### 3.1487 $$\int (d+e x)^2 (a^2+2 a b x+b^2 x^2)^3 \, dx$$

Optimal. Leaf size=65 $\frac{e (a+b x)^8 (b d-a e)}{4 b^3}+\frac{(a+b x)^7 (b d-a e)^2}{7 b^3}+\frac{e^2 (a+b x)^9}{9 b^3}$

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.135757, antiderivative size = 65, 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} $\frac{e (a+b x)^8 (b d-a e)}{4 b^3}+\frac{(a+b x)^7 (b d-a e)^2}{7 b^3}+\frac{e^2 (a+b x)^9}{9 b^3}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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

Mathematica [B]  time = 0.067054, size = 199, normalized size = 3.06 $\frac{1}{252} x \left (126 a^4 b^2 x^2 \left (10 d^2+15 d e x+6 e^2 x^2\right )+84 a^3 b^3 x^3 \left (15 d^2+24 d e x+10 e^2 x^2\right )+36 a^2 b^4 x^4 \left (21 d^2+35 d e x+15 e^2 x^2\right )+126 a^5 b x \left (6 d^2+8 d e x+3 e^2 x^2\right )+84 a^6 \left (3 d^2+3 d e x+e^2 x^2\right )+9 a b^5 x^5 \left (28 d^2+48 d e x+21 e^2 x^2\right )+b^6 x^6 \left (36 d^2+63 d e x+28 e^2 x^2\right )\right )$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(x*(84*a^6*(3*d^2 + 3*d*e*x + e^2*x^2) + 126*a^5*b*x*(6*d^2 + 8*d*e*x + 3*e^2*x^2) + 126*a^4*b^2*x^2*(10*d^2 +
15*d*e*x + 6*e^2*x^2) + 84*a^3*b^3*x^3*(15*d^2 + 24*d*e*x + 10*e^2*x^2) + 36*a^2*b^4*x^4*(21*d^2 + 35*d*e*x +
15*e^2*x^2) + 9*a*b^5*x^5*(28*d^2 + 48*d*e*x + 21*e^2*x^2) + b^6*x^6*(36*d^2 + 63*d*e*x + 28*e^2*x^2)))/252

________________________________________________________________________________________

Maple [B]  time = 0.041, size = 239, normalized size = 3.7 \begin{align*}{\frac{{e}^{2}{b}^{6}{x}^{9}}{9}}+{\frac{ \left ( 6\,{e}^{2}a{b}^{5}+2\,de{b}^{6} \right ){x}^{8}}{8}}+{\frac{ \left ( 15\,{e}^{2}{a}^{2}{b}^{4}+12\,dea{b}^{5}+{b}^{6}{d}^{2} \right ){x}^{7}}{7}}+{\frac{ \left ( 20\,{e}^{2}{a}^{3}{b}^{3}+30\,de{a}^{2}{b}^{4}+6\,{d}^{2}a{b}^{5} \right ){x}^{6}}{6}}+{\frac{ \left ( 15\,{e}^{2}{a}^{4}{b}^{2}+40\,de{a}^{3}{b}^{3}+15\,{d}^{2}{a}^{2}{b}^{4} \right ){x}^{5}}{5}}+{\frac{ \left ( 6\,{e}^{2}{a}^{5}b+30\,de{a}^{4}{b}^{2}+20\,{d}^{2}{a}^{3}{b}^{3} \right ){x}^{4}}{4}}+{\frac{ \left ({e}^{2}{a}^{6}+12\,de{a}^{5}b+15\,{d}^{2}{a}^{4}{b}^{2} \right ){x}^{3}}{3}}+{\frac{ \left ( 2\,de{a}^{6}+6\,{d}^{2}{a}^{5}b \right ){x}^{2}}{2}}+{d}^{2}{a}^{6}x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*e^2*b^6*x^9+1/8*(6*a*b^5*e^2+2*b^6*d*e)*x^8+1/7*(15*a^2*b^4*e^2+12*a*b^5*d*e+b^6*d^2)*x^7+1/6*(20*a^3*b^3*
e^2+30*a^2*b^4*d*e+6*a*b^5*d^2)*x^6+1/5*(15*a^4*b^2*e^2+40*a^3*b^3*d*e+15*a^2*b^4*d^2)*x^5+1/4*(6*a^5*b*e^2+30
*a^4*b^2*d*e+20*a^3*b^3*d^2)*x^4+1/3*(a^6*e^2+12*a^5*b*d*e+15*a^4*b^2*d^2)*x^3+1/2*(2*a^6*d*e+6*a^5*b*d^2)*x^2
+d^2*a^6*x

________________________________________________________________________________________

Maxima [B]  time = 1.09423, size = 316, normalized size = 4.86 \begin{align*} \frac{1}{9} \, b^{6} e^{2} x^{9} + a^{6} d^{2} x + \frac{1}{4} \,{\left (b^{6} d e + 3 \, a b^{5} e^{2}\right )} x^{8} + \frac{1}{7} \,{\left (b^{6} d^{2} + 12 \, a b^{5} d e + 15 \, a^{2} b^{4} e^{2}\right )} x^{7} + \frac{1}{3} \,{\left (3 \, a b^{5} d^{2} + 15 \, a^{2} b^{4} d e + 10 \, a^{3} b^{3} e^{2}\right )} x^{6} +{\left (3 \, a^{2} b^{4} d^{2} + 8 \, a^{3} b^{3} d e + 3 \, a^{4} b^{2} e^{2}\right )} x^{5} + \frac{1}{2} \,{\left (10 \, a^{3} b^{3} d^{2} + 15 \, a^{4} b^{2} d e + 3 \, a^{5} b e^{2}\right )} x^{4} + \frac{1}{3} \,{\left (15 \, a^{4} b^{2} d^{2} + 12 \, a^{5} b d e + a^{6} e^{2}\right )} x^{3} +{\left (3 \, a^{5} b d^{2} + a^{6} d e\right )} x^{2} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*b^6*e^2*x^9 + a^6*d^2*x + 1/4*(b^6*d*e + 3*a*b^5*e^2)*x^8 + 1/7*(b^6*d^2 + 12*a*b^5*d*e + 15*a^2*b^4*e^2)*
x^7 + 1/3*(3*a*b^5*d^2 + 15*a^2*b^4*d*e + 10*a^3*b^3*e^2)*x^6 + (3*a^2*b^4*d^2 + 8*a^3*b^3*d*e + 3*a^4*b^2*e^2
)*x^5 + 1/2*(10*a^3*b^3*d^2 + 15*a^4*b^2*d*e + 3*a^5*b*e^2)*x^4 + 1/3*(15*a^4*b^2*d^2 + 12*a^5*b*d*e + a^6*e^2
)*x^3 + (3*a^5*b*d^2 + a^6*d*e)*x^2

________________________________________________________________________________________

Fricas [B]  time = 1.55068, size = 539, normalized size = 8.29 \begin{align*} \frac{1}{9} x^{9} e^{2} b^{6} + \frac{1}{4} x^{8} e d b^{6} + \frac{3}{4} x^{8} e^{2} b^{5} a + \frac{1}{7} x^{7} d^{2} b^{6} + \frac{12}{7} x^{7} e d b^{5} a + \frac{15}{7} x^{7} e^{2} b^{4} a^{2} + x^{6} d^{2} b^{5} a + 5 x^{6} e d b^{4} a^{2} + \frac{10}{3} x^{6} e^{2} b^{3} a^{3} + 3 x^{5} d^{2} b^{4} a^{2} + 8 x^{5} e d b^{3} a^{3} + 3 x^{5} e^{2} b^{2} a^{4} + 5 x^{4} d^{2} b^{3} a^{3} + \frac{15}{2} x^{4} e d b^{2} a^{4} + \frac{3}{2} x^{4} e^{2} b a^{5} + 5 x^{3} d^{2} b^{2} a^{4} + 4 x^{3} e d b a^{5} + \frac{1}{3} x^{3} e^{2} a^{6} + 3 x^{2} d^{2} b a^{5} + x^{2} e d a^{6} + x d^{2} a^{6} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*x^9*e^2*b^6 + 1/4*x^8*e*d*b^6 + 3/4*x^8*e^2*b^5*a + 1/7*x^7*d^2*b^6 + 12/7*x^7*e*d*b^5*a + 15/7*x^7*e^2*b^
4*a^2 + x^6*d^2*b^5*a + 5*x^6*e*d*b^4*a^2 + 10/3*x^6*e^2*b^3*a^3 + 3*x^5*d^2*b^4*a^2 + 8*x^5*e*d*b^3*a^3 + 3*x
^5*e^2*b^2*a^4 + 5*x^4*d^2*b^3*a^3 + 15/2*x^4*e*d*b^2*a^4 + 3/2*x^4*e^2*b*a^5 + 5*x^3*d^2*b^2*a^4 + 4*x^3*e*d*
b*a^5 + 1/3*x^3*e^2*a^6 + 3*x^2*d^2*b*a^5 + x^2*e*d*a^6 + x*d^2*a^6

________________________________________________________________________________________

Sympy [B]  time = 0.121035, size = 252, normalized size = 3.88 \begin{align*} a^{6} d^{2} x + \frac{b^{6} e^{2} x^{9}}{9} + x^{8} \left (\frac{3 a b^{5} e^{2}}{4} + \frac{b^{6} d e}{4}\right ) + x^{7} \left (\frac{15 a^{2} b^{4} e^{2}}{7} + \frac{12 a b^{5} d e}{7} + \frac{b^{6} d^{2}}{7}\right ) + x^{6} \left (\frac{10 a^{3} b^{3} e^{2}}{3} + 5 a^{2} b^{4} d e + a b^{5} d^{2}\right ) + x^{5} \left (3 a^{4} b^{2} e^{2} + 8 a^{3} b^{3} d e + 3 a^{2} b^{4} d^{2}\right ) + x^{4} \left (\frac{3 a^{5} b e^{2}}{2} + \frac{15 a^{4} b^{2} d e}{2} + 5 a^{3} b^{3} d^{2}\right ) + x^{3} \left (\frac{a^{6} e^{2}}{3} + 4 a^{5} b d e + 5 a^{4} b^{2} d^{2}\right ) + x^{2} \left (a^{6} d e + 3 a^{5} b d^{2}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**6*d**2*x + b**6*e**2*x**9/9 + x**8*(3*a*b**5*e**2/4 + b**6*d*e/4) + x**7*(15*a**2*b**4*e**2/7 + 12*a*b**5*d
*e/7 + b**6*d**2/7) + x**6*(10*a**3*b**3*e**2/3 + 5*a**2*b**4*d*e + a*b**5*d**2) + x**5*(3*a**4*b**2*e**2 + 8*
a**3*b**3*d*e + 3*a**2*b**4*d**2) + x**4*(3*a**5*b*e**2/2 + 15*a**4*b**2*d*e/2 + 5*a**3*b**3*d**2) + x**3*(a**
6*e**2/3 + 4*a**5*b*d*e + 5*a**4*b**2*d**2) + x**2*(a**6*d*e + 3*a**5*b*d**2)

________________________________________________________________________________________

Giac [B]  time = 1.17611, size = 342, normalized size = 5.26 \begin{align*} \frac{1}{9} \, b^{6} x^{9} e^{2} + \frac{1}{4} \, b^{6} d x^{8} e + \frac{1}{7} \, b^{6} d^{2} x^{7} + \frac{3}{4} \, a b^{5} x^{8} e^{2} + \frac{12}{7} \, a b^{5} d x^{7} e + a b^{5} d^{2} x^{6} + \frac{15}{7} \, a^{2} b^{4} x^{7} e^{2} + 5 \, a^{2} b^{4} d x^{6} e + 3 \, a^{2} b^{4} d^{2} x^{5} + \frac{10}{3} \, a^{3} b^{3} x^{6} e^{2} + 8 \, a^{3} b^{3} d x^{5} e + 5 \, a^{3} b^{3} d^{2} x^{4} + 3 \, a^{4} b^{2} x^{5} e^{2} + \frac{15}{2} \, a^{4} b^{2} d x^{4} e + 5 \, a^{4} b^{2} d^{2} x^{3} + \frac{3}{2} \, a^{5} b x^{4} e^{2} + 4 \, a^{5} b d x^{3} e + 3 \, a^{5} b d^{2} x^{2} + \frac{1}{3} \, a^{6} x^{3} e^{2} + a^{6} d x^{2} e + a^{6} d^{2} x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*b^6*x^9*e^2 + 1/4*b^6*d*x^8*e + 1/7*b^6*d^2*x^7 + 3/4*a*b^5*x^8*e^2 + 12/7*a*b^5*d*x^7*e + a*b^5*d^2*x^6 +
15/7*a^2*b^4*x^7*e^2 + 5*a^2*b^4*d*x^6*e + 3*a^2*b^4*d^2*x^5 + 10/3*a^3*b^3*x^6*e^2 + 8*a^3*b^3*d*x^5*e + 5*a
^3*b^3*d^2*x^4 + 3*a^4*b^2*x^5*e^2 + 15/2*a^4*b^2*d*x^4*e + 5*a^4*b^2*d^2*x^3 + 3/2*a^5*b*x^4*e^2 + 4*a^5*b*d*
x^3*e + 3*a^5*b*d^2*x^2 + 1/3*a^6*x^3*e^2 + a^6*d*x^2*e + a^6*d^2*x