3.261 \(\int x^8 \left (a+b x^3\right )^5 \, dx\)

Optimal. Leaf size=53 \[ \frac{a^2 \left (a+b x^3\right )^6}{18 b^3}+\frac{\left (a+b x^3\right )^8}{24 b^3}-\frac{2 a \left (a+b x^3\right )^7}{21 b^3} \]

[Out]

(a^2*(a + b*x^3)^6)/(18*b^3) - (2*a*(a + b*x^3)^7)/(21*b^3) + (a + b*x^3)^8/(24*
b^3)

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Rubi [A]  time = 0.143505, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{a^2 \left (a+b x^3\right )^6}{18 b^3}+\frac{\left (a+b x^3\right )^8}{24 b^3}-\frac{2 a \left (a+b x^3\right )^7}{21 b^3} \]

Antiderivative was successfully verified.

[In]  Int[x^8*(a + b*x^3)^5,x]

[Out]

(a^2*(a + b*x^3)^6)/(18*b^3) - (2*a*(a + b*x^3)^7)/(21*b^3) + (a + b*x^3)^8/(24*
b^3)

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Rubi in Sympy [A]  time = 13.5983, size = 46, normalized size = 0.87 \[ \frac{a^{2} \left (a + b x^{3}\right )^{6}}{18 b^{3}} - \frac{2 a \left (a + b x^{3}\right )^{7}}{21 b^{3}} + \frac{\left (a + b x^{3}\right )^{8}}{24 b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**8*(b*x**3+a)**5,x)

[Out]

a**2*(a + b*x**3)**6/(18*b**3) - 2*a*(a + b*x**3)**7/(21*b**3) + (a + b*x**3)**8
/(24*b**3)

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Mathematica [A]  time = 0.00372556, size = 69, normalized size = 1.3 \[ \frac{a^5 x^9}{9}+\frac{5}{12} a^4 b x^{12}+\frac{2}{3} a^3 b^2 x^{15}+\frac{5}{9} a^2 b^3 x^{18}+\frac{5}{21} a b^4 x^{21}+\frac{b^5 x^{24}}{24} \]

Antiderivative was successfully verified.

[In]  Integrate[x^8*(a + b*x^3)^5,x]

[Out]

(a^5*x^9)/9 + (5*a^4*b*x^12)/12 + (2*a^3*b^2*x^15)/3 + (5*a^2*b^3*x^18)/9 + (5*a
*b^4*x^21)/21 + (b^5*x^24)/24

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Maple [A]  time = 0.002, size = 58, normalized size = 1.1 \[{\frac{{b}^{5}{x}^{24}}{24}}+{\frac{5\,a{b}^{4}{x}^{21}}{21}}+{\frac{5\,{a}^{2}{b}^{3}{x}^{18}}{9}}+{\frac{2\,{a}^{3}{b}^{2}{x}^{15}}{3}}+{\frac{5\,{a}^{4}b{x}^{12}}{12}}+{\frac{{a}^{5}{x}^{9}}{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^8*(b*x^3+a)^5,x)

[Out]

1/24*b^5*x^24+5/21*a*b^4*x^21+5/9*a^2*b^3*x^18+2/3*a^3*b^2*x^15+5/12*a^4*b*x^12+
1/9*a^5*x^9

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Maxima [A]  time = 1.42127, size = 77, normalized size = 1.45 \[ \frac{1}{24} \, b^{5} x^{24} + \frac{5}{21} \, a b^{4} x^{21} + \frac{5}{9} \, a^{2} b^{3} x^{18} + \frac{2}{3} \, a^{3} b^{2} x^{15} + \frac{5}{12} \, a^{4} b x^{12} + \frac{1}{9} \, a^{5} x^{9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a)^5*x^8,x, algorithm="maxima")

[Out]

1/24*b^5*x^24 + 5/21*a*b^4*x^21 + 5/9*a^2*b^3*x^18 + 2/3*a^3*b^2*x^15 + 5/12*a^4
*b*x^12 + 1/9*a^5*x^9

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Fricas [A]  time = 0.189437, size = 1, normalized size = 0.02 \[ \frac{1}{24} x^{24} b^{5} + \frac{5}{21} x^{21} b^{4} a + \frac{5}{9} x^{18} b^{3} a^{2} + \frac{2}{3} x^{15} b^{2} a^{3} + \frac{5}{12} x^{12} b a^{4} + \frac{1}{9} x^{9} a^{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a)^5*x^8,x, algorithm="fricas")

[Out]

1/24*x^24*b^5 + 5/21*x^21*b^4*a + 5/9*x^18*b^3*a^2 + 2/3*x^15*b^2*a^3 + 5/12*x^1
2*b*a^4 + 1/9*x^9*a^5

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Sympy [A]  time = 0.124328, size = 66, normalized size = 1.25 \[ \frac{a^{5} x^{9}}{9} + \frac{5 a^{4} b x^{12}}{12} + \frac{2 a^{3} b^{2} x^{15}}{3} + \frac{5 a^{2} b^{3} x^{18}}{9} + \frac{5 a b^{4} x^{21}}{21} + \frac{b^{5} x^{24}}{24} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**8*(b*x**3+a)**5,x)

[Out]

a**5*x**9/9 + 5*a**4*b*x**12/12 + 2*a**3*b**2*x**15/3 + 5*a**2*b**3*x**18/9 + 5*
a*b**4*x**21/21 + b**5*x**24/24

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GIAC/XCAS [A]  time = 0.215942, size = 77, normalized size = 1.45 \[ \frac{1}{24} \, b^{5} x^{24} + \frac{5}{21} \, a b^{4} x^{21} + \frac{5}{9} \, a^{2} b^{3} x^{18} + \frac{2}{3} \, a^{3} b^{2} x^{15} + \frac{5}{12} \, a^{4} b x^{12} + \frac{1}{9} \, a^{5} x^{9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a)^5*x^8,x, algorithm="giac")

[Out]

1/24*b^5*x^24 + 5/21*a*b^4*x^21 + 5/9*a^2*b^3*x^18 + 2/3*a^3*b^2*x^15 + 5/12*a^4
*b*x^12 + 1/9*a^5*x^9