3.2337 \(\int \left (a+b \sqrt [3]{x}\right )^{15} \, dx\)

Optimal. Leaf size=59 \[ \frac{3 a^2 \left (a+b \sqrt [3]{x}\right )^{16}}{16 b^3}+\frac{\left (a+b \sqrt [3]{x}\right )^{18}}{6 b^3}-\frac{6 a \left (a+b \sqrt [3]{x}\right )^{17}}{17 b^3} \]

[Out]

(3*a^2*(a + b*x^(1/3))^16)/(16*b^3) - (6*a*(a + b*x^(1/3))^17)/(17*b^3) + (a + b
*x^(1/3))^18/(6*b^3)

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Rubi [A]  time = 0.159697, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{3 a^2 \left (a+b \sqrt [3]{x}\right )^{16}}{16 b^3}+\frac{\left (a+b \sqrt [3]{x}\right )^{18}}{6 b^3}-\frac{6 a \left (a+b \sqrt [3]{x}\right )^{17}}{17 b^3} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^(1/3))^15,x]

[Out]

(3*a^2*(a + b*x^(1/3))^16)/(16*b^3) - (6*a*(a + b*x^(1/3))^17)/(17*b^3) + (a + b
*x^(1/3))^18/(6*b^3)

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Rubi in Sympy [A]  time = 27.8305, size = 53, normalized size = 0.9 \[ \frac{3 a^{2} \left (a + b \sqrt [3]{x}\right )^{16}}{16 b^{3}} - \frac{6 a \left (a + b \sqrt [3]{x}\right )^{17}}{17 b^{3}} + \frac{\left (a + b \sqrt [3]{x}\right )^{18}}{6 b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*a**2*(a + b*x**(1/3))**16/(16*b**3) - 6*a*(a + b*x**(1/3))**17/(17*b**3) + (a
+ b*x**(1/3))**18/(6*b**3)

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Mathematica [B]  time = 0.0296419, size = 204, normalized size = 3.46 \[ a^{15} x+\frac{45}{4} a^{14} b x^{4/3}+63 a^{13} b^2 x^{5/3}+\frac{455}{2} a^{12} b^3 x^2+585 a^{11} b^4 x^{7/3}+\frac{9009}{8} a^{10} b^5 x^{8/3}+\frac{5005}{3} a^9 b^6 x^3+\frac{3861}{2} a^8 b^7 x^{10/3}+1755 a^7 b^8 x^{11/3}+\frac{5005}{4} a^6 b^9 x^4+693 a^5 b^{10} x^{13/3}+\frac{585}{2} a^4 b^{11} x^{14/3}+91 a^3 b^{12} x^5+\frac{315}{16} a^2 b^{13} x^{16/3}+\frac{45}{17} a b^{14} x^{17/3}+\frac{b^{15} x^6}{6} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^(1/3))^15,x]

[Out]

a^15*x + (45*a^14*b*x^(4/3))/4 + 63*a^13*b^2*x^(5/3) + (455*a^12*b^3*x^2)/2 + 58
5*a^11*b^4*x^(7/3) + (9009*a^10*b^5*x^(8/3))/8 + (5005*a^9*b^6*x^3)/3 + (3861*a^
8*b^7*x^(10/3))/2 + 1755*a^7*b^8*x^(11/3) + (5005*a^6*b^9*x^4)/4 + 693*a^5*b^10*
x^(13/3) + (585*a^4*b^11*x^(14/3))/2 + 91*a^3*b^12*x^5 + (315*a^2*b^13*x^(16/3))
/16 + (45*a*b^14*x^(17/3))/17 + (b^15*x^6)/6

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Maple [B]  time = 0.033, size = 165, normalized size = 2.8 \[ x{a}^{15}+{\frac{{b}^{15}{x}^{6}}{6}}+{\frac{455\,{x}^{2}{b}^{3}{a}^{12}}{2}}+91\,{a}^{3}{b}^{12}{x}^{5}+{\frac{5005\,{a}^{9}{b}^{6}{x}^{3}}{3}}+{\frac{45\,b{a}^{14}}{4}{x}^{{\frac{4}{3}}}}+63\,{a}^{13}{b}^{2}{x}^{5/3}+585\,{a}^{11}{b}^{4}{x}^{7/3}+{\frac{9009\,{b}^{5}{a}^{10}}{8}{x}^{{\frac{8}{3}}}}+{\frac{3861\,{b}^{7}{a}^{8}}{2}{x}^{{\frac{10}{3}}}}+1755\,{a}^{7}{b}^{8}{x}^{11/3}+{\frac{5005\,{b}^{9}{x}^{4}{a}^{6}}{4}}+693\,{b}^{10}{a}^{5}{x}^{13/3}+{\frac{585\,{b}^{11}{a}^{4}}{2}{x}^{{\frac{14}{3}}}}+{\frac{315\,{b}^{13}{a}^{2}}{16}{x}^{{\frac{16}{3}}}}+{\frac{45\,a{b}^{14}}{17}{x}^{{\frac{17}{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b*x^(1/3))^15,x)

[Out]

x*a^15+1/6*b^15*x^6+455/2*x^2*b^3*a^12+91*a^3*b^12*x^5+5005/3*a^9*b^6*x^3+45/4*b
*a^14*x^(4/3)+63*a^13*b^2*x^(5/3)+585*a^11*b^4*x^(7/3)+9009/8*b^5*a^10*x^(8/3)+3
861/2*b^7*a^8*x^(10/3)+1755*a^7*b^8*x^(11/3)+5005/4*b^9*x^4*a^6+693*b^10*a^5*x^(
13/3)+585/2*b^11*a^4*x^(14/3)+315/16*b^13*a^2*x^(16/3)+45/17*a*b^14*x^(17/3)

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Maxima [A]  time = 1.45593, size = 63, normalized size = 1.07 \[ \frac{{\left (b x^{\frac{1}{3}} + a\right )}^{18}}{6 \, b^{3}} - \frac{6 \,{\left (b x^{\frac{1}{3}} + a\right )}^{17} a}{17 \, b^{3}} + \frac{3 \,{\left (b x^{\frac{1}{3}} + a\right )}^{16} a^{2}}{16 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*(b*x^(1/3) + a)^18/b^3 - 6/17*(b*x^(1/3) + a)^17*a/b^3 + 3/16*(b*x^(1/3) + a
)^16*a^2/b^3

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Fricas [A]  time = 0.213717, size = 232, normalized size = 3.93 \[ \frac{1}{6} \, b^{15} x^{6} + 91 \, a^{3} b^{12} x^{5} + \frac{5005}{4} \, a^{6} b^{9} x^{4} + \frac{5005}{3} \, a^{9} b^{6} x^{3} + \frac{455}{2} \, a^{12} b^{3} x^{2} + a^{15} x + \frac{9}{136} \,{\left (40 \, a b^{14} x^{5} + 4420 \, a^{4} b^{11} x^{4} + 26520 \, a^{7} b^{8} x^{3} + 17017 \, a^{10} b^{5} x^{2} + 952 \, a^{13} b^{2} x\right )} x^{\frac{2}{3}} + \frac{9}{16} \,{\left (35 \, a^{2} b^{13} x^{5} + 1232 \, a^{5} b^{10} x^{4} + 3432 \, a^{8} b^{7} x^{3} + 1040 \, a^{11} b^{4} x^{2} + 20 \, a^{14} b x\right )} x^{\frac{1}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*b^15*x^6 + 91*a^3*b^12*x^5 + 5005/4*a^6*b^9*x^4 + 5005/3*a^9*b^6*x^3 + 455/2
*a^12*b^3*x^2 + a^15*x + 9/136*(40*a*b^14*x^5 + 4420*a^4*b^11*x^4 + 26520*a^7*b^
8*x^3 + 17017*a^10*b^5*x^2 + 952*a^13*b^2*x)*x^(2/3) + 9/16*(35*a^2*b^13*x^5 + 1
232*a^5*b^10*x^4 + 3432*a^8*b^7*x^3 + 1040*a^11*b^4*x^2 + 20*a^14*b*x)*x^(1/3)

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Sympy [A]  time = 7.08213, size = 207, normalized size = 3.51 \[ a^{15} x + \frac{45 a^{14} b x^{\frac{4}{3}}}{4} + 63 a^{13} b^{2} x^{\frac{5}{3}} + \frac{455 a^{12} b^{3} x^{2}}{2} + 585 a^{11} b^{4} x^{\frac{7}{3}} + \frac{9009 a^{10} b^{5} x^{\frac{8}{3}}}{8} + \frac{5005 a^{9} b^{6} x^{3}}{3} + \frac{3861 a^{8} b^{7} x^{\frac{10}{3}}}{2} + 1755 a^{7} b^{8} x^{\frac{11}{3}} + \frac{5005 a^{6} b^{9} x^{4}}{4} + 693 a^{5} b^{10} x^{\frac{13}{3}} + \frac{585 a^{4} b^{11} x^{\frac{14}{3}}}{2} + 91 a^{3} b^{12} x^{5} + \frac{315 a^{2} b^{13} x^{\frac{16}{3}}}{16} + \frac{45 a b^{14} x^{\frac{17}{3}}}{17} + \frac{b^{15} x^{6}}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b*x**(1/3))**15,x)

[Out]

a**15*x + 45*a**14*b*x**(4/3)/4 + 63*a**13*b**2*x**(5/3) + 455*a**12*b**3*x**2/2
+ 585*a**11*b**4*x**(7/3) + 9009*a**10*b**5*x**(8/3)/8 + 5005*a**9*b**6*x**3/3
+ 3861*a**8*b**7*x**(10/3)/2 + 1755*a**7*b**8*x**(11/3) + 5005*a**6*b**9*x**4/4
+ 693*a**5*b**10*x**(13/3) + 585*a**4*b**11*x**(14/3)/2 + 91*a**3*b**12*x**5 + 3
15*a**2*b**13*x**(16/3)/16 + 45*a*b**14*x**(17/3)/17 + b**15*x**6/6

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GIAC/XCAS [A]  time = 0.219076, size = 221, normalized size = 3.75 \[ \frac{1}{6} \, b^{15} x^{6} + \frac{45}{17} \, a b^{14} x^{\frac{17}{3}} + \frac{315}{16} \, a^{2} b^{13} x^{\frac{16}{3}} + 91 \, a^{3} b^{12} x^{5} + \frac{585}{2} \, a^{4} b^{11} x^{\frac{14}{3}} + 693 \, a^{5} b^{10} x^{\frac{13}{3}} + \frac{5005}{4} \, a^{6} b^{9} x^{4} + 1755 \, a^{7} b^{8} x^{\frac{11}{3}} + \frac{3861}{2} \, a^{8} b^{7} x^{\frac{10}{3}} + \frac{5005}{3} \, a^{9} b^{6} x^{3} + \frac{9009}{8} \, a^{10} b^{5} x^{\frac{8}{3}} + 585 \, a^{11} b^{4} x^{\frac{7}{3}} + \frac{455}{2} \, a^{12} b^{3} x^{2} + 63 \, a^{13} b^{2} x^{\frac{5}{3}} + \frac{45}{4} \, a^{14} b x^{\frac{4}{3}} + a^{15} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*b^15*x^6 + 45/17*a*b^14*x^(17/3) + 315/16*a^2*b^13*x^(16/3) + 91*a^3*b^12*x^
5 + 585/2*a^4*b^11*x^(14/3) + 693*a^5*b^10*x^(13/3) + 5005/4*a^6*b^9*x^4 + 1755*
a^7*b^8*x^(11/3) + 3861/2*a^8*b^7*x^(10/3) + 5005/3*a^9*b^6*x^3 + 9009/8*a^10*b^
5*x^(8/3) + 585*a^11*b^4*x^(7/3) + 455/2*a^12*b^3*x^2 + 63*a^13*b^2*x^(5/3) + 45
/4*a^14*b*x^(4/3) + a^15*x