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

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

[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

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Rubi [A]  time = 0.0676504, antiderivative size = 94, normalized size of antiderivative = 1., 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 = 0.0207885, size = 94, normalized size = 1. \[ \frac{2}{9} c^2 d x^9 (3 a d+2 b c)+\frac{1}{5} c^3 x^5 (4 a d+b c)+\frac{1}{17} d^3 x^{17} (a d+4 b c)+\frac{2}{13} c d^2 x^{13} (2 a d+3 b c)+a c^4 x+\frac{1}{21} b d^4 x^{21} \]

Antiderivative was successfully verified.

[In]

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

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Maple [A]  time = 0.001, size = 97, normalized size = 1. \begin{align*}{\frac{b{d}^{4}{x}^{21}}{21}}+{\frac{ \left ( a{d}^{4}+4\,bc{d}^{3} \right ){x}^{17}}{17}}+{\frac{ \left ( 4\,ac{d}^{3}+6\,{c}^{2}{d}^{2}b \right ){x}^{13}}{13}}+{\frac{ \left ( 6\,a{c}^{2}{d}^{2}+4\,{c}^{3}db \right ){x}^{9}}{9}}+{\frac{ \left ( 4\,a{c}^{3}d+b{c}^{4} \right ){x}^{5}}{5}}+a{c}^{4}x \end{align*}

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

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Maxima [A]  time = 0.942229, size = 130, normalized size = 1.38 \begin{align*} \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} \end{align*}

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

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Fricas [A]  time = 1.09905, size = 242, normalized size = 2.57 \begin{align*} \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 \end{align*}

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

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Sympy [A]  time = 0.079749, size = 107, normalized size = 1.14 \begin{align*} 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 ) \end{align*}

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)

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Giac [A]  time = 1.10633, size = 132, normalized size = 1.4 \begin{align*} \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 \end{align*}

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