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

Optimal. Leaf size=101 \[ \frac{1}{3} a^2 A x^3+\frac{1}{6} x^6 \left (2 a B c+2 A b c+b^2 B\right )+\frac{1}{5} x^5 \left (A \left (2 a c+b^2\right )+2 a b B\right )+\frac{1}{4} a x^4 (a B+2 A b)+\frac{1}{7} c x^7 (A c+2 b B)+\frac{1}{8} B c^2 x^8 \]

[Out]

(a^2*A*x^3)/3 + (a*(2*A*b + a*B)*x^4)/4 + ((2*a*b*B + A*(b^2 + 2*a*c))*x^5)/5 + ((b^2*B + 2*A*b*c + 2*a*B*c)*x
^6)/6 + (c*(2*b*B + A*c)*x^7)/7 + (B*c^2*x^8)/8

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Rubi [A]  time = 0.142088, antiderivative size = 101, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048, Rules used = {765} \[ \frac{1}{3} a^2 A x^3+\frac{1}{6} x^6 \left (2 a B c+2 A b c+b^2 B\right )+\frac{1}{5} x^5 \left (A \left (2 a c+b^2\right )+2 a b B\right )+\frac{1}{4} a x^4 (a B+2 A b)+\frac{1}{7} c x^7 (A c+2 b B)+\frac{1}{8} B c^2 x^8 \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^2*A*x^3)/3 + (a*(2*A*b + a*B)*x^4)/4 + ((2*a*b*B + A*(b^2 + 2*a*c))*x^5)/5 + ((b^2*B + 2*A*b*c + 2*a*B*c)*x
^6)/6 + (c*(2*b*B + A*c)*x^7)/7 + (B*c^2*x^8)/8

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^2 (A+B x) \left (a+b x+c x^2\right )^2 \, dx &=\int \left (a^2 A x^2+a (2 A b+a B) x^3+\left (2 a b B+A \left (b^2+2 a c\right )\right ) x^4+\left (b^2 B+2 A b c+2 a B c\right ) x^5+c (2 b B+A c) x^6+B c^2 x^7\right ) \, dx\\ &=\frac{1}{3} a^2 A x^3+\frac{1}{4} a (2 A b+a B) x^4+\frac{1}{5} \left (2 a b B+A \left (b^2+2 a c\right )\right ) x^5+\frac{1}{6} \left (b^2 B+2 A b c+2 a B c\right ) x^6+\frac{1}{7} c (2 b B+A c) x^7+\frac{1}{8} B c^2 x^8\\ \end{align*}

Mathematica [A]  time = 0.026388, size = 101, normalized size = 1. \[ \frac{1}{3} a^2 A x^3+\frac{1}{6} x^6 \left (2 a B c+2 A b c+b^2 B\right )+\frac{1}{5} x^5 \left (2 a A c+2 a b B+A b^2\right )+\frac{1}{4} a x^4 (a B+2 A b)+\frac{1}{7} c x^7 (A c+2 b B)+\frac{1}{8} B c^2 x^8 \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^2*A*x^3)/3 + (a*(2*A*b + a*B)*x^4)/4 + ((A*b^2 + 2*a*b*B + 2*a*A*c)*x^5)/5 + ((b^2*B + 2*A*b*c + 2*a*B*c)*x
^6)/6 + (c*(2*b*B + A*c)*x^7)/7 + (B*c^2*x^8)/8

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Maple [A]  time = 0., size = 94, normalized size = 0.9 \begin{align*}{\frac{B{c}^{2}{x}^{8}}{8}}+{\frac{ \left ( A{c}^{2}+2\,Bcb \right ){x}^{7}}{7}}+{\frac{ \left ( 2\,Abc+B \left ( 2\,ac+{b}^{2} \right ) \right ){x}^{6}}{6}}+{\frac{ \left ( 2\,abB+A \left ( 2\,ac+{b}^{2} \right ) \right ){x}^{5}}{5}}+{\frac{ \left ( 2\,Aab+B{a}^{2} \right ){x}^{4}}{4}}+{\frac{{a}^{2}A{x}^{3}}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*B*c^2*x^8+1/7*(A*c^2+2*B*b*c)*x^7+1/6*(2*A*b*c+B*(2*a*c+b^2))*x^6+1/5*(2*a*b*B+A*(2*a*c+b^2))*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 = 1.21224, size = 126, normalized size = 1.25 \begin{align*} \frac{1}{8} \, B c^{2} x^{8} + \frac{1}{7} \,{\left (2 \, B b c + A c^{2}\right )} x^{7} + \frac{1}{6} \,{\left (B b^{2} + 2 \,{\left (B a + A b\right )} c\right )} x^{6} + \frac{1}{3} \, A a^{2} x^{3} + \frac{1}{5} \,{\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} x^{5} + \frac{1}{4} \,{\left (B a^{2} + 2 \, A a b\right )} x^{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*B*c^2*x^8 + 1/7*(2*B*b*c + A*c^2)*x^7 + 1/6*(B*b^2 + 2*(B*a + A*b)*c)*x^6 + 1/3*A*a^2*x^3 + 1/5*(2*B*a*b +
A*b^2 + 2*A*a*c)*x^5 + 1/4*(B*a^2 + 2*A*a*b)*x^4

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Fricas [A]  time = 1.06926, size = 258, normalized size = 2.55 \begin{align*} \frac{1}{8} x^{8} c^{2} B + \frac{2}{7} x^{7} c b B + \frac{1}{7} x^{7} c^{2} A + \frac{1}{6} x^{6} b^{2} B + \frac{1}{3} x^{6} c a B + \frac{1}{3} x^{6} c b A + \frac{2}{5} x^{5} b a B + \frac{1}{5} x^{5} b^{2} A + \frac{2}{5} x^{5} c a 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 \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*x^8*c^2*B + 2/7*x^7*c*b*B + 1/7*x^7*c^2*A + 1/6*x^6*b^2*B + 1/3*x^6*c*a*B + 1/3*x^6*c*b*A + 2/5*x^5*b*a*B
+ 1/5*x^5*b^2*A + 2/5*x^5*c*a*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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Sympy [A]  time = 0.160985, size = 105, normalized size = 1.04 \begin{align*} \frac{A a^{2} x^{3}}{3} + \frac{B c^{2} x^{8}}{8} + x^{7} \left (\frac{A c^{2}}{7} + \frac{2 B b c}{7}\right ) + x^{6} \left (\frac{A b c}{3} + \frac{B a c}{3} + \frac{B b^{2}}{6}\right ) + x^{5} \left (\frac{2 A a c}{5} + \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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

A*a**2*x**3/3 + B*c**2*x**8/8 + x**7*(A*c**2/7 + 2*B*b*c/7) + x**6*(A*b*c/3 + B*a*c/3 + B*b**2/6) + x**5*(2*A*
a*c/5 + A*b**2/5 + 2*B*a*b/5) + x**4*(A*a*b/2 + B*a**2/4)

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Giac [A]  time = 1.28099, size = 139, normalized size = 1.38 \begin{align*} \frac{1}{8} \, B c^{2} x^{8} + \frac{2}{7} \, B b c x^{7} + \frac{1}{7} \, A c^{2} x^{7} + \frac{1}{6} \, B b^{2} x^{6} + \frac{1}{3} \, B a c x^{6} + \frac{1}{3} \, A b c x^{6} + \frac{2}{5} \, B a b x^{5} + \frac{1}{5} \, A b^{2} x^{5} + \frac{2}{5} \, A a c 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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*B*c^2*x^8 + 2/7*B*b*c*x^7 + 1/7*A*c^2*x^7 + 1/6*B*b^2*x^6 + 1/3*B*a*c*x^6 + 1/3*A*b*c*x^6 + 2/5*B*a*b*x^5
+ 1/5*A*b^2*x^5 + 2/5*A*a*c*x^5 + 1/4*B*a^2*x^4 + 1/2*A*a*b*x^4 + 1/3*A*a^2*x^3