}^{{\frac{16}{3}}}}+{\frac{3\,{b}^{5}}{17}{x}^{{\frac{17}{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b*x^(1/3))^5*x^3,x)

[Out]

1/4*a^5*x^4+15/13*a^4*b*x^(13/3)+15/7*a^3*b^2*x^(14/3)+2*a^2*b^3*x^5+15/16*a*b^4
*x^(16/3)+3/17*b^5*x^(17/3)

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Maxima [A]  time = 1.44093, size = 270, normalized size = 3.6 \[ \frac{3 \,{\left (b x^{\frac{1}{3}} + a\right )}^{17}}{17 \, b^{12}} - \frac{33 \,{\left (b x^{\frac{1}{3}} + a\right )}^{16} a}{16 \, b^{12}} + \frac{11 \,{\left (b x^{\frac{1}{3}} + a\right )}^{15} a^{2}}{b^{12}} - \frac{495 \,{\left (b x^{\frac{1}{3}} + a\right )}^{14} a^{3}}{14 \, b^{12}} + \frac{990 \,{\left (b x^{\frac{1}{3}} + a\right )}^{13} a^{4}}{13 \, b^{12}} - \frac{231 \,{\left (b x^{\frac{1}{3}} + a\right )}^{12} a^{5}}{2 \, b^{12}} + \frac{126 \,{\left (b x^{\frac{1}{3}} + a\right )}^{11} a^{6}}{b^{12}} - \frac{99 \,{\left (b x^{\frac{1}{3}} + a\right )}^{10} a^{7}}{b^{12}} + \frac{55 \,{\left (b x^{\frac{1}{3}} + a\right )}^{9} a^{8}}{b^{12}} - \frac{165 \,{\left (b x^{\frac{1}{3}} + a\right )}^{8} a^{9}}{8 \, b^{12}} + \frac{33 \,{\left (b x^{\frac{1}{3}} + a\right )}^{7} a^{10}}{7 \, b^{12}} - \frac{{\left (b x^{\frac{1}{3}} + a\right )}^{6} a^{11}}{2 \, b^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/17*(b*x^(1/3) + a)^17/b^12 - 33/16*(b*x^(1/3) + a)^16*a/b^12 + 11*(b*x^(1/3) +
a)^15*a^2/b^12 - 495/14*(b*x^(1/3) + a)^14*a^3/b^12 + 990/13*(b*x^(1/3) + a)^13
*a^4/b^12 - 231/2*(b*x^(1/3) + a)^12*a^5/b^12 + 126*(b*x^(1/3) + a)^11*a^6/b^12
- 99*(b*x^(1/3) + a)^10*a^7/b^12 + 55*(b*x^(1/3) + a)^9*a^8/b^12 - 165/8*(b*x^(1
/3) + a)^8*a^9/b^12 + 33/7*(b*x^(1/3) + a)^7*a^10/b^12 - 1/2*(b*x^(1/3) + a)^6*a
^11/b^12

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Fricas [A]  time = 0.212473, size = 93, normalized size = 1.24 \[ 2 \, a^{2} b^{3} x^{5} + \frac{1}{4} \, a^{5} x^{4} + \frac{3}{119} \,{\left (7 \, b^{5} x^{5} + 85 \, a^{3} b^{2} x^{4}\right )} x^{\frac{2}{3}} + \frac{15}{208} \,{\left (13 \, a b^{4} x^{5} + 16 \, a^{4} b x^{4}\right )} x^{\frac{1}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*a^2*b^3*x^5 + 1/4*a^5*x^4 + 3/119*(7*b^5*x^5 + 85*a^3*b^2*x^4)*x^(2/3) + 15/20
8*(13*a*b^4*x^5 + 16*a^4*b*x^4)*x^(1/3)

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Sympy [A]  time = 5.79276, size = 73, normalized size = 0.97 \[ \frac{a^{5} x^{4}}{4} + \frac{15 a^{4} b x^{\frac{13}{3}}}{13} + \frac{15 a^{3} b^{2} x^{\frac{14}{3}}}{7} + 2 a^{2} b^{3} x^{5} + \frac{15 a b^{4} x^{\frac{16}{3}}}{16} + \frac{3 b^{5} x^{\frac{17}{3}}}{17} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**5*x**4/4 + 15*a**4*b*x**(13/3)/13 + 15*a**3*b**2*x**(14/3)/7 + 2*a**2*b**3*x*
*5 + 15*a*b**4*x**(16/3)/16 + 3*b**5*x**(17/3)/17

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GIAC/XCAS [A]  time = 0.250162, size = 77, normalized size = 1.03 \[ \frac{3}{17} \, b^{5} x^{\frac{17}{3}} + \frac{15}{16} \, a b^{4} x^{\frac{16}{3}} + 2 \, a^{2} b^{3} x^{5} + \frac{15}{7} \, a^{3} b^{2} x^{\frac{14}{3}} + \frac{15}{13} \, a^{4} b x^{\frac{13}{3}} + \frac{1}{4} \, a^{5} x^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/17*b^5*x^(17/3) + 15/16*a*b^4*x^(16/3) + 2*a^2*b^3*x^5 + 15/7*a^3*b^2*x^(14/3)
+ 15/13*a^4*b*x^(13/3) + 1/4*a^5*x^4