3.190 \(\int (\text{b1}+\text{c1} x) (a+2 b x+c x^2)^2 \, dx\)

Optimal. Leaf size=96 \[ a^2 \text{b1} x+\frac{1}{2} x^4 \left (a c \text{c1}+2 b^2 \text{c1}+2 b \text{b1} c\right )+\frac{2}{3} x^3 \left (2 a b \text{c1}+a \text{b1} c+2 b^2 \text{b1}\right )+\frac{1}{2} a x^2 (a \text{c1}+4 b \text{b1})+\frac{1}{5} c x^5 (4 b \text{c1}+\text{b1} c)+\frac{1}{6} c^2 \text{c1} x^6 \]

[Out]

a^2*b1*x + (a*(4*b*b1 + a*c1)*x^2)/2 + (2*(2*b^2*b1 + a*b1*c + 2*a*b*c1)*x^3)/3 + ((2*b*b1*c + 2*b^2*c1 + a*c*
c1)*x^4)/2 + (c*(b1*c + 4*b*c1)*x^5)/5 + (c^2*c1*x^6)/6

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Rubi [A]  time = 0.10442, antiderivative size = 96, normalized size of antiderivative = 1., 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 = {631} \[ a^2 \text{b1} x+\frac{1}{2} x^4 \left (a c \text{c1}+2 b^2 \text{c1}+2 b \text{b1} c\right )+\frac{2}{3} x^3 \left (2 a b \text{c1}+a \text{b1} c+2 b^2 \text{b1}\right )+\frac{1}{2} a x^2 (a \text{c1}+4 b \text{b1})+\frac{1}{5} c x^5 (4 b \text{c1}+\text{b1} c)+\frac{1}{6} c^2 \text{c1} x^6 \]

Antiderivative was successfully verified.

[In]

Int[(b1 + c1*x)*(a + 2*b*x + c*x^2)^2,x]

[Out]

a^2*b1*x + (a*(4*b*b1 + a*c1)*x^2)/2 + (2*(2*b^2*b1 + a*b1*c + 2*a*b*c1)*x^3)/3 + ((2*b*b1*c + 2*b^2*c1 + a*c*
c1)*x^4)/2 + (c*(b1*c + 4*b*c1)*x^5)/5 + (c^2*c1*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 (\text{b1}+\text{c1} x) \left (a+2 b x+c x^2\right )^2 \, dx &=\int \left (a^2 \text{b1}+a (4 b \text{b1}+a \text{c1}) x+2 \left (2 b^2 \text{b1}+a \text{b1} c+2 a b \text{c1}\right ) x^2+2 \left (2 b \text{b1} c+2 b^2 \text{c1}+a c \text{c1}\right ) x^3+c (\text{b1} c+4 b \text{c1}) x^4+c^2 \text{c1} x^5\right ) \, dx\\ &=a^2 \text{b1} x+\frac{1}{2} a (4 b \text{b1}+a \text{c1}) x^2+\frac{2}{3} \left (2 b^2 \text{b1}+a \text{b1} c+2 a b \text{c1}\right ) x^3+\frac{1}{2} \left (2 b \text{b1} c+2 b^2 \text{c1}+a c \text{c1}\right ) x^4+\frac{1}{5} c (\text{b1} c+4 b \text{c1}) x^5+\frac{1}{6} c^2 \text{c1} x^6\\ \end{align*}

Mathematica [A]  time = 0.028031, size = 91, normalized size = 0.95 \[ \frac{1}{30} x \left (15 a^2 (2 \text{b1}+\text{c1} x)+5 a x (4 b (3 \text{b1}+2 \text{c1} x)+c x (4 \text{b1}+3 \text{c1} x))+x^2 \left (10 b^2 (4 \text{b1}+3 \text{c1} x)+6 b c x (5 \text{b1}+4 \text{c1} x)+c^2 x^2 (6 \text{b1}+5 \text{c1} x)\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(b1 + c1*x)*(a + 2*b*x + c*x^2)^2,x]

[Out]

(x*(15*a^2*(2*b1 + c1*x) + 5*a*x*(4*b*(3*b1 + 2*c1*x) + c*x*(4*b1 + 3*c1*x)) + x^2*(10*b^2*(4*b1 + 3*c1*x) + 6
*b*c*x*(5*b1 + 4*c1*x) + c^2*x^2*(6*b1 + 5*c1*x))))/30

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Maple [A]  time = 0.001, size = 95, normalized size = 1. \begin{align*}{\frac{{c}^{2}{\it c1}\,{x}^{6}}{6}}+{\frac{ \left ( 4\,{\it c1}\,bc+{\it b1}\,{c}^{2} \right ){x}^{5}}{5}}+{\frac{ \left ( 4\,b{\it b1}\,c+{\it c1}\, \left ( 2\,ac+4\,{b}^{2} \right ) \right ){x}^{4}}{4}}+{\frac{ \left ({\it b1}\, \left ( 2\,ac+4\,{b}^{2} \right ) +4\,ab{\it c1} \right ){x}^{3}}{3}}+{\frac{ \left ({\it c1}\,{a}^{2}+4\,{\it b1}\,ab \right ){x}^{2}}{2}}+{a}^{2}{\it b1}\,x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c1*x+b1)*(c*x^2+2*b*x+a)^2,x)

[Out]

1/6*c^2*c1*x^6+1/5*(4*b*c*c1+b1*c^2)*x^5+1/4*(4*b*b1*c+c1*(2*a*c+4*b^2))*x^4+1/3*(b1*(2*a*c+4*b^2)+4*a*b*c1)*x
^3+1/2*(a^2*c1+4*a*b*b1)*x^2+a^2*b1*x

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Maxima [A]  time = 0.932681, size = 123, normalized size = 1.28 \begin{align*} \frac{1}{6} \, c^{2} c_{1} x^{6} + \frac{1}{5} \,{\left (b_{1} c^{2} + 4 \, b c c_{1}\right )} x^{5} + \frac{1}{2} \,{\left (2 \, b b_{1} c +{\left (2 \, b^{2} + a c\right )} c_{1}\right )} x^{4} + a^{2} b_{1} x + \frac{2}{3} \,{\left (2 \, b^{2} b_{1} + a b_{1} c + 2 \, a b c_{1}\right )} x^{3} + \frac{1}{2} \,{\left (4 \, a b b_{1} + a^{2} c_{1}\right )} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c1*x+b1)*(c*x^2+2*b*x+a)^2,x, algorithm="maxima")

[Out]

1/6*c^2*c1*x^6 + 1/5*(b1*c^2 + 4*b*c*c1)*x^5 + 1/2*(2*b*b1*c + (2*b^2 + a*c)*c1)*x^4 + a^2*b1*x + 2/3*(2*b^2*b
1 + a*b1*c + 2*a*b*c1)*x^3 + 1/2*(4*a*b*b1 + a^2*c1)*x^2

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Fricas [A]  time = 1.56473, size = 252, normalized size = 2.62 \begin{align*} \frac{1}{6} x^{6} c_{1} c^{2} + \frac{1}{5} x^{5} c^{2} b_{1} + \frac{4}{5} x^{5} c_{1} c b + x^{4} c b_{1} b + x^{4} c_{1} b^{2} + \frac{1}{2} x^{4} c_{1} c a + \frac{4}{3} x^{3} b_{1} b^{2} + \frac{2}{3} x^{3} c b_{1} a + \frac{4}{3} x^{3} c_{1} b a + 2 x^{2} b_{1} b a + \frac{1}{2} x^{2} c_{1} a^{2} + x b_{1} a^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c1*x+b1)*(c*x^2+2*b*x+a)^2,x, algorithm="fricas")

[Out]

1/6*x^6*c1*c^2 + 1/5*x^5*c^2*b1 + 4/5*x^5*c1*c*b + x^4*c*b1*b + x^4*c1*b^2 + 1/2*x^4*c1*c*a + 4/3*x^3*b1*b^2 +
2/3*x^3*c*b1*a + 4/3*x^3*c1*b*a + 2*x^2*b1*b*a + 1/2*x^2*c1*a^2 + x*b1*a^2

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Sympy [A]  time = 0.075041, size = 100, normalized size = 1.04 \begin{align*} a^{2} b_{1} x + \frac{c^{2} c_{1} x^{6}}{6} + x^{5} \left (\frac{4 b c c_{1}}{5} + \frac{b_{1} c^{2}}{5}\right ) + x^{4} \left (\frac{a c c_{1}}{2} + b^{2} c_{1} + b b_{1} c\right ) + x^{3} \left (\frac{4 a b c_{1}}{3} + \frac{2 a b_{1} c}{3} + \frac{4 b^{2} b_{1}}{3}\right ) + x^{2} \left (\frac{a^{2} c_{1}}{2} + 2 a b b_{1}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c1*x+b1)*(c*x**2+2*b*x+a)**2,x)

[Out]

a**2*b1*x + c**2*c1*x**6/6 + x**5*(4*b*c*c1/5 + b1*c**2/5) + x**4*(a*c*c1/2 + b**2*c1 + b*b1*c) + x**3*(4*a*b*
c1/3 + 2*a*b1*c/3 + 4*b**2*b1/3) + x**2*(a**2*c1/2 + 2*a*b*b1)

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Giac [A]  time = 1.05815, size = 132, normalized size = 1.38 \begin{align*} \frac{1}{6} \, c^{2} c_{1} x^{6} + \frac{1}{5} \, b_{1} c^{2} x^{5} + \frac{4}{5} \, b c c_{1} x^{5} + b b_{1} c x^{4} + b^{2} c_{1} x^{4} + \frac{1}{2} \, a c c_{1} x^{4} + \frac{4}{3} \, b^{2} b_{1} x^{3} + \frac{2}{3} \, a b_{1} c x^{3} + \frac{4}{3} \, a b c_{1} x^{3} + 2 \, a b b_{1} x^{2} + \frac{1}{2} \, a^{2} c_{1} x^{2} + a^{2} b_{1} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c1*x+b1)*(c*x^2+2*b*x+a)^2,x, algorithm="giac")

[Out]

1/6*c^2*c1*x^6 + 1/5*b1*c^2*x^5 + 4/5*b*c*c1*x^5 + b*b1*c*x^4 + b^2*c1*x^4 + 1/2*a*c*c1*x^4 + 4/3*b^2*b1*x^3 +
2/3*a*b1*c*x^3 + 4/3*a*b*c1*x^3 + 2*a*b*b1*x^2 + 1/2*a^2*c1*x^2 + a^2*b1*x