Integrand size = 36, antiderivative size = 292 \[ \int (d+e x)^m \left (3+2 x+5 x^2\right ) \left (2+x+3 x^2-5 x^3+4 x^4\right ) \, dx=\frac {\left (5 d^2-2 d e+3 e^2\right ) \left (4 d^4+5 d^3 e+3 d^2 e^2-d e^3+2 e^4\right ) (d+e x)^{1+m}}{e^7 (1+m)}-\frac {\left (120 d^5+85 d^4 e+68 d^3 e^2+12 d^2 e^3+42 d e^4-7 e^5\right ) (d+e x)^{2+m}}{e^7 (2+m)}+\frac {\left (300 d^4+170 d^3 e+102 d^2 e^2+12 d e^3+21 e^4\right ) (d+e x)^{3+m}}{e^7 (3+m)}-\frac {2 \left (200 d^3+85 d^2 e+34 d e^2+2 e^3\right ) (d+e x)^{4+m}}{e^7 (4+m)}+\frac {\left (300 d^2+85 d e+17 e^2\right ) (d+e x)^{5+m}}{e^7 (5+m)}-\frac {(120 d+17 e) (d+e x)^{6+m}}{e^7 (6+m)}+\frac {20 (d+e x)^{7+m}}{e^7 (7+m)} \] Output:
(5*d^2-2*d*e+3*e^2)*(4*d^4+5*d^3*e+3*d^2*e^2-d*e^3+2*e^4)*(e*x+d)^(1+m)/e^ 7/(1+m)-(120*d^5+85*d^4*e+68*d^3*e^2+12*d^2*e^3+42*d*e^4-7*e^5)*(e*x+d)^(2 +m)/e^7/(2+m)+(300*d^4+170*d^3*e+102*d^2*e^2+12*d*e^3+21*e^4)*(e*x+d)^(3+m )/e^7/(3+m)-2*(200*d^3+85*d^2*e+34*d*e^2+2*e^3)*(e*x+d)^(4+m)/e^7/(4+m)+(3 00*d^2+85*d*e+17*e^2)*(e*x+d)^(5+m)/e^7/(5+m)-(120*d+17*e)*(e*x+d)^(6+m)/e ^7/(6+m)+20*(e*x+d)^(7+m)/e^7/(7+m)
Time = 0.31 (sec) , antiderivative size = 261, normalized size of antiderivative = 0.89 \[ \int (d+e x)^m \left (3+2 x+5 x^2\right ) \left (2+x+3 x^2-5 x^3+4 x^4\right ) \, dx=\frac {(d+e x)^{1+m} \left (\frac {\left (5 d^2-2 d e+3 e^2\right ) \left (4 d^4+5 d^3 e+3 d^2 e^2-d e^3+2 e^4\right )}{1+m}-\frac {\left (120 d^5+85 d^4 e+68 d^3 e^2+12 d^2 e^3+42 d e^4-7 e^5\right ) (d+e x)}{2+m}+\frac {\left (300 d^4+170 d^3 e+102 d^2 e^2+12 d e^3+21 e^4\right ) (d+e x)^2}{3+m}-\frac {2 \left (200 d^3+85 d^2 e+34 d e^2+2 e^3\right ) (d+e x)^3}{4+m}+\frac {\left (300 d^2+85 d e+17 e^2\right ) (d+e x)^4}{5+m}-\frac {(120 d+17 e) (d+e x)^5}{6+m}+\frac {20 (d+e x)^6}{7+m}\right )}{e^7} \] Input:
Integrate[(d + e*x)^m*(3 + 2*x + 5*x^2)*(2 + x + 3*x^2 - 5*x^3 + 4*x^4),x]
Output:
((d + e*x)^(1 + m)*(((5*d^2 - 2*d*e + 3*e^2)*(4*d^4 + 5*d^3*e + 3*d^2*e^2 - d*e^3 + 2*e^4))/(1 + m) - ((120*d^5 + 85*d^4*e + 68*d^3*e^2 + 12*d^2*e^3 + 42*d*e^4 - 7*e^5)*(d + e*x))/(2 + m) + ((300*d^4 + 170*d^3*e + 102*d^2* e^2 + 12*d*e^3 + 21*e^4)*(d + e*x)^2)/(3 + m) - (2*(200*d^3 + 85*d^2*e + 3 4*d*e^2 + 2*e^3)*(d + e*x)^3)/(4 + m) + ((300*d^2 + 85*d*e + 17*e^2)*(d + e*x)^4)/(5 + m) - ((120*d + 17*e)*(d + e*x)^5)/(6 + m) + (20*(d + e*x)^6)/ (7 + m)))/e^7
Time = 0.84 (sec) , antiderivative size = 292, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {2159, 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.
\(\displaystyle \int \left (5 x^2+2 x+3\right ) \left (4 x^4-5 x^3+3 x^2+x+2\right ) (d+e x)^m \, dx\) |
\(\Big \downarrow \) 2159 |
\(\displaystyle \int \left (\frac {\left (300 d^2+85 d e+17 e^2\right ) (d+e x)^{m+4}}{e^6}-\frac {2 \left (200 d^3+85 d^2 e+34 d e^2+2 e^3\right ) (d+e x)^{m+3}}{e^6}+\frac {\left (300 d^4+170 d^3 e+102 d^2 e^2+12 d e^3+21 e^4\right ) (d+e x)^{m+2}}{e^6}+\frac {\left (-120 d^5-85 d^4 e-68 d^3 e^2-12 d^2 e^3-42 d e^4+7 e^5\right ) (d+e x)^{m+1}}{e^6}+\frac {\left (20 d^6+17 d^5 e+17 d^4 e^2+4 d^3 e^3+21 d^2 e^4-7 d e^5+6 e^6\right ) (d+e x)^m}{e^6}+\frac {(-120 d-17 e) (d+e x)^{m+5}}{e^6}+\frac {20 (d+e x)^{m+6}}{e^6}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\left (300 d^2+85 d e+17 e^2\right ) (d+e x)^{m+5}}{e^7 (m+5)}-\frac {2 \left (200 d^3+85 d^2 e+34 d e^2+2 e^3\right ) (d+e x)^{m+4}}{e^7 (m+4)}+\frac {\left (5 d^2-2 d e+3 e^2\right ) \left (4 d^4+5 d^3 e+3 d^2 e^2-d e^3+2 e^4\right ) (d+e x)^{m+1}}{e^7 (m+1)}+\frac {\left (300 d^4+170 d^3 e+102 d^2 e^2+12 d e^3+21 e^4\right ) (d+e x)^{m+3}}{e^7 (m+3)}-\frac {\left (120 d^5+85 d^4 e+68 d^3 e^2+12 d^2 e^3+42 d e^4-7 e^5\right ) (d+e x)^{m+2}}{e^7 (m+2)}-\frac {(120 d+17 e) (d+e x)^{m+6}}{e^7 (m+6)}+\frac {20 (d+e x)^{m+7}}{e^7 (m+7)}\) |
Input:
Int[(d + e*x)^m*(3 + 2*x + 5*x^2)*(2 + x + 3*x^2 - 5*x^3 + 4*x^4),x]
Output:
((5*d^2 - 2*d*e + 3*e^2)*(4*d^4 + 5*d^3*e + 3*d^2*e^2 - d*e^3 + 2*e^4)*(d + e*x)^(1 + m))/(e^7*(1 + m)) - ((120*d^5 + 85*d^4*e + 68*d^3*e^2 + 12*d^2 *e^3 + 42*d*e^4 - 7*e^5)*(d + e*x)^(2 + m))/(e^7*(2 + m)) + ((300*d^4 + 17 0*d^3*e + 102*d^2*e^2 + 12*d*e^3 + 21*e^4)*(d + e*x)^(3 + m))/(e^7*(3 + m) ) - (2*(200*d^3 + 85*d^2*e + 34*d*e^2 + 2*e^3)*(d + e*x)^(4 + m))/(e^7*(4 + m)) + ((300*d^2 + 85*d*e + 17*e^2)*(d + e*x)^(5 + m))/(e^7*(5 + m)) - (( 120*d + 17*e)*(d + e*x)^(6 + m))/(e^7*(6 + m)) + (20*(d + e*x)^(7 + m))/(e ^7*(7 + m))
Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p _.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*Pq*(a + b*x + c*x^2)^p, x ], x] /; FreeQ[{a, b, c, d, e, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]
Leaf count of result is larger than twice the leaf count of optimal. \(1219\) vs. \(2(292)=584\).
Time = 0.13 (sec) , antiderivative size = 1220, normalized size of antiderivative = 4.18
method | result | size |
norman | \(\text {Expression too large to display}\) | \(1220\) |
gosper | \(\text {Expression too large to display}\) | \(1504\) |
orering | \(\text {Expression too large to display}\) | \(1507\) |
risch | \(\text {Expression too large to display}\) | \(1908\) |
parallelrisch | \(\text {Expression too large to display}\) | \(3215\) |
Input:
int((e*x+d)^m*(5*x^2+2*x+3)*(4*x^4-5*x^3+3*x^2+x+2),x,method=_RETURNVERBOS E)
Output:
d*(6*e^6*m^6-7*d*e^5*m^5+162*e^6*m^5+42*d^2*e^4*m^4-175*d*e^5*m^4+1770*e^6 *m^4+24*d^3*e^3*m^3+924*d^2*e^4*m^3-1715*d*e^5*m^3+9990*e^6*m^3+408*d^4*e^ 2*m^2+432*d^3*e^3*m^2+7518*d^2*e^4*m^2-8225*d*e^5*m^2+30624*e^6*m^2+2040*d ^5*e*m+5304*d^4*e^2*m+2568*d^3*e^3*m+26796*d^2*e^4*m-19278*d*e^5*m+48168*e ^6*m+14400*d^6+14280*d^5*e+17136*d^4*e^2+5040*d^3*e^3+35280*d^2*e^4-17640* d*e^5+30240*e^6)/e^7/(m^7+28*m^6+322*m^5+1960*m^4+6769*m^3+13132*m^2+13068 *m+5040)*exp(m*ln(e*x+d))+(20*d*m-17*e*m-119*e)/e/(m^2+13*m+42)*x^6*exp(m* ln(e*x+d))+(17*d*e^2*m^3-4*e^3*m^3+85*d^2*e*m^2+221*d*e^2*m^2-72*e^3*m^2+6 00*d^3*m+595*d^2*e*m+714*d*e^2*m-428*e^3*m-840*e^3)/e^3/(m^4+22*m^3+179*m^ 2+638*m+840)*x^4*exp(m*ln(e*x+d))+(21*d*e^4*m^5+7*e^5*m^5+12*d^2*e^3*m^4+4 62*d*e^4*m^4+175*e^5*m^4+204*d^3*e^2*m^3+216*d^2*e^3*m^3+3759*d*e^4*m^3+17 15*e^5*m^3+1020*d^4*e*m^2+2652*d^3*e^2*m^2+1284*d^2*e^3*m^2+13398*d*e^4*m^ 2+8225*e^5*m^2+7200*d^5*m+7140*d^4*e*m+8568*d^3*e^2*m+2520*d^2*e^3*m+17640 *d*e^4*m+19278*e^5*m+17640*e^5)/e^5/(m^6+27*m^5+295*m^4+1665*m^3+5104*m^2+ 8028*m+5040)*x^2*exp(m*ln(e*x+d))+20/(7+m)*x^7*exp(m*ln(e*x+d))-(17*d*e*m^ 2-17*e^2*m^2+120*d^2*m+119*d*e*m-221*e^2*m-714*e^2)/e^2/(m^3+18*m^2+107*m+ 210)*x^5*exp(m*ln(e*x+d))-(4*d*e^3*m^4-21*e^4*m^4+68*d^2*e^2*m^3+72*d*e^3* m^3-462*e^4*m^3+340*d^3*e*m^2+884*d^2*e^2*m^2+428*d*e^3*m^2-3759*e^4*m^2+2 400*d^4*m+2380*d^3*e*m+2856*d^2*e^2*m+840*d*e^3*m-13398*e^4*m-17640*e^4)/e ^4/(m^5+25*m^4+245*m^3+1175*m^2+2754*m+2520)*x^3*exp(m*ln(e*x+d))-(-7*d...
Leaf count of result is larger than twice the leaf count of optimal. 1448 vs. \(2 (292) = 584\).
Time = 0.09 (sec) , antiderivative size = 1448, normalized size of antiderivative = 4.96 \[ \int (d+e x)^m \left (3+2 x+5 x^2\right ) \left (2+x+3 x^2-5 x^3+4 x^4\right ) \, dx=\text {Too large to display} \] Input:
integrate((e*x+d)^m*(5*x^2+2*x+3)*(4*x^4-5*x^3+3*x^2+x+2),x, algorithm="fr icas")
Output:
(6*d*e^6*m^6 + 20*(e^7*m^6 + 21*e^7*m^5 + 175*e^7*m^4 + 735*e^7*m^3 + 1624 *e^7*m^2 + 1764*e^7*m + 720*e^7)*x^7 + 14400*d^7 + 14280*d^6*e + 17136*d^5 *e^2 + 5040*d^4*e^3 + 35280*d^3*e^4 - 17640*d^2*e^5 + 30240*d*e^6 - (14280 *e^7 - (20*d*e^6 - 17*e^7)*m^6 - 2*(150*d*e^6 - 187*e^7)*m^5 - 170*(10*d*e ^6 - 19*e^7)*m^4 - 20*(225*d*e^6 - 697*e^7)*m^3 - (5480*d*e^6 - 31433*e^7) *m^2 - 2*(1200*d*e^6 - 17323*e^7)*m)*x^6 - (7*d^2*e^5 - 162*d*e^6)*m^5 + ( 17136*e^7 - 17*(d*e^6 - e^7)*m^6 - (120*d^2*e^5 + 289*d*e^6 - 391*e^7)*m^5 - 3*(400*d^2*e^5 + 595*d*e^6 - 1173*e^7)*m^4 - 5*(840*d^2*e^5 + 1003*d*e^ 6 - 3145*e^7)*m^3 - 2*(3000*d^2*e^5 + 3179*d*e^6 - 18224*e^7)*m^2 - 12*(24 0*d^2*e^5 + 238*d*e^6 - 3417*e^7)*m)*x^5 + (42*d^3*e^4 - 175*d^2*e^5 + 177 0*d*e^6)*m^4 - (5040*e^7 - (17*d*e^6 - 4*e^7)*m^6 - (85*d^2*e^5 + 323*d*e^ 6 - 96*e^7)*m^5 - (600*d^3*e^4 + 1105*d^2*e^5 + 2227*d*e^6 - 904*e^7)*m^4 - (3600*d^3*e^4 + 4505*d^2*e^5 + 6817*d*e^6 - 4224*e^7)*m^3 - 5*(1320*d^3* e^4 + 1411*d^2*e^5 + 1836*d*e^6 - 2036*e^7)*m^2 - 6*(600*d^3*e^4 + 595*d^2 *e^5 + 714*d*e^6 - 1968*e^7)*m)*x^4 + (24*d^4*e^3 + 924*d^3*e^4 - 1715*d^2 *e^5 + 9990*d*e^6)*m^3 + (35280*e^7 - (4*d*e^6 - 21*e^7)*m^6 - (68*d^2*e^5 + 84*d*e^6 - 525*e^7)*m^5 - (340*d^3*e^4 + 1088*d^2*e^5 + 652*d*e^6 - 518 7*e^7)*m^4 - (2400*d^4*e^3 + 3400*d^3*e^4 + 5644*d^2*e^5 + 2268*d*e^6 - 25 599*e^7)*m^3 - 4*(1800*d^4*e^3 + 1955*d^3*e^4 + 2584*d^2*e^5 + 844*d*e^6 - 16338*e^7)*m^2 - 4*(1200*d^4*e^3 + 1190*d^3*e^4 + 1428*d^2*e^5 + 420*d...
Leaf count of result is larger than twice the leaf count of optimal. 26165 vs. \(2 (272) = 544\).
Time = 4.87 (sec) , antiderivative size = 26165, normalized size of antiderivative = 89.61 \[ \int (d+e x)^m \left (3+2 x+5 x^2\right ) \left (2+x+3 x^2-5 x^3+4 x^4\right ) \, dx=\text {Too large to display} \] Input:
integrate((e*x+d)**m*(5*x**2+2*x+3)*(4*x**4-5*x**3+3*x**2+x+2),x)
Output:
Piecewise((d**m*(20*x**7/7 - 17*x**6/6 + 17*x**5/5 - x**4 + 7*x**3 + 7*x** 2/2 + 6*x), Eq(e, 0)), (1200*d**6*log(d/e + x)/(60*d**6*e**7 + 360*d**5*e* *8*x + 900*d**4*e**9*x**2 + 1200*d**3*e**10*x**3 + 900*d**2*e**11*x**4 + 3 60*d*e**12*x**5 + 60*e**13*x**6) + 2940*d**6/(60*d**6*e**7 + 360*d**5*e**8 *x + 900*d**4*e**9*x**2 + 1200*d**3*e**10*x**3 + 900*d**2*e**11*x**4 + 360 *d*e**12*x**5 + 60*e**13*x**6) + 7200*d**5*e*x*log(d/e + x)/(60*d**6*e**7 + 360*d**5*e**8*x + 900*d**4*e**9*x**2 + 1200*d**3*e**10*x**3 + 900*d**2*e **11*x**4 + 360*d*e**12*x**5 + 60*e**13*x**6) + 16440*d**5*e*x/(60*d**6*e* *7 + 360*d**5*e**8*x + 900*d**4*e**9*x**2 + 1200*d**3*e**10*x**3 + 900*d** 2*e**11*x**4 + 360*d*e**12*x**5 + 60*e**13*x**6) + 170*d**5*e/(60*d**6*e** 7 + 360*d**5*e**8*x + 900*d**4*e**9*x**2 + 1200*d**3*e**10*x**3 + 900*d**2 *e**11*x**4 + 360*d*e**12*x**5 + 60*e**13*x**6) + 18000*d**4*e**2*x**2*log (d/e + x)/(60*d**6*e**7 + 360*d**5*e**8*x + 900*d**4*e**9*x**2 + 1200*d**3 *e**10*x**3 + 900*d**2*e**11*x**4 + 360*d*e**12*x**5 + 60*e**13*x**6) + 37 500*d**4*e**2*x**2/(60*d**6*e**7 + 360*d**5*e**8*x + 900*d**4*e**9*x**2 + 1200*d**3*e**10*x**3 + 900*d**2*e**11*x**4 + 360*d*e**12*x**5 + 60*e**13*x **6) + 1020*d**4*e**2*x/(60*d**6*e**7 + 360*d**5*e**8*x + 900*d**4*e**9*x* *2 + 1200*d**3*e**10*x**3 + 900*d**2*e**11*x**4 + 360*d*e**12*x**5 + 60*e* *13*x**6) - 34*d**4*e**2/(60*d**6*e**7 + 360*d**5*e**8*x + 900*d**4*e**9*x **2 + 1200*d**3*e**10*x**3 + 900*d**2*e**11*x**4 + 360*d*e**12*x**5 + 6...
Leaf count of result is larger than twice the leaf count of optimal. 788 vs. \(2 (292) = 584\).
Time = 0.07 (sec) , antiderivative size = 788, normalized size of antiderivative = 2.70 \[ \int (d+e x)^m \left (3+2 x+5 x^2\right ) \left (2+x+3 x^2-5 x^3+4 x^4\right ) \, dx =\text {Too large to display} \] Input:
integrate((e*x+d)^m*(5*x^2+2*x+3)*(4*x^4-5*x^3+3*x^2+x+2),x, algorithm="ma xima")
Output:
7*(e^2*(m + 1)*x^2 + d*e*m*x - d^2)*(e*x + d)^m/((m^2 + 3*m + 2)*e^2) + 6* (e*x + d)^(m + 1)/(e*(m + 1)) + 21*((m^2 + 3*m + 2)*e^3*x^3 + (m^2 + m)*d* e^2*x^2 - 2*d^2*e*m*x + 2*d^3)*(e*x + d)^m/((m^3 + 6*m^2 + 11*m + 6)*e^3) - 4*((m^3 + 6*m^2 + 11*m + 6)*e^4*x^4 + (m^3 + 3*m^2 + 2*m)*d*e^3*x^3 - 3* (m^2 + m)*d^2*e^2*x^2 + 6*d^3*e*m*x - 6*d^4)*(e*x + d)^m/((m^4 + 10*m^3 + 35*m^2 + 50*m + 24)*e^4) + 17*((m^4 + 10*m^3 + 35*m^2 + 50*m + 24)*e^5*x^5 + (m^4 + 6*m^3 + 11*m^2 + 6*m)*d*e^4*x^4 - 4*(m^3 + 3*m^2 + 2*m)*d^2*e^3* x^3 + 12*(m^2 + m)*d^3*e^2*x^2 - 24*d^4*e*m*x + 24*d^5)*(e*x + d)^m/((m^5 + 15*m^4 + 85*m^3 + 225*m^2 + 274*m + 120)*e^5) - 17*((m^5 + 15*m^4 + 85*m ^3 + 225*m^2 + 274*m + 120)*e^6*x^6 + (m^5 + 10*m^4 + 35*m^3 + 50*m^2 + 24 *m)*d*e^5*x^5 - 5*(m^4 + 6*m^3 + 11*m^2 + 6*m)*d^2*e^4*x^4 + 20*(m^3 + 3*m ^2 + 2*m)*d^3*e^3*x^3 - 60*(m^2 + m)*d^4*e^2*x^2 + 120*d^5*e*m*x - 120*d^6 )*(e*x + d)^m/((m^6 + 21*m^5 + 175*m^4 + 735*m^3 + 1624*m^2 + 1764*m + 720 )*e^6) + 20*((m^6 + 21*m^5 + 175*m^4 + 735*m^3 + 1624*m^2 + 1764*m + 720)* e^7*x^7 + (m^6 + 15*m^5 + 85*m^4 + 225*m^3 + 274*m^2 + 120*m)*d*e^6*x^6 - 6*(m^5 + 10*m^4 + 35*m^3 + 50*m^2 + 24*m)*d^2*e^5*x^5 + 30*(m^4 + 6*m^3 + 11*m^2 + 6*m)*d^3*e^4*x^4 - 120*(m^3 + 3*m^2 + 2*m)*d^4*e^3*x^3 + 360*(m^2 + m)*d^5*e^2*x^2 - 720*d^6*e*m*x + 720*d^7)*(e*x + d)^m/((m^7 + 28*m^6 + 322*m^5 + 1960*m^4 + 6769*m^3 + 13132*m^2 + 13068*m + 5040)*e^7)
Leaf count of result is larger than twice the leaf count of optimal. 3099 vs. \(2 (292) = 584\).
Time = 0.22 (sec) , antiderivative size = 3099, normalized size of antiderivative = 10.61 \[ \int (d+e x)^m \left (3+2 x+5 x^2\right ) \left (2+x+3 x^2-5 x^3+4 x^4\right ) \, dx=\text {Too large to display} \] Input:
integrate((e*x+d)^m*(5*x^2+2*x+3)*(4*x^4-5*x^3+3*x^2+x+2),x, algorithm="gi ac")
Output:
(20*(e*x + d)^m*e^7*m^6*x^7 + 20*(e*x + d)^m*d*e^6*m^6*x^6 - 17*(e*x + d)^ m*e^7*m^6*x^6 + 420*(e*x + d)^m*e^7*m^5*x^7 - 17*(e*x + d)^m*d*e^6*m^6*x^5 + 17*(e*x + d)^m*e^7*m^6*x^5 + 300*(e*x + d)^m*d*e^6*m^5*x^6 - 374*(e*x + d)^m*e^7*m^5*x^6 + 3500*(e*x + d)^m*e^7*m^4*x^7 + 17*(e*x + d)^m*d*e^6*m^ 6*x^4 - 4*(e*x + d)^m*e^7*m^6*x^4 - 120*(e*x + d)^m*d^2*e^5*m^5*x^5 - 289* (e*x + d)^m*d*e^6*m^5*x^5 + 391*(e*x + d)^m*e^7*m^5*x^5 + 1700*(e*x + d)^m *d*e^6*m^4*x^6 - 3230*(e*x + d)^m*e^7*m^4*x^6 + 14700*(e*x + d)^m*e^7*m^3* x^7 - 4*(e*x + d)^m*d*e^6*m^6*x^3 + 21*(e*x + d)^m*e^7*m^6*x^3 + 85*(e*x + d)^m*d^2*e^5*m^5*x^4 + 323*(e*x + d)^m*d*e^6*m^5*x^4 - 96*(e*x + d)^m*e^7 *m^5*x^4 - 1200*(e*x + d)^m*d^2*e^5*m^4*x^5 - 1785*(e*x + d)^m*d*e^6*m^4*x ^5 + 3519*(e*x + d)^m*e^7*m^4*x^5 + 4500*(e*x + d)^m*d*e^6*m^3*x^6 - 13940 *(e*x + d)^m*e^7*m^3*x^6 + 32480*(e*x + d)^m*e^7*m^2*x^7 + 21*(e*x + d)^m* d*e^6*m^6*x^2 + 7*(e*x + d)^m*e^7*m^6*x^2 - 68*(e*x + d)^m*d^2*e^5*m^5*x^3 - 84*(e*x + d)^m*d*e^6*m^5*x^3 + 525*(e*x + d)^m*e^7*m^5*x^3 + 600*(e*x + d)^m*d^3*e^4*m^4*x^4 + 1105*(e*x + d)^m*d^2*e^5*m^4*x^4 + 2227*(e*x + d)^ m*d*e^6*m^4*x^4 - 904*(e*x + d)^m*e^7*m^4*x^4 - 4200*(e*x + d)^m*d^2*e^5*m ^3*x^5 - 5015*(e*x + d)^m*d*e^6*m^3*x^5 + 15725*(e*x + d)^m*e^7*m^3*x^5 + 5480*(e*x + d)^m*d*e^6*m^2*x^6 - 31433*(e*x + d)^m*e^7*m^2*x^6 + 35280*(e* x + d)^m*e^7*m*x^7 + 7*(e*x + d)^m*d*e^6*m^6*x + 6*(e*x + d)^m*e^7*m^6*x + 12*(e*x + d)^m*d^2*e^5*m^5*x^2 + 483*(e*x + d)^m*d*e^6*m^5*x^2 + 182*(...
Time = 17.73 (sec) , antiderivative size = 1425, normalized size of antiderivative = 4.88 \[ \int (d+e x)^m \left (3+2 x+5 x^2\right ) \left (2+x+3 x^2-5 x^3+4 x^4\right ) \, dx=\text {Too large to display} \] Input:
int((d + e*x)^m*(2*x + 5*x^2 + 3)*(x + 3*x^2 - 5*x^3 + 4*x^4 + 2),x)
Output:
((d + e*x)^m*(30240*d*e^6 + 14280*d^6*e + 14400*d^7 - 17640*d^2*e^5 + 3528 0*d^3*e^4 + 5040*d^4*e^3 + 17136*d^5*e^2 - 19278*d^2*e^5*m + 26796*d^3*e^4 *m + 2568*d^4*e^3*m + 5304*d^5*e^2*m + 30624*d*e^6*m^2 + 9990*d*e^6*m^3 + 1770*d*e^6*m^4 + 162*d*e^6*m^5 + 6*d*e^6*m^6 - 8225*d^2*e^5*m^2 + 7518*d^3 *e^4*m^2 + 432*d^4*e^3*m^2 + 408*d^5*e^2*m^2 - 1715*d^2*e^5*m^3 + 924*d^3* e^4*m^3 + 24*d^4*e^3*m^3 - 175*d^2*e^5*m^4 + 42*d^3*e^4*m^4 - 7*d^2*e^5*m^ 5 + 48168*d*e^6*m + 2040*d^6*e*m))/(e^7*(13068*m + 13132*m^2 + 6769*m^3 + 1960*m^4 + 322*m^5 + 28*m^6 + m^7 + 5040)) + (20*x^7*(d + e*x)^m*(1764*m + 1624*m^2 + 735*m^3 + 175*m^4 + 21*m^5 + m^6 + 720))/(13068*m + 13132*m^2 + 6769*m^3 + 1960*m^4 + 322*m^5 + 28*m^6 + m^7 + 5040) - (x*(d + e*x)^m*(3 5280*d^2*e^5*m - 30240*e^7 - 30624*e^7*m^2 - 9990*e^7*m^3 - 1770*e^7*m^4 - 162*e^7*m^5 - 6*e^7*m^6 - 48168*e^7*m + 5040*d^3*e^4*m + 17136*d^4*e^3*m + 14280*d^5*e^2*m - 19278*d*e^6*m^2 - 8225*d*e^6*m^3 - 1715*d*e^6*m^4 - 17 5*d*e^6*m^5 - 7*d*e^6*m^6 + 26796*d^2*e^5*m^2 + 2568*d^3*e^4*m^2 + 5304*d^ 4*e^3*m^2 + 2040*d^5*e^2*m^2 + 7518*d^2*e^5*m^3 + 432*d^3*e^4*m^3 + 408*d^ 4*e^3*m^3 + 924*d^2*e^5*m^4 + 24*d^3*e^4*m^4 + 42*d^2*e^5*m^5 - 17640*d*e^ 6*m + 14400*d^6*e*m))/(e^7*(13068*m + 13132*m^2 + 6769*m^3 + 1960*m^4 + 32 2*m^5 + 28*m^6 + m^7 + 5040)) - (x^3*(d + e*x)^m*(3*m + m^2 + 2)*(2400*d^4 *m - 13398*e^4*m - 17640*e^4 - 3759*e^4*m^2 - 462*e^4*m^3 - 21*e^4*m^4 + 2 856*d^2*e^2*m + 428*d*e^3*m^2 + 340*d^3*e*m^2 + 72*d*e^3*m^3 + 4*d*e^3*...
Time = 0.18 (sec) , antiderivative size = 1907, normalized size of antiderivative = 6.53 \[ \int (d+e x)^m \left (3+2 x+5 x^2\right ) \left (2+x+3 x^2-5 x^3+4 x^4\right ) \, dx =\text {Too large to display} \] Input:
int((e*x+d)^m*(5*x^2+2*x+3)*(4*x^4-5*x^3+3*x^2+x+2),x)
Output:
((d + e*x)**m*(14400*d**7 - 14400*d**6*e*m*x + 2040*d**6*e*m + 14280*d**6* e + 7200*d**5*e**2*m**2*x**2 - 2040*d**5*e**2*m**2*x + 408*d**5*e**2*m**2 + 7200*d**5*e**2*m*x**2 - 14280*d**5*e**2*m*x + 5304*d**5*e**2*m + 17136*d **5*e**2 - 2400*d**4*e**3*m**3*x**3 + 1020*d**4*e**3*m**3*x**2 - 408*d**4* e**3*m**3*x + 24*d**4*e**3*m**3 - 7200*d**4*e**3*m**2*x**3 + 8160*d**4*e** 3*m**2*x**2 - 5304*d**4*e**3*m**2*x + 432*d**4*e**3*m**2 - 4800*d**4*e**3* m*x**3 + 7140*d**4*e**3*m*x**2 - 17136*d**4*e**3*m*x + 2568*d**4*e**3*m + 5040*d**4*e**3 + 600*d**3*e**4*m**4*x**4 - 340*d**3*e**4*m**4*x**3 + 204*d **3*e**4*m**4*x**2 - 24*d**3*e**4*m**4*x + 42*d**3*e**4*m**4 + 3600*d**3*e **4*m**3*x**4 - 3400*d**3*e**4*m**3*x**3 + 2856*d**3*e**4*m**3*x**2 - 432* d**3*e**4*m**3*x + 924*d**3*e**4*m**3 + 6600*d**3*e**4*m**2*x**4 - 7820*d* *3*e**4*m**2*x**3 + 11220*d**3*e**4*m**2*x**2 - 2568*d**3*e**4*m**2*x + 75 18*d**3*e**4*m**2 + 3600*d**3*e**4*m*x**4 - 4760*d**3*e**4*m*x**3 + 8568*d **3*e**4*m*x**2 - 5040*d**3*e**4*m*x + 26796*d**3*e**4*m + 35280*d**3*e**4 - 120*d**2*e**5*m**5*x**5 + 85*d**2*e**5*m**5*x**4 - 68*d**2*e**5*m**5*x* *3 + 12*d**2*e**5*m**5*x**2 - 42*d**2*e**5*m**5*x - 7*d**2*e**5*m**5 - 120 0*d**2*e**5*m**4*x**5 + 1105*d**2*e**5*m**4*x**4 - 1088*d**2*e**5*m**4*x** 3 + 228*d**2*e**5*m**4*x**2 - 924*d**2*e**5*m**4*x - 175*d**2*e**5*m**4 - 4200*d**2*e**5*m**3*x**5 + 4505*d**2*e**5*m**3*x**4 - 5644*d**2*e**5*m**3* x**3 + 1500*d**2*e**5*m**3*x**2 - 7518*d**2*e**5*m**3*x - 1715*d**2*e**...