3.2855 \(\int (c+d x)^3 (a+b (c+d x)^3)^3 \, dx\)

Optimal. Leaf size=71 \[ \frac {a^3 (c+d x)^4}{4 d}+\frac {3 a^2 b (c+d x)^7}{7 d}+\frac {3 a b^2 (c+d x)^{10}}{10 d}+\frac {b^3 (c+d x)^{13}}{13 d} \]

[Out]

1/4*a^3*(d*x+c)^4/d+3/7*a^2*b*(d*x+c)^7/d+3/10*a*b^2*(d*x+c)^10/d+1/13*b^3*(d*x+c)^13/d

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Rubi [A]  time = 0.11, antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {372, 270} \[ \frac {3 a^2 b (c+d x)^7}{7 d}+\frac {a^3 (c+d x)^4}{4 d}+\frac {3 a b^2 (c+d x)^{10}}{10 d}+\frac {b^3 (c+d x)^{13}}{13 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^3*(c + d*x)^4)/(4*d) + (3*a^2*b*(c + d*x)^7)/(7*d) + (3*a*b^2*(c + d*x)^10)/(10*d) + (b^3*(c + d*x)^13)/(13
*d)

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rule 372

Int[(u_)^(m_.)*((a_) + (b_.)*(v_)^(n_))^(p_.), x_Symbol] :> Dist[u^m/(Coefficient[v, x, 1]*v^m), Subst[Int[x^m
*(a + b*x^n)^p, x], x, v], x] /; FreeQ[{a, b, m, n, p}, x] && LinearPairQ[u, v, x]

Rubi steps

\begin {align*} \int (c+d x)^3 \left (a+b (c+d x)^3\right )^3 \, dx &=\frac {\operatorname {Subst}\left (\int x^3 \left (a+b x^3\right )^3 \, dx,x,c+d x\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int \left (a^3 x^3+3 a^2 b x^6+3 a b^2 x^9+b^3 x^{12}\right ) \, dx,x,c+d x\right )}{d}\\ &=\frac {a^3 (c+d x)^4}{4 d}+\frac {3 a^2 b (c+d x)^7}{7 d}+\frac {3 a b^2 (c+d x)^{10}}{10 d}+\frac {b^3 (c+d x)^{13}}{13 d}\\ \end {align*}

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Mathematica [B]  time = 0.08, size = 323, normalized size = 4.55 \[ \frac {3}{7} b d^6 x^7 \left (a^2+84 a b c^3+308 b^2 c^6\right )+3 b c d^5 x^6 \left (a^2+21 a b c^3+44 b^2 c^6\right )+\frac {9}{5} b c^2 d^4 x^5 \left (5 a^2+42 a b c^3+55 b^2 c^6\right )+\frac {1}{4} d^3 x^4 \left (a^3+60 a^2 b c^3+252 a b^2 c^6+220 b^3 c^9\right )+c d^2 x^3 \left (a^3+15 a^2 b c^3+36 a b^2 c^6+22 b^3 c^9\right )+\frac {1}{10} b^2 d^9 x^{10} \left (3 a+220 b c^3\right )+b^2 c d^8 x^9 \left (3 a+55 b c^3\right )+\frac {9}{2} b^2 c^2 d^7 x^8 \left (3 a+22 b c^3\right )+c^3 x \left (a+b c^3\right )^3+\frac {3}{2} c^2 d x^2 \left (a+b c^3\right )^2 \left (a+4 b c^3\right )+6 b^3 c^2 d^{10} x^{11}+b^3 c d^{11} x^{12}+\frac {1}{13} b^3 d^{12} x^{13} \]

Antiderivative was successfully verified.

[In]

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

[Out]

c^3*(a + b*c^3)^3*x + (3*c^2*(a + b*c^3)^2*(a + 4*b*c^3)*d*x^2)/2 + c*(a^3 + 15*a^2*b*c^3 + 36*a*b^2*c^6 + 22*
b^3*c^9)*d^2*x^3 + ((a^3 + 60*a^2*b*c^3 + 252*a*b^2*c^6 + 220*b^3*c^9)*d^3*x^4)/4 + (9*b*c^2*(5*a^2 + 42*a*b*c
^3 + 55*b^2*c^6)*d^4*x^5)/5 + 3*b*c*(a^2 + 21*a*b*c^3 + 44*b^2*c^6)*d^5*x^6 + (3*b*(a^2 + 84*a*b*c^3 + 308*b^2
*c^6)*d^6*x^7)/7 + (9*b^2*c^2*(3*a + 22*b*c^3)*d^7*x^8)/2 + b^2*c*(3*a + 55*b*c^3)*d^8*x^9 + (b^2*(3*a + 220*b
*c^3)*d^9*x^10)/10 + 6*b^3*c^2*d^10*x^11 + b^3*c*d^11*x^12 + (b^3*d^12*x^13)/13

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fricas [B]  time = 0.75, size = 442, normalized size = 6.23 \[ \frac {1}{13} x^{13} d^{12} b^{3} + x^{12} d^{11} c b^{3} + 6 x^{11} d^{10} c^{2} b^{3} + 22 x^{10} d^{9} c^{3} b^{3} + 55 x^{9} d^{8} c^{4} b^{3} + 99 x^{8} d^{7} c^{5} b^{3} + 132 x^{7} d^{6} c^{6} b^{3} + \frac {3}{10} x^{10} d^{9} b^{2} a + 132 x^{6} d^{5} c^{7} b^{3} + 3 x^{9} d^{8} c b^{2} a + 99 x^{5} d^{4} c^{8} b^{3} + \frac {27}{2} x^{8} d^{7} c^{2} b^{2} a + 55 x^{4} d^{3} c^{9} b^{3} + 36 x^{7} d^{6} c^{3} b^{2} a + 22 x^{3} d^{2} c^{10} b^{3} + 63 x^{6} d^{5} c^{4} b^{2} a + 6 x^{2} d c^{11} b^{3} + \frac {378}{5} x^{5} d^{4} c^{5} b^{2} a + x c^{12} b^{3} + 63 x^{4} d^{3} c^{6} b^{2} a + \frac {3}{7} x^{7} d^{6} b a^{2} + 36 x^{3} d^{2} c^{7} b^{2} a + 3 x^{6} d^{5} c b a^{2} + \frac {27}{2} x^{2} d c^{8} b^{2} a + 9 x^{5} d^{4} c^{2} b a^{2} + 3 x c^{9} b^{2} a + 15 x^{4} d^{3} c^{3} b a^{2} + 15 x^{3} d^{2} c^{4} b a^{2} + 9 x^{2} d c^{5} b a^{2} + 3 x c^{6} b a^{2} + \frac {1}{4} x^{4} d^{3} a^{3} + x^{3} d^{2} c a^{3} + \frac {3}{2} x^{2} d c^{2} a^{3} + x c^{3} a^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/13*x^13*d^12*b^3 + x^12*d^11*c*b^3 + 6*x^11*d^10*c^2*b^3 + 22*x^10*d^9*c^3*b^3 + 55*x^9*d^8*c^4*b^3 + 99*x^8
*d^7*c^5*b^3 + 132*x^7*d^6*c^6*b^3 + 3/10*x^10*d^9*b^2*a + 132*x^6*d^5*c^7*b^3 + 3*x^9*d^8*c*b^2*a + 99*x^5*d^
4*c^8*b^3 + 27/2*x^8*d^7*c^2*b^2*a + 55*x^4*d^3*c^9*b^3 + 36*x^7*d^6*c^3*b^2*a + 22*x^3*d^2*c^10*b^3 + 63*x^6*
d^5*c^4*b^2*a + 6*x^2*d*c^11*b^3 + 378/5*x^5*d^4*c^5*b^2*a + x*c^12*b^3 + 63*x^4*d^3*c^6*b^2*a + 3/7*x^7*d^6*b
*a^2 + 36*x^3*d^2*c^7*b^2*a + 3*x^6*d^5*c*b*a^2 + 27/2*x^2*d*c^8*b^2*a + 9*x^5*d^4*c^2*b*a^2 + 3*x*c^9*b^2*a +
15*x^4*d^3*c^3*b*a^2 + 15*x^3*d^2*c^4*b*a^2 + 9*x^2*d*c^5*b*a^2 + 3*x*c^6*b*a^2 + 1/4*x^4*d^3*a^3 + x^3*d^2*c
*a^3 + 3/2*x^2*d*c^2*a^3 + x*c^3*a^3

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giac [B]  time = 0.16, size = 442, normalized size = 6.23 \[ \frac {1}{13} \, b^{3} d^{12} x^{13} + b^{3} c d^{11} x^{12} + 6 \, b^{3} c^{2} d^{10} x^{11} + 22 \, b^{3} c^{3} d^{9} x^{10} + 55 \, b^{3} c^{4} d^{8} x^{9} + 99 \, b^{3} c^{5} d^{7} x^{8} + 132 \, b^{3} c^{6} d^{6} x^{7} + \frac {3}{10} \, a b^{2} d^{9} x^{10} + 132 \, b^{3} c^{7} d^{5} x^{6} + 3 \, a b^{2} c d^{8} x^{9} + 99 \, b^{3} c^{8} d^{4} x^{5} + \frac {27}{2} \, a b^{2} c^{2} d^{7} x^{8} + 55 \, b^{3} c^{9} d^{3} x^{4} + 36 \, a b^{2} c^{3} d^{6} x^{7} + 22 \, b^{3} c^{10} d^{2} x^{3} + 63 \, a b^{2} c^{4} d^{5} x^{6} + 6 \, b^{3} c^{11} d x^{2} + \frac {378}{5} \, a b^{2} c^{5} d^{4} x^{5} + b^{3} c^{12} x + 63 \, a b^{2} c^{6} d^{3} x^{4} + \frac {3}{7} \, a^{2} b d^{6} x^{7} + 36 \, a b^{2} c^{7} d^{2} x^{3} + 3 \, a^{2} b c d^{5} x^{6} + \frac {27}{2} \, a b^{2} c^{8} d x^{2} + 9 \, a^{2} b c^{2} d^{4} x^{5} + 3 \, a b^{2} c^{9} x + 15 \, a^{2} b c^{3} d^{3} x^{4} + 15 \, a^{2} b c^{4} d^{2} x^{3} + 9 \, a^{2} b c^{5} d x^{2} + 3 \, a^{2} b c^{6} x + \frac {1}{4} \, a^{3} d^{3} x^{4} + a^{3} c d^{2} x^{3} + \frac {3}{2} \, a^{3} c^{2} d x^{2} + a^{3} c^{3} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/13*b^3*d^12*x^13 + b^3*c*d^11*x^12 + 6*b^3*c^2*d^10*x^11 + 22*b^3*c^3*d^9*x^10 + 55*b^3*c^4*d^8*x^9 + 99*b^3
*c^5*d^7*x^8 + 132*b^3*c^6*d^6*x^7 + 3/10*a*b^2*d^9*x^10 + 132*b^3*c^7*d^5*x^6 + 3*a*b^2*c*d^8*x^9 + 99*b^3*c^
8*d^4*x^5 + 27/2*a*b^2*c^2*d^7*x^8 + 55*b^3*c^9*d^3*x^4 + 36*a*b^2*c^3*d^6*x^7 + 22*b^3*c^10*d^2*x^3 + 63*a*b^
2*c^4*d^5*x^6 + 6*b^3*c^11*d*x^2 + 378/5*a*b^2*c^5*d^4*x^5 + b^3*c^12*x + 63*a*b^2*c^6*d^3*x^4 + 3/7*a^2*b*d^6
*x^7 + 36*a*b^2*c^7*d^2*x^3 + 3*a^2*b*c*d^5*x^6 + 27/2*a*b^2*c^8*d*x^2 + 9*a^2*b*c^2*d^4*x^5 + 3*a*b^2*c^9*x +
15*a^2*b*c^3*d^3*x^4 + 15*a^2*b*c^4*d^2*x^3 + 9*a^2*b*c^5*d*x^2 + 3*a^2*b*c^6*x + 1/4*a^3*d^3*x^4 + a^3*c*d^2
*x^3 + 3/2*a^3*c^2*d*x^2 + a^3*c^3*x

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maple [B]  time = 0.00, size = 1948, normalized size = 27.44 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/13*d^12*b^3*x^13+c*d^11*b^3*x^12+6*c^2*d^10*b^3*x^11+1/10*(136*c^3*b^3*d^9+d^3*((b*c^3+a)*b^2*d^6+63*b^3*c^3
*d^6+b*d^3*(18*b^2*c^3*d^3+2*(b*c^3+a)*b*d^3)))*x^10+1/9*(117*c^4*b^3*d^8+3*c*d^2*((b*c^3+a)*b^2*d^6+63*b^3*c^
3*d^6+b*d^3*(18*b^2*c^3*d^3+2*(b*c^3+a)*b*d^3))+d^3*(6*(b*c^3+a)*b^2*c*d^5+45*b^3*c^4*d^5+3*b*c*d^2*(18*b^2*c^
3*d^3+2*(b*c^3+a)*b*d^3)+b*d^3*(9*b^2*c^4*d^2+6*(b*c^3+a)*b*c*d^2)))*x^9+1/8*(36*c^5*b^3*d^7+3*c^2*d*((b*c^3+a
)*b^2*d^6+63*b^3*c^3*d^6+b*d^3*(18*b^2*c^3*d^3+2*(b*c^3+a)*b*d^3))+3*c*d^2*(6*(b*c^3+a)*b^2*c*d^5+45*b^3*c^4*d
^5+3*b*c*d^2*(18*b^2*c^3*d^3+2*(b*c^3+a)*b*d^3)+b*d^3*(9*b^2*c^4*d^2+6*(b*c^3+a)*b*c*d^2))+d^3*(21*(b*c^3+a)*b
^2*c^2*d^4+3*b*c^2*d*(18*b^2*c^3*d^3+2*(b*c^3+a)*b*d^3)+3*b*c*d^2*(9*b^2*c^4*d^2+6*(b*c^3+a)*b*c*d^2)))*x^8+1/
7*(c^3*((b*c^3+a)*b^2*d^6+63*b^3*c^3*d^6+b*d^3*(18*b^2*c^3*d^3+2*(b*c^3+a)*b*d^3))+3*c^2*d*(6*(b*c^3+a)*b^2*c*
d^5+45*b^3*c^4*d^5+3*b*c*d^2*(18*b^2*c^3*d^3+2*(b*c^3+a)*b*d^3)+b*d^3*(9*b^2*c^4*d^2+6*(b*c^3+a)*b*c*d^2))+3*c
*d^2*(21*(b*c^3+a)*b^2*c^2*d^4+3*b*c^2*d*(18*b^2*c^3*d^3+2*(b*c^3+a)*b*d^3)+3*b*c*d^2*(9*b^2*c^4*d^2+6*(b*c^3+
a)*b*c*d^2))+d^3*((b*c^3+a)*(18*b^2*c^3*d^3+2*(b*c^3+a)*b*d^3)+3*b*c^2*d*(9*b^2*c^4*d^2+6*(b*c^3+a)*b*c*d^2)+1
8*b^2*c^3*d^3*(b*c^3+a)+b*d^3*(b*c^3+a)^2))*x^7+1/6*(c^3*(6*(b*c^3+a)*b^2*c*d^5+45*b^3*c^4*d^5+3*b*c*d^2*(18*b
^2*c^3*d^3+2*(b*c^3+a)*b*d^3)+b*d^3*(9*b^2*c^4*d^2+6*(b*c^3+a)*b*c*d^2))+3*c^2*d*(21*(b*c^3+a)*b^2*c^2*d^4+3*b
*c^2*d*(18*b^2*c^3*d^3+2*(b*c^3+a)*b*d^3)+3*b*c*d^2*(9*b^2*c^4*d^2+6*(b*c^3+a)*b*c*d^2))+3*c*d^2*((b*c^3+a)*(1
8*b^2*c^3*d^3+2*(b*c^3+a)*b*d^3)+3*b*c^2*d*(9*b^2*c^4*d^2+6*(b*c^3+a)*b*c*d^2)+18*b^2*c^3*d^3*(b*c^3+a)+b*d^3*
(b*c^3+a)^2)+d^3*((b*c^3+a)*(9*b^2*c^4*d^2+6*(b*c^3+a)*b*c*d^2)+18*b^2*c^4*d^2*(b*c^3+a)+3*b*c*d^2*(b*c^3+a)^2
))*x^6+1/5*(c^3*(21*(b*c^3+a)*b^2*c^2*d^4+3*b*c^2*d*(18*b^2*c^3*d^3+2*(b*c^3+a)*b*d^3)+3*b*c*d^2*(9*b^2*c^4*d^
2+6*(b*c^3+a)*b*c*d^2))+3*c^2*d*((b*c^3+a)*(18*b^2*c^3*d^3+2*(b*c^3+a)*b*d^3)+3*b*c^2*d*(9*b^2*c^4*d^2+6*(b*c^
3+a)*b*c*d^2)+18*b^2*c^3*d^3*(b*c^3+a)+b*d^3*(b*c^3+a)^2)+3*c*d^2*((b*c^3+a)*(9*b^2*c^4*d^2+6*(b*c^3+a)*b*c*d^
2)+18*b^2*c^4*d^2*(b*c^3+a)+3*b*c*d^2*(b*c^3+a)^2)+9*d^4*(b*c^3+a)^2*b*c^2)*x^5+1/4*(c^3*((b*c^3+a)*(18*b^2*c^
3*d^3+2*(b*c^3+a)*b*d^3)+3*b*c^2*d*(9*b^2*c^4*d^2+6*(b*c^3+a)*b*c*d^2)+18*b^2*c^3*d^3*(b*c^3+a)+b*d^3*(b*c^3+a
)^2)+3*c^2*d*((b*c^3+a)*(9*b^2*c^4*d^2+6*(b*c^3+a)*b*c*d^2)+18*b^2*c^4*d^2*(b*c^3+a)+3*b*c*d^2*(b*c^3+a)^2)+27
*c^3*d^3*(b*c^3+a)^2*b+d^3*(b*c^3+a)^3)*x^4+1/3*(c^3*((b*c^3+a)*(9*b^2*c^4*d^2+6*(b*c^3+a)*b*c*d^2)+18*b^2*c^4
*d^2*(b*c^3+a)+3*b*c*d^2*(b*c^3+a)^2)+27*c^4*d^2*(b*c^3+a)^2*b+3*c*d^2*(b*c^3+a)^3)*x^3+1/2*(9*c^5*(b*c^3+a)^2
*b*d+3*c^2*d*(b*c^3+a)^3)*x^2+c^3*(b*c^3+a)^3*x

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maxima [B]  time = 0.58, size = 359, normalized size = 5.06 \[ \frac {1}{13} \, b^{3} d^{12} x^{13} + b^{3} c d^{11} x^{12} + 6 \, b^{3} c^{2} d^{10} x^{11} + \frac {1}{10} \, {\left (220 \, b^{3} c^{3} + 3 \, a b^{2}\right )} d^{9} x^{10} + {\left (55 \, b^{3} c^{4} + 3 \, a b^{2} c\right )} d^{8} x^{9} + \frac {9}{2} \, {\left (22 \, b^{3} c^{5} + 3 \, a b^{2} c^{2}\right )} d^{7} x^{8} + \frac {3}{7} \, {\left (308 \, b^{3} c^{6} + 84 \, a b^{2} c^{3} + a^{2} b\right )} d^{6} x^{7} + 3 \, {\left (44 \, b^{3} c^{7} + 21 \, a b^{2} c^{4} + a^{2} b c\right )} d^{5} x^{6} + \frac {9}{5} \, {\left (55 \, b^{3} c^{8} + 42 \, a b^{2} c^{5} + 5 \, a^{2} b c^{2}\right )} d^{4} x^{5} + \frac {1}{4} \, {\left (220 \, b^{3} c^{9} + 252 \, a b^{2} c^{6} + 60 \, a^{2} b c^{3} + a^{3}\right )} d^{3} x^{4} + {\left (22 \, b^{3} c^{10} + 36 \, a b^{2} c^{7} + 15 \, a^{2} b c^{4} + a^{3} c\right )} d^{2} x^{3} + \frac {3}{2} \, {\left (4 \, b^{3} c^{11} + 9 \, a b^{2} c^{8} + 6 \, a^{2} b c^{5} + a^{3} c^{2}\right )} d x^{2} + {\left (b^{3} c^{12} + 3 \, a b^{2} c^{9} + 3 \, a^{2} b c^{6} + a^{3} c^{3}\right )} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/13*b^3*d^12*x^13 + b^3*c*d^11*x^12 + 6*b^3*c^2*d^10*x^11 + 1/10*(220*b^3*c^3 + 3*a*b^2)*d^9*x^10 + (55*b^3*c
^4 + 3*a*b^2*c)*d^8*x^9 + 9/2*(22*b^3*c^5 + 3*a*b^2*c^2)*d^7*x^8 + 3/7*(308*b^3*c^6 + 84*a*b^2*c^3 + a^2*b)*d^
6*x^7 + 3*(44*b^3*c^7 + 21*a*b^2*c^4 + a^2*b*c)*d^5*x^6 + 9/5*(55*b^3*c^8 + 42*a*b^2*c^5 + 5*a^2*b*c^2)*d^4*x^
5 + 1/4*(220*b^3*c^9 + 252*a*b^2*c^6 + 60*a^2*b*c^3 + a^3)*d^3*x^4 + (22*b^3*c^10 + 36*a*b^2*c^7 + 15*a^2*b*c^
4 + a^3*c)*d^2*x^3 + 3/2*(4*b^3*c^11 + 9*a*b^2*c^8 + 6*a^2*b*c^5 + a^3*c^2)*d*x^2 + (b^3*c^12 + 3*a*b^2*c^9 +
3*a^2*b*c^6 + a^3*c^3)*x

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mupad [B]  time = 1.28, size = 309, normalized size = 4.35 \[ c^3\,x\,{\left (b\,c^3+a\right )}^3+\frac {d^3\,x^4\,\left (a^3+60\,a^2\,b\,c^3+252\,a\,b^2\,c^6+220\,b^3\,c^9\right )}{4}+\frac {b^3\,d^{12}\,x^{13}}{13}+b^3\,c\,d^{11}\,x^{12}+\frac {3\,b\,d^6\,x^7\,\left (a^2+84\,a\,b\,c^3+308\,b^2\,c^6\right )}{7}+\frac {b^2\,d^9\,x^{10}\,\left (220\,b\,c^3+3\,a\right )}{10}+6\,b^3\,c^2\,d^{10}\,x^{11}+c\,d^2\,x^3\,\left (a^3+15\,a^2\,b\,c^3+36\,a\,b^2\,c^6+22\,b^3\,c^9\right )+\frac {9\,b^2\,c^2\,d^7\,x^8\,\left (22\,b\,c^3+3\,a\right )}{2}+\frac {3\,c^2\,d\,x^2\,{\left (b\,c^3+a\right )}^2\,\left (4\,b\,c^3+a\right )}{2}+\frac {9\,b\,c^2\,d^4\,x^5\,\left (5\,a^2+42\,a\,b\,c^3+55\,b^2\,c^6\right )}{5}+3\,b\,c\,d^5\,x^6\,\left (a^2+21\,a\,b\,c^3+44\,b^2\,c^6\right )+b^2\,c\,d^8\,x^9\,\left (55\,b\,c^3+3\,a\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c^3*x*(a + b*c^3)^3 + (d^3*x^4*(a^3 + 220*b^3*c^9 + 60*a^2*b*c^3 + 252*a*b^2*c^6))/4 + (b^3*d^12*x^13)/13 + b^
3*c*d^11*x^12 + (3*b*d^6*x^7*(a^2 + 308*b^2*c^6 + 84*a*b*c^3))/7 + (b^2*d^9*x^10*(3*a + 220*b*c^3))/10 + 6*b^3
*c^2*d^10*x^11 + c*d^2*x^3*(a^3 + 22*b^3*c^9 + 15*a^2*b*c^3 + 36*a*b^2*c^6) + (9*b^2*c^2*d^7*x^8*(3*a + 22*b*c
^3))/2 + (3*c^2*d*x^2*(a + b*c^3)^2*(a + 4*b*c^3))/2 + (9*b*c^2*d^4*x^5*(5*a^2 + 55*b^2*c^6 + 42*a*b*c^3))/5 +
3*b*c*d^5*x^6*(a^2 + 44*b^2*c^6 + 21*a*b*c^3) + b^2*c*d^8*x^9*(3*a + 55*b*c^3)

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sympy [B]  time = 0.17, size = 437, normalized size = 6.15 \[ 6 b^{3} c^{2} d^{10} x^{11} + b^{3} c d^{11} x^{12} + \frac {b^{3} d^{12} x^{13}}{13} + x^{10} \left (\frac {3 a b^{2} d^{9}}{10} + 22 b^{3} c^{3} d^{9}\right ) + x^{9} \left (3 a b^{2} c d^{8} + 55 b^{3} c^{4} d^{8}\right ) + x^{8} \left (\frac {27 a b^{2} c^{2} d^{7}}{2} + 99 b^{3} c^{5} d^{7}\right ) + x^{7} \left (\frac {3 a^{2} b d^{6}}{7} + 36 a b^{2} c^{3} d^{6} + 132 b^{3} c^{6} d^{6}\right ) + x^{6} \left (3 a^{2} b c d^{5} + 63 a b^{2} c^{4} d^{5} + 132 b^{3} c^{7} d^{5}\right ) + x^{5} \left (9 a^{2} b c^{2} d^{4} + \frac {378 a b^{2} c^{5} d^{4}}{5} + 99 b^{3} c^{8} d^{4}\right ) + x^{4} \left (\frac {a^{3} d^{3}}{4} + 15 a^{2} b c^{3} d^{3} + 63 a b^{2} c^{6} d^{3} + 55 b^{3} c^{9} d^{3}\right ) + x^{3} \left (a^{3} c d^{2} + 15 a^{2} b c^{4} d^{2} + 36 a b^{2} c^{7} d^{2} + 22 b^{3} c^{10} d^{2}\right ) + x^{2} \left (\frac {3 a^{3} c^{2} d}{2} + 9 a^{2} b c^{5} d + \frac {27 a b^{2} c^{8} d}{2} + 6 b^{3} c^{11} d\right ) + x \left (a^{3} c^{3} + 3 a^{2} b c^{6} + 3 a b^{2} c^{9} + b^{3} c^{12}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

6*b**3*c**2*d**10*x**11 + b**3*c*d**11*x**12 + b**3*d**12*x**13/13 + x**10*(3*a*b**2*d**9/10 + 22*b**3*c**3*d*
*9) + x**9*(3*a*b**2*c*d**8 + 55*b**3*c**4*d**8) + x**8*(27*a*b**2*c**2*d**7/2 + 99*b**3*c**5*d**7) + x**7*(3*
a**2*b*d**6/7 + 36*a*b**2*c**3*d**6 + 132*b**3*c**6*d**6) + x**6*(3*a**2*b*c*d**5 + 63*a*b**2*c**4*d**5 + 132*
b**3*c**7*d**5) + x**5*(9*a**2*b*c**2*d**4 + 378*a*b**2*c**5*d**4/5 + 99*b**3*c**8*d**4) + x**4*(a**3*d**3/4 +
15*a**2*b*c**3*d**3 + 63*a*b**2*c**6*d**3 + 55*b**3*c**9*d**3) + x**3*(a**3*c*d**2 + 15*a**2*b*c**4*d**2 + 36
*a*b**2*c**7*d**2 + 22*b**3*c**10*d**2) + x**2*(3*a**3*c**2*d/2 + 9*a**2*b*c**5*d + 27*a*b**2*c**8*d/2 + 6*b**
3*c**11*d) + x*(a**3*c**3 + 3*a**2*b*c**6 + 3*a*b**2*c**9 + b**3*c**12)

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