3.1 \(\int x^2 (A+B x^2) (b x^2+c x^4) \, dx\)

Optimal. Leaf size=33 \[ \frac{1}{7} x^7 (A c+b B)+\frac{1}{5} A b x^5+\frac{1}{9} B c x^9 \]

[Out]

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

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Rubi [A]  time = 0.027026, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {1584, 448} \[ \frac{1}{7} x^7 (A c+b B)+\frac{1}{5} A b x^5+\frac{1}{9} B c x^9 \]

Antiderivative was successfully verified.

[In]

Int[x^2*(A + B*x^2)*(b*x^2 + c*x^4),x]

[Out]

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

Rule 1584

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

Rule 448

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Int[ExpandI
ntegrand[(e*x)^m*(a + b*x^n)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[b*c - a*d, 0] &
& IGtQ[p, 0] && IGtQ[q, 0]

Rubi steps

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

Mathematica [A]  time = 0.0061206, size = 33, normalized size = 1. \[ \frac{1}{7} x^7 (A c+b B)+\frac{1}{5} A b x^5+\frac{1}{9} B c x^9 \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.043, size = 28, normalized size = 0.9 \begin{align*}{\frac{Ab{x}^{5}}{5}}+{\frac{ \left ( Ac+Bb \right ){x}^{7}}{7}}+{\frac{Bc{x}^{9}}{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*A*b*x^5+1/7*(A*c+B*b)*x^7+1/9*B*c*x^9

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Maxima [A]  time = 1.12523, size = 36, normalized size = 1.09 \begin{align*} \frac{1}{9} \, B c x^{9} + \frac{1}{7} \,{\left (B b + A c\right )} x^{7} + \frac{1}{5} \, A b x^{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*B*c*x^9 + 1/7*(B*b + A*c)*x^7 + 1/5*A*b*x^5

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Fricas [A]  time = 0.396868, size = 74, normalized size = 2.24 \begin{align*} \frac{1}{9} x^{9} c B + \frac{1}{7} x^{7} b B + \frac{1}{7} x^{7} c A + \frac{1}{5} x^{5} b A \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*x^9*c*B + 1/7*x^7*b*B + 1/7*x^7*c*A + 1/5*x^5*b*A

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Sympy [A]  time = 0.060546, size = 29, normalized size = 0.88 \begin{align*} \frac{A b x^{5}}{5} + \frac{B c x^{9}}{9} + x^{7} \left (\frac{A c}{7} + \frac{B b}{7}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

A*b*x**5/5 + B*c*x**9/9 + x**7*(A*c/7 + B*b/7)

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Giac [A]  time = 1.27726, size = 39, normalized size = 1.18 \begin{align*} \frac{1}{9} \, B c x^{9} + \frac{1}{7} \, B b x^{7} + \frac{1}{7} \, A c x^{7} + \frac{1}{5} \, A b x^{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*B*c*x^9 + 1/7*B*b*x^7 + 1/7*A*c*x^7 + 1/5*A*b*x^5