Integrand size = 35, antiderivative size = 376 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{17/2}} \, dx=-\frac {2 (b d-a e)^6 \sqrt {a^2+2 a b x+b^2 x^2}}{15 e^7 (a+b x) (d+e x)^{15/2}}+\frac {12 b (b d-a e)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{13 e^7 (a+b x) (d+e x)^{13/2}}-\frac {30 b^2 (b d-a e)^4 \sqrt {a^2+2 a b x+b^2 x^2}}{11 e^7 (a+b x) (d+e x)^{11/2}}+\frac {40 b^3 (b d-a e)^3 \sqrt {a^2+2 a b x+b^2 x^2}}{9 e^7 (a+b x) (d+e x)^{9/2}}-\frac {30 b^4 (b d-a e)^2 \sqrt {a^2+2 a b x+b^2 x^2}}{7 e^7 (a+b x) (d+e x)^{7/2}}+\frac {12 b^5 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}{5 e^7 (a+b x) (d+e x)^{5/2}}-\frac {2 b^6 \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^7 (a+b x) (d+e x)^{3/2}} \] Output:
-2/15*(-a*e+b*d)^6*((b*x+a)^2)^(1/2)/e^7/(b*x+a)/(e*x+d)^(15/2)+12/13*b*(- a*e+b*d)^5*((b*x+a)^2)^(1/2)/e^7/(b*x+a)/(e*x+d)^(13/2)-30/11*b^2*(-a*e+b* d)^4*((b*x+a)^2)^(1/2)/e^7/(b*x+a)/(e*x+d)^(11/2)+40/9*b^3*(-a*e+b*d)^3*(( b*x+a)^2)^(1/2)/e^7/(b*x+a)/(e*x+d)^(9/2)-30/7*b^4*(-a*e+b*d)^2*((b*x+a)^2 )^(1/2)/e^7/(b*x+a)/(e*x+d)^(7/2)+12/5*b^5*(-a*e+b*d)*((b*x+a)^2)^(1/2)/e^ 7/(b*x+a)/(e*x+d)^(5/2)-2/3*b^6*((b*x+a)^2)^(1/2)/e^7/(b*x+a)/(e*x+d)^(3/2 )
Time = 0.34 (sec) , antiderivative size = 309, normalized size of antiderivative = 0.82 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{17/2}} \, dx=-\frac {2 \sqrt {(a+b x)^2} \left (3003 a^6 e^6+1386 a^5 b e^5 (2 d+15 e x)+315 a^4 b^2 e^4 \left (8 d^2+60 d e x+195 e^2 x^2\right )+140 a^3 b^3 e^3 \left (16 d^3+120 d^2 e x+390 d e^2 x^2+715 e^3 x^3\right )+15 a^2 b^4 e^2 \left (128 d^4+960 d^3 e x+3120 d^2 e^2 x^2+5720 d e^3 x^3+6435 e^4 x^4\right )+6 a b^5 e \left (256 d^5+1920 d^4 e x+6240 d^3 e^2 x^2+11440 d^2 e^3 x^3+12870 d e^4 x^4+9009 e^5 x^5\right )+b^6 \left (1024 d^6+7680 d^5 e x+24960 d^4 e^2 x^2+45760 d^3 e^3 x^3+51480 d^2 e^4 x^4+36036 d e^5 x^5+15015 e^6 x^6\right )\right )}{45045 e^7 (a+b x) (d+e x)^{15/2}} \] Input:
Integrate[((a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/(d + e*x)^(17/2),x]
Output:
(-2*Sqrt[(a + b*x)^2]*(3003*a^6*e^6 + 1386*a^5*b*e^5*(2*d + 15*e*x) + 315* a^4*b^2*e^4*(8*d^2 + 60*d*e*x + 195*e^2*x^2) + 140*a^3*b^3*e^3*(16*d^3 + 1 20*d^2*e*x + 390*d*e^2*x^2 + 715*e^3*x^3) + 15*a^2*b^4*e^2*(128*d^4 + 960* d^3*e*x + 3120*d^2*e^2*x^2 + 5720*d*e^3*x^3 + 6435*e^4*x^4) + 6*a*b^5*e*(2 56*d^5 + 1920*d^4*e*x + 6240*d^3*e^2*x^2 + 11440*d^2*e^3*x^3 + 12870*d*e^4 *x^4 + 9009*e^5*x^5) + b^6*(1024*d^6 + 7680*d^5*e*x + 24960*d^4*e^2*x^2 + 45760*d^3*e^3*x^3 + 51480*d^2*e^4*x^4 + 36036*d*e^5*x^5 + 15015*e^6*x^6))) /(45045*e^7*(a + b*x)*(d + e*x)^(15/2))
Time = 0.58 (sec) , antiderivative size = 215, normalized size of antiderivative = 0.57, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.114, 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.
| \(\displaystyle \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{17/2}} \, dx\) |
| \(\Big \downarrow \) 1187 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {b^5 (a+b x)^6}{(d+e x)^{17/2}}dx}{b^5 (a+b x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {(a+b x)^6}{(d+e x)^{17/2}}dx}{a+b x}\) |
| \(\Big \downarrow \) 53 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (\frac {b^6}{e^6 (d+e x)^{5/2}}-\frac {6 (b d-a e) b^5}{e^6 (d+e x)^{7/2}}+\frac {15 (b d-a e)^2 b^4}{e^6 (d+e x)^{9/2}}-\frac {20 (b d-a e)^3 b^3}{e^6 (d+e x)^{11/2}}+\frac {15 (b d-a e)^4 b^2}{e^6 (d+e x)^{13/2}}-\frac {6 (b d-a e)^5 b}{e^6 (d+e x)^{15/2}}+\frac {(a e-b d)^6}{e^6 (d+e x)^{17/2}}\right )dx}{a+b x}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \left (\frac {12 b^5 (b d-a e)}{5 e^7 (d+e x)^{5/2}}-\frac {30 b^4 (b d-a e)^2}{7 e^7 (d+e x)^{7/2}}+\frac {40 b^3 (b d-a e)^3}{9 e^7 (d+e x)^{9/2}}-\frac {30 b^2 (b d-a e)^4}{11 e^7 (d+e x)^{11/2}}+\frac {12 b (b d-a e)^5}{13 e^7 (d+e x)^{13/2}}-\frac {2 (b d-a e)^6}{15 e^7 (d+e x)^{15/2}}-\frac {2 b^6}{3 e^7 (d+e x)^{3/2}}\right )}{a+b x}\) |
Input:
Int[((a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/(d + e*x)^(17/2),x]
Output:
(Sqrt[a^2 + 2*a*b*x + b^2*x^2]*((-2*(b*d - a*e)^6)/(15*e^7*(d + e*x)^(15/2 )) + (12*b*(b*d - a*e)^5)/(13*e^7*(d + e*x)^(13/2)) - (30*b^2*(b*d - a*e)^ 4)/(11*e^7*(d + e*x)^(11/2)) + (40*b^3*(b*d - a*e)^3)/(9*e^7*(d + e*x)^(9/ 2)) - (30*b^4*(b*d - a*e)^2)/(7*e^7*(d + e*x)^(7/2)) + (12*b^5*(b*d - a*e) )/(5*e^7*(d + e*x)^(5/2)) - (2*b^6)/(3*e^7*(d + e*x)^(3/2))))/(a + b*x)
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]
Time = 1.25 (sec) , antiderivative size = 393, normalized size of antiderivative = 1.05
| method | result | size |
| gosper | \(-\frac {2 \left (15015 b^{6} x^{6} e^{6}+54054 x^{5} a \,b^{5} e^{6}+36036 x^{5} b^{6} d \,e^{5}+96525 x^{4} a^{2} b^{4} e^{6}+77220 x^{4} a \,b^{5} d \,e^{5}+51480 x^{4} b^{6} d^{2} e^{4}+100100 x^{3} a^{3} b^{3} e^{6}+85800 x^{3} a^{2} b^{4} d \,e^{5}+68640 x^{3} a \,b^{5} d^{2} e^{4}+45760 x^{3} b^{6} d^{3} e^{3}+61425 x^{2} a^{4} b^{2} e^{6}+54600 x^{2} a^{3} b^{3} d \,e^{5}+46800 x^{2} a^{2} b^{4} d^{2} e^{4}+37440 x^{2} a \,b^{5} d^{3} e^{3}+24960 x^{2} b^{6} d^{4} e^{2}+20790 x \,a^{5} b \,e^{6}+18900 x \,a^{4} b^{2} d \,e^{5}+16800 x \,a^{3} b^{3} d^{2} e^{4}+14400 x \,a^{2} b^{4} d^{3} e^{3}+11520 x a \,b^{5} d^{4} e^{2}+7680 x \,b^{6} d^{5} e +3003 a^{6} e^{6}+2772 a^{5} b d \,e^{5}+2520 a^{4} b^{2} d^{2} e^{4}+2240 a^{3} b^{3} d^{3} e^{3}+1920 a^{2} b^{4} d^{4} e^{2}+1536 a \,b^{5} d^{5} e +1024 b^{6} d^{6}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{45045 \left (e x +d \right )^{\frac {15}{2}} e^{7} \left (b x +a \right )^{5}}\) | \(393\) |
| default | \(-\frac {2 \left (15015 b^{6} x^{6} e^{6}+54054 x^{5} a \,b^{5} e^{6}+36036 x^{5} b^{6} d \,e^{5}+96525 x^{4} a^{2} b^{4} e^{6}+77220 x^{4} a \,b^{5} d \,e^{5}+51480 x^{4} b^{6} d^{2} e^{4}+100100 x^{3} a^{3} b^{3} e^{6}+85800 x^{3} a^{2} b^{4} d \,e^{5}+68640 x^{3} a \,b^{5} d^{2} e^{4}+45760 x^{3} b^{6} d^{3} e^{3}+61425 x^{2} a^{4} b^{2} e^{6}+54600 x^{2} a^{3} b^{3} d \,e^{5}+46800 x^{2} a^{2} b^{4} d^{2} e^{4}+37440 x^{2} a \,b^{5} d^{3} e^{3}+24960 x^{2} b^{6} d^{4} e^{2}+20790 x \,a^{5} b \,e^{6}+18900 x \,a^{4} b^{2} d \,e^{5}+16800 x \,a^{3} b^{3} d^{2} e^{4}+14400 x \,a^{2} b^{4} d^{3} e^{3}+11520 x a \,b^{5} d^{4} e^{2}+7680 x \,b^{6} d^{5} e +3003 a^{6} e^{6}+2772 a^{5} b d \,e^{5}+2520 a^{4} b^{2} d^{2} e^{4}+2240 a^{3} b^{3} d^{3} e^{3}+1920 a^{2} b^{4} d^{4} e^{2}+1536 a \,b^{5} d^{5} e +1024 b^{6} d^{6}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{45045 \left (e x +d \right )^{\frac {15}{2}} e^{7} \left (b x +a \right )^{5}}\) | \(393\) |
| orering | \(-\frac {2 \left (15015 b^{6} x^{6} e^{6}+54054 x^{5} a \,b^{5} e^{6}+36036 x^{5} b^{6} d \,e^{5}+96525 x^{4} a^{2} b^{4} e^{6}+77220 x^{4} a \,b^{5} d \,e^{5}+51480 x^{4} b^{6} d^{2} e^{4}+100100 x^{3} a^{3} b^{3} e^{6}+85800 x^{3} a^{2} b^{4} d \,e^{5}+68640 x^{3} a \,b^{5} d^{2} e^{4}+45760 x^{3} b^{6} d^{3} e^{3}+61425 x^{2} a^{4} b^{2} e^{6}+54600 x^{2} a^{3} b^{3} d \,e^{5}+46800 x^{2} a^{2} b^{4} d^{2} e^{4}+37440 x^{2} a \,b^{5} d^{3} e^{3}+24960 x^{2} b^{6} d^{4} e^{2}+20790 x \,a^{5} b \,e^{6}+18900 x \,a^{4} b^{2} d \,e^{5}+16800 x \,a^{3} b^{3} d^{2} e^{4}+14400 x \,a^{2} b^{4} d^{3} e^{3}+11520 x a \,b^{5} d^{4} e^{2}+7680 x \,b^{6} d^{5} e +3003 a^{6} e^{6}+2772 a^{5} b d \,e^{5}+2520 a^{4} b^{2} d^{2} e^{4}+2240 a^{3} b^{3} d^{3} e^{3}+1920 a^{2} b^{4} d^{4} e^{2}+1536 a \,b^{5} d^{5} e +1024 b^{6} d^{6}\right ) \left (b^{2} x^{2}+2 a b x +a^{2}\right )^{\frac {5}{2}}}{45045 e^{7} \left (b x +a \right )^{5} \left (e x +d \right )^{\frac {15}{2}}}\) | \(402\) |
Input:
int((b*x+a)*(b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^(17/2),x,method=_RETURNVER BOSE)
Output:
-2/45045/(e*x+d)^(15/2)*(15015*b^6*e^6*x^6+54054*a*b^5*e^6*x^5+36036*b^6*d *e^5*x^5+96525*a^2*b^4*e^6*x^4+77220*a*b^5*d*e^5*x^4+51480*b^6*d^2*e^4*x^4 +100100*a^3*b^3*e^6*x^3+85800*a^2*b^4*d*e^5*x^3+68640*a*b^5*d^2*e^4*x^3+45 760*b^6*d^3*e^3*x^3+61425*a^4*b^2*e^6*x^2+54600*a^3*b^3*d*e^5*x^2+46800*a^ 2*b^4*d^2*e^4*x^2+37440*a*b^5*d^3*e^3*x^2+24960*b^6*d^4*e^2*x^2+20790*a^5* b*e^6*x+18900*a^4*b^2*d*e^5*x+16800*a^3*b^3*d^2*e^4*x+14400*a^2*b^4*d^3*e^ 3*x+11520*a*b^5*d^4*e^2*x+7680*b^6*d^5*e*x+3003*a^6*e^6+2772*a^5*b*d*e^5+2 520*a^4*b^2*d^2*e^4+2240*a^3*b^3*d^3*e^3+1920*a^2*b^4*d^4*e^2+1536*a*b^5*d ^5*e+1024*b^6*d^6)*((b*x+a)^2)^(5/2)/e^7/(b*x+a)^5
Time = 0.10 (sec) , antiderivative size = 443, normalized size of antiderivative = 1.18 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{17/2}} \, dx=-\frac {2 \, {\left (15015 \, b^{6} e^{6} x^{6} + 1024 \, b^{6} d^{6} + 1536 \, a b^{5} d^{5} e + 1920 \, a^{2} b^{4} d^{4} e^{2} + 2240 \, a^{3} b^{3} d^{3} e^{3} + 2520 \, a^{4} b^{2} d^{2} e^{4} + 2772 \, a^{5} b d e^{5} + 3003 \, a^{6} e^{6} + 18018 \, {\left (2 \, b^{6} d e^{5} + 3 \, a b^{5} e^{6}\right )} x^{5} + 6435 \, {\left (8 \, b^{6} d^{2} e^{4} + 12 \, a b^{5} d e^{5} + 15 \, a^{2} b^{4} e^{6}\right )} x^{4} + 2860 \, {\left (16 \, b^{6} d^{3} e^{3} + 24 \, a b^{5} d^{2} e^{4} + 30 \, a^{2} b^{4} d e^{5} + 35 \, a^{3} b^{3} e^{6}\right )} x^{3} + 195 \, {\left (128 \, b^{6} d^{4} e^{2} + 192 \, a b^{5} d^{3} e^{3} + 240 \, a^{2} b^{4} d^{2} e^{4} + 280 \, a^{3} b^{3} d e^{5} + 315 \, a^{4} b^{2} e^{6}\right )} x^{2} + 30 \, {\left (256 \, b^{6} d^{5} e + 384 \, a b^{5} d^{4} e^{2} + 480 \, a^{2} b^{4} d^{3} e^{3} + 560 \, a^{3} b^{3} d^{2} e^{4} + 630 \, a^{4} b^{2} d e^{5} + 693 \, a^{5} b e^{6}\right )} x\right )} \sqrt {e x + d}}{45045 \, {\left (e^{15} x^{8} + 8 \, d e^{14} x^{7} + 28 \, d^{2} e^{13} x^{6} + 56 \, d^{3} e^{12} x^{5} + 70 \, d^{4} e^{11} x^{4} + 56 \, d^{5} e^{10} x^{3} + 28 \, d^{6} e^{9} x^{2} + 8 \, d^{7} e^{8} x + d^{8} e^{7}\right )}} \] Input:
integrate((b*x+a)*(b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^(17/2),x, algorithm= "fricas")
Output:
-2/45045*(15015*b^6*e^6*x^6 + 1024*b^6*d^6 + 1536*a*b^5*d^5*e + 1920*a^2*b ^4*d^4*e^2 + 2240*a^3*b^3*d^3*e^3 + 2520*a^4*b^2*d^2*e^4 + 2772*a^5*b*d*e^ 5 + 3003*a^6*e^6 + 18018*(2*b^6*d*e^5 + 3*a*b^5*e^6)*x^5 + 6435*(8*b^6*d^2 *e^4 + 12*a*b^5*d*e^5 + 15*a^2*b^4*e^6)*x^4 + 2860*(16*b^6*d^3*e^3 + 24*a* b^5*d^2*e^4 + 30*a^2*b^4*d*e^5 + 35*a^3*b^3*e^6)*x^3 + 195*(128*b^6*d^4*e^ 2 + 192*a*b^5*d^3*e^3 + 240*a^2*b^4*d^2*e^4 + 280*a^3*b^3*d*e^5 + 315*a^4* b^2*e^6)*x^2 + 30*(256*b^6*d^5*e + 384*a*b^5*d^4*e^2 + 480*a^2*b^4*d^3*e^3 + 560*a^3*b^3*d^2*e^4 + 630*a^4*b^2*d*e^5 + 693*a^5*b*e^6)*x)*sqrt(e*x + d)/(e^15*x^8 + 8*d*e^14*x^7 + 28*d^2*e^13*x^6 + 56*d^3*e^12*x^5 + 70*d^4*e ^11*x^4 + 56*d^5*e^10*x^3 + 28*d^6*e^9*x^2 + 8*d^7*e^8*x + d^8*e^7)
Timed out. \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{17/2}} \, dx=\text {Timed out} \] Input:
integrate((b*x+a)*(b**2*x**2+2*a*b*x+a**2)**(5/2)/(e*x+d)**(17/2),x)
Output:
Timed out
Leaf count of result is larger than twice the leaf count of optimal. 757 vs. \(2 (271) = 542\).
Time = 0.10 (sec) , antiderivative size = 757, normalized size of antiderivative = 2.01 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{17/2}} \, dx =\text {Too large to display} \] Input:
integrate((b*x+a)*(b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^(17/2),x, algorithm= "maxima")
Output:
-2/45045*(9009*b^5*e^5*x^5 + 256*b^5*d^5 + 640*a*b^4*d^4*e + 1120*a^2*b^3* d^3*e^2 + 1680*a^3*b^2*d^2*e^3 + 2310*a^4*b*d*e^4 + 3003*a^5*e^5 + 6435*(2 *b^5*d*e^4 + 5*a*b^4*e^5)*x^4 + 1430*(8*b^5*d^2*e^3 + 20*a*b^4*d*e^4 + 35* a^2*b^3*e^5)*x^3 + 390*(16*b^5*d^3*e^2 + 40*a*b^4*d^2*e^3 + 70*a^2*b^3*d*e ^4 + 105*a^3*b^2*e^5)*x^2 + 15*(128*b^5*d^4*e + 320*a*b^4*d^3*e^2 + 560*a^ 2*b^3*d^2*e^3 + 840*a^3*b^2*d*e^4 + 1155*a^4*b*e^5)*x)*a/((e^13*x^7 + 7*d* e^12*x^6 + 21*d^2*e^11*x^5 + 35*d^3*e^10*x^4 + 35*d^4*e^9*x^3 + 21*d^5*e^8 *x^2 + 7*d^6*e^7*x + d^7*e^6)*sqrt(e*x + d)) - 2/45045*(15015*b^5*e^6*x^6 + 1024*b^5*d^6 + 1280*a*b^4*d^5*e + 1280*a^2*b^3*d^4*e^2 + 1120*a^3*b^2*d^ 3*e^3 + 840*a^4*b*d^2*e^4 + 462*a^5*d*e^5 + 9009*(4*b^5*d*e^5 + 5*a*b^4*e^ 6)*x^5 + 12870*(4*b^5*d^2*e^4 + 5*a*b^4*d*e^5 + 5*a^2*b^3*e^6)*x^4 + 1430* (32*b^5*d^3*e^3 + 40*a*b^4*d^2*e^4 + 40*a^2*b^3*d*e^5 + 35*a^3*b^2*e^6)*x^ 3 + 195*(128*b^5*d^4*e^2 + 160*a*b^4*d^3*e^3 + 160*a^2*b^3*d^2*e^4 + 140*a ^3*b^2*d*e^5 + 105*a^4*b*e^6)*x^2 + 15*(512*b^5*d^5*e + 640*a*b^4*d^4*e^2 + 640*a^2*b^3*d^3*e^3 + 560*a^3*b^2*d^2*e^4 + 420*a^4*b*d*e^5 + 231*a^5*e^ 6)*x)*b/((e^14*x^7 + 7*d*e^13*x^6 + 21*d^2*e^12*x^5 + 35*d^3*e^11*x^4 + 35 *d^4*e^10*x^3 + 21*d^5*e^9*x^2 + 7*d^6*e^8*x + d^7*e^7)*sqrt(e*x + d))
Leaf count of result is larger than twice the leaf count of optimal. 602 vs. \(2 (271) = 542\).
Time = 0.20 (sec) , antiderivative size = 602, normalized size of antiderivative = 1.60 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{17/2}} \, dx =\text {Too large to display} \] Input:
integrate((b*x+a)*(b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^(17/2),x, algorithm= "giac")
Output:
-2/45045*(15015*(e*x + d)^6*b^6*sgn(b*x + a) - 54054*(e*x + d)^5*b^6*d*sgn (b*x + a) + 96525*(e*x + d)^4*b^6*d^2*sgn(b*x + a) - 100100*(e*x + d)^3*b^ 6*d^3*sgn(b*x + a) + 61425*(e*x + d)^2*b^6*d^4*sgn(b*x + a) - 20790*(e*x + d)*b^6*d^5*sgn(b*x + a) + 3003*b^6*d^6*sgn(b*x + a) + 54054*(e*x + d)^5*a *b^5*e*sgn(b*x + a) - 193050*(e*x + d)^4*a*b^5*d*e*sgn(b*x + a) + 300300*( e*x + d)^3*a*b^5*d^2*e*sgn(b*x + a) - 245700*(e*x + d)^2*a*b^5*d^3*e*sgn(b *x + a) + 103950*(e*x + d)*a*b^5*d^4*e*sgn(b*x + a) - 18018*a*b^5*d^5*e*sg n(b*x + a) + 96525*(e*x + d)^4*a^2*b^4*e^2*sgn(b*x + a) - 300300*(e*x + d) ^3*a^2*b^4*d*e^2*sgn(b*x + a) + 368550*(e*x + d)^2*a^2*b^4*d^2*e^2*sgn(b*x + a) - 207900*(e*x + d)*a^2*b^4*d^3*e^2*sgn(b*x + a) + 45045*a^2*b^4*d^4* e^2*sgn(b*x + a) + 100100*(e*x + d)^3*a^3*b^3*e^3*sgn(b*x + a) - 245700*(e *x + d)^2*a^3*b^3*d*e^3*sgn(b*x + a) + 207900*(e*x + d)*a^3*b^3*d^2*e^3*sg n(b*x + a) - 60060*a^3*b^3*d^3*e^3*sgn(b*x + a) + 61425*(e*x + d)^2*a^4*b^ 2*e^4*sgn(b*x + a) - 103950*(e*x + d)*a^4*b^2*d*e^4*sgn(b*x + a) + 45045*a ^4*b^2*d^2*e^4*sgn(b*x + a) + 20790*(e*x + d)*a^5*b*e^5*sgn(b*x + a) - 180 18*a^5*b*d*e^5*sgn(b*x + a) + 3003*a^6*e^6*sgn(b*x + a))/((e*x + d)^(15/2) *e^7)
Time = 12.80 (sec) , antiderivative size = 588, normalized size of antiderivative = 1.56 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{17/2}} \, dx=-\frac {\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}\,\left (\frac {6006\,a^6\,e^6+5544\,a^5\,b\,d\,e^5+5040\,a^4\,b^2\,d^2\,e^4+4480\,a^3\,b^3\,d^3\,e^3+3840\,a^2\,b^4\,d^4\,e^2+3072\,a\,b^5\,d^5\,e+2048\,b^6\,d^6}{45045\,b\,e^{14}}+\frac {2\,b^5\,x^6}{3\,e^8}+\frac {x\,\left (41580\,a^5\,b\,e^6+37800\,a^4\,b^2\,d\,e^5+33600\,a^3\,b^3\,d^2\,e^4+28800\,a^2\,b^4\,d^3\,e^3+23040\,a\,b^5\,d^4\,e^2+15360\,b^6\,d^5\,e\right )}{45045\,b\,e^{14}}+\frac {8\,b^2\,x^3\,\left (35\,a^3\,e^3+30\,a^2\,b\,d\,e^2+24\,a\,b^2\,d^2\,e+16\,b^3\,d^3\right )}{63\,e^{11}}+\frac {2\,b\,x^2\,\left (315\,a^4\,e^4+280\,a^3\,b\,d\,e^3+240\,a^2\,b^2\,d^2\,e^2+192\,a\,b^3\,d^3\,e+128\,b^4\,d^4\right )}{231\,e^{12}}+\frac {4\,b^4\,x^5\,\left (3\,a\,e+2\,b\,d\right )}{5\,e^9}+\frac {2\,b^3\,x^4\,\left (15\,a^2\,e^2+12\,a\,b\,d\,e+8\,b^2\,d^2\right )}{7\,e^{10}}\right )}{x^8\,\sqrt {d+e\,x}+\frac {a\,d^7\,\sqrt {d+e\,x}}{b\,e^7}+\frac {x^7\,\left (a\,e+7\,b\,d\right )\,\sqrt {d+e\,x}}{b\,e}+\frac {7\,d\,x^6\,\left (a\,e+3\,b\,d\right )\,\sqrt {d+e\,x}}{b\,e^2}+\frac {d^6\,x\,\left (7\,a\,e+b\,d\right )\,\sqrt {d+e\,x}}{b\,e^7}+\frac {35\,d^3\,x^4\,\left (a\,e+b\,d\right )\,\sqrt {d+e\,x}}{b\,e^4}+\frac {7\,d^5\,x^2\,\left (3\,a\,e+b\,d\right )\,\sqrt {d+e\,x}}{b\,e^6}+\frac {7\,d^2\,x^5\,\left (3\,a\,e+5\,b\,d\right )\,\sqrt {d+e\,x}}{b\,e^3}+\frac {7\,d^4\,x^3\,\left (5\,a\,e+3\,b\,d\right )\,\sqrt {d+e\,x}}{b\,e^5}} \] Input:
int(((a + b*x)*(a^2 + b^2*x^2 + 2*a*b*x)^(5/2))/(d + e*x)^(17/2),x)
Output:
-((a^2 + b^2*x^2 + 2*a*b*x)^(1/2)*((6006*a^6*e^6 + 2048*b^6*d^6 + 3840*a^2 *b^4*d^4*e^2 + 4480*a^3*b^3*d^3*e^3 + 5040*a^4*b^2*d^2*e^4 + 3072*a*b^5*d^ 5*e + 5544*a^5*b*d*e^5)/(45045*b*e^14) + (2*b^5*x^6)/(3*e^8) + (x*(41580*a ^5*b*e^6 + 15360*b^6*d^5*e + 23040*a*b^5*d^4*e^2 + 37800*a^4*b^2*d*e^5 + 2 8800*a^2*b^4*d^3*e^3 + 33600*a^3*b^3*d^2*e^4))/(45045*b*e^14) + (8*b^2*x^3 *(35*a^3*e^3 + 16*b^3*d^3 + 24*a*b^2*d^2*e + 30*a^2*b*d*e^2))/(63*e^11) + (2*b*x^2*(315*a^4*e^4 + 128*b^4*d^4 + 240*a^2*b^2*d^2*e^2 + 192*a*b^3*d^3* e + 280*a^3*b*d*e^3))/(231*e^12) + (4*b^4*x^5*(3*a*e + 2*b*d))/(5*e^9) + ( 2*b^3*x^4*(15*a^2*e^2 + 8*b^2*d^2 + 12*a*b*d*e))/(7*e^10)))/(x^8*(d + e*x) ^(1/2) + (a*d^7*(d + e*x)^(1/2))/(b*e^7) + (x^7*(a*e + 7*b*d)*(d + e*x)^(1 /2))/(b*e) + (7*d*x^6*(a*e + 3*b*d)*(d + e*x)^(1/2))/(b*e^2) + (d^6*x*(7*a *e + b*d)*(d + e*x)^(1/2))/(b*e^7) + (35*d^3*x^4*(a*e + b*d)*(d + e*x)^(1/ 2))/(b*e^4) + (7*d^5*x^2*(3*a*e + b*d)*(d + e*x)^(1/2))/(b*e^6) + (7*d^2*x ^5*(3*a*e + 5*b*d)*(d + e*x)^(1/2))/(b*e^3) + (7*d^4*x^3*(5*a*e + 3*b*d)*( d + e*x)^(1/2))/(b*e^5))
Time = 0.20 (sec) , antiderivative size = 450, normalized size of antiderivative = 1.20 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{17/2}} \, dx=\frac {-\frac {8}{65} a^{5} b d \,e^{5}-\frac {12}{13} a^{5} b \,e^{6} x -\frac {16}{143} a^{4} b^{2} d^{2} e^{4}-\frac {30}{11} a^{4} b^{2} e^{6} x^{2}-\frac {128}{1287} a^{3} b^{3} d^{3} e^{3}-\frac {40}{9} a^{3} b^{3} e^{6} x^{3}-\frac {256}{3003} a^{2} b^{4} d^{4} e^{2}-\frac {1024}{15015} a \,b^{5} d^{5} e -\frac {12}{5} a \,b^{5} e^{6} x^{5}-\frac {1024}{3003} b^{6} d^{5} e x -\frac {256}{231} b^{6} d^{4} e^{2} x^{2}-\frac {128}{63} b^{6} d^{3} e^{3} x^{3}-\frac {8}{5} b^{6} d \,e^{5} x^{5}-\frac {2}{3} b^{6} e^{6} x^{6}-\frac {120}{143} a^{4} b^{2} d \,e^{5} x -\frac {320}{429} a^{3} b^{3} d^{2} e^{4} x -\frac {80}{33} a^{3} b^{3} d \,e^{5} x^{2}-\frac {640}{1001} a^{2} b^{4} d^{3} e^{3} x -\frac {160}{77} a^{2} b^{4} d^{2} e^{4} x^{2}-\frac {80}{21} a^{2} b^{4} d \,e^{5} x^{3}-\frac {512}{1001} a \,b^{5} d^{4} e^{2} x -\frac {128}{77} a \,b^{5} d^{3} e^{3} x^{2}-\frac {64}{21} a \,b^{5} d^{2} e^{4} x^{3}-\frac {30}{7} a^{2} b^{4} e^{6} x^{4}-\frac {16}{7} b^{6} d^{2} e^{4} x^{4}-\frac {24}{7} a \,b^{5} d \,e^{5} x^{4}-\frac {2}{15} a^{6} e^{6}-\frac {2048}{45045} b^{6} d^{6}}{\sqrt {e x +d}\, e^{7} \left (e^{7} x^{7}+7 d \,e^{6} x^{6}+21 d^{2} e^{5} x^{5}+35 d^{3} e^{4} x^{4}+35 d^{4} e^{3} x^{3}+21 d^{5} e^{2} x^{2}+7 d^{6} e x +d^{7}\right )} \] Input:
int((b*x+a)*(b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^(17/2),x)
Output:
(2*( - 3003*a**6*e**6 - 2772*a**5*b*d*e**5 - 20790*a**5*b*e**6*x - 2520*a* *4*b**2*d**2*e**4 - 18900*a**4*b**2*d*e**5*x - 61425*a**4*b**2*e**6*x**2 - 2240*a**3*b**3*d**3*e**3 - 16800*a**3*b**3*d**2*e**4*x - 54600*a**3*b**3* d*e**5*x**2 - 100100*a**3*b**3*e**6*x**3 - 1920*a**2*b**4*d**4*e**2 - 1440 0*a**2*b**4*d**3*e**3*x - 46800*a**2*b**4*d**2*e**4*x**2 - 85800*a**2*b**4 *d*e**5*x**3 - 96525*a**2*b**4*e**6*x**4 - 1536*a*b**5*d**5*e - 11520*a*b* *5*d**4*e**2*x - 37440*a*b**5*d**3*e**3*x**2 - 68640*a*b**5*d**2*e**4*x**3 - 77220*a*b**5*d*e**5*x**4 - 54054*a*b**5*e**6*x**5 - 1024*b**6*d**6 - 76 80*b**6*d**5*e*x - 24960*b**6*d**4*e**2*x**2 - 45760*b**6*d**3*e**3*x**3 - 51480*b**6*d**2*e**4*x**4 - 36036*b**6*d*e**5*x**5 - 15015*b**6*e**6*x**6 ))/(45045*sqrt(d + e*x)*e**7*(d**7 + 7*d**6*e*x + 21*d**5*e**2*x**2 + 35*d **4*e**3*x**3 + 35*d**3*e**4*x**4 + 21*d**2*e**5*x**5 + 7*d*e**6*x**6 + e* *7*x**7))