3.1.3 \(\int x^2 (A+B x) (b x+c x^2) \, dx\)

Optimal. Leaf size=33 \[ \frac {1}{5} x^5 (A c+b B)+\frac {1}{4} A b x^4+\frac {1}{6} B c x^6 \]

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Rubi [A]  time = 0.04, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {765} \begin {gather*} \frac {1}{5} x^5 (A c+b B)+\frac {1}{4} A b x^4+\frac {1}{6} B c x^6 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^2*(A + B*x)*(b*x + c*x^2),x]

[Out]

(A*b*x^4)/4 + ((b*B + A*c)*x^5)/5 + (B*c*x^6)/6

Rule 765

Int[((e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[Expand
Integrand[(e*x)^m*(f + g*x)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, e, f, g, m}, x] && IntegerQ[p] && (
GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

\begin {align*} \int x^2 (A+B x) \left (b x+c x^2\right ) \, dx &=\int \left (A b x^3+(b B+A c) x^4+B c x^5\right ) \, dx\\ &=\frac {1}{4} A b x^4+\frac {1}{5} (b B+A c) x^5+\frac {1}{6} B c x^6\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 33, normalized size = 1.00 \begin {gather*} \frac {1}{5} x^5 (A c+b B)+\frac {1}{4} A b x^4+\frac {1}{6} B c x^6 \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(A*b*x^4)/4 + ((b*B + A*c)*x^5)/5 + (B*c*x^6)/6

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

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.35, size = 29, normalized size = 0.88 \begin {gather*} \frac {1}{6} x^{6} c B + \frac {1}{5} x^{5} b B + \frac {1}{5} x^{5} c A + \frac {1}{4} x^{4} b A \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*x^6*c*B + 1/5*x^5*b*B + 1/5*x^5*c*A + 1/4*x^4*b*A

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giac [A]  time = 0.15, size = 29, normalized size = 0.88 \begin {gather*} \frac {1}{6} \, B c x^{6} + \frac {1}{5} \, B b x^{5} + \frac {1}{5} \, A c x^{5} + \frac {1}{4} \, A b x^{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*B*c*x^6 + 1/5*B*b*x^5 + 1/5*A*c*x^5 + 1/4*A*b*x^4

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maple [A]  time = 0.05, size = 28, normalized size = 0.85 \begin {gather*} \frac {B c \,x^{6}}{6}+\frac {A b \,x^{4}}{4}+\frac {\left (A c +b B \right ) x^{5}}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*A*b*x^4+1/5*(A*c+B*b)*x^5+1/6*B*c*x^6

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maxima [A]  time = 0.84, size = 27, normalized size = 0.82 \begin {gather*} \frac {1}{6} \, B c x^{6} + \frac {1}{4} \, A b x^{4} + \frac {1}{5} \, {\left (B b + A c\right )} x^{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*B*c*x^6 + 1/4*A*b*x^4 + 1/5*(B*b + A*c)*x^5

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mupad [B]  time = 0.04, size = 28, normalized size = 0.85 \begin {gather*} \frac {B\,c\,x^6}{6}+\left (\frac {A\,c}{5}+\frac {B\,b}{5}\right )\,x^5+\frac {A\,b\,x^4}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(b*x + c*x^2)*(A + B*x),x)

[Out]

x^5*((A*c)/5 + (B*b)/5) + (A*b*x^4)/4 + (B*c*x^6)/6

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sympy [A]  time = 0.06, size = 29, normalized size = 0.88 \begin {gather*} \frac {A b x^{4}}{4} + \frac {B c x^{6}}{6} + x^{5} \left (\frac {A c}{5} + \frac {B b}{5}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

A*b*x**4/4 + B*c*x**6/6 + x**5*(A*c/5 + B*b/5)

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