3.1.13 \(\int (a+b x^3)^2 (A+B x^3) \, dx\)

Optimal. Leaf size=50 \[ a^2 A x+\frac {1}{7} b x^7 (2 a B+A b)+\frac {1}{4} a x^4 (a B+2 A b)+\frac {1}{10} b^2 B x^{10} \]

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Rubi [A]  time = 0.03, antiderivative size = 50, 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} \begin {gather*} a^2 A x+\frac {1}{7} b x^7 (2 a B+A b)+\frac {1}{4} a x^4 (a B+2 A b)+\frac {1}{10} b^2 B x^{10} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x^3)^2*(A + B*x^3),x]

[Out]

a^2*A*x + (a*(2*A*b + a*B)*x^4)/4 + (b*(A*b + 2*a*B)*x^7)/7 + (b^2*B*x^10)/10

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 )^2 \left (A+B x^3\right ) \, dx &=\int \left (a^2 A+a (2 A b+a B) x^3+b (A b+2 a B) x^6+b^2 B x^9\right ) \, dx\\ &=a^2 A x+\frac {1}{4} a (2 A b+a B) x^4+\frac {1}{7} b (A b+2 a B) x^7+\frac {1}{10} b^2 B x^{10}\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 50, normalized size = 1.00 \begin {gather*} a^2 A x+\frac {1}{7} b x^7 (2 a B+A b)+\frac {1}{4} a x^4 (a B+2 A b)+\frac {1}{10} b^2 B x^{10} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^3)^2*(A + B*x^3),x]

[Out]

a^2*A*x + (a*(2*A*b + a*B)*x^4)/4 + (b*(A*b + 2*a*B)*x^7)/7 + (b^2*B*x^10)/10

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a+b x^3\right )^2 \left (A+B x^3\right ) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(a + b*x^3)^2*(A + B*x^3),x]

[Out]

IntegrateAlgebraic[(a + b*x^3)^2*(A + B*x^3), x]

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fricas [A]  time = 0.64, size = 50, normalized size = 1.00 \begin {gather*} \frac {1}{10} x^{10} b^{2} B + \frac {2}{7} x^{7} b a B + \frac {1}{7} x^{7} b^{2} A + \frac {1}{4} x^{4} a^{2} B + \frac {1}{2} x^{4} b a A + x a^{2} A \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^2*(B*x^3+A),x, algorithm="fricas")

[Out]

1/10*x^10*b^2*B + 2/7*x^7*b*a*B + 1/7*x^7*b^2*A + 1/4*x^4*a^2*B + 1/2*x^4*b*a*A + x*a^2*A

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giac [A]  time = 0.17, size = 50, normalized size = 1.00 \begin {gather*} \frac {1}{10} \, B b^{2} x^{10} + \frac {2}{7} \, B a b x^{7} + \frac {1}{7} \, A b^{2} x^{7} + \frac {1}{4} \, B a^{2} x^{4} + \frac {1}{2} \, A a b x^{4} + A a^{2} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^2*(B*x^3+A),x, algorithm="giac")

[Out]

1/10*B*b^2*x^10 + 2/7*B*a*b*x^7 + 1/7*A*b^2*x^7 + 1/4*B*a^2*x^4 + 1/2*A*a*b*x^4 + A*a^2*x

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maple [A]  time = 0.04, size = 49, normalized size = 0.98 \begin {gather*} \frac {B \,b^{2} x^{10}}{10}+\frac {\left (b^{2} A +2 a b B \right ) x^{7}}{7}+A \,a^{2} x +\frac {\left (2 a b A +a^{2} B \right ) x^{4}}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^3+a)^2*(B*x^3+A),x)

[Out]

1/10*b^2*B*x^10+1/7*(A*b^2+2*B*a*b)*x^7+1/4*(2*A*a*b+B*a^2)*x^4+A*a^2*x

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maxima [A]  time = 0.66, size = 48, normalized size = 0.96 \begin {gather*} \frac {1}{10} \, B b^{2} x^{10} + \frac {1}{7} \, {\left (2 \, B a b + A b^{2}\right )} x^{7} + \frac {1}{4} \, {\left (B a^{2} + 2 \, A a b\right )} x^{4} + A a^{2} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^2*(B*x^3+A),x, algorithm="maxima")

[Out]

1/10*B*b^2*x^10 + 1/7*(2*B*a*b + A*b^2)*x^7 + 1/4*(B*a^2 + 2*A*a*b)*x^4 + A*a^2*x

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mupad [B]  time = 0.04, size = 48, normalized size = 0.96 \begin {gather*} x^4\,\left (\frac {B\,a^2}{4}+\frac {A\,b\,a}{2}\right )+x^7\,\left (\frac {A\,b^2}{7}+\frac {2\,B\,a\,b}{7}\right )+\frac {B\,b^2\,x^{10}}{10}+A\,a^2\,x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*x^3)*(a + b*x^3)^2,x)

[Out]

x^4*((B*a^2)/4 + (A*a*b)/2) + x^7*((A*b^2)/7 + (2*B*a*b)/7) + (B*b^2*x^10)/10 + A*a^2*x

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sympy [A]  time = 0.08, size = 51, normalized size = 1.02 \begin {gather*} A a^{2} x + \frac {B b^{2} x^{10}}{10} + x^{7} \left (\frac {A b^{2}}{7} + \frac {2 B a b}{7}\right ) + x^{4} \left (\frac {A a b}{2} + \frac {B a^{2}}{4}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**3+a)**2*(B*x**3+A),x)

[Out]

A*a**2*x + B*b**2*x**10/10 + x**7*(A*b**2/7 + 2*B*a*b/7) + x**4*(A*a*b/2 + B*a**2/4)

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