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*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

________________________________________________________________________________________

Rubi [A]  time = 0.113943, antiderivative size = 154, 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} \[ \frac{1}{17} d^2 x^{17} \left (a^2 d^2+8 a b c d+6 b^2 c^2\right )+\frac{4}{13} c d x^{13} \left (a^2 d^2+3 a b c d+b^2 c^2\right )+\frac{1}{9} c^2 x^9 \left (6 a^2 d^2+8 a b c d+b^2 c^2\right )+a^2 c^4 x+\frac{2}{5} a c^3 x^5 (2 a d+b c)+\frac{2}{21} b d^3 x^{21} (a d+2 b c)+\frac{1}{25} b^2 d^4 x^{25} \]

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 = 0.0314645, size = 154, normalized size = 1. \[ \frac{1}{17} d^2 x^{17} \left (a^2 d^2+8 a b c d+6 b^2 c^2\right )+\frac{4}{13} c d x^{13} \left (a^2 d^2+3 a b c d+b^2 c^2\right )+\frac{1}{9} c^2 x^9 \left (6 a^2 d^2+8 a b c d+b^2 c^2\right )+a^2 c^4 x+\frac{2}{5} a c^3 x^5 (2 a d+b c)+\frac{2}{21} b d^3 x^{21} (a d+2 b c)+\frac{1}{25} b^2 d^4 x^{25} \]

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

________________________________________________________________________________________

Maple [A]  time = 0., size = 163, normalized size = 1.1 \begin{align*}{\frac{{b}^{2}{d}^{4}{x}^{25}}{25}}+{\frac{ \left ( 2\,ab{d}^{4}+4\,{b}^{2}c{d}^{3} \right ){x}^{21}}{21}}+{\frac{ \left ({a}^{2}{d}^{4}+8\,abc{d}^{3}+6\,{b}^{2}{c}^{2}{d}^{2} \right ){x}^{17}}{17}}+{\frac{ \left ( 4\,{a}^{2}c{d}^{3}+12\,ab{c}^{2}{d}^{2}+4\,{b}^{2}{c}^{3}d \right ){x}^{13}}{13}}+{\frac{ \left ( 6\,{a}^{2}{c}^{2}{d}^{2}+8\,ab{c}^{3}d+{b}^{2}{c}^{4} \right ){x}^{9}}{9}}+{\frac{ \left ( 4\,{a}^{2}{c}^{3}d+2\,ab{c}^{4} \right ){x}^{5}}{5}}+{a}^{2}{c}^{4}x \end{align*}

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 = 0.949895, size = 213, normalized size = 1.38 \begin{align*} \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} \end{align*}

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

________________________________________________________________________________________

Fricas [A]  time = 1.11083, size = 413, normalized size = 2.68 \begin{align*} \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} \end{align*}

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

________________________________________________________________________________________

Sympy [A]  time = 0.090721, size = 185, normalized size = 1.2 \begin{align*} 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 ) \end{align*}

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)

________________________________________________________________________________________

Giac [A]  time = 1.11187, size = 234, normalized size = 1.52 \begin{align*} \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 \end{align*}

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