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

Optimal. Leaf size=154 \[ \frac {1}{17} d^2 x^{17} \left (a^2 d^2+8 a b c d+6 b^2 c^2\right )+\frac {4}{13} c d x^{13} \left (a^2 d^2+3 a b c d+b^2 c^2\right )+\frac {1}{9} c^2 x^9 \left (6 a^2 d^2+8 a b c d+b^2 c^2\right )+a^2 c^4 x+\frac {2}{5} a c^3 x^5 (2 a d+b c)+\frac {2}{21} b d^3 x^{21} (a d+2 b c)+\frac {1}{25} b^2 d^4 x^{25} \]

[Out]

a^2*c^4*x+2/5*a*c^3*(2*a*d+b*c)*x^5+1/9*c^2*(6*a^2*d^2+8*a*b*c*d+b^2*c^2)*x^9+4/13*c*d*(a^2*d^2+3*a*b*c*d+b^2*
c^2)*x^13+1/17*d^2*(a^2*d^2+8*a*b*c*d+6*b^2*c^2)*x^17+2/21*b*d^3*(a*d+2*b*c)*x^21+1/25*b^2*d^4*x^25

________________________________________________________________________________________

Rubi [A]  time = 0.11, antiderivative size = 154, 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} \[ \frac {1}{17} d^2 x^{17} \left (a^2 d^2+8 a b c d+6 b^2 c^2\right )+\frac {4}{13} c d x^{13} \left (a^2 d^2+3 a b c d+b^2 c^2\right )+\frac {1}{9} c^2 x^9 \left (6 a^2 d^2+8 a b c d+b^2 c^2\right )+a^2 c^4 x+\frac {2}{5} a c^3 x^5 (2 a d+b c)+\frac {2}{21} b d^3 x^{21} (a d+2 b c)+\frac {1}{25} b^2 d^4 x^{25} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^4)^2*(c + d*x^4)^4,x]

[Out]

a^2*c^4*x + (2*a*c^3*(b*c + 2*a*d)*x^5)/5 + (c^2*(b^2*c^2 + 8*a*b*c*d + 6*a^2*d^2)*x^9)/9 + (4*c*d*(b^2*c^2 +
3*a*b*c*d + a^2*d^2)*x^13)/13 + (d^2*(6*b^2*c^2 + 8*a*b*c*d + a^2*d^2)*x^17)/17 + (2*b*d^3*(2*b*c + a*d)*x^21)
/21 + (b^2*d^4*x^25)/25

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 )^4 \, dx &=\int \left (a^2 c^4+2 a c^3 (b c+2 a d) x^4+c^2 \left (b^2 c^2+8 a b c d+6 a^2 d^2\right ) x^8+4 c d \left (b^2 c^2+3 a b c d+a^2 d^2\right ) x^{12}+d^2 \left (6 b^2 c^2+8 a b c d+a^2 d^2\right ) x^{16}+2 b d^3 (2 b c+a d) x^{20}+b^2 d^4 x^{24}\right ) \, dx\\ &=a^2 c^4 x+\frac {2}{5} a c^3 (b c+2 a d) x^5+\frac {1}{9} c^2 \left (b^2 c^2+8 a b c d+6 a^2 d^2\right ) x^9+\frac {4}{13} c d \left (b^2 c^2+3 a b c d+a^2 d^2\right ) x^{13}+\frac {1}{17} d^2 \left (6 b^2 c^2+8 a b c d+a^2 d^2\right ) x^{17}+\frac {2}{21} b d^3 (2 b c+a d) x^{21}+\frac {1}{25} b^2 d^4 x^{25}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.03, size = 154, normalized size = 1.00 \[ \frac {1}{17} d^2 x^{17} \left (a^2 d^2+8 a b c d+6 b^2 c^2\right )+\frac {4}{13} c d x^{13} \left (a^2 d^2+3 a b c d+b^2 c^2\right )+\frac {1}{9} c^2 x^9 \left (6 a^2 d^2+8 a b c d+b^2 c^2\right )+a^2 c^4 x+\frac {2}{5} a c^3 x^5 (2 a d+b c)+\frac {2}{21} b d^3 x^{21} (a d+2 b c)+\frac {1}{25} b^2 d^4 x^{25} \]

Antiderivative was successfully verified.

[In]

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

[Out]

a^2*c^4*x + (2*a*c^3*(b*c + 2*a*d)*x^5)/5 + (c^2*(b^2*c^2 + 8*a*b*c*d + 6*a^2*d^2)*x^9)/9 + (4*c*d*(b^2*c^2 +
3*a*b*c*d + a^2*d^2)*x^13)/13 + (d^2*(6*b^2*c^2 + 8*a*b*c*d + a^2*d^2)*x^17)/17 + (2*b*d^3*(2*b*c + a*d)*x^21)
/21 + (b^2*d^4*x^25)/25

________________________________________________________________________________________

fricas [A]  time = 0.61, size = 173, normalized size = 1.12 \[ \frac {1}{25} x^{25} d^{4} b^{2} + \frac {4}{21} x^{21} d^{3} c b^{2} + \frac {2}{21} x^{21} d^{4} b a + \frac {6}{17} x^{17} d^{2} c^{2} b^{2} + \frac {8}{17} x^{17} d^{3} c b a + \frac {1}{17} x^{17} d^{4} a^{2} + \frac {4}{13} x^{13} d c^{3} b^{2} + \frac {12}{13} x^{13} d^{2} c^{2} b a + \frac {4}{13} x^{13} d^{3} c a^{2} + \frac {1}{9} x^{9} c^{4} b^{2} + \frac {8}{9} x^{9} d c^{3} b a + \frac {2}{3} x^{9} d^{2} c^{2} a^{2} + \frac {2}{5} x^{5} c^{4} b a + \frac {4}{5} x^{5} d c^{3} a^{2} + x c^{4} a^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/25*x^25*d^4*b^2 + 4/21*x^21*d^3*c*b^2 + 2/21*x^21*d^4*b*a + 6/17*x^17*d^2*c^2*b^2 + 8/17*x^17*d^3*c*b*a + 1/
17*x^17*d^4*a^2 + 4/13*x^13*d*c^3*b^2 + 12/13*x^13*d^2*c^2*b*a + 4/13*x^13*d^3*c*a^2 + 1/9*x^9*c^4*b^2 + 8/9*x
^9*d*c^3*b*a + 2/3*x^9*d^2*c^2*a^2 + 2/5*x^5*c^4*b*a + 4/5*x^5*d*c^3*a^2 + x*c^4*a^2

________________________________________________________________________________________

giac [A]  time = 0.15, size = 173, normalized size = 1.12 \[ \frac {1}{25} \, b^{2} d^{4} x^{25} + \frac {4}{21} \, b^{2} c d^{3} x^{21} + \frac {2}{21} \, a b d^{4} x^{21} + \frac {6}{17} \, b^{2} c^{2} d^{2} x^{17} + \frac {8}{17} \, a b c d^{3} x^{17} + \frac {1}{17} \, a^{2} d^{4} x^{17} + \frac {4}{13} \, b^{2} c^{3} d x^{13} + \frac {12}{13} \, a b c^{2} d^{2} x^{13} + \frac {4}{13} \, a^{2} c d^{3} x^{13} + \frac {1}{9} \, b^{2} c^{4} x^{9} + \frac {8}{9} \, a b c^{3} d x^{9} + \frac {2}{3} \, a^{2} c^{2} d^{2} x^{9} + \frac {2}{5} \, a b c^{4} x^{5} + \frac {4}{5} \, a^{2} c^{3} d x^{5} + a^{2} c^{4} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/25*b^2*d^4*x^25 + 4/21*b^2*c*d^3*x^21 + 2/21*a*b*d^4*x^21 + 6/17*b^2*c^2*d^2*x^17 + 8/17*a*b*c*d^3*x^17 + 1/
17*a^2*d^4*x^17 + 4/13*b^2*c^3*d*x^13 + 12/13*a*b*c^2*d^2*x^13 + 4/13*a^2*c*d^3*x^13 + 1/9*b^2*c^4*x^9 + 8/9*a
*b*c^3*d*x^9 + 2/3*a^2*c^2*d^2*x^9 + 2/5*a*b*c^4*x^5 + 4/5*a^2*c^3*d*x^5 + a^2*c^4*x

________________________________________________________________________________________

maple [A]  time = 0.04, size = 163, normalized size = 1.06 \[ \frac {b^{2} d^{4} x^{25}}{25}+\frac {\left (2 a b \,d^{4}+4 b^{2} c \,d^{3}\right ) x^{21}}{21}+\frac {\left (a^{2} d^{4}+8 a b c \,d^{3}+6 b^{2} c^{2} d^{2}\right ) x^{17}}{17}+\frac {\left (4 a^{2} c \,d^{3}+12 a b \,c^{2} d^{2}+4 b^{2} c^{3} d \right ) x^{13}}{13}+\frac {\left (6 a^{2} c^{2} d^{2}+8 a b \,c^{3} d +b^{2} c^{4}\right ) x^{9}}{9}+a^{2} c^{4} x +\frac {\left (4 a^{2} c^{3} d +2 a b \,c^{4}\right ) x^{5}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/25*b^2*d^4*x^25+1/21*(2*a*b*d^4+4*b^2*c*d^3)*x^21+1/17*(a^2*d^4+8*a*b*c*d^3+6*b^2*c^2*d^2)*x^17+1/13*(4*a^2*
c*d^3+12*a*b*c^2*d^2+4*b^2*c^3*d)*x^13+1/9*(6*a^2*c^2*d^2+8*a*b*c^3*d+b^2*c^4)*x^9+1/5*(4*a^2*c^3*d+2*a*b*c^4)
*x^5+a^2*c^4*x

________________________________________________________________________________________

maxima [A]  time = 0.55, size = 158, normalized size = 1.03 \[ \frac {1}{25} \, b^{2} d^{4} x^{25} + \frac {2}{21} \, {\left (2 \, b^{2} c d^{3} + a b d^{4}\right )} x^{21} + \frac {1}{17} \, {\left (6 \, b^{2} c^{2} d^{2} + 8 \, a b c d^{3} + a^{2} d^{4}\right )} x^{17} + \frac {4}{13} \, {\left (b^{2} c^{3} d + 3 \, a b c^{2} d^{2} + a^{2} c d^{3}\right )} x^{13} + \frac {1}{9} \, {\left (b^{2} c^{4} + 8 \, a b c^{3} d + 6 \, a^{2} c^{2} d^{2}\right )} x^{9} + a^{2} c^{4} x + \frac {2}{5} \, {\left (a b c^{4} + 2 \, a^{2} c^{3} d\right )} x^{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/25*b^2*d^4*x^25 + 2/21*(2*b^2*c*d^3 + a*b*d^4)*x^21 + 1/17*(6*b^2*c^2*d^2 + 8*a*b*c*d^3 + a^2*d^4)*x^17 + 4/
13*(b^2*c^3*d + 3*a*b*c^2*d^2 + a^2*c*d^3)*x^13 + 1/9*(b^2*c^4 + 8*a*b*c^3*d + 6*a^2*c^2*d^2)*x^9 + a^2*c^4*x
+ 2/5*(a*b*c^4 + 2*a^2*c^3*d)*x^5

________________________________________________________________________________________

mupad [B]  time = 0.07, size = 146, normalized size = 0.95 \[ x^9\,\left (\frac {2\,a^2\,c^2\,d^2}{3}+\frac {8\,a\,b\,c^3\,d}{9}+\frac {b^2\,c^4}{9}\right )+x^{17}\,\left (\frac {a^2\,d^4}{17}+\frac {8\,a\,b\,c\,d^3}{17}+\frac {6\,b^2\,c^2\,d^2}{17}\right )+a^2\,c^4\,x+\frac {b^2\,d^4\,x^{25}}{25}+\frac {2\,a\,c^3\,x^5\,\left (2\,a\,d+b\,c\right )}{5}+\frac {2\,b\,d^3\,x^{21}\,\left (a\,d+2\,b\,c\right )}{21}+\frac {4\,c\,d\,x^{13}\,\left (a^2\,d^2+3\,a\,b\,c\,d+b^2\,c^2\right )}{13} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^4)^2*(c + d*x^4)^4,x)

[Out]

x^9*((b^2*c^4)/9 + (2*a^2*c^2*d^2)/3 + (8*a*b*c^3*d)/9) + x^17*((a^2*d^4)/17 + (6*b^2*c^2*d^2)/17 + (8*a*b*c*d
^3)/17) + a^2*c^4*x + (b^2*d^4*x^25)/25 + (2*a*c^3*x^5*(2*a*d + b*c))/5 + (2*b*d^3*x^21*(a*d + 2*b*c))/21 + (4
*c*d*x^13*(a^2*d^2 + b^2*c^2 + 3*a*b*c*d))/13

________________________________________________________________________________________

sympy [A]  time = 0.12, size = 185, normalized size = 1.20 \[ a^{2} c^{4} x + \frac {b^{2} d^{4} x^{25}}{25} + x^{21} \left (\frac {2 a b d^{4}}{21} + \frac {4 b^{2} c d^{3}}{21}\right ) + x^{17} \left (\frac {a^{2} d^{4}}{17} + \frac {8 a b c d^{3}}{17} + \frac {6 b^{2} c^{2} d^{2}}{17}\right ) + x^{13} \left (\frac {4 a^{2} c d^{3}}{13} + \frac {12 a b c^{2} d^{2}}{13} + \frac {4 b^{2} c^{3} d}{13}\right ) + x^{9} \left (\frac {2 a^{2} c^{2} d^{2}}{3} + \frac {8 a b c^{3} d}{9} + \frac {b^{2} c^{4}}{9}\right ) + x^{5} \left (\frac {4 a^{2} c^{3} d}{5} + \frac {2 a b c^{4}}{5}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**2*c**4*x + b**2*d**4*x**25/25 + x**21*(2*a*b*d**4/21 + 4*b**2*c*d**3/21) + x**17*(a**2*d**4/17 + 8*a*b*c*d*
*3/17 + 6*b**2*c**2*d**2/17) + x**13*(4*a**2*c*d**3/13 + 12*a*b*c**2*d**2/13 + 4*b**2*c**3*d/13) + x**9*(2*a**
2*c**2*d**2/3 + 8*a*b*c**3*d/9 + b**2*c**4/9) + x**5*(4*a**2*c**3*d/5 + 2*a*b*c**4/5)

________________________________________________________________________________________