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

Optimal. Leaf size=30 \[ \frac {b^2 x^5}{5}+\frac {2}{7} b c x^7+\frac {c^2 x^9}{9} \]

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Rubi [A]  time = 0.01, antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {1593, 270} \begin {gather*} \frac {b^2 x^5}{5}+\frac {2}{7} b c x^7+\frac {c^2 x^9}{9} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(b*x^2 + c*x^4)^2,x]

[Out]

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

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rule 1593

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rubi steps

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

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Mathematica [A]  time = 0.00, size = 30, normalized size = 1.00 \begin {gather*} \frac {b^2 x^5}{5}+\frac {2}{7} b c x^7+\frac {c^2 x^9}{9} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(b*x^2 + c*x^4)^2,x]

[Out]

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

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

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.62, size = 24, normalized size = 0.80 \begin {gather*} \frac {1}{9} x^{9} c^{2} + \frac {2}{7} x^{7} c b + \frac {1}{5} x^{5} b^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.16, size = 24, normalized size = 0.80 \begin {gather*} \frac {1}{9} \, c^{2} x^{9} + \frac {2}{7} \, b c x^{7} + \frac {1}{5} \, b^{2} x^{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.00, size = 25, normalized size = 0.83 \begin {gather*} \frac {1}{9} c^{2} x^{9}+\frac {2}{7} b c \,x^{7}+\frac {1}{5} b^{2} x^{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.38, size = 24, normalized size = 0.80 \begin {gather*} \frac {1}{9} \, c^{2} x^{9} + \frac {2}{7} \, b c x^{7} + \frac {1}{5} \, b^{2} x^{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.04, size = 24, normalized size = 0.80 \begin {gather*} \frac {b^2\,x^5}{5}+\frac {2\,b\,c\,x^7}{7}+\frac {c^2\,x^9}{9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2 + c*x^4)^2,x)

[Out]

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

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sympy [A]  time = 0.07, size = 26, normalized size = 0.87 \begin {gather*} \frac {b^{2} x^{5}}{5} + \frac {2 b c x^{7}}{7} + \frac {c^{2} x^{9}}{9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b**2*x**5/5 + 2*b*c*x**7/7 + c**2*x**9/9

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