3.1.1 \(\int x^2 (a+b x^3) (A+B x^3) \, dx\) [1]

Optimal. Leaf size=33 \[ \frac {1}{3} a A x^3+\frac {1}{6} (A b+a B) x^6+\frac {1}{9} b B x^9 \]

[Out]

1/3*a*A*x^3+1/6*(A*b+B*a)*x^6+1/9*b*B*x^9

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Rubi [A]
time = 0.02, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {455, 45} \begin {gather*} \frac {1}{6} x^6 (a B+A b)+\frac {1}{3} a A x^3+\frac {1}{9} b B x^9 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^2*(a + b*x^3)*(A + B*x^3),x]

[Out]

(a*A*x^3)/3 + ((A*b + a*B)*x^6)/6 + (b*B*x^9)/9

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 455

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && EqQ[m
- n + 1, 0]

Rubi steps

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

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Mathematica [A]
time = 0.01, size = 33, normalized size = 1.00 \begin {gather*} \frac {1}{3} a A x^3+\frac {1}{6} (A b+a B) x^6+\frac {1}{9} b B x^9 \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(a*A*x^3)/3 + ((A*b + a*B)*x^6)/6 + (b*B*x^9)/9

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Maple [A]
time = 0.12, size = 28, normalized size = 0.85

method	result	size
default	\(\frac {a A \,x^{3}}{3}+\frac {\left (A b +B a \right ) x^{6}}{6}+\frac {b B \,x^{9}}{9}\)	\(28\)
norman	\(\frac {b B \,x^{9}}{9}+\left (\frac {A b}{6}+\frac {B a}{6}\right ) x^{6}+\frac {a A \,x^{3}}{3}\)	\(29\)
gosper	\(\frac {1}{9} b B \,x^{9}+\frac {1}{6} x^{6} A b +\frac {1}{6} x^{6} B a +\frac {1}{3} a A \,x^{3}\)	\(30\)
risch	\(\frac {1}{9} b B \,x^{9}+\frac {1}{6} x^{6} A b +\frac {1}{6} x^{6} B a +\frac {1}{3} a A \,x^{3}\)	\(30\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*a*A*x^3+1/6*(A*b+B*a)*x^6+1/9*b*B*x^9

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Maxima [A]
time = 0.30, size = 27, normalized size = 0.82 \begin {gather*} \frac {1}{9} \, B b x^{9} + \frac {1}{6} \, {\left (B a + A b\right )} x^{6} + \frac {1}{3} \, A a x^{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*B*b*x^9 + 1/6*(B*a + A*b)*x^6 + 1/3*A*a*x^3

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Fricas [A]
time = 2.23, size = 27, normalized size = 0.82 \begin {gather*} \frac {1}{9} \, B b x^{9} + \frac {1}{6} \, {\left (B a + A b\right )} x^{6} + \frac {1}{3} \, A a x^{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*B*b*x^9 + 1/6*(B*a + A*b)*x^6 + 1/3*A*a*x^3

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Sympy [A]
time = 0.01, size = 29, normalized size = 0.88 \begin {gather*} \frac {A a x^{3}}{3} + \frac {B b x^{9}}{9} + x^{6} \left (\frac {A b}{6} + \frac {B a}{6}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

A*a*x**3/3 + B*b*x**9/9 + x**6*(A*b/6 + B*a/6)

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Giac [A]
time = 0.64, size = 29, normalized size = 0.88 \begin {gather*} \frac {1}{9} \, B b x^{9} + \frac {1}{6} \, B a x^{6} + \frac {1}{6} \, A b x^{6} + \frac {1}{3} \, A a x^{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*B*b*x^9 + 1/6*B*a*x^6 + 1/6*A*b*x^6 + 1/3*A*a*x^3

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Mupad [B]
time = 0.20, size = 28, normalized size = 0.85 \begin {gather*} \frac {B\,b\,x^9}{9}+\left (\frac {A\,b}{6}+\frac {B\,a}{6}\right )\,x^6+\frac {A\,a\,x^3}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(A + B*x^3)*(a + b*x^3),x)

[Out]

x^6*((A*b)/6 + (B*a)/6) + (A*a*x^3)/3 + (B*b*x^9)/9

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