3.192 \(\int (\text{b1}+\text{c1} x) (a+2 b x+c x^2)^4 \, dx\)

Optimal. Leaf size=263 \[ \frac{1}{3} x^6 \left (3 a^2 c^2 \text{c1}+24 a b^2 c \text{c1}+12 a b \text{b1} c^2+16 b^3 \text{b1} c+8 b^4 \text{c1}\right )+\frac{2}{5} x^5 \left (12 a^2 b c \text{c1}+3 a^2 \text{b1} c^2+24 a b^2 \text{b1} c+16 a b^3 \text{c1}+8 b^4 \text{b1}\right )+a x^4 \left (a^2 c \text{c1}+6 a b^2 \text{c1}+6 a b \text{b1} c+8 b^3 \text{b1}\right )+\frac{4}{3} a^2 x^3 \left (2 a b \text{c1}+a \text{b1} c+6 b^2 \text{b1}\right )+\frac{1}{2} a^3 x^2 (a \text{c1}+8 b \text{b1})+a^4 \text{b1} x+\frac{1}{2} c^2 x^8 \left (a c \text{c1}+6 b^2 \text{c1}+2 b \text{b1} c\right )+\frac{4}{7} c x^7 \left (6 a b c \text{c1}+a \text{b1} c^2+6 b^2 \text{b1} c+8 b^3 \text{c1}\right )+\frac{1}{9} c^3 x^9 (8 b \text{c1}+\text{b1} c)+\frac{1}{10} c^4 \text{c1} x^{10} \]

[Out]

a^4*b1*x + (a^3*(8*b*b1 + a*c1)*x^2)/2 + (4*a^2*(6*b^2*b1 + a*b1*c + 2*a*b*c1)*x^3)/3 + a*(8*b^3*b1 + 6*a*b*b1
*c + 6*a*b^2*c1 + a^2*c*c1)*x^4 + (2*(8*b^4*b1 + 24*a*b^2*b1*c + 3*a^2*b1*c^2 + 16*a*b^3*c1 + 12*a^2*b*c*c1)*x
^5)/5 + ((16*b^3*b1*c + 12*a*b*b1*c^2 + 8*b^4*c1 + 24*a*b^2*c*c1 + 3*a^2*c^2*c1)*x^6)/3 + (4*c*(6*b^2*b1*c + a
*b1*c^2 + 8*b^3*c1 + 6*a*b*c*c1)*x^7)/7 + (c^2*(2*b*b1*c + 6*b^2*c1 + a*c*c1)*x^8)/2 + (c^3*(b1*c + 8*b*c1)*x^
9)/9 + (c^4*c1*x^10)/10

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Rubi [A]  time = 0.337079, antiderivative size = 263, 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} \[ \frac{1}{3} x^6 \left (3 a^2 c^2 \text{c1}+24 a b^2 c \text{c1}+12 a b \text{b1} c^2+16 b^3 \text{b1} c+8 b^4 \text{c1}\right )+\frac{2}{5} x^5 \left (12 a^2 b c \text{c1}+3 a^2 \text{b1} c^2+24 a b^2 \text{b1} c+16 a b^3 \text{c1}+8 b^4 \text{b1}\right )+a x^4 \left (a^2 c \text{c1}+6 a b^2 \text{c1}+6 a b \text{b1} c+8 b^3 \text{b1}\right )+\frac{4}{3} a^2 x^3 \left (2 a b \text{c1}+a \text{b1} c+6 b^2 \text{b1}\right )+\frac{1}{2} a^3 x^2 (a \text{c1}+8 b \text{b1})+a^4 \text{b1} x+\frac{1}{2} c^2 x^8 \left (a c \text{c1}+6 b^2 \text{c1}+2 b \text{b1} c\right )+\frac{4}{7} c x^7 \left (6 a b c \text{c1}+a \text{b1} c^2+6 b^2 \text{b1} c+8 b^3 \text{c1}\right )+\frac{1}{9} c^3 x^9 (8 b \text{c1}+\text{b1} c)+\frac{1}{10} c^4 \text{c1} x^{10} \]

Antiderivative was successfully verified.

[In]

Int[(b1 + c1*x)*(a + 2*b*x + c*x^2)^4,x]

[Out]

a^4*b1*x + (a^3*(8*b*b1 + a*c1)*x^2)/2 + (4*a^2*(6*b^2*b1 + a*b1*c + 2*a*b*c1)*x^3)/3 + a*(8*b^3*b1 + 6*a*b*b1
*c + 6*a*b^2*c1 + a^2*c*c1)*x^4 + (2*(8*b^4*b1 + 24*a*b^2*b1*c + 3*a^2*b1*c^2 + 16*a*b^3*c1 + 12*a^2*b*c*c1)*x
^5)/5 + ((16*b^3*b1*c + 12*a*b*b1*c^2 + 8*b^4*c1 + 24*a*b^2*c*c1 + 3*a^2*c^2*c1)*x^6)/3 + (4*c*(6*b^2*b1*c + a
*b1*c^2 + 8*b^3*c1 + 6*a*b*c*c1)*x^7)/7 + (c^2*(2*b*b1*c + 6*b^2*c1 + a*c*c1)*x^8)/2 + (c^3*(b1*c + 8*b*c1)*x^
9)/9 + (c^4*c1*x^10)/10

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 )^4 \, dx &=\int \left (a^4 \text{b1}+a^3 (8 b \text{b1}+a \text{c1}) x+4 a^2 \left (6 b^2 \text{b1}+a \text{b1} c+2 a b \text{c1}\right ) x^2+4 a \left (8 b^3 \text{b1}+6 a b \text{b1} c+6 a b^2 \text{c1}+a^2 c \text{c1}\right ) x^3+2 \left (8 b^4 \text{b1}+24 a b^2 \text{b1} c+3 a^2 \text{b1} c^2+16 a b^3 \text{c1}+12 a^2 b c \text{c1}\right ) x^4+2 \left (16 b^3 \text{b1} c+12 a b \text{b1} c^2+8 b^4 \text{c1}+24 a b^2 c \text{c1}+3 a^2 c^2 \text{c1}\right ) x^5+4 c \left (6 b^2 \text{b1} c+a \text{b1} c^2+8 b^3 \text{c1}+6 a b c \text{c1}\right ) x^6+4 c^2 \left (2 b \text{b1} c+6 b^2 \text{c1}+a c \text{c1}\right ) x^7+c^3 (\text{b1} c+8 b \text{c1}) x^8+c^4 \text{c1} x^9\right ) \, dx\\ &=a^4 \text{b1} x+\frac{1}{2} a^3 (8 b \text{b1}+a \text{c1}) x^2+\frac{4}{3} a^2 \left (6 b^2 \text{b1}+a \text{b1} c+2 a b \text{c1}\right ) x^3+a \left (8 b^3 \text{b1}+6 a b \text{b1} c+6 a b^2 \text{c1}+a^2 c \text{c1}\right ) x^4+\frac{2}{5} \left (8 b^4 \text{b1}+24 a b^2 \text{b1} c+3 a^2 \text{b1} c^2+16 a b^3 \text{c1}+12 a^2 b c \text{c1}\right ) x^5+\frac{1}{3} \left (16 b^3 \text{b1} c+12 a b \text{b1} c^2+8 b^4 \text{c1}+24 a b^2 c \text{c1}+3 a^2 c^2 \text{c1}\right ) x^6+\frac{4}{7} c \left (6 b^2 \text{b1} c+a \text{b1} c^2+8 b^3 \text{c1}+6 a b c \text{c1}\right ) x^7+\frac{1}{2} c^2 \left (2 b \text{b1} c+6 b^2 \text{c1}+a c \text{c1}\right ) x^8+\frac{1}{9} c^3 (\text{b1} c+8 b \text{c1}) x^9+\frac{1}{10} c^4 \text{c1} x^{10}\\ \end{align*}

Mathematica [A]  time = 0.0594917, size = 263, normalized size = 1. \[ \frac{1}{3} x^6 \left (3 a^2 c^2 \text{c1}+24 a b^2 c \text{c1}+12 a b \text{b1} c^2+16 b^3 \text{b1} c+8 b^4 \text{c1}\right )+\frac{2}{5} x^5 \left (12 a^2 b c \text{c1}+3 a^2 \text{b1} c^2+24 a b^2 \text{b1} c+16 a b^3 \text{c1}+8 b^4 \text{b1}\right )+a x^4 \left (a^2 c \text{c1}+6 a b^2 \text{c1}+6 a b \text{b1} c+8 b^3 \text{b1}\right )+\frac{4}{3} a^2 x^3 \left (2 a b \text{c1}+a \text{b1} c+6 b^2 \text{b1}\right )+\frac{1}{2} a^3 x^2 (a \text{c1}+8 b \text{b1})+a^4 \text{b1} x+\frac{1}{2} c^2 x^8 \left (a c \text{c1}+6 b^2 \text{c1}+2 b \text{b1} c\right )+\frac{4}{7} c x^7 \left (6 a b c \text{c1}+a \text{b1} c^2+6 b^2 \text{b1} c+8 b^3 \text{c1}\right )+\frac{1}{9} c^3 x^9 (8 b \text{c1}+\text{b1} c)+\frac{1}{10} c^4 \text{c1} x^{10} \]

Antiderivative was successfully verified.

[In]

Integrate[(b1 + c1*x)*(a + 2*b*x + c*x^2)^4,x]

[Out]

a^4*b1*x + (a^3*(8*b*b1 + a*c1)*x^2)/2 + (4*a^2*(6*b^2*b1 + a*b1*c + 2*a*b*c1)*x^3)/3 + a*(8*b^3*b1 + 6*a*b*b1
*c + 6*a*b^2*c1 + a^2*c*c1)*x^4 + (2*(8*b^4*b1 + 24*a*b^2*b1*c + 3*a^2*b1*c^2 + 16*a*b^3*c1 + 12*a^2*b*c*c1)*x
^5)/5 + ((16*b^3*b1*c + 12*a*b*b1*c^2 + 8*b^4*c1 + 24*a*b^2*c*c1 + 3*a^2*c^2*c1)*x^6)/3 + (4*c*(6*b^2*b1*c + a
*b1*c^2 + 8*b^3*c1 + 6*a*b*c*c1)*x^7)/7 + (c^2*(2*b*b1*c + 6*b^2*c1 + a*c*c1)*x^8)/2 + (c^3*(b1*c + 8*b*c1)*x^
9)/9 + (c^4*c1*x^10)/10

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Maple [A]  time = 0.001, size = 363, normalized size = 1.4 \begin{align*}{\frac{{c}^{4}{\it c1}\,{x}^{10}}{10}}+{\frac{ \left ( 8\,{\it c1}\,b{c}^{3}+{\it b1}\,{c}^{4} \right ){x}^{9}}{9}}+{\frac{ \left ( 8\,{\it b1}\,b{c}^{3}+{\it c1}\, \left ( 2\, \left ( 2\,ac+4\,{b}^{2} \right ){c}^{2}+16\,{b}^{2}{c}^{2} \right ) \right ){x}^{8}}{8}}+{\frac{ \left ({\it b1}\, \left ( 2\, \left ( 2\,ac+4\,{b}^{2} \right ){c}^{2}+16\,{b}^{2}{c}^{2} \right ) +{\it c1}\, \left ( 8\,ab{c}^{2}+8\, \left ( 2\,ac+4\,{b}^{2} \right ) bc \right ) \right ){x}^{7}}{7}}+{\frac{ \left ({\it b1}\, \left ( 8\,ab{c}^{2}+8\, \left ( 2\,ac+4\,{b}^{2} \right ) bc \right ) +{\it c1}\, \left ( 2\,{a}^{2}{c}^{2}+32\,a{b}^{2}c+ \left ( 2\,ac+4\,{b}^{2} \right ) ^{2} \right ) \right ){x}^{6}}{6}}+{\frac{ \left ({\it b1}\, \left ( 2\,{a}^{2}{c}^{2}+32\,a{b}^{2}c+ \left ( 2\,ac+4\,{b}^{2} \right ) ^{2} \right ) +{\it c1}\, \left ( 8\,{a}^{2}bc+8\,ab \left ( 2\,ac+4\,{b}^{2} \right ) \right ) \right ){x}^{5}}{5}}+{\frac{ \left ({\it b1}\, \left ( 8\,{a}^{2}bc+8\,ab \left ( 2\,ac+4\,{b}^{2} \right ) \right ) +{\it c1}\, \left ( 2\,{a}^{2} \left ( 2\,ac+4\,{b}^{2} \right ) +16\,{b}^{2}{a}^{2} \right ) \right ){x}^{4}}{4}}+{\frac{ \left ({\it b1}\, \left ( 2\,{a}^{2} \left ( 2\,ac+4\,{b}^{2} \right ) +16\,{b}^{2}{a}^{2} \right ) +8\,{\it c1}\,{a}^{3}b \right ){x}^{3}}{3}}+{\frac{ \left ({\it c1}\,{a}^{4}+8\,{\it b1}\,{a}^{3}b \right ){x}^{2}}{2}}+{a}^{4}{\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)^4,x)

[Out]

1/10*c^4*c1*x^10+1/9*(8*b*c^3*c1+b1*c^4)*x^9+1/8*(8*b1*b*c^3+c1*(2*(2*a*c+4*b^2)*c^2+16*b^2*c^2))*x^8+1/7*(b1*
(2*(2*a*c+4*b^2)*c^2+16*b^2*c^2)+c1*(8*a*b*c^2+8*(2*a*c+4*b^2)*b*c))*x^7+1/6*(b1*(8*a*b*c^2+8*(2*a*c+4*b^2)*b*
c)+c1*(2*a^2*c^2+32*a*b^2*c+(2*a*c+4*b^2)^2))*x^6+1/5*(b1*(2*a^2*c^2+32*a*b^2*c+(2*a*c+4*b^2)^2)+c1*(8*a^2*b*c
+8*a*b*(2*a*c+4*b^2)))*x^5+1/4*(b1*(8*a^2*b*c+8*a*b*(2*a*c+4*b^2))+c1*(2*a^2*(2*a*c+4*b^2)+16*b^2*a^2))*x^4+1/
3*(b1*(2*a^2*(2*a*c+4*b^2)+16*b^2*a^2)+8*c1*a^3*b)*x^3+1/2*(a^4*c1+8*a^3*b*b1)*x^2+a^4*b1*x

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Maxima [A]  time = 0.936922, size = 369, normalized size = 1.4 \begin{align*} \frac{1}{10} \, c^{4} c_{1} x^{10} + \frac{1}{9} \,{\left (b_{1} c^{4} + 8 \, b c^{3} c_{1}\right )} x^{9} + \frac{1}{2} \,{\left (2 \, b b_{1} c^{3} +{\left (6 \, b^{2} c^{2} + a c^{3}\right )} c_{1}\right )} x^{8} + \frac{4}{7} \,{\left (6 \, b^{2} b_{1} c^{2} + a b_{1} c^{3} + 2 \,{\left (4 \, b^{3} c + 3 \, a b c^{2}\right )} c_{1}\right )} x^{7} + \frac{1}{3} \,{\left (16 \, b^{3} b_{1} c + 12 \, a b b_{1} c^{2} +{\left (8 \, b^{4} + 24 \, a b^{2} c + 3 \, a^{2} c^{2}\right )} c_{1}\right )} x^{6} + a^{4} b_{1} x + \frac{2}{5} \,{\left (8 \, b^{4} b_{1} + 24 \, a b^{2} b_{1} c + 3 \, a^{2} b_{1} c^{2} + 4 \,{\left (4 \, a b^{3} + 3 \, a^{2} b c\right )} c_{1}\right )} x^{5} +{\left (8 \, a b^{3} b_{1} + 6 \, a^{2} b b_{1} c +{\left (6 \, a^{2} b^{2} + a^{3} c\right )} c_{1}\right )} x^{4} + \frac{4}{3} \,{\left (6 \, a^{2} b^{2} b_{1} + a^{3} b_{1} c + 2 \, a^{3} b c_{1}\right )} x^{3} + \frac{1}{2} \,{\left (8 \, a^{3} b b_{1} + a^{4} 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)^4,x, algorithm="maxima")

[Out]

1/10*c^4*c1*x^10 + 1/9*(b1*c^4 + 8*b*c^3*c1)*x^9 + 1/2*(2*b*b1*c^3 + (6*b^2*c^2 + a*c^3)*c1)*x^8 + 4/7*(6*b^2*
b1*c^2 + a*b1*c^3 + 2*(4*b^3*c + 3*a*b*c^2)*c1)*x^7 + 1/3*(16*b^3*b1*c + 12*a*b*b1*c^2 + (8*b^4 + 24*a*b^2*c +
3*a^2*c^2)*c1)*x^6 + a^4*b1*x + 2/5*(8*b^4*b1 + 24*a*b^2*b1*c + 3*a^2*b1*c^2 + 4*(4*a*b^3 + 3*a^2*b*c)*c1)*x^
5 + (8*a*b^3*b1 + 6*a^2*b*b1*c + (6*a^2*b^2 + a^3*c)*c1)*x^4 + 4/3*(6*a^2*b^2*b1 + a^3*b1*c + 2*a^3*b*c1)*x^3
+ 1/2*(8*a^3*b*b1 + a^4*c1)*x^2

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Fricas [A]  time = 1.5445, size = 752, normalized size = 2.86 \begin{align*} \frac{1}{10} x^{10} c_{1} c^{4} + \frac{1}{9} x^{9} c^{4} b_{1} + \frac{8}{9} x^{9} c_{1} c^{3} b + x^{8} c^{3} b_{1} b + 3 x^{8} c_{1} c^{2} b^{2} + \frac{1}{2} x^{8} c_{1} c^{3} a + \frac{24}{7} x^{7} c^{2} b_{1} b^{2} + \frac{32}{7} x^{7} c_{1} c b^{3} + \frac{4}{7} x^{7} c^{3} b_{1} a + \frac{24}{7} x^{7} c_{1} c^{2} b a + \frac{16}{3} x^{6} c b_{1} b^{3} + \frac{8}{3} x^{6} c_{1} b^{4} + 4 x^{6} c^{2} b_{1} b a + 8 x^{6} c_{1} c b^{2} a + x^{6} c_{1} c^{2} a^{2} + \frac{16}{5} x^{5} b_{1} b^{4} + \frac{48}{5} x^{5} c b_{1} b^{2} a + \frac{32}{5} x^{5} c_{1} b^{3} a + \frac{6}{5} x^{5} c^{2} b_{1} a^{2} + \frac{24}{5} x^{5} c_{1} c b a^{2} + 8 x^{4} b_{1} b^{3} a + 6 x^{4} c b_{1} b a^{2} + 6 x^{4} c_{1} b^{2} a^{2} + x^{4} c_{1} c a^{3} + 8 x^{3} b_{1} b^{2} a^{2} + \frac{4}{3} x^{3} c b_{1} a^{3} + \frac{8}{3} x^{3} c_{1} b a^{3} + 4 x^{2} b_{1} b a^{3} + \frac{1}{2} x^{2} c_{1} a^{4} + x b_{1} a^{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c1*x+b1)*(c*x^2+2*b*x+a)^4,x, algorithm="fricas")

[Out]

1/10*x^10*c1*c^4 + 1/9*x^9*c^4*b1 + 8/9*x^9*c1*c^3*b + x^8*c^3*b1*b + 3*x^8*c1*c^2*b^2 + 1/2*x^8*c1*c^3*a + 24
/7*x^7*c^2*b1*b^2 + 32/7*x^7*c1*c*b^3 + 4/7*x^7*c^3*b1*a + 24/7*x^7*c1*c^2*b*a + 16/3*x^6*c*b1*b^3 + 8/3*x^6*c
1*b^4 + 4*x^6*c^2*b1*b*a + 8*x^6*c1*c*b^2*a + x^6*c1*c^2*a^2 + 16/5*x^5*b1*b^4 + 48/5*x^5*c*b1*b^2*a + 32/5*x^
5*c1*b^3*a + 6/5*x^5*c^2*b1*a^2 + 24/5*x^5*c1*c*b*a^2 + 8*x^4*b1*b^3*a + 6*x^4*c*b1*b*a^2 + 6*x^4*c1*b^2*a^2 +
x^4*c1*c*a^3 + 8*x^3*b1*b^2*a^2 + 4/3*x^3*c*b1*a^3 + 8/3*x^3*c1*b*a^3 + 4*x^2*b1*b*a^3 + 1/2*x^2*c1*a^4 + x*b
1*a^4

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Sympy [A]  time = 0.108007, size = 313, normalized size = 1.19 \begin{align*} a^{4} b_{1} x + \frac{c^{4} c_{1} x^{10}}{10} + x^{9} \left (\frac{8 b c^{3} c_{1}}{9} + \frac{b_{1} c^{4}}{9}\right ) + x^{8} \left (\frac{a c^{3} c_{1}}{2} + 3 b^{2} c^{2} c_{1} + b b_{1} c^{3}\right ) + x^{7} \left (\frac{24 a b c^{2} c_{1}}{7} + \frac{4 a b_{1} c^{3}}{7} + \frac{32 b^{3} c c_{1}}{7} + \frac{24 b^{2} b_{1} c^{2}}{7}\right ) + x^{6} \left (a^{2} c^{2} c_{1} + 8 a b^{2} c c_{1} + 4 a b b_{1} c^{2} + \frac{8 b^{4} c_{1}}{3} + \frac{16 b^{3} b_{1} c}{3}\right ) + x^{5} \left (\frac{24 a^{2} b c c_{1}}{5} + \frac{6 a^{2} b_{1} c^{2}}{5} + \frac{32 a b^{3} c_{1}}{5} + \frac{48 a b^{2} b_{1} c}{5} + \frac{16 b^{4} b_{1}}{5}\right ) + x^{4} \left (a^{3} c c_{1} + 6 a^{2} b^{2} c_{1} + 6 a^{2} b b_{1} c + 8 a b^{3} b_{1}\right ) + x^{3} \left (\frac{8 a^{3} b c_{1}}{3} + \frac{4 a^{3} b_{1} c}{3} + 8 a^{2} b^{2} b_{1}\right ) + x^{2} \left (\frac{a^{4} c_{1}}{2} + 4 a^{3} 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)**4,x)

[Out]

a**4*b1*x + c**4*c1*x**10/10 + x**9*(8*b*c**3*c1/9 + b1*c**4/9) + x**8*(a*c**3*c1/2 + 3*b**2*c**2*c1 + b*b1*c*
*3) + x**7*(24*a*b*c**2*c1/7 + 4*a*b1*c**3/7 + 32*b**3*c*c1/7 + 24*b**2*b1*c**2/7) + x**6*(a**2*c**2*c1 + 8*a*
b**2*c*c1 + 4*a*b*b1*c**2 + 8*b**4*c1/3 + 16*b**3*b1*c/3) + x**5*(24*a**2*b*c*c1/5 + 6*a**2*b1*c**2/5 + 32*a*b
**3*c1/5 + 48*a*b**2*b1*c/5 + 16*b**4*b1/5) + x**4*(a**3*c*c1 + 6*a**2*b**2*c1 + 6*a**2*b*b1*c + 8*a*b**3*b1)
+ x**3*(8*a**3*b*c1/3 + 4*a**3*b1*c/3 + 8*a**2*b**2*b1) + x**2*(a**4*c1/2 + 4*a**3*b*b1)

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Giac [A]  time = 1.05069, size = 414, normalized size = 1.57 \begin{align*} \frac{1}{10} \, c^{4} c_{1} x^{10} + \frac{1}{9} \, b_{1} c^{4} x^{9} + \frac{8}{9} \, b c^{3} c_{1} x^{9} + b b_{1} c^{3} x^{8} + 3 \, b^{2} c^{2} c_{1} x^{8} + \frac{1}{2} \, a c^{3} c_{1} x^{8} + \frac{24}{7} \, b^{2} b_{1} c^{2} x^{7} + \frac{4}{7} \, a b_{1} c^{3} x^{7} + \frac{32}{7} \, b^{3} c c_{1} x^{7} + \frac{24}{7} \, a b c^{2} c_{1} x^{7} + \frac{16}{3} \, b^{3} b_{1} c x^{6} + 4 \, a b b_{1} c^{2} x^{6} + \frac{8}{3} \, b^{4} c_{1} x^{6} + 8 \, a b^{2} c c_{1} x^{6} + a^{2} c^{2} c_{1} x^{6} + \frac{16}{5} \, b^{4} b_{1} x^{5} + \frac{48}{5} \, a b^{2} b_{1} c x^{5} + \frac{6}{5} \, a^{2} b_{1} c^{2} x^{5} + \frac{32}{5} \, a b^{3} c_{1} x^{5} + \frac{24}{5} \, a^{2} b c c_{1} x^{5} + 8 \, a b^{3} b_{1} x^{4} + 6 \, a^{2} b b_{1} c x^{4} + 6 \, a^{2} b^{2} c_{1} x^{4} + a^{3} c c_{1} x^{4} + 8 \, a^{2} b^{2} b_{1} x^{3} + \frac{4}{3} \, a^{3} b_{1} c x^{3} + \frac{8}{3} \, a^{3} b c_{1} x^{3} + 4 \, a^{3} b b_{1} x^{2} + \frac{1}{2} \, a^{4} c_{1} x^{2} + a^{4} 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)^4,x, algorithm="giac")

[Out]

1/10*c^4*c1*x^10 + 1/9*b1*c^4*x^9 + 8/9*b*c^3*c1*x^9 + b*b1*c^3*x^8 + 3*b^2*c^2*c1*x^8 + 1/2*a*c^3*c1*x^8 + 24
/7*b^2*b1*c^2*x^7 + 4/7*a*b1*c^3*x^7 + 32/7*b^3*c*c1*x^7 + 24/7*a*b*c^2*c1*x^7 + 16/3*b^3*b1*c*x^6 + 4*a*b*b1*
c^2*x^6 + 8/3*b^4*c1*x^6 + 8*a*b^2*c*c1*x^6 + a^2*c^2*c1*x^6 + 16/5*b^4*b1*x^5 + 48/5*a*b^2*b1*c*x^5 + 6/5*a^2
*b1*c^2*x^5 + 32/5*a*b^3*c1*x^5 + 24/5*a^2*b*c*c1*x^5 + 8*a*b^3*b1*x^4 + 6*a^2*b*b1*c*x^4 + 6*a^2*b^2*c1*x^4 +
a^3*c*c1*x^4 + 8*a^2*b^2*b1*x^3 + 4/3*a^3*b1*c*x^3 + 8/3*a^3*b*c1*x^3 + 4*a^3*b*b1*x^2 + 1/2*a^4*c1*x^2 + a^4
*b1*x