3.251 \(\int x^3 (A+B x) (a+c x^2) \, dx\)

Optimal. Leaf size=37 \[ \frac{1}{4} a A x^4+\frac{1}{5} a B x^5+\frac{1}{6} A c x^6+\frac{1}{7} B c x^7 \]

[Out]

(a*A*x^4)/4 + (a*B*x^5)/5 + (A*c*x^6)/6 + (B*c*x^7)/7

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Rubi [A]  time = 0.035158, antiderivative size = 37, normalized size of antiderivative = 1., 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 = {766} \[ \frac{1}{4} a A x^4+\frac{1}{5} a B x^5+\frac{1}{6} A c x^6+\frac{1}{7} B c x^7 \]

Antiderivative was successfully verified.

[In]

Int[x^3*(A + B*x)*(a + c*x^2),x]

[Out]

(a*A*x^4)/4 + (a*B*x^5)/5 + (A*c*x^6)/6 + (B*c*x^7)/7

Rule 766

Int[((e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(e*x
)^m*(f + g*x)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, e, f, g, m}, x] && IGtQ[p, 0]

Rubi steps

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

Mathematica [A]  time = 0.002291, size = 37, normalized size = 1. \[ \frac{1}{4} a A x^4+\frac{1}{5} a B x^5+\frac{1}{6} A c x^6+\frac{1}{7} B c x^7 \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a*A*x^4)/4 + (a*B*x^5)/5 + (A*c*x^6)/6 + (B*c*x^7)/7

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Maple [A]  time = 0.001, size = 30, normalized size = 0.8 \begin{align*}{\frac{aA{x}^{4}}{4}}+{\frac{aB{x}^{5}}{5}}+{\frac{Ac{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+a),x)

[Out]

1/4*a*A*x^4+1/5*a*B*x^5+1/6*A*c*x^6+1/7*B*c*x^7

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Maxima [A]  time = 1.05521, size = 39, normalized size = 1.05 \begin{align*} \frac{1}{7} \, B c x^{7} + \frac{1}{6} \, A c x^{6} + \frac{1}{5} \, B a x^{5} + \frac{1}{4} \, A a x^{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*B*c*x^7 + 1/6*A*c*x^6 + 1/5*B*a*x^5 + 1/4*A*a*x^4

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Fricas [A]  time = 1.30428, size = 74, normalized size = 2. \begin{align*} \frac{1}{7} x^{7} c B + \frac{1}{6} x^{6} c A + \frac{1}{5} x^{5} a B + \frac{1}{4} x^{4} a A \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*x^7*c*B + 1/6*x^6*c*A + 1/5*x^5*a*B + 1/4*x^4*a*A

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Sympy [A]  time = 0.061884, size = 32, normalized size = 0.86 \begin{align*} \frac{A a x^{4}}{4} + \frac{A c x^{6}}{6} + \frac{B a x^{5}}{5} + \frac{B c x^{7}}{7} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

A*a*x**4/4 + A*c*x**6/6 + B*a*x**5/5 + B*c*x**7/7

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Giac [A]  time = 1.09268, size = 39, normalized size = 1.05 \begin{align*} \frac{1}{7} \, B c x^{7} + \frac{1}{6} \, A c x^{6} + \frac{1}{5} \, B a x^{5} + \frac{1}{4} \, A a x^{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*B*c*x^7 + 1/6*A*c*x^6 + 1/5*B*a*x^5 + 1/4*A*a*x^4