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

Optimal. Leaf size=82 \[ \frac{1}{5} x^5 \left (a^2 d^2+4 a b c d+b^2 c^2\right )+a^2 c^2 x+\frac{2}{7} b d x^7 (a d+b c)+\frac{2}{3} a c x^3 (a d+b c)+\frac{1}{9} b^2 d^2 x^9 \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0387515, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053, Rules used = {373} \[ \frac{1}{5} x^5 \left (a^2 d^2+4 a b c d+b^2 c^2\right )+a^2 c^2 x+\frac{2}{7} b d x^7 (a d+b c)+\frac{2}{3} a c x^3 (a d+b c)+\frac{1}{9} b^2 d^2 x^9 \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^2*(c + d*x^2)^2,x]

[Out]

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

Rule 373

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

Rubi steps

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

Mathematica [A]  time = 0.0158499, size = 82, normalized size = 1. \[ \frac{1}{5} x^5 \left (a^2 d^2+4 a b c d+b^2 c^2\right )+a^2 c^2 x+\frac{2}{7} b d x^7 (a d+b c)+\frac{2}{3} a c x^3 (a d+b c)+\frac{1}{9} b^2 d^2 x^9 \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0., size = 87, normalized size = 1.1 \begin{align*}{\frac{{b}^{2}{d}^{2}{x}^{9}}{9}}+{\frac{ \left ( 2\,ab{d}^{2}+2\,{b}^{2}cd \right ){x}^{7}}{7}}+{\frac{ \left ({a}^{2}{d}^{2}+4\,cabd+{b}^{2}{c}^{2} \right ){x}^{5}}{5}}+{\frac{ \left ( 2\,{a}^{2}cd+2\,ab{c}^{2} \right ){x}^{3}}{3}}+{a}^{2}{c}^{2}x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.21545, size = 111, normalized size = 1.35 \begin{align*} \frac{1}{9} \, b^{2} d^{2} x^{9} + \frac{2}{7} \,{\left (b^{2} c d + a b d^{2}\right )} x^{7} + \frac{1}{5} \,{\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} x^{5} + a^{2} c^{2} x + \frac{2}{3} \,{\left (a b c^{2} + a^{2} c d\right )} x^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.14746, size = 209, normalized size = 2.55 \begin{align*} \frac{1}{9} x^{9} d^{2} b^{2} + \frac{2}{7} x^{7} d c b^{2} + \frac{2}{7} x^{7} d^{2} b a + \frac{1}{5} x^{5} c^{2} b^{2} + \frac{4}{5} x^{5} d c b a + \frac{1}{5} x^{5} d^{2} a^{2} + \frac{2}{3} x^{3} c^{2} b a + \frac{2}{3} x^{3} d c a^{2} + x c^{2} a^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [A]  time = 0.075106, size = 97, normalized size = 1.18 \begin{align*} a^{2} c^{2} x + \frac{b^{2} d^{2} x^{9}}{9} + x^{7} \left (\frac{2 a b d^{2}}{7} + \frac{2 b^{2} c d}{7}\right ) + x^{5} \left (\frac{a^{2} d^{2}}{5} + \frac{4 a b c d}{5} + \frac{b^{2} c^{2}}{5}\right ) + x^{3} \left (\frac{2 a^{2} c d}{3} + \frac{2 a b c^{2}}{3}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 1.1322, size = 123, normalized size = 1.5 \begin{align*} \frac{1}{9} \, b^{2} d^{2} x^{9} + \frac{2}{7} \, b^{2} c d x^{7} + \frac{2}{7} \, a b d^{2} x^{7} + \frac{1}{5} \, b^{2} c^{2} x^{5} + \frac{4}{5} \, a b c d x^{5} + \frac{1}{5} \, a^{2} d^{2} x^{5} + \frac{2}{3} \, a b c^{2} x^{3} + \frac{2}{3} \, a^{2} c d x^{3} + a^{2} c^{2} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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