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

Optimal. Leaf size=33 \[ \frac {1}{4} x^4 (a B+A b)+\frac {1}{2} a A x^2+\frac {1}{6} b B x^6 \]

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Rubi [A]  time = 0.03, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {444, 43} \begin {gather*} \frac {1}{4} x^4 (a B+A b)+\frac {1}{2} a A x^2+\frac {1}{6} b B x^6 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x*(a + b*x^2)*(A + B*x^2),x]

[Out]

(a*A*x^2)/2 + ((A*b + a*B)*x^4)/4 + (b*B*x^6)/6

Rule 43

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 444

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

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Mathematica [A]  time = 0.02, size = 33, normalized size = 1.00 \begin {gather*} \frac {1}{4} x^4 (a B+A b)+\frac {1}{2} a A x^2+\frac {1}{6} b B x^6 \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(a*A*x^2)/2 + ((A*b + a*B)*x^4)/4 + (b*B*x^6)/6

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

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.37, size = 29, normalized size = 0.88 \begin {gather*} \frac {1}{6} x^{6} b B + \frac {1}{4} x^{4} a B + \frac {1}{4} x^{4} b A + \frac {1}{2} x^{2} a A \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*x^6*b*B + 1/4*x^4*a*B + 1/4*x^4*b*A + 1/2*x^2*a*A

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giac [A]  time = 0.38, size = 29, normalized size = 0.88 \begin {gather*} \frac {1}{6} \, B b x^{6} + \frac {1}{4} \, B a x^{4} + \frac {1}{4} \, A b x^{4} + \frac {1}{2} \, A a x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*B*b*x^6 + 1/4*B*a*x^4 + 1/4*A*b*x^4 + 1/2*A*a*x^2

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maple [A]  time = 0.00, size = 28, normalized size = 0.85 \begin {gather*} \frac {B b \,x^{6}}{6}+\frac {A a \,x^{2}}{2}+\frac {\left (A b +B a \right ) x^{4}}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*a*A*x^2+1/4*(A*b+B*a)*x^4+1/6*b*B*x^6

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maxima [A]  time = 1.35, size = 27, normalized size = 0.82 \begin {gather*} \frac {1}{6} \, B b x^{6} + \frac {1}{4} \, {\left (B a + A b\right )} x^{4} + \frac {1}{2} \, A a x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*B*b*x^6 + 1/4*(B*a + A*b)*x^4 + 1/2*A*a*x^2

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mupad [B]  time = 0.04, size = 28, normalized size = 0.85 \begin {gather*} \frac {B\,b\,x^6}{6}+\left (\frac {A\,b}{4}+\frac {B\,a}{4}\right )\,x^4+\frac {A\,a\,x^2}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(A + B*x^2)*(a + b*x^2),x)

[Out]

x^4*((A*b)/4 + (B*a)/4) + (A*a*x^2)/2 + (B*b*x^6)/6

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sympy [A]  time = 0.07, size = 29, normalized size = 0.88 \begin {gather*} \frac {A a x^{2}}{2} + \frac {B b x^{6}}{6} + x^{4} \left (\frac {A b}{4} + \frac {B a}{4}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

A*a*x**2/2 + B*b*x**6/6 + x**4*(A*b/4 + B*a/4)

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