3.523 \(\int x^2 \left (a^2+2 a b x^n+b^2 x^{2 n}\right )^{3/2} \, dx\)

Optimal. Leaf size=212 \[ \frac{3 a^2 b^2 x^{n+3} \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{(n+3) \left (a b+b^2 x^n\right )}+\frac{b^4 x^{3 (n+1)} \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{3 (n+1) \left (a b+b^2 x^n\right )}+\frac{3 a b^3 x^{2 n+3} \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{(2 n+3) \left (a b+b^2 x^n\right )}+\frac{a^3 x^3 \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{3 \left (a+b x^n\right )} \]

[Out]

(a^3*x^3*Sqrt[a^2 + 2*a*b*x^n + b^2*x^(2*n)])/(3*(a + b*x^n)) + (b^4*x^(3*(1 + n
))*Sqrt[a^2 + 2*a*b*x^n + b^2*x^(2*n)])/(3*(1 + n)*(a*b + b^2*x^n)) + (3*a^2*b^2
*x^(3 + n)*Sqrt[a^2 + 2*a*b*x^n + b^2*x^(2*n)])/((3 + n)*(a*b + b^2*x^n)) + (3*a
*b^3*x^(3 + 2*n)*Sqrt[a^2 + 2*a*b*x^n + b^2*x^(2*n)])/((3 + 2*n)*(a*b + b^2*x^n)
)

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Rubi [A]  time = 0.160002, antiderivative size = 212, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071 \[ \frac{3 a^2 b^2 x^{n+3} \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{(n+3) \left (a b+b^2 x^n\right )}+\frac{b^4 x^{3 (n+1)} \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{3 (n+1) \left (a b+b^2 x^n\right )}+\frac{3 a b^3 x^{2 n+3} \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{(2 n+3) \left (a b+b^2 x^n\right )}+\frac{a^3 x^3 \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{3 \left (a+b x^n\right )} \]

Antiderivative was successfully verified.

[In]  Int[x^2*(a^2 + 2*a*b*x^n + b^2*x^(2*n))^(3/2),x]

[Out]

(a^3*x^3*Sqrt[a^2 + 2*a*b*x^n + b^2*x^(2*n)])/(3*(a + b*x^n)) + (b^4*x^(3*(1 + n
))*Sqrt[a^2 + 2*a*b*x^n + b^2*x^(2*n)])/(3*(1 + n)*(a*b + b^2*x^n)) + (3*a^2*b^2
*x^(3 + n)*Sqrt[a^2 + 2*a*b*x^n + b^2*x^(2*n)])/((3 + n)*(a*b + b^2*x^n)) + (3*a
*b^3*x^(3 + 2*n)*Sqrt[a^2 + 2*a*b*x^n + b^2*x^(2*n)])/((3 + 2*n)*(a*b + b^2*x^n)
)

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Rubi in Sympy [A]  time = 29.5476, size = 197, normalized size = 0.93 \[ \frac{a^{3} b x^{3} \sqrt{a^{2} + 2 a b x^{n} + b^{2} x^{2 n}}}{3 \left (a b + b^{2} x^{n}\right )} + \frac{3 a^{2} b^{2} x^{n + 3} \sqrt{a^{2} + 2 a b x^{n} + b^{2} x^{2 n}}}{\left (n + 3\right ) \left (a b + b^{2} x^{n}\right )} + \frac{3 a b^{3} x^{2 n + 3} \sqrt{a^{2} + 2 a b x^{n} + b^{2} x^{2 n}}}{\left (2 n + 3\right ) \left (a b + b^{2} x^{n}\right )} + \frac{b^{4} x^{3 n + 3} \sqrt{a^{2} + 2 a b x^{n} + b^{2} x^{2 n}}}{3 \left (n + 1\right ) \left (a b + b^{2} x^{n}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**2*(a**2+2*a*b*x**n+b**2*x**(2*n))**(3/2),x)

[Out]

a**3*b*x**3*sqrt(a**2 + 2*a*b*x**n + b**2*x**(2*n))/(3*(a*b + b**2*x**n)) + 3*a*
*2*b**2*x**(n + 3)*sqrt(a**2 + 2*a*b*x**n + b**2*x**(2*n))/((n + 3)*(a*b + b**2*
x**n)) + 3*a*b**3*x**(2*n + 3)*sqrt(a**2 + 2*a*b*x**n + b**2*x**(2*n))/((2*n + 3
)*(a*b + b**2*x**n)) + b**4*x**(3*n + 3)*sqrt(a**2 + 2*a*b*x**n + b**2*x**(2*n))
/(3*(n + 1)*(a*b + b**2*x**n))

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Mathematica [A]  time = 0.106807, size = 123, normalized size = 0.58 \[ \frac{x^3 \sqrt{\left (a+b x^n\right )^2} \left (a^3 \left (2 n^3+11 n^2+18 n+9\right )+9 a^2 b \left (2 n^2+5 n+3\right ) x^n+9 a b^2 \left (n^2+4 n+3\right ) x^{2 n}+b^3 \left (2 n^2+9 n+9\right ) x^{3 n}\right )}{3 (n+1) (n+3) (2 n+3) \left (a+b x^n\right )} \]

Antiderivative was successfully verified.

[In]  Integrate[x^2*(a^2 + 2*a*b*x^n + b^2*x^(2*n))^(3/2),x]

[Out]

(x^3*Sqrt[(a + b*x^n)^2]*(a^3*(9 + 18*n + 11*n^2 + 2*n^3) + 9*a^2*b*(3 + 5*n + 2
*n^2)*x^n + 9*a*b^2*(3 + 4*n + n^2)*x^(2*n) + b^3*(9 + 9*n + 2*n^2)*x^(3*n)))/(3
*(1 + n)*(3 + n)*(3 + 2*n)*(a + b*x^n))

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Maple [A]  time = 0.025, size = 146, normalized size = 0.7 \[{\frac{{x}^{3}{a}^{3}}{3\,a+3\,b{x}^{n}}\sqrt{ \left ( a+b{x}^{n} \right ) ^{2}}}+{\frac{{b}^{3}{x}^{3} \left ({x}^{n} \right ) ^{3}}{ \left ( 3\,a+3\,b{x}^{n} \right ) \left ( 1+n \right ) }\sqrt{ \left ( a+b{x}^{n} \right ) ^{2}}}+3\,{\frac{\sqrt{ \left ( a+b{x}^{n} \right ) ^{2}}a{b}^{2}{x}^{3} \left ({x}^{n} \right ) ^{2}}{ \left ( a+b{x}^{n} \right ) \left ( 3+2\,n \right ) }}+3\,{\frac{\sqrt{ \left ( a+b{x}^{n} \right ) ^{2}}{a}^{2}b{x}^{3}{x}^{n}}{ \left ( a+b{x}^{n} \right ) \left ( 3+n \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2*(a^2+2*a*b*x^n+b^2*x^(2*n))^(3/2),x)

[Out]

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

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Maxima [A]  time = 0.754999, size = 146, normalized size = 0.69 \[ \frac{{\left (2 \, n^{2} + 9 \, n + 9\right )} b^{3} x^{3} x^{3 \, n} + 9 \,{\left (n^{2} + 4 \, n + 3\right )} a b^{2} x^{3} x^{2 \, n} + 9 \,{\left (2 \, n^{2} + 5 \, n + 3\right )} a^{2} b x^{3} x^{n} +{\left (2 \, n^{3} + 11 \, n^{2} + 18 \, n + 9\right )} a^{3} x^{3}}{3 \,{\left (2 \, n^{3} + 11 \, n^{2} + 18 \, n + 9\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^(2*n) + 2*a*b*x^n + a^2)^(3/2)*x^2,x, algorithm="maxima")

[Out]

1/3*((2*n^2 + 9*n + 9)*b^3*x^3*x^(3*n) + 9*(n^2 + 4*n + 3)*a*b^2*x^3*x^(2*n) + 9
*(2*n^2 + 5*n + 3)*a^2*b*x^3*x^n + (2*n^3 + 11*n^2 + 18*n + 9)*a^3*x^3)/(2*n^3 +
11*n^2 + 18*n + 9)

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Fricas [A]  time = 0.271024, size = 194, normalized size = 0.92 \[ \frac{{\left (2 \, b^{3} n^{2} + 9 \, b^{3} n + 9 \, b^{3}\right )} x^{3} x^{3 \, n} + 9 \,{\left (a b^{2} n^{2} + 4 \, a b^{2} n + 3 \, a b^{2}\right )} x^{3} x^{2 \, n} + 9 \,{\left (2 \, a^{2} b n^{2} + 5 \, a^{2} b n + 3 \, a^{2} b\right )} x^{3} x^{n} +{\left (2 \, a^{3} n^{3} + 11 \, a^{3} n^{2} + 18 \, a^{3} n + 9 \, a^{3}\right )} x^{3}}{3 \,{\left (2 \, n^{3} + 11 \, n^{2} + 18 \, n + 9\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^(2*n) + 2*a*b*x^n + a^2)^(3/2)*x^2,x, algorithm="fricas")

[Out]

1/3*((2*b^3*n^2 + 9*b^3*n + 9*b^3)*x^3*x^(3*n) + 9*(a*b^2*n^2 + 4*a*b^2*n + 3*a*
b^2)*x^3*x^(2*n) + 9*(2*a^2*b*n^2 + 5*a^2*b*n + 3*a^2*b)*x^3*x^n + (2*a^3*n^3 +
11*a^3*n^2 + 18*a^3*n + 9*a^3)*x^3)/(2*n^3 + 11*n^2 + 18*n + 9)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2*(a**2+2*a*b*x**n+b**2*x**(2*n))**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.290506, size = 410, normalized size = 1.93 \[ \frac{2 \, a^{3} n^{3} x^{3}{\rm sign}\left (b x^{n} + a\right ) + 2 \, b^{3} n^{2} x^{3} e^{\left (3 \, n{\rm ln}\left (x\right )\right )}{\rm sign}\left (b x^{n} + a\right ) + 9 \, a b^{2} n^{2} x^{3} e^{\left (2 \, n{\rm ln}\left (x\right )\right )}{\rm sign}\left (b x^{n} + a\right ) + 18 \, a^{2} b n^{2} x^{3} e^{\left (n{\rm ln}\left (x\right )\right )}{\rm sign}\left (b x^{n} + a\right ) + 11 \, a^{3} n^{2} x^{3}{\rm sign}\left (b x^{n} + a\right ) + 9 \, b^{3} n x^{3} e^{\left (3 \, n{\rm ln}\left (x\right )\right )}{\rm sign}\left (b x^{n} + a\right ) + 36 \, a b^{2} n x^{3} e^{\left (2 \, n{\rm ln}\left (x\right )\right )}{\rm sign}\left (b x^{n} + a\right ) + 45 \, a^{2} b n x^{3} e^{\left (n{\rm ln}\left (x\right )\right )}{\rm sign}\left (b x^{n} + a\right ) + 18 \, a^{3} n x^{3}{\rm sign}\left (b x^{n} + a\right ) + 9 \, b^{3} x^{3} e^{\left (3 \, n{\rm ln}\left (x\right )\right )}{\rm sign}\left (b x^{n} + a\right ) + 27 \, a b^{2} x^{3} e^{\left (2 \, n{\rm ln}\left (x\right )\right )}{\rm sign}\left (b x^{n} + a\right ) + 27 \, a^{2} b x^{3} e^{\left (n{\rm ln}\left (x\right )\right )}{\rm sign}\left (b x^{n} + a\right ) + 9 \, a^{3} x^{3}{\rm sign}\left (b x^{n} + a\right )}{3 \,{\left (2 \, n^{3} + 11 \, n^{2} + 18 \, n + 9\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^(2*n) + 2*a*b*x^n + a^2)^(3/2)*x^2,x, algorithm="giac")

[Out]

1/3*(2*a^3*n^3*x^3*sign(b*x^n + a) + 2*b^3*n^2*x^3*e^(3*n*ln(x))*sign(b*x^n + a)
+ 9*a*b^2*n^2*x^3*e^(2*n*ln(x))*sign(b*x^n + a) + 18*a^2*b*n^2*x^3*e^(n*ln(x))*
sign(b*x^n + a) + 11*a^3*n^2*x^3*sign(b*x^n + a) + 9*b^3*n*x^3*e^(3*n*ln(x))*sig
n(b*x^n + a) + 36*a*b^2*n*x^3*e^(2*n*ln(x))*sign(b*x^n + a) + 45*a^2*b*n*x^3*e^(
n*ln(x))*sign(b*x^n + a) + 18*a^3*n*x^3*sign(b*x^n + a) + 9*b^3*x^3*e^(3*n*ln(x)
)*sign(b*x^n + a) + 27*a*b^2*x^3*e^(2*n*ln(x))*sign(b*x^n + a) + 27*a^2*b*x^3*e^
(n*ln(x))*sign(b*x^n + a) + 9*a^3*x^3*sign(b*x^n + a))/(2*n^3 + 11*n^2 + 18*n +
9)