3.147 \(\int (a+b x^4) (c+d x^4)^3 \, dx\)

Optimal. Leaf size=70 \[ \frac{1}{5} c^2 x^5 (3 a d+b c)+\frac{1}{13} d^2 x^{13} (a d+3 b c)+\frac{1}{3} c d x^9 (a d+b c)+a c^3 x+\frac{1}{17} b d^3 x^{17} \]

[Out]

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

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Rubi [A]  time = 0.0487288, antiderivative size = 70, 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} \[ \frac{1}{5} c^2 x^5 (3 a d+b c)+\frac{1}{13} d^2 x^{13} (a d+3 b c)+\frac{1}{3} c d x^9 (a d+b c)+a c^3 x+\frac{1}{17} b d^3 x^{17} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^4)*(c + d*x^4)^3,x]

[Out]

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

Mathematica [A]  time = 0.0153898, size = 70, normalized size = 1. \[ \frac{1}{5} c^2 x^5 (3 a d+b c)+\frac{1}{13} d^2 x^{13} (a d+3 b c)+\frac{1}{3} c d x^9 (a d+b c)+a c^3 x+\frac{1}{17} b d^3 x^{17} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^4)*(c + d*x^4)^3,x]

[Out]

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

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Maple [A]  time = 0., size = 73, normalized size = 1. \begin{align*}{\frac{b{d}^{3}{x}^{17}}{17}}+{\frac{ \left ( a{d}^{3}+3\,bc{d}^{2} \right ){x}^{13}}{13}}+{\frac{ \left ( 3\,ac{d}^{2}+3\,b{c}^{2}d \right ){x}^{9}}{9}}+{\frac{ \left ( 3\,a{c}^{2}d+b{c}^{3} \right ){x}^{5}}{5}}+a{c}^{3}x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^4+a)*(d*x^4+c)^3,x)

[Out]

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

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Maxima [A]  time = 0.940268, size = 95, normalized size = 1.36 \begin{align*} \frac{1}{17} \, b d^{3} x^{17} + \frac{1}{13} \,{\left (3 \, b c d^{2} + a d^{3}\right )} x^{13} + \frac{1}{3} \,{\left (b c^{2} d + a c d^{2}\right )} x^{9} + \frac{1}{5} \,{\left (b c^{3} + 3 \, a c^{2} d\right )} x^{5} + a c^{3} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^4+a)*(d*x^4+c)^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.02544, size = 182, normalized size = 2.6 \begin{align*} \frac{1}{17} x^{17} d^{3} b + \frac{3}{13} x^{13} d^{2} c b + \frac{1}{13} x^{13} d^{3} a + \frac{1}{3} x^{9} d c^{2} b + \frac{1}{3} x^{9} d^{2} c a + \frac{1}{5} x^{5} c^{3} b + \frac{3}{5} x^{5} d c^{2} a + x c^{3} a \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^4+a)*(d*x^4+c)^3,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.073041, size = 76, normalized size = 1.09 \begin{align*} a c^{3} x + \frac{b d^{3} x^{17}}{17} + x^{13} \left (\frac{a d^{3}}{13} + \frac{3 b c d^{2}}{13}\right ) + x^{9} \left (\frac{a c d^{2}}{3} + \frac{b c^{2} d}{3}\right ) + x^{5} \left (\frac{3 a c^{2} d}{5} + \frac{b c^{3}}{5}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**4+a)*(d*x**4+c)**3,x)

[Out]

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

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Giac [A]  time = 1.10574, size = 100, normalized size = 1.43 \begin{align*} \frac{1}{17} \, b d^{3} x^{17} + \frac{3}{13} \, b c d^{2} x^{13} + \frac{1}{13} \, a d^{3} x^{13} + \frac{1}{3} \, b c^{2} d x^{9} + \frac{1}{3} \, a c d^{2} x^{9} + \frac{1}{5} \, b c^{3} x^{5} + \frac{3}{5} \, a c^{2} d x^{5} + a c^{3} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^4+a)*(d*x^4+c)^3,x, algorithm="giac")

[Out]

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