### 3.1486 $$\int (d+e x)^3 (a^2+2 a b x+b^2 x^2)^3 \, dx$$

Optimal. Leaf size=92 $\frac{e^2 (a+b x)^9 (b d-a e)}{3 b^4}+\frac{3 e (a+b x)^8 (b d-a e)^2}{8 b^4}+\frac{(a+b x)^7 (b d-a e)^3}{7 b^4}+\frac{e^3 (a+b x)^{10}}{10 b^4}$

[Out]

((b*d - a*e)^3*(a + b*x)^7)/(7*b^4) + (3*e*(b*d - a*e)^2*(a + b*x)^8)/(8*b^4) + (e^2*(b*d - a*e)*(a + b*x)^9)/
(3*b^4) + (e^3*(a + b*x)^10)/(10*b^4)

________________________________________________________________________________________

Rubi [A]  time = 0.191523, 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} $\frac{e^2 (a+b x)^9 (b d-a e)}{3 b^4}+\frac{3 e (a+b x)^8 (b d-a e)^2}{8 b^4}+\frac{(a+b x)^7 (b d-a e)^3}{7 b^4}+\frac{e^3 (a+b x)^{10}}{10 b^4}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

Mathematica [B]  time = 0.0813186, size = 276, normalized size = 3. $\frac{1}{840} x \left (210 a^4 b^2 x^2 \left (45 d^2 e x+20 d^3+36 d e^2 x^2+10 e^3 x^3\right )+120 a^3 b^3 x^3 \left (84 d^2 e x+35 d^3+70 d e^2 x^2+20 e^3 x^3\right )+45 a^2 b^4 x^4 \left (140 d^2 e x+56 d^3+120 d e^2 x^2+35 e^3 x^3\right )+252 a^5 b x \left (20 d^2 e x+10 d^3+15 d e^2 x^2+4 e^3 x^3\right )+210 a^6 \left (6 d^2 e x+4 d^3+4 d e^2 x^2+e^3 x^3\right )+10 a b^5 x^5 \left (216 d^2 e x+84 d^3+189 d e^2 x^2+56 e^3 x^3\right )+b^6 x^6 \left (315 d^2 e x+120 d^3+280 d e^2 x^2+84 e^3 x^3\right )\right )$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(x*(210*a^6*(4*d^3 + 6*d^2*e*x + 4*d*e^2*x^2 + e^3*x^3) + 252*a^5*b*x*(10*d^3 + 20*d^2*e*x + 15*d*e^2*x^2 + 4*
e^3*x^3) + 210*a^4*b^2*x^2*(20*d^3 + 45*d^2*e*x + 36*d*e^2*x^2 + 10*e^3*x^3) + 120*a^3*b^3*x^3*(35*d^3 + 84*d^
2*e*x + 70*d*e^2*x^2 + 20*e^3*x^3) + 45*a^2*b^4*x^4*(56*d^3 + 140*d^2*e*x + 120*d*e^2*x^2 + 35*e^3*x^3) + 10*a
*b^5*x^5*(84*d^3 + 216*d^2*e*x + 189*d*e^2*x^2 + 56*e^3*x^3) + b^6*x^6*(120*d^3 + 315*d^2*e*x + 280*d*e^2*x^2
+ 84*e^3*x^3)))/840

________________________________________________________________________________________

Maple [B]  time = 0.04, size = 333, normalized size = 3.6 \begin{align*}{\frac{{e}^{3}{b}^{6}{x}^{10}}{10}}+{\frac{ \left ( 6\,{e}^{3}a{b}^{5}+3\,d{e}^{2}{b}^{6} \right ){x}^{9}}{9}}+{\frac{ \left ( 15\,{e}^{3}{a}^{2}{b}^{4}+18\,d{e}^{2}a{b}^{5}+3\,{d}^{2}e{b}^{6} \right ){x}^{8}}{8}}+{\frac{ \left ( 20\,{e}^{3}{a}^{3}{b}^{3}+45\,d{e}^{2}{a}^{2}{b}^{4}+18\,{d}^{2}ea{b}^{5}+{d}^{3}{b}^{6} \right ){x}^{7}}{7}}+{\frac{ \left ( 15\,{e}^{3}{a}^{4}{b}^{2}+60\,d{e}^{2}{a}^{3}{b}^{3}+45\,{d}^{2}e{a}^{2}{b}^{4}+6\,{d}^{3}a{b}^{5} \right ){x}^{6}}{6}}+{\frac{ \left ( 6\,{e}^{3}{a}^{5}b+45\,d{e}^{2}{a}^{4}{b}^{2}+60\,{d}^{2}e{a}^{3}{b}^{3}+15\,{d}^{3}{a}^{2}{b}^{4} \right ){x}^{5}}{5}}+{\frac{ \left ({e}^{3}{a}^{6}+18\,d{e}^{2}{a}^{5}b+45\,{d}^{2}e{a}^{4}{b}^{2}+20\,{d}^{3}{a}^{3}{b}^{3} \right ){x}^{4}}{4}}+{\frac{ \left ( 3\,d{e}^{2}{a}^{6}+18\,{d}^{2}e{a}^{5}b+15\,{d}^{3}{a}^{4}{b}^{2} \right ){x}^{3}}{3}}+{\frac{ \left ( 3\,{d}^{2}e{a}^{6}+6\,{d}^{3}{a}^{5}b \right ){x}^{2}}{2}}+{d}^{3}{a}^{6}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)^3,x)

[Out]

1/10*e^3*b^6*x^10+1/9*(6*a*b^5*e^3+3*b^6*d*e^2)*x^9+1/8*(15*a^2*b^4*e^3+18*a*b^5*d*e^2+3*b^6*d^2*e)*x^8+1/7*(2
0*a^3*b^3*e^3+45*a^2*b^4*d*e^2+18*a*b^5*d^2*e+b^6*d^3)*x^7+1/6*(15*a^4*b^2*e^3+60*a^3*b^3*d*e^2+45*a^2*b^4*d^2
*e+6*a*b^5*d^3)*x^6+1/5*(6*a^5*b*e^3+45*a^4*b^2*d*e^2+60*a^3*b^3*d^2*e+15*a^2*b^4*d^3)*x^5+1/4*(a^6*e^3+18*a^5
*b*d*e^2+45*a^4*b^2*d^2*e+20*a^3*b^3*d^3)*x^4+1/3*(3*a^6*d*e^2+18*a^5*b*d^2*e+15*a^4*b^2*d^3)*x^3+1/2*(3*a^6*d
^2*e+6*a^5*b*d^3)*x^2+d^3*a^6*x

________________________________________________________________________________________

Maxima [B]  time = 1.07234, size = 441, normalized size = 4.79 \begin{align*} \frac{1}{10} \, b^{6} e^{3} x^{10} + a^{6} d^{3} x + \frac{1}{3} \,{\left (b^{6} d e^{2} + 2 \, a b^{5} e^{3}\right )} x^{9} + \frac{3}{8} \,{\left (b^{6} d^{2} e + 6 \, a b^{5} d e^{2} + 5 \, a^{2} b^{4} e^{3}\right )} x^{8} + \frac{1}{7} \,{\left (b^{6} d^{3} + 18 \, a b^{5} d^{2} e + 45 \, a^{2} b^{4} d e^{2} + 20 \, a^{3} b^{3} e^{3}\right )} x^{7} + \frac{1}{2} \,{\left (2 \, a b^{5} d^{3} + 15 \, a^{2} b^{4} d^{2} e + 20 \, a^{3} b^{3} d e^{2} + 5 \, a^{4} b^{2} e^{3}\right )} x^{6} + \frac{3}{5} \,{\left (5 \, a^{2} b^{4} d^{3} + 20 \, a^{3} b^{3} d^{2} e + 15 \, a^{4} b^{2} d e^{2} + 2 \, a^{5} b e^{3}\right )} x^{5} + \frac{1}{4} \,{\left (20 \, a^{3} b^{3} d^{3} + 45 \, a^{4} b^{2} d^{2} e + 18 \, a^{5} b d e^{2} + a^{6} e^{3}\right )} x^{4} +{\left (5 \, a^{4} b^{2} d^{3} + 6 \, a^{5} b d^{2} e + a^{6} d e^{2}\right )} x^{3} + \frac{3}{2} \,{\left (2 \, a^{5} b d^{3} + a^{6} 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)^3,x, algorithm="maxima")

[Out]

1/10*b^6*e^3*x^10 + a^6*d^3*x + 1/3*(b^6*d*e^2 + 2*a*b^5*e^3)*x^9 + 3/8*(b^6*d^2*e + 6*a*b^5*d*e^2 + 5*a^2*b^4
*e^3)*x^8 + 1/7*(b^6*d^3 + 18*a*b^5*d^2*e + 45*a^2*b^4*d*e^2 + 20*a^3*b^3*e^3)*x^7 + 1/2*(2*a*b^5*d^3 + 15*a^2
*b^4*d^2*e + 20*a^3*b^3*d*e^2 + 5*a^4*b^2*e^3)*x^6 + 3/5*(5*a^2*b^4*d^3 + 20*a^3*b^3*d^2*e + 15*a^4*b^2*d*e^2
+ 2*a^5*b*e^3)*x^5 + 1/4*(20*a^3*b^3*d^3 + 45*a^4*b^2*d^2*e + 18*a^5*b*d*e^2 + a^6*e^3)*x^4 + (5*a^4*b^2*d^3 +
6*a^5*b*d^2*e + a^6*d*e^2)*x^3 + 3/2*(2*a^5*b*d^3 + a^6*d^2*e)*x^2

________________________________________________________________________________________

Fricas [B]  time = 1.62168, size = 776, normalized size = 8.43 \begin{align*} \frac{1}{10} x^{10} e^{3} b^{6} + \frac{1}{3} x^{9} e^{2} d b^{6} + \frac{2}{3} x^{9} e^{3} b^{5} a + \frac{3}{8} x^{8} e d^{2} b^{6} + \frac{9}{4} x^{8} e^{2} d b^{5} a + \frac{15}{8} x^{8} e^{3} b^{4} a^{2} + \frac{1}{7} x^{7} d^{3} b^{6} + \frac{18}{7} x^{7} e d^{2} b^{5} a + \frac{45}{7} x^{7} e^{2} d b^{4} a^{2} + \frac{20}{7} x^{7} e^{3} b^{3} a^{3} + x^{6} d^{3} b^{5} a + \frac{15}{2} x^{6} e d^{2} b^{4} a^{2} + 10 x^{6} e^{2} d b^{3} a^{3} + \frac{5}{2} x^{6} e^{3} b^{2} a^{4} + 3 x^{5} d^{3} b^{4} a^{2} + 12 x^{5} e d^{2} b^{3} a^{3} + 9 x^{5} e^{2} d b^{2} a^{4} + \frac{6}{5} x^{5} e^{3} b a^{5} + 5 x^{4} d^{3} b^{3} a^{3} + \frac{45}{4} x^{4} e d^{2} b^{2} a^{4} + \frac{9}{2} x^{4} e^{2} d b a^{5} + \frac{1}{4} x^{4} e^{3} a^{6} + 5 x^{3} d^{3} b^{2} a^{4} + 6 x^{3} e d^{2} b a^{5} + x^{3} e^{2} d a^{6} + 3 x^{2} d^{3} b a^{5} + \frac{3}{2} x^{2} e d^{2} a^{6} + x d^{3} a^{6} \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)^3,x, algorithm="fricas")

[Out]

1/10*x^10*e^3*b^6 + 1/3*x^9*e^2*d*b^6 + 2/3*x^9*e^3*b^5*a + 3/8*x^8*e*d^2*b^6 + 9/4*x^8*e^2*d*b^5*a + 15/8*x^8
*e^3*b^4*a^2 + 1/7*x^7*d^3*b^6 + 18/7*x^7*e*d^2*b^5*a + 45/7*x^7*e^2*d*b^4*a^2 + 20/7*x^7*e^3*b^3*a^3 + x^6*d^
3*b^5*a + 15/2*x^6*e*d^2*b^4*a^2 + 10*x^6*e^2*d*b^3*a^3 + 5/2*x^6*e^3*b^2*a^4 + 3*x^5*d^3*b^4*a^2 + 12*x^5*e*d
^2*b^3*a^3 + 9*x^5*e^2*d*b^2*a^4 + 6/5*x^5*e^3*b*a^5 + 5*x^4*d^3*b^3*a^3 + 45/4*x^4*e*d^2*b^2*a^4 + 9/2*x^4*e^
2*d*b*a^5 + 1/4*x^4*e^3*a^6 + 5*x^3*d^3*b^2*a^4 + 6*x^3*e*d^2*b*a^5 + x^3*e^2*d*a^6 + 3*x^2*d^3*b*a^5 + 3/2*x^
2*e*d^2*a^6 + x*d^3*a^6

________________________________________________________________________________________

Sympy [B]  time = 0.125044, size = 364, normalized size = 3.96 \begin{align*} a^{6} d^{3} x + \frac{b^{6} e^{3} x^{10}}{10} + x^{9} \left (\frac{2 a b^{5} e^{3}}{3} + \frac{b^{6} d e^{2}}{3}\right ) + x^{8} \left (\frac{15 a^{2} b^{4} e^{3}}{8} + \frac{9 a b^{5} d e^{2}}{4} + \frac{3 b^{6} d^{2} e}{8}\right ) + x^{7} \left (\frac{20 a^{3} b^{3} e^{3}}{7} + \frac{45 a^{2} b^{4} d e^{2}}{7} + \frac{18 a b^{5} d^{2} e}{7} + \frac{b^{6} d^{3}}{7}\right ) + x^{6} \left (\frac{5 a^{4} b^{2} e^{3}}{2} + 10 a^{3} b^{3} d e^{2} + \frac{15 a^{2} b^{4} d^{2} e}{2} + a b^{5} d^{3}\right ) + x^{5} \left (\frac{6 a^{5} b e^{3}}{5} + 9 a^{4} b^{2} d e^{2} + 12 a^{3} b^{3} d^{2} e + 3 a^{2} b^{4} d^{3}\right ) + x^{4} \left (\frac{a^{6} e^{3}}{4} + \frac{9 a^{5} b d e^{2}}{2} + \frac{45 a^{4} b^{2} d^{2} e}{4} + 5 a^{3} b^{3} d^{3}\right ) + x^{3} \left (a^{6} d e^{2} + 6 a^{5} b d^{2} e + 5 a^{4} b^{2} d^{3}\right ) + x^{2} \left (\frac{3 a^{6} d^{2} e}{2} + 3 a^{5} 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)**3,x)

[Out]

a**6*d**3*x + b**6*e**3*x**10/10 + x**9*(2*a*b**5*e**3/3 + b**6*d*e**2/3) + x**8*(15*a**2*b**4*e**3/8 + 9*a*b*
*5*d*e**2/4 + 3*b**6*d**2*e/8) + x**7*(20*a**3*b**3*e**3/7 + 45*a**2*b**4*d*e**2/7 + 18*a*b**5*d**2*e/7 + b**6
*d**3/7) + x**6*(5*a**4*b**2*e**3/2 + 10*a**3*b**3*d*e**2 + 15*a**2*b**4*d**2*e/2 + a*b**5*d**3) + x**5*(6*a**
5*b*e**3/5 + 9*a**4*b**2*d*e**2 + 12*a**3*b**3*d**2*e + 3*a**2*b**4*d**3) + x**4*(a**6*e**3/4 + 9*a**5*b*d*e**
2/2 + 45*a**4*b**2*d**2*e/4 + 5*a**3*b**3*d**3) + x**3*(a**6*d*e**2 + 6*a**5*b*d**2*e + 5*a**4*b**2*d**3) + x*
*2*(3*a**6*d**2*e/2 + 3*a**5*b*d**3)

________________________________________________________________________________________

Giac [B]  time = 1.1808, size = 479, normalized size = 5.21 \begin{align*} \frac{1}{10} \, b^{6} x^{10} e^{3} + \frac{1}{3} \, b^{6} d x^{9} e^{2} + \frac{3}{8} \, b^{6} d^{2} x^{8} e + \frac{1}{7} \, b^{6} d^{3} x^{7} + \frac{2}{3} \, a b^{5} x^{9} e^{3} + \frac{9}{4} \, a b^{5} d x^{8} e^{2} + \frac{18}{7} \, a b^{5} d^{2} x^{7} e + a b^{5} d^{3} x^{6} + \frac{15}{8} \, a^{2} b^{4} x^{8} e^{3} + \frac{45}{7} \, a^{2} b^{4} d x^{7} e^{2} + \frac{15}{2} \, a^{2} b^{4} d^{2} x^{6} e + 3 \, a^{2} b^{4} d^{3} x^{5} + \frac{20}{7} \, a^{3} b^{3} x^{7} e^{3} + 10 \, a^{3} b^{3} d x^{6} e^{2} + 12 \, a^{3} b^{3} d^{2} x^{5} e + 5 \, a^{3} b^{3} d^{3} x^{4} + \frac{5}{2} \, a^{4} b^{2} x^{6} e^{3} + 9 \, a^{4} b^{2} d x^{5} e^{2} + \frac{45}{4} \, a^{4} b^{2} d^{2} x^{4} e + 5 \, a^{4} b^{2} d^{3} x^{3} + \frac{6}{5} \, a^{5} b x^{5} e^{3} + \frac{9}{2} \, a^{5} b d x^{4} e^{2} + 6 \, a^{5} b d^{2} x^{3} e + 3 \, a^{5} b d^{3} x^{2} + \frac{1}{4} \, a^{6} x^{4} e^{3} + a^{6} d x^{3} e^{2} + \frac{3}{2} \, a^{6} d^{2} x^{2} e + a^{6} 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)^3,x, algorithm="giac")

[Out]

1/10*b^6*x^10*e^3 + 1/3*b^6*d*x^9*e^2 + 3/8*b^6*d^2*x^8*e + 1/7*b^6*d^3*x^7 + 2/3*a*b^5*x^9*e^3 + 9/4*a*b^5*d*
x^8*e^2 + 18/7*a*b^5*d^2*x^7*e + a*b^5*d^3*x^6 + 15/8*a^2*b^4*x^8*e^3 + 45/7*a^2*b^4*d*x^7*e^2 + 15/2*a^2*b^4*
d^2*x^6*e + 3*a^2*b^4*d^3*x^5 + 20/7*a^3*b^3*x^7*e^3 + 10*a^3*b^3*d*x^6*e^2 + 12*a^3*b^3*d^2*x^5*e + 5*a^3*b^3
*d^3*x^4 + 5/2*a^4*b^2*x^6*e^3 + 9*a^4*b^2*d*x^5*e^2 + 45/4*a^4*b^2*d^2*x^4*e + 5*a^4*b^2*d^3*x^3 + 6/5*a^5*b*
x^5*e^3 + 9/2*a^5*b*d*x^4*e^2 + 6*a^5*b*d^2*x^3*e + 3*a^5*b*d^3*x^2 + 1/4*a^6*x^4*e^3 + a^6*d*x^3*e^2 + 3/2*a^
6*d^2*x^2*e + a^6*d^3*x