3.2321 \(\int \left (a+b \sqrt [3]{x}\right )^{10} \, dx\)

Optimal. Leaf size=59 \[ \frac{3 a^2 \left (a+b \sqrt [3]{x}\right )^{11}}{11 b^3}+\frac{3 \left (a+b \sqrt [3]{x}\right )^{13}}{13 b^3}-\frac{a \left (a+b \sqrt [3]{x}\right )^{12}}{2 b^3} \]

[Out]

(3*a^2*(a + b*x^(1/3))^11)/(11*b^3) - (a*(a + b*x^(1/3))^12)/(2*b^3) + (3*(a + b
*x^(1/3))^13)/(13*b^3)

_______________________________________________________________________________________

Rubi [A]  time = 0.107089, 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 \[ \frac{3 a^2 \left (a+b \sqrt [3]{x}\right )^{11}}{11 b^3}+\frac{3 \left (a+b \sqrt [3]{x}\right )^{13}}{13 b^3}-\frac{a \left (a+b \sqrt [3]{x}\right )^{12}}{2 b^3} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^(1/3))^10,x]

[Out]

(3*a^2*(a + b*x^(1/3))^11)/(11*b^3) - (a*(a + b*x^(1/3))^12)/(2*b^3) + (3*(a + b
*x^(1/3))^13)/(13*b^3)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 17.6768, size = 53, normalized size = 0.9 \[ \frac{3 a^{2} \left (a + b \sqrt [3]{x}\right )^{11}}{11 b^{3}} - \frac{a \left (a + b \sqrt [3]{x}\right )^{12}}{2 b^{3}} + \frac{3 \left (a + b \sqrt [3]{x}\right )^{13}}{13 b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b*x**(1/3))**10,x)

[Out]

3*a**2*(a + b*x**(1/3))**11/(11*b**3) - a*(a + b*x**(1/3))**12/(2*b**3) + 3*(a +
b*x**(1/3))**13/(13*b**3)

_______________________________________________________________________________________

Mathematica [B]  time = 0.0195622, size = 133, normalized size = 2.25 \[ a^{10} x+\frac{15}{2} a^9 b x^{4/3}+27 a^8 b^2 x^{5/3}+60 a^7 b^3 x^2+90 a^6 b^4 x^{7/3}+\frac{189}{2} a^5 b^5 x^{8/3}+70 a^4 b^6 x^3+36 a^3 b^7 x^{10/3}+\frac{135}{11} a^2 b^8 x^{11/3}+\frac{5}{2} a b^9 x^4+\frac{3}{13} b^{10} x^{13/3} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^(1/3))^10,x]

[Out]

a^10*x + (15*a^9*b*x^(4/3))/2 + 27*a^8*b^2*x^(5/3) + 60*a^7*b^3*x^2 + 90*a^6*b^4
*x^(7/3) + (189*a^5*b^5*x^(8/3))/2 + 70*a^4*b^6*x^3 + 36*a^3*b^7*x^(10/3) + (135
*a^2*b^8*x^(11/3))/11 + (5*a*b^9*x^4)/2 + (3*b^10*x^(13/3))/13

_______________________________________________________________________________________

Maple [B]  time = 0.004, size = 110, normalized size = 1.9 \[ x{a}^{10}+{\frac{3\,{b}^{10}}{13}{x}^{{\frac{13}{3}}}}+{\frac{5\,a{b}^{9}{x}^{4}}{2}}+{\frac{135\,{a}^{2}{b}^{8}}{11}{x}^{{\frac{11}{3}}}}+36\,{a}^{3}{b}^{7}{x}^{10/3}+70\,{a}^{4}{b}^{6}{x}^{3}+{\frac{189\,{a}^{5}{b}^{5}}{2}{x}^{{\frac{8}{3}}}}+90\,{a}^{6}{b}^{4}{x}^{7/3}+60\,{a}^{7}{b}^{3}{x}^{2}+27\,{a}^{8}{b}^{2}{x}^{5/3}+{\frac{15\,{a}^{9}b}{2}{x}^{{\frac{4}{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b*x^(1/3))^10,x)

[Out]

x*a^10+3/13*b^10*x^(13/3)+5/2*a*b^9*x^4+135/11*a^2*b^8*x^(11/3)+36*a^3*b^7*x^(10
/3)+70*a^4*b^6*x^3+189/2*a^5*b^5*x^(8/3)+90*a^6*b^4*x^(7/3)+60*a^7*b^3*x^2+27*a^
8*b^2*x^(5/3)+15/2*a^9*b*x^(4/3)

_______________________________________________________________________________________

Maxima [A]  time = 1.44466, size = 63, normalized size = 1.07 \[ \frac{3 \,{\left (b x^{\frac{1}{3}} + a\right )}^{13}}{13 \, b^{3}} - \frac{{\left (b x^{\frac{1}{3}} + a\right )}^{12} a}{2 \, b^{3}} + \frac{3 \,{\left (b x^{\frac{1}{3}} + a\right )}^{11} a^{2}}{11 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^(1/3) + a)^10,x, algorithm="maxima")

[Out]

3/13*(b*x^(1/3) + a)^13/b^3 - 1/2*(b*x^(1/3) + a)^12*a/b^3 + 3/11*(b*x^(1/3) + a
)^11*a^2/b^3

_______________________________________________________________________________________

Fricas [A]  time = 0.238159, size = 158, normalized size = 2.68 \[ \frac{5}{2} \, a b^{9} x^{4} + 70 \, a^{4} b^{6} x^{3} + 60 \, a^{7} b^{3} x^{2} + a^{10} x + \frac{27}{22} \,{\left (10 \, a^{2} b^{8} x^{3} + 77 \, a^{5} b^{5} x^{2} + 22 \, a^{8} b^{2} x\right )} x^{\frac{2}{3}} + \frac{3}{26} \,{\left (2 \, b^{10} x^{4} + 312 \, a^{3} b^{7} x^{3} + 780 \, a^{6} b^{4} x^{2} + 65 \, a^{9} b x\right )} x^{\frac{1}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^(1/3) + a)^10,x, algorithm="fricas")

[Out]

5/2*a*b^9*x^4 + 70*a^4*b^6*x^3 + 60*a^7*b^3*x^2 + a^10*x + 27/22*(10*a^2*b^8*x^3
+ 77*a^5*b^5*x^2 + 22*a^8*b^2*x)*x^(2/3) + 3/26*(2*b^10*x^4 + 312*a^3*b^7*x^3 +
780*a^6*b^4*x^2 + 65*a^9*b*x)*x^(1/3)

_______________________________________________________________________________________

Sympy [A]  time = 2.80363, size = 136, normalized size = 2.31 \[ a^{10} x + \frac{15 a^{9} b x^{\frac{4}{3}}}{2} + 27 a^{8} b^{2} x^{\frac{5}{3}} + 60 a^{7} b^{3} x^{2} + 90 a^{6} b^{4} x^{\frac{7}{3}} + \frac{189 a^{5} b^{5} x^{\frac{8}{3}}}{2} + 70 a^{4} b^{6} x^{3} + 36 a^{3} b^{7} x^{\frac{10}{3}} + \frac{135 a^{2} b^{8} x^{\frac{11}{3}}}{11} + \frac{5 a b^{9} x^{4}}{2} + \frac{3 b^{10} x^{\frac{13}{3}}}{13} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b*x**(1/3))**10,x)

[Out]

a**10*x + 15*a**9*b*x**(4/3)/2 + 27*a**8*b**2*x**(5/3) + 60*a**7*b**3*x**2 + 90*
a**6*b**4*x**(7/3) + 189*a**5*b**5*x**(8/3)/2 + 70*a**4*b**6*x**3 + 36*a**3*b**7
*x**(10/3) + 135*a**2*b**8*x**(11/3)/11 + 5*a*b**9*x**4/2 + 3*b**10*x**(13/3)/13

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.25986, size = 147, normalized size = 2.49 \[ \frac{3}{13} \, b^{10} x^{\frac{13}{3}} + \frac{5}{2} \, a b^{9} x^{4} + \frac{135}{11} \, a^{2} b^{8} x^{\frac{11}{3}} + 36 \, a^{3} b^{7} x^{\frac{10}{3}} + 70 \, a^{4} b^{6} x^{3} + \frac{189}{2} \, a^{5} b^{5} x^{\frac{8}{3}} + 90 \, a^{6} b^{4} x^{\frac{7}{3}} + 60 \, a^{7} b^{3} x^{2} + 27 \, a^{8} b^{2} x^{\frac{5}{3}} + \frac{15}{2} \, a^{9} b x^{\frac{4}{3}} + a^{10} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^(1/3) + a)^10,x, algorithm="giac")

[Out]

3/13*b^10*x^(13/3) + 5/2*a*b^9*x^4 + 135/11*a^2*b^8*x^(11/3) + 36*a^3*b^7*x^(10/
3) + 70*a^4*b^6*x^3 + 189/2*a^5*b^5*x^(8/3) + 90*a^6*b^4*x^(7/3) + 60*a^7*b^3*x^
2 + 27*a^8*b^2*x^(5/3) + 15/2*a^9*b*x^(4/3) + a^10*x