3.1783 \(\int (a+b x)^3 (a c+(b c+a d) x+b d x^2)^3 \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.219694, antiderivative size = 92, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069, Rules used = {626, 43} \[ \frac{d^2 (a+b x)^9 (b c-a d)}{3 b^4}+\frac{3 d (a+b x)^8 (b c-a d)^2}{8 b^4}+\frac{(a+b x)^7 (b c-a d)^3}{7 b^4}+\frac{d^3 (a+b x)^{10}}{10 b^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 626

Int[((d_) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[(d + e*x)^(m + p)*(a
/d + (c*x)/e)^p, x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 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 (a+b x)^3 \left (a c+(b c+a d) x+b d x^2\right )^3 \, dx &=\int (a+b x)^6 (c+d x)^3 \, dx\\ &=\int \left (\frac{(b c-a d)^3 (a+b x)^6}{b^3}+\frac{3 d (b c-a d)^2 (a+b x)^7}{b^3}+\frac{3 d^2 (b c-a d) (a+b x)^8}{b^3}+\frac{d^3 (a+b x)^9}{b^3}\right ) \, dx\\ &=\frac{(b c-a d)^3 (a+b x)^7}{7 b^4}+\frac{3 d (b c-a d)^2 (a+b x)^8}{8 b^4}+\frac{d^2 (b c-a d) (a+b x)^9}{3 b^4}+\frac{d^3 (a+b x)^{10}}{10 b^4}\\ \end{align*}

Mathematica [B]  time = 0.0824648, size = 276, normalized size = 3. \[ \frac{1}{840} x \left (210 a^4 b^2 x^2 \left (45 c^2 d x+20 c^3+36 c d^2 x^2+10 d^3 x^3\right )+120 a^3 b^3 x^3 \left (84 c^2 d x+35 c^3+70 c d^2 x^2+20 d^3 x^3\right )+45 a^2 b^4 x^4 \left (140 c^2 d x+56 c^3+120 c d^2 x^2+35 d^3 x^3\right )+252 a^5 b x \left (20 c^2 d x+10 c^3+15 c d^2 x^2+4 d^3 x^3\right )+210 a^6 \left (6 c^2 d x+4 c^3+4 c d^2 x^2+d^3 x^3\right )+10 a b^5 x^5 \left (216 c^2 d x+84 c^3+189 c d^2 x^2+56 d^3 x^3\right )+b^6 x^6 \left (315 c^2 d x+120 c^3+280 c d^2 x^2+84 d^3 x^3\right )\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.04, size = 811, normalized size = 8.8 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.04754, size = 441, normalized size = 4.79 \begin{align*} \frac{1}{10} \, b^{6} d^{3} x^{10} + a^{6} c^{3} x + \frac{1}{3} \,{\left (b^{6} c d^{2} + 2 \, a b^{5} d^{3}\right )} x^{9} + \frac{3}{8} \,{\left (b^{6} c^{2} d + 6 \, a b^{5} c d^{2} + 5 \, a^{2} b^{4} d^{3}\right )} x^{8} + \frac{1}{7} \,{\left (b^{6} c^{3} + 18 \, a b^{5} c^{2} d + 45 \, a^{2} b^{4} c d^{2} + 20 \, a^{3} b^{3} d^{3}\right )} x^{7} + \frac{1}{2} \,{\left (2 \, a b^{5} c^{3} + 15 \, a^{2} b^{4} c^{2} d + 20 \, a^{3} b^{3} c d^{2} + 5 \, a^{4} b^{2} d^{3}\right )} x^{6} + \frac{3}{5} \,{\left (5 \, a^{2} b^{4} c^{3} + 20 \, a^{3} b^{3} c^{2} d + 15 \, a^{4} b^{2} c d^{2} + 2 \, a^{5} b d^{3}\right )} x^{5} + \frac{1}{4} \,{\left (20 \, a^{3} b^{3} c^{3} + 45 \, a^{4} b^{2} c^{2} d + 18 \, a^{5} b c d^{2} + a^{6} d^{3}\right )} x^{4} +{\left (5 \, a^{4} b^{2} c^{3} + 6 \, a^{5} b c^{2} d + a^{6} c d^{2}\right )} x^{3} + \frac{3}{2} \,{\left (2 \, a^{5} b c^{3} + a^{6} c^{2} d\right )} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 1.28594, size = 776, normalized size = 8.43 \begin{align*} \frac{1}{10} x^{10} d^{3} b^{6} + \frac{1}{3} x^{9} d^{2} c b^{6} + \frac{2}{3} x^{9} d^{3} b^{5} a + \frac{3}{8} x^{8} d c^{2} b^{6} + \frac{9}{4} x^{8} d^{2} c b^{5} a + \frac{15}{8} x^{8} d^{3} b^{4} a^{2} + \frac{1}{7} x^{7} c^{3} b^{6} + \frac{18}{7} x^{7} d c^{2} b^{5} a + \frac{45}{7} x^{7} d^{2} c b^{4} a^{2} + \frac{20}{7} x^{7} d^{3} b^{3} a^{3} + x^{6} c^{3} b^{5} a + \frac{15}{2} x^{6} d c^{2} b^{4} a^{2} + 10 x^{6} d^{2} c b^{3} a^{3} + \frac{5}{2} x^{6} d^{3} b^{2} a^{4} + 3 x^{5} c^{3} b^{4} a^{2} + 12 x^{5} d c^{2} b^{3} a^{3} + 9 x^{5} d^{2} c b^{2} a^{4} + \frac{6}{5} x^{5} d^{3} b a^{5} + 5 x^{4} c^{3} b^{3} a^{3} + \frac{45}{4} x^{4} d c^{2} b^{2} a^{4} + \frac{9}{2} x^{4} d^{2} c b a^{5} + \frac{1}{4} x^{4} d^{3} a^{6} + 5 x^{3} c^{3} b^{2} a^{4} + 6 x^{3} d c^{2} b a^{5} + x^{3} d^{2} c a^{6} + 3 x^{2} c^{3} b a^{5} + \frac{3}{2} x^{2} d c^{2} a^{6} + x c^{3} a^{6} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 0.245507, size = 364, normalized size = 3.96 \begin{align*} a^{6} c^{3} x + \frac{b^{6} d^{3} x^{10}}{10} + x^{9} \left (\frac{2 a b^{5} d^{3}}{3} + \frac{b^{6} c d^{2}}{3}\right ) + x^{8} \left (\frac{15 a^{2} b^{4} d^{3}}{8} + \frac{9 a b^{5} c d^{2}}{4} + \frac{3 b^{6} c^{2} d}{8}\right ) + x^{7} \left (\frac{20 a^{3} b^{3} d^{3}}{7} + \frac{45 a^{2} b^{4} c d^{2}}{7} + \frac{18 a b^{5} c^{2} d}{7} + \frac{b^{6} c^{3}}{7}\right ) + x^{6} \left (\frac{5 a^{4} b^{2} d^{3}}{2} + 10 a^{3} b^{3} c d^{2} + \frac{15 a^{2} b^{4} c^{2} d}{2} + a b^{5} c^{3}\right ) + x^{5} \left (\frac{6 a^{5} b d^{3}}{5} + 9 a^{4} b^{2} c d^{2} + 12 a^{3} b^{3} c^{2} d + 3 a^{2} b^{4} c^{3}\right ) + x^{4} \left (\frac{a^{6} d^{3}}{4} + \frac{9 a^{5} b c d^{2}}{2} + \frac{45 a^{4} b^{2} c^{2} d}{4} + 5 a^{3} b^{3} c^{3}\right ) + x^{3} \left (a^{6} c d^{2} + 6 a^{5} b c^{2} d + 5 a^{4} b^{2} c^{3}\right ) + x^{2} \left (\frac{3 a^{6} c^{2} d}{2} + 3 a^{5} b c^{3}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.21896, size = 489, normalized size = 5.32 \begin{align*} \frac{1}{10} \, b^{6} d^{3} x^{10} + \frac{1}{3} \, b^{6} c d^{2} x^{9} + \frac{2}{3} \, a b^{5} d^{3} x^{9} + \frac{3}{8} \, b^{6} c^{2} d x^{8} + \frac{9}{4} \, a b^{5} c d^{2} x^{8} + \frac{15}{8} \, a^{2} b^{4} d^{3} x^{8} + \frac{1}{7} \, b^{6} c^{3} x^{7} + \frac{18}{7} \, a b^{5} c^{2} d x^{7} + \frac{45}{7} \, a^{2} b^{4} c d^{2} x^{7} + \frac{20}{7} \, a^{3} b^{3} d^{3} x^{7} + a b^{5} c^{3} x^{6} + \frac{15}{2} \, a^{2} b^{4} c^{2} d x^{6} + 10 \, a^{3} b^{3} c d^{2} x^{6} + \frac{5}{2} \, a^{4} b^{2} d^{3} x^{6} + 3 \, a^{2} b^{4} c^{3} x^{5} + 12 \, a^{3} b^{3} c^{2} d x^{5} + 9 \, a^{4} b^{2} c d^{2} x^{5} + \frac{6}{5} \, a^{5} b d^{3} x^{5} + 5 \, a^{3} b^{3} c^{3} x^{4} + \frac{45}{4} \, a^{4} b^{2} c^{2} d x^{4} + \frac{9}{2} \, a^{5} b c d^{2} x^{4} + \frac{1}{4} \, a^{6} d^{3} x^{4} + 5 \, a^{4} b^{2} c^{3} x^{3} + 6 \, a^{5} b c^{2} d x^{3} + a^{6} c d^{2} x^{3} + 3 \, a^{5} b c^{3} x^{2} + \frac{3}{2} \, a^{6} c^{2} d x^{2} + a^{6} c^{3} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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