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

Optimal. Leaf size=28 \[ \frac {1}{5} x^5 (a d+b c)+a c x+\frac {1}{9} b d x^9 \]

[Out]

a*c*x+1/5*(a*d+b*c)*x^5+1/9*b*d*x^9

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Rubi [A]  time = 0.01, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {373} \[ \frac {1}{5} x^5 (a d+b c)+a c x+\frac {1}{9} b d x^9 \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^4)*(c + d*x^4),x]

[Out]

a*c*x + ((b*c + a*d)*x^5)/5 + (b*d*x^9)/9

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

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Mathematica [A]  time = 0.01, size = 28, normalized size = 1.00 \[ \frac {1}{5} x^5 (a d+b c)+a c x+\frac {1}{9} b d x^9 \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^4)*(c + d*x^4),x]

[Out]

a*c*x + ((b*c + a*d)*x^5)/5 + (b*d*x^9)/9

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fricas [A]  time = 0.84, size = 26, normalized size = 0.93 \[ \frac {1}{9} x^{9} d b + \frac {1}{5} x^{5} c b + \frac {1}{5} x^{5} d a + x c a \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*x^9*d*b + 1/5*x^5*c*b + 1/5*x^5*d*a + x*c*a

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giac [A]  time = 0.15, size = 26, normalized size = 0.93 \[ \frac {1}{9} \, b d x^{9} + \frac {1}{5} \, b c x^{5} + \frac {1}{5} \, a d x^{5} + a c x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*b*d*x^9 + 1/5*b*c*x^5 + 1/5*a*d*x^5 + a*c*x

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maple [A]  time = 0.04, size = 25, normalized size = 0.89 \[ \frac {b d \,x^{9}}{9}+\frac {\left (a d +b c \right ) x^{5}}{5}+a c x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*c*x+1/5*(a*d+b*c)*x^5+1/9*b*d*x^9

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maxima [A]  time = 0.49, size = 24, normalized size = 0.86 \[ \frac {1}{9} \, b d x^{9} + \frac {1}{5} \, {\left (b c + a d\right )} x^{5} + a c x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*b*d*x^9 + 1/5*(b*c + a*d)*x^5 + a*c*x

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mupad [B]  time = 0.04, size = 25, normalized size = 0.89 \[ \frac {b\,d\,x^9}{9}+\left (\frac {a\,d}{5}+\frac {b\,c}{5}\right )\,x^5+a\,c\,x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^4)*(c + d*x^4),x)

[Out]

x^5*((a*d)/5 + (b*c)/5) + a*c*x + (b*d*x^9)/9

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sympy [A]  time = 0.06, size = 26, normalized size = 0.93 \[ a c x + \frac {b d x^{9}}{9} + x^{5} \left (\frac {a d}{5} + \frac {b c}{5}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*c*x + b*d*x**9/9 + x**5*(a*d/5 + b*c/5)

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