3.3 \(\int (a+b x^3) (A+B x^3) \, dx\)

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

[Out]

a*A*x + ((A*b + a*B)*x^4)/4 + (b*B*x^7)/7

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Rubi [A]  time = 0.0129807, 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} \[ \frac{1}{4} x^4 (a B+A b)+a A x+\frac{1}{7} b B x^7 \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^3)*(A + B*x^3),x]

[Out]

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

Mathematica [A]  time = 0.0054895, size = 28, normalized size = 1. \[ \frac{1}{4} x^4 (a B+A b)+a A x+\frac{1}{7} b B x^7 \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^3)*(A + B*x^3),x]

[Out]

a*A*x + ((A*b + a*B)*x^4)/4 + (b*B*x^7)/7

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Maple [A]  time = 0., size = 25, normalized size = 0.9 \begin{align*} aAx+{\frac{ \left ( Ab+Ba \right ){x}^{4}}{4}}+{\frac{bB{x}^{7}}{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^3+a)*(B*x^3+A),x)

[Out]

a*A*x+1/4*(A*b+B*a)*x^4+1/7*b*B*x^7

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Maxima [A]  time = 1.26996, size = 32, normalized size = 1.14 \begin{align*} \frac{1}{7} \, B b x^{7} + \frac{1}{4} \,{\left (B a + A b\right )} x^{4} + A a x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)*(B*x^3+A),x, algorithm="maxima")

[Out]

1/7*B*b*x^7 + 1/4*(B*a + A*b)*x^4 + A*a*x

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Fricas [A]  time = 1.26144, size = 66, normalized size = 2.36 \begin{align*} \frac{1}{7} x^{7} b B + \frac{1}{4} x^{4} a B + \frac{1}{4} x^{4} b A + x a A \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)*(B*x^3+A),x, algorithm="fricas")

[Out]

1/7*x^7*b*B + 1/4*x^4*a*B + 1/4*x^4*b*A + x*a*A

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Sympy [A]  time = 0.057488, size = 26, normalized size = 0.93 \begin{align*} A a x + \frac{B b x^{7}}{7} + x^{4} \left (\frac{A b}{4} + \frac{B a}{4}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**3+a)*(B*x**3+A),x)

[Out]

A*a*x + B*b*x**7/7 + x**4*(A*b/4 + B*a/4)

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Giac [A]  time = 1.15625, size = 35, normalized size = 1.25 \begin{align*} \frac{1}{7} \, B b x^{7} + \frac{1}{4} \, B a x^{4} + \frac{1}{4} \, A b x^{4} + A a x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)*(B*x^3+A),x, algorithm="giac")

[Out]

1/7*B*b*x^7 + 1/4*B*a*x^4 + 1/4*A*b*x^4 + A*a*x