3.191 \(\int (\text{b1}+\text{c1} x) (a+2 b x+c x^2)^3 \, dx\)

Optimal. Leaf size=167 \[ \frac{1}{4} x^4 \left (3 a^2 c \text{c1}+12 a b^2 \text{c1}+12 a b \text{b1} c+8 b^3 \text{b1}\right )+\frac{1}{2} a^2 x^2 (a \text{c1}+6 b \text{b1})+a^3 \text{b1} x+\frac{1}{5} x^5 \left (12 a b c \text{c1}+3 a \text{b1} c^2+12 b^2 \text{b1} c+8 b^3 \text{c1}\right )+\frac{1}{2} c x^6 \left (a c \text{c1}+4 b^2 \text{c1}+2 b \text{b1} c\right )+a x^3 \left (2 a b \text{c1}+a \text{b1} c+4 b^2 \text{b1}\right )+\frac{1}{7} c^2 x^7 (6 b \text{c1}+\text{b1} c)+\frac{1}{8} c^3 \text{c1} x^8 \]

[Out]

a^3*b1*x + (a^2*(6*b*b1 + a*c1)*x^2)/2 + a*(4*b^2*b1 + a*b1*c + 2*a*b*c1)*x^3 + ((8*b^3*b1 + 12*a*b*b1*c + 12*
a*b^2*c1 + 3*a^2*c*c1)*x^4)/4 + ((12*b^2*b1*c + 3*a*b1*c^2 + 8*b^3*c1 + 12*a*b*c*c1)*x^5)/5 + (c*(2*b*b1*c + 4
*b^2*c1 + a*c*c1)*x^6)/2 + (c^2*(b1*c + 6*b*c1)*x^7)/7 + (c^3*c1*x^8)/8

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Rubi [A]  time = 0.187678, antiderivative size = 167, 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} \[ \frac{1}{4} x^4 \left (3 a^2 c \text{c1}+12 a b^2 \text{c1}+12 a b \text{b1} c+8 b^3 \text{b1}\right )+\frac{1}{2} a^2 x^2 (a \text{c1}+6 b \text{b1})+a^3 \text{b1} x+\frac{1}{5} x^5 \left (12 a b c \text{c1}+3 a \text{b1} c^2+12 b^2 \text{b1} c+8 b^3 \text{c1}\right )+\frac{1}{2} c x^6 \left (a c \text{c1}+4 b^2 \text{c1}+2 b \text{b1} c\right )+a x^3 \left (2 a b \text{c1}+a \text{b1} c+4 b^2 \text{b1}\right )+\frac{1}{7} c^2 x^7 (6 b \text{c1}+\text{b1} c)+\frac{1}{8} c^3 \text{c1} x^8 \]

Antiderivative was successfully verified.

[In]

Int[(b1 + c1*x)*(a + 2*b*x + c*x^2)^3,x]

[Out]

a^3*b1*x + (a^2*(6*b*b1 + a*c1)*x^2)/2 + a*(4*b^2*b1 + a*b1*c + 2*a*b*c1)*x^3 + ((8*b^3*b1 + 12*a*b*b1*c + 12*
a*b^2*c1 + 3*a^2*c*c1)*x^4)/4 + ((12*b^2*b1*c + 3*a*b1*c^2 + 8*b^3*c1 + 12*a*b*c*c1)*x^5)/5 + (c*(2*b*b1*c + 4
*b^2*c1 + a*c*c1)*x^6)/2 + (c^2*(b1*c + 6*b*c1)*x^7)/7 + (c^3*c1*x^8)/8

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 )^3 \, dx &=\int \left (a^3 \text{b1}+a^2 (6 b \text{b1}+a \text{c1}) x+3 a \left (4 b^2 \text{b1}+a \text{b1} c+2 a b \text{c1}\right ) x^2+\left (8 b^3 \text{b1}+12 a b \text{b1} c+12 a b^2 \text{c1}+3 a^2 c \text{c1}\right ) x^3+\left (12 b^2 \text{b1} c+3 a \text{b1} c^2+8 b^3 \text{c1}+12 a b c \text{c1}\right ) x^4+3 c \left (2 b \text{b1} c+4 b^2 \text{c1}+a c \text{c1}\right ) x^5+c^2 (\text{b1} c+6 b \text{c1}) x^6+c^3 \text{c1} x^7\right ) \, dx\\ &=a^3 \text{b1} x+\frac{1}{2} a^2 (6 b \text{b1}+a \text{c1}) x^2+a \left (4 b^2 \text{b1}+a \text{b1} c+2 a b \text{c1}\right ) x^3+\frac{1}{4} \left (8 b^3 \text{b1}+12 a b \text{b1} c+12 a b^2 \text{c1}+3 a^2 c \text{c1}\right ) x^4+\frac{1}{5} \left (12 b^2 \text{b1} c+3 a \text{b1} c^2+8 b^3 \text{c1}+12 a b c \text{c1}\right ) x^5+\frac{1}{2} c \left (2 b \text{b1} c+4 b^2 \text{c1}+a c \text{c1}\right ) x^6+\frac{1}{7} c^2 (\text{b1} c+6 b \text{c1}) x^7+\frac{1}{8} c^3 \text{c1} x^8\\ \end{align*}

Mathematica [A]  time = 0.0329432, size = 167, normalized size = 1. \[ \frac{1}{4} x^4 \left (3 a^2 c \text{c1}+12 a b^2 \text{c1}+12 a b \text{b1} c+8 b^3 \text{b1}\right )+\frac{1}{2} a^2 x^2 (a \text{c1}+6 b \text{b1})+a^3 \text{b1} x+\frac{1}{5} x^5 \left (12 a b c \text{c1}+3 a \text{b1} c^2+12 b^2 \text{b1} c+8 b^3 \text{c1}\right )+\frac{1}{2} c x^6 \left (a c \text{c1}+4 b^2 \text{c1}+2 b \text{b1} c\right )+a x^3 \left (2 a b \text{c1}+a \text{b1} c+4 b^2 \text{b1}\right )+\frac{1}{7} c^2 x^7 (6 b \text{c1}+\text{b1} c)+\frac{1}{8} c^3 \text{c1} x^8 \]

Antiderivative was successfully verified.

[In]

Integrate[(b1 + c1*x)*(a + 2*b*x + c*x^2)^3,x]

[Out]

a^3*b1*x + (a^2*(6*b*b1 + a*c1)*x^2)/2 + a*(4*b^2*b1 + a*b1*c + 2*a*b*c1)*x^3 + ((8*b^3*b1 + 12*a*b*b1*c + 12*
a*b^2*c1 + 3*a^2*c*c1)*x^4)/4 + ((12*b^2*b1*c + 3*a*b1*c^2 + 8*b^3*c1 + 12*a*b*c*c1)*x^5)/5 + (c*(2*b*b1*c + 4
*b^2*c1 + a*c*c1)*x^6)/2 + (c^2*(b1*c + 6*b*c1)*x^7)/7 + (c^3*c1*x^8)/8

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Maple [A]  time = 0.001, size = 237, normalized size = 1.4 \begin{align*}{\frac{{c}^{3}{\it c1}\,{x}^{8}}{8}}+{\frac{ \left ( 6\,{\it c1}\,b{c}^{2}+{\it b1}\,{c}^{3} \right ){x}^{7}}{7}}+{\frac{ \left ( 6\,{\it b1}\,b{c}^{2}+{\it c1}\, \left ( a{c}^{2}+8\,{b}^{2}c+c \left ( 2\,ac+4\,{b}^{2} \right ) \right ) \right ){x}^{6}}{6}}+{\frac{ \left ({\it b1}\, \left ( a{c}^{2}+8\,{b}^{2}c+c \left ( 2\,ac+4\,{b}^{2} \right ) \right ) +{\it c1}\, \left ( 8\,abc+2\,b \left ( 2\,ac+4\,{b}^{2} \right ) \right ) \right ){x}^{5}}{5}}+{\frac{ \left ({\it b1}\, \left ( 8\,abc+2\,b \left ( 2\,ac+4\,{b}^{2} \right ) \right ) +{\it c1}\, \left ( a \left ( 2\,ac+4\,{b}^{2} \right ) +8\,{b}^{2}a+c{a}^{2} \right ) \right ){x}^{4}}{4}}+{\frac{ \left ({\it b1}\, \left ( a \left ( 2\,ac+4\,{b}^{2} \right ) +8\,{b}^{2}a+c{a}^{2} \right ) +6\,{\it c1}\,{a}^{2}b \right ){x}^{3}}{3}}+{\frac{ \left ({\it c1}\,{a}^{3}+6\,{\it b1}\,{a}^{2}b \right ){x}^{2}}{2}}+{a}^{3}{\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)^3,x)

[Out]

1/8*c^3*c1*x^8+1/7*(6*b*c^2*c1+b1*c^3)*x^7+1/6*(6*b1*b*c^2+c1*(a*c^2+8*b^2*c+c*(2*a*c+4*b^2)))*x^6+1/5*(b1*(a*
c^2+8*b^2*c+c*(2*a*c+4*b^2))+c1*(8*a*b*c+2*b*(2*a*c+4*b^2)))*x^5+1/4*(b1*(8*a*b*c+2*b*(2*a*c+4*b^2))+c1*(a*(2*
a*c+4*b^2)+8*b^2*a+c*a^2))*x^4+1/3*(b1*(a*(2*a*c+4*b^2)+8*b^2*a+c*a^2)+6*c1*a^2*b)*x^3+1/2*(a^3*c1+6*a^2*b*b1)
*x^2+a^3*b1*x

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Maxima [A]  time = 0.933209, size = 231, normalized size = 1.38 \begin{align*} \frac{1}{8} \, c^{3} c_{1} x^{8} + \frac{1}{7} \,{\left (b_{1} c^{3} + 6 \, b c^{2} c_{1}\right )} x^{7} + \frac{1}{2} \,{\left (2 \, b b_{1} c^{2} +{\left (4 \, b^{2} c + a c^{2}\right )} c_{1}\right )} x^{6} + \frac{1}{5} \,{\left (12 \, b^{2} b_{1} c + 3 \, a b_{1} c^{2} + 4 \,{\left (2 \, b^{3} + 3 \, a b c\right )} c_{1}\right )} x^{5} + a^{3} b_{1} x + \frac{1}{4} \,{\left (8 \, b^{3} b_{1} + 12 \, a b b_{1} c + 3 \,{\left (4 \, a b^{2} + a^{2} c\right )} c_{1}\right )} x^{4} +{\left (4 \, a b^{2} b_{1} + a^{2} b_{1} c + 2 \, a^{2} b c_{1}\right )} x^{3} + \frac{1}{2} \,{\left (6 \, a^{2} b b_{1} + a^{3} 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)^3,x, algorithm="maxima")

[Out]

1/8*c^3*c1*x^8 + 1/7*(b1*c^3 + 6*b*c^2*c1)*x^7 + 1/2*(2*b*b1*c^2 + (4*b^2*c + a*c^2)*c1)*x^6 + 1/5*(12*b^2*b1*
c + 3*a*b1*c^2 + 4*(2*b^3 + 3*a*b*c)*c1)*x^5 + a^3*b1*x + 1/4*(8*b^3*b1 + 12*a*b*b1*c + 3*(4*a*b^2 + a^2*c)*c1
)*x^4 + (4*a*b^2*b1 + a^2*b1*c + 2*a^2*b*c1)*x^3 + 1/2*(6*a^2*b*b1 + a^3*c1)*x^2

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Fricas [A]  time = 1.51364, size = 460, normalized size = 2.75 \begin{align*} \frac{1}{8} x^{8} c_{1} c^{3} + \frac{1}{7} x^{7} c^{3} b_{1} + \frac{6}{7} x^{7} c_{1} c^{2} b + x^{6} c^{2} b_{1} b + 2 x^{6} c_{1} c b^{2} + \frac{1}{2} x^{6} c_{1} c^{2} a + \frac{12}{5} x^{5} c b_{1} b^{2} + \frac{8}{5} x^{5} c_{1} b^{3} + \frac{3}{5} x^{5} c^{2} b_{1} a + \frac{12}{5} x^{5} c_{1} c b a + 2 x^{4} b_{1} b^{3} + 3 x^{4} c b_{1} b a + 3 x^{4} c_{1} b^{2} a + \frac{3}{4} x^{4} c_{1} c a^{2} + 4 x^{3} b_{1} b^{2} a + x^{3} c b_{1} a^{2} + 2 x^{3} c_{1} b a^{2} + 3 x^{2} b_{1} b a^{2} + \frac{1}{2} x^{2} c_{1} a^{3} + x b_{1} a^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c1*x+b1)*(c*x^2+2*b*x+a)^3,x, algorithm="fricas")

[Out]

1/8*x^8*c1*c^3 + 1/7*x^7*c^3*b1 + 6/7*x^7*c1*c^2*b + x^6*c^2*b1*b + 2*x^6*c1*c*b^2 + 1/2*x^6*c1*c^2*a + 12/5*x
^5*c*b1*b^2 + 8/5*x^5*c1*b^3 + 3/5*x^5*c^2*b1*a + 12/5*x^5*c1*c*b*a + 2*x^4*b1*b^3 + 3*x^4*c*b1*b*a + 3*x^4*c1
*b^2*a + 3/4*x^4*c1*c*a^2 + 4*x^3*b1*b^2*a + x^3*c*b1*a^2 + 2*x^3*c1*b*a^2 + 3*x^2*b1*b*a^2 + 1/2*x^2*c1*a^3 +
x*b1*a^3

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Sympy [A]  time = 0.087628, size = 189, normalized size = 1.13 \begin{align*} a^{3} b_{1} x + \frac{c^{3} c_{1} x^{8}}{8} + x^{7} \left (\frac{6 b c^{2} c_{1}}{7} + \frac{b_{1} c^{3}}{7}\right ) + x^{6} \left (\frac{a c^{2} c_{1}}{2} + 2 b^{2} c c_{1} + b b_{1} c^{2}\right ) + x^{5} \left (\frac{12 a b c c_{1}}{5} + \frac{3 a b_{1} c^{2}}{5} + \frac{8 b^{3} c_{1}}{5} + \frac{12 b^{2} b_{1} c}{5}\right ) + x^{4} \left (\frac{3 a^{2} c c_{1}}{4} + 3 a b^{2} c_{1} + 3 a b b_{1} c + 2 b^{3} b_{1}\right ) + x^{3} \left (2 a^{2} b c_{1} + a^{2} b_{1} c + 4 a b^{2} b_{1}\right ) + x^{2} \left (\frac{a^{3} c_{1}}{2} + 3 a^{2} 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)**3,x)

[Out]

a**3*b1*x + c**3*c1*x**8/8 + x**7*(6*b*c**2*c1/7 + b1*c**3/7) + x**6*(a*c**2*c1/2 + 2*b**2*c*c1 + b*b1*c**2) +
x**5*(12*a*b*c*c1/5 + 3*a*b1*c**2/5 + 8*b**3*c1/5 + 12*b**2*b1*c/5) + x**4*(3*a**2*c*c1/4 + 3*a*b**2*c1 + 3*a
*b*b1*c + 2*b**3*b1) + x**3*(2*a**2*b*c1 + a**2*b1*c + 4*a*b**2*b1) + x**2*(a**3*c1/2 + 3*a**2*b*b1)

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Giac [A]  time = 1.06303, size = 254, normalized size = 1.52 \begin{align*} \frac{1}{8} \, c^{3} c_{1} x^{8} + \frac{1}{7} \, b_{1} c^{3} x^{7} + \frac{6}{7} \, b c^{2} c_{1} x^{7} + b b_{1} c^{2} x^{6} + 2 \, b^{2} c c_{1} x^{6} + \frac{1}{2} \, a c^{2} c_{1} x^{6} + \frac{12}{5} \, b^{2} b_{1} c x^{5} + \frac{3}{5} \, a b_{1} c^{2} x^{5} + \frac{8}{5} \, b^{3} c_{1} x^{5} + \frac{12}{5} \, a b c c_{1} x^{5} + 2 \, b^{3} b_{1} x^{4} + 3 \, a b b_{1} c x^{4} + 3 \, a b^{2} c_{1} x^{4} + \frac{3}{4} \, a^{2} c c_{1} x^{4} + 4 \, a b^{2} b_{1} x^{3} + a^{2} b_{1} c x^{3} + 2 \, a^{2} b c_{1} x^{3} + 3 \, a^{2} b b_{1} x^{2} + \frac{1}{2} \, a^{3} c_{1} x^{2} + a^{3} 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)^3,x, algorithm="giac")

[Out]

1/8*c^3*c1*x^8 + 1/7*b1*c^3*x^7 + 6/7*b*c^2*c1*x^7 + b*b1*c^2*x^6 + 2*b^2*c*c1*x^6 + 1/2*a*c^2*c1*x^6 + 12/5*b
^2*b1*c*x^5 + 3/5*a*b1*c^2*x^5 + 8/5*b^3*c1*x^5 + 12/5*a*b*c*c1*x^5 + 2*b^3*b1*x^4 + 3*a*b*b1*c*x^4 + 3*a*b^2*
c1*x^4 + 3/4*a^2*c*c1*x^4 + 4*a*b^2*b1*x^3 + a^2*b1*c*x^3 + 2*a^2*b*c1*x^3 + 3*a^2*b*b1*x^2 + 1/2*a^3*c1*x^2 +
a^3*b1*x