3.1517 \(\int (b+2 c x) (d+e x) (a+b x+c x^2)^3 \, dx\)

Optimal. Leaf size=251 \[ \frac{1}{5} x^5 \left (6 a^2 c^2 e+12 a b^2 c e+15 a b c^2 d+5 b^3 c d+b^4 e\right )+\frac{1}{4} x^4 \left (9 a^2 b c e+6 a^2 c^2 d+12 a b^2 c d+3 a b^3 e+b^4 d\right )+\frac{1}{3} a x^3 \left (2 a^2 c e+3 a b^2 e+9 a b c d+3 b^3 d\right )+\frac{1}{2} a^2 x^2 \left (a b e+2 a c d+3 b^2 d\right )+a^3 b d x+\frac{1}{7} c^2 x^7 \left (6 a c e+9 b^2 e+7 b c d\right )+\frac{1}{6} c x^6 \left (15 a b c e+6 a c^2 d+9 b^2 c d+5 b^3 e\right )+\frac{1}{8} c^3 x^8 (7 b e+2 c d)+\frac{2}{9} c^4 e x^9 \]

[Out]

a^3*b*d*x + (a^2*(3*b^2*d + 2*a*c*d + a*b*e)*x^2)/2 + (a*(3*b^3*d + 9*a*b*c*d + 3*a*b^2*e + 2*a^2*c*e)*x^3)/3
+ ((b^4*d + 12*a*b^2*c*d + 6*a^2*c^2*d + 3*a*b^3*e + 9*a^2*b*c*e)*x^4)/4 + ((5*b^3*c*d + 15*a*b*c^2*d + b^4*e
+ 12*a*b^2*c*e + 6*a^2*c^2*e)*x^5)/5 + (c*(9*b^2*c*d + 6*a*c^2*d + 5*b^3*e + 15*a*b*c*e)*x^6)/6 + (c^2*(7*b*c*
d + 9*b^2*e + 6*a*c*e)*x^7)/7 + (c^3*(2*c*d + 7*b*e)*x^8)/8 + (2*c^4*e*x^9)/9

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Rubi [A]  time = 0.229176, antiderivative size = 251, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042, Rules used = {771} \[ \frac{1}{5} x^5 \left (6 a^2 c^2 e+12 a b^2 c e+15 a b c^2 d+5 b^3 c d+b^4 e\right )+\frac{1}{4} x^4 \left (9 a^2 b c e+6 a^2 c^2 d+12 a b^2 c d+3 a b^3 e+b^4 d\right )+\frac{1}{3} a x^3 \left (2 a^2 c e+3 a b^2 e+9 a b c d+3 b^3 d\right )+\frac{1}{2} a^2 x^2 \left (a b e+2 a c d+3 b^2 d\right )+a^3 b d x+\frac{1}{7} c^2 x^7 \left (6 a c e+9 b^2 e+7 b c d\right )+\frac{1}{6} c x^6 \left (15 a b c e+6 a c^2 d+9 b^2 c d+5 b^3 e\right )+\frac{1}{8} c^3 x^8 (7 b e+2 c d)+\frac{2}{9} c^4 e x^9 \]

Antiderivative was successfully verified.

[In]

Int[(b + 2*c*x)*(d + e*x)*(a + b*x + c*x^2)^3,x]

[Out]

a^3*b*d*x + (a^2*(3*b^2*d + 2*a*c*d + a*b*e)*x^2)/2 + (a*(3*b^3*d + 9*a*b*c*d + 3*a*b^2*e + 2*a^2*c*e)*x^3)/3
+ ((b^4*d + 12*a*b^2*c*d + 6*a^2*c^2*d + 3*a*b^3*e + 9*a^2*b*c*e)*x^4)/4 + ((5*b^3*c*d + 15*a*b*c^2*d + b^4*e
+ 12*a*b^2*c*e + 6*a^2*c^2*e)*x^5)/5 + (c*(9*b^2*c*d + 6*a*c^2*d + 5*b^3*e + 15*a*b*c*e)*x^6)/6 + (c^2*(7*b*c*
d + 9*b^2*e + 6*a*c*e)*x^7)/7 + (c^3*(2*c*d + 7*b*e)*x^8)/8 + (2*c^4*e*x^9)/9

Rule 771

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> In
t[ExpandIntegrand[(d + e*x)^m*(f + g*x)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && N
eQ[b^2 - 4*a*c, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

\begin{align*} \int (b+2 c x) (d+e x) \left (a+b x+c x^2\right )^3 \, dx &=\int \left (a^3 b d+a^2 \left (3 b^2 d+2 a c d+a b e\right ) x+a \left (3 b^3 d+9 a b c d+3 a b^2 e+2 a^2 c e\right ) x^2+\left (b^4 d+12 a b^2 c d+6 a^2 c^2 d+3 a b^3 e+9 a^2 b c e\right ) x^3+\left (5 b^3 c d+15 a b c^2 d+b^4 e+12 a b^2 c e+6 a^2 c^2 e\right ) x^4+c \left (9 b^2 c d+6 a c^2 d+5 b^3 e+15 a b c e\right ) x^5+c^2 \left (7 b c d+9 b^2 e+6 a c e\right ) x^6+c^3 (2 c d+7 b e) x^7+2 c^4 e x^8\right ) \, dx\\ &=a^3 b d x+\frac{1}{2} a^2 \left (3 b^2 d+2 a c d+a b e\right ) x^2+\frac{1}{3} a \left (3 b^3 d+9 a b c d+3 a b^2 e+2 a^2 c e\right ) x^3+\frac{1}{4} \left (b^4 d+12 a b^2 c d+6 a^2 c^2 d+3 a b^3 e+9 a^2 b c e\right ) x^4+\frac{1}{5} \left (5 b^3 c d+15 a b c^2 d+b^4 e+12 a b^2 c e+6 a^2 c^2 e\right ) x^5+\frac{1}{6} c \left (9 b^2 c d+6 a c^2 d+5 b^3 e+15 a b c e\right ) x^6+\frac{1}{7} c^2 \left (7 b c d+9 b^2 e+6 a c e\right ) x^7+\frac{1}{8} c^3 (2 c d+7 b e) x^8+\frac{2}{9} c^4 e x^9\\ \end{align*}

Mathematica [A]  time = 0.0613576, size = 251, normalized size = 1. \[ \frac{1}{5} x^5 \left (6 a^2 c^2 e+12 a b^2 c e+15 a b c^2 d+5 b^3 c d+b^4 e\right )+\frac{1}{4} x^4 \left (9 a^2 b c e+6 a^2 c^2 d+12 a b^2 c d+3 a b^3 e+b^4 d\right )+\frac{1}{3} a x^3 \left (2 a^2 c e+3 a b^2 e+9 a b c d+3 b^3 d\right )+\frac{1}{2} a^2 x^2 \left (a b e+2 a c d+3 b^2 d\right )+a^3 b d x+\frac{1}{7} c^2 x^7 \left (6 a c e+9 b^2 e+7 b c d\right )+\frac{1}{6} c x^6 \left (15 a b c e+6 a c^2 d+9 b^2 c d+5 b^3 e\right )+\frac{1}{8} c^3 x^8 (7 b e+2 c d)+\frac{2}{9} c^4 e x^9 \]

Antiderivative was successfully verified.

[In]

Integrate[(b + 2*c*x)*(d + e*x)*(a + b*x + c*x^2)^3,x]

[Out]

a^3*b*d*x + (a^2*(3*b^2*d + 2*a*c*d + a*b*e)*x^2)/2 + (a*(3*b^3*d + 9*a*b*c*d + 3*a*b^2*e + 2*a^2*c*e)*x^3)/3
+ ((b^4*d + 12*a*b^2*c*d + 6*a^2*c^2*d + 3*a*b^3*e + 9*a^2*b*c*e)*x^4)/4 + ((5*b^3*c*d + 15*a*b*c^2*d + b^4*e
+ 12*a*b^2*c*e + 6*a^2*c^2*e)*x^5)/5 + (c*(9*b^2*c*d + 6*a*c^2*d + 5*b^3*e + 15*a*b*c*e)*x^6)/6 + (c^2*(7*b*c*
d + 9*b^2*e + 6*a*c*e)*x^7)/7 + (c^3*(2*c*d + 7*b*e)*x^8)/8 + (2*c^4*e*x^9)/9

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Maple [A]  time = 0.003, size = 386, normalized size = 1.5 \begin{align*}{\frac{2\,{c}^{4}e{x}^{9}}{9}}+{\frac{ \left ( \left ( be+2\,cd \right ){c}^{3}+6\,{c}^{3}eb \right ){x}^{8}}{8}}+{\frac{ \left ( bd{c}^{3}+3\, \left ( be+2\,cd \right ) b{c}^{2}+2\,ce \left ( a{c}^{2}+2\,{b}^{2}c+c \left ( 2\,ac+{b}^{2} \right ) \right ) \right ){x}^{7}}{7}}+{\frac{ \left ( 3\,{b}^{2}d{c}^{2}+ \left ( be+2\,cd \right ) \left ( a{c}^{2}+2\,{b}^{2}c+c \left ( 2\,ac+{b}^{2} \right ) \right ) +2\,ce \left ( b \left ( 2\,ac+{b}^{2} \right ) +4\,abc \right ) \right ){x}^{6}}{6}}+{\frac{ \left ( bd \left ( a{c}^{2}+2\,{b}^{2}c+c \left ( 2\,ac+{b}^{2} \right ) \right ) + \left ( be+2\,cd \right ) \left ( b \left ( 2\,ac+{b}^{2} \right ) +4\,abc \right ) +2\,ce \left ( a \left ( 2\,ac+{b}^{2} \right ) +2\,{b}^{2}a+c{a}^{2} \right ) \right ){x}^{5}}{5}}+{\frac{ \left ( bd \left ( b \left ( 2\,ac+{b}^{2} \right ) +4\,abc \right ) + \left ( be+2\,cd \right ) \left ( a \left ( 2\,ac+{b}^{2} \right ) +2\,{b}^{2}a+c{a}^{2} \right ) +6\,{a}^{2}bce \right ){x}^{4}}{4}}+{\frac{ \left ( bd \left ( a \left ( 2\,ac+{b}^{2} \right ) +2\,{b}^{2}a+c{a}^{2} \right ) +3\, \left ( be+2\,cd \right ){a}^{2}b+2\,ce{a}^{3} \right ){x}^{3}}{3}}+{\frac{ \left ( 3\,{b}^{2}d{a}^{2}+ \left ( be+2\,cd \right ){a}^{3} \right ){x}^{2}}{2}}+{a}^{3}bdx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*c*x+b)*(e*x+d)*(c*x^2+b*x+a)^3,x)

[Out]

2/9*c^4*e*x^9+1/8*((b*e+2*c*d)*c^3+6*c^3*e*b)*x^8+1/7*(b*d*c^3+3*(b*e+2*c*d)*b*c^2+2*c*e*(a*c^2+2*b^2*c+c*(2*a
*c+b^2)))*x^7+1/6*(3*b^2*d*c^2+(b*e+2*c*d)*(a*c^2+2*b^2*c+c*(2*a*c+b^2))+2*c*e*(b*(2*a*c+b^2)+4*a*b*c))*x^6+1/
5*(b*d*(a*c^2+2*b^2*c+c*(2*a*c+b^2))+(b*e+2*c*d)*(b*(2*a*c+b^2)+4*a*b*c)+2*c*e*(a*(2*a*c+b^2)+2*b^2*a+c*a^2))*
x^5+1/4*(b*d*(b*(2*a*c+b^2)+4*a*b*c)+(b*e+2*c*d)*(a*(2*a*c+b^2)+2*b^2*a+c*a^2)+6*a^2*b*c*e)*x^4+1/3*(b*d*(a*(2
*a*c+b^2)+2*b^2*a+c*a^2)+3*(b*e+2*c*d)*a^2*b+2*c*e*a^3)*x^3+1/2*(3*b^2*d*a^2+(b*e+2*c*d)*a^3)*x^2+a^3*b*d*x

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Maxima [A]  time = 0.988689, size = 352, normalized size = 1.4 \begin{align*} \frac{2}{9} \, c^{4} e x^{9} + \frac{1}{8} \,{\left (2 \, c^{4} d + 7 \, b c^{3} e\right )} x^{8} + \frac{1}{7} \,{\left (7 \, b c^{3} d + 3 \,{\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} e\right )} x^{7} + \frac{1}{6} \,{\left (3 \,{\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d + 5 \,{\left (b^{3} c + 3 \, a b c^{2}\right )} e\right )} x^{6} + a^{3} b d x + \frac{1}{5} \,{\left (5 \,{\left (b^{3} c + 3 \, a b c^{2}\right )} d +{\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} e\right )} x^{5} + \frac{1}{4} \,{\left ({\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d + 3 \,{\left (a b^{3} + 3 \, a^{2} b c\right )} e\right )} x^{4} + \frac{1}{3} \,{\left (3 \,{\left (a b^{3} + 3 \, a^{2} b c\right )} d +{\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} e\right )} x^{3} + \frac{1}{2} \,{\left (a^{3} b e +{\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d\right )} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*x+b)*(e*x+d)*(c*x^2+b*x+a)^3,x, algorithm="maxima")

[Out]

2/9*c^4*e*x^9 + 1/8*(2*c^4*d + 7*b*c^3*e)*x^8 + 1/7*(7*b*c^3*d + 3*(3*b^2*c^2 + 2*a*c^3)*e)*x^7 + 1/6*(3*(3*b^
2*c^2 + 2*a*c^3)*d + 5*(b^3*c + 3*a*b*c^2)*e)*x^6 + a^3*b*d*x + 1/5*(5*(b^3*c + 3*a*b*c^2)*d + (b^4 + 12*a*b^2
*c + 6*a^2*c^2)*e)*x^5 + 1/4*((b^4 + 12*a*b^2*c + 6*a^2*c^2)*d + 3*(a*b^3 + 3*a^2*b*c)*e)*x^4 + 1/3*(3*(a*b^3
+ 3*a^2*b*c)*d + (3*a^2*b^2 + 2*a^3*c)*e)*x^3 + 1/2*(a^3*b*e + (3*a^2*b^2 + 2*a^3*c)*d)*x^2

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Fricas [A]  time = 1.28043, size = 653, normalized size = 2.6 \begin{align*} \frac{2}{9} x^{9} e c^{4} + \frac{1}{4} x^{8} d c^{4} + \frac{7}{8} x^{8} e c^{3} b + x^{7} d c^{3} b + \frac{9}{7} x^{7} e c^{2} b^{2} + \frac{6}{7} x^{7} e c^{3} a + \frac{3}{2} x^{6} d c^{2} b^{2} + \frac{5}{6} x^{6} e c b^{3} + x^{6} d c^{3} a + \frac{5}{2} x^{6} e c^{2} b a + x^{5} d c b^{3} + \frac{1}{5} x^{5} e b^{4} + 3 x^{5} d c^{2} b a + \frac{12}{5} x^{5} e c b^{2} a + \frac{6}{5} x^{5} e c^{2} a^{2} + \frac{1}{4} x^{4} d b^{4} + 3 x^{4} d c b^{2} a + \frac{3}{4} x^{4} e b^{3} a + \frac{3}{2} x^{4} d c^{2} a^{2} + \frac{9}{4} x^{4} e c b a^{2} + x^{3} d b^{3} a + 3 x^{3} d c b a^{2} + x^{3} e b^{2} a^{2} + \frac{2}{3} x^{3} e c a^{3} + \frac{3}{2} x^{2} d b^{2} a^{2} + x^{2} d c a^{3} + \frac{1}{2} x^{2} e b a^{3} + x d b a^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/9*x^9*e*c^4 + 1/4*x^8*d*c^4 + 7/8*x^8*e*c^3*b + x^7*d*c^3*b + 9/7*x^7*e*c^2*b^2 + 6/7*x^7*e*c^3*a + 3/2*x^6*
d*c^2*b^2 + 5/6*x^6*e*c*b^3 + x^6*d*c^3*a + 5/2*x^6*e*c^2*b*a + x^5*d*c*b^3 + 1/5*x^5*e*b^4 + 3*x^5*d*c^2*b*a
+ 12/5*x^5*e*c*b^2*a + 6/5*x^5*e*c^2*a^2 + 1/4*x^4*d*b^4 + 3*x^4*d*c*b^2*a + 3/4*x^4*e*b^3*a + 3/2*x^4*d*c^2*a
^2 + 9/4*x^4*e*c*b*a^2 + x^3*d*b^3*a + 3*x^3*d*c*b*a^2 + x^3*e*b^2*a^2 + 2/3*x^3*e*c*a^3 + 3/2*x^2*d*b^2*a^2 +
x^2*d*c*a^3 + 1/2*x^2*e*b*a^3 + x*d*b*a^3

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Sympy [A]  time = 0.104308, size = 291, normalized size = 1.16 \begin{align*} a^{3} b d x + \frac{2 c^{4} e x^{9}}{9} + x^{8} \left (\frac{7 b c^{3} e}{8} + \frac{c^{4} d}{4}\right ) + x^{7} \left (\frac{6 a c^{3} e}{7} + \frac{9 b^{2} c^{2} e}{7} + b c^{3} d\right ) + x^{6} \left (\frac{5 a b c^{2} e}{2} + a c^{3} d + \frac{5 b^{3} c e}{6} + \frac{3 b^{2} c^{2} d}{2}\right ) + x^{5} \left (\frac{6 a^{2} c^{2} e}{5} + \frac{12 a b^{2} c e}{5} + 3 a b c^{2} d + \frac{b^{4} e}{5} + b^{3} c d\right ) + x^{4} \left (\frac{9 a^{2} b c e}{4} + \frac{3 a^{2} c^{2} d}{2} + \frac{3 a b^{3} e}{4} + 3 a b^{2} c d + \frac{b^{4} d}{4}\right ) + x^{3} \left (\frac{2 a^{3} c e}{3} + a^{2} b^{2} e + 3 a^{2} b c d + a b^{3} d\right ) + x^{2} \left (\frac{a^{3} b e}{2} + a^{3} c d + \frac{3 a^{2} b^{2} d}{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*x+b)*(e*x+d)*(c*x**2+b*x+a)**3,x)

[Out]

a**3*b*d*x + 2*c**4*e*x**9/9 + x**8*(7*b*c**3*e/8 + c**4*d/4) + x**7*(6*a*c**3*e/7 + 9*b**2*c**2*e/7 + b*c**3*
d) + x**6*(5*a*b*c**2*e/2 + a*c**3*d + 5*b**3*c*e/6 + 3*b**2*c**2*d/2) + x**5*(6*a**2*c**2*e/5 + 12*a*b**2*c*e
/5 + 3*a*b*c**2*d + b**4*e/5 + b**3*c*d) + x**4*(9*a**2*b*c*e/4 + 3*a**2*c**2*d/2 + 3*a*b**3*e/4 + 3*a*b**2*c*
d + b**4*d/4) + x**3*(2*a**3*c*e/3 + a**2*b**2*e + 3*a**2*b*c*d + a*b**3*d) + x**2*(a**3*b*e/2 + a**3*c*d + 3*
a**2*b**2*d/2)

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Giac [A]  time = 1.17882, size = 405, normalized size = 1.61 \begin{align*} \frac{2}{9} \, c^{4} x^{9} e + \frac{1}{4} \, c^{4} d x^{8} + \frac{7}{8} \, b c^{3} x^{8} e + b c^{3} d x^{7} + \frac{9}{7} \, b^{2} c^{2} x^{7} e + \frac{6}{7} \, a c^{3} x^{7} e + \frac{3}{2} \, b^{2} c^{2} d x^{6} + a c^{3} d x^{6} + \frac{5}{6} \, b^{3} c x^{6} e + \frac{5}{2} \, a b c^{2} x^{6} e + b^{3} c d x^{5} + 3 \, a b c^{2} d x^{5} + \frac{1}{5} \, b^{4} x^{5} e + \frac{12}{5} \, a b^{2} c x^{5} e + \frac{6}{5} \, a^{2} c^{2} x^{5} e + \frac{1}{4} \, b^{4} d x^{4} + 3 \, a b^{2} c d x^{4} + \frac{3}{2} \, a^{2} c^{2} d x^{4} + \frac{3}{4} \, a b^{3} x^{4} e + \frac{9}{4} \, a^{2} b c x^{4} e + a b^{3} d x^{3} + 3 \, a^{2} b c d x^{3} + a^{2} b^{2} x^{3} e + \frac{2}{3} \, a^{3} c x^{3} e + \frac{3}{2} \, a^{2} b^{2} d x^{2} + a^{3} c d x^{2} + \frac{1}{2} \, a^{3} b x^{2} e + a^{3} b d x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*x+b)*(e*x+d)*(c*x^2+b*x+a)^3,x, algorithm="giac")

[Out]

2/9*c^4*x^9*e + 1/4*c^4*d*x^8 + 7/8*b*c^3*x^8*e + b*c^3*d*x^7 + 9/7*b^2*c^2*x^7*e + 6/7*a*c^3*x^7*e + 3/2*b^2*
c^2*d*x^6 + a*c^3*d*x^6 + 5/6*b^3*c*x^6*e + 5/2*a*b*c^2*x^6*e + b^3*c*d*x^5 + 3*a*b*c^2*d*x^5 + 1/5*b^4*x^5*e
+ 12/5*a*b^2*c*x^5*e + 6/5*a^2*c^2*x^5*e + 1/4*b^4*d*x^4 + 3*a*b^2*c*d*x^4 + 3/2*a^2*c^2*d*x^4 + 3/4*a*b^3*x^4
*e + 9/4*a^2*b*c*x^4*e + a*b^3*d*x^3 + 3*a^2*b*c*d*x^3 + a^2*b^2*x^3*e + 2/3*a^3*c*x^3*e + 3/2*a^2*b^2*d*x^2 +
a^3*c*d*x^2 + 1/2*a^3*b*x^2*e + a^3*b*d*x