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

Optimal. Leaf size=55 \[ \frac {1}{3} a^2 A x^3+\frac {1}{5} b x^5 (2 a B+A b)+\frac {1}{4} a x^4 (a B+2 A b)+\frac {1}{6} b^2 B x^6 \]

[Out]

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

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Rubi [A]  time = 0.03, antiderivative size = 55, 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 = {76} \[ \frac {1}{3} a^2 A x^3+\frac {1}{5} b x^5 (2 a B+A b)+\frac {1}{4} a x^4 (a B+2 A b)+\frac {1}{6} b^2 B x^6 \]

Antiderivative was successfully verified.

[In]

Int[x^2*(a + b*x)^2*(A + B*x),x]

[Out]

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

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

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Mathematica [A]  time = 0.01, size = 50, normalized size = 0.91 \[ \frac {1}{60} x^3 \left (5 a^2 (4 A+3 B x)+6 a b x (5 A+4 B x)+2 b^2 x^2 (6 A+5 B x)\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x^3*(5*a^2*(4*A + 3*B*x) + 6*a*b*x*(5*A + 4*B*x) + 2*b^2*x^2*(6*A + 5*B*x)))/60

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fricas [A]  time = 0.47, size = 53, normalized size = 0.96 \[ \frac {1}{6} x^{6} b^{2} B + \frac {2}{5} x^{5} b a B + \frac {1}{5} x^{5} b^{2} A + \frac {1}{4} x^{4} a^{2} B + \frac {1}{2} x^{4} b a A + \frac {1}{3} x^{3} a^{2} A \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 1.24, size = 53, normalized size = 0.96 \[ \frac {1}{6} \, B b^{2} x^{6} + \frac {2}{5} \, B a b x^{5} + \frac {1}{5} \, A b^{2} x^{5} + \frac {1}{4} \, B a^{2} x^{4} + \frac {1}{2} \, A a b x^{4} + \frac {1}{3} \, A a^{2} x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.00, size = 52, normalized size = 0.95 \[ \frac {B \,b^{2} x^{6}}{6}+\frac {A \,a^{2} x^{3}}{3}+\frac {\left (b^{2} A +2 a b B \right ) x^{5}}{5}+\frac {\left (2 a b A +a^{2} B \right ) x^{4}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.00, size = 51, normalized size = 0.93 \[ \frac {1}{6} \, B b^{2} x^{6} + \frac {1}{3} \, A a^{2} x^{3} + \frac {1}{5} \, {\left (2 \, B a b + A b^{2}\right )} x^{5} + \frac {1}{4} \, {\left (B a^{2} + 2 \, A a b\right )} x^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.06, size = 51, normalized size = 0.93 \[ x^4\,\left (\frac {B\,a^2}{4}+\frac {A\,b\,a}{2}\right )+x^5\,\left (\frac {A\,b^2}{5}+\frac {2\,B\,a\,b}{5}\right )+\frac {A\,a^2\,x^3}{3}+\frac {B\,b^2\,x^6}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(A + B*x)*(a + b*x)^2,x)

[Out]

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

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sympy [A]  time = 0.09, size = 54, normalized size = 0.98 \[ \frac {A a^{2} x^{3}}{3} + \frac {B b^{2} x^{6}}{6} + x^{5} \left (\frac {A b^{2}}{5} + \frac {2 B a b}{5}\right ) + x^{4} \left (\frac {A a b}{2} + \frac {B a^{2}}{4}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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