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

Optimal. Leaf size=94 \[ \frac {1}{5} c^3 x^5 (4 a d+b c)+\frac {2}{9} c^2 d x^9 (3 a d+2 b c)+\frac {1}{17} d^3 x^{17} (a d+4 b c)+\frac {2}{13} c d^2 x^{13} (2 a d+3 b c)+a c^4 x+\frac {1}{21} b d^4 x^{21} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.07, antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {373} \[ \frac {2}{9} c^2 d x^9 (3 a d+2 b c)+\frac {1}{5} c^3 x^5 (4 a d+b c)+\frac {1}{17} d^3 x^{17} (a d+4 b c)+\frac {2}{13} c d^2 x^{13} (2 a d+3 b c)+a c^4 x+\frac {1}{21} b d^4 x^{21} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^4)*(c + d*x^4)^4,x]

[Out]

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

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

________________________________________________________________________________________

Mathematica [A]  time = 0.02, size = 94, normalized size = 1.00 \[ \frac {1}{5} c^3 x^5 (4 a d+b c)+\frac {2}{9} c^2 d x^9 (3 a d+2 b c)+\frac {1}{17} d^3 x^{17} (a d+4 b c)+\frac {2}{13} c d^2 x^{13} (2 a d+3 b c)+a c^4 x+\frac {1}{21} b d^4 x^{21} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 0.90, size = 98, normalized size = 1.04 \[ \frac {1}{21} x^{21} d^{4} b + \frac {4}{17} x^{17} d^{3} c b + \frac {1}{17} x^{17} d^{4} a + \frac {6}{13} x^{13} d^{2} c^{2} b + \frac {4}{13} x^{13} d^{3} c a + \frac {4}{9} x^{9} d c^{3} b + \frac {2}{3} x^{9} d^{2} c^{2} a + \frac {1}{5} x^{5} c^{4} b + \frac {4}{5} x^{5} d c^{3} a + x c^{4} a \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

giac [A]  time = 0.16, size = 98, normalized size = 1.04 \[ \frac {1}{21} \, b d^{4} x^{21} + \frac {4}{17} \, b c d^{3} x^{17} + \frac {1}{17} \, a d^{4} x^{17} + \frac {6}{13} \, b c^{2} d^{2} x^{13} + \frac {4}{13} \, a c d^{3} x^{13} + \frac {4}{9} \, b c^{3} d x^{9} + \frac {2}{3} \, a c^{2} d^{2} x^{9} + \frac {1}{5} \, b c^{4} x^{5} + \frac {4}{5} \, a c^{3} d x^{5} + a c^{4} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [A]  time = 0.04, size = 97, normalized size = 1.03 \[ \frac {b \,d^{4} x^{21}}{21}+\frac {\left (a \,d^{4}+4 b c \,d^{3}\right ) x^{17}}{17}+\frac {\left (4 a c \,d^{3}+6 c^{2} d^{2} b \right ) x^{13}}{13}+\frac {\left (6 a \,c^{2} d^{2}+4 c^{3} d b \right ) x^{9}}{9}+a \,c^{4} x +\frac {\left (4 a \,c^{3} d +b \,c^{4}\right ) x^{5}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 0.65, size = 96, normalized size = 1.02 \[ \frac {1}{21} \, b d^{4} x^{21} + \frac {1}{17} \, {\left (4 \, b c d^{3} + a d^{4}\right )} x^{17} + \frac {2}{13} \, {\left (3 \, b c^{2} d^{2} + 2 \, a c d^{3}\right )} x^{13} + \frac {2}{9} \, {\left (2 \, b c^{3} d + 3 \, a c^{2} d^{2}\right )} x^{9} + a c^{4} x + \frac {1}{5} \, {\left (b c^{4} + 4 \, a c^{3} d\right )} x^{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

mupad [B]  time = 1.30, size = 88, normalized size = 0.94 \[ x^5\,\left (\frac {b\,c^4}{5}+\frac {4\,a\,d\,c^3}{5}\right )+x^{17}\,\left (\frac {a\,d^4}{17}+\frac {4\,b\,c\,d^3}{17}\right )+\frac {b\,d^4\,x^{21}}{21}+a\,c^4\,x+\frac {2\,c^2\,d\,x^9\,\left (3\,a\,d+2\,b\,c\right )}{9}+\frac {2\,c\,d^2\,x^{13}\,\left (2\,a\,d+3\,b\,c\right )}{13} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^4)*(c + d*x^4)^4,x)

[Out]

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

________________________________________________________________________________________

sympy [A]  time = 0.09, size = 107, normalized size = 1.14 \[ a c^{4} x + \frac {b d^{4} x^{21}}{21} + x^{17} \left (\frac {a d^{4}}{17} + \frac {4 b c d^{3}}{17}\right ) + x^{13} \left (\frac {4 a c d^{3}}{13} + \frac {6 b c^{2} d^{2}}{13}\right ) + x^{9} \left (\frac {2 a c^{2} d^{2}}{3} + \frac {4 b c^{3} d}{9}\right ) + x^{5} \left (\frac {4 a c^{3} d}{5} + \frac {b c^{4}}{5}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________