Integrand size = 31, antiderivative size = 239 \[ \int (a+b x) (d+e x)^m \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=-\frac {(b d-a e)^7 (d+e x)^{1+m}}{e^8 (1+m)}+\frac {7 b (b d-a e)^6 (d+e x)^{2+m}}{e^8 (2+m)}-\frac {21 b^2 (b d-a e)^5 (d+e x)^{3+m}}{e^8 (3+m)}+\frac {35 b^3 (b d-a e)^4 (d+e x)^{4+m}}{e^8 (4+m)}-\frac {35 b^4 (b d-a e)^3 (d+e x)^{5+m}}{e^8 (5+m)}+\frac {21 b^5 (b d-a e)^2 (d+e x)^{6+m}}{e^8 (6+m)}-\frac {7 b^6 (b d-a e) (d+e x)^{7+m}}{e^8 (7+m)}+\frac {b^7 (d+e x)^{8+m}}{e^8 (8+m)} \]
-(-a*e+b*d)^7*(e*x+d)^(1+m)/e^8/(1+m)+7*b*(-a*e+b*d)^6*(e*x+d)^(2+m)/e^8/( 2+m)-21*b^2*(-a*e+b*d)^5*(e*x+d)^(3+m)/e^8/(3+m)+35*b^3*(-a*e+b*d)^4*(e*x+ d)^(4+m)/e^8/(4+m)-35*b^4*(-a*e+b*d)^3*(e*x+d)^(5+m)/e^8/(5+m)+21*b^5*(-a* e+b*d)^2*(e*x+d)^(6+m)/e^8/(6+m)-7*b^6*(-a*e+b*d)*(e*x+d)^(7+m)/e^8/(7+m)+ b^7*(e*x+d)^(8+m)/e^8/(8+m)
Time = 0.22 (sec) , antiderivative size = 203, normalized size of antiderivative = 0.85 \[ \int (a+b x) (d+e x)^m \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\frac {(d+e x)^{1+m} \left (-\frac {(b d-a e)^7}{1+m}+\frac {7 b (b d-a e)^6 (d+e x)}{2+m}-\frac {21 b^2 (b d-a e)^5 (d+e x)^2}{3+m}+\frac {35 b^3 (b d-a e)^4 (d+e x)^3}{4+m}-\frac {35 b^4 (b d-a e)^3 (d+e x)^4}{5+m}+\frac {21 b^5 (b d-a e)^2 (d+e x)^5}{6+m}-\frac {7 b^6 (b d-a e) (d+e x)^6}{7+m}+\frac {b^7 (d+e x)^7}{8+m}\right )}{e^8} \]
((d + e*x)^(1 + m)*(-((b*d - a*e)^7/(1 + m)) + (7*b*(b*d - a*e)^6*(d + e*x ))/(2 + m) - (21*b^2*(b*d - a*e)^5*(d + e*x)^2)/(3 + m) + (35*b^3*(b*d - a *e)^4*(d + e*x)^3)/(4 + m) - (35*b^4*(b*d - a*e)^3*(d + e*x)^4)/(5 + m) + (21*b^5*(b*d - a*e)^2*(d + e*x)^5)/(6 + m) - (7*b^6*(b*d - a*e)*(d + e*x)^ 6)/(7 + m) + (b^7*(d + e*x)^7)/(8 + m)))/e^8
Time = 0.42 (sec) , antiderivative size = 239, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.129, Rules used = {1184, 27, 53, 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.
\(\displaystyle \int (a+b x) \left (a^2+2 a b x+b^2 x^2\right )^3 (d+e x)^m \, dx\) |
\(\Big \downarrow \) 1184 |
\(\displaystyle \frac {\int b^6 (a+b x)^7 (d+e x)^mdx}{b^6}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \int (a+b x)^7 (d+e x)^mdx\) |
\(\Big \downarrow \) 53 |
\(\displaystyle \int \left (-\frac {7 b^6 (b d-a e) (d+e x)^{m+6}}{e^7}+\frac {21 b^5 (b d-a e)^2 (d+e x)^{m+5}}{e^7}-\frac {35 b^4 (b d-a e)^3 (d+e x)^{m+4}}{e^7}+\frac {35 b^3 (b d-a e)^4 (d+e x)^{m+3}}{e^7}-\frac {21 b^2 (b d-a e)^5 (d+e x)^{m+2}}{e^7}+\frac {(a e-b d)^7 (d+e x)^m}{e^7}+\frac {7 b (b d-a e)^6 (d+e x)^{m+1}}{e^7}+\frac {b^7 (d+e x)^{m+7}}{e^7}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {7 b^6 (b d-a e) (d+e x)^{m+7}}{e^8 (m+7)}+\frac {21 b^5 (b d-a e)^2 (d+e x)^{m+6}}{e^8 (m+6)}-\frac {35 b^4 (b d-a e)^3 (d+e x)^{m+5}}{e^8 (m+5)}+\frac {35 b^3 (b d-a e)^4 (d+e x)^{m+4}}{e^8 (m+4)}-\frac {21 b^2 (b d-a e)^5 (d+e x)^{m+3}}{e^8 (m+3)}-\frac {(b d-a e)^7 (d+e x)^{m+1}}{e^8 (m+1)}+\frac {7 b (b d-a e)^6 (d+e x)^{m+2}}{e^8 (m+2)}+\frac {b^7 (d+e x)^{m+8}}{e^8 (m+8)}\) |
-(((b*d - a*e)^7*(d + e*x)^(1 + m))/(e^8*(1 + m))) + (7*b*(b*d - a*e)^6*(d + e*x)^(2 + m))/(e^8*(2 + m)) - (21*b^2*(b*d - a*e)^5*(d + e*x)^(3 + m))/ (e^8*(3 + m)) + (35*b^3*(b*d - a*e)^4*(d + e*x)^(4 + m))/(e^8*(4 + m)) - ( 35*b^4*(b*d - a*e)^3*(d + e*x)^(5 + m))/(e^8*(5 + m)) + (21*b^5*(b*d - a*e )^2*(d + e*x)^(6 + m))/(e^8*(6 + m)) - (7*b^6*(b*d - a*e)*(d + e*x)^(7 + m ))/(e^8*(7 + m)) + (b^7*(d + e*x)^(8 + m))/(e^8*(8 + m))
3.22.46.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0] && LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_) + (b_.)*(x_ ) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[1/c^p Int[(d + e*x)^m*(f + g*x )^n*(b/2 + c*x)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x] && E qQ[b^2 - 4*a*c, 0] && IntegerQ[p]
Leaf count of result is larger than twice the leaf count of optimal. \(2379\) vs. \(2(239)=478\).
Time = 0.48 (sec) , antiderivative size = 2380, normalized size of antiderivative = 9.96
method | result | size |
norman | \(\text {Expression too large to display}\) | \(2380\) |
gosper | \(\text {Expression too large to display}\) | \(3244\) |
risch | \(\text {Expression too large to display}\) | \(3943\) |
parallelrisch | \(\text {Expression too large to display}\) | \(5797\) |
b^7/(8+m)*x^8*exp(m*ln(e*x+d))+d*(a^7*e^7*m^7+35*a^7*e^7*m^6-7*a^6*b*d*e^6 *m^6+511*a^7*e^7*m^5-231*a^6*b*d*e^6*m^5+42*a^5*b^2*d^2*e^5*m^5+4025*a^7*e ^7*m^4-3115*a^6*b*d*e^6*m^4+1260*a^5*b^2*d^2*e^5*m^4-210*a^4*b^3*d^3*e^4*m ^4+18424*a^7*e^7*m^3-21945*a^6*b*d*e^6*m^3+14910*a^5*b^2*d^2*e^5*m^3-5460* a^4*b^3*d^3*e^4*m^3+840*a^3*b^4*d^4*e^3*m^3+48860*a^7*e^7*m^2-85078*a^6*b* d*e^6*m^2+86940*a^5*b^2*d^2*e^5*m^2-52710*a^4*b^3*d^3*e^4*m^2+17640*a^3*b^ 4*d^4*e^3*m^2-2520*a^2*b^5*d^5*e^2*m^2+69264*a^7*e^7*m-171864*a^6*b*d*e^6* m+249648*a^5*b^2*d^2*e^5*m-223860*a^4*b^3*d^3*e^4*m+122640*a^3*b^4*d^4*e^3 *m-37800*a^2*b^5*d^5*e^2*m+5040*a*b^6*d^6*e*m+40320*a^7*e^7-141120*a^6*b*d *e^6+282240*a^5*b^2*d^2*e^5-352800*a^4*b^3*d^3*e^4+282240*a^3*b^4*d^4*e^3- 141120*a^2*b^5*d^5*e^2+40320*a*b^6*d^6*e-5040*b^7*d^7)/e^8/(m^8+36*m^7+546 *m^6+4536*m^5+22449*m^4+67284*m^3+118124*m^2+109584*m+40320)*exp(m*ln(e*x+ d))+(a^7*e^7*m^7+7*a^6*b*d*e^6*m^7+35*a^7*e^7*m^6+231*a^6*b*d*e^6*m^6-42*a ^5*b^2*d^2*e^5*m^6+511*a^7*e^7*m^5+3115*a^6*b*d*e^6*m^5-1260*a^5*b^2*d^2*e ^5*m^5+210*a^4*b^3*d^3*e^4*m^5+4025*a^7*e^7*m^4+21945*a^6*b*d*e^6*m^4-1491 0*a^5*b^2*d^2*e^5*m^4+5460*a^4*b^3*d^3*e^4*m^4-840*a^3*b^4*d^4*e^3*m^4+184 24*a^7*e^7*m^3+85078*a^6*b*d*e^6*m^3-86940*a^5*b^2*d^2*e^5*m^3+52710*a^4*b ^3*d^3*e^4*m^3-17640*a^3*b^4*d^4*e^3*m^3+2520*a^2*b^5*d^5*e^2*m^3+48860*a^ 7*e^7*m^2+171864*a^6*b*d*e^6*m^2-249648*a^5*b^2*d^2*e^5*m^2+223860*a^4*b^3 *d^3*e^4*m^2-122640*a^3*b^4*d^4*e^3*m^2+37800*a^2*b^5*d^5*e^2*m^2-5040*...
Leaf count of result is larger than twice the leaf count of optimal. 3201 vs. \(2 (239) = 478\).
Time = 0.43 (sec) , antiderivative size = 3201, normalized size of antiderivative = 13.39 \[ \int (a+b x) (d+e x)^m \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\text {Too large to display} \]
(a^7*d*e^7*m^7 - 5040*b^7*d^8 + 40320*a*b^6*d^7*e - 141120*a^2*b^5*d^6*e^2 + 282240*a^3*b^4*d^5*e^3 - 352800*a^4*b^3*d^4*e^4 + 282240*a^5*b^2*d^3*e^ 5 - 141120*a^6*b*d^2*e^6 + 40320*a^7*d*e^7 + (b^7*e^8*m^7 + 28*b^7*e^8*m^6 + 322*b^7*e^8*m^5 + 1960*b^7*e^8*m^4 + 6769*b^7*e^8*m^3 + 13132*b^7*e^8*m ^2 + 13068*b^7*e^8*m + 5040*b^7*e^8)*x^8 + (40320*a*b^6*e^8 + (b^7*d*e^7 + 7*a*b^6*e^8)*m^7 + 7*(3*b^7*d*e^7 + 29*a*b^6*e^8)*m^6 + 7*(25*b^7*d*e^7 + 343*a*b^6*e^8)*m^5 + 245*(3*b^7*d*e^7 + 61*a*b^6*e^8)*m^4 + 56*(29*b^7*d* e^7 + 938*a*b^6*e^8)*m^3 + 196*(9*b^7*d*e^7 + 527*a*b^6*e^8)*m^2 + 144*(5* b^7*d*e^7 + 721*a*b^6*e^8)*m)*x^7 - 7*(a^6*b*d^2*e^6 - 5*a^7*d*e^7)*m^6 + 7*(20160*a^2*b^5*e^8 + (a*b^6*d*e^7 + 3*a^2*b^5*e^8)*m^7 - (b^7*d^2*e^6 - 23*a*b^6*d*e^7 - 90*a^2*b^5*e^8)*m^6 - (15*b^7*d^2*e^6 - 205*a*b^6*d*e^7 - 1098*a^2*b^5*e^8)*m^5 - 5*(17*b^7*d^2*e^6 - 181*a*b^6*d*e^7 - 1404*a^2*b^ 5*e^8)*m^4 - (225*b^7*d^2*e^6 - 2074*a*b^6*d*e^7 - 25227*a^2*b^5*e^8)*m^3 - 2*(137*b^7*d^2*e^6 - 1156*a*b^6*d*e^7 - 25245*a^2*b^5*e^8)*m^2 - 24*(5*b ^7*d^2*e^6 - 40*a*b^6*d*e^7 - 2143*a^2*b^5*e^8)*m)*x^6 + 7*(6*a^5*b^2*d^3* e^5 - 33*a^6*b*d^2*e^6 + 73*a^7*d*e^7)*m^5 + 7*(40320*a^3*b^4*e^8 + (3*a^2 *b^5*d*e^7 + 5*a^3*b^4*e^8)*m^7 - (6*a*b^6*d^2*e^6 - 75*a^2*b^5*d*e^7 - 15 5*a^3*b^4*e^8)*m^6 + (6*b^7*d^3*e^5 - 108*a*b^6*d^2*e^6 + 723*a^2*b^5*d*e^ 7 + 1955*a^3*b^4*e^8)*m^5 + 5*(12*b^7*d^3*e^5 - 138*a*b^6*d^2*e^6 + 681*a^ 2*b^5*d*e^7 + 2581*a^3*b^4*e^8)*m^4 + 2*(105*b^7*d^3*e^5 - 990*a*b^6*d^...
Leaf count of result is larger than twice the leaf count of optimal. 45812 vs. \(2 (211) = 422\).
Time = 10.00 (sec) , antiderivative size = 45812, normalized size of antiderivative = 191.68 \[ \int (a+b x) (d+e x)^m \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\text {Too large to display} \]
Piecewise((d**m*(a**7*x + 7*a**6*b*x**2/2 + 7*a**5*b**2*x**3 + 35*a**4*b** 3*x**4/4 + 7*a**3*b**4*x**5 + 7*a**2*b**5*x**6/2 + a*b**6*x**7 + b**7*x**8 /8), Eq(e, 0)), (-60*a**7*e**7/(420*d**7*e**8 + 2940*d**6*e**9*x + 8820*d* *5*e**10*x**2 + 14700*d**4*e**11*x**3 + 14700*d**3*e**12*x**4 + 8820*d**2* e**13*x**5 + 2940*d*e**14*x**6 + 420*e**15*x**7) - 70*a**6*b*d*e**6/(420*d **7*e**8 + 2940*d**6*e**9*x + 8820*d**5*e**10*x**2 + 14700*d**4*e**11*x**3 + 14700*d**3*e**12*x**4 + 8820*d**2*e**13*x**5 + 2940*d*e**14*x**6 + 420* e**15*x**7) - 490*a**6*b*e**7*x/(420*d**7*e**8 + 2940*d**6*e**9*x + 8820*d **5*e**10*x**2 + 14700*d**4*e**11*x**3 + 14700*d**3*e**12*x**4 + 8820*d**2 *e**13*x**5 + 2940*d*e**14*x**6 + 420*e**15*x**7) - 84*a**5*b**2*d**2*e**5 /(420*d**7*e**8 + 2940*d**6*e**9*x + 8820*d**5*e**10*x**2 + 14700*d**4*e** 11*x**3 + 14700*d**3*e**12*x**4 + 8820*d**2*e**13*x**5 + 2940*d*e**14*x**6 + 420*e**15*x**7) - 588*a**5*b**2*d*e**6*x/(420*d**7*e**8 + 2940*d**6*e** 9*x + 8820*d**5*e**10*x**2 + 14700*d**4*e**11*x**3 + 14700*d**3*e**12*x**4 + 8820*d**2*e**13*x**5 + 2940*d*e**14*x**6 + 420*e**15*x**7) - 1764*a**5* b**2*e**7*x**2/(420*d**7*e**8 + 2940*d**6*e**9*x + 8820*d**5*e**10*x**2 + 14700*d**4*e**11*x**3 + 14700*d**3*e**12*x**4 + 8820*d**2*e**13*x**5 + 294 0*d*e**14*x**6 + 420*e**15*x**7) - 105*a**4*b**3*d**3*e**4/(420*d**7*e**8 + 2940*d**6*e**9*x + 8820*d**5*e**10*x**2 + 14700*d**4*e**11*x**3 + 14700* d**3*e**12*x**4 + 8820*d**2*e**13*x**5 + 2940*d*e**14*x**6 + 420*e**15*...
Leaf count of result is larger than twice the leaf count of optimal. 1108 vs. \(2 (239) = 478\).
Time = 0.23 (sec) , antiderivative size = 1108, normalized size of antiderivative = 4.64 \[ \int (a+b x) (d+e x)^m \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\text {Too large to display} \]
7*(e^2*(m + 1)*x^2 + d*e*m*x - d^2)*(e*x + d)^m*a^6*b/((m^2 + 3*m + 2)*e^2 ) + (e*x + d)^(m + 1)*a^7/(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*a^5*b^2/((m^3 + 6*m^2 + 11*m + 6)*e^3) + 35*((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* a^4*b^3/((m^4 + 10*m^3 + 35*m^2 + 50*m + 24)*e^4) + 35*((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 + 2 4*d^5)*(e*x + d)^m*a^3*b^4/((m^5 + 15*m^4 + 85*m^3 + 225*m^2 + 274*m + 120 )*e^5) + 21*((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*a^2*b^5/((m^6 + 21*m^5 + 17 5*m^4 + 735*m^3 + 1624*m^2 + 1764*m + 720)*e^6) + 7*((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*a*b^6/((m^7 + 28*m^6 + 322*m^5 + 1960*m^4 + 6769*m^ 3 + 13132*m^2 + 13068*m + 5040)*e^7) + ((m^7 + 28*m^6 + 322*m^5 + 1960*...
Leaf count of result is larger than twice the leaf count of optimal. 5641 vs. \(2 (239) = 478\).
Time = 0.30 (sec) , antiderivative size = 5641, normalized size of antiderivative = 23.60 \[ \int (a+b x) (d+e x)^m \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\text {Too large to display} \]
((e*x + d)^m*b^7*e^8*m^7*x^8 + (e*x + d)^m*b^7*d*e^7*m^7*x^7 + 7*(e*x + d) ^m*a*b^6*e^8*m^7*x^7 + 28*(e*x + d)^m*b^7*e^8*m^6*x^8 + 7*(e*x + d)^m*a*b^ 6*d*e^7*m^7*x^6 + 21*(e*x + d)^m*a^2*b^5*e^8*m^7*x^6 + 21*(e*x + d)^m*b^7* d*e^7*m^6*x^7 + 203*(e*x + d)^m*a*b^6*e^8*m^6*x^7 + 322*(e*x + d)^m*b^7*e^ 8*m^5*x^8 + 21*(e*x + d)^m*a^2*b^5*d*e^7*m^7*x^5 + 35*(e*x + d)^m*a^3*b^4* e^8*m^7*x^5 - 7*(e*x + d)^m*b^7*d^2*e^6*m^6*x^6 + 161*(e*x + d)^m*a*b^6*d* e^7*m^6*x^6 + 630*(e*x + d)^m*a^2*b^5*e^8*m^6*x^6 + 175*(e*x + d)^m*b^7*d* e^7*m^5*x^7 + 2401*(e*x + d)^m*a*b^6*e^8*m^5*x^7 + 1960*(e*x + d)^m*b^7*e^ 8*m^4*x^8 + 35*(e*x + d)^m*a^3*b^4*d*e^7*m^7*x^4 + 35*(e*x + d)^m*a^4*b^3* e^8*m^7*x^4 - 42*(e*x + d)^m*a*b^6*d^2*e^6*m^6*x^5 + 525*(e*x + d)^m*a^2*b ^5*d*e^7*m^6*x^5 + 1085*(e*x + d)^m*a^3*b^4*e^8*m^6*x^5 - 105*(e*x + d)^m* b^7*d^2*e^6*m^5*x^6 + 1435*(e*x + d)^m*a*b^6*d*e^7*m^5*x^6 + 7686*(e*x + d )^m*a^2*b^5*e^8*m^5*x^6 + 735*(e*x + d)^m*b^7*d*e^7*m^4*x^7 + 14945*(e*x + d)^m*a*b^6*e^8*m^4*x^7 + 6769*(e*x + d)^m*b^7*e^8*m^3*x^8 + 35*(e*x + d)^ m*a^4*b^3*d*e^7*m^7*x^3 + 21*(e*x + d)^m*a^5*b^2*e^8*m^7*x^3 - 105*(e*x + d)^m*a^2*b^5*d^2*e^6*m^6*x^4 + 945*(e*x + d)^m*a^3*b^4*d*e^7*m^6*x^4 + 112 0*(e*x + d)^m*a^4*b^3*e^8*m^6*x^4 + 42*(e*x + d)^m*b^7*d^3*e^5*m^5*x^5 - 7 56*(e*x + d)^m*a*b^6*d^2*e^6*m^5*x^5 + 5061*(e*x + d)^m*a^2*b^5*d*e^7*m^5* x^5 + 13685*(e*x + d)^m*a^3*b^4*e^8*m^5*x^5 - 595*(e*x + d)^m*b^7*d^2*e^6* m^4*x^6 + 6335*(e*x + d)^m*a*b^6*d*e^7*m^4*x^6 + 49140*(e*x + d)^m*a^2*...
Time = 12.70 (sec) , antiderivative size = 2653, normalized size of antiderivative = 11.10 \[ \int (a+b x) (d+e x)^m \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\text {Too large to display} \]
((d + e*x)^m*(40320*a^7*d*e^7 - 5040*b^7*d^8 - 141120*a^6*b*d^2*e^6 + 4886 0*a^7*d*e^7*m^2 + 18424*a^7*d*e^7*m^3 + 4025*a^7*d*e^7*m^4 + 511*a^7*d*e^7 *m^5 + 35*a^7*d*e^7*m^6 + a^7*d*e^7*m^7 - 141120*a^2*b^5*d^6*e^2 + 282240* a^3*b^4*d^5*e^3 - 352800*a^4*b^3*d^4*e^4 + 282240*a^5*b^2*d^3*e^5 + 40320* a*b^6*d^7*e + 69264*a^7*d*e^7*m + 5040*a*b^6*d^7*e*m - 2520*a^2*b^5*d^6*e^ 2*m^2 + 17640*a^3*b^4*d^5*e^3*m^2 - 52710*a^4*b^3*d^4*e^4*m^2 + 86940*a^5* b^2*d^3*e^5*m^2 + 840*a^3*b^4*d^5*e^3*m^3 - 5460*a^4*b^3*d^4*e^4*m^3 + 149 10*a^5*b^2*d^3*e^5*m^3 - 210*a^4*b^3*d^4*e^4*m^4 + 1260*a^5*b^2*d^3*e^5*m^ 4 + 42*a^5*b^2*d^3*e^5*m^5 - 171864*a^6*b*d^2*e^6*m - 37800*a^2*b^5*d^6*e^ 2*m + 122640*a^3*b^4*d^5*e^3*m - 223860*a^4*b^3*d^4*e^4*m + 249648*a^5*b^2 *d^3*e^5*m - 85078*a^6*b*d^2*e^6*m^2 - 21945*a^6*b*d^2*e^6*m^3 - 3115*a^6* b*d^2*e^6*m^4 - 231*a^6*b*d^2*e^6*m^5 - 7*a^6*b*d^2*e^6*m^6))/(e^8*(109584 *m + 118124*m^2 + 67284*m^3 + 22449*m^4 + 4536*m^5 + 546*m^6 + 36*m^7 + m^ 8 + 40320)) + (x*(d + e*x)^m*(40320*a^7*e^8 + 69264*a^7*e^8*m + 48860*a^7* e^8*m^2 + 18424*a^7*e^8*m^3 + 4025*a^7*e^8*m^4 + 511*a^7*e^8*m^5 + 35*a^7* e^8*m^6 + a^7*e^8*m^7 + 5040*b^7*d^7*e*m + 141120*a^6*b*d*e^7*m + 37800*a^ 2*b^5*d^5*e^3*m^2 - 122640*a^3*b^4*d^4*e^4*m^2 + 223860*a^4*b^3*d^3*e^5*m^ 2 - 249648*a^5*b^2*d^2*e^6*m^2 + 2520*a^2*b^5*d^5*e^3*m^3 - 17640*a^3*b^4* d^4*e^4*m^3 + 52710*a^4*b^3*d^3*e^5*m^3 - 86940*a^5*b^2*d^2*e^6*m^3 - 840* a^3*b^4*d^4*e^4*m^4 + 5460*a^4*b^3*d^3*e^5*m^4 - 14910*a^5*b^2*d^2*e^6*...