3.1496 \(\int (b+2 c x) (d+e x)^2 (a+b x+c x^2) \, dx\)

Optimal. Leaf size=124 \[ \frac{(d+e x)^4 \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{4 e^4}-\frac{(d+e x)^3 (2 c d-b e) \left (a e^2-b d e+c d^2\right )}{3 e^4}-\frac{3 c (d+e x)^5 (2 c d-b e)}{5 e^4}+\frac{c^2 (d+e x)^6}{3 e^4} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.109109, antiderivative size = 124, 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} \[ \frac{(d+e x)^4 \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{4 e^4}-\frac{(d+e x)^3 (2 c d-b e) \left (a e^2-b d e+c d^2\right )}{3 e^4}-\frac{3 c (d+e x)^5 (2 c d-b e)}{5 e^4}+\frac{c^2 (d+e x)^6}{3 e^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.0391408, size = 133, normalized size = 1.07 \[ \frac{1}{4} x^4 \left (2 a c e^2+b^2 e^2+6 b c d e+2 c^2 d^2\right )+\frac{1}{3} x^3 \left (a b e^2+4 a c d e+2 b^2 d e+3 b c d^2\right )+\frac{1}{2} d x^2 \left (2 a b e+2 a c d+b^2 d\right )+a b d^2 x+\frac{1}{5} c e x^5 (3 b e+4 c d)+\frac{1}{3} c^2 e^2 x^6 \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0., size = 152, normalized size = 1.2 \begin{align*}{\frac{{c}^{2}{e}^{2}{x}^{6}}{3}}+{\frac{ \left ( \left ( b{e}^{2}+4\,cde \right ) c+2\,c{e}^{2}b \right ){x}^{5}}{5}}+{\frac{ \left ( \left ( 2\,bde+2\,c{d}^{2} \right ) c+ \left ( b{e}^{2}+4\,cde \right ) b+2\,ac{e}^{2} \right ){x}^{4}}{4}}+{\frac{ \left ( b{d}^{2}c+ \left ( 2\,bde+2\,c{d}^{2} \right ) b+ \left ( b{e}^{2}+4\,cde \right ) a \right ){x}^{3}}{3}}+{\frac{ \left ({b}^{2}{d}^{2}+ \left ( 2\,bde+2\,c{d}^{2} \right ) a \right ){x}^{2}}{2}}+b{d}^{2}ax \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.00476, size = 170, normalized size = 1.37 \begin{align*} \frac{1}{3} \, c^{2} e^{2} x^{6} + \frac{1}{5} \,{\left (4 \, c^{2} d e + 3 \, b c e^{2}\right )} x^{5} + a b d^{2} x + \frac{1}{4} \,{\left (2 \, c^{2} d^{2} + 6 \, b c d e +{\left (b^{2} + 2 \, a c\right )} e^{2}\right )} x^{4} + \frac{1}{3} \,{\left (3 \, b c d^{2} + a b e^{2} + 2 \,{\left (b^{2} + 2 \, a c\right )} d e\right )} x^{3} + \frac{1}{2} \,{\left (2 \, a b d e +{\left (b^{2} + 2 \, a c\right )} d^{2}\right )} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.36309, size = 339, normalized size = 2.73 \begin{align*} \frac{1}{3} x^{6} e^{2} c^{2} + \frac{4}{5} x^{5} e d c^{2} + \frac{3}{5} x^{5} e^{2} c b + \frac{1}{2} x^{4} d^{2} c^{2} + \frac{3}{2} x^{4} e d c b + \frac{1}{4} x^{4} e^{2} b^{2} + \frac{1}{2} x^{4} e^{2} c a + x^{3} d^{2} c b + \frac{2}{3} x^{3} e d b^{2} + \frac{4}{3} x^{3} e d c a + \frac{1}{3} x^{3} e^{2} b a + \frac{1}{2} x^{2} d^{2} b^{2} + x^{2} d^{2} c a + x^{2} e d b a + x d^{2} b a \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [A]  time = 0.085777, size = 146, normalized size = 1.18 \begin{align*} a b d^{2} x + \frac{c^{2} e^{2} x^{6}}{3} + x^{5} \left (\frac{3 b c e^{2}}{5} + \frac{4 c^{2} d e}{5}\right ) + x^{4} \left (\frac{a c e^{2}}{2} + \frac{b^{2} e^{2}}{4} + \frac{3 b c d e}{2} + \frac{c^{2} d^{2}}{2}\right ) + x^{3} \left (\frac{a b e^{2}}{3} + \frac{4 a c d e}{3} + \frac{2 b^{2} d e}{3} + b c d^{2}\right ) + x^{2} \left (a b d e + a c d^{2} + \frac{b^{2} d^{2}}{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 1.1937, size = 197, normalized size = 1.59 \begin{align*} \frac{1}{3} \, c^{2} x^{6} e^{2} + \frac{4}{5} \, c^{2} d x^{5} e + \frac{1}{2} \, c^{2} d^{2} x^{4} + \frac{3}{5} \, b c x^{5} e^{2} + \frac{3}{2} \, b c d x^{4} e + b c d^{2} x^{3} + \frac{1}{4} \, b^{2} x^{4} e^{2} + \frac{1}{2} \, a c x^{4} e^{2} + \frac{2}{3} \, b^{2} d x^{3} e + \frac{4}{3} \, a c d x^{3} e + \frac{1}{2} \, b^{2} d^{2} x^{2} + a c d^{2} x^{2} + \frac{1}{3} \, a b x^{3} e^{2} + a b d x^{2} e + a b d^{2} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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