3.599 \(\int x^8 \left (a+b x^3\right )^p \, dx\)

Optimal. Leaf size=74 \[ \frac{a^2 \left (a+b x^3\right )^{p+1}}{3 b^3 (p+1)}-\frac{2 a \left (a+b x^3\right )^{p+2}}{3 b^3 (p+2)}+\frac{\left (a+b x^3\right )^{p+3}}{3 b^3 (p+3)} \]

[Out]

(a^2*(a + b*x^3)^(1 + p))/(3*b^3*(1 + p)) - (2*a*(a + b*x^3)^(2 + p))/(3*b^3*(2
+ p)) + (a + b*x^3)^(3 + p)/(3*b^3*(3 + p))

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Rubi [A]  time = 0.096864, antiderivative size = 74, 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 \[ \frac{a^2 \left (a+b x^3\right )^{p+1}}{3 b^3 (p+1)}-\frac{2 a \left (a+b x^3\right )^{p+2}}{3 b^3 (p+2)}+\frac{\left (a+b x^3\right )^{p+3}}{3 b^3 (p+3)} \]

Antiderivative was successfully verified.

[In]  Int[x^8*(a + b*x^3)^p,x]

[Out]

(a^2*(a + b*x^3)^(1 + p))/(3*b^3*(1 + p)) - (2*a*(a + b*x^3)^(2 + p))/(3*b^3*(2
+ p)) + (a + b*x^3)^(3 + p)/(3*b^3*(3 + p))

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Rubi in Sympy [A]  time = 16.566, size = 61, normalized size = 0.82 \[ \frac{a^{2} \left (a + b x^{3}\right )^{p + 1}}{3 b^{3} \left (p + 1\right )} - \frac{2 a \left (a + b x^{3}\right )^{p + 2}}{3 b^{3} \left (p + 2\right )} + \frac{\left (a + b x^{3}\right )^{p + 3}}{3 b^{3} \left (p + 3\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**2*(a + b*x**3)**(p + 1)/(3*b**3*(p + 1)) - 2*a*(a + b*x**3)**(p + 2)/(3*b**3*
(p + 2)) + (a + b*x**3)**(p + 3)/(3*b**3*(p + 3))

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Mathematica [A]  time = 0.0458337, size = 64, normalized size = 0.86 \[ \frac{\left (a+b x^3\right )^{p+1} \left (2 a^2-2 a b (p+1) x^3+b^2 \left (p^2+3 p+2\right ) x^6\right )}{3 b^3 (p+1) (p+2) (p+3)} \]

Antiderivative was successfully verified.

[In]  Integrate[x^8*(a + b*x^3)^p,x]

[Out]

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

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Maple [A]  time = 0.008, size = 80, normalized size = 1.1 \[{\frac{ \left ( b{x}^{3}+a \right ) ^{1+p} \left ({b}^{2}{p}^{2}{x}^{6}+3\,{b}^{2}p{x}^{6}+2\,{b}^{2}{x}^{6}-2\,abp{x}^{3}-2\,ab{x}^{3}+2\,{a}^{2} \right ) }{3\,{b}^{3} \left ({p}^{3}+6\,{p}^{2}+11\,p+6 \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^8*(b*x^3+a)^p,x)

[Out]

1/3*(b*x^3+a)^(1+p)*(b^2*p^2*x^6+3*b^2*p*x^6+2*b^2*x^6-2*a*b*p*x^3-2*a*b*x^3+2*a
^2)/b^3/(p^3+6*p^2+11*p+6)

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Maxima [A]  time = 1.44993, size = 99, normalized size = 1.34 \[ \frac{{\left ({\left (p^{2} + 3 \, p + 2\right )} b^{3} x^{9} +{\left (p^{2} + p\right )} a b^{2} x^{6} - 2 \, a^{2} b p x^{3} + 2 \, a^{3}\right )}{\left (b x^{3} + a\right )}^{p}}{3 \,{\left (p^{3} + 6 \, p^{2} + 11 \, p + 6\right )} b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*((p^2 + 3*p + 2)*b^3*x^9 + (p^2 + p)*a*b^2*x^6 - 2*a^2*b*p*x^3 + 2*a^3)*(b*x
^3 + a)^p/((p^3 + 6*p^2 + 11*p + 6)*b^3)

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Fricas [A]  time = 0.243509, size = 132, normalized size = 1.78 \[ \frac{{\left ({\left (b^{3} p^{2} + 3 \, b^{3} p + 2 \, b^{3}\right )} x^{9} - 2 \, a^{2} b p x^{3} +{\left (a b^{2} p^{2} + a b^{2} p\right )} x^{6} + 2 \, a^{3}\right )}{\left (b x^{3} + a\right )}^{p}}{3 \,{\left (b^{3} p^{3} + 6 \, b^{3} p^{2} + 11 \, b^{3} p + 6 \, b^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*((b^3*p^2 + 3*b^3*p + 2*b^3)*x^9 - 2*a^2*b*p*x^3 + (a*b^2*p^2 + a*b^2*p)*x^6
+ 2*a^3)*(b*x^3 + a)^p/(b^3*p^3 + 6*b^3*p^2 + 11*b^3*p + 6*b^3)

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Sympy [A]  time = 90.7707, size = 1368, normalized size = 18.49 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**8*(b*x**3+a)**p,x)

[Out]

Piecewise((a**p*x**9/9, Eq(b, 0)), (2*a**2*log(-(-1)**(1/3)*a**(1/3)*(1/b)**(1/3
) + x)/(6*a**2*b**3 + 12*a*b**4*x**3 + 6*b**5*x**6) + 2*a**2*log(4*(-1)**(2/3)*a
**(2/3)*(1/b)**(2/3) + 4*(-1)**(1/3)*a**(1/3)*x*(1/b)**(1/3) + 4*x**2)/(6*a**2*b
**3 + 12*a*b**4*x**3 + 6*b**5*x**6) - 4*a**2*log(2)/(6*a**2*b**3 + 12*a*b**4*x**
3 + 6*b**5*x**6) + a**2/(6*a**2*b**3 + 12*a*b**4*x**3 + 6*b**5*x**6) + 4*a*b*x**
3*log(-(-1)**(1/3)*a**(1/3)*(1/b)**(1/3) + x)/(6*a**2*b**3 + 12*a*b**4*x**3 + 6*
b**5*x**6) + 4*a*b*x**3*log(4*(-1)**(2/3)*a**(2/3)*(1/b)**(2/3) + 4*(-1)**(1/3)*
a**(1/3)*x*(1/b)**(1/3) + 4*x**2)/(6*a**2*b**3 + 12*a*b**4*x**3 + 6*b**5*x**6) -
8*a*b*x**3*log(2)/(6*a**2*b**3 + 12*a*b**4*x**3 + 6*b**5*x**6) + 2*b**2*x**6*lo
g(-(-1)**(1/3)*a**(1/3)*(1/b)**(1/3) + x)/(6*a**2*b**3 + 12*a*b**4*x**3 + 6*b**5
*x**6) + 2*b**2*x**6*log(4*(-1)**(2/3)*a**(2/3)*(1/b)**(2/3) + 4*(-1)**(1/3)*a**
(1/3)*x*(1/b)**(1/3) + 4*x**2)/(6*a**2*b**3 + 12*a*b**4*x**3 + 6*b**5*x**6) - 4*
b**2*x**6*log(2)/(6*a**2*b**3 + 12*a*b**4*x**3 + 6*b**5*x**6) - 2*b**2*x**6/(6*a
**2*b**3 + 12*a*b**4*x**3 + 6*b**5*x**6), Eq(p, -3)), (-2*a**2*log(-(-1)**(1/3)*
a**(1/3)*(1/b)**(1/3) + x)/(3*a*b**3 + 3*b**4*x**3) - 2*a**2*log(4*(-1)**(2/3)*a
**(2/3)*(1/b)**(2/3) + 4*(-1)**(1/3)*a**(1/3)*x*(1/b)**(1/3) + 4*x**2)/(3*a*b**3
+ 3*b**4*x**3) - 2*a**2/(3*a*b**3 + 3*b**4*x**3) + 4*a**2*log(2)/(3*a*b**3 + 3*
b**4*x**3) - 2*a*b*x**3*log(-(-1)**(1/3)*a**(1/3)*(1/b)**(1/3) + x)/(3*a*b**3 +
3*b**4*x**3) - 2*a*b*x**3*log(4*(-1)**(2/3)*a**(2/3)*(1/b)**(2/3) + 4*(-1)**(1/3
)*a**(1/3)*x*(1/b)**(1/3) + 4*x**2)/(3*a*b**3 + 3*b**4*x**3) + 4*a*b*x**3*log(2)
/(3*a*b**3 + 3*b**4*x**3) + b**2*x**6/(3*a*b**3 + 3*b**4*x**3), Eq(p, -2)), (a**
2*log(-(-1)**(1/3)*a**(1/3)*(1/b)**(1/3) + x)/(3*b**3) + a**2*log(4*(-1)**(2/3)*
a**(2/3)*(1/b)**(2/3) + 4*(-1)**(1/3)*a**(1/3)*x*(1/b)**(1/3) + 4*x**2)/(3*b**3)
- a*x**3/(3*b**2) + x**6/(6*b), Eq(p, -1)), (2*a**3*(a + b*x**3)**p/(3*b**3*p**
3 + 18*b**3*p**2 + 33*b**3*p + 18*b**3) - 2*a**2*b*p*x**3*(a + b*x**3)**p/(3*b**
3*p**3 + 18*b**3*p**2 + 33*b**3*p + 18*b**3) + a*b**2*p**2*x**6*(a + b*x**3)**p/
(3*b**3*p**3 + 18*b**3*p**2 + 33*b**3*p + 18*b**3) + a*b**2*p*x**6*(a + b*x**3)*
*p/(3*b**3*p**3 + 18*b**3*p**2 + 33*b**3*p + 18*b**3) + b**3*p**2*x**9*(a + b*x*
*3)**p/(3*b**3*p**3 + 18*b**3*p**2 + 33*b**3*p + 18*b**3) + 3*b**3*p*x**9*(a + b
*x**3)**p/(3*b**3*p**3 + 18*b**3*p**2 + 33*b**3*p + 18*b**3) + 2*b**3*x**9*(a +
b*x**3)**p/(3*b**3*p**3 + 18*b**3*p**2 + 33*b**3*p + 18*b**3), True))

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GIAC/XCAS [A]  time = 0.219965, size = 336, normalized size = 4.54 \[ \frac{{\left (b x^{3} + a\right )}^{3} p^{2} e^{\left (p{\rm ln}\left (b x^{3} + a\right )\right )} - 2 \,{\left (b x^{3} + a\right )}^{2} a p^{2} e^{\left (p{\rm ln}\left (b x^{3} + a\right )\right )} +{\left (b x^{3} + a\right )} a^{2} p^{2} e^{\left (p{\rm ln}\left (b x^{3} + a\right )\right )} + 3 \,{\left (b x^{3} + a\right )}^{3} p e^{\left (p{\rm ln}\left (b x^{3} + a\right )\right )} - 8 \,{\left (b x^{3} + a\right )}^{2} a p e^{\left (p{\rm ln}\left (b x^{3} + a\right )\right )} + 5 \,{\left (b x^{3} + a\right )} a^{2} p e^{\left (p{\rm ln}\left (b x^{3} + a\right )\right )} + 2 \,{\left (b x^{3} + a\right )}^{3} e^{\left (p{\rm ln}\left (b x^{3} + a\right )\right )} - 6 \,{\left (b x^{3} + a\right )}^{2} a e^{\left (p{\rm ln}\left (b x^{3} + a\right )\right )} + 6 \,{\left (b x^{3} + a\right )} a^{2} e^{\left (p{\rm ln}\left (b x^{3} + a\right )\right )}}{3 \,{\left (b^{2} p^{3} + 6 \, b^{2} p^{2} + 11 \, b^{2} p + 6 \, b^{2}\right )} b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*((b*x^3 + a)^3*p^2*e^(p*ln(b*x^3 + a)) - 2*(b*x^3 + a)^2*a*p^2*e^(p*ln(b*x^3
+ a)) + (b*x^3 + a)*a^2*p^2*e^(p*ln(b*x^3 + a)) + 3*(b*x^3 + a)^3*p*e^(p*ln(b*x
^3 + a)) - 8*(b*x^3 + a)^2*a*p*e^(p*ln(b*x^3 + a)) + 5*(b*x^3 + a)*a^2*p*e^(p*ln
(b*x^3 + a)) + 2*(b*x^3 + a)^3*e^(p*ln(b*x^3 + a)) - 6*(b*x^3 + a)^2*a*e^(p*ln(b
*x^3 + a)) + 6*(b*x^3 + a)*a^2*e^(p*ln(b*x^3 + a)))/((b^2*p^3 + 6*b^2*p^2 + 11*b
^2*p + 6*b^2)*b)