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

Optimal. Leaf size=54 \[ \frac {a^2 x^3}{3}+\frac {1}{7} x^7 \left (2 a c+b^2\right )+\frac {2}{5} a b x^5+\frac {2}{9} b c x^9+\frac {c^2 x^{11}}{11} \]

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Rubi [A]  time = 0.03, antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {1108} \begin {gather*} \frac {a^2 x^3}{3}+\frac {1}{7} x^7 \left (2 a c+b^2\right )+\frac {2}{5} a b x^5+\frac {2}{9} b c x^9+\frac {c^2 x^{11}}{11} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^2*(a + b*x^2 + c*x^4)^2,x]

[Out]

(a^2*x^3)/3 + (2*a*b*x^5)/5 + ((b^2 + 2*a*c)*x^7)/7 + (2*b*c*x^9)/9 + (c^2*x^11)/11

Rule 1108

Int[((d_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d*x)^m*(a
+ b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[p, 0] &&  !IntegerQ[(m + 1)/2]

Rubi steps

\begin {align*} \int x^2 \left (a+b x^2+c x^4\right )^2 \, dx &=\int \left (a^2 x^2+2 a b x^4+\left (b^2+2 a c\right ) x^6+2 b c x^8+c^2 x^{10}\right ) \, dx\\ &=\frac {a^2 x^3}{3}+\frac {2}{5} a b x^5+\frac {1}{7} \left (b^2+2 a c\right ) x^7+\frac {2}{9} b c x^9+\frac {c^2 x^{11}}{11}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 54, normalized size = 1.00 \begin {gather*} \frac {a^2 x^3}{3}+\frac {1}{7} x^7 \left (2 a c+b^2\right )+\frac {2}{5} a b x^5+\frac {2}{9} b c x^9+\frac {c^2 x^{11}}{11} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(a^2*x^3)/3 + (2*a*b*x^5)/5 + ((b^2 + 2*a*c)*x^7)/7 + (2*b*c*x^9)/9 + (c^2*x^11)/11

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^2 \left (a+b x^2+c x^4\right )^2 \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 1.03, size = 46, normalized size = 0.85 \begin {gather*} \frac {1}{11} x^{11} c^{2} + \frac {2}{9} x^{9} c b + \frac {1}{7} x^{7} b^{2} + \frac {2}{7} x^{7} c a + \frac {2}{5} x^{5} b a + \frac {1}{3} x^{3} a^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/11*x^11*c^2 + 2/9*x^9*c*b + 1/7*x^7*b^2 + 2/7*x^7*c*a + 2/5*x^5*b*a + 1/3*x^3*a^2

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giac [A]  time = 0.15, size = 46, normalized size = 0.85 \begin {gather*} \frac {1}{11} \, c^{2} x^{11} + \frac {2}{9} \, b c x^{9} + \frac {1}{7} \, b^{2} x^{7} + \frac {2}{7} \, a c x^{7} + \frac {2}{5} \, a b x^{5} + \frac {1}{3} \, a^{2} x^{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/11*c^2*x^11 + 2/9*b*c*x^9 + 1/7*b^2*x^7 + 2/7*a*c*x^7 + 2/5*a*b*x^5 + 1/3*a^2*x^3

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maple [A]  time = 0.00, size = 45, normalized size = 0.83 \begin {gather*} \frac {c^{2} x^{11}}{11}+\frac {2 b c \,x^{9}}{9}+\frac {2 a b \,x^{5}}{5}+\frac {\left (2 a c +b^{2}\right ) x^{7}}{7}+\frac {a^{2} x^{3}}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*a^2*x^3+2/5*a*b*x^5+1/7*(2*a*c+b^2)*x^7+2/9*b*c*x^9+1/11*c^2*x^11

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maxima [A]  time = 1.31, size = 44, normalized size = 0.81 \begin {gather*} \frac {1}{11} \, c^{2} x^{11} + \frac {2}{9} \, b c x^{9} + \frac {1}{7} \, {\left (b^{2} + 2 \, a c\right )} x^{7} + \frac {2}{5} \, a b x^{5} + \frac {1}{3} \, a^{2} x^{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/11*c^2*x^11 + 2/9*b*c*x^9 + 1/7*(b^2 + 2*a*c)*x^7 + 2/5*a*b*x^5 + 1/3*a^2*x^3

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mupad [B]  time = 0.03, size = 45, normalized size = 0.83 \begin {gather*} x^7\,\left (\frac {b^2}{7}+\frac {2\,a\,c}{7}\right )+\frac {a^2\,x^3}{3}+\frac {c^2\,x^{11}}{11}+\frac {2\,a\,b\,x^5}{5}+\frac {2\,b\,c\,x^9}{9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(a + b*x^2 + c*x^4)^2,x)

[Out]

x^7*((2*a*c)/7 + b^2/7) + (a^2*x^3)/3 + (c^2*x^11)/11 + (2*a*b*x^5)/5 + (2*b*c*x^9)/9

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sympy [A]  time = 0.08, size = 51, normalized size = 0.94 \begin {gather*} \frac {a^{2} x^{3}}{3} + \frac {2 a b x^{5}}{5} + \frac {2 b c x^{9}}{9} + \frac {c^{2} x^{11}}{11} + x^{7} \left (\frac {2 a c}{7} + \frac {b^{2}}{7}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**2*x**3/3 + 2*a*b*x**5/5 + 2*b*c*x**9/9 + c**2*x**11/11 + x**7*(2*a*c/7 + b**2/7)

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