### 3.233 $$\int (d+e x)^2 (b x+c x^2)^2 \, dx$$

Optimal. Leaf size=87 $\frac{1}{5} x^5 \left (b^2 e^2+4 b c d e+c^2 d^2\right )+\frac{1}{3} b^2 d^2 x^3+\frac{1}{3} c e x^6 (b e+c d)+\frac{1}{2} b d x^4 (b e+c d)+\frac{1}{7} c^2 e^2 x^7$

[Out]

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

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Rubi [A]  time = 0.0764796, antiderivative size = 87, 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} $\frac{1}{5} x^5 \left (b^2 e^2+4 b c d e+c^2 d^2\right )+\frac{1}{3} b^2 d^2 x^3+\frac{1}{3} c e x^6 (b e+c d)+\frac{1}{2} b d x^4 (b e+c d)+\frac{1}{7} c^2 e^2 x^7$

Antiderivative was successfully veriﬁed.

[In]

Int[(d + e*x)^2*(b*x + c*x^2)^2,x]

[Out]

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

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

Mathematica [A]  time = 0.0141169, size = 87, normalized size = 1. $\frac{1}{5} x^5 \left (b^2 e^2+4 b c d e+c^2 d^2\right )+\frac{1}{3} b^2 d^2 x^3+\frac{1}{3} c e x^6 (b e+c d)+\frac{1}{2} b d x^4 (b e+c d)+\frac{1}{7} c^2 e^2 x^7$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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Maple [A]  time = 0.045, size = 90, normalized size = 1. \begin{align*}{\frac{{c}^{2}{e}^{2}{x}^{7}}{7}}+{\frac{ \left ( 2\,{e}^{2}bc+2\,de{c}^{2} \right ){x}^{6}}{6}}+{\frac{ \left ({b}^{2}{e}^{2}+4\,bcde+{c}^{2}{d}^{2} \right ){x}^{5}}{5}}+{\frac{ \left ( 2\,{b}^{2}de+2\,{d}^{2}bc \right ){x}^{4}}{4}}+{\frac{{b}^{2}{d}^{2}{x}^{3}}{3}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.11869, size = 115, normalized size = 1.32 \begin{align*} \frac{1}{7} \, c^{2} e^{2} x^{7} + \frac{1}{3} \, b^{2} d^{2} x^{3} + \frac{1}{3} \,{\left (c^{2} d e + b c e^{2}\right )} x^{6} + \frac{1}{5} \,{\left (c^{2} d^{2} + 4 \, b c d e + b^{2} e^{2}\right )} x^{5} + \frac{1}{2} \,{\left (b c d^{2} + b^{2} d e\right )} 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)^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.40058, size = 217, normalized size = 2.49 \begin{align*} \frac{1}{7} x^{7} e^{2} c^{2} + \frac{1}{3} x^{6} e d c^{2} + \frac{1}{3} x^{6} e^{2} c b + \frac{1}{5} x^{5} d^{2} c^{2} + \frac{4}{5} x^{5} e d c b + \frac{1}{5} x^{5} e^{2} b^{2} + \frac{1}{2} x^{4} d^{2} c b + \frac{1}{2} x^{4} e d b^{2} + \frac{1}{3} x^{3} d^{2} b^{2} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.266324, size = 94, normalized size = 1.08 \begin{align*} \frac{b^{2} d^{2} x^{3}}{3} + \frac{c^{2} e^{2} x^{7}}{7} + x^{6} \left (\frac{b c e^{2}}{3} + \frac{c^{2} d e}{3}\right ) + x^{5} \left (\frac{b^{2} e^{2}}{5} + \frac{4 b c d e}{5} + \frac{c^{2} d^{2}}{5}\right ) + x^{4} \left (\frac{b^{2} d e}{2} + \frac{b c d^{2}}{2}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.16886, size = 127, normalized size = 1.46 \begin{align*} \frac{1}{7} \, c^{2} x^{7} e^{2} + \frac{1}{3} \, c^{2} d x^{6} e + \frac{1}{5} \, c^{2} d^{2} x^{5} + \frac{1}{3} \, b c x^{6} e^{2} + \frac{4}{5} \, b c d x^{5} e + \frac{1}{2} \, b c d^{2} x^{4} + \frac{1}{5} \, b^{2} x^{5} e^{2} + \frac{1}{2} \, b^{2} d x^{4} e + \frac{1}{3} \, b^{2} d^{2} x^{3} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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