3.516 \(\int x^2 \sqrt{a^2+2 a b x^n+b^2 x^{2 n}} \, dx\)

Optimal. Leaf size=93 \[ \frac{b^2 x^{n+3} \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{(n+3) \left (a b+b^2 x^n\right )}+\frac{a x^3 \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{3 \left (a+b x^n\right )} \]

[Out]

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

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Rubi [A]  time = 0.0283172, antiderivative size = 93, 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, Rules used = {1355, 14} \[ \frac{b^2 x^{n+3} \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{(n+3) \left (a b+b^2 x^n\right )}+\frac{a x^3 \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{3 \left (a+b x^n\right )} \]

Antiderivative was successfully verified.

[In]

Int[x^2*Sqrt[a^2 + 2*a*b*x^n + b^2*x^(2*n)],x]

[Out]

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

Rule 1355

Int[((d_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.))^(p_), x_Symbol] :> Dist[(a + b*x^n + c*x^
(2*n))^FracPart[p]/(c^IntPart[p]*(b/2 + c*x^n)^(2*FracPart[p])), Int[(d*x)^m*(b/2 + c*x^n)^(2*p), x], x] /; Fr
eeQ[{a, b, c, d, m, n, p}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p - 1/2]

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rubi steps

\begin{align*} \int x^2 \sqrt{a^2+2 a b x^n+b^2 x^{2 n}} \, dx &=\frac{\sqrt{a^2+2 a b x^n+b^2 x^{2 n}} \int x^2 \left (a b+b^2 x^n\right ) \, dx}{a b+b^2 x^n}\\ &=\frac{\sqrt{a^2+2 a b x^n+b^2 x^{2 n}} \int \left (a b x^2+b^2 x^{2+n}\right ) \, dx}{a b+b^2 x^n}\\ &=\frac{a x^3 \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{3 \left (a+b x^n\right )}+\frac{b^2 x^{3+n} \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{(3+n) \left (a b+b^2 x^n\right )}\\ \end{align*}

Mathematica [A]  time = 0.023155, size = 46, normalized size = 0.49 \[ \frac{x^3 \sqrt{\left (a+b x^n\right )^2} \left (a (n+3)+3 b x^n\right )}{3 (n+3) \left (a+b x^n\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2*Sqrt[a^2 + 2*a*b*x^n + b^2*x^(2*n)],x]

[Out]

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

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Maple [A]  time = 0.015, size = 61, normalized size = 0.7 \begin{align*}{\frac{a{x}^{3}}{3\,a+3\,b{x}^{n}}\sqrt{ \left ( a+b{x}^{n} \right ) ^{2}}}+{\frac{b{x}^{3}{x}^{n}}{ \left ( a+b{x}^{n} \right ) \left ( 3+n \right ) }\sqrt{ \left ( a+b{x}^{n} \right ) ^{2}}} \end{align*}

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))^(1/2),x)

[Out]

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

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Maxima [A]  time = 1.0194, size = 34, normalized size = 0.37 \begin{align*} \frac{3 \, b x^{3} x^{n} + a{\left (n + 3\right )} x^{3}}{3 \,{\left (n + 3\right )}} \end{align*}

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))^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.53308, size = 61, normalized size = 0.66 \begin{align*} \frac{3 \, b x^{3} x^{n} +{\left (a n + 3 \, a\right )} x^{3}}{3 \,{\left (n + 3\right )}} \end{align*}

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))^(1/2),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{2} \sqrt{\left (a + b x^{n}\right )^{2}}\, dx \end{align*}

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))**(1/2),x)

[Out]

Integral(x**2*sqrt((a + b*x**n)**2), x)

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Giac [A]  time = 1.12569, size = 72, normalized size = 0.77 \begin{align*} \frac{3 \, b x^{3} x^{n} \mathrm{sgn}\left (b x^{n} + a\right ) + a n x^{3} \mathrm{sgn}\left (b x^{n} + a\right ) + 3 \, a x^{3} \mathrm{sgn}\left (b x^{n} + a\right )}{3 \,{\left (n + 3\right )}} \end{align*}

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))^(1/2),x, algorithm="giac")

[Out]

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