3.2108 \(\int (d+e x)^3 (a+b x+c x^2) \, dx\)

Optimal. Leaf size=69 \[ \frac{(d+e x)^4 \left (a e^2-b d e+c d^2\right )}{4 e^3}-\frac{(d+e x)^5 (2 c d-b e)}{5 e^3}+\frac{c (d+e x)^6}{6 e^3} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0691618, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056, Rules used = {698} \[ \frac{(d+e x)^4 \left (a e^2-b d e+c d^2\right )}{4 e^3}-\frac{(d+e x)^5 (2 c d-b e)}{5 e^3}+\frac{c (d+e x)^6}{6 e^3} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^3*(a + b*x + c*x^2),x]

[Out]

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

Rule 698

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

Rubi steps

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

Mathematica [A]  time = 0.0245141, size = 104, normalized size = 1.51 \[ \frac{1}{4} e x^4 \left (a e^2+3 b d e+3 c d^2\right )+\frac{1}{3} d x^3 \left (3 a e^2+3 b d e+c d^2\right )+\frac{1}{2} d^2 x^2 (3 a e+b d)+a d^3 x+\frac{1}{5} e^2 x^5 (b e+3 c d)+\frac{1}{6} c e^3 x^6 \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.04, size = 103, normalized size = 1.5 \begin{align*}{\frac{{e}^{3}c{x}^{6}}{6}}+{\frac{ \left ({e}^{3}b+3\,d{e}^{2}c \right ){x}^{5}}{5}}+{\frac{ \left ( a{e}^{3}+3\,d{e}^{2}b+3\,{d}^{2}ec \right ){x}^{4}}{4}}+{\frac{ \left ( 3\,ad{e}^{2}+3\,{d}^{2}eb+c{d}^{3} \right ){x}^{3}}{3}}+{\frac{ \left ( 3\,{d}^{2}ea+{d}^{3}b \right ){x}^{2}}{2}}+{d}^{3}ax \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 0.975367, size = 138, normalized size = 2. \begin{align*} \frac{1}{6} \, c e^{3} x^{6} + \frac{1}{5} \,{\left (3 \, c d e^{2} + b e^{3}\right )} x^{5} + a d^{3} x + \frac{1}{4} \,{\left (3 \, c d^{2} e + 3 \, b d e^{2} + a e^{3}\right )} x^{4} + \frac{1}{3} \,{\left (c d^{3} + 3 \, b d^{2} e + 3 \, a d e^{2}\right )} x^{3} + \frac{1}{2} \,{\left (b d^{3} + 3 \, a d^{2} e\right )} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.75739, size = 255, normalized size = 3.7 \begin{align*} \frac{1}{6} x^{6} e^{3} c + \frac{3}{5} x^{5} e^{2} d c + \frac{1}{5} x^{5} e^{3} b + \frac{3}{4} x^{4} e d^{2} c + \frac{3}{4} x^{4} e^{2} d b + \frac{1}{4} x^{4} e^{3} a + \frac{1}{3} x^{3} d^{3} c + x^{3} e d^{2} b + x^{3} e^{2} d a + \frac{1}{2} x^{2} d^{3} b + \frac{3}{2} x^{2} e d^{2} a + x d^{3} a \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [A]  time = 0.07866, size = 110, normalized size = 1.59 \begin{align*} a d^{3} x + \frac{c e^{3} x^{6}}{6} + x^{5} \left (\frac{b e^{3}}{5} + \frac{3 c d e^{2}}{5}\right ) + x^{4} \left (\frac{a e^{3}}{4} + \frac{3 b d e^{2}}{4} + \frac{3 c d^{2} e}{4}\right ) + x^{3} \left (a d e^{2} + b d^{2} e + \frac{c d^{3}}{3}\right ) + x^{2} \left (\frac{3 a d^{2} e}{2} + \frac{b d^{3}}{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 1.12474, size = 144, normalized size = 2.09 \begin{align*} \frac{1}{6} \, c x^{6} e^{3} + \frac{3}{5} \, c d x^{5} e^{2} + \frac{3}{4} \, c d^{2} x^{4} e + \frac{1}{3} \, c d^{3} x^{3} + \frac{1}{5} \, b x^{5} e^{3} + \frac{3}{4} \, b d x^{4} e^{2} + b d^{2} x^{3} e + \frac{1}{2} \, b d^{3} x^{2} + \frac{1}{4} \, a x^{4} e^{3} + a d x^{3} e^{2} + \frac{3}{2} \, a d^{2} x^{2} e + a d^{3} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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