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

________________________________________________________________________________________

Rubi [A]  time = 0.05, antiderivative size = 82, normalized size of antiderivative = 1.00, 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 = 0.02, size = 82, normalized size = 1.00 \[ \frac {1}{9} x^9 \left (a^2 d^2+4 a b c d+b^2 c^2\right )+a^2 c^2 x+\frac {2}{13} b d x^{13} (a d+b c)+\frac {2}{5} a c x^5 (a d+b c)+\frac {1}{17} b^2 d^2 x^{17} \]

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

________________________________________________________________________________________

fricas [A]  time = 0.99, size = 91, normalized size = 1.11 \[ \frac {1}{17} x^{17} d^{2} b^{2} + \frac {2}{13} x^{13} d c b^{2} + \frac {2}{13} x^{13} d^{2} b a + \frac {1}{9} x^{9} c^{2} b^{2} + \frac {4}{9} x^{9} d c b a + \frac {1}{9} x^{9} d^{2} a^{2} + \frac {2}{5} x^{5} c^{2} b a + \frac {2}{5} x^{5} d c a^{2} + x c^{2} a^{2} \]

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

________________________________________________________________________________________

giac [A]  time = 0.17, size = 91, normalized size = 1.11 \[ \frac {1}{17} \, b^{2} d^{2} x^{17} + \frac {2}{13} \, b^{2} c d x^{13} + \frac {2}{13} \, a b d^{2} x^{13} + \frac {1}{9} \, b^{2} c^{2} x^{9} + \frac {4}{9} \, a b c d x^{9} + \frac {1}{9} \, a^{2} d^{2} x^{9} + \frac {2}{5} \, a b c^{2} x^{5} + \frac {2}{5} \, a^{2} c d x^{5} + a^{2} c^{2} x \]

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

________________________________________________________________________________________

maple [A]  time = 0.04, size = 87, normalized size = 1.06 \[ \frac {b^{2} d^{2} x^{17}}{17}+\frac {\left (2 a b \,d^{2}+2 b^{2} c d \right ) x^{13}}{13}+\frac {\left (a^{2} d^{2}+4 a b c d +b^{2} c^{2}\right ) x^{9}}{9}+a^{2} c^{2} x +\frac {\left (2 a^{2} c d +2 a b \,c^{2}\right ) x^{5}}{5} \]

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

________________________________________________________________________________________

maxima [A]  time = 0.69, size = 82, normalized size = 1.00 \[ \frac {1}{17} \, b^{2} d^{2} x^{17} + \frac {2}{13} \, {\left (b^{2} c d + a b d^{2}\right )} x^{13} + \frac {1}{9} \, {\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} x^{9} + \frac {2}{5} \, {\left (a b c^{2} + a^{2} c d\right )} x^{5} + a^{2} c^{2} x \]

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

________________________________________________________________________________________

mupad [B]  time = 0.05, size = 75, normalized size = 0.91 \[ x^9\,\left (\frac {a^2\,d^2}{9}+\frac {4\,a\,b\,c\,d}{9}+\frac {b^2\,c^2}{9}\right )+a^2\,c^2\,x+\frac {b^2\,d^2\,x^{17}}{17}+\frac {2\,a\,c\,x^5\,\left (a\,d+b\,c\right )}{5}+\frac {2\,b\,d\,x^{13}\,\left (a\,d+b\,c\right )}{13} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^4)^2*(c + d*x^4)^2,x)

[Out]

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

________________________________________________________________________________________

sympy [A]  time = 0.09, size = 97, normalized size = 1.18 \[ a^{2} c^{2} x + \frac {b^{2} d^{2} x^{17}}{17} + x^{13} \left (\frac {2 a b d^{2}}{13} + \frac {2 b^{2} c d}{13}\right ) + x^{9} \left (\frac {a^{2} d^{2}}{9} + \frac {4 a b c d}{9} + \frac {b^{2} c^{2}}{9}\right ) + x^{5} \left (\frac {2 a^{2} c d}{5} + \frac {2 a b c^{2}}{5}\right ) \]

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)

________________________________________________________________________________________