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

Optimal. Leaf size=211 \[ \frac{3 a^2 b^2 x^{n+2} \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{(n+2) \left (a b+b^2 x^n\right )}+\frac{b^4 x^{3 n+2} \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{(3 n+2) \left (a b+b^2 x^n\right )}+\frac{3 a b^3 x^{2 (n+1)} \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{2 (n+1) \left (a b+b^2 x^n\right )}+\frac{a^3 x^2 \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{2 \left (a+b x^n\right )} \]

[Out]

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

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Rubi [A]  time = 0.139376, antiderivative size = 211, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ \frac{3 a^2 b^2 x^{n+2} \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{(n+2) \left (a b+b^2 x^n\right )}+\frac{b^4 x^{3 n+2} \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{(3 n+2) \left (a b+b^2 x^n\right )}+\frac{3 a b^3 x^{2 (n+1)} \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{2 (n+1) \left (a b+b^2 x^n\right )}+\frac{a^3 x^2 \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{2 \left (a+b x^n\right )} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{a^{3} b \sqrt{a^{2} + 2 a b x^{n} + b^{2} x^{2 n}} \int x\, dx}{a b + b^{2} x^{n}} + \frac{3 a^{2} b^{2} x^{n + 2} \sqrt{a^{2} + 2 a b x^{n} + b^{2} x^{2 n}}}{\left (n + 2\right ) \left (a b + b^{2} x^{n}\right )} + \frac{3 a b^{3} x^{2 n + 2} \sqrt{a^{2} + 2 a b x^{n} + b^{2} x^{2 n}}}{2 \left (n + 1\right ) \left (a b + b^{2} x^{n}\right )} + \frac{b^{4} x^{3 n + 2} \sqrt{a^{2} + 2 a b x^{n} + b^{2} x^{2 n}}}{\left (3 n + 2\right ) \left (a b + b^{2} x^{n}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.106605, size = 124, normalized size = 0.59 \[ \frac{x^2 \sqrt{\left (a+b x^n\right )^2} \left (a^3 \left (3 n^3+11 n^2+12 n+4\right )+6 a^2 b \left (3 n^2+5 n+2\right ) x^n+3 a b^2 \left (3 n^2+8 n+4\right ) x^{2 n}+2 b^3 \left (n^2+3 n+2\right ) x^{3 n}\right )}{2 (n+1) (n+2) (3 n+2) \left (a+b x^n\right )} \]

Antiderivative was successfully verified.

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

[Out]

(x^2*Sqrt[(a + b*x^n)^2]*(a^3*(4 + 12*n + 11*n^2 + 3*n^3) + 6*a^2*b*(2 + 5*n + 3
*n^2)*x^n + 3*a*b^2*(4 + 8*n + 3*n^2)*x^(2*n) + 2*b^3*(2 + 3*n + n^2)*x^(3*n)))/
(2*(1 + n)*(2 + n)*(2 + 3*n)*(a + b*x^n))

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Maple [A]  time = 0.025, size = 145, normalized size = 0.7 \[{\frac{{x}^{2}{a}^{3}}{2\,a+2\,b{x}^{n}}\sqrt{ \left ( a+b{x}^{n} \right ) ^{2}}}+{\frac{{b}^{3}{x}^{2} \left ({x}^{n} \right ) ^{3}}{ \left ( a+b{x}^{n} \right ) \left ( 2+3\,n \right ) }\sqrt{ \left ( a+b{x}^{n} \right ) ^{2}}}+{\frac{3\,a{b}^{2}{x}^{2} \left ({x}^{n} \right ) ^{2}}{ \left ( 2\,a+2\,b{x}^{n} \right ) \left ( 1+n \right ) }\sqrt{ \left ( a+b{x}^{n} \right ) ^{2}}}+3\,{\frac{\sqrt{ \left ( a+b{x}^{n} \right ) ^{2}}{a}^{2}b{x}^{2}{x}^{n}}{ \left ( a+b{x}^{n} \right ) \left ( 2+n \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.754804, size = 147, normalized size = 0.7 \[ \frac{2 \,{\left (n^{2} + 3 \, n + 2\right )} b^{3} x^{2} x^{3 \, n} + 3 \,{\left (3 \, n^{2} + 8 \, n + 4\right )} a b^{2} x^{2} x^{2 \, n} + 6 \,{\left (3 \, n^{2} + 5 \, n + 2\right )} a^{2} b x^{2} x^{n} +{\left (3 \, n^{3} + 11 \, n^{2} + 12 \, n + 4\right )} a^{3} x^{2}}{2 \,{\left (3 \, n^{3} + 11 \, n^{2} + 12 \, n + 4\right )}} \]

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,x, algorithm="maxima")

[Out]

1/2*(2*(n^2 + 3*n + 2)*b^3*x^2*x^(3*n) + 3*(3*n^2 + 8*n + 4)*a*b^2*x^2*x^(2*n) +
6*(3*n^2 + 5*n + 2)*a^2*b*x^2*x^n + (3*n^3 + 11*n^2 + 12*n + 4)*a^3*x^2)/(3*n^3
+ 11*n^2 + 12*n + 4)

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Fricas [A]  time = 0.283687, size = 196, normalized size = 0.93 \[ \frac{2 \,{\left (b^{3} n^{2} + 3 \, b^{3} n + 2 \, b^{3}\right )} x^{2} x^{3 \, n} + 3 \,{\left (3 \, a b^{2} n^{2} + 8 \, a b^{2} n + 4 \, a b^{2}\right )} x^{2} x^{2 \, n} + 6 \,{\left (3 \, a^{2} b n^{2} + 5 \, a^{2} b n + 2 \, a^{2} b\right )} x^{2} x^{n} +{\left (3 \, a^{3} n^{3} + 11 \, a^{3} n^{2} + 12 \, a^{3} n + 4 \, a^{3}\right )} x^{2}}{2 \,{\left (3 \, n^{3} + 11 \, n^{2} + 12 \, n + 4\right )}} \]

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,x, algorithm="fricas")

[Out]

1/2*(2*(b^3*n^2 + 3*b^3*n + 2*b^3)*x^2*x^(3*n) + 3*(3*a*b^2*n^2 + 8*a*b^2*n + 4*
a*b^2)*x^2*x^(2*n) + 6*(3*a^2*b*n^2 + 5*a^2*b*n + 2*a^2*b)*x^2*x^n + (3*a^3*n^3
+ 11*a^3*n^2 + 12*a^3*n + 4*a^3)*x^2)/(3*n^3 + 11*n^2 + 12*n + 4)

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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*(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 = 0.290692, size = 410, normalized size = 1.94 \[ \frac{3 \, a^{3} n^{3} x^{2}{\rm sign}\left (b x^{n} + a\right ) + 2 \, b^{3} n^{2} x^{2} e^{\left (3 \, n{\rm ln}\left (x\right )\right )}{\rm sign}\left (b x^{n} + a\right ) + 9 \, a b^{2} n^{2} x^{2} e^{\left (2 \, n{\rm ln}\left (x\right )\right )}{\rm sign}\left (b x^{n} + a\right ) + 18 \, a^{2} b n^{2} x^{2} e^{\left (n{\rm ln}\left (x\right )\right )}{\rm sign}\left (b x^{n} + a\right ) + 11 \, a^{3} n^{2} x^{2}{\rm sign}\left (b x^{n} + a\right ) + 6 \, b^{3} n x^{2} e^{\left (3 \, n{\rm ln}\left (x\right )\right )}{\rm sign}\left (b x^{n} + a\right ) + 24 \, a b^{2} n x^{2} e^{\left (2 \, n{\rm ln}\left (x\right )\right )}{\rm sign}\left (b x^{n} + a\right ) + 30 \, a^{2} b n x^{2} e^{\left (n{\rm ln}\left (x\right )\right )}{\rm sign}\left (b x^{n} + a\right ) + 12 \, a^{3} n x^{2}{\rm sign}\left (b x^{n} + a\right ) + 4 \, b^{3} x^{2} e^{\left (3 \, n{\rm ln}\left (x\right )\right )}{\rm sign}\left (b x^{n} + a\right ) + 12 \, a b^{2} x^{2} e^{\left (2 \, n{\rm ln}\left (x\right )\right )}{\rm sign}\left (b x^{n} + a\right ) + 12 \, a^{2} b x^{2} e^{\left (n{\rm ln}\left (x\right )\right )}{\rm sign}\left (b x^{n} + a\right ) + 4 \, a^{3} x^{2}{\rm sign}\left (b x^{n} + a\right )}{2 \,{\left (3 \, n^{3} + 11 \, n^{2} + 12 \, n + 4\right )}} \]

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,x, algorithm="giac")

[Out]

1/2*(3*a^3*n^3*x^2*sign(b*x^n + a) + 2*b^3*n^2*x^2*e^(3*n*ln(x))*sign(b*x^n + a)
+ 9*a*b^2*n^2*x^2*e^(2*n*ln(x))*sign(b*x^n + a) + 18*a^2*b*n^2*x^2*e^(n*ln(x))*
sign(b*x^n + a) + 11*a^3*n^2*x^2*sign(b*x^n + a) + 6*b^3*n*x^2*e^(3*n*ln(x))*sig
n(b*x^n + a) + 24*a*b^2*n*x^2*e^(2*n*ln(x))*sign(b*x^n + a) + 30*a^2*b*n*x^2*e^(
n*ln(x))*sign(b*x^n + a) + 12*a^3*n*x^2*sign(b*x^n + a) + 4*b^3*x^2*e^(3*n*ln(x)
)*sign(b*x^n + a) + 12*a*b^2*x^2*e^(2*n*ln(x))*sign(b*x^n + a) + 12*a^2*b*x^2*e^
(n*ln(x))*sign(b*x^n + a) + 4*a^3*x^2*sign(b*x^n + a))/(3*n^3 + 11*n^2 + 12*n +
4)