3.410 \(\int x^2 (a^2+2 a b x^2+b^2 x^4) \, dx\)

Optimal. Leaf size=30 \[ \frac {a^2 x^3}{3}+\frac {2}{5} a b x^5+\frac {b^2 x^7}{7} \]

[Out]

1/3*a^2*x^3+2/5*a*b*x^5+1/7*b^2*x^7

________________________________________________________________________________________

Rubi [A]  time = 0.01, antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {14} \[ \frac {a^2 x^3}{3}+\frac {2}{5} a b x^5+\frac {b^2 x^7}{7} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^2*x^3)/3 + (2*a*b*x^5)/5 + (b^2*x^7)/7

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 \left (a^2+2 a b x^2+b^2 x^4\right ) \, dx &=\int \left (a^2 x^2+2 a b x^4+b^2 x^6\right ) \, dx\\ &=\frac {a^2 x^3}{3}+\frac {2}{5} a b x^5+\frac {b^2 x^7}{7}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.00, size = 30, normalized size = 1.00 \[ \frac {a^2 x^3}{3}+\frac {2}{5} a b x^5+\frac {b^2 x^7}{7} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^2*x^3)/3 + (2*a*b*x^5)/5 + (b^2*x^7)/7

________________________________________________________________________________________

fricas [A]  time = 0.74, size = 24, normalized size = 0.80 \[ \frac {1}{7} x^{7} b^{2} + \frac {2}{5} x^{5} b a + \frac {1}{3} x^{3} a^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(b^2*x^4+2*a*b*x^2+a^2),x, algorithm="fricas")

[Out]

1/7*x^7*b^2 + 2/5*x^5*b*a + 1/3*x^3*a^2

________________________________________________________________________________________

giac [A]  time = 0.15, size = 24, normalized size = 0.80 \[ \frac {1}{7} \, b^{2} x^{7} + \frac {2}{5} \, a b x^{5} + \frac {1}{3} \, a^{2} x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(b^2*x^4+2*a*b*x^2+a^2),x, algorithm="giac")

[Out]

1/7*b^2*x^7 + 2/5*a*b*x^5 + 1/3*a^2*x^3

________________________________________________________________________________________

maple [A]  time = 0.00, size = 25, normalized size = 0.83 \[ \frac {1}{7} b^{2} x^{7}+\frac {2}{5} a b \,x^{5}+\frac {1}{3} a^{2} x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(b^2*x^4+2*a*b*x^2+a^2),x)

[Out]

1/3*a^2*x^3+2/5*a*b*x^5+1/7*b^2*x^7

________________________________________________________________________________________

maxima [A]  time = 1.33, size = 24, normalized size = 0.80 \[ \frac {1}{7} \, b^{2} x^{7} + \frac {2}{5} \, a b x^{5} + \frac {1}{3} \, a^{2} x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(b^2*x^4+2*a*b*x^2+a^2),x, algorithm="maxima")

[Out]

1/7*b^2*x^7 + 2/5*a*b*x^5 + 1/3*a^2*x^3

________________________________________________________________________________________

mupad [B]  time = 0.03, size = 24, normalized size = 0.80 \[ \frac {a^2\,x^3}{3}+\frac {2\,a\,b\,x^5}{5}+\frac {b^2\,x^7}{7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(a^2 + b^2*x^4 + 2*a*b*x^2),x)

[Out]

(a^2*x^3)/3 + (b^2*x^7)/7 + (2*a*b*x^5)/5

________________________________________________________________________________________

sympy [A]  time = 0.07, size = 26, normalized size = 0.87 \[ \frac {a^{2} x^{3}}{3} + \frac {2 a b x^{5}}{5} + \frac {b^{2} x^{7}}{7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*(b**2*x**4+2*a*b*x**2+a**2),x)

[Out]

a**2*x**3/3 + 2*a*b*x**5/5 + b**2*x**7/7

________________________________________________________________________________________