Integrand size = 28, antiderivative size = 191 \[ \int (d x)^m \left (a x^n+b x^{1+n}+c x^{2+n}\right )^3 \, dx=\frac {3 a \left (b^2+a c\right ) x^{3 (1+n)} (d x)^m}{3+m+3 n}+\frac {3 b c^2 x^{3 (2+n)} (d x)^m}{6+m+3 n}+\frac {a^3 x^{1+3 n} (d x)^m}{1+m+3 n}+\frac {3 a^2 b x^{2+3 n} (d x)^m}{2+m+3 n}+\frac {b \left (b^2+6 a c\right ) x^{4+3 n} (d x)^m}{4+m+3 n}+\frac {3 c \left (b^2+a c\right ) x^{5+3 n} (d x)^m}{5+m+3 n}+\frac {c^3 x^{7+3 n} (d x)^m}{7+m+3 n} \] Output:
3*a*(a*c+b^2)*x^(3+3*n)*(d*x)^m/(3+m+3*n)+3*b*c^2*x^(6+3*n)*(d*x)^m/(6+m+3 *n)+a^3*x^(1+3*n)*(d*x)^m/(1+m+3*n)+3*a^2*b*x^(2+3*n)*(d*x)^m/(2+m+3*n)+b* (6*a*c+b^2)*x^(4+3*n)*(d*x)^m/(4+m+3*n)+3*c*(a*c+b^2)*x^(5+3*n)*(d*x)^m/(5 +m+3*n)+c^3*x^(7+3*n)*(d*x)^m/(7+m+3*n)
Time = 1.15 (sec) , antiderivative size = 305, normalized size of antiderivative = 1.60 \[ \int (d x)^m \left (a x^n+b x^{1+n}+c x^{2+n}\right )^3 \, dx=\frac {x^{1+3 n} (d x)^m \left (-\frac {3 x (b (11+m+3 n)+2 c (6+m+3 n) x) (a+x (b+c x))^2}{(6+m+3 n) (7+m+3 n)}+(a+x (b+c x))^3+\frac {6 x \left (a b \left (-b^2 (1+m+3 n)-\frac {20 a c (5+m+3 n)}{2+m+3 n}+4 a c (6+m+3 n)\right )-\frac {\left (10 a b^2 c (5+m+3 n)+\left (b^2 (3+m+3 n)-2 a c (4+m+3 n)\right ) \left (b^2 (1+m+3 n)-4 a c (6+m+3 n)\right )\right ) x}{3+m+3 n}+\left (b^3 (1+m+3 n)-10 a b c (5+m+3 n)-4 a b c (6+m+3 n)+c (4+m+3 n) \left (b^2 (1+m+3 n)-4 a c (6+m+3 n)\right ) x\right ) (a+x (b+c x))\right )}{c (4+m+3 n) (5+m+3 n) (6+m+3 n) (7+m+3 n)}\right )}{1+m+3 n} \] Input:
Integrate[(d*x)^m*(a*x^n + b*x^(1 + n) + c*x^(2 + n))^3,x]
Output:
(x^(1 + 3*n)*(d*x)^m*((-3*x*(b*(11 + m + 3*n) + 2*c*(6 + m + 3*n)*x)*(a + x*(b + c*x))^2)/((6 + m + 3*n)*(7 + m + 3*n)) + (a + x*(b + c*x))^3 + (6*x *(a*b*(-(b^2*(1 + m + 3*n)) - (20*a*c*(5 + m + 3*n))/(2 + m + 3*n) + 4*a*c *(6 + m + 3*n)) - ((10*a*b^2*c*(5 + m + 3*n) + (b^2*(3 + m + 3*n) - 2*a*c* (4 + m + 3*n))*(b^2*(1 + m + 3*n) - 4*a*c*(6 + m + 3*n)))*x)/(3 + m + 3*n) + (b^3*(1 + m + 3*n) - 10*a*b*c*(5 + m + 3*n) - 4*a*b*c*(6 + m + 3*n) + c *(4 + m + 3*n)*(b^2*(1 + m + 3*n) - 4*a*c*(6 + m + 3*n))*x)*(a + x*(b + c* x))))/(c*(4 + m + 3*n)*(5 + m + 3*n)*(6 + m + 3*n)*(7 + m + 3*n))))/(1 + m + 3*n)
Time = 0.36 (sec) , antiderivative size = 174, normalized size of antiderivative = 0.91, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2028, 30, 1140, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int (d x)^m \left (a x^n+b x^{n+1}+c x^{n+2}\right )^3 \, dx\) |
| \(\Big \downarrow \) 2028 |
| \(\displaystyle \int x^{3 n} (d x)^m \left (a+b x+c x^2\right )^3dx\) |
| \(\Big \downarrow \) 30 |
| \(\displaystyle x^{-m} (d x)^m \int x^{m+3 n} \left (c x^2+b x+a\right )^3dx\) |
| \(\Big \downarrow \) 1140 |
| \(\displaystyle x^{-m} (d x)^m \int \left (a^3 x^{m+3 n}+3 a^2 b x^{m+3 n+1}+3 a \left (b^2+a c\right ) x^{m+3 n+2}+b \left (b^2+6 a c\right ) x^{m+3 n+3}+3 c \left (b^2+a c\right ) x^{m+3 n+4}+3 b c^2 x^{m+3 n+5}+c^3 x^{m+3 n+6}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle x^{-m} (d x)^m \left (\frac {a^3 x^{m+3 n+1}}{m+3 n+1}+\frac {3 a^2 b x^{m+3 n+2}}{m+3 n+2}+\frac {3 a \left (a c+b^2\right ) x^{m+3 n+3}}{m+3 n+3}+\frac {b \left (6 a c+b^2\right ) x^{m+3 n+4}}{m+3 n+4}+\frac {3 c \left (a c+b^2\right ) x^{m+3 n+5}}{m+3 n+5}+\frac {3 b c^2 x^{m+3 n+6}}{m+3 n+6}+\frac {c^3 x^{m+3 n+7}}{m+3 n+7}\right )\) |
Input:
Int[(d*x)^m*(a*x^n + b*x^(1 + n) + c*x^(2 + n))^3,x]
Output:
((d*x)^m*((a^3*x^(1 + m + 3*n))/(1 + m + 3*n) + (3*a^2*b*x^(2 + m + 3*n))/ (2 + m + 3*n) + (3*a*(b^2 + a*c)*x^(3 + m + 3*n))/(3 + m + 3*n) + (b*(b^2 + 6*a*c)*x^(4 + m + 3*n))/(4 + m + 3*n) + (3*c*(b^2 + a*c)*x^(5 + m + 3*n) )/(5 + m + 3*n) + (3*b*c^2*x^(6 + m + 3*n))/(6 + m + 3*n) + (c^3*x^(7 + m + 3*n))/(7 + m + 3*n)))/x^m
Int[(u_.)*((a_.)*(x_))^(m_.)*((b_.)*(x_)^(i_.))^(p_), x_Symbol] :> Simp[b^I ntPart[p]*((b*x^i)^FracPart[p]/(a^(i*IntPart[p])*(a*x)^(i*FracPart[p]))) Int[u*(a*x)^(m + i*p), x], x] /; FreeQ[{a, b, i, m, p}, x] && IntegerQ[i] & & !IntegerQ[p]
Int[((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x _Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && IGtQ[p, 0]
Int[(Fx_.)*((a_.)*(x_)^(r_.) + (b_.)*(x_)^(s_.) + (c_.)*(x_)^(t_.))^(p_.), x_Symbol] :> Int[x^(p*r)*(a + b*x^(s - r) + c*x^(t - r))^p*Fx, x] /; FreeQ[ {a, b, c, r, s, t}, x] && IntegerQ[p] && PosQ[s - r] && PosQ[t - r] && !(E qQ[p, 1] && EqQ[u, 1])
Leaf count of result is larger than twice the leaf count of optimal. \(3383\) vs. \(2(191)=382\).
Time = 20.48 (sec) , antiderivative size = 3384, normalized size of antiderivative = 17.72
| method | result | size |
| orering | \(\text {Expression too large to display}\) | \(3384\) |
| risch | \(\text {Expression too large to display}\) | \(3396\) |
| parallelrisch | \(\text {Expression too large to display}\) | \(7012\) |
Input:
int((d*x)^m*(x^n*a+b*x^(1+n)+c*x^(2+n))^3,x,method=_RETURNVERBOSE)
Output:
(c^3*m^6*x^6+18*c^3*m^5*n*x^6+135*c^3*m^4*n^2*x^6+540*c^3*m^3*n^3*x^6+1215 *c^3*m^2*n^4*x^6+1458*c^3*m*n^5*x^6+729*c^3*n^6*x^6+3*b*c^2*m^6*x^5+54*b*c ^2*m^5*n*x^5+405*b*c^2*m^4*n^2*x^5+1620*b*c^2*m^3*n^3*x^5+3645*b*c^2*m^2*n ^4*x^5+4374*b*c^2*m*n^5*x^5+2187*b*c^2*n^6*x^5+21*c^3*m^5*x^6+315*c^3*m^4* n*x^6+1890*c^3*m^3*n^2*x^6+5670*c^3*m^2*n^3*x^6+8505*c^3*m*n^4*x^6+5103*c^ 3*n^5*x^6+3*a*c^2*m^6*x^4+54*a*c^2*m^5*n*x^4+405*a*c^2*m^4*n^2*x^4+1620*a* c^2*m^3*n^3*x^4+3645*a*c^2*m^2*n^4*x^4+4374*a*c^2*m*n^5*x^4+2187*a*c^2*n^6 *x^4+3*b^2*c*m^6*x^4+54*b^2*c*m^5*n*x^4+405*b^2*c*m^4*n^2*x^4+1620*b^2*c*m ^3*n^3*x^4+3645*b^2*c*m^2*n^4*x^4+4374*b^2*c*m*n^5*x^4+2187*b^2*c*n^6*x^4+ 66*b*c^2*m^5*x^5+990*b*c^2*m^4*n*x^5+5940*b*c^2*m^3*n^2*x^5+17820*b*c^2*m^ 2*n^3*x^5+26730*b*c^2*m*n^4*x^5+16038*b*c^2*n^5*x^5+175*c^3*m^4*x^6+2100*c ^3*m^3*n*x^6+9450*c^3*m^2*n^2*x^6+18900*c^3*m*n^3*x^6+14175*c^3*n^4*x^6+6* a*b*c*m^6*x^3+108*a*b*c*m^5*n*x^3+810*a*b*c*m^4*n^2*x^3+3240*a*b*c*m^3*n^3 *x^3+7290*a*b*c*m^2*n^4*x^3+8748*a*b*c*m*n^5*x^3+4374*a*b*c*n^6*x^3+69*a*c ^2*m^5*x^4+1035*a*c^2*m^4*n*x^4+6210*a*c^2*m^3*n^2*x^4+18630*a*c^2*m^2*n^3 *x^4+27945*a*c^2*m*n^4*x^4+16767*a*c^2*n^5*x^4+b^3*m^6*x^3+18*b^3*m^5*n*x^ 3+135*b^3*m^4*n^2*x^3+540*b^3*m^3*n^3*x^3+1215*b^3*m^2*n^4*x^3+1458*b^3*m* n^5*x^3+729*b^3*n^6*x^3+69*b^2*c*m^5*x^4+1035*b^2*c*m^4*n*x^4+6210*b^2*c*m ^3*n^2*x^4+18630*b^2*c*m^2*n^3*x^4+27945*b^2*c*m*n^4*x^4+16767*b^2*c*n^5*x ^4+570*b*c^2*m^4*x^5+6840*b*c^2*m^3*n*x^5+30780*b*c^2*m^2*n^2*x^5+61560...
Leaf count of result is larger than twice the leaf count of optimal. 2421 vs. \(2 (191) = 382\).
Time = 0.16 (sec) , antiderivative size = 2421, normalized size of antiderivative = 12.68 \[ \int (d x)^m \left (a x^n+b x^{1+n}+c x^{2+n}\right )^3 \, dx=\text {Too large to display} \] Input:
integrate((d*x)^m*(a*x^n+b*x^(1+n)+c*x^(2+n))^3,x, algorithm="fricas")
Output:
(a^3*m^6 + 729*a^3*n^6 + 27*a^3*m^5 + 295*a^3*m^4 + (c^3*m^6 + 729*c^3*n^6 + 21*c^3*m^5 + 175*c^3*m^4 + 735*c^3*m^3 + 729*(2*c^3*m + 7*c^3)*n^5 + 16 24*c^3*m^2 + 405*(3*c^3*m^2 + 21*c^3*m + 35*c^3)*n^4 + 1764*c^3*m + 135*(4 *c^3*m^3 + 42*c^3*m^2 + 140*c^3*m + 147*c^3)*n^3 + 720*c^3 + 9*(15*c^3*m^4 + 210*c^3*m^3 + 1050*c^3*m^2 + 2205*c^3*m + 1624*c^3)*n^2 + 3*(6*c^3*m^5 + 105*c^3*m^4 + 700*c^3*m^3 + 2205*c^3*m^2 + 3248*c^3*m + 1764*c^3)*n)*x^6 + 1665*a^3*m^3 + 729*(2*a^3*m + 9*a^3)*n^5 + 3*(b*c^2*m^6 + 729*b*c^2*n^6 + 22*b*c^2*m^5 + 190*b*c^2*m^4 + 820*b*c^2*m^3 + 486*(3*b*c^2*m + 11*b*c^ 2)*n^5 + 1849*b*c^2*m^2 + 405*(3*b*c^2*m^2 + 22*b*c^2*m + 38*b*c^2)*n^4 + 2038*b*c^2*m + 540*(b*c^2*m^3 + 11*b*c^2*m^2 + 38*b*c^2*m + 41*b*c^2)*n^3 + 840*b*c^2 + 9*(15*b*c^2*m^4 + 220*b*c^2*m^3 + 1140*b*c^2*m^2 + 2460*b*c^ 2*m + 1849*b*c^2)*n^2 + 6*(3*b*c^2*m^5 + 55*b*c^2*m^4 + 380*b*c^2*m^3 + 12 30*b*c^2*m^2 + 1849*b*c^2*m + 1019*b*c^2)*n)*x^5 + 5104*a^3*m^2 + 405*(3*a ^3*m^2 + 27*a^3*m + 59*a^3)*n^4 + 3*((b^2*c + a*c^2)*m^6 + 729*(b^2*c + a* c^2)*n^6 + 23*(b^2*c + a*c^2)*m^5 + 243*(23*b^2*c + 23*a*c^2 + 6*(b^2*c + a*c^2)*m)*n^5 + 207*(b^2*c + a*c^2)*m^4 + 81*(207*b^2*c + 207*a*c^2 + 15*( b^2*c + a*c^2)*m^2 + 115*(b^2*c + a*c^2)*m)*n^4 + 925*(b^2*c + a*c^2)*m^3 + 27*(20*(b^2*c + a*c^2)*m^3 + 925*b^2*c + 925*a*c^2 + 230*(b^2*c + a*c^2) *m^2 + 828*(b^2*c + a*c^2)*m)*n^3 + 1008*b^2*c + 1008*a*c^2 + 2144*(b^2*c + a*c^2)*m^2 + 9*(15*(b^2*c + a*c^2)*m^4 + 230*(b^2*c + a*c^2)*m^3 + 21...
Timed out. \[ \int (d x)^m \left (a x^n+b x^{1+n}+c x^{2+n}\right )^3 \, dx=\text {Timed out} \] Input:
integrate((d*x)**m*(a*x**n+b*x**(1+n)+c*x**(2+n))**3,x)
Output:
Timed out
Time = 0.07 (sec) , antiderivative size = 302, normalized size of antiderivative = 1.58 \[ \int (d x)^m \left (a x^n+b x^{1+n}+c x^{2+n}\right )^3 \, dx=\frac {c^{3} d^{m} x^{7} e^{\left (m \log \left (x\right ) + 3 \, n \log \left (x\right )\right )}}{m + 3 \, n + 7} + \frac {3 \, b c^{2} d^{m} x^{6} e^{\left (m \log \left (x\right ) + 3 \, n \log \left (x\right )\right )}}{m + 3 \, n + 6} + \frac {3 \, b^{2} c d^{m} x^{5} e^{\left (m \log \left (x\right ) + 3 \, n \log \left (x\right )\right )}}{m + 3 \, n + 5} + \frac {3 \, a c^{2} d^{m} x^{5} e^{\left (m \log \left (x\right ) + 3 \, n \log \left (x\right )\right )}}{m + 3 \, n + 5} + \frac {b^{3} d^{m} x^{4} e^{\left (m \log \left (x\right ) + 3 \, n \log \left (x\right )\right )}}{m + 3 \, n + 4} + \frac {6 \, a b c d^{m} x^{4} e^{\left (m \log \left (x\right ) + 3 \, n \log \left (x\right )\right )}}{m + 3 \, n + 4} + \frac {3 \, a b^{2} d^{m} x^{3} e^{\left (m \log \left (x\right ) + 3 \, n \log \left (x\right )\right )}}{m + 3 \, n + 3} + \frac {3 \, a^{2} c d^{m} x^{3} e^{\left (m \log \left (x\right ) + 3 \, n \log \left (x\right )\right )}}{m + 3 \, n + 3} + \frac {3 \, a^{2} b d^{m} x^{2} e^{\left (m \log \left (x\right ) + 3 \, n \log \left (x\right )\right )}}{m + 3 \, n + 2} + \frac {a^{3} d^{m} x e^{\left (m \log \left (x\right ) + 3 \, n \log \left (x\right )\right )}}{m + 3 \, n + 1} \] Input:
integrate((d*x)^m*(a*x^n+b*x^(1+n)+c*x^(2+n))^3,x, algorithm="maxima")
Output:
c^3*d^m*x^7*e^(m*log(x) + 3*n*log(x))/(m + 3*n + 7) + 3*b*c^2*d^m*x^6*e^(m *log(x) + 3*n*log(x))/(m + 3*n + 6) + 3*b^2*c*d^m*x^5*e^(m*log(x) + 3*n*lo g(x))/(m + 3*n + 5) + 3*a*c^2*d^m*x^5*e^(m*log(x) + 3*n*log(x))/(m + 3*n + 5) + b^3*d^m*x^4*e^(m*log(x) + 3*n*log(x))/(m + 3*n + 4) + 6*a*b*c*d^m*x^ 4*e^(m*log(x) + 3*n*log(x))/(m + 3*n + 4) + 3*a*b^2*d^m*x^3*e^(m*log(x) + 3*n*log(x))/(m + 3*n + 3) + 3*a^2*c*d^m*x^3*e^(m*log(x) + 3*n*log(x))/(m + 3*n + 3) + 3*a^2*b*d^m*x^2*e^(m*log(x) + 3*n*log(x))/(m + 3*n + 2) + a^3* d^m*x*e^(m*log(x) + 3*n*log(x))/(m + 3*n + 1)
Leaf count of result is larger than twice the leaf count of optimal. 7783 vs. \(2 (191) = 382\).
Time = 0.30 (sec) , antiderivative size = 7783, normalized size of antiderivative = 40.75 \[ \int (d x)^m \left (a x^n+b x^{1+n}+c x^{2+n}\right )^3 \, dx=\text {Too large to display} \] Input:
integrate((d*x)^m*(a*x^n+b*x^(1+n)+c*x^(2+n))^3,x, algorithm="giac")
Output:
(c^3*m^6*x^7*x^(3*n)*e^(m*log(d) + m*log(x)) + 18*c^3*m^5*n*x^7*x^(3*n)*e^ (m*log(d) + m*log(x)) + 135*c^3*m^4*n^2*x^7*x^(3*n)*e^(m*log(d) + m*log(x) ) + 540*c^3*m^3*n^3*x^7*x^(3*n)*e^(m*log(d) + m*log(x)) + 1215*c^3*m^2*n^4 *x^7*x^(3*n)*e^(m*log(d) + m*log(x)) + 1458*c^3*m*n^5*x^7*x^(3*n)*e^(m*log (d) + m*log(x)) + 729*c^3*n^6*x^7*x^(3*n)*e^(m*log(d) + m*log(x)) + 3*b*c^ 2*m^6*x^6*x^(3*n)*e^(m*log(d) + m*log(x)) + 54*b*c^2*m^5*n*x^6*x^(3*n)*e^( m*log(d) + m*log(x)) + 405*b*c^2*m^4*n^2*x^6*x^(3*n)*e^(m*log(d) + m*log(x )) + 1620*b*c^2*m^3*n^3*x^6*x^(3*n)*e^(m*log(d) + m*log(x)) + 3645*b*c^2*m ^2*n^4*x^6*x^(3*n)*e^(m*log(d) + m*log(x)) + 4374*b*c^2*m*n^5*x^6*x^(3*n)* e^(m*log(d) + m*log(x)) + 2187*b*c^2*n^6*x^6*x^(3*n)*e^(m*log(d) + m*log(x )) + 21*c^3*m^5*x^7*x^(3*n)*e^(m*log(d) + m*log(x)) + 315*c^3*m^4*n*x^7*x^ (3*n)*e^(m*log(d) + m*log(x)) + 1890*c^3*m^3*n^2*x^7*x^(3*n)*e^(m*log(d) + m*log(x)) + 5670*c^3*m^2*n^3*x^7*x^(3*n)*e^(m*log(d) + m*log(x)) + 8505*c ^3*m*n^4*x^7*x^(3*n)*e^(m*log(d) + m*log(x)) + 5103*c^3*n^5*x^7*x^(3*n)*e^ (m*log(d) + m*log(x)) + 3*b^2*c*m^6*x^5*x^(3*n)*e^(m*log(d) + m*log(x)) + 3*a*c^2*m^6*x^5*x^(3*n)*e^(m*log(d) + m*log(x)) + 54*b^2*c*m^5*n*x^5*x^(3* n)*e^(m*log(d) + m*log(x)) + 54*a*c^2*m^5*n*x^5*x^(3*n)*e^(m*log(d) + m*lo g(x)) + 405*b^2*c*m^4*n^2*x^5*x^(3*n)*e^(m*log(d) + m*log(x)) + 405*a*c^2* m^4*n^2*x^5*x^(3*n)*e^(m*log(d) + m*log(x)) + 1620*b^2*c*m^3*n^3*x^5*x^(3* n)*e^(m*log(d) + m*log(x)) + 1620*a*c^2*m^3*n^3*x^5*x^(3*n)*e^(m*log(d)...
Time = 13.67 (sec) , antiderivative size = 3890, normalized size of antiderivative = 20.37 \[ \int (d x)^m \left (a x^n+b x^{1+n}+c x^{2+n}\right )^3 \, dx=\text {Too large to display} \] Input:
int((d*x)^m*(a*x^n + b*x^(n + 1) + c*x^(n + 2))^3,x)
Output:
(a^3*x*x^(3*n)*(d*x)^m*(8028*m + 24084*n + 30624*m*n + 44955*m*n^2 + 14985 *m^2*n + 31860*m*n^3 + 3540*m^3*n + 10935*m*n^4 + 405*m^4*n + 1458*m*n^5 + 18*m^5*n + 5104*m^2 + 1665*m^3 + 295*m^4 + 27*m^5 + m^6 + 45936*n^2 + 449 55*n^3 + 23895*n^4 + 6561*n^5 + 729*n^6 + 15930*m^2*n^2 + 7290*m^2*n^3 + 2 430*m^3*n^2 + 1215*m^2*n^4 + 540*m^3*n^3 + 135*m^4*n^2 + 5040))/(13068*m + 39204*n + 78792*m*n + 182763*m*n^2 + 60921*m^2*n + 211680*m*n^3 + 23520*m ^3*n + 130410*m*n^4 + 4830*m^4*n + 40824*m*n^5 + 504*m^5*n + 5103*m*n^6 + 21*m^6*n + 13132*m^2 + 6769*m^3 + 1960*m^4 + 322*m^5 + 28*m^6 + m^7 + 1181 88*n^2 + 182763*n^3 + 158760*n^4 + 78246*n^5 + 20412*n^6 + 2187*n^7 + 1058 40*m^2*n^2 + 86940*m^2*n^3 + 28980*m^3*n^2 + 34020*m^2*n^4 + 15120*m^3*n^3 + 3780*m^4*n^2 + 5103*m^2*n^5 + 2835*m^3*n^4 + 945*m^4*n^3 + 189*m^5*n^2 + 5040) + (b^3*x*x^(3*n + 3)*(d*x)^m*(2952*m + 8856*n + 15270*m*n + 28512* m*n^2 + 9504*m^2*n + 24408*m*n^3 + 2712*m^3*n + 9720*m*n^4 + 360*m^4*n + 1 458*m*n^5 + 18*m^5*n + 2545*m^2 + 1056*m^3 + 226*m^4 + 24*m^5 + m^6 + 2290 5*n^2 + 28512*n^3 + 18306*n^4 + 5832*n^5 + 729*n^6 + 12204*m^2*n^2 + 6480* m^2*n^3 + 2160*m^3*n^2 + 1215*m^2*n^4 + 540*m^3*n^3 + 135*m^4*n^2 + 1260)) /(13068*m + 39204*n + 78792*m*n + 182763*m*n^2 + 60921*m^2*n + 211680*m*n^ 3 + 23520*m^3*n + 130410*m*n^4 + 4830*m^4*n + 40824*m*n^5 + 504*m^5*n + 51 03*m*n^6 + 21*m^6*n + 13132*m^2 + 6769*m^3 + 1960*m^4 + 322*m^5 + 28*m^6 + m^7 + 118188*n^2 + 182763*n^3 + 158760*n^4 + 78246*n^5 + 20412*n^6 + 2...
Time = 0.23 (sec) , antiderivative size = 3510, normalized size of antiderivative = 18.38 \[ \int (d x)^m \left (a x^n+b x^{1+n}+c x^{2+n}\right )^3 \, dx =\text {Too large to display} \] Input:
int((d*x)^m*(a*x^n+b*x^(1+n)+c*x^(2+n))^3,x)
Output:
(x**(m + 3*n)*d**m*x*(a**3*m**6 + 18*a**3*m**5*n + 27*a**3*m**5 + 135*a**3 *m**4*n**2 + 405*a**3*m**4*n + 295*a**3*m**4 + 540*a**3*m**3*n**3 + 2430*a **3*m**3*n**2 + 3540*a**3*m**3*n + 1665*a**3*m**3 + 1215*a**3*m**2*n**4 + 7290*a**3*m**2*n**3 + 15930*a**3*m**2*n**2 + 14985*a**3*m**2*n + 5104*a**3 *m**2 + 1458*a**3*m*n**5 + 10935*a**3*m*n**4 + 31860*a**3*m*n**3 + 44955*a **3*m*n**2 + 30624*a**3*m*n + 8028*a**3*m + 729*a**3*n**6 + 6561*a**3*n**5 + 23895*a**3*n**4 + 44955*a**3*n**3 + 45936*a**3*n**2 + 24084*a**3*n + 50 40*a**3 + 3*a**2*b*m**6*x + 54*a**2*b*m**5*n*x + 78*a**2*b*m**5*x + 405*a* *2*b*m**4*n**2*x + 1170*a**2*b*m**4*n*x + 810*a**2*b*m**4*x + 1620*a**2*b* m**3*n**3*x + 7020*a**2*b*m**3*n**2*x + 9720*a**2*b*m**3*n*x + 4260*a**2*b *m**3*x + 3645*a**2*b*m**2*n**4*x + 21060*a**2*b*m**2*n**3*x + 43740*a**2* b*m**2*n**2*x + 38340*a**2*b*m**2*n*x + 11787*a**2*b*m**2*x + 4374*a**2*b* m*n**5*x + 31590*a**2*b*m*n**4*x + 87480*a**2*b*m*n**3*x + 115020*a**2*b*m *n**2*x + 70722*a**2*b*m*n*x + 15822*a**2*b*m*x + 2187*a**2*b*n**6*x + 189 54*a**2*b*n**5*x + 65610*a**2*b*n**4*x + 115020*a**2*b*n**3*x + 106083*a** 2*b*n**2*x + 47466*a**2*b*n*x + 7560*a**2*b*x + 3*a**2*c*m**6*x**2 + 54*a* *2*c*m**5*n*x**2 + 75*a**2*c*m**5*x**2 + 405*a**2*c*m**4*n**2*x**2 + 1125* a**2*c*m**4*n*x**2 + 741*a**2*c*m**4*x**2 + 1620*a**2*c*m**3*n**3*x**2 + 6 750*a**2*c*m**3*n**2*x**2 + 8892*a**2*c*m**3*n*x**2 + 3657*a**2*c*m**3*x** 2 + 3645*a**2*c*m**2*n**4*x**2 + 20250*a**2*c*m**2*n**3*x**2 + 40014*a*...