3.2309 \(\int \left (a+b \sqrt [3]{x}\right )^5 \, dx\)

Optimal. Leaf size=59 \[ \frac{a^2 \left (a+b \sqrt [3]{x}\right )^6}{2 b^3}+\frac{3 \left (a+b \sqrt [3]{x}\right )^8}{8 b^3}-\frac{6 a \left (a+b \sqrt [3]{x}\right )^7}{7 b^3} \]

[Out]

(a^2*(a + b*x^(1/3))^6)/(2*b^3) - (6*a*(a + b*x^(1/3))^7)/(7*b^3) + (3*(a + b*x^
(1/3))^8)/(8*b^3)

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Rubi [A]  time = 0.080046, 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 \[ \frac{a^2 \left (a+b \sqrt [3]{x}\right )^6}{2 b^3}+\frac{3 \left (a+b \sqrt [3]{x}\right )^8}{8 b^3}-\frac{6 a \left (a+b \sqrt [3]{x}\right )^7}{7 b^3} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^(1/3))^5,x]

[Out]

(a^2*(a + b*x^(1/3))^6)/(2*b^3) - (6*a*(a + b*x^(1/3))^7)/(7*b^3) + (3*(a + b*x^
(1/3))^8)/(8*b^3)

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Rubi in Sympy [A]  time = 12.1733, size = 53, normalized size = 0.9 \[ \frac{a^{2} \left (a + b \sqrt [3]{x}\right )^{6}}{2 b^{3}} - \frac{6 a \left (a + b \sqrt [3]{x}\right )^{7}}{7 b^{3}} + \frac{3 \left (a + b \sqrt [3]{x}\right )^{8}}{8 b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**2*(a + b*x**(1/3))**6/(2*b**3) - 6*a*(a + b*x**(1/3))**7/(7*b**3) + 3*(a + b*
x**(1/3))**8/(8*b**3)

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Mathematica [A]  time = 0.0126144, size = 68, normalized size = 1.15 \[ a^5 x+\frac{15}{4} a^4 b x^{4/3}+6 a^3 b^2 x^{5/3}+5 a^2 b^3 x^2+\frac{15}{7} a b^4 x^{7/3}+\frac{3}{8} b^5 x^{8/3} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^(1/3))^5,x]

[Out]

a^5*x + (15*a^4*b*x^(4/3))/4 + 6*a^3*b^2*x^(5/3) + 5*a^2*b^3*x^2 + (15*a*b^4*x^(
7/3))/7 + (3*b^5*x^(8/3))/8

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Maple [A]  time = 0.003, size = 55, normalized size = 0.9 \[ x{a}^{5}+{\frac{3\,{b}^{5}}{8}{x}^{{\frac{8}{3}}}}+{\frac{15\,a{b}^{4}}{7}{x}^{{\frac{7}{3}}}}+5\,{a}^{2}{b}^{3}{x}^{2}+6\,{a}^{3}{b}^{2}{x}^{5/3}+{\frac{15\,{a}^{4}b}{4}{x}^{{\frac{4}{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b*x^(1/3))^5,x)

[Out]

x*a^5+3/8*b^5*x^(8/3)+15/7*a*b^4*x^(7/3)+5*a^2*b^3*x^2+6*a^3*b^2*x^(5/3)+15/4*a^
4*b*x^(4/3)

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Maxima [A]  time = 1.43557, size = 73, normalized size = 1.24 \[ \frac{3}{8} \, b^{5} x^{\frac{8}{3}} + \frac{15}{7} \, a b^{4} x^{\frac{7}{3}} + 5 \, a^{2} b^{3} x^{2} + 6 \, a^{3} b^{2} x^{\frac{5}{3}} + \frac{15}{4} \, a^{4} b x^{\frac{4}{3}} + a^{5} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/8*b^5*x^(8/3) + 15/7*a*b^4*x^(7/3) + 5*a^2*b^3*x^2 + 6*a^3*b^2*x^(5/3) + 15/4*
a^4*b*x^(4/3) + a^5*x

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Fricas [A]  time = 0.208988, size = 82, normalized size = 1.39 \[ 5 \, a^{2} b^{3} x^{2} + a^{5} x + \frac{3}{8} \,{\left (b^{5} x^{2} + 16 \, a^{3} b^{2} x\right )} x^{\frac{2}{3}} + \frac{15}{28} \,{\left (4 \, a b^{4} x^{2} + 7 \, a^{4} b x\right )} x^{\frac{1}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

5*a^2*b^3*x^2 + a^5*x + 3/8*(b^5*x^2 + 16*a^3*b^2*x)*x^(2/3) + 15/28*(4*a*b^4*x^
2 + 7*a^4*b*x)*x^(1/3)

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Sympy [A]  time = 1.42529, size = 68, normalized size = 1.15 \[ a^{5} x + \frac{15 a^{4} b x^{\frac{4}{3}}}{4} + 6 a^{3} b^{2} x^{\frac{5}{3}} + 5 a^{2} b^{3} x^{2} + \frac{15 a b^{4} x^{\frac{7}{3}}}{7} + \frac{3 b^{5} x^{\frac{8}{3}}}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b*x**(1/3))**5,x)

[Out]

a**5*x + 15*a**4*b*x**(4/3)/4 + 6*a**3*b**2*x**(5/3) + 5*a**2*b**3*x**2 + 15*a*b
**4*x**(7/3)/7 + 3*b**5*x**(8/3)/8

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GIAC/XCAS [A]  time = 0.229454, size = 73, normalized size = 1.24 \[ \frac{3}{8} \, b^{5} x^{\frac{8}{3}} + \frac{15}{7} \, a b^{4} x^{\frac{7}{3}} + 5 \, a^{2} b^{3} x^{2} + 6 \, a^{3} b^{2} x^{\frac{5}{3}} + \frac{15}{4} \, a^{4} b x^{\frac{4}{3}} + a^{5} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/8*b^5*x^(8/3) + 15/7*a*b^4*x^(7/3) + 5*a^2*b^3*x^2 + 6*a^3*b^2*x^(5/3) + 15/4*
a^4*b*x^(4/3) + a^5*x