### 3.231 $$\int (d+e x)^4 (b x+c x^2)^2 \, dx$$

Optimal. Leaf size=137 $\frac{(d+e x)^7 \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )}{7 e^5}+\frac{d^2 (d+e x)^5 (c d-b e)^2}{5 e^5}-\frac{c (d+e x)^8 (2 c d-b e)}{4 e^5}-\frac{d (d+e x)^6 (c d-b e) (2 c d-b e)}{3 e^5}+\frac{c^2 (d+e x)^9}{9 e^5}$

[Out]

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

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Rubi [A]  time = 0.140859, antiderivative size = 137, 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} $\frac{(d+e x)^7 \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )}{7 e^5}+\frac{d^2 (d+e x)^5 (c d-b e)^2}{5 e^5}-\frac{c (d+e x)^8 (2 c d-b e)}{4 e^5}-\frac{d (d+e x)^6 (c d-b e) (2 c d-b e)}{3 e^5}+\frac{c^2 (d+e x)^9}{9 e^5}$

Antiderivative was successfully veriﬁed.

[In]

Int[(d + e*x)^4*(b*x + c*x^2)^2,x]

[Out]

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

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

Mathematica [A]  time = 0.0255949, size = 159, normalized size = 1.16 $\frac{1}{7} e^2 x^7 \left (b^2 e^2+8 b c d e+6 c^2 d^2\right )+\frac{2}{3} d e x^6 \left (b^2 e^2+3 b c d e+c^2 d^2\right )+\frac{1}{5} d^2 x^5 \left (6 b^2 e^2+8 b c d e+c^2 d^2\right )+\frac{1}{3} b^2 d^4 x^3+\frac{1}{2} b d^3 x^4 (2 b e+c d)+\frac{1}{4} c e^3 x^8 (b e+2 c d)+\frac{1}{9} c^2 e^4 x^9$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(d + e*x)^4*(b*x + c*x^2)^2,x]

[Out]

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

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Maple [A]  time = 0.044, size = 166, normalized size = 1.2 \begin{align*}{\frac{{e}^{4}{c}^{2}{x}^{9}}{9}}+{\frac{ \left ( 2\,{e}^{4}bc+4\,d{e}^{3}{c}^{2} \right ){x}^{8}}{8}}+{\frac{ \left ({e}^{4}{b}^{2}+8\,d{e}^{3}bc+6\,{d}^{2}{e}^{2}{c}^{2} \right ){x}^{7}}{7}}+{\frac{ \left ( 4\,d{e}^{3}{b}^{2}+12\,{d}^{2}{e}^{2}bc+4\,{d}^{3}e{c}^{2} \right ){x}^{6}}{6}}+{\frac{ \left ( 6\,{d}^{2}{e}^{2}{b}^{2}+8\,{d}^{3}ebc+{c}^{2}{d}^{4} \right ){x}^{5}}{5}}+{\frac{ \left ( 4\,{d}^{3}e{b}^{2}+2\,{d}^{4}bc \right ){x}^{4}}{4}}+{\frac{{d}^{4}{b}^{2}{x}^{3}}{3}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^4*(c*x^2+b*x)^2,x)

[Out]

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

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Maxima [A]  time = 1.16143, size = 217, normalized size = 1.58 \begin{align*} \frac{1}{9} \, c^{2} e^{4} x^{9} + \frac{1}{3} \, b^{2} d^{4} x^{3} + \frac{1}{4} \,{\left (2 \, c^{2} d e^{3} + b c e^{4}\right )} x^{8} + \frac{1}{7} \,{\left (6 \, c^{2} d^{2} e^{2} + 8 \, b c d e^{3} + b^{2} e^{4}\right )} x^{7} + \frac{2}{3} \,{\left (c^{2} d^{3} e + 3 \, b c d^{2} e^{2} + b^{2} d e^{3}\right )} x^{6} + \frac{1}{5} \,{\left (c^{2} d^{4} + 8 \, b c d^{3} e + 6 \, b^{2} d^{2} e^{2}\right )} x^{5} + \frac{1}{2} \,{\left (b c d^{4} + 2 \, b^{2} d^{3} e\right )} x^{4} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^4*(c*x^2+b*x)^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.38888, size = 387, normalized size = 2.82 \begin{align*} \frac{1}{9} x^{9} e^{4} c^{2} + \frac{1}{2} x^{8} e^{3} d c^{2} + \frac{1}{4} x^{8} e^{4} c b + \frac{6}{7} x^{7} e^{2} d^{2} c^{2} + \frac{8}{7} x^{7} e^{3} d c b + \frac{1}{7} x^{7} e^{4} b^{2} + \frac{2}{3} x^{6} e d^{3} c^{2} + 2 x^{6} e^{2} d^{2} c b + \frac{2}{3} x^{6} e^{3} d b^{2} + \frac{1}{5} x^{5} d^{4} c^{2} + \frac{8}{5} x^{5} e d^{3} c b + \frac{6}{5} x^{5} e^{2} d^{2} b^{2} + \frac{1}{2} x^{4} d^{4} c b + x^{4} e d^{3} b^{2} + \frac{1}{3} x^{3} d^{4} b^{2} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^4*(c*x^2+b*x)^2,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.374781, size = 178, normalized size = 1.3 \begin{align*} \frac{b^{2} d^{4} x^{3}}{3} + \frac{c^{2} e^{4} x^{9}}{9} + x^{8} \left (\frac{b c e^{4}}{4} + \frac{c^{2} d e^{3}}{2}\right ) + x^{7} \left (\frac{b^{2} e^{4}}{7} + \frac{8 b c d e^{3}}{7} + \frac{6 c^{2} d^{2} e^{2}}{7}\right ) + x^{6} \left (\frac{2 b^{2} d e^{3}}{3} + 2 b c d^{2} e^{2} + \frac{2 c^{2} d^{3} e}{3}\right ) + x^{5} \left (\frac{6 b^{2} d^{2} e^{2}}{5} + \frac{8 b c d^{3} e}{5} + \frac{c^{2} d^{4}}{5}\right ) + x^{4} \left (b^{2} d^{3} e + \frac{b c d^{4}}{2}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**4*(c*x**2+b*x)**2,x)

[Out]

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

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Giac [A]  time = 1.25349, size = 228, normalized size = 1.66 \begin{align*} \frac{1}{9} \, c^{2} x^{9} e^{4} + \frac{1}{2} \, c^{2} d x^{8} e^{3} + \frac{6}{7} \, c^{2} d^{2} x^{7} e^{2} + \frac{2}{3} \, c^{2} d^{3} x^{6} e + \frac{1}{5} \, c^{2} d^{4} x^{5} + \frac{1}{4} \, b c x^{8} e^{4} + \frac{8}{7} \, b c d x^{7} e^{3} + 2 \, b c d^{2} x^{6} e^{2} + \frac{8}{5} \, b c d^{3} x^{5} e + \frac{1}{2} \, b c d^{4} x^{4} + \frac{1}{7} \, b^{2} x^{7} e^{4} + \frac{2}{3} \, b^{2} d x^{6} e^{3} + \frac{6}{5} \, b^{2} d^{2} x^{5} e^{2} + b^{2} d^{3} x^{4} e + \frac{1}{3} \, b^{2} d^{4} x^{3} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^4*(c*x^2+b*x)^2,x, algorithm="giac")

[Out]

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