3.2353 \(\int x (d+e x)^3 (a+b x+c x^2) \, dx\)

Optimal. Leaf size=103 \[ \frac{(d+e x)^5 \left (3 c d^2-e (2 b d-a e)\right )}{5 e^4}-\frac{d (d+e x)^4 \left (a e^2-b d e+c d^2\right )}{4 e^4}-\frac{(d+e x)^6 (3 c d-b e)}{6 e^4}+\frac{c (d+e x)^7}{7 e^4} \]

[Out]

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

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Rubi [A]  time = 0.10269, antiderivative size = 103, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053, Rules used = {771} \[ \frac{(d+e x)^5 \left (3 c d^2-e (2 b d-a e)\right )}{5 e^4}-\frac{d (d+e x)^4 \left (a e^2-b d e+c d^2\right )}{4 e^4}-\frac{(d+e x)^6 (3 c d-b e)}{6 e^4}+\frac{c (d+e x)^7}{7 e^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.029562, size = 109, normalized size = 1.06 \[ \frac{1}{5} e x^5 \left (a e^2+3 b d e+3 c d^2\right )+\frac{1}{4} d x^4 \left (3 a e^2+3 b d e+c d^2\right )+\frac{1}{3} d^2 x^3 (3 a e+b d)+\frac{1}{2} a d^3 x^2+\frac{1}{6} e^2 x^6 (b e+3 c d)+\frac{1}{7} c e^3 x^7 \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0., size = 106, normalized size = 1. \begin{align*}{\frac{{e}^{3}c{x}^{7}}{7}}+{\frac{ \left ( b{e}^{3}+3\,cd{e}^{2} \right ){x}^{6}}{6}}+{\frac{ \left ({e}^{3}a+3\,d{e}^{2}b+3\,{d}^{2}ec \right ){x}^{5}}{5}}+{\frac{ \left ( 3\,d{e}^{2}a+3\,{d}^{2}eb+{d}^{3}c \right ){x}^{4}}{4}}+{\frac{ \left ( 3\,{d}^{2}ea+{d}^{3}b \right ){x}^{3}}{3}}+{\frac{{d}^{3}a{x}^{2}}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.02754, size = 142, normalized size = 1.38 \begin{align*} \frac{1}{7} \, c e^{3} x^{7} + \frac{1}{6} \,{\left (3 \, c d e^{2} + b e^{3}\right )} x^{6} + \frac{1}{2} \, a d^{3} x^{2} + \frac{1}{5} \,{\left (3 \, c d^{2} e + 3 \, b d e^{2} + a e^{3}\right )} x^{5} + \frac{1}{4} \,{\left (c d^{3} + 3 \, b d^{2} e + 3 \, a d e^{2}\right )} x^{4} + \frac{1}{3} \,{\left (b d^{3} + 3 \, a d^{2} e\right )} x^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.16339, size = 269, normalized size = 2.61 \begin{align*} \frac{1}{7} x^{7} e^{3} c + \frac{1}{2} x^{6} e^{2} d c + \frac{1}{6} x^{6} e^{3} b + \frac{3}{5} x^{5} e d^{2} c + \frac{3}{5} x^{5} e^{2} d b + \frac{1}{5} x^{5} e^{3} a + \frac{1}{4} x^{4} d^{3} c + \frac{3}{4} x^{4} e d^{2} b + \frac{3}{4} x^{4} e^{2} d a + \frac{1}{3} x^{3} d^{3} b + x^{3} e d^{2} a + \frac{1}{2} x^{2} d^{3} a \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.078766, size = 116, normalized size = 1.13 \begin{align*} \frac{a d^{3} x^{2}}{2} + \frac{c e^{3} x^{7}}{7} + x^{6} \left (\frac{b e^{3}}{6} + \frac{c d e^{2}}{2}\right ) + x^{5} \left (\frac{a e^{3}}{5} + \frac{3 b d e^{2}}{5} + \frac{3 c d^{2} e}{5}\right ) + x^{4} \left (\frac{3 a d e^{2}}{4} + \frac{3 b d^{2} e}{4} + \frac{c d^{3}}{4}\right ) + x^{3} \left (a d^{2} e + \frac{b d^{3}}{3}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.06855, size = 150, normalized size = 1.46 \begin{align*} \frac{1}{7} \, c x^{7} e^{3} + \frac{1}{2} \, c d x^{6} e^{2} + \frac{3}{5} \, c d^{2} x^{5} e + \frac{1}{4} \, c d^{3} x^{4} + \frac{1}{6} \, b x^{6} e^{3} + \frac{3}{5} \, b d x^{5} e^{2} + \frac{3}{4} \, b d^{2} x^{4} e + \frac{1}{3} \, b d^{3} x^{3} + \frac{1}{5} \, a x^{5} e^{3} + \frac{3}{4} \, a d x^{4} e^{2} + a d^{2} x^{3} e + \frac{1}{2} \, a d^{3} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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