3.1507 \(\int (b+2 c x) (d+e x) (a+b x+c x^2)^2 \, dx\)

Optimal. Leaf size=153 \[ \frac{1}{3} x^3 \left (2 a^2 c e+2 a b^2 e+6 a b c d+b^3 d\right )+a^2 b d x+\frac{1}{4} x^4 \left (6 a b c e+4 a c^2 d+4 b^2 c d+b^3 e\right )+\frac{1}{5} c x^5 \left (4 a c e+4 b^2 e+5 b c d\right )+\frac{1}{2} a x^2 \left (a b e+2 a c d+2 b^2 d\right )+\frac{1}{6} c^2 x^6 (5 b e+2 c d)+\frac{2}{7} c^3 e x^7 \]

[Out]

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

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Rubi [A]  time = 0.130695, antiderivative size = 153, 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} \[ \frac{1}{3} x^3 \left (2 a^2 c e+2 a b^2 e+6 a b c d+b^3 d\right )+a^2 b d x+\frac{1}{4} x^4 \left (6 a b c e+4 a c^2 d+4 b^2 c d+b^3 e\right )+\frac{1}{5} c x^5 \left (4 a c e+4 b^2 e+5 b c d\right )+\frac{1}{2} a x^2 \left (a b e+2 a c d+2 b^2 d\right )+\frac{1}{6} c^2 x^6 (5 b e+2 c d)+\frac{2}{7} c^3 e x^7 \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.0447514, size = 153, normalized size = 1. \[ \frac{1}{3} x^3 \left (2 a^2 c e+2 a b^2 e+6 a b c d+b^3 d\right )+a^2 b d x+\frac{1}{4} x^4 \left (6 a b c e+4 a c^2 d+4 b^2 c d+b^3 e\right )+\frac{1}{5} c x^5 \left (4 a c e+4 b^2 e+5 b c d\right )+\frac{1}{2} a x^2 \left (a b e+2 a c d+2 b^2 d\right )+\frac{1}{6} c^2 x^6 (5 b e+2 c d)+\frac{2}{7} c^3 e x^7 \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0., size = 176, normalized size = 1.2 \begin{align*}{\frac{2\,{c}^{3}e{x}^{7}}{7}}+{\frac{ \left ( \left ( be+2\,cd \right ){c}^{2}+4\,{c}^{2}eb \right ){x}^{6}}{6}}+{\frac{ \left ( bd{c}^{2}+2\, \left ( be+2\,cd \right ) bc+2\,ce \left ( 2\,ac+{b}^{2} \right ) \right ){x}^{5}}{5}}+{\frac{ \left ( 2\,{b}^{2}cd+ \left ( be+2\,cd \right ) \left ( 2\,ac+{b}^{2} \right ) +4\,abce \right ){x}^{4}}{4}}+{\frac{ \left ( bd \left ( 2\,ac+{b}^{2} \right ) +2\, \left ( be+2\,cd \right ) ab+2\,{a}^{2}ce \right ){x}^{3}}{3}}+{\frac{ \left ( 2\,{b}^{2}da+ \left ( be+2\,cd \right ){a}^{2} \right ){x}^{2}}{2}}+{a}^{2}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)^2,x)

[Out]

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

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Maxima [A]  time = 1.2092, size = 204, normalized size = 1.33 \begin{align*} \frac{2}{7} \, c^{3} e x^{7} + \frac{1}{6} \,{\left (2 \, c^{3} d + 5 \, b c^{2} e\right )} x^{6} + \frac{1}{5} \,{\left (5 \, b c^{2} d + 4 \,{\left (b^{2} c + a c^{2}\right )} e\right )} x^{5} + a^{2} b d x + \frac{1}{4} \,{\left (4 \,{\left (b^{2} c + a c^{2}\right )} d +{\left (b^{3} + 6 \, a b c\right )} e\right )} x^{4} + \frac{1}{3} \,{\left ({\left (b^{3} + 6 \, a b c\right )} d + 2 \,{\left (a b^{2} + a^{2} c\right )} e\right )} x^{3} + \frac{1}{2} \,{\left (a^{2} b e + 2 \,{\left (a b^{2} + a^{2} 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)^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.79418, size = 387, normalized size = 2.53 \begin{align*} \frac{2}{7} x^{7} e c^{3} + \frac{1}{3} x^{6} d c^{3} + \frac{5}{6} x^{6} e c^{2} b + x^{5} d c^{2} b + \frac{4}{5} x^{5} e c b^{2} + \frac{4}{5} x^{5} e c^{2} a + x^{4} d c b^{2} + \frac{1}{4} x^{4} e b^{3} + x^{4} d c^{2} a + \frac{3}{2} x^{4} e c b a + \frac{1}{3} x^{3} d b^{3} + 2 x^{3} d c b a + \frac{2}{3} x^{3} e b^{2} a + \frac{2}{3} x^{3} e c a^{2} + x^{2} d b^{2} a + x^{2} d c a^{2} + \frac{1}{2} x^{2} e b a^{2} + x d b a^{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)^2,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.092897, size = 168, normalized size = 1.1 \begin{align*} a^{2} b d x + \frac{2 c^{3} e x^{7}}{7} + x^{6} \left (\frac{5 b c^{2} e}{6} + \frac{c^{3} d}{3}\right ) + x^{5} \left (\frac{4 a c^{2} e}{5} + \frac{4 b^{2} c e}{5} + b c^{2} d\right ) + x^{4} \left (\frac{3 a b c e}{2} + a c^{2} d + \frac{b^{3} e}{4} + b^{2} c d\right ) + x^{3} \left (\frac{2 a^{2} c e}{3} + \frac{2 a b^{2} e}{3} + 2 a b c d + \frac{b^{3} d}{3}\right ) + x^{2} \left (\frac{a^{2} b e}{2} + a^{2} c d + a b^{2} d\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)**2,x)

[Out]

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

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Giac [A]  time = 1.2096, size = 238, normalized size = 1.56 \begin{align*} \frac{2}{7} \, c^{3} x^{7} e + \frac{1}{3} \, c^{3} d x^{6} + \frac{5}{6} \, b c^{2} x^{6} e + b c^{2} d x^{5} + \frac{4}{5} \, b^{2} c x^{5} e + \frac{4}{5} \, a c^{2} x^{5} e + b^{2} c d x^{4} + a c^{2} d x^{4} + \frac{1}{4} \, b^{3} x^{4} e + \frac{3}{2} \, a b c x^{4} e + \frac{1}{3} \, b^{3} d x^{3} + 2 \, a b c d x^{3} + \frac{2}{3} \, a b^{2} x^{3} e + \frac{2}{3} \, a^{2} c x^{3} e + a b^{2} d x^{2} + a^{2} c d x^{2} + \frac{1}{2} \, a^{2} b x^{2} e + a^{2} 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)^2,x, algorithm="giac")

[Out]

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