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

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Rubi [A]  time = 0.05, antiderivative size = 70, 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} \[ \frac {1}{5} c^2 x^5 (3 a d+b c)+\frac {1}{13} d^2 x^{13} (a d+3 b c)+\frac {1}{3} c d x^9 (a d+b c)+a c^3 x+\frac {1}{17} b d^3 x^{17} \]

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*}

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Mathematica [A]  time = 0.02, size = 70, normalized size = 1.00 \[ \frac {1}{5} c^2 x^5 (3 a d+b c)+\frac {1}{13} d^2 x^{13} (a d+3 b c)+\frac {1}{3} c d x^9 (a d+b c)+a c^3 x+\frac {1}{17} b d^3 x^{17} \]

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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fricas [A]  time = 0.96, size = 74, normalized size = 1.06 \[ \frac {1}{17} x^{17} d^{3} b + \frac {3}{13} x^{13} d^{2} c b + \frac {1}{13} x^{13} d^{3} a + \frac {1}{3} x^{9} d c^{2} b + \frac {1}{3} x^{9} d^{2} c a + \frac {1}{5} x^{5} c^{3} b + \frac {3}{5} x^{5} d c^{2} a + x c^{3} a \]

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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giac [A]  time = 0.15, size = 74, normalized size = 1.06 \[ \frac {1}{17} \, b d^{3} x^{17} + \frac {3}{13} \, b c d^{2} x^{13} + \frac {1}{13} \, a d^{3} x^{13} + \frac {1}{3} \, b c^{2} d x^{9} + \frac {1}{3} \, a c d^{2} x^{9} + \frac {1}{5} \, b c^{3} x^{5} + \frac {3}{5} \, a c^{2} d x^{5} + a c^{3} x \]

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

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maple [A]  time = 0.04, size = 73, normalized size = 1.04 \[ \frac {b \,d^{3} x^{17}}{17}+\frac {\left (a \,d^{3}+3 b c \,d^{2}\right ) x^{13}}{13}+\frac {\left (3 a c \,d^{2}+3 b \,c^{2} d \right ) x^{9}}{9}+a \,c^{3} x +\frac {\left (3 a \,c^{2} d +b \,c^{3}\right ) x^{5}}{5} \]

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 = 0.65, size = 70, normalized size = 1.00 \[ \frac {1}{17} \, b d^{3} x^{17} + \frac {1}{13} \, {\left (3 \, b c d^{2} + a d^{3}\right )} x^{13} + \frac {1}{3} \, {\left (b c^{2} d + a c d^{2}\right )} x^{9} + \frac {1}{5} \, {\left (b c^{3} + 3 \, a c^{2} d\right )} x^{5} + a c^{3} x \]

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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mupad [B]  time = 1.24, size = 66, normalized size = 0.94 \[ x^5\,\left (\frac {b\,c^3}{5}+\frac {3\,a\,d\,c^2}{5}\right )+x^{13}\,\left (\frac {a\,d^3}{13}+\frac {3\,b\,c\,d^2}{13}\right )+\frac {b\,d^3\,x^{17}}{17}+a\,c^3\,x+\frac {c\,d\,x^9\,\left (a\,d+b\,c\right )}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^4)*(c + d*x^4)^3,x)

[Out]

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

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sympy [A]  time = 0.08, size = 76, normalized size = 1.09 \[ a c^{3} x + \frac {b d^{3} x^{17}}{17} + x^{13} \left (\frac {a d^{3}}{13} + \frac {3 b c d^{2}}{13}\right ) + x^{9} \left (\frac {a c d^{2}}{3} + \frac {b c^{2} d}{3}\right ) + x^{5} \left (\frac {3 a c^{2} d}{5} + \frac {b c^{3}}{5}\right ) \]

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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