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

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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} \[ \frac {1}{4} x^4 (a B+A b)+a A x+\frac {1}{7} b B x^7 \]

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*}

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Mathematica [A]  time = 0.01, size = 28, normalized size = 1.00 \[ \frac {1}{4} x^4 (a B+A b)+a A x+\frac {1}{7} b B x^7 \]

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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fricas [A]  time = 0.94, size = 26, normalized size = 0.93 \[ \frac {1}{7} x^{7} b B + \frac {1}{4} x^{4} a B + \frac {1}{4} x^{4} b A + x a A \]

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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giac [A]  time = 0.15, size = 26, normalized size = 0.93 \[ \frac {1}{7} \, B b x^{7} + \frac {1}{4} \, B a x^{4} + \frac {1}{4} \, A b x^{4} + A a x \]

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

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maple [A]  time = 0.04, size = 25, normalized size = 0.89 \[ \frac {B b \,x^{7}}{7}+\frac {\left (A b +B a \right ) x^{4}}{4}+A a x \]

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 = 0.56, size = 24, normalized size = 0.86 \[ \frac {1}{7} \, B b x^{7} + \frac {1}{4} \, {\left (B a + A b\right )} x^{4} + A a x \]

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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mupad [B]  time = 0.03, size = 25, normalized size = 0.89 \[ \frac {B\,b\,x^7}{7}+\left (\frac {A\,b}{4}+\frac {B\,a}{4}\right )\,x^4+A\,a\,x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*x^3)*(a + b*x^3),x)

[Out]

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

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sympy [A]  time = 0.06, size = 26, normalized size = 0.93 \[ A a x + \frac {B b x^{7}}{7} + x^{4} \left (\frac {A b}{4} + \frac {B a}{4}\right ) \]

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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