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]

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

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Rubi [A]  time = 0.22, antiderivative size = 92, normalized size of antiderivative = 1.00, 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 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])

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]

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*}

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Mathematica [B]  time = 0.08, size = 276, normalized size = 3.00 \[ \frac {1}{840} x \left (210 a^6 \left (4 c^3+6 c^2 d x+4 c d^2 x^2+d^3 x^3\right )+252 a^5 b x \left (10 c^3+20 c^2 d x+15 c d^2 x^2+4 d^3 x^3\right )+210 a^4 b^2 x^2 \left (20 c^3+45 c^2 d x+36 c d^2 x^2+10 d^3 x^3\right )+120 a^3 b^3 x^3 \left (35 c^3+84 c^2 d x+70 c d^2 x^2+20 d^3 x^3\right )+45 a^2 b^4 x^4 \left (56 c^3+140 c^2 d x+120 c d^2 x^2+35 d^3 x^3\right )+10 a b^5 x^5 \left (84 c^3+216 c^2 d x+189 c d^2 x^2+56 d^3 x^3\right )+b^6 x^6 \left (120 c^3+315 c^2 d x+280 c d^2 x^2+84 d^3 x^3\right )\right ) \]

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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fricas [B]  time = 1.25, size = 362, normalized size = 3.93 \[ \frac {1}{10} x^{10} d^{3} b^{6} + \frac {1}{3} x^{9} d^{2} c b^{6} + \frac {2}{3} x^{9} d^{3} b^{5} a + \frac {3}{8} x^{8} d c^{2} b^{6} + \frac {9}{4} x^{8} d^{2} c b^{5} a + \frac {15}{8} x^{8} d^{3} b^{4} a^{2} + \frac {1}{7} x^{7} c^{3} b^{6} + \frac {18}{7} x^{7} d c^{2} b^{5} a + \frac {45}{7} x^{7} d^{2} c b^{4} a^{2} + \frac {20}{7} x^{7} d^{3} b^{3} a^{3} + x^{6} c^{3} b^{5} a + \frac {15}{2} x^{6} d c^{2} b^{4} a^{2} + 10 x^{6} d^{2} c b^{3} a^{3} + \frac {5}{2} x^{6} d^{3} b^{2} a^{4} + 3 x^{5} c^{3} b^{4} a^{2} + 12 x^{5} d c^{2} b^{3} a^{3} + 9 x^{5} d^{2} c b^{2} a^{4} + \frac {6}{5} x^{5} d^{3} b a^{5} + 5 x^{4} c^{3} b^{3} a^{3} + \frac {45}{4} x^{4} d c^{2} b^{2} a^{4} + \frac {9}{2} x^{4} d^{2} c b a^{5} + \frac {1}{4} x^{4} d^{3} a^{6} + 5 x^{3} c^{3} b^{2} a^{4} + 6 x^{3} d c^{2} b a^{5} + x^{3} d^{2} c a^{6} + 3 x^{2} c^{3} b a^{5} + \frac {3}{2} x^{2} d c^{2} a^{6} + x c^{3} a^{6} \]

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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giac [B]  time = 0.17, size = 362, normalized size = 3.93 \[ \frac {1}{10} \, b^{6} d^{3} x^{10} + \frac {1}{3} \, b^{6} c d^{2} x^{9} + \frac {2}{3} \, a b^{5} d^{3} x^{9} + \frac {3}{8} \, b^{6} c^{2} d x^{8} + \frac {9}{4} \, a b^{5} c d^{2} x^{8} + \frac {15}{8} \, a^{2} b^{4} d^{3} x^{8} + \frac {1}{7} \, b^{6} c^{3} x^{7} + \frac {18}{7} \, a b^{5} c^{2} d x^{7} + \frac {45}{7} \, a^{2} b^{4} c d^{2} x^{7} + \frac {20}{7} \, a^{3} b^{3} d^{3} x^{7} + a b^{5} c^{3} x^{6} + \frac {15}{2} \, a^{2} b^{4} c^{2} d x^{6} + 10 \, a^{3} b^{3} c d^{2} x^{6} + \frac {5}{2} \, a^{4} b^{2} d^{3} x^{6} + 3 \, a^{2} b^{4} c^{3} x^{5} + 12 \, a^{3} b^{3} c^{2} d x^{5} + 9 \, a^{4} b^{2} c d^{2} x^{5} + \frac {6}{5} \, a^{5} b d^{3} x^{5} + 5 \, a^{3} b^{3} c^{3} x^{4} + \frac {45}{4} \, a^{4} b^{2} c^{2} d x^{4} + \frac {9}{2} \, a^{5} b c d^{2} x^{4} + \frac {1}{4} \, a^{6} d^{3} x^{4} + 5 \, a^{4} b^{2} c^{3} x^{3} + 6 \, a^{5} b c^{2} d x^{3} + a^{6} c d^{2} x^{3} + 3 \, a^{5} b c^{3} x^{2} + \frac {3}{2} \, a^{6} c^{2} d x^{2} + a^{6} c^{3} x \]

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

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maple [B]  time = 0.04, size = 811, normalized size = 8.82 \[ \frac {b^{6} d^{3} x^{10}}{10}+a^{6} c^{3} x +\frac {\left (3 a \,b^{5} d^{3}+3 \left (a d +b c \right ) b^{5} d^{2}\right ) x^{9}}{9}+\frac {\left (3 a^{2} b^{4} d^{3}+9 \left (a d +b c \right ) a \,b^{4} d^{2}+\left (a \,b^{2} c \,d^{2}+2 \left (a d +b c \right )^{2} b d +\left (2 a b c d +\left (a d +b c \right )^{2}\right ) b d \right ) b^{3}\right ) x^{8}}{8}+\frac {\left (a^{3} b^{3} d^{3}+9 \left (a d +b c \right ) a^{2} b^{3} d^{2}+3 \left (a \,b^{2} c \,d^{2}+2 \left (a d +b c \right )^{2} b d +\left (2 a b c d +\left (a d +b c \right )^{2}\right ) b d \right ) a \,b^{2}+\left (4 \left (a d +b c \right ) a b c d +\left (a d +b c \right ) \left (2 a b c d +\left (a d +b c \right )^{2}\right )\right ) b^{3}\right ) x^{7}}{7}+\frac {\left (3 \left (a d +b c \right ) a^{3} b^{2} d^{2}+3 \left (a \,b^{2} c \,d^{2}+2 \left (a d +b c \right )^{2} b d +\left (2 a b c d +\left (a d +b c \right )^{2}\right ) b d \right ) a^{2} b +3 \left (4 \left (a d +b c \right ) a b c d +\left (a d +b c \right ) \left (2 a b c d +\left (a d +b c \right )^{2}\right )\right ) a \,b^{2}+\left (a^{2} b \,c^{2} d +\left (2 a b c d +\left (a d +b c \right )^{2}\right ) a c +2 \left (a d +b c \right )^{2} a c \right ) b^{3}\right ) x^{6}}{6}+\frac {\left (3 \left (a d +b c \right ) a^{2} b^{3} c^{2}+\left (a \,b^{2} c \,d^{2}+2 \left (a d +b c \right )^{2} b d +\left (2 a b c d +\left (a d +b c \right )^{2}\right ) b d \right ) a^{3}+3 \left (4 \left (a d +b c \right ) a b c d +\left (a d +b c \right ) \left (2 a b c d +\left (a d +b c \right )^{2}\right )\right ) a^{2} b +3 \left (a^{2} b \,c^{2} d +\left (2 a b c d +\left (a d +b c \right )^{2}\right ) a c +2 \left (a d +b c \right )^{2} a c \right ) a \,b^{2}\right ) x^{5}}{5}+\frac {\left (a^{3} b^{3} c^{3}+9 \left (a d +b c \right ) a^{3} b^{2} c^{2}+\left (4 \left (a d +b c \right ) a b c d +\left (a d +b c \right ) \left (2 a b c d +\left (a d +b c \right )^{2}\right )\right ) a^{3}+3 \left (a^{2} b \,c^{2} d +\left (2 a b c d +\left (a d +b c \right )^{2}\right ) a c +2 \left (a d +b c \right )^{2} a c \right ) a^{2} b \right ) x^{4}}{4}+\frac {\left (3 a^{4} b^{2} c^{3}+9 \left (a d +b c \right ) a^{4} b \,c^{2}+\left (a^{2} b \,c^{2} d +\left (2 a b c d +\left (a d +b c \right )^{2}\right ) a c +2 \left (a d +b c \right )^{2} a c \right ) a^{3}\right ) x^{3}}{3}+\frac {\left (3 a^{5} b \,c^{3}+3 \left (a d +b c \right ) a^{5} c^{2}\right ) x^{2}}{2} \]

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

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maxima [B]  time = 1.10, size = 327, normalized size = 3.55 \[ \frac {1}{10} \, b^{6} d^{3} x^{10} + a^{6} c^{3} x + \frac {1}{3} \, {\left (b^{6} c d^{2} + 2 \, a b^{5} d^{3}\right )} x^{9} + \frac {3}{8} \, {\left (b^{6} c^{2} d + 6 \, a b^{5} c d^{2} + 5 \, a^{2} b^{4} d^{3}\right )} x^{8} + \frac {1}{7} \, {\left (b^{6} c^{3} + 18 \, a b^{5} c^{2} d + 45 \, a^{2} b^{4} c d^{2} + 20 \, a^{3} b^{3} d^{3}\right )} x^{7} + \frac {1}{2} \, {\left (2 \, a b^{5} c^{3} + 15 \, a^{2} b^{4} c^{2} d + 20 \, a^{3} b^{3} c d^{2} + 5 \, a^{4} b^{2} d^{3}\right )} x^{6} + \frac {3}{5} \, {\left (5 \, a^{2} b^{4} c^{3} + 20 \, a^{3} b^{3} c^{2} d + 15 \, a^{4} b^{2} c d^{2} + 2 \, a^{5} b d^{3}\right )} x^{5} + \frac {1}{4} \, {\left (20 \, a^{3} b^{3} c^{3} + 45 \, a^{4} b^{2} c^{2} d + 18 \, a^{5} b c d^{2} + a^{6} d^{3}\right )} x^{4} + {\left (5 \, a^{4} b^{2} c^{3} + 6 \, a^{5} b c^{2} d + a^{6} c d^{2}\right )} x^{3} + \frac {3}{2} \, {\left (2 \, a^{5} b c^{3} + a^{6} c^{2} d\right )} x^{2} \]

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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mupad [B]  time = 0.67, size = 308, normalized size = 3.35 \[ x^5\,\left (\frac {6\,a^5\,b\,d^3}{5}+9\,a^4\,b^2\,c\,d^2+12\,a^3\,b^3\,c^2\,d+3\,a^2\,b^4\,c^3\right )+x^6\,\left (\frac {5\,a^4\,b^2\,d^3}{2}+10\,a^3\,b^3\,c\,d^2+\frac {15\,a^2\,b^4\,c^2\,d}{2}+a\,b^5\,c^3\right )+x^4\,\left (\frac {a^6\,d^3}{4}+\frac {9\,a^5\,b\,c\,d^2}{2}+\frac {45\,a^4\,b^2\,c^2\,d}{4}+5\,a^3\,b^3\,c^3\right )+x^7\,\left (\frac {20\,a^3\,b^3\,d^3}{7}+\frac {45\,a^2\,b^4\,c\,d^2}{7}+\frac {18\,a\,b^5\,c^2\,d}{7}+\frac {b^6\,c^3}{7}\right )+a^6\,c^3\,x+\frac {b^6\,d^3\,x^{10}}{10}+\frac {3\,a^5\,c^2\,x^2\,\left (a\,d+2\,b\,c\right )}{2}+\frac {b^5\,d^2\,x^9\,\left (2\,a\,d+b\,c\right )}{3}+a^4\,c\,x^3\,\left (a^2\,d^2+6\,a\,b\,c\,d+5\,b^2\,c^2\right )+\frac {3\,b^4\,d\,x^8\,\left (5\,a^2\,d^2+6\,a\,b\,c\,d+b^2\,c^2\right )}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [B]  time = 0.16, size = 364, normalized size = 3.96 \[ a^{6} c^{3} x + \frac {b^{6} d^{3} x^{10}}{10} + x^{9} \left (\frac {2 a b^{5} d^{3}}{3} + \frac {b^{6} c d^{2}}{3}\right ) + x^{8} \left (\frac {15 a^{2} b^{4} d^{3}}{8} + \frac {9 a b^{5} c d^{2}}{4} + \frac {3 b^{6} c^{2} d}{8}\right ) + x^{7} \left (\frac {20 a^{3} b^{3} d^{3}}{7} + \frac {45 a^{2} b^{4} c d^{2}}{7} + \frac {18 a b^{5} c^{2} d}{7} + \frac {b^{6} c^{3}}{7}\right ) + x^{6} \left (\frac {5 a^{4} b^{2} d^{3}}{2} + 10 a^{3} b^{3} c d^{2} + \frac {15 a^{2} b^{4} c^{2} d}{2} + a b^{5} c^{3}\right ) + x^{5} \left (\frac {6 a^{5} b d^{3}}{5} + 9 a^{4} b^{2} c d^{2} + 12 a^{3} b^{3} c^{2} d + 3 a^{2} b^{4} c^{3}\right ) + x^{4} \left (\frac {a^{6} d^{3}}{4} + \frac {9 a^{5} b c d^{2}}{2} + \frac {45 a^{4} b^{2} c^{2} d}{4} + 5 a^{3} b^{3} c^{3}\right ) + x^{3} \left (a^{6} c d^{2} + 6 a^{5} b c^{2} d + 5 a^{4} b^{2} c^{3}\right ) + x^{2} \left (\frac {3 a^{6} c^{2} d}{2} + 3 a^{5} b c^{3}\right ) \]

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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