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

Optimal. Leaf size=33 \[ \frac{1}{5} x^5 (a B+A b)+\frac{1}{3} a A x^3+\frac{1}{7} b B x^7 \]

[Out]

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

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Rubi [A]  time = 0.0195923, antiderivative size = 33, normalized size of antiderivative = 1., 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 = {448} \[ \frac{1}{5} x^5 (a B+A b)+\frac{1}{3} a A x^3+\frac{1}{7} b B x^7 \]

Antiderivative was successfully verified.

[In]

Int[x^2*(a + b*x^2)*(A + B*x^2),x]

[Out]

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

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 (A+B x^2\right ) \, dx &=\int \left (a A x^2+(A b+a B) x^4+b B x^6\right ) \, dx\\ &=\frac{1}{3} a A x^3+\frac{1}{5} (A b+a B) x^5+\frac{1}{7} b B x^7\\ \end{align*}

Mathematica [A]  time = 0.0055308, size = 33, normalized size = 1. \[ \frac{1}{5} x^5 (a B+A b)+\frac{1}{3} a A x^3+\frac{1}{7} b B x^7 \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.001, size = 28, normalized size = 0.9 \begin{align*}{\frac{aA{x}^{3}}{3}}+{\frac{ \left ( Ab+Ba \right ){x}^{5}}{5}}+{\frac{bB{x}^{7}}{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*a*A*x^3+1/5*(A*b+B*a)*x^5+1/7*b*B*x^7

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Maxima [A]  time = 0.991062, size = 36, normalized size = 1.09 \begin{align*} \frac{1}{7} \, B b x^{7} + \frac{1}{5} \,{\left (B a + A b\right )} x^{5} + \frac{1}{3} \, A a x^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*B*b*x^7 + 1/5*(B*a + A*b)*x^5 + 1/3*A*a*x^3

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Fricas [A]  time = 1.24705, size = 74, normalized size = 2.24 \begin{align*} \frac{1}{7} x^{7} b B + \frac{1}{5} x^{5} a B + \frac{1}{5} x^{5} b A + \frac{1}{3} x^{3} a A \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*x^7*b*B + 1/5*x^5*a*B + 1/5*x^5*b*A + 1/3*x^3*a*A

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Sympy [A]  time = 0.057648, size = 29, normalized size = 0.88 \begin{align*} \frac{A a x^{3}}{3} + \frac{B b x^{7}}{7} + x^{5} \left (\frac{A b}{5} + \frac{B a}{5}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*(b*x**2+a)*(B*x**2+A),x)

[Out]

A*a*x**3/3 + B*b*x**7/7 + x**5*(A*b/5 + B*a/5)

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Giac [A]  time = 1.12168, size = 39, normalized size = 1.18 \begin{align*} \frac{1}{7} \, B b x^{7} + \frac{1}{5} \, B a x^{5} + \frac{1}{5} \, A b x^{5} + \frac{1}{3} \, A a x^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*B*b*x^7 + 1/5*B*a*x^5 + 1/5*A*b*x^5 + 1/3*A*a*x^3