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

Optimal. Leaf size=47 \[ \frac {1}{5} x^5 (a B+A b)+\frac {1}{4} a A x^4+\frac {1}{6} x^6 (A c+b B)+\frac {1}{7} B c x^7 \]

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Rubi [A]  time = 0.06, antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {765} \begin {gather*} \frac {1}{5} x^5 (a B+A b)+\frac {1}{4} a A x^4+\frac {1}{6} x^6 (A c+b B)+\frac {1}{7} B c x^7 \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Mathematica [A]  time = 0.01, size = 47, normalized size = 1.00 \begin {gather*} \frac {1}{5} x^5 (a B+A b)+\frac {1}{4} a A x^4+\frac {1}{6} x^6 (A c+b B)+\frac {1}{7} B c x^7 \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^3 (A+B x) \left (a+b x+c x^2\right ) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.34, size = 43, normalized size = 0.91 \begin {gather*} \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} a B + \frac {1}{5} x^{5} b A + \frac {1}{4} x^{4} a A \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(B*x+A)*(c*x^2+b*x+a),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*a*B + 1/5*x^5*b*A + 1/4*x^4*a*A

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giac [A]  time = 0.17, size = 43, normalized size = 0.91 \begin {gather*} \frac {1}{7} \, B c x^{7} + \frac {1}{6} \, B b x^{6} + \frac {1}{6} \, A c x^{6} + \frac {1}{5} \, B a x^{5} + \frac {1}{5} \, A b x^{5} + \frac {1}{4} \, A a x^{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.04, size = 40, normalized size = 0.85 \begin {gather*} \frac {B c \,x^{7}}{7}+\frac {A a \,x^{4}}{4}+\frac {\left (A c +b B \right ) x^{6}}{6}+\frac {\left (A b +B a \right ) x^{5}}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.48, size = 39, normalized size = 0.83 \begin {gather*} \frac {1}{7} \, B c x^{7} + \frac {1}{6} \, {\left (B b + A c\right )} x^{6} + \frac {1}{4} \, A a x^{4} + \frac {1}{5} \, {\left (B a + A b\right )} x^{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.05, size = 41, normalized size = 0.87 \begin {gather*} \frac {B\,c\,x^7}{7}+\left (\frac {A\,c}{6}+\frac {B\,b}{6}\right )\,x^6+\left (\frac {A\,b}{5}+\frac {B\,a}{5}\right )\,x^5+\frac {A\,a\,x^4}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.07, size = 42, normalized size = 0.89 \begin {gather*} \frac {A a x^{4}}{4} + \frac {B c x^{7}}{7} + x^{6} \left (\frac {A c}{6} + \frac {B b}{6}\right ) + x^{5} \left (\frac {A b}{5} + \frac {B a}{5}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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