3.583 \(\int x^m (a+b x^3)^8 \, dx\)
Optimal. Leaf size=151 \[ \frac{28 a^6 b^2 x^{m+7}}{m+7}+\frac{56 a^5 b^3 x^{m+10}}{m+10}+\frac{70 a^4 b^4 x^{m+13}}{m+13}+\frac{56 a^3 b^5 x^{m+16}}{m+16}+\frac{28 a^2 b^6 x^{m+19}}{m+19}+\frac{8 a^7 b x^{m+4}}{m+4}+\frac{a^8 x^{m+1}}{m+1}+\frac{8 a b^7 x^{m+22}}{m+22}+\frac{b^8 x^{m+25}}{m+25} \]
[Out]
(a^8*x^(1 + m))/(1 + m) + (8*a^7*b*x^(4 + m))/(4 + m) + (28*a^6*b^2*x^(7 + m))/(7 + m) + (56*a^5*b^3*x^(10 + m
))/(10 + m) + (70*a^4*b^4*x^(13 + m))/(13 + m) + (56*a^3*b^5*x^(16 + m))/(16 + m) + (28*a^2*b^6*x^(19 + m))/(1
9 + m) + (8*a*b^7*x^(22 + m))/(22 + m) + (b^8*x^(25 + m))/(25 + m)
________________________________________________________________________________________
Rubi [A] time = 0.0756078, antiderivative size = 151, normalized size of antiderivative = 1.,
number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used =
{270} \[ \frac{28 a^6 b^2 x^{m+7}}{m+7}+\frac{56 a^5 b^3 x^{m+10}}{m+10}+\frac{70 a^4 b^4 x^{m+13}}{m+13}+\frac{56 a^3 b^5 x^{m+16}}{m+16}+\frac{28 a^2 b^6 x^{m+19}}{m+19}+\frac{8 a^7 b x^{m+4}}{m+4}+\frac{a^8 x^{m+1}}{m+1}+\frac{8 a b^7 x^{m+22}}{m+22}+\frac{b^8 x^{m+25}}{m+25} \]
Antiderivative was successfully verified.
[In]
Int[x^m*(a + b*x^3)^8,x]
[Out]
(a^8*x^(1 + m))/(1 + m) + (8*a^7*b*x^(4 + m))/(4 + m) + (28*a^6*b^2*x^(7 + m))/(7 + m) + (56*a^5*b^3*x^(10 + m
))/(10 + m) + (70*a^4*b^4*x^(13 + m))/(13 + m) + (56*a^3*b^5*x^(16 + m))/(16 + m) + (28*a^2*b^6*x^(19 + m))/(1
9 + m) + (8*a*b^7*x^(22 + m))/(22 + m) + (b^8*x^(25 + m))/(25 + m)
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]
Rubi steps
\begin{align*} \int x^m \left (a+b x^3\right )^8 \, dx &=\int \left (a^8 x^m+8 a^7 b x^{3+m}+28 a^6 b^2 x^{6+m}+56 a^5 b^3 x^{9+m}+70 a^4 b^4 x^{12+m}+56 a^3 b^5 x^{15+m}+28 a^2 b^6 x^{18+m}+8 a b^7 x^{21+m}+b^8 x^{24+m}\right ) \, dx\\ &=\frac{a^8 x^{1+m}}{1+m}+\frac{8 a^7 b x^{4+m}}{4+m}+\frac{28 a^6 b^2 x^{7+m}}{7+m}+\frac{56 a^5 b^3 x^{10+m}}{10+m}+\frac{70 a^4 b^4 x^{13+m}}{13+m}+\frac{56 a^3 b^5 x^{16+m}}{16+m}+\frac{28 a^2 b^6 x^{19+m}}{19+m}+\frac{8 a b^7 x^{22+m}}{22+m}+\frac{b^8 x^{25+m}}{25+m}\\ \end{align*}
Mathematica [A] time = 0.0923884, size = 136, normalized size = 0.9 \[ x^{m+1} \left (\frac{28 a^2 b^6 x^{18}}{m+19}+\frac{56 a^3 b^5 x^{15}}{m+16}+\frac{70 a^4 b^4 x^{12}}{m+13}+\frac{56 a^5 b^3 x^9}{m+10}+\frac{28 a^6 b^2 x^6}{m+7}+\frac{8 a^7 b x^3}{m+4}+\frac{a^8}{m+1}+\frac{8 a b^7 x^{21}}{m+22}+\frac{b^8 x^{24}}{m+25}\right ) \]
Antiderivative was successfully verified.
[In]
Integrate[x^m*(a + b*x^3)^8,x]
[Out]
x^(1 + m)*(a^8/(1 + m) + (8*a^7*b*x^3)/(4 + m) + (28*a^6*b^2*x^6)/(7 + m) + (56*a^5*b^3*x^9)/(10 + m) + (70*a^
4*b^4*x^12)/(13 + m) + (56*a^3*b^5*x^15)/(16 + m) + (28*a^2*b^6*x^18)/(19 + m) + (8*a*b^7*x^21)/(22 + m) + (b^
8*x^24)/(25 + m))
________________________________________________________________________________________
Maple [B] time = 0.01, size = 1023, normalized size = 6.8 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x^m*(b*x^3+a)^8,x)
[Out]
x^(1+m)*(b^8*m^8*x^24+92*b^8*m^7*x^24+3514*b^8*m^6*x^24+8*a*b^7*m^8*x^21+72128*b^8*m^5*x^24+760*a*b^7*m^7*x^21
+859369*b^8*m^4*x^24+29792*a*b^7*m^6*x^21+5974388*b^8*m^3*x^24+28*a^2*b^6*m^8*x^18+624400*a*b^7*m^5*x^21+22963
996*b^8*m^2*x^24+2744*a^2*b^6*m^7*x^18+7563752*a*b^7*m^4*x^21+42124592*b^8*m*x^24+110656*a^2*b^6*m^6*x^18+5326
6360*a*b^7*m^3*x^21+24344320*b^8*x^24+56*a^3*b^5*m^8*x^15+2376920*a^2*b^6*m^5*x^18+206729648*a*b^7*m^2*x^21+56
56*a^3*b^5*m^7*x^15+29390452*a^2*b^6*m^4*x^18+381743680*a*b^7*m*x^21+235088*a^3*b^5*m^6*x^15+210422576*a^2*b^6
*m^3*x^18+221312000*a*b^7*x^21+70*a^4*b^4*m^8*x^12+5197360*a^3*b^5*m^5*x^15+827034544*a^2*b^6*m^2*x^18+7280*a^
4*b^4*m^7*x^12+65946104*a^3*b^5*m^4*x^15+1540629440*a^2*b^6*m*x^18+312340*a^4*b^4*m^6*x^12+482544664*a^3*b^5*m
^3*x^15+896896000*a^2*b^6*x^18+56*a^5*b^3*m^8*x^9+7138040*a^4*b^4*m^5*x^12+1929412352*a^3*b^5*m^2*x^15+5992*a^
5*b^3*m^7*x^9+93585310*a^4*b^4*m^4*x^12+3637973920*a^3*b^5*m*x^15+265664*a^5*b^3*m^6*x^9+705493880*a^4*b^4*m^3
*x^12+2130128000*a^3*b^5*x^15+28*a^6*b^2*m^8*x^6+6302128*a^5*b^3*m^5*x^9+2891238280*a^4*b^4*m^2*x^12+3080*a^6*
b^2*m^7*x^6+86082584*a^5*b^3*m^4*x^9+5549616800*a^4*b^4*m*x^12+141232*a^6*b^2*m^6*x^6+676856488*a^5*b^3*m^3*x^
9+3277120000*a^4*b^4*x^12+8*a^7*b*m^8*x^3+3490760*a^6*b^2*m^5*x^6+2881562096*a^5*b^3*m^2*x^9+904*a^7*b*m^7*x^3
+50116612*a^6*b^2*m^4*x^6+5692950592*a^5*b^3*m*x^9+42896*a^7*b*m^6*x^3+418024880*a^6*b^2*m^3*x^6+3408204800*a^
5*b^3*x^9+a^8*m^8+1108240*a^7*b*m^5*x^3+1898889328*a^6*b^2*m^2*x^6+116*a^8*m^7+16867592*a^7*b*m^4*x^3+39620604
80*a^6*b^2*m*x^6+5698*a^8*m^6+152198536*a^7*b*m^3*x^3+2434432000*a^6*b^2*x^6+154280*a^8*m^5+769795424*a^7*b*m^
2*x^3+2508289*a^8*m^4+1850614240*a^7*b*m*x^3+24950324*a^8*m^3+1217216000*a^7*b*x^3+147373372*a^8*m^2+468851120
*a^8*m+608608000*a^8)/(25+m)/(22+m)/(16+m)/(19+m)/(10+m)/(13+m)/(7+m)/(4+m)/(1+m)
________________________________________________________________________________________
Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^m*(b*x^3+a)^8,x, algorithm="maxima")
[Out]
Exception raised: ValueError
________________________________________________________________________________________
Fricas [B] time = 1.66643, size = 2317, normalized size = 15.34 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^m*(b*x^3+a)^8,x, algorithm="fricas")
[Out]
((b^8*m^8 + 92*b^8*m^7 + 3514*b^8*m^6 + 72128*b^8*m^5 + 859369*b^8*m^4 + 5974388*b^8*m^3 + 22963996*b^8*m^2 +
42124592*b^8*m + 24344320*b^8)*x^25 + 8*(a*b^7*m^8 + 95*a*b^7*m^7 + 3724*a*b^7*m^6 + 78050*a*b^7*m^5 + 945469*
a*b^7*m^4 + 6658295*a*b^7*m^3 + 25841206*a*b^7*m^2 + 47717960*a*b^7*m + 27664000*a*b^7)*x^22 + 28*(a^2*b^6*m^8
+ 98*a^2*b^6*m^7 + 3952*a^2*b^6*m^6 + 84890*a^2*b^6*m^5 + 1049659*a^2*b^6*m^4 + 7515092*a^2*b^6*m^3 + 2953694
8*a^2*b^6*m^2 + 55022480*a^2*b^6*m + 32032000*a^2*b^6)*x^19 + 56*(a^3*b^5*m^8 + 101*a^3*b^5*m^7 + 4198*a^3*b^5
*m^6 + 92810*a^3*b^5*m^5 + 1177609*a^3*b^5*m^4 + 8616869*a^3*b^5*m^3 + 34453792*a^3*b^5*m^2 + 64963820*a^3*b^5
*m + 38038000*a^3*b^5)*x^16 + 70*(a^4*b^4*m^8 + 104*a^4*b^4*m^7 + 4462*a^4*b^4*m^6 + 101972*a^4*b^4*m^5 + 1336
933*a^4*b^4*m^4 + 10078484*a^4*b^4*m^3 + 41303404*a^4*b^4*m^2 + 79280240*a^4*b^4*m + 46816000*a^4*b^4)*x^13 +
56*(a^5*b^3*m^8 + 107*a^5*b^3*m^7 + 4744*a^5*b^3*m^6 + 112538*a^5*b^3*m^5 + 1537189*a^5*b^3*m^4 + 12086723*a^5
*b^3*m^3 + 51456466*a^5*b^3*m^2 + 101659832*a^5*b^3*m + 60860800*a^5*b^3)*x^10 + 28*(a^6*b^2*m^8 + 110*a^6*b^2
*m^7 + 5044*a^6*b^2*m^6 + 124670*a^6*b^2*m^5 + 1789879*a^6*b^2*m^4 + 14929460*a^6*b^2*m^3 + 67817476*a^6*b^2*m
^2 + 141502160*a^6*b^2*m + 86944000*a^6*b^2)*x^7 + 8*(a^7*b*m^8 + 113*a^7*b*m^7 + 5362*a^7*b*m^6 + 138530*a^7*
b*m^5 + 2108449*a^7*b*m^4 + 19024817*a^7*b*m^3 + 96224428*a^7*b*m^2 + 231326780*a^7*b*m + 152152000*a^7*b)*x^4
+ (a^8*m^8 + 116*a^8*m^7 + 5698*a^8*m^6 + 154280*a^8*m^5 + 2508289*a^8*m^4 + 24950324*a^8*m^3 + 147373372*a^8
*m^2 + 468851120*a^8*m + 608608000*a^8)*x)*x^m/(m^9 + 117*m^8 + 5814*m^7 + 159978*m^6 + 2662569*m^5 + 27458613
*m^4 + 172323696*m^3 + 616224492*m^2 + 1077459120*m + 608608000)
________________________________________________________________________________________
Sympy [A] time = 61.4025, size = 5902, normalized size = 39.09 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x**m*(b*x**3+a)**8,x)
[Out]
Piecewise((-a**8/(24*x**24) - 8*a**7*b/(21*x**21) - 14*a**6*b**2/(9*x**18) - 56*a**5*b**3/(15*x**15) - 35*a**4
*b**4/(6*x**12) - 56*a**3*b**5/(9*x**9) - 14*a**2*b**6/(3*x**6) - 8*a*b**7/(3*x**3) + b**8*log(x), Eq(m, -25))
, (-a**8/(21*x**21) - 4*a**7*b/(9*x**18) - 28*a**6*b**2/(15*x**15) - 14*a**5*b**3/(3*x**12) - 70*a**4*b**4/(9*
x**9) - 28*a**3*b**5/(3*x**6) - 28*a**2*b**6/(3*x**3) + 8*a*b**7*log(x) + b**8*x**3/3, Eq(m, -22)), (-a**8/(18
*x**18) - 8*a**7*b/(15*x**15) - 7*a**6*b**2/(3*x**12) - 56*a**5*b**3/(9*x**9) - 35*a**4*b**4/(3*x**6) - 56*a**
3*b**5/(3*x**3) + 28*a**2*b**6*log(x) + 8*a*b**7*x**3/3 + b**8*x**6/6, Eq(m, -19)), (-a**8/(15*x**15) - 2*a**7
*b/(3*x**12) - 28*a**6*b**2/(9*x**9) - 28*a**5*b**3/(3*x**6) - 70*a**4*b**4/(3*x**3) + 56*a**3*b**5*log(x) + 2
8*a**2*b**6*x**3/3 + 4*a*b**7*x**6/3 + b**8*x**9/9, Eq(m, -16)), (-a**8/(12*x**12) - 8*a**7*b/(9*x**9) - 14*a*
*6*b**2/(3*x**6) - 56*a**5*b**3/(3*x**3) + 70*a**4*b**4*log(x) + 56*a**3*b**5*x**3/3 + 14*a**2*b**6*x**6/3 + 8
*a*b**7*x**9/9 + b**8*x**12/12, Eq(m, -13)), (-a**8/(9*x**9) - 4*a**7*b/(3*x**6) - 28*a**6*b**2/(3*x**3) + 56*
a**5*b**3*log(x) + 70*a**4*b**4*x**3/3 + 28*a**3*b**5*x**6/3 + 28*a**2*b**6*x**9/9 + 2*a*b**7*x**12/3 + b**8*x
**15/15, Eq(m, -10)), (-a**8/(6*x**6) - 8*a**7*b/(3*x**3) + 28*a**6*b**2*log(x) + 56*a**5*b**3*x**3/3 + 35*a**
4*b**4*x**6/3 + 56*a**3*b**5*x**9/9 + 7*a**2*b**6*x**12/3 + 8*a*b**7*x**15/15 + b**8*x**18/18, Eq(m, -7)), (-a
**8/(3*x**3) + 8*a**7*b*log(x) + 28*a**6*b**2*x**3/3 + 28*a**5*b**3*x**6/3 + 70*a**4*b**4*x**9/9 + 14*a**3*b**
5*x**12/3 + 28*a**2*b**6*x**15/15 + 4*a*b**7*x**18/9 + b**8*x**21/21, Eq(m, -4)), (a**8*log(x) + 8*a**7*b*x**3
/3 + 14*a**6*b**2*x**6/3 + 56*a**5*b**3*x**9/9 + 35*a**4*b**4*x**12/6 + 56*a**3*b**5*x**15/15 + 14*a**2*b**6*x
**18/9 + 8*a*b**7*x**21/21 + b**8*x**24/24, Eq(m, -1)), (a**8*m**8*x*x**m/(m**9 + 117*m**8 + 5814*m**7 + 15997
8*m**6 + 2662569*m**5 + 27458613*m**4 + 172323696*m**3 + 616224492*m**2 + 1077459120*m + 608608000) + 116*a**8
*m**7*x*x**m/(m**9 + 117*m**8 + 5814*m**7 + 159978*m**6 + 2662569*m**5 + 27458613*m**4 + 172323696*m**3 + 6162
24492*m**2 + 1077459120*m + 608608000) + 5698*a**8*m**6*x*x**m/(m**9 + 117*m**8 + 5814*m**7 + 159978*m**6 + 26
62569*m**5 + 27458613*m**4 + 172323696*m**3 + 616224492*m**2 + 1077459120*m + 608608000) + 154280*a**8*m**5*x*
x**m/(m**9 + 117*m**8 + 5814*m**7 + 159978*m**6 + 2662569*m**5 + 27458613*m**4 + 172323696*m**3 + 616224492*m*
*2 + 1077459120*m + 608608000) + 2508289*a**8*m**4*x*x**m/(m**9 + 117*m**8 + 5814*m**7 + 159978*m**6 + 2662569
*m**5 + 27458613*m**4 + 172323696*m**3 + 616224492*m**2 + 1077459120*m + 608608000) + 24950324*a**8*m**3*x*x**
m/(m**9 + 117*m**8 + 5814*m**7 + 159978*m**6 + 2662569*m**5 + 27458613*m**4 + 172323696*m**3 + 616224492*m**2
+ 1077459120*m + 608608000) + 147373372*a**8*m**2*x*x**m/(m**9 + 117*m**8 + 5814*m**7 + 159978*m**6 + 2662569*
m**5 + 27458613*m**4 + 172323696*m**3 + 616224492*m**2 + 1077459120*m + 608608000) + 468851120*a**8*m*x*x**m/(
m**9 + 117*m**8 + 5814*m**7 + 159978*m**6 + 2662569*m**5 + 27458613*m**4 + 172323696*m**3 + 616224492*m**2 + 1
077459120*m + 608608000) + 608608000*a**8*x*x**m/(m**9 + 117*m**8 + 5814*m**7 + 159978*m**6 + 2662569*m**5 + 2
7458613*m**4 + 172323696*m**3 + 616224492*m**2 + 1077459120*m + 608608000) + 8*a**7*b*m**8*x**4*x**m/(m**9 + 1
17*m**8 + 5814*m**7 + 159978*m**6 + 2662569*m**5 + 27458613*m**4 + 172323696*m**3 + 616224492*m**2 + 107745912
0*m + 608608000) + 904*a**7*b*m**7*x**4*x**m/(m**9 + 117*m**8 + 5814*m**7 + 159978*m**6 + 2662569*m**5 + 27458
613*m**4 + 172323696*m**3 + 616224492*m**2 + 1077459120*m + 608608000) + 42896*a**7*b*m**6*x**4*x**m/(m**9 + 1
17*m**8 + 5814*m**7 + 159978*m**6 + 2662569*m**5 + 27458613*m**4 + 172323696*m**3 + 616224492*m**2 + 107745912
0*m + 608608000) + 1108240*a**7*b*m**5*x**4*x**m/(m**9 + 117*m**8 + 5814*m**7 + 159978*m**6 + 2662569*m**5 + 2
7458613*m**4 + 172323696*m**3 + 616224492*m**2 + 1077459120*m + 608608000) + 16867592*a**7*b*m**4*x**4*x**m/(m
**9 + 117*m**8 + 5814*m**7 + 159978*m**6 + 2662569*m**5 + 27458613*m**4 + 172323696*m**3 + 616224492*m**2 + 10
77459120*m + 608608000) + 152198536*a**7*b*m**3*x**4*x**m/(m**9 + 117*m**8 + 5814*m**7 + 159978*m**6 + 2662569
*m**5 + 27458613*m**4 + 172323696*m**3 + 616224492*m**2 + 1077459120*m + 608608000) + 769795424*a**7*b*m**2*x*
*4*x**m/(m**9 + 117*m**8 + 5814*m**7 + 159978*m**6 + 2662569*m**5 + 27458613*m**4 + 172323696*m**3 + 616224492
*m**2 + 1077459120*m + 608608000) + 1850614240*a**7*b*m*x**4*x**m/(m**9 + 117*m**8 + 5814*m**7 + 159978*m**6 +
2662569*m**5 + 27458613*m**4 + 172323696*m**3 + 616224492*m**2 + 1077459120*m + 608608000) + 1217216000*a**7*
b*x**4*x**m/(m**9 + 117*m**8 + 5814*m**7 + 159978*m**6 + 2662569*m**5 + 27458613*m**4 + 172323696*m**3 + 61622
4492*m**2 + 1077459120*m + 608608000) + 28*a**6*b**2*m**8*x**7*x**m/(m**9 + 117*m**8 + 5814*m**7 + 159978*m**6
+ 2662569*m**5 + 27458613*m**4 + 172323696*m**3 + 616224492*m**2 + 1077459120*m + 608608000) + 3080*a**6*b**2
*m**7*x**7*x**m/(m**9 + 117*m**8 + 5814*m**7 + 159978*m**6 + 2662569*m**5 + 27458613*m**4 + 172323696*m**3 + 6
16224492*m**2 + 1077459120*m + 608608000) + 141232*a**6*b**2*m**6*x**7*x**m/(m**9 + 117*m**8 + 5814*m**7 + 159
978*m**6 + 2662569*m**5 + 27458613*m**4 + 172323696*m**3 + 616224492*m**2 + 1077459120*m + 608608000) + 349076
0*a**6*b**2*m**5*x**7*x**m/(m**9 + 117*m**8 + 5814*m**7 + 159978*m**6 + 2662569*m**5 + 27458613*m**4 + 1723236
96*m**3 + 616224492*m**2 + 1077459120*m + 608608000) + 50116612*a**6*b**2*m**4*x**7*x**m/(m**9 + 117*m**8 + 58
14*m**7 + 159978*m**6 + 2662569*m**5 + 27458613*m**4 + 172323696*m**3 + 616224492*m**2 + 1077459120*m + 608608
000) + 418024880*a**6*b**2*m**3*x**7*x**m/(m**9 + 117*m**8 + 5814*m**7 + 159978*m**6 + 2662569*m**5 + 27458613
*m**4 + 172323696*m**3 + 616224492*m**2 + 1077459120*m + 608608000) + 1898889328*a**6*b**2*m**2*x**7*x**m/(m**
9 + 117*m**8 + 5814*m**7 + 159978*m**6 + 2662569*m**5 + 27458613*m**4 + 172323696*m**3 + 616224492*m**2 + 1077
459120*m + 608608000) + 3962060480*a**6*b**2*m*x**7*x**m/(m**9 + 117*m**8 + 5814*m**7 + 159978*m**6 + 2662569*
m**5 + 27458613*m**4 + 172323696*m**3 + 616224492*m**2 + 1077459120*m + 608608000) + 2434432000*a**6*b**2*x**7
*x**m/(m**9 + 117*m**8 + 5814*m**7 + 159978*m**6 + 2662569*m**5 + 27458613*m**4 + 172323696*m**3 + 616224492*m
**2 + 1077459120*m + 608608000) + 56*a**5*b**3*m**8*x**10*x**m/(m**9 + 117*m**8 + 5814*m**7 + 159978*m**6 + 26
62569*m**5 + 27458613*m**4 + 172323696*m**3 + 616224492*m**2 + 1077459120*m + 608608000) + 5992*a**5*b**3*m**7
*x**10*x**m/(m**9 + 117*m**8 + 5814*m**7 + 159978*m**6 + 2662569*m**5 + 27458613*m**4 + 172323696*m**3 + 61622
4492*m**2 + 1077459120*m + 608608000) + 265664*a**5*b**3*m**6*x**10*x**m/(m**9 + 117*m**8 + 5814*m**7 + 159978
*m**6 + 2662569*m**5 + 27458613*m**4 + 172323696*m**3 + 616224492*m**2 + 1077459120*m + 608608000) + 6302128*a
**5*b**3*m**5*x**10*x**m/(m**9 + 117*m**8 + 5814*m**7 + 159978*m**6 + 2662569*m**5 + 27458613*m**4 + 172323696
*m**3 + 616224492*m**2 + 1077459120*m + 608608000) + 86082584*a**5*b**3*m**4*x**10*x**m/(m**9 + 117*m**8 + 581
4*m**7 + 159978*m**6 + 2662569*m**5 + 27458613*m**4 + 172323696*m**3 + 616224492*m**2 + 1077459120*m + 6086080
00) + 676856488*a**5*b**3*m**3*x**10*x**m/(m**9 + 117*m**8 + 5814*m**7 + 159978*m**6 + 2662569*m**5 + 27458613
*m**4 + 172323696*m**3 + 616224492*m**2 + 1077459120*m + 608608000) + 2881562096*a**5*b**3*m**2*x**10*x**m/(m*
*9 + 117*m**8 + 5814*m**7 + 159978*m**6 + 2662569*m**5 + 27458613*m**4 + 172323696*m**3 + 616224492*m**2 + 107
7459120*m + 608608000) + 5692950592*a**5*b**3*m*x**10*x**m/(m**9 + 117*m**8 + 5814*m**7 + 159978*m**6 + 266256
9*m**5 + 27458613*m**4 + 172323696*m**3 + 616224492*m**2 + 1077459120*m + 608608000) + 3408204800*a**5*b**3*x*
*10*x**m/(m**9 + 117*m**8 + 5814*m**7 + 159978*m**6 + 2662569*m**5 + 27458613*m**4 + 172323696*m**3 + 61622449
2*m**2 + 1077459120*m + 608608000) + 70*a**4*b**4*m**8*x**13*x**m/(m**9 + 117*m**8 + 5814*m**7 + 159978*m**6 +
2662569*m**5 + 27458613*m**4 + 172323696*m**3 + 616224492*m**2 + 1077459120*m + 608608000) + 7280*a**4*b**4*m
**7*x**13*x**m/(m**9 + 117*m**8 + 5814*m**7 + 159978*m**6 + 2662569*m**5 + 27458613*m**4 + 172323696*m**3 + 61
6224492*m**2 + 1077459120*m + 608608000) + 312340*a**4*b**4*m**6*x**13*x**m/(m**9 + 117*m**8 + 5814*m**7 + 159
978*m**6 + 2662569*m**5 + 27458613*m**4 + 172323696*m**3 + 616224492*m**2 + 1077459120*m + 608608000) + 713804
0*a**4*b**4*m**5*x**13*x**m/(m**9 + 117*m**8 + 5814*m**7 + 159978*m**6 + 2662569*m**5 + 27458613*m**4 + 172323
696*m**3 + 616224492*m**2 + 1077459120*m + 608608000) + 93585310*a**4*b**4*m**4*x**13*x**m/(m**9 + 117*m**8 +
5814*m**7 + 159978*m**6 + 2662569*m**5 + 27458613*m**4 + 172323696*m**3 + 616224492*m**2 + 1077459120*m + 6086
08000) + 705493880*a**4*b**4*m**3*x**13*x**m/(m**9 + 117*m**8 + 5814*m**7 + 159978*m**6 + 2662569*m**5 + 27458
613*m**4 + 172323696*m**3 + 616224492*m**2 + 1077459120*m + 608608000) + 2891238280*a**4*b**4*m**2*x**13*x**m/
(m**9 + 117*m**8 + 5814*m**7 + 159978*m**6 + 2662569*m**5 + 27458613*m**4 + 172323696*m**3 + 616224492*m**2 +
1077459120*m + 608608000) + 5549616800*a**4*b**4*m*x**13*x**m/(m**9 + 117*m**8 + 5814*m**7 + 159978*m**6 + 266
2569*m**5 + 27458613*m**4 + 172323696*m**3 + 616224492*m**2 + 1077459120*m + 608608000) + 3277120000*a**4*b**4
*x**13*x**m/(m**9 + 117*m**8 + 5814*m**7 + 159978*m**6 + 2662569*m**5 + 27458613*m**4 + 172323696*m**3 + 61622
4492*m**2 + 1077459120*m + 608608000) + 56*a**3*b**5*m**8*x**16*x**m/(m**9 + 117*m**8 + 5814*m**7 + 159978*m**
6 + 2662569*m**5 + 27458613*m**4 + 172323696*m**3 + 616224492*m**2 + 1077459120*m + 608608000) + 5656*a**3*b**
5*m**7*x**16*x**m/(m**9 + 117*m**8 + 5814*m**7 + 159978*m**6 + 2662569*m**5 + 27458613*m**4 + 172323696*m**3 +
616224492*m**2 + 1077459120*m + 608608000) + 235088*a**3*b**5*m**6*x**16*x**m/(m**9 + 117*m**8 + 5814*m**7 +
159978*m**6 + 2662569*m**5 + 27458613*m**4 + 172323696*m**3 + 616224492*m**2 + 1077459120*m + 608608000) + 519
7360*a**3*b**5*m**5*x**16*x**m/(m**9 + 117*m**8 + 5814*m**7 + 159978*m**6 + 2662569*m**5 + 27458613*m**4 + 172
323696*m**3 + 616224492*m**2 + 1077459120*m + 608608000) + 65946104*a**3*b**5*m**4*x**16*x**m/(m**9 + 117*m**8
+ 5814*m**7 + 159978*m**6 + 2662569*m**5 + 27458613*m**4 + 172323696*m**3 + 616224492*m**2 + 1077459120*m + 6
08608000) + 482544664*a**3*b**5*m**3*x**16*x**m/(m**9 + 117*m**8 + 5814*m**7 + 159978*m**6 + 2662569*m**5 + 27
458613*m**4 + 172323696*m**3 + 616224492*m**2 + 1077459120*m + 608608000) + 1929412352*a**3*b**5*m**2*x**16*x*
*m/(m**9 + 117*m**8 + 5814*m**7 + 159978*m**6 + 2662569*m**5 + 27458613*m**4 + 172323696*m**3 + 616224492*m**2
+ 1077459120*m + 608608000) + 3637973920*a**3*b**5*m*x**16*x**m/(m**9 + 117*m**8 + 5814*m**7 + 159978*m**6 +
2662569*m**5 + 27458613*m**4 + 172323696*m**3 + 616224492*m**2 + 1077459120*m + 608608000) + 2130128000*a**3*b
**5*x**16*x**m/(m**9 + 117*m**8 + 5814*m**7 + 159978*m**6 + 2662569*m**5 + 27458613*m**4 + 172323696*m**3 + 61
6224492*m**2 + 1077459120*m + 608608000) + 28*a**2*b**6*m**8*x**19*x**m/(m**9 + 117*m**8 + 5814*m**7 + 159978*
m**6 + 2662569*m**5 + 27458613*m**4 + 172323696*m**3 + 616224492*m**2 + 1077459120*m + 608608000) + 2744*a**2*
b**6*m**7*x**19*x**m/(m**9 + 117*m**8 + 5814*m**7 + 159978*m**6 + 2662569*m**5 + 27458613*m**4 + 172323696*m**
3 + 616224492*m**2 + 1077459120*m + 608608000) + 110656*a**2*b**6*m**6*x**19*x**m/(m**9 + 117*m**8 + 5814*m**7
+ 159978*m**6 + 2662569*m**5 + 27458613*m**4 + 172323696*m**3 + 616224492*m**2 + 1077459120*m + 608608000) +
2376920*a**2*b**6*m**5*x**19*x**m/(m**9 + 117*m**8 + 5814*m**7 + 159978*m**6 + 2662569*m**5 + 27458613*m**4 +
172323696*m**3 + 616224492*m**2 + 1077459120*m + 608608000) + 29390452*a**2*b**6*m**4*x**19*x**m/(m**9 + 117*m
**8 + 5814*m**7 + 159978*m**6 + 2662569*m**5 + 27458613*m**4 + 172323696*m**3 + 616224492*m**2 + 1077459120*m
+ 608608000) + 210422576*a**2*b**6*m**3*x**19*x**m/(m**9 + 117*m**8 + 5814*m**7 + 159978*m**6 + 2662569*m**5 +
27458613*m**4 + 172323696*m**3 + 616224492*m**2 + 1077459120*m + 608608000) + 827034544*a**2*b**6*m**2*x**19*
x**m/(m**9 + 117*m**8 + 5814*m**7 + 159978*m**6 + 2662569*m**5 + 27458613*m**4 + 172323696*m**3 + 616224492*m*
*2 + 1077459120*m + 608608000) + 1540629440*a**2*b**6*m*x**19*x**m/(m**9 + 117*m**8 + 5814*m**7 + 159978*m**6
+ 2662569*m**5 + 27458613*m**4 + 172323696*m**3 + 616224492*m**2 + 1077459120*m + 608608000) + 896896000*a**2*
b**6*x**19*x**m/(m**9 + 117*m**8 + 5814*m**7 + 159978*m**6 + 2662569*m**5 + 27458613*m**4 + 172323696*m**3 + 6
16224492*m**2 + 1077459120*m + 608608000) + 8*a*b**7*m**8*x**22*x**m/(m**9 + 117*m**8 + 5814*m**7 + 159978*m**
6 + 2662569*m**5 + 27458613*m**4 + 172323696*m**3 + 616224492*m**2 + 1077459120*m + 608608000) + 760*a*b**7*m*
*7*x**22*x**m/(m**9 + 117*m**8 + 5814*m**7 + 159978*m**6 + 2662569*m**5 + 27458613*m**4 + 172323696*m**3 + 616
224492*m**2 + 1077459120*m + 608608000) + 29792*a*b**7*m**6*x**22*x**m/(m**9 + 117*m**8 + 5814*m**7 + 159978*m
**6 + 2662569*m**5 + 27458613*m**4 + 172323696*m**3 + 616224492*m**2 + 1077459120*m + 608608000) + 624400*a*b*
*7*m**5*x**22*x**m/(m**9 + 117*m**8 + 5814*m**7 + 159978*m**6 + 2662569*m**5 + 27458613*m**4 + 172323696*m**3
+ 616224492*m**2 + 1077459120*m + 608608000) + 7563752*a*b**7*m**4*x**22*x**m/(m**9 + 117*m**8 + 5814*m**7 + 1
59978*m**6 + 2662569*m**5 + 27458613*m**4 + 172323696*m**3 + 616224492*m**2 + 1077459120*m + 608608000) + 5326
6360*a*b**7*m**3*x**22*x**m/(m**9 + 117*m**8 + 5814*m**7 + 159978*m**6 + 2662569*m**5 + 27458613*m**4 + 172323
696*m**3 + 616224492*m**2 + 1077459120*m + 608608000) + 206729648*a*b**7*m**2*x**22*x**m/(m**9 + 117*m**8 + 58
14*m**7 + 159978*m**6 + 2662569*m**5 + 27458613*m**4 + 172323696*m**3 + 616224492*m**2 + 1077459120*m + 608608
000) + 381743680*a*b**7*m*x**22*x**m/(m**9 + 117*m**8 + 5814*m**7 + 159978*m**6 + 2662569*m**5 + 27458613*m**4
+ 172323696*m**3 + 616224492*m**2 + 1077459120*m + 608608000) + 221312000*a*b**7*x**22*x**m/(m**9 + 117*m**8
+ 5814*m**7 + 159978*m**6 + 2662569*m**5 + 27458613*m**4 + 172323696*m**3 + 616224492*m**2 + 1077459120*m + 60
8608000) + b**8*m**8*x**25*x**m/(m**9 + 117*m**8 + 5814*m**7 + 159978*m**6 + 2662569*m**5 + 27458613*m**4 + 17
2323696*m**3 + 616224492*m**2 + 1077459120*m + 608608000) + 92*b**8*m**7*x**25*x**m/(m**9 + 117*m**8 + 5814*m*
*7 + 159978*m**6 + 2662569*m**5 + 27458613*m**4 + 172323696*m**3 + 616224492*m**2 + 1077459120*m + 608608000)
+ 3514*b**8*m**6*x**25*x**m/(m**9 + 117*m**8 + 5814*m**7 + 159978*m**6 + 2662569*m**5 + 27458613*m**4 + 172323
696*m**3 + 616224492*m**2 + 1077459120*m + 608608000) + 72128*b**8*m**5*x**25*x**m/(m**9 + 117*m**8 + 5814*m**
7 + 159978*m**6 + 2662569*m**5 + 27458613*m**4 + 172323696*m**3 + 616224492*m**2 + 1077459120*m + 608608000) +
859369*b**8*m**4*x**25*x**m/(m**9 + 117*m**8 + 5814*m**7 + 159978*m**6 + 2662569*m**5 + 27458613*m**4 + 17232
3696*m**3 + 616224492*m**2 + 1077459120*m + 608608000) + 5974388*b**8*m**3*x**25*x**m/(m**9 + 117*m**8 + 5814*
m**7 + 159978*m**6 + 2662569*m**5 + 27458613*m**4 + 172323696*m**3 + 616224492*m**2 + 1077459120*m + 608608000
) + 22963996*b**8*m**2*x**25*x**m/(m**9 + 117*m**8 + 5814*m**7 + 159978*m**6 + 2662569*m**5 + 27458613*m**4 +
172323696*m**3 + 616224492*m**2 + 1077459120*m + 608608000) + 42124592*b**8*m*x**25*x**m/(m**9 + 117*m**8 + 58
14*m**7 + 159978*m**6 + 2662569*m**5 + 27458613*m**4 + 172323696*m**3 + 616224492*m**2 + 1077459120*m + 608608
000) + 24344320*b**8*x**25*x**m/(m**9 + 117*m**8 + 5814*m**7 + 159978*m**6 + 2662569*m**5 + 27458613*m**4 + 17
2323696*m**3 + 616224492*m**2 + 1077459120*m + 608608000), True))
________________________________________________________________________________________
Giac [B] time = 1.19303, size = 1713, normalized size = 11.34 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^m*(b*x^3+a)^8,x, algorithm="giac")
[Out]
(b^8*m^8*x^25*x^m + 92*b^8*m^7*x^25*x^m + 3514*b^8*m^6*x^25*x^m + 8*a*b^7*m^8*x^22*x^m + 72128*b^8*m^5*x^25*x^
m + 760*a*b^7*m^7*x^22*x^m + 859369*b^8*m^4*x^25*x^m + 29792*a*b^7*m^6*x^22*x^m + 5974388*b^8*m^3*x^25*x^m + 2
8*a^2*b^6*m^8*x^19*x^m + 624400*a*b^7*m^5*x^22*x^m + 22963996*b^8*m^2*x^25*x^m + 2744*a^2*b^6*m^7*x^19*x^m + 7
563752*a*b^7*m^4*x^22*x^m + 42124592*b^8*m*x^25*x^m + 110656*a^2*b^6*m^6*x^19*x^m + 53266360*a*b^7*m^3*x^22*x^
m + 24344320*b^8*x^25*x^m + 56*a^3*b^5*m^8*x^16*x^m + 2376920*a^2*b^6*m^5*x^19*x^m + 206729648*a*b^7*m^2*x^22*
x^m + 5656*a^3*b^5*m^7*x^16*x^m + 29390452*a^2*b^6*m^4*x^19*x^m + 381743680*a*b^7*m*x^22*x^m + 235088*a^3*b^5*
m^6*x^16*x^m + 210422576*a^2*b^6*m^3*x^19*x^m + 221312000*a*b^7*x^22*x^m + 70*a^4*b^4*m^8*x^13*x^m + 5197360*a
^3*b^5*m^5*x^16*x^m + 827034544*a^2*b^6*m^2*x^19*x^m + 7280*a^4*b^4*m^7*x^13*x^m + 65946104*a^3*b^5*m^4*x^16*x
^m + 1540629440*a^2*b^6*m*x^19*x^m + 312340*a^4*b^4*m^6*x^13*x^m + 482544664*a^3*b^5*m^3*x^16*x^m + 896896000*
a^2*b^6*x^19*x^m + 56*a^5*b^3*m^8*x^10*x^m + 7138040*a^4*b^4*m^5*x^13*x^m + 1929412352*a^3*b^5*m^2*x^16*x^m +
5992*a^5*b^3*m^7*x^10*x^m + 93585310*a^4*b^4*m^4*x^13*x^m + 3637973920*a^3*b^5*m*x^16*x^m + 265664*a^5*b^3*m^6
*x^10*x^m + 705493880*a^4*b^4*m^3*x^13*x^m + 2130128000*a^3*b^5*x^16*x^m + 28*a^6*b^2*m^8*x^7*x^m + 6302128*a^
5*b^3*m^5*x^10*x^m + 2891238280*a^4*b^4*m^2*x^13*x^m + 3080*a^6*b^2*m^7*x^7*x^m + 86082584*a^5*b^3*m^4*x^10*x^
m + 5549616800*a^4*b^4*m*x^13*x^m + 141232*a^6*b^2*m^6*x^7*x^m + 676856488*a^5*b^3*m^3*x^10*x^m + 3277120000*a
^4*b^4*x^13*x^m + 8*a^7*b*m^8*x^4*x^m + 3490760*a^6*b^2*m^5*x^7*x^m + 2881562096*a^5*b^3*m^2*x^10*x^m + 904*a^
7*b*m^7*x^4*x^m + 50116612*a^6*b^2*m^4*x^7*x^m + 5692950592*a^5*b^3*m*x^10*x^m + 42896*a^7*b*m^6*x^4*x^m + 418
024880*a^6*b^2*m^3*x^7*x^m + 3408204800*a^5*b^3*x^10*x^m + a^8*m^8*x*x^m + 1108240*a^7*b*m^5*x^4*x^m + 1898889
328*a^6*b^2*m^2*x^7*x^m + 116*a^8*m^7*x*x^m + 16867592*a^7*b*m^4*x^4*x^m + 3962060480*a^6*b^2*m*x^7*x^m + 5698
*a^8*m^6*x*x^m + 152198536*a^7*b*m^3*x^4*x^m + 2434432000*a^6*b^2*x^7*x^m + 154280*a^8*m^5*x*x^m + 769795424*a
^7*b*m^2*x^4*x^m + 2508289*a^8*m^4*x*x^m + 1850614240*a^7*b*m*x^4*x^m + 24950324*a^8*m^3*x*x^m + 1217216000*a^
7*b*x^4*x^m + 147373372*a^8*m^2*x*x^m + 468851120*a^8*m*x*x^m + 608608000*a^8*x*x^m)/(m^9 + 117*m^8 + 5814*m^7
+ 159978*m^6 + 2662569*m^5 + 27458613*m^4 + 172323696*m^3 + 616224492*m^2 + 1077459120*m + 608608000)