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

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

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Rubi [A]  time = 0.13, antiderivative size = 153, normalized size of antiderivative = 1.00, 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} \begin {gather*} \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 \end {gather*}

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*}

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Mathematica [A]  time = 0.04, size = 153, normalized size = 1.00 \begin {gather*} \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}{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}{4} x^4 \left (6 a b c e+4 a c^2 d+b^3 e+4 b^2 c d\right )+\frac {1}{6} c^2 x^6 (5 b e+2 c d)+\frac {2}{7} c^3 e x^7 \end {gather*}

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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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int (b+2 c x) (d+e x) \left (a+b x+c x^2\right )^2 \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.37, size = 167, normalized size = 1.09 \begin {gather*} \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 {gather*}

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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giac [A]  time = 0.19, size = 176, normalized size = 1.15 \begin {gather*} \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 {gather*}

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

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maple [A]  time = 0.04, size = 176, normalized size = 1.15 \begin {gather*} \frac {2 c^{3} e \,x^{7}}{7}+\frac {\left (4 b \,c^{2} e +\left (b e +2 c d \right ) c^{2}\right ) x^{6}}{6}+a^{2} b d x +\frac {\left (b \,c^{2} d +2 \left (b e +2 c d \right ) b c +2 \left (2 a c +b^{2}\right ) c e \right ) x^{5}}{5}+\frac {\left (4 a b c e +2 b^{2} c d +\left (b e +2 c d \right ) \left (2 a c +b^{2}\right )\right ) x^{4}}{4}+\frac {\left (2 a^{2} c e +2 \left (b e +2 c d \right ) a b +\left (2 a c +b^{2}\right ) b d \right ) x^{3}}{3}+\frac {\left (2 a \,b^{2} d +\left (b e +2 c d \right ) a^{2}\right ) x^{2}}{2} \end {gather*}

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*c^2*d+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 = 0.55, size = 151, normalized size = 0.99 \begin {gather*} \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 {gather*}

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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mupad [B]  time = 0.06, size = 144, normalized size = 0.94 \begin {gather*} x^6\,\left (\frac {d\,c^3}{3}+\frac {5\,b\,e\,c^2}{6}\right )+x^2\,\left (\frac {e\,a^2\,b}{2}+c\,d\,a^2+d\,a\,b^2\right )+x^5\,\left (\frac {4\,e\,b^2\,c}{5}+d\,b\,c^2+\frac {4\,a\,e\,c^2}{5}\right )+x^3\,\left (\frac {2\,c\,e\,a^2}{3}+\frac {2\,e\,a\,b^2}{3}+2\,c\,d\,a\,b+\frac {d\,b^3}{3}\right )+x^4\,\left (\frac {e\,b^3}{4}+d\,b^2\,c+\frac {3\,a\,e\,b\,c}{2}+a\,d\,c^2\right )+\frac {2\,c^3\,e\,x^7}{7}+a^2\,b\,d\,x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.10, size = 168, normalized size = 1.10 \begin {gather*} 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 {gather*}

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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