3.143 \(\int x^2 (a+b x^2)^2 (c+d x^2) \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0296691, 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} \[ \frac{1}{3} a^2 c x^3+\frac{1}{7} b x^7 (2 a d+b c)+\frac{1}{5} a x^5 (a d+2 b c)+\frac{1}{9} b^2 d x^9 \]

Antiderivative was successfully verified.

[In]

Int[x^2*(a + b*x^2)^2*(c + d*x^2),x]

[Out]

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

Mathematica [A]  time = 0.0074007, size = 55, normalized size = 1. \[ \frac{1}{3} a^2 c x^3+\frac{1}{7} b x^7 (2 a d+b c)+\frac{1}{5} a x^5 (a d+2 b c)+\frac{1}{9} b^2 d x^9 \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.00241, size = 69, normalized size = 1.25 \begin{align*} \frac{1}{9} \, b^{2} d x^{9} + \frac{1}{7} \,{\left (b^{2} c + 2 \, a b d\right )} x^{7} + \frac{1}{3} \, a^{2} c x^{3} + \frac{1}{5} \,{\left (2 \, a b c + a^{2} d\right )} x^{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.1323, size = 128, normalized size = 2.33 \begin{align*} \frac{1}{9} x^{9} d b^{2} + \frac{1}{7} x^{7} c b^{2} + \frac{2}{7} x^{7} d b a + \frac{2}{5} x^{5} c b a + \frac{1}{5} x^{5} d a^{2} + \frac{1}{3} x^{3} c a^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.067026, size = 56, normalized size = 1.02 \begin{align*} \frac{a^{2} c x^{3}}{3} + \frac{b^{2} d x^{9}}{9} + x^{7} \left (\frac{2 a b d}{7} + \frac{b^{2} c}{7}\right ) + x^{5} \left (\frac{a^{2} d}{5} + \frac{2 a b c}{5}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.18801, size = 72, normalized size = 1.31 \begin{align*} \frac{1}{9} \, b^{2} d x^{9} + \frac{1}{7} \, b^{2} c x^{7} + \frac{2}{7} \, a b d x^{7} + \frac{2}{5} \, a b c x^{5} + \frac{1}{5} \, a^{2} d x^{5} + \frac{1}{3} \, a^{2} c x^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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