### 3.1466 $$\int (d+e x)^3 (a^2+2 a b x+b^2 x^2)^2 \, dx$$

Optimal. Leaf size=92 $\frac{3 e^2 (a+b x)^7 (b d-a e)}{7 b^4}+\frac{e (a+b x)^6 (b d-a e)^2}{2 b^4}+\frac{(a+b x)^5 (b d-a e)^3}{5 b^4}+\frac{e^3 (a+b x)^8}{8 b^4}$

[Out]

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

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Rubi [A]  time = 0.130013, antiderivative size = 92, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 26, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.077, Rules used = {27, 43} $\frac{3 e^2 (a+b x)^7 (b d-a e)}{7 b^4}+\frac{e (a+b x)^6 (b d-a e)^2}{2 b^4}+\frac{(a+b x)^5 (b d-a e)^3}{5 b^4}+\frac{e^3 (a+b x)^8}{8 b^4}$

Antiderivative was successfully veriﬁed.

[In]

Int[(d + e*x)^3*(a^2 + 2*a*b*x + b^2*x^2)^2,x]

[Out]

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

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

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

Mathematica [B]  time = 0.0311339, size = 217, normalized size = 2.36 $\frac{1}{2} b^2 e x^6 \left (2 a^2 e^2+4 a b d e+b^2 d^2\right )+\frac{1}{5} b x^5 \left (18 a^2 b d e^2+4 a^3 e^3+12 a b^2 d^2 e+b^3 d^3\right )+\frac{1}{4} a x^4 \left (12 a^2 b d e^2+a^3 e^3+18 a b^2 d^2 e+4 b^3 d^3\right )+a^2 d x^3 \left (a^2 e^2+4 a b d e+2 b^2 d^2\right )+\frac{1}{2} a^3 d^2 x^2 (3 a e+4 b d)+a^4 d^3 x+\frac{1}{7} b^3 e^2 x^7 (4 a e+3 b d)+\frac{1}{8} b^4 e^3 x^8$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

a^4*d^3*x + (a^3*d^2*(4*b*d + 3*a*e)*x^2)/2 + a^2*d*(2*b^2*d^2 + 4*a*b*d*e + a^2*e^2)*x^3 + (a*(4*b^3*d^3 + 18
*a*b^2*d^2*e + 12*a^2*b*d*e^2 + a^3*e^3)*x^4)/4 + (b*(b^3*d^3 + 12*a*b^2*d^2*e + 18*a^2*b*d*e^2 + 4*a^3*e^3)*x
^5)/5 + (b^2*e*(b^2*d^2 + 4*a*b*d*e + 2*a^2*e^2)*x^6)/2 + (b^3*e^2*(3*b*d + 4*a*e)*x^7)/7 + (b^4*e^3*x^8)/8

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Maple [B]  time = 0.041, size = 229, normalized size = 2.5 \begin{align*}{\frac{{e}^{3}{b}^{4}{x}^{8}}{8}}+{\frac{ \left ( 4\,{e}^{3}a{b}^{3}+3\,d{e}^{2}{b}^{4} \right ){x}^{7}}{7}}+{\frac{ \left ( 6\,{e}^{3}{b}^{2}{a}^{2}+12\,d{e}^{2}a{b}^{3}+3\,{d}^{2}e{b}^{4} \right ){x}^{6}}{6}}+{\frac{ \left ( 4\,{e}^{3}{a}^{3}b+18\,d{e}^{2}{b}^{2}{a}^{2}+12\,{d}^{2}ea{b}^{3}+{d}^{3}{b}^{4} \right ){x}^{5}}{5}}+{\frac{ \left ({e}^{3}{a}^{4}+12\,d{e}^{2}{a}^{3}b+18\,{d}^{2}e{b}^{2}{a}^{2}+4\,{d}^{3}a{b}^{3} \right ){x}^{4}}{4}}+{\frac{ \left ( 3\,d{e}^{2}{a}^{4}+12\,{d}^{2}e{a}^{3}b+6\,{d}^{3}{b}^{2}{a}^{2} \right ){x}^{3}}{3}}+{\frac{ \left ( 3\,{d}^{2}e{a}^{4}+4\,{d}^{3}{a}^{3}b \right ){x}^{2}}{2}}+{a}^{4}{d}^{3}x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*e^3*b^4*x^8+1/7*(4*a*b^3*e^3+3*b^4*d*e^2)*x^7+1/6*(6*a^2*b^2*e^3+12*a*b^3*d*e^2+3*b^4*d^2*e)*x^6+1/5*(4*a^
3*b*e^3+18*a^2*b^2*d*e^2+12*a*b^3*d^2*e+b^4*d^3)*x^5+1/4*(a^4*e^3+12*a^3*b*d*e^2+18*a^2*b^2*d^2*e+4*a*b^3*d^3)
*x^4+1/3*(3*a^4*d*e^2+12*a^3*b*d^2*e+6*a^2*b^2*d^3)*x^3+1/2*(3*a^4*d^2*e+4*a^3*b*d^3)*x^2+a^4*d^3*x

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Maxima [B]  time = 1.18706, size = 304, normalized size = 3.3 \begin{align*} \frac{1}{8} \, b^{4} e^{3} x^{8} + a^{4} d^{3} x + \frac{1}{7} \,{\left (3 \, b^{4} d e^{2} + 4 \, a b^{3} e^{3}\right )} x^{7} + \frac{1}{2} \,{\left (b^{4} d^{2} e + 4 \, a b^{3} d e^{2} + 2 \, a^{2} b^{2} e^{3}\right )} x^{6} + \frac{1}{5} \,{\left (b^{4} d^{3} + 12 \, a b^{3} d^{2} e + 18 \, a^{2} b^{2} d e^{2} + 4 \, a^{3} b e^{3}\right )} x^{5} + \frac{1}{4} \,{\left (4 \, a b^{3} d^{3} + 18 \, a^{2} b^{2} d^{2} e + 12 \, a^{3} b d e^{2} + a^{4} e^{3}\right )} x^{4} +{\left (2 \, a^{2} b^{2} d^{3} + 4 \, a^{3} b d^{2} e + a^{4} d e^{2}\right )} x^{3} + \frac{1}{2} \,{\left (4 \, a^{3} b d^{3} + 3 \, a^{4} d^{2} e\right )} x^{2} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*b^4*e^3*x^8 + a^4*d^3*x + 1/7*(3*b^4*d*e^2 + 4*a*b^3*e^3)*x^7 + 1/2*(b^4*d^2*e + 4*a*b^3*d*e^2 + 2*a^2*b^2
*e^3)*x^6 + 1/5*(b^4*d^3 + 12*a*b^3*d^2*e + 18*a^2*b^2*d*e^2 + 4*a^3*b*e^3)*x^5 + 1/4*(4*a*b^3*d^3 + 18*a^2*b^
2*d^2*e + 12*a^3*b*d*e^2 + a^4*e^3)*x^4 + (2*a^2*b^2*d^3 + 4*a^3*b*d^2*e + a^4*d*e^2)*x^3 + 1/2*(4*a^3*b*d^3 +
3*a^4*d^2*e)*x^2

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Fricas [B]  time = 1.51139, size = 520, normalized size = 5.65 \begin{align*} \frac{1}{8} x^{8} e^{3} b^{4} + \frac{3}{7} x^{7} e^{2} d b^{4} + \frac{4}{7} x^{7} e^{3} b^{3} a + \frac{1}{2} x^{6} e d^{2} b^{4} + 2 x^{6} e^{2} d b^{3} a + x^{6} e^{3} b^{2} a^{2} + \frac{1}{5} x^{5} d^{3} b^{4} + \frac{12}{5} x^{5} e d^{2} b^{3} a + \frac{18}{5} x^{5} e^{2} d b^{2} a^{2} + \frac{4}{5} x^{5} e^{3} b a^{3} + x^{4} d^{3} b^{3} a + \frac{9}{2} x^{4} e d^{2} b^{2} a^{2} + 3 x^{4} e^{2} d b a^{3} + \frac{1}{4} x^{4} e^{3} a^{4} + 2 x^{3} d^{3} b^{2} a^{2} + 4 x^{3} e d^{2} b a^{3} + x^{3} e^{2} d a^{4} + 2 x^{2} d^{3} b a^{3} + \frac{3}{2} x^{2} e d^{2} a^{4} + x d^{3} a^{4} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*x^8*e^3*b^4 + 3/7*x^7*e^2*d*b^4 + 4/7*x^7*e^3*b^3*a + 1/2*x^6*e*d^2*b^4 + 2*x^6*e^2*d*b^3*a + x^6*e^3*b^2*
a^2 + 1/5*x^5*d^3*b^4 + 12/5*x^5*e*d^2*b^3*a + 18/5*x^5*e^2*d*b^2*a^2 + 4/5*x^5*e^3*b*a^3 + x^4*d^3*b^3*a + 9/
2*x^4*e*d^2*b^2*a^2 + 3*x^4*e^2*d*b*a^3 + 1/4*x^4*e^3*a^4 + 2*x^3*d^3*b^2*a^2 + 4*x^3*e*d^2*b*a^3 + x^3*e^2*d*
a^4 + 2*x^2*d^3*b*a^3 + 3/2*x^2*e*d^2*a^4 + x*d^3*a^4

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Sympy [B]  time = 0.109806, size = 243, normalized size = 2.64 \begin{align*} a^{4} d^{3} x + \frac{b^{4} e^{3} x^{8}}{8} + x^{7} \left (\frac{4 a b^{3} e^{3}}{7} + \frac{3 b^{4} d e^{2}}{7}\right ) + x^{6} \left (a^{2} b^{2} e^{3} + 2 a b^{3} d e^{2} + \frac{b^{4} d^{2} e}{2}\right ) + x^{5} \left (\frac{4 a^{3} b e^{3}}{5} + \frac{18 a^{2} b^{2} d e^{2}}{5} + \frac{12 a b^{3} d^{2} e}{5} + \frac{b^{4} d^{3}}{5}\right ) + x^{4} \left (\frac{a^{4} e^{3}}{4} + 3 a^{3} b d e^{2} + \frac{9 a^{2} b^{2} d^{2} e}{2} + a b^{3} d^{3}\right ) + x^{3} \left (a^{4} d e^{2} + 4 a^{3} b d^{2} e + 2 a^{2} b^{2} d^{3}\right ) + x^{2} \left (\frac{3 a^{4} d^{2} e}{2} + 2 a^{3} b d^{3}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**4*d**3*x + b**4*e**3*x**8/8 + x**7*(4*a*b**3*e**3/7 + 3*b**4*d*e**2/7) + x**6*(a**2*b**2*e**3 + 2*a*b**3*d*
e**2 + b**4*d**2*e/2) + x**5*(4*a**3*b*e**3/5 + 18*a**2*b**2*d*e**2/5 + 12*a*b**3*d**2*e/5 + b**4*d**3/5) + x*
*4*(a**4*e**3/4 + 3*a**3*b*d*e**2 + 9*a**2*b**2*d**2*e/2 + a*b**3*d**3) + x**3*(a**4*d*e**2 + 4*a**3*b*d**2*e
+ 2*a**2*b**2*d**3) + x**2*(3*a**4*d**2*e/2 + 2*a**3*b*d**3)

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Giac [B]  time = 1.16648, size = 324, normalized size = 3.52 \begin{align*} \frac{1}{8} \, b^{4} x^{8} e^{3} + \frac{3}{7} \, b^{4} d x^{7} e^{2} + \frac{1}{2} \, b^{4} d^{2} x^{6} e + \frac{1}{5} \, b^{4} d^{3} x^{5} + \frac{4}{7} \, a b^{3} x^{7} e^{3} + 2 \, a b^{3} d x^{6} e^{2} + \frac{12}{5} \, a b^{3} d^{2} x^{5} e + a b^{3} d^{3} x^{4} + a^{2} b^{2} x^{6} e^{3} + \frac{18}{5} \, a^{2} b^{2} d x^{5} e^{2} + \frac{9}{2} \, a^{2} b^{2} d^{2} x^{4} e + 2 \, a^{2} b^{2} d^{3} x^{3} + \frac{4}{5} \, a^{3} b x^{5} e^{3} + 3 \, a^{3} b d x^{4} e^{2} + 4 \, a^{3} b d^{2} x^{3} e + 2 \, a^{3} b d^{3} x^{2} + \frac{1}{4} \, a^{4} x^{4} e^{3} + a^{4} d x^{3} e^{2} + \frac{3}{2} \, a^{4} d^{2} x^{2} e + a^{4} d^{3} x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*b^4*x^8*e^3 + 3/7*b^4*d*x^7*e^2 + 1/2*b^4*d^2*x^6*e + 1/5*b^4*d^3*x^5 + 4/7*a*b^3*x^7*e^3 + 2*a*b^3*d*x^6*
e^2 + 12/5*a*b^3*d^2*x^5*e + a*b^3*d^3*x^4 + a^2*b^2*x^6*e^3 + 18/5*a^2*b^2*d*x^5*e^2 + 9/2*a^2*b^2*d^2*x^4*e
+ 2*a^2*b^2*d^3*x^3 + 4/5*a^3*b*x^5*e^3 + 3*a^3*b*d*x^4*e^2 + 4*a^3*b*d^2*x^3*e + 2*a^3*b*d^3*x^2 + 1/4*a^4*x^
4*e^3 + a^4*d*x^3*e^2 + 3/2*a^4*d^2*x^2*e + a^4*d^3*x