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

Optimal. Leaf size=55 \[ \frac{1}{3} a^2 A x^3+\frac{1}{7} b x^7 (2 a B+A b)+\frac{1}{5} a x^5 (a B+2 A b)+\frac{1}{9} b^2 B x^9 \]

[Out]

(a^2*A*x^3)/3 + (a*(2*A*b + a*B)*x^5)/5 + (b*(A*b + 2*a*B)*x^7)/7 + (b^2*B*x^9)/9

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Rubi [A]  time = 0.0361749, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {448} \[ \frac{1}{3} a^2 A x^3+\frac{1}{7} b x^7 (2 a B+A b)+\frac{1}{5} a x^5 (a B+2 A b)+\frac{1}{9} b^2 B x^9 \]

Antiderivative was successfully verified.

[In]

Int[x^2*(a + b*x^2)^2*(A + B*x^2),x]

[Out]

(a^2*A*x^3)/3 + (a*(2*A*b + a*B)*x^5)/5 + (b*(A*b + 2*a*B)*x^7)/7 + (b^2*B*x^9)/9

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

Mathematica [A]  time = 0.0077699, size = 55, normalized size = 1. \[ \frac{1}{3} a^2 A x^3+\frac{1}{7} b x^7 (2 a B+A b)+\frac{1}{5} a x^5 (a B+2 A b)+\frac{1}{9} b^2 B x^9 \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^2*A*x^3)/3 + (a*(2*A*b + a*B)*x^5)/5 + (b*(A*b + 2*a*B)*x^7)/7 + (b^2*B*x^9)/9

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Maple [A]  time = 0.001, size = 52, normalized size = 1. \begin{align*}{\frac{{b}^{2}B{x}^{9}}{9}}+{\frac{ \left ({b}^{2}A+2\,abB \right ){x}^{7}}{7}}+{\frac{ \left ( 2\,abA+{a}^{2}B \right ){x}^{5}}{5}}+{\frac{{a}^{2}A{x}^{3}}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*b^2*B*x^9+1/7*(A*b^2+2*B*a*b)*x^7+1/5*(2*A*a*b+B*a^2)*x^5+1/3*a^2*A*x^3

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Maxima [A]  time = 1.02242, size = 69, normalized size = 1.25 \begin{align*} \frac{1}{9} \, B b^{2} x^{9} + \frac{1}{7} \,{\left (2 \, B a b + A b^{2}\right )} x^{7} + \frac{1}{3} \, A a^{2} x^{3} + \frac{1}{5} \,{\left (B a^{2} + 2 \, A a b\right )} x^{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*B*b^2*x^9 + 1/7*(2*B*a*b + A*b^2)*x^7 + 1/3*A*a^2*x^3 + 1/5*(B*a^2 + 2*A*a*b)*x^5

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Fricas [A]  time = 1.25053, size = 128, normalized size = 2.33 \begin{align*} \frac{1}{9} x^{9} b^{2} B + \frac{2}{7} x^{7} b a B + \frac{1}{7} x^{7} b^{2} A + \frac{1}{5} x^{5} a^{2} B + \frac{2}{5} x^{5} b a A + \frac{1}{3} x^{3} a^{2} A \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*x^9*b^2*B + 2/7*x^7*b*a*B + 1/7*x^7*b^2*A + 1/5*x^5*a^2*B + 2/5*x^5*b*a*A + 1/3*x^3*a^2*A

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Sympy [A]  time = 0.065141, size = 56, normalized size = 1.02 \begin{align*} \frac{A a^{2} x^{3}}{3} + \frac{B b^{2} x^{9}}{9} + x^{7} \left (\frac{A b^{2}}{7} + \frac{2 B a b}{7}\right ) + x^{5} \left (\frac{2 A a b}{5} + \frac{B a^{2}}{5}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

A*a**2*x**3/3 + B*b**2*x**9/9 + x**7*(A*b**2/7 + 2*B*a*b/7) + x**5*(2*A*a*b/5 + B*a**2/5)

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Giac [A]  time = 1.11797, size = 72, normalized size = 1.31 \begin{align*} \frac{1}{9} \, B b^{2} x^{9} + \frac{2}{7} \, B a b x^{7} + \frac{1}{7} \, A b^{2} x^{7} + \frac{1}{5} \, B a^{2} x^{5} + \frac{2}{5} \, A a b x^{5} + \frac{1}{3} \, A a^{2} x^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*B*b^2*x^9 + 2/7*B*a*b*x^7 + 1/7*A*b^2*x^7 + 1/5*B*a^2*x^5 + 2/5*A*a*b*x^5 + 1/3*A*a^2*x^3