### 3.245 $$\int (d+e x)^3 (b x+c x^2)^3 \, dx$$

Optimal. Leaf size=162 $\frac{3}{8} c e x^8 \left (b^2 e^2+3 b c d e+c^2 d^2\right )+\frac{1}{7} x^7 (b e+c d) \left (b^2 e^2+8 b c d e+c^2 d^2\right )+\frac{1}{2} b d x^6 \left (b^2 e^2+3 b c d e+c^2 d^2\right )+\frac{3}{5} b^2 d^2 x^5 (b e+c d)+\frac{1}{4} b^3 d^3 x^4+\frac{1}{3} c^2 e^2 x^9 (b e+c d)+\frac{1}{10} c^3 e^3 x^{10}$

[Out]

(b^3*d^3*x^4)/4 + (3*b^2*d^2*(c*d + b*e)*x^5)/5 + (b*d*(c^2*d^2 + 3*b*c*d*e + b^2*e^2)*x^6)/2 + ((c*d + b*e)*(
c^2*d^2 + 8*b*c*d*e + b^2*e^2)*x^7)/7 + (3*c*e*(c^2*d^2 + 3*b*c*d*e + b^2*e^2)*x^8)/8 + (c^2*e^2*(c*d + b*e)*x
^9)/3 + (c^3*e^3*x^10)/10

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Rubi [A]  time = 0.15603, antiderivative size = 162, 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} $\frac{3}{8} c e x^8 \left (b^2 e^2+3 b c d e+c^2 d^2\right )+\frac{1}{7} x^7 (b e+c d) \left (b^2 e^2+8 b c d e+c^2 d^2\right )+\frac{1}{2} b d x^6 \left (b^2 e^2+3 b c d e+c^2 d^2\right )+\frac{3}{5} b^2 d^2 x^5 (b e+c d)+\frac{1}{4} b^3 d^3 x^4+\frac{1}{3} c^2 e^2 x^9 (b e+c d)+\frac{1}{10} c^3 e^3 x^{10}$

Antiderivative was successfully veriﬁed.

[In]

Int[(d + e*x)^3*(b*x + c*x^2)^3,x]

[Out]

(b^3*d^3*x^4)/4 + (3*b^2*d^2*(c*d + b*e)*x^5)/5 + (b*d*(c^2*d^2 + 3*b*c*d*e + b^2*e^2)*x^6)/2 + ((c*d + b*e)*(
c^2*d^2 + 8*b*c*d*e + b^2*e^2)*x^7)/7 + (3*c*e*(c^2*d^2 + 3*b*c*d*e + b^2*e^2)*x^8)/8 + (c^2*e^2*(c*d + b*e)*x
^9)/3 + (c^3*e^3*x^10)/10

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 )^3 \, dx &=\int \left (b^3 d^3 x^3+3 b^2 d^2 (c d+b e) x^4+3 b d \left (c^2 d^2+3 b c d e+b^2 e^2\right ) x^5+(c d+b e) \left (c^2 d^2+8 b c d e+b^2 e^2\right ) x^6+3 c e \left (c^2 d^2+3 b c d e+b^2 e^2\right ) x^7+3 c^2 e^2 (c d+b e) x^8+c^3 e^3 x^9\right ) \, dx\\ &=\frac{1}{4} b^3 d^3 x^4+\frac{3}{5} b^2 d^2 (c d+b e) x^5+\frac{1}{2} b d \left (c^2 d^2+3 b c d e+b^2 e^2\right ) x^6+\frac{1}{7} (c d+b e) \left (c^2 d^2+8 b c d e+b^2 e^2\right ) x^7+\frac{3}{8} c e \left (c^2 d^2+3 b c d e+b^2 e^2\right ) x^8+\frac{1}{3} c^2 e^2 (c d+b e) x^9+\frac{1}{10} c^3 e^3 x^{10}\\ \end{align*}

Mathematica [A]  time = 0.0257193, size = 169, normalized size = 1.04 $\frac{3}{8} c e x^8 \left (b^2 e^2+3 b c d e+c^2 d^2\right )+\frac{1}{7} x^7 \left (9 b^2 c d e^2+b^3 e^3+9 b c^2 d^2 e+c^3 d^3\right )+\frac{1}{2} b d x^6 \left (b^2 e^2+3 b c d e+c^2 d^2\right )+\frac{3}{5} b^2 d^2 x^5 (b e+c d)+\frac{1}{4} b^3 d^3 x^4+\frac{1}{3} c^2 e^2 x^9 (b e+c d)+\frac{1}{10} c^3 e^3 x^{10}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(b^3*d^3*x^4)/4 + (3*b^2*d^2*(c*d + b*e)*x^5)/5 + (b*d*(c^2*d^2 + 3*b*c*d*e + b^2*e^2)*x^6)/2 + ((c^3*d^3 + 9*
b*c^2*d^2*e + 9*b^2*c*d*e^2 + b^3*e^3)*x^7)/7 + (3*c*e*(c^2*d^2 + 3*b*c*d*e + b^2*e^2)*x^8)/8 + (c^2*e^2*(c*d
+ b*e)*x^9)/3 + (c^3*e^3*x^10)/10

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Maple [A]  time = 0.042, size = 180, normalized size = 1.1 \begin{align*}{\frac{{c}^{3}{e}^{3}{x}^{10}}{10}}+{\frac{ \left ( 3\,{e}^{3}b{c}^{2}+3\,d{e}^{2}{c}^{3} \right ){x}^{9}}{9}}+{\frac{ \left ( 3\,{e}^{3}{b}^{2}c+9\,d{e}^{2}b{c}^{2}+3\,{d}^{2}e{c}^{3} \right ){x}^{8}}{8}}+{\frac{ \left ({b}^{3}{e}^{3}+9\,{b}^{2}cd{e}^{2}+9\,b{c}^{2}{d}^{2}e+{c}^{3}{d}^{3} \right ){x}^{7}}{7}}+{\frac{ \left ( 3\,{b}^{3}d{e}^{2}+9\,{d}^{2}e{b}^{2}c+3\,{d}^{3}b{c}^{2} \right ){x}^{6}}{6}}+{\frac{ \left ( 3\,{d}^{2}e{b}^{3}+3\,{d}^{3}{b}^{2}c \right ){x}^{5}}{5}}+{\frac{{b}^{3}{d}^{3}{x}^{4}}{4}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*c^3*e^3*x^10+1/9*(3*b*c^2*e^3+3*c^3*d*e^2)*x^9+1/8*(3*b^2*c*e^3+9*b*c^2*d*e^2+3*c^3*d^2*e)*x^8+1/7*(b^3*e
^3+9*b^2*c*d*e^2+9*b*c^2*d^2*e+c^3*d^3)*x^7+1/6*(3*b^3*d*e^2+9*b^2*c*d^2*e+3*b*c^2*d^3)*x^6+1/5*(3*b^3*d^2*e+3
*b^2*c*d^3)*x^5+1/4*b^3*d^3*x^4

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Maxima [A]  time = 1.14228, size = 231, normalized size = 1.43 \begin{align*} \frac{1}{10} \, c^{3} e^{3} x^{10} + \frac{1}{4} \, b^{3} d^{3} x^{4} + \frac{1}{3} \,{\left (c^{3} d e^{2} + b c^{2} e^{3}\right )} x^{9} + \frac{3}{8} \,{\left (c^{3} d^{2} e + 3 \, b c^{2} d e^{2} + b^{2} c e^{3}\right )} x^{8} + \frac{1}{7} \,{\left (c^{3} d^{3} + 9 \, b c^{2} d^{2} e + 9 \, b^{2} c d e^{2} + b^{3} e^{3}\right )} x^{7} + \frac{1}{2} \,{\left (b c^{2} d^{3} + 3 \, b^{2} c d^{2} e + b^{3} d e^{2}\right )} x^{6} + \frac{3}{5} \,{\left (b^{2} c d^{3} + b^{3} d^{2} e\right )} x^{5} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*c^3*e^3*x^10 + 1/4*b^3*d^3*x^4 + 1/3*(c^3*d*e^2 + b*c^2*e^3)*x^9 + 3/8*(c^3*d^2*e + 3*b*c^2*d*e^2 + b^2*c
*e^3)*x^8 + 1/7*(c^3*d^3 + 9*b*c^2*d^2*e + 9*b^2*c*d*e^2 + b^3*e^3)*x^7 + 1/2*(b*c^2*d^3 + 3*b^2*c*d^2*e + b^3
*d*e^2)*x^6 + 3/5*(b^2*c*d^3 + b^3*d^2*e)*x^5

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Fricas [A]  time = 1.37637, size = 433, normalized size = 2.67 \begin{align*} \frac{1}{10} x^{10} e^{3} c^{3} + \frac{1}{3} x^{9} e^{2} d c^{3} + \frac{1}{3} x^{9} e^{3} c^{2} b + \frac{3}{8} x^{8} e d^{2} c^{3} + \frac{9}{8} x^{8} e^{2} d c^{2} b + \frac{3}{8} x^{8} e^{3} c b^{2} + \frac{1}{7} x^{7} d^{3} c^{3} + \frac{9}{7} x^{7} e d^{2} c^{2} b + \frac{9}{7} x^{7} e^{2} d c b^{2} + \frac{1}{7} x^{7} e^{3} b^{3} + \frac{1}{2} x^{6} d^{3} c^{2} b + \frac{3}{2} x^{6} e d^{2} c b^{2} + \frac{1}{2} x^{6} e^{2} d b^{3} + \frac{3}{5} x^{5} d^{3} c b^{2} + \frac{3}{5} x^{5} e d^{2} b^{3} + \frac{1}{4} x^{4} d^{3} b^{3} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*x^10*e^3*c^3 + 1/3*x^9*e^2*d*c^3 + 1/3*x^9*e^3*c^2*b + 3/8*x^8*e*d^2*c^3 + 9/8*x^8*e^2*d*c^2*b + 3/8*x^8*
e^3*c*b^2 + 1/7*x^7*d^3*c^3 + 9/7*x^7*e*d^2*c^2*b + 9/7*x^7*e^2*d*c*b^2 + 1/7*x^7*e^3*b^3 + 1/2*x^6*d^3*c^2*b
+ 3/2*x^6*e*d^2*c*b^2 + 1/2*x^6*e^2*d*b^3 + 3/5*x^5*d^3*c*b^2 + 3/5*x^5*e*d^2*b^3 + 1/4*x^4*d^3*b^3

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Sympy [A]  time = 0.335741, size = 199, normalized size = 1.23 \begin{align*} \frac{b^{3} d^{3} x^{4}}{4} + \frac{c^{3} e^{3} x^{10}}{10} + x^{9} \left (\frac{b c^{2} e^{3}}{3} + \frac{c^{3} d e^{2}}{3}\right ) + x^{8} \left (\frac{3 b^{2} c e^{3}}{8} + \frac{9 b c^{2} d e^{2}}{8} + \frac{3 c^{3} d^{2} e}{8}\right ) + x^{7} \left (\frac{b^{3} e^{3}}{7} + \frac{9 b^{2} c d e^{2}}{7} + \frac{9 b c^{2} d^{2} e}{7} + \frac{c^{3} d^{3}}{7}\right ) + x^{6} \left (\frac{b^{3} d e^{2}}{2} + \frac{3 b^{2} c d^{2} e}{2} + \frac{b c^{2} d^{3}}{2}\right ) + x^{5} \left (\frac{3 b^{3} d^{2} e}{5} + \frac{3 b^{2} c d^{3}}{5}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b**3*d**3*x**4/4 + c**3*e**3*x**10/10 + x**9*(b*c**2*e**3/3 + c**3*d*e**2/3) + x**8*(3*b**2*c*e**3/8 + 9*b*c**
2*d*e**2/8 + 3*c**3*d**2*e/8) + x**7*(b**3*e**3/7 + 9*b**2*c*d*e**2/7 + 9*b*c**2*d**2*e/7 + c**3*d**3/7) + x**
6*(b**3*d*e**2/2 + 3*b**2*c*d**2*e/2 + b*c**2*d**3/2) + x**5*(3*b**3*d**2*e/5 + 3*b**2*c*d**3/5)

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Giac [A]  time = 1.2647, size = 255, normalized size = 1.57 \begin{align*} \frac{1}{10} \, c^{3} x^{10} e^{3} + \frac{1}{3} \, c^{3} d x^{9} e^{2} + \frac{3}{8} \, c^{3} d^{2} x^{8} e + \frac{1}{7} \, c^{3} d^{3} x^{7} + \frac{1}{3} \, b c^{2} x^{9} e^{3} + \frac{9}{8} \, b c^{2} d x^{8} e^{2} + \frac{9}{7} \, b c^{2} d^{2} x^{7} e + \frac{1}{2} \, b c^{2} d^{3} x^{6} + \frac{3}{8} \, b^{2} c x^{8} e^{3} + \frac{9}{7} \, b^{2} c d x^{7} e^{2} + \frac{3}{2} \, b^{2} c d^{2} x^{6} e + \frac{3}{5} \, b^{2} c d^{3} x^{5} + \frac{1}{7} \, b^{3} x^{7} e^{3} + \frac{1}{2} \, b^{3} d x^{6} e^{2} + \frac{3}{5} \, b^{3} d^{2} x^{5} e + \frac{1}{4} \, b^{3} d^{3} 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)^3,x, algorithm="giac")

[Out]

1/10*c^3*x^10*e^3 + 1/3*c^3*d*x^9*e^2 + 3/8*c^3*d^2*x^8*e + 1/7*c^3*d^3*x^7 + 1/3*b*c^2*x^9*e^3 + 9/8*b*c^2*d*
x^8*e^2 + 9/7*b*c^2*d^2*x^7*e + 1/2*b*c^2*d^3*x^6 + 3/8*b^2*c*x^8*e^3 + 9/7*b^2*c*d*x^7*e^2 + 3/2*b^2*c*d^2*x^
6*e + 3/5*b^2*c*d^3*x^5 + 1/7*b^3*x^7*e^3 + 1/2*b^3*d*x^6*e^2 + 3/5*b^3*d^2*x^5*e + 1/4*b^3*d^3*x^4