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

Optimal. Leaf size=33 \[ \frac {1}{4} x^4 (A c+b B)+\frac {1}{3} A b x^3+\frac {1}{5} B c x^5 \]

[Out]

1/3*A*b*x^3+1/4*(A*c+B*b)*x^4+1/5*B*c*x^5

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Rubi [A]  time = 0.03, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {765} \[ \frac {1}{4} x^4 (A c+b B)+\frac {1}{3} A b x^3+\frac {1}{5} B c x^5 \]

Antiderivative was successfully verified.

[In]

Int[x*(A + B*x)*(b*x + c*x^2),x]

[Out]

(A*b*x^3)/3 + ((b*B + A*c)*x^4)/4 + (B*c*x^5)/5

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

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Mathematica [A]  time = 0.00, size = 33, normalized size = 1.00 \[ \frac {1}{4} x^4 (A c+b B)+\frac {1}{3} A b x^3+\frac {1}{5} B c x^5 \]

Antiderivative was successfully verified.

[In]

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

[Out]

(A*b*x^3)/3 + ((b*B + A*c)*x^4)/4 + (B*c*x^5)/5

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fricas [A]  time = 0.85, size = 29, normalized size = 0.88 \[ \frac {1}{5} x^{5} c B + \frac {1}{4} x^{4} b B + \frac {1}{4} x^{4} c A + \frac {1}{3} x^{3} b A \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*x^5*c*B + 1/4*x^4*b*B + 1/4*x^4*c*A + 1/3*x^3*b*A

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giac [A]  time = 0.15, size = 29, normalized size = 0.88 \[ \frac {1}{5} \, B c x^{5} + \frac {1}{4} \, B b x^{4} + \frac {1}{4} \, A c x^{4} + \frac {1}{3} \, A b x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*B*c*x^5 + 1/4*B*b*x^4 + 1/4*A*c*x^4 + 1/3*A*b*x^3

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maple [A]  time = 0.04, size = 28, normalized size = 0.85 \[ \frac {B c \,x^{5}}{5}+\frac {A b \,x^{3}}{3}+\frac {\left (A c +b B \right ) x^{4}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*A*b*x^3+1/4*(A*c+B*b)*x^4+1/5*B*c*x^5

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maxima [A]  time = 0.81, size = 27, normalized size = 0.82 \[ \frac {1}{5} \, B c x^{5} + \frac {1}{3} \, A b x^{3} + \frac {1}{4} \, {\left (B b + A c\right )} x^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*B*c*x^5 + 1/3*A*b*x^3 + 1/4*(B*b + A*c)*x^4

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mupad [B]  time = 0.04, size = 28, normalized size = 0.85 \[ \frac {B\,c\,x^5}{5}+\left (\frac {A\,c}{4}+\frac {B\,b}{4}\right )\,x^4+\frac {A\,b\,x^3}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(b*x + c*x^2)*(A + B*x),x)

[Out]

x^4*((A*c)/4 + (B*b)/4) + (A*b*x^3)/3 + (B*c*x^5)/5

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sympy [A]  time = 0.07, size = 29, normalized size = 0.88 \[ \frac {A b x^{3}}{3} + \frac {B c x^{5}}{5} + x^{4} \left (\frac {A c}{4} + \frac {B b}{4}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

A*b*x**3/3 + B*c*x**5/5 + x**4*(A*c/4 + B*b/4)

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