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

Optimal. Leaf size=127 \[ \frac{1}{6} e x^6 \left (b^2 e^2+6 b c d e+3 c^2 d^2\right )+\frac{1}{5} d x^5 \left (3 b^2 e^2+6 b c d e+c^2 d^2\right )+\frac{1}{3} b^2 d^3 x^3+\frac{1}{4} b d^2 x^4 (3 b e+2 c d)+\frac{1}{7} c e^2 x^7 (2 b e+3 c d)+\frac{1}{8} c^2 e^3 x^8 \]

[Out]

(b^2*d^3*x^3)/3 + (b*d^2*(2*c*d + 3*b*e)*x^4)/4 + (d*(c^2*d^2 + 6*b*c*d*e + 3*b^2*e^2)*x^5)/5 + (e*(3*c^2*d^2
+ 6*b*c*d*e + b^2*e^2)*x^6)/6 + (c*e^2*(3*c*d + 2*b*e)*x^7)/7 + (c^2*e^3*x^8)/8

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Rubi [A]  time = 0.0928371, antiderivative size = 127, 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 = {698} \[ \frac{1}{6} e x^6 \left (b^2 e^2+6 b c d e+3 c^2 d^2\right )+\frac{1}{5} d x^5 \left (3 b^2 e^2+6 b c d e+c^2 d^2\right )+\frac{1}{3} b^2 d^3 x^3+\frac{1}{4} b d^2 x^4 (3 b e+2 c d)+\frac{1}{7} c e^2 x^7 (2 b e+3 c d)+\frac{1}{8} c^2 e^3 x^8 \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^3*(b*x + c*x^2)^2,x]

[Out]

(b^2*d^3*x^3)/3 + (b*d^2*(2*c*d + 3*b*e)*x^4)/4 + (d*(c^2*d^2 + 6*b*c*d*e + 3*b^2*e^2)*x^5)/5 + (e*(3*c^2*d^2
+ 6*b*c*d*e + b^2*e^2)*x^6)/6 + (c*e^2*(3*c*d + 2*b*e)*x^7)/7 + (c^2*e^3*x^8)/8

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 (b x+c x^2\right )^2 \, dx &=\int \left (b^2 d^3 x^2+b d^2 (2 c d+3 b e) x^3+d \left (c^2 d^2+6 b c d e+3 b^2 e^2\right ) x^4+e \left (3 c^2 d^2+6 b c d e+b^2 e^2\right ) x^5+c e^2 (3 c d+2 b e) x^6+c^2 e^3 x^7\right ) \, dx\\ &=\frac{1}{3} b^2 d^3 x^3+\frac{1}{4} b d^2 (2 c d+3 b e) x^4+\frac{1}{5} d \left (c^2 d^2+6 b c d e+3 b^2 e^2\right ) x^5+\frac{1}{6} e \left (3 c^2 d^2+6 b c d e+b^2 e^2\right ) x^6+\frac{1}{7} c e^2 (3 c d+2 b e) x^7+\frac{1}{8} c^2 e^3 x^8\\ \end{align*}

Mathematica [A]  time = 0.0174962, size = 127, normalized size = 1. \[ \frac{1}{6} e x^6 \left (b^2 e^2+6 b c d e+3 c^2 d^2\right )+\frac{1}{5} d x^5 \left (3 b^2 e^2+6 b c d e+c^2 d^2\right )+\frac{1}{3} b^2 d^3 x^3+\frac{1}{4} b d^2 x^4 (3 b e+2 c d)+\frac{1}{7} c e^2 x^7 (2 b e+3 c d)+\frac{1}{8} c^2 e^3 x^8 \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b^2*d^3*x^3)/3 + (b*d^2*(2*c*d + 3*b*e)*x^4)/4 + (d*(c^2*d^2 + 6*b*c*d*e + 3*b^2*e^2)*x^5)/5 + (e*(3*c^2*d^2
+ 6*b*c*d*e + b^2*e^2)*x^6)/6 + (c*e^2*(3*c*d + 2*b*e)*x^7)/7 + (c^2*e^3*x^8)/8

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Maple [A]  time = 0.042, size = 128, normalized size = 1. \begin{align*}{\frac{{c}^{2}{e}^{3}{x}^{8}}{8}}+{\frac{ \left ( 2\,{e}^{3}bc+3\,d{e}^{2}{c}^{2} \right ){x}^{7}}{7}}+{\frac{ \left ({e}^{3}{b}^{2}+6\,d{e}^{2}bc+3\,{d}^{2}e{c}^{2} \right ){x}^{6}}{6}}+{\frac{ \left ( 3\,d{e}^{2}{b}^{2}+6\,{d}^{2}ebc+{c}^{2}{d}^{3} \right ){x}^{5}}{5}}+{\frac{ \left ( 3\,{d}^{2}e{b}^{2}+2\,{d}^{3}bc \right ){x}^{4}}{4}}+{\frac{{b}^{2}{d}^{3}{x}^{3}}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*c^2*e^3*x^8+1/7*(2*b*c*e^3+3*c^2*d*e^2)*x^7+1/6*(b^2*e^3+6*b*c*d*e^2+3*c^2*d^2*e)*x^6+1/5*(3*b^2*d*e^2+6*b
*c*d^2*e+c^2*d^3)*x^5+1/4*(3*b^2*d^2*e+2*b*c*d^3)*x^4+1/3*b^2*d^3*x^3

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Maxima [A]  time = 1.15568, size = 171, normalized size = 1.35 \begin{align*} \frac{1}{8} \, c^{2} e^{3} x^{8} + \frac{1}{3} \, b^{2} d^{3} x^{3} + \frac{1}{7} \,{\left (3 \, c^{2} d e^{2} + 2 \, b c e^{3}\right )} x^{7} + \frac{1}{6} \,{\left (3 \, c^{2} d^{2} e + 6 \, b c d e^{2} + b^{2} e^{3}\right )} x^{6} + \frac{1}{5} \,{\left (c^{2} d^{3} + 6 \, b c d^{2} e + 3 \, b^{2} d e^{2}\right )} x^{5} + \frac{1}{4} \,{\left (2 \, b c d^{3} + 3 \, b^{2} d^{2} e\right )} x^{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*c^2*e^3*x^8 + 1/3*b^2*d^3*x^3 + 1/7*(3*c^2*d*e^2 + 2*b*c*e^3)*x^7 + 1/6*(3*c^2*d^2*e + 6*b*c*d*e^2 + b^2*e
^3)*x^6 + 1/5*(c^2*d^3 + 6*b*c*d^2*e + 3*b^2*d*e^2)*x^5 + 1/4*(2*b*c*d^3 + 3*b^2*d^2*e)*x^4

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Fricas [A]  time = 1.55294, size = 301, normalized size = 2.37 \begin{align*} \frac{1}{8} x^{8} e^{3} c^{2} + \frac{3}{7} x^{7} e^{2} d c^{2} + \frac{2}{7} x^{7} e^{3} c b + \frac{1}{2} x^{6} e d^{2} c^{2} + x^{6} e^{2} d c b + \frac{1}{6} x^{6} e^{3} b^{2} + \frac{1}{5} x^{5} d^{3} c^{2} + \frac{6}{5} x^{5} e d^{2} c b + \frac{3}{5} x^{5} e^{2} d b^{2} + \frac{1}{2} x^{4} d^{3} c b + \frac{3}{4} x^{4} e d^{2} b^{2} + \frac{1}{3} x^{3} d^{3} b^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*x^8*e^3*c^2 + 3/7*x^7*e^2*d*c^2 + 2/7*x^7*e^3*c*b + 1/2*x^6*e*d^2*c^2 + x^6*e^2*d*c*b + 1/6*x^6*e^3*b^2 +
1/5*x^5*d^3*c^2 + 6/5*x^5*e*d^2*c*b + 3/5*x^5*e^2*d*b^2 + 1/2*x^4*d^3*c*b + 3/4*x^4*e*d^2*b^2 + 1/3*x^3*d^3*b^
2

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Sympy [A]  time = 0.234868, size = 138, normalized size = 1.09 \begin{align*} \frac{b^{2} d^{3} x^{3}}{3} + \frac{c^{2} e^{3} x^{8}}{8} + x^{7} \left (\frac{2 b c e^{3}}{7} + \frac{3 c^{2} d e^{2}}{7}\right ) + x^{6} \left (\frac{b^{2} e^{3}}{6} + b c d e^{2} + \frac{c^{2} d^{2} e}{2}\right ) + x^{5} \left (\frac{3 b^{2} d e^{2}}{5} + \frac{6 b c d^{2} e}{5} + \frac{c^{2} d^{3}}{5}\right ) + x^{4} \left (\frac{3 b^{2} d^{2} e}{4} + \frac{b c 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)**2,x)

[Out]

b**2*d**3*x**3/3 + c**2*e**3*x**8/8 + x**7*(2*b*c*e**3/7 + 3*c**2*d*e**2/7) + x**6*(b**2*e**3/6 + b*c*d*e**2 +
c**2*d**2*e/2) + x**5*(3*b**2*d*e**2/5 + 6*b*c*d**2*e/5 + c**2*d**3/5) + x**4*(3*b**2*d**2*e/4 + b*c*d**3/2)

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Giac [A]  time = 1.26225, size = 177, normalized size = 1.39 \begin{align*} \frac{1}{8} \, c^{2} x^{8} e^{3} + \frac{3}{7} \, c^{2} d x^{7} e^{2} + \frac{1}{2} \, c^{2} d^{2} x^{6} e + \frac{1}{5} \, c^{2} d^{3} x^{5} + \frac{2}{7} \, b c x^{7} e^{3} + b c d x^{6} e^{2} + \frac{6}{5} \, b c d^{2} x^{5} e + \frac{1}{2} \, b c d^{3} x^{4} + \frac{1}{6} \, b^{2} x^{6} e^{3} + \frac{3}{5} \, b^{2} d x^{5} e^{2} + \frac{3}{4} \, b^{2} d^{2} x^{4} e + \frac{1}{3} \, b^{2} d^{3} x^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*c^2*x^8*e^3 + 3/7*c^2*d*x^7*e^2 + 1/2*c^2*d^2*x^6*e + 1/5*c^2*d^3*x^5 + 2/7*b*c*x^7*e^3 + b*c*d*x^6*e^2 +
6/5*b*c*d^2*x^5*e + 1/2*b*c*d^3*x^4 + 1/6*b^2*x^6*e^3 + 3/5*b^2*d*x^5*e^2 + 3/4*b^2*d^2*x^4*e + 1/3*b^2*d^3*x^
3