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

Optimal. Leaf size=87 \[ \frac{1}{7} x^7 \left (a^2 d^2+4 a b c d+b^2 c^2\right )+\frac{1}{3} a^2 c^2 x^3+\frac{2}{9} b d x^9 (a d+b c)+\frac{2}{5} a c x^5 (a d+b c)+\frac{1}{11} b^2 d^2 x^{11} \]

[Out]

(a^2*c^2*x^3)/3 + (2*a*c*(b*c + a*d)*x^5)/5 + ((b^2*c^2 + 4*a*b*c*d + a^2*d^2)*x^7)/7 + (2*b*d*(b*c + a*d)*x^9
)/9 + (b^2*d^2*x^11)/11

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Rubi [A]  time = 0.052869, antiderivative size = 87, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used = {448} \[ \frac{1}{7} x^7 \left (a^2 d^2+4 a b c d+b^2 c^2\right )+\frac{1}{3} a^2 c^2 x^3+\frac{2}{9} b d x^9 (a d+b c)+\frac{2}{5} a c x^5 (a d+b c)+\frac{1}{11} b^2 d^2 x^{11} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^2*c^2*x^3)/3 + (2*a*c*(b*c + a*d)*x^5)/5 + ((b^2*c^2 + 4*a*b*c*d + a^2*d^2)*x^7)/7 + (2*b*d*(b*c + a*d)*x^9
)/9 + (b^2*d^2*x^11)/11

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 )^2 \, dx &=\int \left (a^2 c^2 x^2+2 a c (b c+a d) x^4+\left (b^2 c^2+4 a b c d+a^2 d^2\right ) x^6+2 b d (b c+a d) x^8+b^2 d^2 x^{10}\right ) \, dx\\ &=\frac{1}{3} a^2 c^2 x^3+\frac{2}{5} a c (b c+a d) x^5+\frac{1}{7} \left (b^2 c^2+4 a b c d+a^2 d^2\right ) x^7+\frac{2}{9} b d (b c+a d) x^9+\frac{1}{11} b^2 d^2 x^{11}\\ \end{align*}

Mathematica [A]  time = 0.0176405, size = 87, normalized size = 1. \[ \frac{1}{7} x^7 \left (a^2 d^2+4 a b c d+b^2 c^2\right )+\frac{1}{3} a^2 c^2 x^3+\frac{2}{9} b d x^9 (a d+b c)+\frac{2}{5} a c x^5 (a d+b c)+\frac{1}{11} b^2 d^2 x^{11} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^2*c^2*x^3)/3 + (2*a*c*(b*c + a*d)*x^5)/5 + ((b^2*c^2 + 4*a*b*c*d + a^2*d^2)*x^7)/7 + (2*b*d*(b*c + a*d)*x^9
)/9 + (b^2*d^2*x^11)/11

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Maple [A]  time = 0., size = 90, normalized size = 1. \begin{align*}{\frac{{b}^{2}{d}^{2}{x}^{11}}{11}}+{\frac{ \left ( 2\,ab{d}^{2}+2\,{b}^{2}cd \right ){x}^{9}}{9}}+{\frac{ \left ({a}^{2}{d}^{2}+4\,cabd+{b}^{2}{c}^{2} \right ){x}^{7}}{7}}+{\frac{ \left ( 2\,{a}^{2}cd+2\,ab{c}^{2} \right ){x}^{5}}{5}}+{\frac{{a}^{2}{c}^{2}{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)^2,x)

[Out]

1/11*b^2*d^2*x^11+1/9*(2*a*b*d^2+2*b^2*c*d)*x^9+1/7*(a^2*d^2+4*a*b*c*d+b^2*c^2)*x^7+1/5*(2*a^2*c*d+2*a*b*c^2)*
x^5+1/3*a^2*c^2*x^3

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Maxima [A]  time = 1.00862, size = 115, normalized size = 1.32 \begin{align*} \frac{1}{11} \, b^{2} d^{2} x^{11} + \frac{2}{9} \,{\left (b^{2} c d + a b d^{2}\right )} x^{9} + \frac{1}{7} \,{\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} x^{7} + \frac{1}{3} \, a^{2} c^{2} x^{3} + \frac{2}{5} \,{\left (a b c^{2} + a^{2} c 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)^2,x, algorithm="maxima")

[Out]

1/11*b^2*d^2*x^11 + 2/9*(b^2*c*d + a*b*d^2)*x^9 + 1/7*(b^2*c^2 + 4*a*b*c*d + a^2*d^2)*x^7 + 1/3*a^2*c^2*x^3 +
2/5*(a*b*c^2 + a^2*c*d)*x^5

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Fricas [A]  time = 1.0268, size = 220, normalized size = 2.53 \begin{align*} \frac{1}{11} x^{11} d^{2} b^{2} + \frac{2}{9} x^{9} d c b^{2} + \frac{2}{9} x^{9} d^{2} b a + \frac{1}{7} x^{7} c^{2} b^{2} + \frac{4}{7} x^{7} d c b a + \frac{1}{7} x^{7} d^{2} a^{2} + \frac{2}{5} x^{5} c^{2} b a + \frac{2}{5} x^{5} d c a^{2} + \frac{1}{3} x^{3} c^{2} 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)^2,x, algorithm="fricas")

[Out]

1/11*x^11*d^2*b^2 + 2/9*x^9*d*c*b^2 + 2/9*x^9*d^2*b*a + 1/7*x^7*c^2*b^2 + 4/7*x^7*d*c*b*a + 1/7*x^7*d^2*a^2 +
2/5*x^5*c^2*b*a + 2/5*x^5*d*c*a^2 + 1/3*x^3*c^2*a^2

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Sympy [A]  time = 0.076705, size = 100, normalized size = 1.15 \begin{align*} \frac{a^{2} c^{2} x^{3}}{3} + \frac{b^{2} d^{2} x^{11}}{11} + x^{9} \left (\frac{2 a b d^{2}}{9} + \frac{2 b^{2} c d}{9}\right ) + x^{7} \left (\frac{a^{2} d^{2}}{7} + \frac{4 a b c d}{7} + \frac{b^{2} c^{2}}{7}\right ) + x^{5} \left (\frac{2 a^{2} c d}{5} + \frac{2 a b c^{2}}{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)**2,x)

[Out]

a**2*c**2*x**3/3 + b**2*d**2*x**11/11 + x**9*(2*a*b*d**2/9 + 2*b**2*c*d/9) + x**7*(a**2*d**2/7 + 4*a*b*c*d/7 +
b**2*c**2/7) + x**5*(2*a**2*c*d/5 + 2*a*b*c**2/5)

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Giac [A]  time = 1.14725, size = 127, normalized size = 1.46 \begin{align*} \frac{1}{11} \, b^{2} d^{2} x^{11} + \frac{2}{9} \, b^{2} c d x^{9} + \frac{2}{9} \, a b d^{2} x^{9} + \frac{1}{7} \, b^{2} c^{2} x^{7} + \frac{4}{7} \, a b c d x^{7} + \frac{1}{7} \, a^{2} d^{2} x^{7} + \frac{2}{5} \, a b c^{2} x^{5} + \frac{2}{5} \, a^{2} c d x^{5} + \frac{1}{3} \, a^{2} c^{2} 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)^2,x, algorithm="giac")

[Out]

1/11*b^2*d^2*x^11 + 2/9*b^2*c*d*x^9 + 2/9*a*b*d^2*x^9 + 1/7*b^2*c^2*x^7 + 4/7*a*b*c*d*x^7 + 1/7*a^2*d^2*x^7 +
2/5*a*b*c^2*x^5 + 2/5*a^2*c*d*x^5 + 1/3*a^2*c^2*x^3