3.125 \(\int x^{11} \left (a^2+2 a b x^3+b^2 x^6\right )^p \, dx\)

Optimal. Leaf size=172 \[ \frac{\left (a+b x^3\right )^4 \left (a^2+2 a b x^3+b^2 x^6\right )^p}{6 b^4 (p+2)}-\frac{a \left (a+b x^3\right )^3 \left (a^2+2 a b x^3+b^2 x^6\right )^p}{b^4 (2 p+3)}+\frac{a^2 \left (a+b x^3\right )^2 \left (a^2+2 a b x^3+b^2 x^6\right )^p}{2 b^4 (p+1)}-\frac{a^3 \left (a+b x^3\right ) \left (a^2+2 a b x^3+b^2 x^6\right )^p}{3 b^4 (2 p+1)} \]

[Out]

-(a^3*(a + b*x^3)*(a^2 + 2*a*b*x^3 + b^2*x^6)^p)/(3*b^4*(1 + 2*p)) + (a^2*(a + b
*x^3)^2*(a^2 + 2*a*b*x^3 + b^2*x^6)^p)/(2*b^4*(1 + p)) - (a*(a + b*x^3)^3*(a^2 +
2*a*b*x^3 + b^2*x^6)^p)/(b^4*(3 + 2*p)) + ((a + b*x^3)^4*(a^2 + 2*a*b*x^3 + b^2
*x^6)^p)/(6*b^4*(2 + p))

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Rubi [A]  time = 0.22608, antiderivative size = 172, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ \frac{\left (a+b x^3\right )^4 \left (a^2+2 a b x^3+b^2 x^6\right )^p}{6 b^4 (p+2)}-\frac{a \left (a+b x^3\right )^3 \left (a^2+2 a b x^3+b^2 x^6\right )^p}{b^4 (2 p+3)}+\frac{a^2 \left (a+b x^3\right )^2 \left (a^2+2 a b x^3+b^2 x^6\right )^p}{2 b^4 (p+1)}-\frac{a^3 \left (a+b x^3\right ) \left (a^2+2 a b x^3+b^2 x^6\right )^p}{3 b^4 (2 p+1)} \]

Antiderivative was successfully verified.

[In]  Int[x^11*(a^2 + 2*a*b*x^3 + b^2*x^6)^p,x]

[Out]

-(a^3*(a + b*x^3)*(a^2 + 2*a*b*x^3 + b^2*x^6)^p)/(3*b^4*(1 + 2*p)) + (a^2*(a + b
*x^3)^2*(a^2 + 2*a*b*x^3 + b^2*x^6)^p)/(2*b^4*(1 + p)) - (a*(a + b*x^3)^3*(a^2 +
2*a*b*x^3 + b^2*x^6)^p)/(b^4*(3 + 2*p)) + ((a + b*x^3)^4*(a^2 + 2*a*b*x^3 + b^2
*x^6)^p)/(6*b^4*(2 + p))

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Rubi in Sympy [A]  time = 45.8687, size = 184, normalized size = 1.07 \[ - \frac{a^{3} \left (2 a + 2 b x^{3}\right ) \left (a^{2} + 2 a b x^{3} + b^{2} x^{6}\right )^{p}}{2 b^{4} \left (p + 2\right ) \left (2 p + 1\right ) \left (2 p + 3\right )} + \frac{a^{2} \left (a^{2} + 2 a b x^{3} + b^{2} x^{6}\right )^{p + 1}}{2 b^{4} \left (p + 1\right ) \left (p + 2\right ) \left (2 p + 3\right )} - \frac{a x^{6} \left (2 a + 2 b x^{3}\right ) \left (a^{2} + 2 a b x^{3} + b^{2} x^{6}\right )^{p}}{4 b^{2} \left (p + 2\right ) \left (2 p + 3\right )} + \frac{x^{9} \left (2 a + 2 b x^{3}\right ) \left (a^{2} + 2 a b x^{3} + b^{2} x^{6}\right )^{p}}{12 b \left (p + 2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**11*(b**2*x**6+2*a*b*x**3+a**2)**p,x)

[Out]

-a**3*(2*a + 2*b*x**3)*(a**2 + 2*a*b*x**3 + b**2*x**6)**p/(2*b**4*(p + 2)*(2*p +
1)*(2*p + 3)) + a**2*(a**2 + 2*a*b*x**3 + b**2*x**6)**(p + 1)/(2*b**4*(p + 1)*(
p + 2)*(2*p + 3)) - a*x**6*(2*a + 2*b*x**3)*(a**2 + 2*a*b*x**3 + b**2*x**6)**p/(
4*b**2*(p + 2)*(2*p + 3)) + x**9*(2*a + 2*b*x**3)*(a**2 + 2*a*b*x**3 + b**2*x**6
)**p/(12*b*(p + 2))

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Mathematica [A]  time = 0.0725597, size = 110, normalized size = 0.64 \[ \frac{\left (a+b x^3\right ) \left (\left (a+b x^3\right )^2\right )^p \left (-3 a^3+3 a^2 b (2 p+1) x^3-3 a b^2 \left (2 p^2+3 p+1\right ) x^6+b^3 \left (4 p^3+12 p^2+11 p+3\right ) x^9\right )}{6 b^4 (p+1) (p+2) (2 p+1) (2 p+3)} \]

Antiderivative was successfully verified.

[In]  Integrate[x^11*(a^2 + 2*a*b*x^3 + b^2*x^6)^p,x]

[Out]

((a + b*x^3)*((a + b*x^3)^2)^p*(-3*a^3 + 3*a^2*b*(1 + 2*p)*x^3 - 3*a*b^2*(1 + 3*
p + 2*p^2)*x^6 + b^3*(3 + 11*p + 12*p^2 + 4*p^3)*x^9))/(6*b^4*(1 + p)*(2 + p)*(1
+ 2*p)*(3 + 2*p))

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Maple [A]  time = 0.012, size = 150, normalized size = 0.9 \[ -{\frac{ \left ({b}^{2}{x}^{6}+2\,ab{x}^{3}+{a}^{2} \right ) ^{p} \left ( -4\,{b}^{3}{p}^{3}{x}^{9}-12\,{b}^{3}{p}^{2}{x}^{9}-11\,{b}^{3}p{x}^{9}-3\,{b}^{3}{x}^{9}+6\,a{b}^{2}{p}^{2}{x}^{6}+9\,a{b}^{2}p{x}^{6}+3\,a{x}^{6}{b}^{2}-6\,{a}^{2}bp{x}^{3}-3\,{x}^{3}{a}^{2}b+3\,{a}^{3} \right ) \left ( b{x}^{3}+a \right ) }{6\,{b}^{4} \left ( 4\,{p}^{4}+20\,{p}^{3}+35\,{p}^{2}+25\,p+6 \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^11*(b^2*x^6+2*a*b*x^3+a^2)^p,x)

[Out]

-1/6*(b^2*x^6+2*a*b*x^3+a^2)^p*(-4*b^3*p^3*x^9-12*b^3*p^2*x^9-11*b^3*p*x^9-3*b^3
*x^9+6*a*b^2*p^2*x^6+9*a*b^2*p*x^6+3*a*b^2*x^6-6*a^2*b*p*x^3-3*a^2*b*x^3+3*a^3)*
(b*x^3+a)/b^4/(4*p^4+20*p^3+35*p^2+25*p+6)

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Maxima [A]  time = 0.794362, size = 155, normalized size = 0.9 \[ \frac{{\left ({\left (4 \, p^{3} + 12 \, p^{2} + 11 \, p + 3\right )} b^{4} x^{12} + 2 \,{\left (2 \, p^{3} + 3 \, p^{2} + p\right )} a b^{3} x^{9} - 3 \,{\left (2 \, p^{2} + p\right )} a^{2} b^{2} x^{6} + 6 \, a^{3} b p x^{3} - 3 \, a^{4}\right )}{\left (b x^{3} + a\right )}^{2 \, p}}{6 \,{\left (4 \, p^{4} + 20 \, p^{3} + 35 \, p^{2} + 25 \, p + 6\right )} b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^6 + 2*a*b*x^3 + a^2)^p*x^11,x, algorithm="maxima")

[Out]

1/6*((4*p^3 + 12*p^2 + 11*p + 3)*b^4*x^12 + 2*(2*p^3 + 3*p^2 + p)*a*b^3*x^9 - 3*
(2*p^2 + p)*a^2*b^2*x^6 + 6*a^3*b*p*x^3 - 3*a^4)*(b*x^3 + a)^(2*p)/((4*p^4 + 20*
p^3 + 35*p^2 + 25*p + 6)*b^4)

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Fricas [A]  time = 0.268128, size = 220, normalized size = 1.28 \[ \frac{{\left ({\left (4 \, b^{4} p^{3} + 12 \, b^{4} p^{2} + 11 \, b^{4} p + 3 \, b^{4}\right )} x^{12} + 2 \,{\left (2 \, a b^{3} p^{3} + 3 \, a b^{3} p^{2} + a b^{3} p\right )} x^{9} + 6 \, a^{3} b p x^{3} - 3 \,{\left (2 \, a^{2} b^{2} p^{2} + a^{2} b^{2} p\right )} x^{6} - 3 \, a^{4}\right )}{\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{p}}{6 \,{\left (4 \, b^{4} p^{4} + 20 \, b^{4} p^{3} + 35 \, b^{4} p^{2} + 25 \, b^{4} p + 6 \, b^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^6 + 2*a*b*x^3 + a^2)^p*x^11,x, algorithm="fricas")

[Out]

1/6*((4*b^4*p^3 + 12*b^4*p^2 + 11*b^4*p + 3*b^4)*x^12 + 2*(2*a*b^3*p^3 + 3*a*b^3
*p^2 + a*b^3*p)*x^9 + 6*a^3*b*p*x^3 - 3*(2*a^2*b^2*p^2 + a^2*b^2*p)*x^6 - 3*a^4)
*(b^2*x^6 + 2*a*b*x^3 + a^2)^p/(4*b^4*p^4 + 20*b^4*p^3 + 35*b^4*p^2 + 25*b^4*p +
6*b^4)

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**11*(b**2*x**6+2*a*b*x**3+a**2)**p,x)

[Out]

Exception raised: TypeError

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GIAC/XCAS [A]  time = 0.296167, size = 536, normalized size = 3.12 \[ \frac{4 \, b^{4} p^{3} x^{12} e^{\left (p{\rm ln}\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )\right )} + 12 \, b^{4} p^{2} x^{12} e^{\left (p{\rm ln}\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )\right )} + 11 \, b^{4} p x^{12} e^{\left (p{\rm ln}\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )\right )} + 4 \, a b^{3} p^{3} x^{9} e^{\left (p{\rm ln}\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )\right )} + 3 \, b^{4} x^{12} e^{\left (p{\rm ln}\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )\right )} + 6 \, a b^{3} p^{2} x^{9} e^{\left (p{\rm ln}\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )\right )} + 2 \, a b^{3} p x^{9} e^{\left (p{\rm ln}\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )\right )} - 6 \, a^{2} b^{2} p^{2} x^{6} e^{\left (p{\rm ln}\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )\right )} - 3 \, a^{2} b^{2} p x^{6} e^{\left (p{\rm ln}\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )\right )} + 6 \, a^{3} b p x^{3} e^{\left (p{\rm ln}\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )\right )} - 3 \, a^{4} e^{\left (p{\rm ln}\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )\right )}}{6 \,{\left (4 \, b^{4} p^{4} + 20 \, b^{4} p^{3} + 35 \, b^{4} p^{2} + 25 \, b^{4} p + 6 \, b^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^6 + 2*a*b*x^3 + a^2)^p*x^11,x, algorithm="giac")

[Out]

1/6*(4*b^4*p^3*x^12*e^(p*ln(b^2*x^6 + 2*a*b*x^3 + a^2)) + 12*b^4*p^2*x^12*e^(p*l
n(b^2*x^6 + 2*a*b*x^3 + a^2)) + 11*b^4*p*x^12*e^(p*ln(b^2*x^6 + 2*a*b*x^3 + a^2)
) + 4*a*b^3*p^3*x^9*e^(p*ln(b^2*x^6 + 2*a*b*x^3 + a^2)) + 3*b^4*x^12*e^(p*ln(b^2
*x^6 + 2*a*b*x^3 + a^2)) + 6*a*b^3*p^2*x^9*e^(p*ln(b^2*x^6 + 2*a*b*x^3 + a^2)) +
2*a*b^3*p*x^9*e^(p*ln(b^2*x^6 + 2*a*b*x^3 + a^2)) - 6*a^2*b^2*p^2*x^6*e^(p*ln(b
^2*x^6 + 2*a*b*x^3 + a^2)) - 3*a^2*b^2*p*x^6*e^(p*ln(b^2*x^6 + 2*a*b*x^3 + a^2))
+ 6*a^3*b*p*x^3*e^(p*ln(b^2*x^6 + 2*a*b*x^3 + a^2)) - 3*a^4*e^(p*ln(b^2*x^6 + 2
*a*b*x^3 + a^2)))/(4*b^4*p^4 + 20*b^4*p^3 + 35*b^4*p^2 + 25*b^4*p + 6*b^4)