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

Optimal. Leaf size=96 \[ a^2 A x+\frac{1}{4} x^4 \left (2 a B c+2 A b c+b^2 B\right )+\frac{1}{3} x^3 \left (A \left (2 a c+b^2\right )+2 a b B\right )+\frac{1}{2} a x^2 (a B+2 A b)+\frac{1}{5} c x^5 (A c+2 b B)+\frac{1}{6} B c^2 x^6 \]

[Out]

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

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Rubi [A]  time = 0.0693599, antiderivative size = 96, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056, Rules used = {631} \[ a^2 A x+\frac{1}{4} x^4 \left (2 a B c+2 A b c+b^2 B\right )+\frac{1}{3} x^3 \left (A \left (2 a c+b^2\right )+2 a b B\right )+\frac{1}{2} a x^2 (a B+2 A b)+\frac{1}{5} c x^5 (A c+2 b B)+\frac{1}{6} B c^2 x^6 \]

Antiderivative was successfully verified.

[In]

Int[(A + B*x)*(a + b*x + c*x^2)^2,x]

[Out]

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

Rule 631

Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)
*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[2*c*d - b*e, 0] && IntegerQ[p] && (GtQ[p, 0]
|| EqQ[a, 0])

Rubi steps

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

Mathematica [A]  time = 0.0191913, size = 96, normalized size = 1. \[ a^2 A x+\frac{1}{4} x^4 \left (2 a B c+2 A b c+b^2 B\right )+\frac{1}{3} x^3 \left (2 a A c+2 a b B+A b^2\right )+\frac{1}{2} a x^2 (a B+2 A b)+\frac{1}{5} c x^5 (A c+2 b B)+\frac{1}{6} B c^2 x^6 \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.002, size = 91, normalized size = 1. \begin{align*}{\frac{B{c}^{2}{x}^{6}}{6}}+{\frac{ \left ( A{c}^{2}+2\,Bcb \right ){x}^{5}}{5}}+{\frac{ \left ( 2\,Abc+B \left ( 2\,ac+{b}^{2} \right ) \right ){x}^{4}}{4}}+{\frac{ \left ( 2\,abB+A \left ( 2\,ac+{b}^{2} \right ) \right ){x}^{3}}{3}}+{\frac{ \left ( 2\,Aab+B{a}^{2} \right ){x}^{2}}{2}}+{a}^{2}Ax \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.10264, size = 122, normalized size = 1.27 \begin{align*} \frac{1}{6} \, B c^{2} x^{6} + \frac{1}{5} \,{\left (2 \, B b c + A c^{2}\right )} x^{5} + \frac{1}{4} \,{\left (B b^{2} + 2 \,{\left (B a + A b\right )} c\right )} x^{4} + A a^{2} x + \frac{1}{3} \,{\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} x^{3} + \frac{1}{2} \,{\left (B a^{2} + 2 \, A a b\right )} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.14998, size = 244, normalized size = 2.54 \begin{align*} \frac{1}{6} x^{6} c^{2} B + \frac{2}{5} x^{5} c b B + \frac{1}{5} x^{5} c^{2} A + \frac{1}{4} x^{4} b^{2} B + \frac{1}{2} x^{4} c a B + \frac{1}{2} x^{4} c b A + \frac{2}{3} x^{3} b a B + \frac{1}{3} x^{3} b^{2} A + \frac{2}{3} x^{3} c a A + \frac{1}{2} x^{2} a^{2} B + x^{2} b a A + x a^{2} A \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.145895, size = 100, normalized size = 1.04 \begin{align*} A a^{2} x + \frac{B c^{2} x^{6}}{6} + x^{5} \left (\frac{A c^{2}}{5} + \frac{2 B b c}{5}\right ) + x^{4} \left (\frac{A b c}{2} + \frac{B a c}{2} + \frac{B b^{2}}{4}\right ) + x^{3} \left (\frac{2 A a c}{3} + \frac{A b^{2}}{3} + \frac{2 B a b}{3}\right ) + x^{2} \left (A a b + \frac{B a^{2}}{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.20326, size = 134, normalized size = 1.4 \begin{align*} \frac{1}{6} \, B c^{2} x^{6} + \frac{2}{5} \, B b c x^{5} + \frac{1}{5} \, A c^{2} x^{5} + \frac{1}{4} \, B b^{2} x^{4} + \frac{1}{2} \, B a c x^{4} + \frac{1}{2} \, A b c x^{4} + \frac{2}{3} \, B a b x^{3} + \frac{1}{3} \, A b^{2} x^{3} + \frac{2}{3} \, A a c x^{3} + \frac{1}{2} \, B a^{2} x^{2} + A a b x^{2} + A a^{2} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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