3.161 \(\int (a+b x)^n \left (c+d x^3\right )^3 \, dx\)
Optimal. Leaf size=337 \[ -\frac{18 a d^2 \left (b^3 c-7 a^3 d\right ) (a+b x)^{n+6}}{b^{10} (n+6)}+\frac{3 d^2 \left (b^3 c-28 a^3 d\right ) (a+b x)^{n+7}}{b^{10} (n+7)}+\frac{\left (b^3 c-a^3 d\right )^3 (a+b x)^{n+1}}{b^{10} (n+1)}-\frac{9 a d \left (b^3 c-4 a^3 d\right ) \left (b^3 c-a^3 d\right ) (a+b x)^{n+3}}{b^{10} (n+3)}+\frac{36 a^2 d^3 (a+b x)^{n+8}}{b^{10} (n+8)}+\frac{3 d \left (28 a^6 d^2-20 a^3 b^3 c d+b^6 c^2\right ) (a+b x)^{n+4}}{b^{10} (n+4)}+\frac{9 a^2 d^2 \left (5 b^3 c-14 a^3 d\right ) (a+b x)^{n+5}}{b^{10} (n+5)}+\frac{9 a^2 d \left (b^3 c-a^3 d\right )^2 (a+b x)^{n+2}}{b^{10} (n+2)}-\frac{9 a d^3 (a+b x)^{n+9}}{b^{10} (n+9)}+\frac{d^3 (a+b x)^{n+10}}{b^{10} (n+10)} \]
[Out]
((b^3*c - a^3*d)^3*(a + b*x)^(1 + n))/(b^10*(1 + n)) + (9*a^2*d*(b^3*c - a^3*d)^
2*(a + b*x)^(2 + n))/(b^10*(2 + n)) - (9*a*d*(b^3*c - 4*a^3*d)*(b^3*c - a^3*d)*(
a + b*x)^(3 + n))/(b^10*(3 + n)) + (3*d*(b^6*c^2 - 20*a^3*b^3*c*d + 28*a^6*d^2)*
(a + b*x)^(4 + n))/(b^10*(4 + n)) + (9*a^2*d^2*(5*b^3*c - 14*a^3*d)*(a + b*x)^(5
+ n))/(b^10*(5 + n)) - (18*a*d^2*(b^3*c - 7*a^3*d)*(a + b*x)^(6 + n))/(b^10*(6
+ n)) + (3*d^2*(b^3*c - 28*a^3*d)*(a + b*x)^(7 + n))/(b^10*(7 + n)) + (36*a^2*d^
3*(a + b*x)^(8 + n))/(b^10*(8 + n)) - (9*a*d^3*(a + b*x)^(9 + n))/(b^10*(9 + n))
+ (d^3*(a + b*x)^(10 + n))/(b^10*(10 + n))
_______________________________________________________________________________________
Rubi [A] time = 0.466382, antiderivative size = 337, normalized size of antiderivative = 1.,
number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059
\[ -\frac{18 a d^2 \left (b^3 c-7 a^3 d\right ) (a+b x)^{n+6}}{b^{10} (n+6)}+\frac{3 d^2 \left (b^3 c-28 a^3 d\right ) (a+b x)^{n+7}}{b^{10} (n+7)}+\frac{\left (b^3 c-a^3 d\right )^3 (a+b x)^{n+1}}{b^{10} (n+1)}-\frac{9 a d \left (b^3 c-4 a^3 d\right ) \left (b^3 c-a^3 d\right ) (a+b x)^{n+3}}{b^{10} (n+3)}+\frac{36 a^2 d^3 (a+b x)^{n+8}}{b^{10} (n+8)}+\frac{3 d \left (28 a^6 d^2-20 a^3 b^3 c d+b^6 c^2\right ) (a+b x)^{n+4}}{b^{10} (n+4)}+\frac{9 a^2 d^2 \left (5 b^3 c-14 a^3 d\right ) (a+b x)^{n+5}}{b^{10} (n+5)}+\frac{9 a^2 d \left (b^3 c-a^3 d\right )^2 (a+b x)^{n+2}}{b^{10} (n+2)}-\frac{9 a d^3 (a+b x)^{n+9}}{b^{10} (n+9)}+\frac{d^3 (a+b x)^{n+10}}{b^{10} (n+10)} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x)^n*(c + d*x^3)^3,x]
[Out]
((b^3*c - a^3*d)^3*(a + b*x)^(1 + n))/(b^10*(1 + n)) + (9*a^2*d*(b^3*c - a^3*d)^
2*(a + b*x)^(2 + n))/(b^10*(2 + n)) - (9*a*d*(b^3*c - 4*a^3*d)*(b^3*c - a^3*d)*(
a + b*x)^(3 + n))/(b^10*(3 + n)) + (3*d*(b^6*c^2 - 20*a^3*b^3*c*d + 28*a^6*d^2)*
(a + b*x)^(4 + n))/(b^10*(4 + n)) + (9*a^2*d^2*(5*b^3*c - 14*a^3*d)*(a + b*x)^(5
+ n))/(b^10*(5 + n)) - (18*a*d^2*(b^3*c - 7*a^3*d)*(a + b*x)^(6 + n))/(b^10*(6
+ n)) + (3*d^2*(b^3*c - 28*a^3*d)*(a + b*x)^(7 + n))/(b^10*(7 + n)) + (36*a^2*d^
3*(a + b*x)^(8 + n))/(b^10*(8 + n)) - (9*a*d^3*(a + b*x)^(9 + n))/(b^10*(9 + n))
+ (d^3*(a + b*x)^(10 + n))/(b^10*(10 + n))
_______________________________________________________________________________________
Rubi in Sympy [A] time = 100.789, size = 316, normalized size = 0.94 \[ \frac{36 a^{2} d^{3} \left (a + b x\right )^{n + 8}}{b^{10} \left (n + 8\right )} - \frac{9 a^{2} d^{2} \left (a + b x\right )^{n + 5} \left (14 a^{3} d - 5 b^{3} c\right )}{b^{10} \left (n + 5\right )} + \frac{9 a^{2} d \left (a + b x\right )^{n + 2} \left (a^{3} d - b^{3} c\right )^{2}}{b^{10} \left (n + 2\right )} - \frac{9 a d^{3} \left (a + b x\right )^{n + 9}}{b^{10} \left (n + 9\right )} + \frac{18 a d^{2} \left (a + b x\right )^{n + 6} \left (7 a^{3} d - b^{3} c\right )}{b^{10} \left (n + 6\right )} - \frac{9 a d \left (a + b x\right )^{n + 3} \left (a^{3} d - b^{3} c\right ) \left (4 a^{3} d - b^{3} c\right )}{b^{10} \left (n + 3\right )} + \frac{d^{3} \left (a + b x\right )^{n + 10}}{b^{10} \left (n + 10\right )} - \frac{3 d^{2} \left (a + b x\right )^{n + 7} \left (28 a^{3} d - b^{3} c\right )}{b^{10} \left (n + 7\right )} + \frac{3 d \left (a + b x\right )^{n + 4} \left (28 a^{6} d^{2} - 20 a^{3} b^{3} c d + b^{6} c^{2}\right )}{b^{10} \left (n + 4\right )} - \frac{\left (a + b x\right )^{n + 1} \left (a^{3} d - b^{3} c\right )^{3}}{b^{10} \left (n + 1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)**n*(d*x**3+c)**3,x)
[Out]
36*a**2*d**3*(a + b*x)**(n + 8)/(b**10*(n + 8)) - 9*a**2*d**2*(a + b*x)**(n + 5)
*(14*a**3*d - 5*b**3*c)/(b**10*(n + 5)) + 9*a**2*d*(a + b*x)**(n + 2)*(a**3*d -
b**3*c)**2/(b**10*(n + 2)) - 9*a*d**3*(a + b*x)**(n + 9)/(b**10*(n + 9)) + 18*a*
d**2*(a + b*x)**(n + 6)*(7*a**3*d - b**3*c)/(b**10*(n + 6)) - 9*a*d*(a + b*x)**(
n + 3)*(a**3*d - b**3*c)*(4*a**3*d - b**3*c)/(b**10*(n + 3)) + d**3*(a + b*x)**(
n + 10)/(b**10*(n + 10)) - 3*d**2*(a + b*x)**(n + 7)*(28*a**3*d - b**3*c)/(b**10
*(n + 7)) + 3*d*(a + b*x)**(n + 4)*(28*a**6*d**2 - 20*a**3*b**3*c*d + b**6*c**2)
/(b**10*(n + 4)) - (a + b*x)**(n + 1)*(a**3*d - b**3*c)**3/(b**10*(n + 1))
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Mathematica [B] time = 1.09458, size = 706, normalized size = 2.09 \[ \frac{(a+b x)^{n+1} \left (-362880 a^9 d^3+362880 a^8 b d^3 (n+1) x-181440 a^7 b^2 d^3 \left (n^2+3 n+2\right ) x^2+2160 a^6 b^3 d^2 \left (c \left (n^3+27 n^2+242 n+720\right )+28 d \left (n^3+6 n^2+11 n+6\right ) x^3\right )-2160 a^5 b^4 d^2 (n+1) x \left (c \left (n^3+27 n^2+242 n+720\right )+7 d \left (n^3+9 n^2+26 n+24\right ) x^3\right )+216 a^4 b^5 d^2 \left (n^2+3 n+2\right ) x^2 \left (5 c \left (n^3+27 n^2+242 n+720\right )+14 d \left (n^3+12 n^2+47 n+60\right ) x^3\right )-18 a^3 b^6 d \left (c^2 \left (n^6+45 n^5+835 n^4+8175 n^3+44524 n^2+127860 n+151200\right )+20 c d \left (n^6+33 n^5+415 n^4+2475 n^3+7144 n^2+9372 n+4320\right ) x^3+28 d^2 \left (n^6+21 n^5+175 n^4+735 n^3+1624 n^2+1764 n+720\right ) x^6\right )+18 a^2 b^7 d (n+1) x \left (c^2 \left (n^6+45 n^5+835 n^4+8175 n^3+44524 n^2+127860 n+151200\right )+5 c d \left (n^6+36 n^5+511 n^4+3624 n^3+13420 n^2+24528 n+17280\right ) x^3+4 d^2 \left (n^6+27 n^5+295 n^4+1665 n^3+5104 n^2+8028 n+5040\right ) x^6\right )-9 a b^8 d \left (n^4+16 n^3+81 n^2+146 n+80\right ) x^2 \left (c^2 \left (n^4+32 n^3+379 n^2+1968 n+3780\right )+2 c d \left (n^4+26 n^3+235 n^2+858 n+1080\right ) x^3+d^2 \left (n^4+20 n^3+145 n^2+450 n+504\right ) x^6\right )+b^9 \left (n^6+33 n^5+435 n^4+2915 n^3+10404 n^2+18612 n+12960\right ) \left (c^3 \left (n^3+21 n^2+138 n+280\right )+3 c^2 d \left (n^3+18 n^2+87 n+70\right ) x^3+3 c d^2 \left (n^3+15 n^2+54 n+40\right ) x^6+d^3 \left (n^3+12 n^2+39 n+28\right ) x^9\right )\right )}{b^{10} (n+1) (n+2) (n+3) (n+4) (n+5) (n+6) (n+7) (n+8) (n+9) (n+10)} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x)^n*(c + d*x^3)^3,x]
[Out]
((a + b*x)^(1 + n)*(-362880*a^9*d^3 + 362880*a^8*b*d^3*(1 + n)*x - 181440*a^7*b^
2*d^3*(2 + 3*n + n^2)*x^2 + 2160*a^6*b^3*d^2*(c*(720 + 242*n + 27*n^2 + n^3) + 2
8*d*(6 + 11*n + 6*n^2 + n^3)*x^3) - 2160*a^5*b^4*d^2*(1 + n)*x*(c*(720 + 242*n +
27*n^2 + n^3) + 7*d*(24 + 26*n + 9*n^2 + n^3)*x^3) + 216*a^4*b^5*d^2*(2 + 3*n +
n^2)*x^2*(5*c*(720 + 242*n + 27*n^2 + n^3) + 14*d*(60 + 47*n + 12*n^2 + n^3)*x^
3) - 9*a*b^8*d*(80 + 146*n + 81*n^2 + 16*n^3 + n^4)*x^2*(c^2*(3780 + 1968*n + 37
9*n^2 + 32*n^3 + n^4) + 2*c*d*(1080 + 858*n + 235*n^2 + 26*n^3 + n^4)*x^3 + d^2*
(504 + 450*n + 145*n^2 + 20*n^3 + n^4)*x^6) - 18*a^3*b^6*d*(c^2*(151200 + 127860
*n + 44524*n^2 + 8175*n^3 + 835*n^4 + 45*n^5 + n^6) + 20*c*d*(4320 + 9372*n + 71
44*n^2 + 2475*n^3 + 415*n^4 + 33*n^5 + n^6)*x^3 + 28*d^2*(720 + 1764*n + 1624*n^
2 + 735*n^3 + 175*n^4 + 21*n^5 + n^6)*x^6) + 18*a^2*b^7*d*(1 + n)*x*(c^2*(151200
+ 127860*n + 44524*n^2 + 8175*n^3 + 835*n^4 + 45*n^5 + n^6) + 5*c*d*(17280 + 24
528*n + 13420*n^2 + 3624*n^3 + 511*n^4 + 36*n^5 + n^6)*x^3 + 4*d^2*(5040 + 8028*
n + 5104*n^2 + 1665*n^3 + 295*n^4 + 27*n^5 + n^6)*x^6) + b^9*(12960 + 18612*n +
10404*n^2 + 2915*n^3 + 435*n^4 + 33*n^5 + n^6)*(c^3*(280 + 138*n + 21*n^2 + n^3)
+ 3*c^2*d*(70 + 87*n + 18*n^2 + n^3)*x^3 + 3*c*d^2*(40 + 54*n + 15*n^2 + n^3)*x
^6 + d^3*(28 + 39*n + 12*n^2 + n^3)*x^9)))/(b^10*(1 + n)*(2 + n)*(3 + n)*(4 + n)
*(5 + n)*(6 + n)*(7 + n)*(8 + n)*(9 + n)*(10 + n))
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Maple [B] time = 0.028, size = 2280, normalized size = 6.8 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)^n*(d*x^3+c)^3,x)
[Out]
-(b*x+a)^(1+n)*(-b^9*d^3*n^9*x^9-45*b^9*d^3*n^8*x^9+9*a*b^8*d^3*n^8*x^8-870*b^9*
d^3*n^7*x^9+324*a*b^8*d^3*n^7*x^8-3*b^9*c*d^2*n^9*x^6-9450*b^9*d^3*n^6*x^9-72*a^
2*b^7*d^3*n^7*x^7+4914*a*b^8*d^3*n^6*x^8-144*b^9*c*d^2*n^8*x^6-63273*b^9*d^3*n^5
*x^9-2016*a^2*b^7*d^3*n^6*x^7+18*a*b^8*c*d^2*n^8*x^5+40824*a*b^8*d^3*n^5*x^8-295
2*b^9*c*d^2*n^7*x^6-269325*b^9*d^3*n^4*x^9+504*a^3*b^6*d^3*n^6*x^6-23184*a^2*b^7
*d^3*n^5*x^7+756*a*b^8*c*d^2*n^7*x^5+202041*a*b^8*d^3*n^4*x^8-3*b^9*c^2*d*n^9*x^
3-33786*b^9*c*d^2*n^6*x^6-723680*b^9*d^3*n^3*x^9+10584*a^3*b^6*d^3*n^5*x^6-90*a^
2*b^7*c*d^2*n^7*x^4-141120*a^2*b^7*d^3*n^4*x^7+13176*a*b^8*c*d^2*n^6*x^5+605556*
a*b^8*d^3*n^3*x^8-153*b^9*c^2*d*n^8*x^3-236817*b^9*c*d^2*n^5*x^6-1172700*b^9*d^3
*n^2*x^9-3024*a^4*b^5*d^3*n^5*x^5+88200*a^3*b^6*d^3*n^4*x^6-3330*a^2*b^7*c*d^2*n
^6*x^4-487368*a^2*b^7*d^3*n^3*x^7+9*a*b^8*c^2*d*n^8*x^2+123660*a*b^8*c*d^2*n^5*x
^5+1063116*a*b^8*d^3*n^2*x^8-3348*b^9*c^2*d*n^7*x^3-1048446*b^9*c*d^2*n^4*x^6-10
26576*b^9*d^3*n*x^9-45360*a^4*b^5*d^3*n^4*x^5+360*a^3*b^6*c*d^2*n^6*x^3+370440*a
^3*b^6*d^3*n^3*x^6-49230*a^2*b^7*c*d^2*n^5*x^4-945504*a^2*b^7*d^3*n^2*x^7+432*a*
b^8*c^2*d*n^7*x^2+678942*a*b^8*c*d^2*n^4*x^5+986256*a*b^8*d^3*n*x^8-b^9*c^3*n^9-
41058*b^9*c^2*d*n^6*x^3-2911668*b^9*c*d^2*n^3*x^6-362880*b^9*d^3*x^9+15120*a^5*b
^4*d^3*n^4*x^4-257040*a^4*b^5*d^3*n^3*x^5+11880*a^3*b^6*c*d^2*n^5*x^3+818496*a^3
*b^6*d^3*n^2*x^6-18*a^2*b^7*c^2*d*n^7*x-372150*a^2*b^7*c*d^2*n^4*x^4-940896*a^2*
b^7*d^3*n*x^7+8748*a*b^8*c^2*d*n^6*x^2+2217024*a*b^8*c*d^2*n^3*x^5+362880*a*b^8*
d^3*x^8-54*b^9*c^3*n^8-309087*b^9*c^2*d*n^5*x^3-4846824*b^9*c*d^2*n^2*x^6+151200
*a^5*b^4*d^3*n^3*x^4-1080*a^4*b^5*c*d^2*n^5*x^2-680400*a^4*b^5*d^3*n^2*x^5+14940
0*a^3*b^6*c*d^2*n^4*x^3+889056*a^3*b^6*d^3*n*x^6-828*a^2*b^7*c^2*d*n^6*x-1533960
*a^2*b^7*c*d^2*n^3*x^4-362880*a^2*b^7*d^3*x^7+96930*a*b^8*c^2*d*n^5*x^2+4167864*
a*b^8*c*d^2*n^2*x^5-1266*b^9*c^3*n^7-1469817*b^9*c^2*d*n^4*x^3-4332960*b^9*c*d^2
*n*x^6-60480*a^6*b^3*d^3*n^3*x^3+529200*a^5*b^4*d^3*n^2*x^4-32400*a^4*b^5*c*d^2*
n^4*x^2-828576*a^4*b^5*d^3*n*x^5+18*a^3*b^6*c^2*d*n^6+891000*a^3*b^6*c*d^2*n^3*x
^3+362880*a^3*b^6*d^3*x^6-15840*a^2*b^7*c^2*d*n^5*x-3415320*a^2*b^7*c*d^2*n^2*x^
4+636471*a*b^8*c^2*d*n^4*x^2+4073760*a*b^8*c*d^2*n*x^5-16884*b^9*c^3*n^6-4371522
*b^9*c^2*d*n^3*x^3-1555200*b^9*c*d^2*x^6-362880*a^6*b^3*d^3*n^2*x^3+2160*a^5*b^4
*c*d^2*n^4*x+756000*a^5*b^4*d^3*n*x^4-351000*a^4*b^5*c*d^2*n^3*x^2-362880*a^4*b^
5*d^3*x^5+810*a^3*b^6*c^2*d*n^5+2571840*a^3*b^6*c*d^2*n^2*x^3-162180*a^2*b^7*c^2
*d*n^4*x-3762720*a^2*b^7*c*d^2*n*x^4+2500038*a*b^8*c^2*d*n^3*x^2+1555200*a*b^8*c
*d^2*x^5-140889*b^9*c^3*n^5-7742412*b^9*c^2*d*n^2*x^3+181440*a^7*b^2*d^3*n^2*x^2
-665280*a^6*b^3*d^3*n*x^3+60480*a^5*b^4*c*d^2*n^3*x+362880*a^5*b^4*d^3*x^4-16200
00*a^4*b^5*c*d^2*n^2*x^2+15030*a^3*b^6*c^2*d*n^4+3373920*a^3*b^6*c*d^2*n*x^3-948
582*a^2*b^7*c^2*d*n^3*x-1555200*a^2*b^7*c*d^2*x^4+5614452*a*b^8*c^2*d*n^2*x^2-76
1166*b^9*c^3*n^4-7291080*b^9*c^2*d*n*x^3+544320*a^7*b^2*d^3*n*x^2-2160*a^6*b^3*c
*d^2*n^3-362880*a^6*b^3*d^3*x^3+581040*a^5*b^4*c*d^2*n^2*x-2855520*a^4*b^5*c*d^2
*n*x^2+147150*a^3*b^6*c^2*d*n^3+1555200*a^3*b^6*c*d^2*x^3-3102912*a^2*b^7*c^2*d*
n^2*x+6383880*a*b^8*c^2*d*n*x^2-2655764*b^9*c^3*n^3-2721600*b^9*c^2*d*x^3-362880
*a^8*b*d^3*n*x+362880*a^7*b^2*d^3*x^2-58320*a^6*b^3*c*d^2*n^2+2077920*a^5*b^4*c*
d^2*n*x-1555200*a^4*b^5*c*d^2*x^2+801432*a^3*b^6*c^2*d*n^2-5023080*a^2*b^7*c^2*d
*n*x+2721600*a*b^8*c^2*d*x^2-5753736*b^9*c^3*n^2-362880*a^8*b*d^3*x-522720*a^6*b
^3*c*d^2*n+1555200*a^5*b^4*c*d^2*x+2301480*a^3*b^6*c^2*d*n-2721600*a^2*b^7*c^2*d
*x-6999840*b^9*c^3*n+362880*a^9*d^3-1555200*a^6*b^3*c*d^2+2721600*a^3*b^6*c^2*d-
3628800*b^9*c^3)/b^10/(n^10+55*n^9+1320*n^8+18150*n^7+157773*n^6+902055*n^5+3416
930*n^4+8409500*n^3+12753576*n^2+10628640*n+3628800)
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^3 + c)^3*(b*x + a)^n,x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [A] time = 0.319317, size = 3123, normalized size = 9.27 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^3 + c)^3*(b*x + a)^n,x, algorithm="fricas")
[Out]
(a*b^9*c^3*n^9 + 54*a*b^9*c^3*n^8 + 1266*a*b^9*c^3*n^7 + 3628800*a*b^9*c^3 - 272
1600*a^4*b^6*c^2*d + 1555200*a^7*b^3*c*d^2 - 362880*a^10*d^3 + (b^10*d^3*n^9 + 4
5*b^10*d^3*n^8 + 870*b^10*d^3*n^7 + 9450*b^10*d^3*n^6 + 63273*b^10*d^3*n^5 + 269
325*b^10*d^3*n^4 + 723680*b^10*d^3*n^3 + 1172700*b^10*d^3*n^2 + 1026576*b^10*d^3
*n + 362880*b^10*d^3)*x^10 + (a*b^9*d^3*n^9 + 36*a*b^9*d^3*n^8 + 546*a*b^9*d^3*n
^7 + 4536*a*b^9*d^3*n^6 + 22449*a*b^9*d^3*n^5 + 67284*a*b^9*d^3*n^4 + 118124*a*b
^9*d^3*n^3 + 109584*a*b^9*d^3*n^2 + 40320*a*b^9*d^3*n)*x^9 - 9*(a^2*b^8*d^3*n^8
+ 28*a^2*b^8*d^3*n^7 + 322*a^2*b^8*d^3*n^6 + 1960*a^2*b^8*d^3*n^5 + 6769*a^2*b^8
*d^3*n^4 + 13132*a^2*b^8*d^3*n^3 + 13068*a^2*b^8*d^3*n^2 + 5040*a^2*b^8*d^3*n)*x
^8 + 3*(b^10*c*d^2*n^9 + 48*b^10*c*d^2*n^8 + 518400*b^10*c*d^2 + 24*(41*b^10*c*d
^2 + a^3*b^7*d^3)*n^7 + 6*(1877*b^10*c*d^2 + 84*a^3*b^7*d^3)*n^6 + 21*(3759*b^10
*c*d^2 + 200*a^3*b^7*d^3)*n^5 + 42*(8321*b^10*c*d^2 + 420*a^3*b^7*d^3)*n^4 + 4*(
242639*b^10*c*d^2 + 9744*a^3*b^7*d^3)*n^3 + 72*(22439*b^10*c*d^2 + 588*a^3*b^7*d
^3)*n^2 + 1440*(1003*b^10*c*d^2 + 12*a^3*b^7*d^3)*n)*x^7 + 18*(938*a*b^9*c^3 - a
^4*b^6*c^2*d)*n^6 + 3*(a*b^9*c*d^2*n^9 + 42*a*b^9*c*d^2*n^8 + 732*a*b^9*c*d^2*n^
7 + 6*(1145*a*b^9*c*d^2 - 28*a^4*b^6*d^3)*n^6 + 9*(4191*a*b^9*c*d^2 - 280*a^4*b^
6*d^3)*n^5 + 24*(5132*a*b^9*c*d^2 - 595*a^4*b^6*d^3)*n^4 + 4*(57887*a*b^9*c*d^2
- 9450*a^4*b^6*d^3)*n^3 + 48*(4715*a*b^9*c*d^2 - 959*a^4*b^6*d^3)*n^2 + 2880*(30
*a*b^9*c*d^2 - 7*a^4*b^6*d^3)*n)*x^6 + 3*(46963*a*b^9*c^3 - 270*a^4*b^6*c^2*d)*n
^5 - 18*(a^2*b^8*c*d^2*n^8 + 37*a^2*b^8*c*d^2*n^7 + 547*a^2*b^8*c*d^2*n^6 + (413
5*a^2*b^8*c*d^2 - 168*a^5*b^5*d^3)*n^5 + 4*(4261*a^2*b^8*c*d^2 - 420*a^5*b^5*d^3
)*n^4 + 4*(9487*a^2*b^8*c*d^2 - 1470*a^5*b^5*d^3)*n^3 + 48*(871*a^2*b^8*c*d^2 -
175*a^5*b^5*d^3)*n^2 + 576*(30*a^2*b^8*c*d^2 - 7*a^5*b^5*d^3)*n)*x^5 + 18*(42287
*a*b^9*c^3 - 835*a^4*b^6*c^2*d)*n^4 + 3*(b^10*c^2*d*n^9 + 51*b^10*c^2*d*n^8 + 90
7200*b^10*c^2*d + 6*(186*b^10*c^2*d + 5*a^3*b^7*c*d^2)*n^7 + 6*(2281*b^10*c^2*d
+ 165*a^3*b^7*c*d^2)*n^6 + 3*(34343*b^10*c^2*d + 4150*a^3*b^7*c*d^2)*n^5 + 3*(16
3313*b^10*c^2*d + 24750*a^3*b^7*c*d^2 - 1680*a^6*b^4*d^3)*n^4 + 2*(728587*b^10*c
^2*d + 107160*a^3*b^7*c*d^2 - 15120*a^6*b^4*d^3)*n^3 + 36*(71689*b^10*c^2*d + 78
10*a^3*b^7*c*d^2 - 1540*a^6*b^4*d^3)*n^2 + 360*(6751*b^10*c^2*d + 360*a^3*b^7*c*
d^2 - 84*a^6*b^4*d^3)*n)*x^4 + 2*(1327882*a*b^9*c^3 - 73575*a^4*b^6*c^2*d + 1080
*a^7*b^3*c*d^2)*n^3 + 3*(a*b^9*c^2*d*n^9 + 48*a*b^9*c^2*d*n^8 + 972*a*b^9*c^2*d*
n^7 + 30*(359*a*b^9*c^2*d - 4*a^4*b^6*c*d^2)*n^6 + 3*(23573*a*b^9*c^2*d - 1200*a
^4*b^6*c*d^2)*n^5 + 6*(46297*a*b^9*c^2*d - 6500*a^4*b^6*c*d^2)*n^4 + 4*(155957*a
*b^9*c^2*d - 45000*a^4*b^6*c*d^2 + 5040*a^7*b^3*d^3)*n^3 + 120*(5911*a*b^9*c^2*d
- 2644*a^4*b^6*c*d^2 + 504*a^7*b^3*d^3)*n^2 + 2880*(105*a*b^9*c^2*d - 60*a^4*b^
6*c*d^2 + 14*a^7*b^3*d^3)*n)*x^3 + 72*(79913*a*b^9*c^3 - 11131*a^4*b^6*c^2*d + 8
10*a^7*b^3*c*d^2)*n^2 - 9*(a^2*b^8*c^2*d*n^8 + 46*a^2*b^8*c^2*d*n^7 + 880*a^2*b^
8*c^2*d*n^6 + 10*(901*a^2*b^8*c^2*d - 12*a^5*b^5*c*d^2)*n^5 + (52699*a^2*b^8*c^2
*d - 3360*a^5*b^5*c*d^2)*n^4 + 8*(21548*a^2*b^8*c^2*d - 4035*a^5*b^5*c*d^2)*n^3
+ 60*(4651*a^2*b^8*c^2*d - 1924*a^5*b^5*c*d^2 + 336*a^8*b^2*d^3)*n^2 + 1440*(105
*a^2*b^8*c^2*d - 60*a^5*b^5*c*d^2 + 14*a^8*b^2*d^3)*n)*x^2 + 360*(19444*a*b^9*c^
3 - 6393*a^4*b^6*c^2*d + 1452*a^7*b^3*c*d^2)*n + (b^10*c^3*n^9 + 54*b^10*c^3*n^8
+ 3628800*b^10*c^3 + 6*(211*b^10*c^3 + 3*a^3*b^7*c^2*d)*n^7 + 18*(938*b^10*c^3
+ 45*a^3*b^7*c^2*d)*n^6 + 3*(46963*b^10*c^3 + 5010*a^3*b^7*c^2*d)*n^5 + 18*(4228
7*b^10*c^3 + 8175*a^3*b^7*c^2*d - 120*a^6*b^4*c*d^2)*n^4 + 4*(663941*b^10*c^3 +
200358*a^3*b^7*c^2*d - 14580*a^6*b^4*c*d^2)*n^3 + 72*(79913*b^10*c^3 + 31965*a^3
*b^7*c^2*d - 7260*a^6*b^4*c*d^2)*n^2 + 1440*(4861*b^10*c^3 + 1890*a^3*b^7*c^2*d
- 1080*a^6*b^4*c*d^2 + 252*a^9*b*d^3)*n)*x)*(b*x + a)^n/(b^10*n^10 + 55*b^10*n^9
+ 1320*b^10*n^8 + 18150*b^10*n^7 + 157773*b^10*n^6 + 902055*b^10*n^5 + 3416930*
b^10*n^4 + 8409500*b^10*n^3 + 12753576*b^10*n^2 + 10628640*b^10*n + 3628800*b^10
)
_______________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x+a)**n*(d*x**3+c)**3,x)
[Out]
Timed out
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.281307, size = 1, normalized size = 0. \[ \mathit{Done} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^3 + c)^3*(b*x + a)^n,x, algorithm="giac")
[Out]
Done