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

Optimal. Leaf size=28 \[ \frac{1}{4} x^4 (a d+b c)+a c x+\frac{1}{7} b d x^7 \]

[Out]

a*c*x + ((b*c + a*d)*x^4)/4 + (b*d*x^7)/7

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Rubi [A]  time = 0.0126816, antiderivative size = 28, normalized size of antiderivative = 1., 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} \[ \frac{1}{4} x^4 (a d+b c)+a c x+\frac{1}{7} b d x^7 \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^3)*(c + d*x^3),x]

[Out]

a*c*x + ((b*c + a*d)*x^4)/4 + (b*d*x^7)/7

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^3\right ) \left (c+d x^3\right ) \, dx &=\int \left (a c+(b c+a d) x^3+b d x^6\right ) \, dx\\ &=a c x+\frac{1}{4} (b c+a d) x^4+\frac{1}{7} b d x^7\\ \end{align*}

Mathematica [A]  time = 0.0050431, size = 28, normalized size = 1. \[ \frac{1}{4} x^4 (a d+b c)+a c x+\frac{1}{7} b d x^7 \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^3)*(c + d*x^3),x]

[Out]

a*c*x + ((b*c + a*d)*x^4)/4 + (b*d*x^7)/7

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Maple [A]  time = 0., size = 25, normalized size = 0.9 \begin{align*} acx+{\frac{ \left ( ad+bc \right ){x}^{4}}{4}}+{\frac{bd{x}^{7}}{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*c*x+1/4*(a*d+b*c)*x^4+1/7*b*d*x^7

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Maxima [A]  time = 0.959284, size = 32, normalized size = 1.14 \begin{align*} \frac{1}{7} \, b d x^{7} + \frac{1}{4} \,{\left (b c + a d\right )} x^{4} + a c x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*b*d*x^7 + 1/4*(b*c + a*d)*x^4 + a*c*x

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Fricas [A]  time = 1.39328, size = 66, normalized size = 2.36 \begin{align*} \frac{1}{7} x^{7} d b + \frac{1}{4} x^{4} c b + \frac{1}{4} x^{4} d a + x c a \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*x^7*d*b + 1/4*x^4*c*b + 1/4*x^4*d*a + x*c*a

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Sympy [A]  time = 0.075002, size = 26, normalized size = 0.93 \begin{align*} a c x + \frac{b d x^{7}}{7} + x^{4} \left (\frac{a d}{4} + \frac{b c}{4}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*c*x + b*d*x**7/7 + x**4*(a*d/4 + b*c/4)

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Giac [A]  time = 1.11432, size = 35, normalized size = 1.25 \begin{align*} \frac{1}{7} \, b d x^{7} + \frac{1}{4} \, b c x^{4} + \frac{1}{4} \, a d x^{4} + a c x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*b*d*x^7 + 1/4*b*c*x^4 + 1/4*a*d*x^4 + a*c*x