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

Optimal. Leaf size=55 \[ \frac {1}{4} a^2 A x^4+\frac {1}{6} b x^6 (2 a B+A b)+\frac {1}{5} a x^5 (a B+2 A b)+\frac {1}{7} b^2 B x^7 \]

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Rubi [A]  time = 0.04, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {27, 76} \begin {gather*} \frac {1}{4} a^2 A x^4+\frac {1}{6} b x^6 (2 a B+A b)+\frac {1}{5} a x^5 (a B+2 A b)+\frac {1}{7} b^2 B x^7 \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(a^2*A*x^4)/4 + (a*(2*A*b + a*B)*x^5)/5 + (b*(A*b + 2*a*B)*x^6)/6 + (b^2*B*x^7)/7

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 76

Int[((d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_))*((e_) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*
x)*(d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && IGtQ[p, 0] && (NeQ[n, -1] || EqQ[p, 1]) && N
eQ[b*e + a*f, 0] && ( !IntegerQ[n] || LtQ[9*p + 5*n, 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && Rational
Q[a, b, d, e, f])) && (NeQ[n + p + 3, 0] || EqQ[p, 1])

Rubi steps

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

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Mathematica [A]  time = 0.01, size = 55, normalized size = 1.00 \begin {gather*} \frac {1}{4} a^2 A x^4+\frac {1}{6} b x^6 (2 a B+A b)+\frac {1}{5} a x^5 (a B+2 A b)+\frac {1}{7} b^2 B x^7 \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(a^2*A*x^4)/4 + (a*(2*A*b + a*B)*x^5)/5 + (b*(A*b + 2*a*B)*x^6)/6 + (b^2*B*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^2+2 a b x+b^2 x^2\right ) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.37, size = 53, normalized size = 0.96 \begin {gather*} \frac {1}{7} x^{7} b^{2} B + \frac {1}{3} x^{6} b a B + \frac {1}{6} x^{6} b^{2} A + \frac {1}{5} x^{5} a^{2} B + \frac {2}{5} x^{5} b a A + \frac {1}{4} x^{4} a^{2} A \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*x^7*b^2*B + 1/3*x^6*b*a*B + 1/6*x^6*b^2*A + 1/5*x^5*a^2*B + 2/5*x^5*b*a*A + 1/4*x^4*a^2*A

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giac [A]  time = 0.21, size = 53, normalized size = 0.96 \begin {gather*} \frac {1}{7} \, B b^{2} x^{7} + \frac {1}{3} \, B a b x^{6} + \frac {1}{6} \, A b^{2} x^{6} + \frac {1}{5} \, B a^{2} x^{5} + \frac {2}{5} \, A a b x^{5} + \frac {1}{4} \, A a^{2} x^{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*B*b^2*x^7 + 1/3*B*a*b*x^6 + 1/6*A*b^2*x^6 + 1/5*B*a^2*x^5 + 2/5*A*a*b*x^5 + 1/4*A*a^2*x^4

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maple [A]  time = 0.05, size = 52, normalized size = 0.95 \begin {gather*} \frac {B \,b^{2} x^{7}}{7}+\frac {A \,a^{2} x^{4}}{4}+\frac {\left (A \,b^{2}+2 a b B \right ) x^{6}}{6}+\frac {\left (2 A a b +a^{2} B \right ) x^{5}}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*B*b^2*x^7+1/6*(A*b^2+2*B*a*b)*x^6+1/5*(2*A*a*b+B*a^2)*x^5+1/4*A*a^2*x^4

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maxima [A]  time = 0.56, size = 51, normalized size = 0.93 \begin {gather*} \frac {1}{7} \, B b^{2} x^{7} + \frac {1}{4} \, A a^{2} x^{4} + \frac {1}{6} \, {\left (2 \, B a b + A b^{2}\right )} x^{6} + \frac {1}{5} \, {\left (B a^{2} + 2 \, 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)*(b^2*x^2+2*a*b*x+a^2),x, algorithm="maxima")

[Out]

1/7*B*b^2*x^7 + 1/4*A*a^2*x^4 + 1/6*(2*B*a*b + A*b^2)*x^6 + 1/5*(B*a^2 + 2*A*a*b)*x^5

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mupad [B]  time = 0.05, size = 51, normalized size = 0.93 \begin {gather*} x^5\,\left (\frac {B\,a^2}{5}+\frac {2\,A\,b\,a}{5}\right )+x^6\,\left (\frac {A\,b^2}{6}+\frac {B\,a\,b}{3}\right )+\frac {A\,a^2\,x^4}{4}+\frac {B\,b^2\,x^7}{7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^5*((B*a^2)/5 + (2*A*a*b)/5) + x^6*((A*b^2)/6 + (B*a*b)/3) + (A*a^2*x^4)/4 + (B*b^2*x^7)/7

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sympy [A]  time = 0.07, size = 54, normalized size = 0.98 \begin {gather*} \frac {A a^{2} x^{4}}{4} + \frac {B b^{2} x^{7}}{7} + x^{6} \left (\frac {A b^{2}}{6} + \frac {B a b}{3}\right ) + x^{5} \left (\frac {2 A a b}{5} + \frac {B a^{2}}{5}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

A*a**2*x**4/4 + B*b**2*x**7/7 + x**6*(A*b**2/6 + B*a*b/3) + x**5*(2*A*a*b/5 + B*a**2/5)

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