3.155 \(\int (a+b x^4)^2 (c+d x^4)^2 \, dx\)

Optimal. Leaf size=82 \[ \frac{1}{9} x^9 \left (a^2 d^2+4 a b c d+b^2 c^2\right )+a^2 c^2 x+\frac{2}{13} b d x^{13} (a d+b c)+\frac{2}{5} a c x^5 (a d+b c)+\frac{1}{17} b^2 d^2 x^{17} \]

[Out]

a^2*c^2*x + (2*a*c*(b*c + a*d)*x^5)/5 + ((b^2*c^2 + 4*a*b*c*d + a^2*d^2)*x^9)/9 + (2*b*d*(b*c + a*d)*x^13)/13
+ (b^2*d^2*x^17)/17

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Rubi [A]  time = 0.0491069, 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} \[ \frac{1}{9} x^9 \left (a^2 d^2+4 a b c d+b^2 c^2\right )+a^2 c^2 x+\frac{2}{13} b d x^{13} (a d+b c)+\frac{2}{5} a c x^5 (a d+b c)+\frac{1}{17} b^2 d^2 x^{17} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^4)^2*(c + d*x^4)^2,x]

[Out]

a^2*c^2*x + (2*a*c*(b*c + a*d)*x^5)/5 + ((b^2*c^2 + 4*a*b*c*d + a^2*d^2)*x^9)/9 + (2*b*d*(b*c + a*d)*x^13)/13
+ (b^2*d^2*x^17)/17

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^4\right )^2 \left (c+d x^4\right )^2 \, dx &=\int \left (a^2 c^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^8+2 b d (b c+a d) x^{12}+b^2 d^2 x^{16}\right ) \, dx\\ &=a^2 c^2 x+\frac{2}{5} a c (b c+a d) x^5+\frac{1}{9} \left (b^2 c^2+4 a b c d+a^2 d^2\right ) x^9+\frac{2}{13} b d (b c+a d) x^{13}+\frac{1}{17} b^2 d^2 x^{17}\\ \end{align*}

Mathematica [A]  time = 0.0166878, size = 82, normalized size = 1. \[ \frac{1}{9} x^9 \left (a^2 d^2+4 a b c d+b^2 c^2\right )+a^2 c^2 x+\frac{2}{13} b d x^{13} (a d+b c)+\frac{2}{5} a c x^5 (a d+b c)+\frac{1}{17} b^2 d^2 x^{17} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^4)^2*(c + d*x^4)^2,x]

[Out]

a^2*c^2*x + (2*a*c*(b*c + a*d)*x^5)/5 + ((b^2*c^2 + 4*a*b*c*d + a^2*d^2)*x^9)/9 + (2*b*d*(b*c + a*d)*x^13)/13
+ (b^2*d^2*x^17)/17

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Maple [A]  time = 0.001, size = 87, normalized size = 1.1 \begin{align*}{\frac{{b}^{2}{d}^{2}{x}^{17}}{17}}+{\frac{ \left ( 2\,ab{d}^{2}+2\,{b}^{2}cd \right ){x}^{13}}{13}}+{\frac{ \left ({a}^{2}{d}^{2}+4\,cabd+{b}^{2}{c}^{2} \right ){x}^{9}}{9}}+{\frac{ \left ( 2\,{a}^{2}cd+2\,ab{c}^{2} \right ){x}^{5}}{5}}+{a}^{2}{c}^{2}x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^4+a)^2*(d*x^4+c)^2,x)

[Out]

1/17*b^2*d^2*x^17+1/13*(2*a*b*d^2+2*b^2*c*d)*x^13+1/9*(a^2*d^2+4*a*b*c*d+b^2*c^2)*x^9+1/5*(2*a^2*c*d+2*a*b*c^2
)*x^5+a^2*c^2*x

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Maxima [A]  time = 0.94481, size = 111, normalized size = 1.35 \begin{align*} \frac{1}{17} \, b^{2} d^{2} x^{17} + \frac{2}{13} \,{\left (b^{2} c d + a b d^{2}\right )} x^{13} + \frac{1}{9} \,{\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} x^{9} + \frac{2}{5} \,{\left (a b c^{2} + a^{2} c d\right )} x^{5} + a^{2} c^{2} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^4+a)^2*(d*x^4+c)^2,x, algorithm="maxima")

[Out]

1/17*b^2*d^2*x^17 + 2/13*(b^2*c*d + a*b*d^2)*x^13 + 1/9*(b^2*c^2 + 4*a*b*c*d + a^2*d^2)*x^9 + 2/5*(a*b*c^2 + a
^2*c*d)*x^5 + a^2*c^2*x

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Fricas [A]  time = 1.08849, size = 217, normalized size = 2.65 \begin{align*} \frac{1}{17} x^{17} d^{2} b^{2} + \frac{2}{13} x^{13} d c b^{2} + \frac{2}{13} x^{13} d^{2} b a + \frac{1}{9} x^{9} c^{2} b^{2} + \frac{4}{9} x^{9} d c b a + \frac{1}{9} x^{9} d^{2} a^{2} + \frac{2}{5} x^{5} c^{2} b a + \frac{2}{5} x^{5} 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^4+a)^2*(d*x^4+c)^2,x, algorithm="fricas")

[Out]

1/17*x^17*d^2*b^2 + 2/13*x^13*d*c*b^2 + 2/13*x^13*d^2*b*a + 1/9*x^9*c^2*b^2 + 4/9*x^9*d*c*b*a + 1/9*x^9*d^2*a^
2 + 2/5*x^5*c^2*b*a + 2/5*x^5*d*c*a^2 + x*c^2*a^2

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Sympy [A]  time = 0.076386, size = 97, normalized size = 1.18 \begin{align*} a^{2} c^{2} x + \frac{b^{2} d^{2} x^{17}}{17} + x^{13} \left (\frac{2 a b d^{2}}{13} + \frac{2 b^{2} c d}{13}\right ) + x^{9} \left (\frac{a^{2} d^{2}}{9} + \frac{4 a b c d}{9} + \frac{b^{2} c^{2}}{9}\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((b*x**4+a)**2*(d*x**4+c)**2,x)

[Out]

a**2*c**2*x + b**2*d**2*x**17/17 + x**13*(2*a*b*d**2/13 + 2*b**2*c*d/13) + x**9*(a**2*d**2/9 + 4*a*b*c*d/9 + b
**2*c**2/9) + x**5*(2*a**2*c*d/5 + 2*a*b*c**2/5)

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Giac [A]  time = 1.09573, size = 123, normalized size = 1.5 \begin{align*} \frac{1}{17} \, b^{2} d^{2} x^{17} + \frac{2}{13} \, b^{2} c d x^{13} + \frac{2}{13} \, a b d^{2} x^{13} + \frac{1}{9} \, b^{2} c^{2} x^{9} + \frac{4}{9} \, a b c d x^{9} + \frac{1}{9} \, a^{2} d^{2} x^{9} + \frac{2}{5} \, a b c^{2} x^{5} + \frac{2}{5} \, a^{2} c d x^{5} + a^{2} c^{2} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^4+a)^2*(d*x^4+c)^2,x, algorithm="giac")

[Out]

1/17*b^2*d^2*x^17 + 2/13*b^2*c*d*x^13 + 2/13*a*b*d^2*x^13 + 1/9*b^2*c^2*x^9 + 4/9*a*b*c*d*x^9 + 1/9*a^2*d^2*x^
9 + 2/5*a*b*c^2*x^5 + 2/5*a^2*c*d*x^5 + a^2*c^2*x