3.2 \(\int x^3 (A+B x) (b x+c x^2) \, dx\)

Optimal. Leaf size=33 \[ \frac{1}{6} x^6 (A c+b B)+\frac{1}{5} A b x^5+\frac{1}{7} B c x^7 \]

[Out]

(A*b*x^5)/5 + ((b*B + A*c)*x^6)/6 + (B*c*x^7)/7

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Rubi [A]  time = 0.0441337, antiderivative size = 33, normalized size of antiderivative = 1., 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} \[ \frac{1}{6} x^6 (A c+b B)+\frac{1}{5} A b x^5+\frac{1}{7} B c x^7 \]

Antiderivative was successfully verified.

[In]

Int[x^3*(A + B*x)*(b*x + c*x^2),x]

[Out]

(A*b*x^5)/5 + ((b*B + A*c)*x^6)/6 + (B*c*x^7)/7

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^3 (A+B x) \left (b x+c x^2\right ) \, dx &=\int \left (A b x^4+(b B+A c) x^5+B c x^6\right ) \, dx\\ &=\frac{1}{5} A b x^5+\frac{1}{6} (b B+A c) x^6+\frac{1}{7} B c x^7\\ \end{align*}

Mathematica [A]  time = 0.0053291, size = 33, normalized size = 1. \[ \frac{1}{6} x^6 (A c+b B)+\frac{1}{5} A b x^5+\frac{1}{7} B c x^7 \]

Antiderivative was successfully verified.

[In]

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

[Out]

(A*b*x^5)/5 + ((b*B + A*c)*x^6)/6 + (B*c*x^7)/7

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Maple [A]  time = 0.002, size = 28, normalized size = 0.9 \begin{align*}{\frac{Ab{x}^{5}}{5}}+{\frac{ \left ( Ac+bB \right ){x}^{6}}{6}}+{\frac{Bc{x}^{7}}{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*A*b*x^5+1/6*(A*c+B*b)*x^6+1/7*B*c*x^7

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Maxima [A]  time = 0.963759, size = 36, normalized size = 1.09 \begin{align*} \frac{1}{7} \, B c x^{7} + \frac{1}{5} \, A b x^{5} + \frac{1}{6} \,{\left (B b + A c\right )} x^{6} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*B*c*x^7 + 1/5*A*b*x^5 + 1/6*(B*b + A*c)*x^6

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Fricas [A]  time = 1.60246, size = 74, normalized size = 2.24 \begin{align*} \frac{1}{7} x^{7} c B + \frac{1}{6} x^{6} b B + \frac{1}{6} x^{6} c A + \frac{1}{5} x^{5} b A \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*x^7*c*B + 1/6*x^6*b*B + 1/6*x^6*c*A + 1/5*x^5*b*A

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Sympy [A]  time = 0.062981, size = 29, normalized size = 0.88 \begin{align*} \frac{A b x^{5}}{5} + \frac{B c x^{7}}{7} + x^{6} \left (\frac{A c}{6} + \frac{B b}{6}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

A*b*x**5/5 + B*c*x**7/7 + x**6*(A*c/6 + B*b/6)

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Giac [A]  time = 1.15303, size = 39, normalized size = 1.18 \begin{align*} \frac{1}{7} \, B c x^{7} + \frac{1}{6} \, B b x^{6} + \frac{1}{6} \, A c x^{6} + \frac{1}{5} \, A b x^{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*B*c*x^7 + 1/6*B*b*x^6 + 1/6*A*c*x^6 + 1/5*A*b*x^5