### 3.246 $$\int (d+e x)^2 (b x+c x^2)^3 \, dx$$

Optimal. Leaf size=127 $\frac{1}{7} c x^7 \left (3 b^2 e^2+6 b c d e+c^2 d^2\right )+\frac{1}{6} b x^6 \left (b^2 e^2+6 b c d e+3 c^2 d^2\right )+\frac{1}{5} b^2 d x^5 (2 b e+3 c d)+\frac{1}{4} b^3 d^2 x^4+\frac{1}{8} c^2 e x^8 (3 b e+2 c d)+\frac{1}{9} c^3 e^2 x^9$

[Out]

(b^3*d^2*x^4)/4 + (b^2*d*(3*c*d + 2*b*e)*x^5)/5 + (b*(3*c^2*d^2 + 6*b*c*d*e + b^2*e^2)*x^6)/6 + (c*(c^2*d^2 +
6*b*c*d*e + 3*b^2*e^2)*x^7)/7 + (c^2*e*(2*c*d + 3*b*e)*x^8)/8 + (c^3*e^2*x^9)/9

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Rubi [A]  time = 0.109911, 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} $\frac{1}{7} c x^7 \left (3 b^2 e^2+6 b c d e+c^2 d^2\right )+\frac{1}{6} b x^6 \left (b^2 e^2+6 b c d e+3 c^2 d^2\right )+\frac{1}{5} b^2 d x^5 (2 b e+3 c d)+\frac{1}{4} b^3 d^2 x^4+\frac{1}{8} c^2 e x^8 (3 b e+2 c d)+\frac{1}{9} c^3 e^2 x^9$

Antiderivative was successfully veriﬁed.

[In]

Int[(d + e*x)^2*(b*x + c*x^2)^3,x]

[Out]

(b^3*d^2*x^4)/4 + (b^2*d*(3*c*d + 2*b*e)*x^5)/5 + (b*(3*c^2*d^2 + 6*b*c*d*e + b^2*e^2)*x^6)/6 + (c*(c^2*d^2 +
6*b*c*d*e + 3*b^2*e^2)*x^7)/7 + (c^2*e*(2*c*d + 3*b*e)*x^8)/8 + (c^3*e^2*x^9)/9

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)^2 \left (b x+c x^2\right )^3 \, dx &=\int \left (b^3 d^2 x^3+b^2 d (3 c d+2 b e) x^4+b \left (3 c^2 d^2+6 b c d e+b^2 e^2\right ) x^5+c \left (c^2 d^2+6 b c d e+3 b^2 e^2\right ) x^6+c^2 e (2 c d+3 b e) x^7+c^3 e^2 x^8\right ) \, dx\\ &=\frac{1}{4} b^3 d^2 x^4+\frac{1}{5} b^2 d (3 c d+2 b e) x^5+\frac{1}{6} b \left (3 c^2 d^2+6 b c d e+b^2 e^2\right ) x^6+\frac{1}{7} c \left (c^2 d^2+6 b c d e+3 b^2 e^2\right ) x^7+\frac{1}{8} c^2 e (2 c d+3 b e) x^8+\frac{1}{9} c^3 e^2 x^9\\ \end{align*}

Mathematica [A]  time = 0.0239765, size = 127, normalized size = 1. $\frac{1}{7} c x^7 \left (3 b^2 e^2+6 b c d e+c^2 d^2\right )+\frac{1}{6} b x^6 \left (b^2 e^2+6 b c d e+3 c^2 d^2\right )+\frac{1}{5} b^2 d x^5 (2 b e+3 c d)+\frac{1}{4} b^3 d^2 x^4+\frac{1}{8} c^2 e x^8 (3 b e+2 c d)+\frac{1}{9} c^3 e^2 x^9$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(b^3*d^2*x^4)/4 + (b^2*d*(3*c*d + 2*b*e)*x^5)/5 + (b*(3*c^2*d^2 + 6*b*c*d*e + b^2*e^2)*x^6)/6 + (c*(c^2*d^2 +
6*b*c*d*e + 3*b^2*e^2)*x^7)/7 + (c^2*e*(2*c*d + 3*b*e)*x^8)/8 + (c^3*e^2*x^9)/9

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Maple [A]  time = 0.043, size = 128, normalized size = 1. \begin{align*}{\frac{{c}^{3}{e}^{2}{x}^{9}}{9}}+{\frac{ \left ( 3\,{e}^{2}b{c}^{2}+2\,de{c}^{3} \right ){x}^{8}}{8}}+{\frac{ \left ( 3\,{e}^{2}{b}^{2}c+6\,deb{c}^{2}+{d}^{2}{c}^{3} \right ){x}^{7}}{7}}+{\frac{ \left ({b}^{3}{e}^{2}+6\,{b}^{2}cde+3\,{d}^{2}b{c}^{2} \right ){x}^{6}}{6}}+{\frac{ \left ( 2\,de{b}^{3}+3\,{d}^{2}{b}^{2}c \right ){x}^{5}}{5}}+{\frac{{b}^{3}{d}^{2}{x}^{4}}{4}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*c^3*e^2*x^9+1/8*(3*b*c^2*e^2+2*c^3*d*e)*x^8+1/7*(3*b^2*c*e^2+6*b*c^2*d*e+c^3*d^2)*x^7+1/6*(b^3*e^2+6*b^2*c
*d*e+3*b*c^2*d^2)*x^6+1/5*(2*b^3*d*e+3*b^2*c*d^2)*x^5+1/4*b^3*d^2*x^4

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Maxima [A]  time = 1.13131, size = 171, normalized size = 1.35 \begin{align*} \frac{1}{9} \, c^{3} e^{2} x^{9} + \frac{1}{4} \, b^{3} d^{2} x^{4} + \frac{1}{8} \,{\left (2 \, c^{3} d e + 3 \, b c^{2} e^{2}\right )} x^{8} + \frac{1}{7} \,{\left (c^{3} d^{2} + 6 \, b c^{2} d e + 3 \, b^{2} c e^{2}\right )} x^{7} + \frac{1}{6} \,{\left (3 \, b c^{2} d^{2} + 6 \, b^{2} c d e + b^{3} e^{2}\right )} x^{6} + \frac{1}{5} \,{\left (3 \, b^{2} c d^{2} + 2 \, b^{3} d e\right )} x^{5} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*c^3*e^2*x^9 + 1/4*b^3*d^2*x^4 + 1/8*(2*c^3*d*e + 3*b*c^2*e^2)*x^8 + 1/7*(c^3*d^2 + 6*b*c^2*d*e + 3*b^2*c*e
^2)*x^7 + 1/6*(3*b*c^2*d^2 + 6*b^2*c*d*e + b^3*e^2)*x^6 + 1/5*(3*b^2*c*d^2 + 2*b^3*d*e)*x^5

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Fricas [A]  time = 1.42058, size = 301, normalized size = 2.37 \begin{align*} \frac{1}{9} x^{9} e^{2} c^{3} + \frac{1}{4} x^{8} e d c^{3} + \frac{3}{8} x^{8} e^{2} c^{2} b + \frac{1}{7} x^{7} d^{2} c^{3} + \frac{6}{7} x^{7} e d c^{2} b + \frac{3}{7} x^{7} e^{2} c b^{2} + \frac{1}{2} x^{6} d^{2} c^{2} b + x^{6} e d c b^{2} + \frac{1}{6} x^{6} e^{2} b^{3} + \frac{3}{5} x^{5} d^{2} c b^{2} + \frac{2}{5} x^{5} e d b^{3} + \frac{1}{4} x^{4} d^{2} b^{3} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*x^9*e^2*c^3 + 1/4*x^8*e*d*c^3 + 3/8*x^8*e^2*c^2*b + 1/7*x^7*d^2*c^3 + 6/7*x^7*e*d*c^2*b + 3/7*x^7*e^2*c*b^
2 + 1/2*x^6*d^2*c^2*b + x^6*e*d*c*b^2 + 1/6*x^6*e^2*b^3 + 3/5*x^5*d^2*c*b^2 + 2/5*x^5*e*d*b^3 + 1/4*x^4*d^2*b^
3

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Sympy [A]  time = 0.237614, size = 138, normalized size = 1.09 \begin{align*} \frac{b^{3} d^{2} x^{4}}{4} + \frac{c^{3} e^{2} x^{9}}{9} + x^{8} \left (\frac{3 b c^{2} e^{2}}{8} + \frac{c^{3} d e}{4}\right ) + x^{7} \left (\frac{3 b^{2} c e^{2}}{7} + \frac{6 b c^{2} d e}{7} + \frac{c^{3} d^{2}}{7}\right ) + x^{6} \left (\frac{b^{3} e^{2}}{6} + b^{2} c d e + \frac{b c^{2} d^{2}}{2}\right ) + x^{5} \left (\frac{2 b^{3} d e}{5} + \frac{3 b^{2} c d^{2}}{5}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b**3*d**2*x**4/4 + c**3*e**2*x**9/9 + x**8*(3*b*c**2*e**2/8 + c**3*d*e/4) + x**7*(3*b**2*c*e**2/7 + 6*b*c**2*d
*e/7 + c**3*d**2/7) + x**6*(b**3*e**2/6 + b**2*c*d*e + b*c**2*d**2/2) + x**5*(2*b**3*d*e/5 + 3*b**2*c*d**2/5)

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Giac [A]  time = 1.31292, size = 181, normalized size = 1.43 \begin{align*} \frac{1}{9} \, c^{3} x^{9} e^{2} + \frac{1}{4} \, c^{3} d x^{8} e + \frac{1}{7} \, c^{3} d^{2} x^{7} + \frac{3}{8} \, b c^{2} x^{8} e^{2} + \frac{6}{7} \, b c^{2} d x^{7} e + \frac{1}{2} \, b c^{2} d^{2} x^{6} + \frac{3}{7} \, b^{2} c x^{7} e^{2} + b^{2} c d x^{6} e + \frac{3}{5} \, b^{2} c d^{2} x^{5} + \frac{1}{6} \, b^{3} x^{6} e^{2} + \frac{2}{5} \, b^{3} d x^{5} e + \frac{1}{4} \, b^{3} d^{2} x^{4} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*c^3*x^9*e^2 + 1/4*c^3*d*x^8*e + 1/7*c^3*d^2*x^7 + 3/8*b*c^2*x^8*e^2 + 6/7*b*c^2*d*x^7*e + 1/2*b*c^2*d^2*x^
6 + 3/7*b^2*c*x^7*e^2 + b^2*c*d*x^6*e + 3/5*b^2*c*d^2*x^5 + 1/6*b^3*x^6*e^2 + 2/5*b^3*d*x^5*e + 1/4*b^3*d^2*x^
4