3.2906 \(\int (c+d x)^3 \left (a+b (c+d x)^4\right )^3 \, dx\)
Optimal. Leaf size=23 \[ \frac{\left (a+b (c+d x)^4\right )^4}{16 b d} \]
[Out]
(a + b*(c + d*x)^4)^4/(16*b*d)
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Rubi [A] time = 0.0430844, antiderivative size = 23, normalized size of antiderivative = 1.,
number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048
\[ \frac{\left (a+b (c+d x)^4\right )^4}{16 b d} \]
Antiderivative was successfully verified.
[In] Int[(c + d*x)^3*(a + b*(c + d*x)^4)^3,x]
[Out]
(a + b*(c + d*x)^4)^4/(16*b*d)
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Rubi in Sympy [A] time = 5.99503, size = 15, normalized size = 0.65 \[ \frac{\left (a + b \left (c + d x\right )^{4}\right )^{4}}{16 b d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((d*x+c)**3*(a+b*(d*x+c)**4)**3,x)
[Out]
(a + b*(c + d*x)**4)**4/(16*b*d)
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Mathematica [B] time = 0.157606, size = 308, normalized size = 13.39 \[ \frac{1}{16} x \left (4 c^3+6 c^2 d x+4 c d^2 x^2+d^3 x^3\right ) \left (4 a^3+6 a^2 b \left (2 c^4+4 c^3 d x+6 c^2 d^2 x^2+4 c d^3 x^3+d^4 x^4\right )+4 a b^2 \left (3 c^8+12 c^7 d x+34 c^6 d^2 x^2+60 c^5 d^3 x^3+71 c^4 d^4 x^4+56 c^3 d^5 x^5+28 c^2 d^6 x^6+8 c d^7 x^7+d^8 x^8\right )+b^3 \left (4 c^{12}+24 c^{11} d x+100 c^{10} d^2 x^2+280 c^9 d^3 x^3+566 c^8 d^4 x^4+848 c^7 d^5 x^5+952 c^6 d^6 x^6+800 c^5 d^7 x^7+496 c^4 d^8 x^8+220 c^3 d^9 x^9+66 c^2 d^{10} x^{10}+12 c d^{11} x^{11}+d^{12} x^{12}\right )\right ) \]
Antiderivative was successfully verified.
[In] Integrate[(c + d*x)^3*(a + b*(c + d*x)^4)^3,x]
[Out]
(x*(4*c^3 + 6*c^2*d*x + 4*c*d^2*x^2 + d^3*x^3)*(4*a^3 + 6*a^2*b*(2*c^4 + 4*c^3*d
*x + 6*c^2*d^2*x^2 + 4*c*d^3*x^3 + d^4*x^4) + 4*a*b^2*(3*c^8 + 12*c^7*d*x + 34*c
^6*d^2*x^2 + 60*c^5*d^3*x^3 + 71*c^4*d^4*x^4 + 56*c^3*d^5*x^5 + 28*c^2*d^6*x^6 +
8*c*d^7*x^7 + d^8*x^8) + b^3*(4*c^12 + 24*c^11*d*x + 100*c^10*d^2*x^2 + 280*c^9
*d^3*x^3 + 566*c^8*d^4*x^4 + 848*c^7*d^5*x^5 + 952*c^6*d^6*x^6 + 800*c^5*d^7*x^7
+ 496*c^4*d^8*x^8 + 220*c^3*d^9*x^9 + 66*c^2*d^10*x^10 + 12*c*d^11*x^11 + d^12*
x^12)))/16
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Maple [B] time = 0.003, size = 3262, normalized size = 141.8 \[ \text{output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((d*x+c)^3*(a+b*(d*x+c)^4)^3,x)
[Out]
1/16*d^15*b^3*x^16+c*d^14*b^3*x^15+15/2*c^2*d^13*b^3*x^14+35*c^3*b^3*d^12*x^13+1
/12*(870*c^4*b^3*d^11+d^3*((b*c^4+a)*b^2*d^8+424*c^4*b^3*d^8+b*d^4*(2*(b*c^4+a)*
b*d^4+68*c^4*d^4*b^2)))*x^12+1/11*(726*c^5*d^10*b^3+3*c*d^2*((b*c^4+a)*b^2*d^8+4
24*c^4*b^3*d^8+b*d^4*(2*(b*c^4+a)*b*d^4+68*c^4*d^4*b^2))+d^3*(8*(b*c^4+a)*b^2*c*
d^7+448*c^5*d^7*b^3+4*b*c*d^3*(2*(b*c^4+a)*b*d^4+68*c^4*d^4*b^2)+b*d^4*(8*(b*c^4
+a)*b*c*d^3+48*c^5*d^3*b^2)))*x^11+1/10*(220*c^6*b^3*d^9+3*c^2*d*((b*c^4+a)*b^2*
d^8+424*c^4*b^3*d^8+b*d^4*(2*(b*c^4+a)*b*d^4+68*c^4*d^4*b^2))+3*c*d^2*(8*(b*c^4+
a)*b^2*c*d^7+448*c^5*d^7*b^3+4*b*c*d^3*(2*(b*c^4+a)*b*d^4+68*c^4*d^4*b^2)+b*d^4*
(8*(b*c^4+a)*b*c*d^3+48*c^5*d^3*b^2))+d^3*(28*(b*c^4+a)*c^2*d^6*b^2+224*c^6*d^6*
b^3+6*c^2*d^2*b*(2*(b*c^4+a)*b*d^4+68*c^4*d^4*b^2)+4*b*c*d^3*(8*(b*c^4+a)*b*c*d^
3+48*c^5*d^3*b^2)+b*d^4*(12*(b*c^4+a)*c^2*d^2*b+16*c^6*d^2*b^2)))*x^10+1/9*(c^3*
((b*c^4+a)*b^2*d^8+424*c^4*b^3*d^8+b*d^4*(2*(b*c^4+a)*b*d^4+68*c^4*d^4*b^2))+3*c
^2*d*(8*(b*c^4+a)*b^2*c*d^7+448*c^5*d^7*b^3+4*b*c*d^3*(2*(b*c^4+a)*b*d^4+68*c^4*
d^4*b^2)+b*d^4*(8*(b*c^4+a)*b*c*d^3+48*c^5*d^3*b^2))+3*c*d^2*(28*(b*c^4+a)*c^2*d
^6*b^2+224*c^6*d^6*b^3+6*c^2*d^2*b*(2*(b*c^4+a)*b*d^4+68*c^4*d^4*b^2)+4*b*c*d^3*
(8*(b*c^4+a)*b*c*d^3+48*c^5*d^3*b^2)+b*d^4*(12*(b*c^4+a)*c^2*d^2*b+16*c^6*d^2*b^
2))+d^3*(64*(b*c^4+a)*c^3*d^5*b^2+4*c^3*d*b*(2*(b*c^4+a)*b*d^4+68*c^4*d^4*b^2)+6
*c^2*d^2*b*(8*(b*c^4+a)*b*c*d^3+48*c^5*d^3*b^2)+4*b*c*d^3*(12*(b*c^4+a)*c^2*d^2*
b+16*c^6*d^2*b^2)))*x^9+1/8*(c^3*(8*(b*c^4+a)*b^2*c*d^7+448*c^5*d^7*b^3+4*b*c*d^
3*(2*(b*c^4+a)*b*d^4+68*c^4*d^4*b^2)+b*d^4*(8*(b*c^4+a)*b*c*d^3+48*c^5*d^3*b^2))
+3*c^2*d*(28*(b*c^4+a)*c^2*d^6*b^2+224*c^6*d^6*b^3+6*c^2*d^2*b*(2*(b*c^4+a)*b*d^
4+68*c^4*d^4*b^2)+4*b*c*d^3*(8*(b*c^4+a)*b*c*d^3+48*c^5*d^3*b^2)+b*d^4*(12*(b*c^
4+a)*c^2*d^2*b+16*c^6*d^2*b^2))+3*c*d^2*(64*(b*c^4+a)*c^3*d^5*b^2+4*c^3*d*b*(2*(
b*c^4+a)*b*d^4+68*c^4*d^4*b^2)+6*c^2*d^2*b*(8*(b*c^4+a)*b*c*d^3+48*c^5*d^3*b^2)+
4*b*c*d^3*(12*(b*c^4+a)*c^2*d^2*b+16*c^6*d^2*b^2))+d^3*((b*c^4+a)*(2*(b*c^4+a)*b
*d^4+68*c^4*d^4*b^2)+4*c^3*d*b*(8*(b*c^4+a)*b*c*d^3+48*c^5*d^3*b^2)+6*c^2*d^2*b*
(12*(b*c^4+a)*c^2*d^2*b+16*c^6*d^2*b^2)+32*b^2*c^4*d^4*(b*c^4+a)+b*d^4*(b*c^4+a)
^2))*x^8+1/7*(c^3*(28*(b*c^4+a)*c^2*d^6*b^2+224*c^6*d^6*b^3+6*c^2*d^2*b*(2*(b*c^
4+a)*b*d^4+68*c^4*d^4*b^2)+4*b*c*d^3*(8*(b*c^4+a)*b*c*d^3+48*c^5*d^3*b^2)+b*d^4*
(12*(b*c^4+a)*c^2*d^2*b+16*c^6*d^2*b^2))+3*c^2*d*(64*(b*c^4+a)*c^3*d^5*b^2+4*c^3
*d*b*(2*(b*c^4+a)*b*d^4+68*c^4*d^4*b^2)+6*c^2*d^2*b*(8*(b*c^4+a)*b*c*d^3+48*c^5*
d^3*b^2)+4*b*c*d^3*(12*(b*c^4+a)*c^2*d^2*b+16*c^6*d^2*b^2))+3*c*d^2*((b*c^4+a)*(
2*(b*c^4+a)*b*d^4+68*c^4*d^4*b^2)+4*c^3*d*b*(8*(b*c^4+a)*b*c*d^3+48*c^5*d^3*b^2)
+6*c^2*d^2*b*(12*(b*c^4+a)*c^2*d^2*b+16*c^6*d^2*b^2)+32*b^2*c^4*d^4*(b*c^4+a)+b*
d^4*(b*c^4+a)^2)+d^3*((b*c^4+a)*(8*(b*c^4+a)*b*c*d^3+48*c^5*d^3*b^2)+4*c^3*d*b*(
12*(b*c^4+a)*c^2*d^2*b+16*c^6*d^2*b^2)+48*c^5*d^3*b^2*(b*c^4+a)+4*b*c*d^3*(b*c^4
+a)^2))*x^7+1/6*(c^3*(64*(b*c^4+a)*c^3*d^5*b^2+4*c^3*d*b*(2*(b*c^4+a)*b*d^4+68*c
^4*d^4*b^2)+6*c^2*d^2*b*(8*(b*c^4+a)*b*c*d^3+48*c^5*d^3*b^2)+4*b*c*d^3*(12*(b*c^
4+a)*c^2*d^2*b+16*c^6*d^2*b^2))+3*c^2*d*((b*c^4+a)*(2*(b*c^4+a)*b*d^4+68*c^4*d^4
*b^2)+4*c^3*d*b*(8*(b*c^4+a)*b*c*d^3+48*c^5*d^3*b^2)+6*c^2*d^2*b*(12*(b*c^4+a)*c
^2*d^2*b+16*c^6*d^2*b^2)+32*b^2*c^4*d^4*(b*c^4+a)+b*d^4*(b*c^4+a)^2)+3*c*d^2*((b
*c^4+a)*(8*(b*c^4+a)*b*c*d^3+48*c^5*d^3*b^2)+4*c^3*d*b*(12*(b*c^4+a)*c^2*d^2*b+1
6*c^6*d^2*b^2)+48*c^5*d^3*b^2*(b*c^4+a)+4*b*c*d^3*(b*c^4+a)^2)+d^3*((b*c^4+a)*(1
2*(b*c^4+a)*c^2*d^2*b+16*c^6*d^2*b^2)+32*c^6*d^2*b^2*(b*c^4+a)+6*c^2*d^2*b*(b*c^
4+a)^2))*x^6+1/5*(c^3*((b*c^4+a)*(2*(b*c^4+a)*b*d^4+68*c^4*d^4*b^2)+4*c^3*d*b*(8
*(b*c^4+a)*b*c*d^3+48*c^5*d^3*b^2)+6*c^2*d^2*b*(12*(b*c^4+a)*c^2*d^2*b+16*c^6*d^
2*b^2)+32*b^2*c^4*d^4*(b*c^4+a)+b*d^4*(b*c^4+a)^2)+3*c^2*d*((b*c^4+a)*(8*(b*c^4+
a)*b*c*d^3+48*c^5*d^3*b^2)+4*c^3*d*b*(12*(b*c^4+a)*c^2*d^2*b+16*c^6*d^2*b^2)+48*
c^5*d^3*b^2*(b*c^4+a)+4*b*c*d^3*(b*c^4+a)^2)+3*c*d^2*((b*c^4+a)*(12*(b*c^4+a)*c^
2*d^2*b+16*c^6*d^2*b^2)+32*c^6*d^2*b^2*(b*c^4+a)+6*c^2*d^2*b*(b*c^4+a)^2)+12*d^4
*(b*c^4+a)^2*c^3*b)*x^5+1/4*(c^3*((b*c^4+a)*(8*(b*c^4+a)*b*c*d^3+48*c^5*d^3*b^2)
+4*c^3*d*b*(12*(b*c^4+a)*c^2*d^2*b+16*c^6*d^2*b^2)+48*c^5*d^3*b^2*(b*c^4+a)+4*b*
c*d^3*(b*c^4+a)^2)+3*c^2*d*((b*c^4+a)*(12*(b*c^4+a)*c^2*d^2*b+16*c^6*d^2*b^2)+32
*c^6*d^2*b^2*(b*c^4+a)+6*c^2*d^2*b*(b*c^4+a)^2)+36*c^4*d^3*(b*c^4+a)^2*b+d^3*(b*
c^4+a)^3)*x^4+1/3*(c^3*((b*c^4+a)*(12*(b*c^4+a)*c^2*d^2*b+16*c^6*d^2*b^2)+32*c^6
*d^2*b^2*(b*c^4+a)+6*c^2*d^2*b*(b*c^4+a)^2)+36*c^5*d^2*(b*c^4+a)^2*b+3*c*d^2*(b*
c^4+a)^3)*x^3+1/2*(12*c^6*(b*c^4+a)^2*d*b+3*c^2*d*(b*c^4+a)^3)*x^2+c^3*(b*c^4+a)
^3*x
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Maxima [A] time = 1.34622, size = 28, normalized size = 1.22 \[ \frac{{\left ({\left (d x + c\right )}^{4} b + a\right )}^{4}}{16 \, b d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(((d*x + c)^4*b + a)^3*(d*x + c)^3,x, algorithm="maxima")
[Out]
1/16*((d*x + c)^4*b + a)^4/(b*d)
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Fricas [A] time = 0.182283, size = 1, normalized size = 0.04 \[ \frac{1}{16} x^{16} d^{15} b^{3} + x^{15} d^{14} c b^{3} + \frac{15}{2} x^{14} d^{13} c^{2} b^{3} + 35 x^{13} d^{12} c^{3} b^{3} + \frac{455}{4} x^{12} d^{11} c^{4} b^{3} + 273 x^{11} d^{10} c^{5} b^{3} + \frac{1001}{2} x^{10} d^{9} c^{6} b^{3} + 715 x^{9} d^{8} c^{7} b^{3} + \frac{6435}{8} x^{8} d^{7} c^{8} b^{3} + \frac{1}{4} x^{12} d^{11} b^{2} a + 715 x^{7} d^{6} c^{9} b^{3} + 3 x^{11} d^{10} c b^{2} a + \frac{1001}{2} x^{6} d^{5} c^{10} b^{3} + \frac{33}{2} x^{10} d^{9} c^{2} b^{2} a + 273 x^{5} d^{4} c^{11} b^{3} + 55 x^{9} d^{8} c^{3} b^{2} a + \frac{455}{4} x^{4} d^{3} c^{12} b^{3} + \frac{495}{4} x^{8} d^{7} c^{4} b^{2} a + 35 x^{3} d^{2} c^{13} b^{3} + 198 x^{7} d^{6} c^{5} b^{2} a + \frac{15}{2} x^{2} d c^{14} b^{3} + 231 x^{6} d^{5} c^{6} b^{2} a + x c^{15} b^{3} + 198 x^{5} d^{4} c^{7} b^{2} a + \frac{495}{4} x^{4} d^{3} c^{8} b^{2} a + \frac{3}{8} x^{8} d^{7} b a^{2} + 55 x^{3} d^{2} c^{9} b^{2} a + 3 x^{7} d^{6} c b a^{2} + \frac{33}{2} x^{2} d c^{10} b^{2} a + \frac{21}{2} x^{6} d^{5} c^{2} b a^{2} + 3 x c^{11} b^{2} a + 21 x^{5} d^{4} c^{3} b a^{2} + \frac{105}{4} x^{4} d^{3} c^{4} b a^{2} + 21 x^{3} d^{2} c^{5} b a^{2} + \frac{21}{2} x^{2} d c^{6} b a^{2} + 3 x c^{7} b a^{2} + \frac{1}{4} x^{4} d^{3} a^{3} + x^{3} d^{2} c a^{3} + \frac{3}{2} x^{2} d c^{2} a^{3} + x c^{3} a^{3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(((d*x + c)^4*b + a)^3*(d*x + c)^3,x, algorithm="fricas")
[Out]
1/16*x^16*d^15*b^3 + x^15*d^14*c*b^3 + 15/2*x^14*d^13*c^2*b^3 + 35*x^13*d^12*c^3
*b^3 + 455/4*x^12*d^11*c^4*b^3 + 273*x^11*d^10*c^5*b^3 + 1001/2*x^10*d^9*c^6*b^3
+ 715*x^9*d^8*c^7*b^3 + 6435/8*x^8*d^7*c^8*b^3 + 1/4*x^12*d^11*b^2*a + 715*x^7*
d^6*c^9*b^3 + 3*x^11*d^10*c*b^2*a + 1001/2*x^6*d^5*c^10*b^3 + 33/2*x^10*d^9*c^2*
b^2*a + 273*x^5*d^4*c^11*b^3 + 55*x^9*d^8*c^3*b^2*a + 455/4*x^4*d^3*c^12*b^3 + 4
95/4*x^8*d^7*c^4*b^2*a + 35*x^3*d^2*c^13*b^3 + 198*x^7*d^6*c^5*b^2*a + 15/2*x^2*
d*c^14*b^3 + 231*x^6*d^5*c^6*b^2*a + x*c^15*b^3 + 198*x^5*d^4*c^7*b^2*a + 495/4*
x^4*d^3*c^8*b^2*a + 3/8*x^8*d^7*b*a^2 + 55*x^3*d^2*c^9*b^2*a + 3*x^7*d^6*c*b*a^2
+ 33/2*x^2*d*c^10*b^2*a + 21/2*x^6*d^5*c^2*b*a^2 + 3*x*c^11*b^2*a + 21*x^5*d^4*
c^3*b*a^2 + 105/4*x^4*d^3*c^4*b*a^2 + 21*x^3*d^2*c^5*b*a^2 + 21/2*x^2*d*c^6*b*a^
2 + 3*x*c^7*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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Sympy [A] time = 0.471586, size = 541, normalized size = 23.52 \[ 35 b^{3} c^{3} d^{12} x^{13} + \frac{15 b^{3} c^{2} d^{13} x^{14}}{2} + b^{3} c d^{14} x^{15} + \frac{b^{3} d^{15} x^{16}}{16} + x^{12} \left (\frac{a b^{2} d^{11}}{4} + \frac{455 b^{3} c^{4} d^{11}}{4}\right ) + x^{11} \left (3 a b^{2} c d^{10} + 273 b^{3} c^{5} d^{10}\right ) + x^{10} \left (\frac{33 a b^{2} c^{2} d^{9}}{2} + \frac{1001 b^{3} c^{6} d^{9}}{2}\right ) + x^{9} \left (55 a b^{2} c^{3} d^{8} + 715 b^{3} c^{7} d^{8}\right ) + x^{8} \left (\frac{3 a^{2} b d^{7}}{8} + \frac{495 a b^{2} c^{4} d^{7}}{4} + \frac{6435 b^{3} c^{8} d^{7}}{8}\right ) + x^{7} \left (3 a^{2} b c d^{6} + 198 a b^{2} c^{5} d^{6} + 715 b^{3} c^{9} d^{6}\right ) + x^{6} \left (\frac{21 a^{2} b c^{2} d^{5}}{2} + 231 a b^{2} c^{6} d^{5} + \frac{1001 b^{3} c^{10} d^{5}}{2}\right ) + x^{5} \left (21 a^{2} b c^{3} d^{4} + 198 a b^{2} c^{7} d^{4} + 273 b^{3} c^{11} d^{4}\right ) + x^{4} \left (\frac{a^{3} d^{3}}{4} + \frac{105 a^{2} b c^{4} d^{3}}{4} + \frac{495 a b^{2} c^{8} d^{3}}{4} + \frac{455 b^{3} c^{12} d^{3}}{4}\right ) + x^{3} \left (a^{3} c d^{2} + 21 a^{2} b c^{5} d^{2} + 55 a b^{2} c^{9} d^{2} + 35 b^{3} c^{13} d^{2}\right ) + x^{2} \left (\frac{3 a^{3} c^{2} d}{2} + \frac{21 a^{2} b c^{6} d}{2} + \frac{33 a b^{2} c^{10} d}{2} + \frac{15 b^{3} c^{14} d}{2}\right ) + x \left (a^{3} c^{3} + 3 a^{2} b c^{7} + 3 a b^{2} c^{11} + b^{3} c^{15}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x+c)**3*(a+b*(d*x+c)**4)**3,x)
[Out]
35*b**3*c**3*d**12*x**13 + 15*b**3*c**2*d**13*x**14/2 + b**3*c*d**14*x**15 + b**
3*d**15*x**16/16 + x**12*(a*b**2*d**11/4 + 455*b**3*c**4*d**11/4) + x**11*(3*a*b
**2*c*d**10 + 273*b**3*c**5*d**10) + x**10*(33*a*b**2*c**2*d**9/2 + 1001*b**3*c*
*6*d**9/2) + x**9*(55*a*b**2*c**3*d**8 + 715*b**3*c**7*d**8) + x**8*(3*a**2*b*d*
*7/8 + 495*a*b**2*c**4*d**7/4 + 6435*b**3*c**8*d**7/8) + x**7*(3*a**2*b*c*d**6 +
198*a*b**2*c**5*d**6 + 715*b**3*c**9*d**6) + x**6*(21*a**2*b*c**2*d**5/2 + 231*
a*b**2*c**6*d**5 + 1001*b**3*c**10*d**5/2) + x**5*(21*a**2*b*c**3*d**4 + 198*a*b
**2*c**7*d**4 + 273*b**3*c**11*d**4) + x**4*(a**3*d**3/4 + 105*a**2*b*c**4*d**3/
4 + 495*a*b**2*c**8*d**3/4 + 455*b**3*c**12*d**3/4) + x**3*(a**3*c*d**2 + 21*a**
2*b*c**5*d**2 + 55*a*b**2*c**9*d**2 + 35*b**3*c**13*d**2) + x**2*(3*a**3*c**2*d/
2 + 21*a**2*b*c**6*d/2 + 33*a*b**2*c**10*d/2 + 15*b**3*c**14*d/2) + x*(a**3*c**3
+ 3*a**2*b*c**7 + 3*a*b**2*c**11 + b**3*c**15)
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.216352, size = 28, normalized size = 1.22 \[ \frac{{\left ({\left (d x + c\right )}^{4} b + a\right )}^{4}}{16 \, b d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(((d*x + c)^4*b + a)^3*(d*x + c)^3,x, algorithm="giac")
[Out]
1/16*((d*x + c)^4*b + a)^4/(b*d)