Integrand size = 33, antiderivative size = 395 \[ \int (a+b x) (d+e x)^m \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\frac {(b d-a e)^6 (d+e x)^{1+m} \sqrt {a^2+2 a b x+b^2 x^2}}{e^7 (1+m) (a+b x)}-\frac {6 b (b d-a e)^5 (d+e x)^{2+m} \sqrt {a^2+2 a b x+b^2 x^2}}{e^7 (2+m) (a+b x)}+\frac {15 b^2 (b d-a e)^4 (d+e x)^{3+m} \sqrt {a^2+2 a b x+b^2 x^2}}{e^7 (3+m) (a+b x)}-\frac {20 b^3 (b d-a e)^3 (d+e x)^{4+m} \sqrt {a^2+2 a b x+b^2 x^2}}{e^7 (4+m) (a+b x)}+\frac {15 b^4 (b d-a e)^2 (d+e x)^{5+m} \sqrt {a^2+2 a b x+b^2 x^2}}{e^7 (5+m) (a+b x)}-\frac {6 b^5 (b d-a e) (d+e x)^{6+m} \sqrt {a^2+2 a b x+b^2 x^2}}{e^7 (6+m) (a+b x)}+\frac {b^6 (d+e x)^{7+m} \sqrt {a^2+2 a b x+b^2 x^2}}{e^7 (7+m) (a+b x)} \]
(-a*e+b*d)^6*(e*x+d)^(1+m)*((b*x+a)^2)^(1/2)/e^7/(1+m)/(b*x+a)-6*b*(-a*e+b *d)^5*(e*x+d)^(2+m)*((b*x+a)^2)^(1/2)/e^7/(2+m)/(b*x+a)+15*b^2*(-a*e+b*d)^ 4*(e*x+d)^(3+m)*((b*x+a)^2)^(1/2)/e^7/(3+m)/(b*x+a)-20*b^3*(-a*e+b*d)^3*(e *x+d)^(4+m)*((b*x+a)^2)^(1/2)/e^7/(4+m)/(b*x+a)+15*b^4*(-a*e+b*d)^2*(e*x+d )^(5+m)*((b*x+a)^2)^(1/2)/e^7/(5+m)/(b*x+a)-6*b^5*(-a*e+b*d)*(e*x+d)^(6+m) *((b*x+a)^2)^(1/2)/e^7/(6+m)/(b*x+a)+b^6*(e*x+d)^(7+m)*((b*x+a)^2)^(1/2)/e ^7/(7+m)/(b*x+a)
Time = 0.13 (sec) , antiderivative size = 193, normalized size of antiderivative = 0.49 \[ \int (a+b x) (d+e x)^m \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\frac {\sqrt {(a+b x)^2} (d+e x)^{1+m} \left (\frac {(b d-a e)^6}{1+m}-\frac {6 b (b d-a e)^5 (d+e x)}{2+m}+\frac {15 b^2 (b d-a e)^4 (d+e x)^2}{3+m}-\frac {20 b^3 (b d-a e)^3 (d+e x)^3}{4+m}+\frac {15 b^4 (b d-a e)^2 (d+e x)^4}{5+m}-\frac {6 b^5 (b d-a e) (d+e x)^5}{6+m}+\frac {b^6 (d+e x)^6}{7+m}\right )}{e^7 (a+b x)} \]
(Sqrt[(a + b*x)^2]*(d + e*x)^(1 + m)*((b*d - a*e)^6/(1 + m) - (6*b*(b*d - a*e)^5*(d + e*x))/(2 + m) + (15*b^2*(b*d - a*e)^4*(d + e*x)^2)/(3 + m) - ( 20*b^3*(b*d - a*e)^3*(d + e*x)^3)/(4 + m) + (15*b^4*(b*d - a*e)^2*(d + e*x )^4)/(5 + m) - (6*b^5*(b*d - a*e)*(d + e*x)^5)/(6 + m) + (b^6*(d + e*x)^6) /(7 + m)))/(e^7*(a + b*x))
Time = 0.40 (sec) , antiderivative size = 234, normalized size of antiderivative = 0.59, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.121, Rules used = {1187, 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.
| \(\displaystyle \int (a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2} (d+e x)^m \, dx\) |
| \(\Big \downarrow \) 1187 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int b^5 (a+b x)^6 (d+e x)^mdx}{b^5 (a+b x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int (a+b x)^6 (d+e x)^mdx}{a+b x}\) |
| \(\Big \downarrow \) 53 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (\frac {(a e-b d)^6 (d+e x)^m}{e^6}-\frac {6 b (b d-a e)^5 (d+e x)^{m+1}}{e^6}+\frac {15 b^2 (b d-a e)^4 (d+e x)^{m+2}}{e^6}-\frac {20 b^3 (b d-a e)^3 (d+e x)^{m+3}}{e^6}+\frac {15 b^4 (b d-a e)^2 (d+e x)^{m+4}}{e^6}-\frac {6 b^5 (b d-a e) (d+e x)^{m+5}}{e^6}+\frac {b^6 (d+e x)^{m+6}}{e^6}\right )dx}{a+b x}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \left (-\frac {6 b^5 (b d-a e) (d+e x)^{m+6}}{e^7 (m+6)}+\frac {15 b^4 (b d-a e)^2 (d+e x)^{m+5}}{e^7 (m+5)}-\frac {20 b^3 (b d-a e)^3 (d+e x)^{m+4}}{e^7 (m+4)}+\frac {15 b^2 (b d-a e)^4 (d+e x)^{m+3}}{e^7 (m+3)}+\frac {(b d-a e)^6 (d+e x)^{m+1}}{e^7 (m+1)}-\frac {6 b (b d-a e)^5 (d+e x)^{m+2}}{e^7 (m+2)}+\frac {b^6 (d+e x)^{m+7}}{e^7 (m+7)}\right )}{a+b x}\) |
(Sqrt[a^2 + 2*a*b*x + b^2*x^2]*(((b*d - a*e)^6*(d + e*x)^(1 + m))/(e^7*(1 + m)) - (6*b*(b*d - a*e)^5*(d + e*x)^(2 + m))/(e^7*(2 + m)) + (15*b^2*(b*d - a*e)^4*(d + e*x)^(3 + m))/(e^7*(3 + m)) - (20*b^3*(b*d - a*e)^3*(d + e* x)^(4 + m))/(e^7*(4 + m)) + (15*b^4*(b*d - a*e)^2*(d + e*x)^(5 + m))/(e^7* (5 + m)) - (6*b^5*(b*d - a*e)*(d + e*x)^(6 + m))/(e^7*(6 + m)) + (b^6*(d + e*x)^(7 + m))/(e^7*(7 + m))))/(a + b*x)
3.22.51.3.1 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[(a + b*x + c*x^2)^FracPart[p]/(c^ IntPart[p]*(b/2 + c*x)^(2*FracPart[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, p}, x] && EqQ[b^2 - 4*a*c, 0] && !IntegerQ[p]
Leaf count of result is larger than twice the leaf count of optimal. \(2172\) vs. \(2(318)=636\).
Time = 0.41 (sec) , antiderivative size = 2173, normalized size of antiderivative = 5.50
| method | result | size |
| gosper | \(\text {Expression too large to display}\) | \(2173\) |
| risch | \(\text {Expression too large to display}\) | \(2703\) |
1/e^7*(e*x+d)^(1+m)/(b*x+a)^5*((b*x+a)^2)^(5/2)/(m^7+28*m^6+322*m^5+1960*m ^4+6769*m^3+13132*m^2+13068*m+5040)*(b^6*e^6*m^6*x^6+6*a*b^5*e^6*m^6*x^5+2 1*b^6*e^6*m^5*x^6+15*a^2*b^4*e^6*m^6*x^4+132*a*b^5*e^6*m^5*x^5-6*b^6*d*e^5 *m^5*x^5+175*b^6*e^6*m^4*x^6+20*a^3*b^3*e^6*m^6*x^3+345*a^2*b^4*e^6*m^5*x^ 4-30*a*b^5*d*e^5*m^5*x^4+1140*a*b^5*e^6*m^4*x^5-90*b^6*d*e^5*m^4*x^5+735*b ^6*e^6*m^3*x^6+15*a^4*b^2*e^6*m^6*x^2+480*a^3*b^3*e^6*m^5*x^3-60*a^2*b^4*d *e^5*m^5*x^3+3105*a^2*b^4*e^6*m^4*x^4-510*a*b^5*d*e^5*m^4*x^4+4920*a*b^5*e ^6*m^3*x^5+30*b^6*d^2*e^4*m^4*x^4-510*b^6*d*e^5*m^3*x^5+1624*b^6*e^6*m^2*x ^6+6*a^5*b*e^6*m^6*x+375*a^4*b^2*e^6*m^5*x^2-60*a^3*b^3*d*e^5*m^5*x^2+4520 *a^3*b^3*e^6*m^4*x^3-1140*a^2*b^4*d*e^5*m^4*x^3+13875*a^2*b^4*e^6*m^3*x^4+ 120*a*b^5*d^2*e^4*m^4*x^3-3150*a*b^5*d*e^5*m^3*x^4+11094*a*b^5*e^6*m^2*x^5 +300*b^6*d^2*e^4*m^3*x^4-1350*b^6*d*e^5*m^2*x^5+1764*b^6*e^6*m*x^6+a^6*e^6 *m^6+156*a^5*b*e^6*m^5*x-30*a^4*b^2*d*e^5*m^5*x+3705*a^4*b^2*e^6*m^4*x^2-1 260*a^3*b^3*d*e^5*m^4*x^2+21120*a^3*b^3*e^6*m^3*x^3+180*a^2*b^4*d^2*e^4*m^ 4*x^2-7860*a^2*b^4*d*e^5*m^3*x^3+32160*a^2*b^4*e^6*m^2*x^4+1560*a*b^5*d^2* e^4*m^3*x^3-8850*a*b^5*d*e^5*m^2*x^4+12228*a*b^5*e^6*m*x^5-120*b^6*d^3*e^3 *m^3*x^3+1050*b^6*d^2*e^4*m^2*x^4-1644*b^6*d*e^5*m*x^5+720*b^6*e^6*x^6+27* a^6*e^6*m^5-6*a^5*b*d*e^5*m^5+1620*a^5*b*e^6*m^4*x-690*a^4*b^2*d*e^5*m^4*x +18285*a^4*b^2*e^6*m^3*x^2+120*a^3*b^3*d^2*e^4*m^4*x-9780*a^3*b^3*d*e^5*m^ 3*x^2+50900*a^3*b^3*e^6*m^2*x^3+2880*a^2*b^4*d^2*e^4*m^3*x^2-24060*a^2*...
Leaf count of result is larger than twice the leaf count of optimal. 2230 vs. \(2 (318) = 636\).
Time = 0.48 (sec) , antiderivative size = 2230, normalized size of antiderivative = 5.65 \[ \int (a+b x) (d+e x)^m \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\text {Too large to display} \]
(a^6*d*e^6*m^6 + 720*b^6*d^7 - 5040*a*b^5*d^6*e + 15120*a^2*b^4*d^5*e^2 - 25200*a^3*b^3*d^4*e^3 + 25200*a^4*b^2*d^3*e^4 - 15120*a^5*b*d^2*e^5 + 5040 *a^6*d*e^6 + (b^6*e^7*m^6 + 21*b^6*e^7*m^5 + 175*b^6*e^7*m^4 + 735*b^6*e^7 *m^3 + 1624*b^6*e^7*m^2 + 1764*b^6*e^7*m + 720*b^6*e^7)*x^7 + (5040*a*b^5* e^7 + (b^6*d*e^6 + 6*a*b^5*e^7)*m^6 + 3*(5*b^6*d*e^6 + 44*a*b^5*e^7)*m^5 + 5*(17*b^6*d*e^6 + 228*a*b^5*e^7)*m^4 + 15*(15*b^6*d*e^6 + 328*a*b^5*e^7)* m^3 + 2*(137*b^6*d*e^6 + 5547*a*b^5*e^7)*m^2 + 12*(10*b^6*d*e^6 + 1019*a*b ^5*e^7)*m)*x^6 - 3*(2*a^5*b*d^2*e^5 - 9*a^6*d*e^6)*m^5 + 3*(5040*a^2*b^4*e ^7 + (2*a*b^5*d*e^6 + 5*a^2*b^4*e^7)*m^6 - (2*b^6*d^2*e^5 - 34*a*b^5*d*e^6 - 115*a^2*b^4*e^7)*m^5 - 5*(4*b^6*d^2*e^5 - 42*a*b^5*d*e^6 - 207*a^2*b^4* e^7)*m^4 - 5*(14*b^6*d^2*e^5 - 118*a*b^5*d*e^6 - 925*a^2*b^4*e^7)*m^3 - 4* (25*b^6*d^2*e^5 - 187*a*b^5*d*e^6 - 2680*a^2*b^4*e^7)*m^2 - 12*(4*b^6*d^2* e^5 - 28*a*b^5*d*e^6 - 1005*a^2*b^4*e^7)*m)*x^5 + 5*(6*a^4*b^2*d^3*e^4 - 3 0*a^5*b*d^2*e^5 + 59*a^6*d*e^6)*m^4 + 5*(5040*a^3*b^3*e^7 + (3*a^2*b^4*d*e ^6 + 4*a^3*b^3*e^7)*m^6 - 3*(2*a*b^5*d^2*e^5 - 19*a^2*b^4*d*e^6 - 32*a^3*b ^3*e^7)*m^5 + (6*b^6*d^3*e^4 - 78*a*b^5*d^2*e^5 + 393*a^2*b^4*d*e^6 + 904* a^3*b^3*e^7)*m^4 + 3*(12*b^6*d^3*e^4 - 106*a*b^5*d^2*e^5 + 401*a^2*b^4*d*e ^6 + 1408*a^3*b^3*e^7)*m^3 + 2*(33*b^6*d^3*e^4 - 249*a*b^5*d^2*e^5 + 810*a ^2*b^4*d*e^6 + 5090*a^3*b^3*e^7)*m^2 + 36*(b^6*d^3*e^4 - 7*a*b^5*d^2*e^5 + 21*a^2*b^4*d*e^6 + 328*a^3*b^3*e^7)*m)*x^4 - 15*(8*a^3*b^3*d^4*e^3 - 4...
Exception generated. \[ \int (a+b x) (d+e x)^m \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\text {Exception raised: HeuristicGCDFailed} \]
Leaf count of result is larger than twice the leaf count of optimal. 1864 vs. \(2 (318) = 636\).
Time = 0.22 (sec) , antiderivative size = 1864, normalized size of antiderivative = 4.72 \[ \int (a+b x) (d+e x)^m \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\text {Too large to display} \]
((m^5 + 15*m^4 + 85*m^3 + 225*m^2 + 274*m + 120)*b^5*e^6*x^6 - 60*(m^2 + 1 1*m + 30)*a^2*b^3*d^4*e^2 + 20*(m^3 + 15*m^2 + 74*m + 120)*a^3*b^2*d^3*e^3 - 5*(m^4 + 18*m^3 + 119*m^2 + 342*m + 360)*a^4*b*d^2*e^4 + (m^5 + 20*m^4 + 155*m^3 + 580*m^2 + 1044*m + 720)*a^5*d*e^5 + 120*a*b^4*d^5*e*(m + 6) - 120*b^5*d^6 + ((m^5 + 10*m^4 + 35*m^3 + 50*m^2 + 24*m)*b^5*d*e^5 + 5*(m^5 + 16*m^4 + 95*m^3 + 260*m^2 + 324*m + 144)*a*b^4*e^6)*x^5 - 5*((m^4 + 6*m^ 3 + 11*m^2 + 6*m)*b^5*d^2*e^4 - (m^5 + 12*m^4 + 47*m^3 + 72*m^2 + 36*m)*a* b^4*d*e^5 - 2*(m^5 + 17*m^4 + 107*m^3 + 307*m^2 + 396*m + 180)*a^2*b^3*e^6 )*x^4 + 10*(2*(m^3 + 3*m^2 + 2*m)*b^5*d^3*e^3 - 2*(m^4 + 9*m^3 + 20*m^2 + 12*m)*a*b^4*d^2*e^4 + (m^5 + 14*m^4 + 65*m^3 + 112*m^2 + 60*m)*a^2*b^3*d*e ^5 + (m^5 + 18*m^4 + 121*m^3 + 372*m^2 + 508*m + 240)*a^3*b^2*e^6)*x^3 - 5 *(12*(m^2 + m)*b^5*d^4*e^2 - 12*(m^3 + 7*m^2 + 6*m)*a*b^4*d^3*e^3 + 6*(m^4 + 12*m^3 + 41*m^2 + 30*m)*a^2*b^3*d^2*e^4 - 2*(m^5 + 16*m^4 + 89*m^3 + 19 4*m^2 + 120*m)*a^3*b^2*d*e^5 - (m^5 + 19*m^4 + 137*m^3 + 461*m^2 + 702*m + 360)*a^4*b*e^6)*x^2 - (120*(m^2 + 6*m)*a*b^4*d^4*e^2 - 60*(m^3 + 11*m^2 + 30*m)*a^2*b^3*d^3*e^3 + 20*(m^4 + 15*m^3 + 74*m^2 + 120*m)*a^3*b^2*d^2*e^ 4 - 5*(m^5 + 18*m^4 + 119*m^3 + 342*m^2 + 360*m)*a^4*b*d*e^5 - (m^5 + 20*m ^4 + 155*m^3 + 580*m^2 + 1044*m + 720)*a^5*e^6 - 120*b^5*d^5*e*m)*x)*(e*x + d)^m*a/((m^6 + 21*m^5 + 175*m^4 + 735*m^3 + 1624*m^2 + 1764*m + 720)*e^6 ) + ((m^6 + 21*m^5 + 175*m^4 + 735*m^3 + 1624*m^2 + 1764*m + 720)*b^5*e...
Leaf count of result is larger than twice the leaf count of optimal. 4886 vs. \(2 (318) = 636\).
Time = 0.37 (sec) , antiderivative size = 4886, normalized size of antiderivative = 12.37 \[ \int (a+b x) (d+e x)^m \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\text {Too large to display} \]
((e*x + d)^m*b^6*e^7*m^6*x^7*sgn(b*x + a) + (e*x + d)^m*b^6*d*e^6*m^6*x^6* sgn(b*x + a) + 6*(e*x + d)^m*a*b^5*e^7*m^6*x^6*sgn(b*x + a) + 21*(e*x + d) ^m*b^6*e^7*m^5*x^7*sgn(b*x + a) + 6*(e*x + d)^m*a*b^5*d*e^6*m^6*x^5*sgn(b* x + a) + 15*(e*x + d)^m*a^2*b^4*e^7*m^6*x^5*sgn(b*x + a) + 15*(e*x + d)^m* b^6*d*e^6*m^5*x^6*sgn(b*x + a) + 132*(e*x + d)^m*a*b^5*e^7*m^5*x^6*sgn(b*x + a) + 175*(e*x + d)^m*b^6*e^7*m^4*x^7*sgn(b*x + a) + 15*(e*x + d)^m*a^2* b^4*d*e^6*m^6*x^4*sgn(b*x + a) + 20*(e*x + d)^m*a^3*b^3*e^7*m^6*x^4*sgn(b* x + a) - 6*(e*x + d)^m*b^6*d^2*e^5*m^5*x^5*sgn(b*x + a) + 102*(e*x + d)^m* a*b^5*d*e^6*m^5*x^5*sgn(b*x + a) + 345*(e*x + d)^m*a^2*b^4*e^7*m^5*x^5*sgn (b*x + a) + 85*(e*x + d)^m*b^6*d*e^6*m^4*x^6*sgn(b*x + a) + 1140*(e*x + d) ^m*a*b^5*e^7*m^4*x^6*sgn(b*x + a) + 735*(e*x + d)^m*b^6*e^7*m^3*x^7*sgn(b* x + a) + 20*(e*x + d)^m*a^3*b^3*d*e^6*m^6*x^3*sgn(b*x + a) + 15*(e*x + d)^ m*a^4*b^2*e^7*m^6*x^3*sgn(b*x + a) - 30*(e*x + d)^m*a*b^5*d^2*e^5*m^5*x^4* sgn(b*x + a) + 285*(e*x + d)^m*a^2*b^4*d*e^6*m^5*x^4*sgn(b*x + a) + 480*(e *x + d)^m*a^3*b^3*e^7*m^5*x^4*sgn(b*x + a) - 60*(e*x + d)^m*b^6*d^2*e^5*m^ 4*x^5*sgn(b*x + a) + 630*(e*x + d)^m*a*b^5*d*e^6*m^4*x^5*sgn(b*x + a) + 31 05*(e*x + d)^m*a^2*b^4*e^7*m^4*x^5*sgn(b*x + a) + 225*(e*x + d)^m*b^6*d*e^ 6*m^3*x^6*sgn(b*x + a) + 4920*(e*x + d)^m*a*b^5*e^7*m^3*x^6*sgn(b*x + a) + 1624*(e*x + d)^m*b^6*e^7*m^2*x^7*sgn(b*x + a) + 15*(e*x + d)^m*a^4*b^2*d* e^6*m^6*x^2*sgn(b*x + a) + 6*(e*x + d)^m*a^5*b*e^7*m^6*x^2*sgn(b*x + a)...
Timed out. \[ \int (a+b x) (d+e x)^m \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\int \left (a+b\,x\right )\,{\left (d+e\,x\right )}^m\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2} \,d x \]