3.798 \(\int x^7 \left (a^2+2 a b x^2+b^2 x^4\right )^p \, dx\)

Optimal. Leaf size=174 \[ \frac{\left (a+b x^2\right )^4 \left (a^2+2 a b x^2+b^2 x^4\right )^p}{4 b^4 (p+2)}-\frac{3 a \left (a+b x^2\right )^3 \left (a^2+2 a b x^2+b^2 x^4\right )^p}{2 b^4 (2 p+3)}+\frac{3 a^2 \left (a+b x^2\right )^2 \left (a^2+2 a b x^2+b^2 x^4\right )^p}{4 b^4 (p+1)}-\frac{a^3 \left (a+b x^2\right ) \left (a^2+2 a b x^2+b^2 x^4\right )^p}{2 b^4 (2 p+1)} \]

[Out]

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

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Rubi [A]  time = 0.263751, antiderivative size = 174, 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 \[ \frac{\left (a+b x^2\right )^4 \left (a^2+2 a b x^2+b^2 x^4\right )^p}{4 b^4 (p+2)}-\frac{3 a \left (a+b x^2\right )^3 \left (a^2+2 a b x^2+b^2 x^4\right )^p}{2 b^4 (2 p+3)}+\frac{3 a^2 \left (a+b x^2\right )^2 \left (a^2+2 a b x^2+b^2 x^4\right )^p}{4 b^4 (p+1)}-\frac{a^3 \left (a+b x^2\right ) \left (a^2+2 a b x^2+b^2 x^4\right )^p}{2 b^4 (2 p+1)} \]

Antiderivative was successfully verified.

[In]  Int[x^7*(a^2 + 2*a*b*x^2 + b^2*x^4)^p,x]

[Out]

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

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Rubi in Sympy [A]  time = 43.4795, size = 189, normalized size = 1.09 \[ - \frac{3 a^{3} \left (2 a + 2 b x^{2}\right ) \left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{p}}{4 b^{4} \left (p + 2\right ) \left (2 p + 1\right ) \left (2 p + 3\right )} + \frac{3 a^{2} \left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{p + 1}}{4 b^{4} \left (p + 1\right ) \left (p + 2\right ) \left (2 p + 3\right )} - \frac{3 a x^{4} \left (2 a + 2 b x^{2}\right ) \left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{p}}{8 b^{2} \left (p + 2\right ) \left (2 p + 3\right )} + \frac{x^{6} \left (2 a + 2 b x^{2}\right ) \left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{p}}{8 b \left (p + 2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**7*(b**2*x**4+2*a*b*x**2+a**2)**p,x)

[Out]

-3*a**3*(2*a + 2*b*x**2)*(a**2 + 2*a*b*x**2 + b**2*x**4)**p/(4*b**4*(p + 2)*(2*p
+ 1)*(2*p + 3)) + 3*a**2*(a**2 + 2*a*b*x**2 + b**2*x**4)**(p + 1)/(4*b**4*(p +
1)*(p + 2)*(2*p + 3)) - 3*a*x**4*(2*a + 2*b*x**2)*(a**2 + 2*a*b*x**2 + b**2*x**4
)**p/(8*b**2*(p + 2)*(2*p + 3)) + x**6*(2*a + 2*b*x**2)*(a**2 + 2*a*b*x**2 + b**
2*x**4)**p/(8*b*(p + 2))

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Mathematica [A]  time = 0.0776516, size = 110, normalized size = 0.63 \[ \frac{\left (a+b x^2\right ) \left (\left (a+b x^2\right )^2\right )^p \left (-3 a^3+3 a^2 b (2 p+1) x^2-3 a b^2 \left (2 p^2+3 p+1\right ) x^4+b^3 \left (4 p^3+12 p^2+11 p+3\right ) x^6\right )}{4 b^4 (p+1) (p+2) (2 p+1) (2 p+3)} \]

Antiderivative was successfully verified.

[In]  Integrate[x^7*(a^2 + 2*a*b*x^2 + b^2*x^4)^p,x]

[Out]

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

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Maple [A]  time = 0.011, size = 150, normalized size = 0.9 \[ -{\frac{ \left ({b}^{2}{x}^{4}+2\,ab{x}^{2}+{a}^{2} \right ) ^{p} \left ( -4\,{b}^{3}{p}^{3}{x}^{6}-12\,{b}^{3}{p}^{2}{x}^{6}-11\,{b}^{3}p{x}^{6}+6\,a{b}^{2}{p}^{2}{x}^{4}-3\,{b}^{3}{x}^{6}+9\,a{b}^{2}p{x}^{4}+3\,a{x}^{4}{b}^{2}-6\,{a}^{2}bp{x}^{2}-3\,{a}^{2}b{x}^{2}+3\,{a}^{3} \right ) \left ( b{x}^{2}+a \right ) }{4\,{b}^{4} \left ( 4\,{p}^{4}+20\,{p}^{3}+35\,{p}^{2}+25\,p+6 \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^7*(b^2*x^4+2*a*b*x^2+a^2)^p,x)

[Out]

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

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Maxima [A]  time = 0.692871, size = 155, normalized size = 0.89 \[ \frac{{\left ({\left (4 \, p^{3} + 12 \, p^{2} + 11 \, p + 3\right )} b^{4} x^{8} + 2 \,{\left (2 \, p^{3} + 3 \, p^{2} + p\right )} a b^{3} x^{6} - 3 \,{\left (2 \, p^{2} + p\right )} a^{2} b^{2} x^{4} + 6 \, a^{3} b p x^{2} - 3 \, a^{4}\right )}{\left (b x^{2} + a\right )}^{2 \, p}}{4 \,{\left (4 \, p^{4} + 20 \, p^{3} + 35 \, p^{2} + 25 \, p + 6\right )} b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^4 + 2*a*b*x^2 + a^2)^p*x^7,x, algorithm="maxima")

[Out]

1/4*((4*p^3 + 12*p^2 + 11*p + 3)*b^4*x^8 + 2*(2*p^3 + 3*p^2 + p)*a*b^3*x^6 - 3*(
2*p^2 + p)*a^2*b^2*x^4 + 6*a^3*b*p*x^2 - 3*a^4)*(b*x^2 + a)^(2*p)/((4*p^4 + 20*p
^3 + 35*p^2 + 25*p + 6)*b^4)

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Fricas [A]  time = 0.289315, size = 220, normalized size = 1.26 \[ \frac{{\left ({\left (4 \, b^{4} p^{3} + 12 \, b^{4} p^{2} + 11 \, b^{4} p + 3 \, b^{4}\right )} x^{8} + 6 \, a^{3} b p x^{2} + 2 \,{\left (2 \, a b^{3} p^{3} + 3 \, a b^{3} p^{2} + a b^{3} p\right )} x^{6} - 3 \,{\left (2 \, a^{2} b^{2} p^{2} + a^{2} b^{2} p\right )} x^{4} - 3 \, a^{4}\right )}{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{p}}{4 \,{\left (4 \, b^{4} p^{4} + 20 \, b^{4} p^{3} + 35 \, b^{4} p^{2} + 25 \, b^{4} p + 6 \, b^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^4 + 2*a*b*x^2 + a^2)^p*x^7,x, algorithm="fricas")

[Out]

1/4*((4*b^4*p^3 + 12*b^4*p^2 + 11*b^4*p + 3*b^4)*x^8 + 6*a^3*b*p*x^2 + 2*(2*a*b^
3*p^3 + 3*a*b^3*p^2 + a*b^3*p)*x^6 - 3*(2*a^2*b^2*p^2 + a^2*b^2*p)*x^4 - 3*a^4)*
(b^2*x^4 + 2*a*b*x^2 + 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**7*(b**2*x**4+2*a*b*x**2+a**2)**p,x)

[Out]

Exception raised: TypeError

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GIAC/XCAS [A]  time = 0.272281, size = 536, normalized size = 3.08 \[ \frac{4 \, b^{4} p^{3} x^{8} e^{\left (p{\rm ln}\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )\right )} + 12 \, b^{4} p^{2} x^{8} e^{\left (p{\rm ln}\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )\right )} + 4 \, a b^{3} p^{3} x^{6} e^{\left (p{\rm ln}\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )\right )} + 11 \, b^{4} p x^{8} e^{\left (p{\rm ln}\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )\right )} + 6 \, a b^{3} p^{2} x^{6} e^{\left (p{\rm ln}\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )\right )} + 3 \, b^{4} x^{8} e^{\left (p{\rm ln}\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )\right )} + 2 \, a b^{3} p x^{6} e^{\left (p{\rm ln}\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )\right )} - 6 \, a^{2} b^{2} p^{2} x^{4} e^{\left (p{\rm ln}\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )\right )} - 3 \, a^{2} b^{2} p x^{4} e^{\left (p{\rm ln}\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )\right )} + 6 \, a^{3} b p x^{2} e^{\left (p{\rm ln}\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )\right )} - 3 \, a^{4} e^{\left (p{\rm ln}\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )\right )}}{4 \,{\left (4 \, b^{4} p^{4} + 20 \, b^{4} p^{3} + 35 \, b^{4} p^{2} + 25 \, b^{4} p + 6 \, b^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^4 + 2*a*b*x^2 + a^2)^p*x^7,x, algorithm="giac")

[Out]

1/4*(4*b^4*p^3*x^8*e^(p*ln(b^2*x^4 + 2*a*b*x^2 + a^2)) + 12*b^4*p^2*x^8*e^(p*ln(
b^2*x^4 + 2*a*b*x^2 + a^2)) + 4*a*b^3*p^3*x^6*e^(p*ln(b^2*x^4 + 2*a*b*x^2 + a^2)
) + 11*b^4*p*x^8*e^(p*ln(b^2*x^4 + 2*a*b*x^2 + a^2)) + 6*a*b^3*p^2*x^6*e^(p*ln(b
^2*x^4 + 2*a*b*x^2 + a^2)) + 3*b^4*x^8*e^(p*ln(b^2*x^4 + 2*a*b*x^2 + a^2)) + 2*a
*b^3*p*x^6*e^(p*ln(b^2*x^4 + 2*a*b*x^2 + a^2)) - 6*a^2*b^2*p^2*x^4*e^(p*ln(b^2*x
^4 + 2*a*b*x^2 + a^2)) - 3*a^2*b^2*p*x^4*e^(p*ln(b^2*x^4 + 2*a*b*x^2 + a^2)) + 6
*a^3*b*p*x^2*e^(p*ln(b^2*x^4 + 2*a*b*x^2 + a^2)) - 3*a^4*e^(p*ln(b^2*x^4 + 2*a*b
*x^2 + a^2)))/(4*b^4*p^4 + 20*b^4*p^3 + 35*b^4*p^2 + 25*b^4*p + 6*b^4)