Integrand size = 35, antiderivative size = 370 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{11/2}} \, dx=-\frac {2 (b d-a e)^6 \sqrt {a^2+2 a b x+b^2 x^2}}{9 e^7 (a+b x) (d+e x)^{9/2}}+\frac {12 b (b d-a e)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{7 e^7 (a+b x) (d+e x)^{7/2}}-\frac {6 b^2 (b d-a e)^4 \sqrt {a^2+2 a b x+b^2 x^2}}{e^7 (a+b x) (d+e x)^{5/2}}+\frac {40 b^3 (b d-a e)^3 \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^7 (a+b x) (d+e x)^{3/2}}-\frac {30 b^4 (b d-a e)^2 \sqrt {a^2+2 a b x+b^2 x^2}}{e^7 (a+b x) \sqrt {d+e x}}-\frac {12 b^5 (b d-a e) \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}{e^7 (a+b x)}+\frac {2 b^6 (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^7 (a+b x)} \] Output:
-2/9*(-a*e+b*d)^6*((b*x+a)^2)^(1/2)/e^7/(b*x+a)/(e*x+d)^(9/2)+12/7*b*(-a*e +b*d)^5*((b*x+a)^2)^(1/2)/e^7/(b*x+a)/(e*x+d)^(7/2)-6*b^2*(-a*e+b*d)^4*((b *x+a)^2)^(1/2)/e^7/(b*x+a)/(e*x+d)^(5/2)+40/3*b^3*(-a*e+b*d)^3*((b*x+a)^2) ^(1/2)/e^7/(b*x+a)/(e*x+d)^(3/2)-30*b^4*(-a*e+b*d)^2*((b*x+a)^2)^(1/2)/e^7 /(b*x+a)/(e*x+d)^(1/2)-12*b^5*(-a*e+b*d)*(e*x+d)^(1/2)*((b*x+a)^2)^(1/2)/e ^7/(b*x+a)+2/3*b^6*(e*x+d)^(3/2)*((b*x+a)^2)^(1/2)/e^7/(b*x+a)
Time = 0.30 (sec) , antiderivative size = 309, normalized size of antiderivative = 0.84 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{11/2}} \, dx=-\frac {2 \sqrt {(a+b x)^2} \left (7 a^6 e^6+6 a^5 b e^5 (2 d+9 e x)+3 a^4 b^2 e^4 \left (8 d^2+36 d e x+63 e^2 x^2\right )+4 a^3 b^3 e^3 \left (16 d^3+72 d^2 e x+126 d e^2 x^2+105 e^3 x^3\right )+3 a^2 b^4 e^2 \left (128 d^4+576 d^3 e x+1008 d^2 e^2 x^2+840 d e^3 x^3+315 e^4 x^4\right )-6 a b^5 e \left (256 d^5+1152 d^4 e x+2016 d^3 e^2 x^2+1680 d^2 e^3 x^3+630 d e^4 x^4+63 e^5 x^5\right )+b^6 \left (1024 d^6+4608 d^5 e x+8064 d^4 e^2 x^2+6720 d^3 e^3 x^3+2520 d^2 e^4 x^4+252 d e^5 x^5-21 e^6 x^6\right )\right )}{63 e^7 (a+b x) (d+e x)^{9/2}} \] Input:
Integrate[((a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/(d + e*x)^(11/2),x]
Output:
(-2*Sqrt[(a + b*x)^2]*(7*a^6*e^6 + 6*a^5*b*e^5*(2*d + 9*e*x) + 3*a^4*b^2*e ^4*(8*d^2 + 36*d*e*x + 63*e^2*x^2) + 4*a^3*b^3*e^3*(16*d^3 + 72*d^2*e*x + 126*d*e^2*x^2 + 105*e^3*x^3) + 3*a^2*b^4*e^2*(128*d^4 + 576*d^3*e*x + 1008 *d^2*e^2*x^2 + 840*d*e^3*x^3 + 315*e^4*x^4) - 6*a*b^5*e*(256*d^5 + 1152*d^ 4*e*x + 2016*d^3*e^2*x^2 + 1680*d^2*e^3*x^3 + 630*d*e^4*x^4 + 63*e^5*x^5) + b^6*(1024*d^6 + 4608*d^5*e*x + 8064*d^4*e^2*x^2 + 6720*d^3*e^3*x^3 + 252 0*d^2*e^4*x^4 + 252*d*e^5*x^5 - 21*e^6*x^6)))/(63*e^7*(a + b*x)*(d + e*x)^ (9/2))
Time = 0.59 (sec) , antiderivative size = 209, normalized size of antiderivative = 0.56, 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.
| \(\displaystyle \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{11/2}} \, dx\) |
| \(\Big \downarrow \) 1187 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {b^5 (a+b x)^6}{(d+e x)^{11/2}}dx}{b^5 (a+b x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {(a+b x)^6}{(d+e x)^{11/2}}dx}{a+b x}\) |
| \(\Big \downarrow \) 53 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (\frac {\sqrt {d+e x} b^6}{e^6}-\frac {6 (b d-a e) b^5}{e^6 \sqrt {d+e x}}+\frac {15 (b d-a e)^2 b^4}{e^6 (d+e x)^{3/2}}-\frac {20 (b d-a e)^3 b^3}{e^6 (d+e x)^{5/2}}+\frac {15 (b d-a e)^4 b^2}{e^6 (d+e x)^{7/2}}-\frac {6 (b d-a e)^5 b}{e^6 (d+e x)^{9/2}}+\frac {(a e-b d)^6}{e^6 (d+e x)^{11/2}}\right )dx}{a+b x}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \left (-\frac {12 b^5 \sqrt {d+e x} (b d-a e)}{e^7}-\frac {30 b^4 (b d-a e)^2}{e^7 \sqrt {d+e x}}+\frac {40 b^3 (b d-a e)^3}{3 e^7 (d+e x)^{3/2}}-\frac {6 b^2 (b d-a e)^4}{e^7 (d+e x)^{5/2}}+\frac {12 b (b d-a e)^5}{7 e^7 (d+e x)^{7/2}}-\frac {2 (b d-a e)^6}{9 e^7 (d+e x)^{9/2}}+\frac {2 b^6 (d+e x)^{3/2}}{3 e^7}\right )}{a+b x}\) |
Input:
Int[((a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/(d + e*x)^(11/2),x]
Output:
(Sqrt[a^2 + 2*a*b*x + b^2*x^2]*((-2*(b*d - a*e)^6)/(9*e^7*(d + e*x)^(9/2)) + (12*b*(b*d - a*e)^5)/(7*e^7*(d + e*x)^(7/2)) - (6*b^2*(b*d - a*e)^4)/(e ^7*(d + e*x)^(5/2)) + (40*b^3*(b*d - a*e)^3)/(3*e^7*(d + e*x)^(3/2)) - (30 *b^4*(b*d - a*e)^2)/(e^7*Sqrt[d + e*x]) - (12*b^5*(b*d - a*e)*Sqrt[d + e*x ])/e^7 + (2*b^6*(d + e*x)^(3/2))/(3*e^7)))/(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 = 1.30 (sec) , antiderivative size = 308, normalized size of antiderivative = 0.83
| method | result | size |
| risch | \(\frac {2 b^{5} \left (b e x +18 a e -17 b d \right ) \sqrt {e x +d}\, \sqrt {\left (b x +a \right )^{2}}}{3 e^{7} \left (b x +a \right )}-\frac {2 \left (945 b^{4} x^{4} e^{4}+420 x^{3} a \,b^{3} e^{4}+3360 x^{3} b^{4} d \,e^{3}+189 x^{2} a^{2} b^{2} e^{4}+882 x^{2} a \,b^{3} d \,e^{3}+4599 x^{2} b^{4} d^{2} e^{2}+54 x \,a^{3} b \,e^{4}+216 x \,a^{2} b^{2} d \,e^{3}+666 x a \,b^{3} d^{2} e^{2}+2844 x \,b^{4} d^{3} e +7 a^{4} e^{4}+26 a^{3} b d \,e^{3}+69 a^{2} b^{2} d^{2} e^{2}+176 a \,b^{3} d^{3} e +667 b^{4} d^{4}\right ) \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) \sqrt {\left (b x +a \right )^{2}}}{63 e^{7} \sqrt {e x +d}\, \left (e^{4} x^{4}+4 d \,e^{3} x^{3}+6 d^{2} e^{2} x^{2}+4 d^{3} e x +d^{4}\right ) \left (b x +a \right )}\) | \(308\) |
| gosper | \(-\frac {2 \left (-21 b^{6} x^{6} e^{6}-378 x^{5} a \,b^{5} e^{6}+252 x^{5} b^{6} d \,e^{5}+945 x^{4} a^{2} b^{4} e^{6}-3780 x^{4} a \,b^{5} d \,e^{5}+2520 x^{4} b^{6} d^{2} e^{4}+420 x^{3} a^{3} b^{3} e^{6}+2520 x^{3} a^{2} b^{4} d \,e^{5}-10080 x^{3} a \,b^{5} d^{2} e^{4}+6720 x^{3} b^{6} d^{3} e^{3}+189 x^{2} a^{4} b^{2} e^{6}+504 x^{2} a^{3} b^{3} d \,e^{5}+3024 x^{2} a^{2} b^{4} d^{2} e^{4}-12096 x^{2} a \,b^{5} d^{3} e^{3}+8064 x^{2} b^{6} d^{4} e^{2}+54 x \,a^{5} b \,e^{6}+108 x \,a^{4} b^{2} d \,e^{5}+288 x \,a^{3} b^{3} d^{2} e^{4}+1728 x \,a^{2} b^{4} d^{3} e^{3}-6912 x a \,b^{5} d^{4} e^{2}+4608 x \,b^{6} d^{5} e +7 a^{6} e^{6}+12 a^{5} b d \,e^{5}+24 a^{4} b^{2} d^{2} e^{4}+64 a^{3} b^{3} d^{3} e^{3}+384 a^{2} b^{4} d^{4} e^{2}-1536 a \,b^{5} d^{5} e +1024 b^{6} d^{6}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{63 \left (e x +d \right )^{\frac {9}{2}} e^{7} \left (b x +a \right )^{5}}\) | \(393\) |
| default | \(-\frac {2 \left (-21 b^{6} x^{6} e^{6}-378 x^{5} a \,b^{5} e^{6}+252 x^{5} b^{6} d \,e^{5}+945 x^{4} a^{2} b^{4} e^{6}-3780 x^{4} a \,b^{5} d \,e^{5}+2520 x^{4} b^{6} d^{2} e^{4}+420 x^{3} a^{3} b^{3} e^{6}+2520 x^{3} a^{2} b^{4} d \,e^{5}-10080 x^{3} a \,b^{5} d^{2} e^{4}+6720 x^{3} b^{6} d^{3} e^{3}+189 x^{2} a^{4} b^{2} e^{6}+504 x^{2} a^{3} b^{3} d \,e^{5}+3024 x^{2} a^{2} b^{4} d^{2} e^{4}-12096 x^{2} a \,b^{5} d^{3} e^{3}+8064 x^{2} b^{6} d^{4} e^{2}+54 x \,a^{5} b \,e^{6}+108 x \,a^{4} b^{2} d \,e^{5}+288 x \,a^{3} b^{3} d^{2} e^{4}+1728 x \,a^{2} b^{4} d^{3} e^{3}-6912 x a \,b^{5} d^{4} e^{2}+4608 x \,b^{6} d^{5} e +7 a^{6} e^{6}+12 a^{5} b d \,e^{5}+24 a^{4} b^{2} d^{2} e^{4}+64 a^{3} b^{3} d^{3} e^{3}+384 a^{2} b^{4} d^{4} e^{2}-1536 a \,b^{5} d^{5} e +1024 b^{6} d^{6}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{63 \left (e x +d \right )^{\frac {9}{2}} e^{7} \left (b x +a \right )^{5}}\) | \(393\) |
| orering | \(-\frac {2 \left (-21 b^{6} x^{6} e^{6}-378 x^{5} a \,b^{5} e^{6}+252 x^{5} b^{6} d \,e^{5}+945 x^{4} a^{2} b^{4} e^{6}-3780 x^{4} a \,b^{5} d \,e^{5}+2520 x^{4} b^{6} d^{2} e^{4}+420 x^{3} a^{3} b^{3} e^{6}+2520 x^{3} a^{2} b^{4} d \,e^{5}-10080 x^{3} a \,b^{5} d^{2} e^{4}+6720 x^{3} b^{6} d^{3} e^{3}+189 x^{2} a^{4} b^{2} e^{6}+504 x^{2} a^{3} b^{3} d \,e^{5}+3024 x^{2} a^{2} b^{4} d^{2} e^{4}-12096 x^{2} a \,b^{5} d^{3} e^{3}+8064 x^{2} b^{6} d^{4} e^{2}+54 x \,a^{5} b \,e^{6}+108 x \,a^{4} b^{2} d \,e^{5}+288 x \,a^{3} b^{3} d^{2} e^{4}+1728 x \,a^{2} b^{4} d^{3} e^{3}-6912 x a \,b^{5} d^{4} e^{2}+4608 x \,b^{6} d^{5} e +7 a^{6} e^{6}+12 a^{5} b d \,e^{5}+24 a^{4} b^{2} d^{2} e^{4}+64 a^{3} b^{3} d^{3} e^{3}+384 a^{2} b^{4} d^{4} e^{2}-1536 a \,b^{5} d^{5} e +1024 b^{6} d^{6}\right ) \left (b^{2} x^{2}+2 a b x +a^{2}\right )^{\frac {5}{2}}}{63 e^{7} \left (b x +a \right )^{5} \left (e x +d \right )^{\frac {9}{2}}}\) | \(402\) |
Input:
int((b*x+a)*(b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^(11/2),x,method=_RETURNVER BOSE)
Output:
2/3*b^5*(b*e*x+18*a*e-17*b*d)*(e*x+d)^(1/2)/e^7*((b*x+a)^2)^(1/2)/(b*x+a)- 2/63*(945*b^4*e^4*x^4+420*a*b^3*e^4*x^3+3360*b^4*d*e^3*x^3+189*a^2*b^2*e^4 *x^2+882*a*b^3*d*e^3*x^2+4599*b^4*d^2*e^2*x^2+54*a^3*b*e^4*x+216*a^2*b^2*d *e^3*x+666*a*b^3*d^2*e^2*x+2844*b^4*d^3*e*x+7*a^4*e^4+26*a^3*b*d*e^3+69*a^ 2*b^2*d^2*e^2+176*a*b^3*d^3*e+667*b^4*d^4)*(a^2*e^2-2*a*b*d*e+b^2*d^2)/e^7 /(e*x+d)^(1/2)/(e^4*x^4+4*d*e^3*x^3+6*d^2*e^2*x^2+4*d^3*e*x+d^4)*((b*x+a)^ 2)^(1/2)/(b*x+a)
Time = 0.10 (sec) , antiderivative size = 409, normalized size of antiderivative = 1.11 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{11/2}} \, dx=\frac {2 \, {\left (21 \, b^{6} e^{6} x^{6} - 1024 \, b^{6} d^{6} + 1536 \, a b^{5} d^{5} e - 384 \, a^{2} b^{4} d^{4} e^{2} - 64 \, a^{3} b^{3} d^{3} e^{3} - 24 \, a^{4} b^{2} d^{2} e^{4} - 12 \, a^{5} b d e^{5} - 7 \, a^{6} e^{6} - 126 \, {\left (2 \, b^{6} d e^{5} - 3 \, a b^{5} e^{6}\right )} x^{5} - 315 \, {\left (8 \, b^{6} d^{2} e^{4} - 12 \, a b^{5} d e^{5} + 3 \, a^{2} b^{4} e^{6}\right )} x^{4} - 420 \, {\left (16 \, b^{6} d^{3} e^{3} - 24 \, a b^{5} d^{2} e^{4} + 6 \, a^{2} b^{4} d e^{5} + a^{3} b^{3} e^{6}\right )} x^{3} - 63 \, {\left (128 \, b^{6} d^{4} e^{2} - 192 \, a b^{5} d^{3} e^{3} + 48 \, a^{2} b^{4} d^{2} e^{4} + 8 \, a^{3} b^{3} d e^{5} + 3 \, a^{4} b^{2} e^{6}\right )} x^{2} - 18 \, {\left (256 \, b^{6} d^{5} e - 384 \, a b^{5} d^{4} e^{2} + 96 \, a^{2} b^{4} d^{3} e^{3} + 16 \, a^{3} b^{3} d^{2} e^{4} + 6 \, a^{4} b^{2} d e^{5} + 3 \, a^{5} b e^{6}\right )} x\right )} \sqrt {e x + d}}{63 \, {\left (e^{12} x^{5} + 5 \, d e^{11} x^{4} + 10 \, d^{2} e^{10} x^{3} + 10 \, d^{3} e^{9} x^{2} + 5 \, d^{4} e^{8} x + d^{5} e^{7}\right )}} \] Input:
integrate((b*x+a)*(b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^(11/2),x, algorithm= "fricas")
Output:
2/63*(21*b^6*e^6*x^6 - 1024*b^6*d^6 + 1536*a*b^5*d^5*e - 384*a^2*b^4*d^4*e ^2 - 64*a^3*b^3*d^3*e^3 - 24*a^4*b^2*d^2*e^4 - 12*a^5*b*d*e^5 - 7*a^6*e^6 - 126*(2*b^6*d*e^5 - 3*a*b^5*e^6)*x^5 - 315*(8*b^6*d^2*e^4 - 12*a*b^5*d*e^ 5 + 3*a^2*b^4*e^6)*x^4 - 420*(16*b^6*d^3*e^3 - 24*a*b^5*d^2*e^4 + 6*a^2*b^ 4*d*e^5 + a^3*b^3*e^6)*x^3 - 63*(128*b^6*d^4*e^2 - 192*a*b^5*d^3*e^3 + 48* a^2*b^4*d^2*e^4 + 8*a^3*b^3*d*e^5 + 3*a^4*b^2*e^6)*x^2 - 18*(256*b^6*d^5*e - 384*a*b^5*d^4*e^2 + 96*a^2*b^4*d^3*e^3 + 16*a^3*b^3*d^2*e^4 + 6*a^4*b^2 *d*e^5 + 3*a^5*b*e^6)*x)*sqrt(e*x + d)/(e^12*x^5 + 5*d*e^11*x^4 + 10*d^2*e ^10*x^3 + 10*d^3*e^9*x^2 + 5*d^4*e^8*x + d^5*e^7)
Timed out. \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{11/2}} \, dx=\text {Timed out} \] Input:
integrate((b*x+a)*(b**2*x**2+2*a*b*x+a**2)**(5/2)/(e*x+d)**(11/2),x)
Output:
Timed out
Leaf count of result is larger than twice the leaf count of optimal. 687 vs. \(2 (271) = 542\).
Time = 0.09 (sec) , antiderivative size = 687, normalized size of antiderivative = 1.86 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{11/2}} \, dx=\frac {2 \, {\left (63 \, b^{5} e^{5} x^{5} + 256 \, b^{5} d^{5} - 128 \, a b^{4} d^{4} e - 32 \, a^{2} b^{3} d^{3} e^{2} - 16 \, a^{3} b^{2} d^{2} e^{3} - 10 \, a^{4} b d e^{4} - 7 \, a^{5} e^{5} + 315 \, {\left (2 \, b^{5} d e^{4} - a b^{4} e^{5}\right )} x^{4} + 210 \, {\left (8 \, b^{5} d^{2} e^{3} - 4 \, a b^{4} d e^{4} - a^{2} b^{3} e^{5}\right )} x^{3} + 126 \, {\left (16 \, b^{5} d^{3} e^{2} - 8 \, a b^{4} d^{2} e^{3} - 2 \, a^{2} b^{3} d e^{4} - a^{3} b^{2} e^{5}\right )} x^{2} + 9 \, {\left (128 \, b^{5} d^{4} e - 64 \, a b^{4} d^{3} e^{2} - 16 \, a^{2} b^{3} d^{2} e^{3} - 8 \, a^{3} b^{2} d e^{4} - 5 \, a^{4} b e^{5}\right )} x\right )} a}{63 \, {\left (e^{10} x^{4} + 4 \, d e^{9} x^{3} + 6 \, d^{2} e^{8} x^{2} + 4 \, d^{3} e^{7} x + d^{4} e^{6}\right )} \sqrt {e x + d}} + \frac {2 \, {\left (21 \, b^{5} e^{6} x^{6} - 1024 \, b^{5} d^{6} + 1280 \, a b^{4} d^{5} e - 256 \, a^{2} b^{3} d^{4} e^{2} - 32 \, a^{3} b^{2} d^{3} e^{3} - 8 \, a^{4} b d^{2} e^{4} - 2 \, a^{5} d e^{5} - 63 \, {\left (4 \, b^{5} d e^{5} - 5 \, a b^{4} e^{6}\right )} x^{5} - 630 \, {\left (4 \, b^{5} d^{2} e^{4} - 5 \, a b^{4} d e^{5} + a^{2} b^{3} e^{6}\right )} x^{4} - 210 \, {\left (32 \, b^{5} d^{3} e^{3} - 40 \, a b^{4} d^{2} e^{4} + 8 \, a^{2} b^{3} d e^{5} + a^{3} b^{2} e^{6}\right )} x^{3} - 63 \, {\left (128 \, b^{5} d^{4} e^{2} - 160 \, a b^{4} d^{3} e^{3} + 32 \, a^{2} b^{3} d^{2} e^{4} + 4 \, a^{3} b^{2} d e^{5} + a^{4} b e^{6}\right )} x^{2} - 9 \, {\left (512 \, b^{5} d^{5} e - 640 \, a b^{4} d^{4} e^{2} + 128 \, a^{2} b^{3} d^{3} e^{3} + 16 \, a^{3} b^{2} d^{2} e^{4} + 4 \, a^{4} b d e^{5} + a^{5} e^{6}\right )} x\right )} b}{63 \, {\left (e^{11} x^{4} + 4 \, d e^{10} x^{3} + 6 \, d^{2} e^{9} x^{2} + 4 \, d^{3} e^{8} x + d^{4} e^{7}\right )} \sqrt {e x + d}} \] Input:
integrate((b*x+a)*(b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^(11/2),x, algorithm= "maxima")
Output:
2/63*(63*b^5*e^5*x^5 + 256*b^5*d^5 - 128*a*b^4*d^4*e - 32*a^2*b^3*d^3*e^2 - 16*a^3*b^2*d^2*e^3 - 10*a^4*b*d*e^4 - 7*a^5*e^5 + 315*(2*b^5*d*e^4 - a*b ^4*e^5)*x^4 + 210*(8*b^5*d^2*e^3 - 4*a*b^4*d*e^4 - a^2*b^3*e^5)*x^3 + 126* (16*b^5*d^3*e^2 - 8*a*b^4*d^2*e^3 - 2*a^2*b^3*d*e^4 - a^3*b^2*e^5)*x^2 + 9 *(128*b^5*d^4*e - 64*a*b^4*d^3*e^2 - 16*a^2*b^3*d^2*e^3 - 8*a^3*b^2*d*e^4 - 5*a^4*b*e^5)*x)*a/((e^10*x^4 + 4*d*e^9*x^3 + 6*d^2*e^8*x^2 + 4*d^3*e^7*x + d^4*e^6)*sqrt(e*x + d)) + 2/63*(21*b^5*e^6*x^6 - 1024*b^5*d^6 + 1280*a* b^4*d^5*e - 256*a^2*b^3*d^4*e^2 - 32*a^3*b^2*d^3*e^3 - 8*a^4*b*d^2*e^4 - 2 *a^5*d*e^5 - 63*(4*b^5*d*e^5 - 5*a*b^4*e^6)*x^5 - 630*(4*b^5*d^2*e^4 - 5*a *b^4*d*e^5 + a^2*b^3*e^6)*x^4 - 210*(32*b^5*d^3*e^3 - 40*a*b^4*d^2*e^4 + 8 *a^2*b^3*d*e^5 + a^3*b^2*e^6)*x^3 - 63*(128*b^5*d^4*e^2 - 160*a*b^4*d^3*e^ 3 + 32*a^2*b^3*d^2*e^4 + 4*a^3*b^2*d*e^5 + a^4*b*e^6)*x^2 - 9*(512*b^5*d^5 *e - 640*a*b^4*d^4*e^2 + 128*a^2*b^3*d^3*e^3 + 16*a^3*b^2*d^2*e^4 + 4*a^4* b*d*e^5 + a^5*e^6)*x)*b/((e^11*x^4 + 4*d*e^10*x^3 + 6*d^2*e^9*x^2 + 4*d^3* e^8*x + d^4*e^7)*sqrt(e*x + d))
Leaf count of result is larger than twice the leaf count of optimal. 616 vs. \(2 (271) = 542\).
Time = 0.21 (sec) , antiderivative size = 616, normalized size of antiderivative = 1.66 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{11/2}} \, dx =\text {Too large to display} \] Input:
integrate((b*x+a)*(b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^(11/2),x, algorithm= "giac")
Output:
-2/63*(945*(e*x + d)^4*b^6*d^2*sgn(b*x + a) - 420*(e*x + d)^3*b^6*d^3*sgn( b*x + a) + 189*(e*x + d)^2*b^6*d^4*sgn(b*x + a) - 54*(e*x + d)*b^6*d^5*sgn (b*x + a) + 7*b^6*d^6*sgn(b*x + a) - 1890*(e*x + d)^4*a*b^5*d*e*sgn(b*x + a) + 1260*(e*x + d)^3*a*b^5*d^2*e*sgn(b*x + a) - 756*(e*x + d)^2*a*b^5*d^3 *e*sgn(b*x + a) + 270*(e*x + d)*a*b^5*d^4*e*sgn(b*x + a) - 42*a*b^5*d^5*e* sgn(b*x + a) + 945*(e*x + d)^4*a^2*b^4*e^2*sgn(b*x + a) - 1260*(e*x + d)^3 *a^2*b^4*d*e^2*sgn(b*x + a) + 1134*(e*x + d)^2*a^2*b^4*d^2*e^2*sgn(b*x + a ) - 540*(e*x + d)*a^2*b^4*d^3*e^2*sgn(b*x + a) + 105*a^2*b^4*d^4*e^2*sgn(b *x + a) + 420*(e*x + d)^3*a^3*b^3*e^3*sgn(b*x + a) - 756*(e*x + d)^2*a^3*b ^3*d*e^3*sgn(b*x + a) + 540*(e*x + d)*a^3*b^3*d^2*e^3*sgn(b*x + a) - 140*a ^3*b^3*d^3*e^3*sgn(b*x + a) + 189*(e*x + d)^2*a^4*b^2*e^4*sgn(b*x + a) - 2 70*(e*x + d)*a^4*b^2*d*e^4*sgn(b*x + a) + 105*a^4*b^2*d^2*e^4*sgn(b*x + a) + 54*(e*x + d)*a^5*b*e^5*sgn(b*x + a) - 42*a^5*b*d*e^5*sgn(b*x + a) + 7*a ^6*e^6*sgn(b*x + a))/((e*x + d)^(9/2)*e^7) + 2/3*((e*x + d)^(3/2)*b^6*e^14 *sgn(b*x + a) - 18*sqrt(e*x + d)*b^6*d*e^14*sgn(b*x + a) + 18*sqrt(e*x + d )*a*b^5*e^15*sgn(b*x + a))/e^21
Time = 12.75 (sec) , antiderivative size = 508, normalized size of antiderivative = 1.37 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{11/2}} \, dx=-\frac {\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}\,\left (\frac {14\,a^6\,e^6+24\,a^5\,b\,d\,e^5+48\,a^4\,b^2\,d^2\,e^4+128\,a^3\,b^3\,d^3\,e^3+768\,a^2\,b^4\,d^4\,e^2-3072\,a\,b^5\,d^5\,e+2048\,b^6\,d^6}{63\,b\,e^{11}}-\frac {2\,b^5\,x^6}{3\,e^5}+\frac {x\,\left (108\,a^5\,b\,e^6+216\,a^4\,b^2\,d\,e^5+576\,a^3\,b^3\,d^2\,e^4+3456\,a^2\,b^4\,d^3\,e^3-13824\,a\,b^5\,d^4\,e^2+9216\,b^6\,d^5\,e\right )}{63\,b\,e^{11}}+\frac {40\,b^2\,x^3\,\left (a^3\,e^3+6\,a^2\,b\,d\,e^2-24\,a\,b^2\,d^2\,e+16\,b^3\,d^3\right )}{3\,e^8}+\frac {2\,b\,x^2\,\left (3\,a^4\,e^4+8\,a^3\,b\,d\,e^3+48\,a^2\,b^2\,d^2\,e^2-192\,a\,b^3\,d^3\,e+128\,b^4\,d^4\right )}{e^9}-\frac {4\,b^4\,x^5\,\left (3\,a\,e-2\,b\,d\right )}{e^6}+\frac {10\,b^3\,x^4\,\left (3\,a^2\,e^2-12\,a\,b\,d\,e+8\,b^2\,d^2\right )}{e^7}\right )}{x^5\,\sqrt {d+e\,x}+\frac {a\,d^4\,\sqrt {d+e\,x}}{b\,e^4}+\frac {x^4\,\left (63\,a\,e^{11}+252\,b\,d\,e^{10}\right )\,\sqrt {d+e\,x}}{63\,b\,e^{11}}+\frac {2\,d\,x^3\,\left (2\,a\,e+3\,b\,d\right )\,\sqrt {d+e\,x}}{b\,e^2}+\frac {d^3\,x\,\left (4\,a\,e+b\,d\right )\,\sqrt {d+e\,x}}{b\,e^4}+\frac {2\,d^2\,x^2\,\left (3\,a\,e+2\,b\,d\right )\,\sqrt {d+e\,x}}{b\,e^3}} \] Input:
int(((a + b*x)*(a^2 + b^2*x^2 + 2*a*b*x)^(5/2))/(d + e*x)^(11/2),x)
Output:
-((a^2 + b^2*x^2 + 2*a*b*x)^(1/2)*((14*a^6*e^6 + 2048*b^6*d^6 + 768*a^2*b^ 4*d^4*e^2 + 128*a^3*b^3*d^3*e^3 + 48*a^4*b^2*d^2*e^4 - 3072*a*b^5*d^5*e + 24*a^5*b*d*e^5)/(63*b*e^11) - (2*b^5*x^6)/(3*e^5) + (x*(108*a^5*b*e^6 + 92 16*b^6*d^5*e - 13824*a*b^5*d^4*e^2 + 216*a^4*b^2*d*e^5 + 3456*a^2*b^4*d^3* e^3 + 576*a^3*b^3*d^2*e^4))/(63*b*e^11) + (40*b^2*x^3*(a^3*e^3 + 16*b^3*d^ 3 - 24*a*b^2*d^2*e + 6*a^2*b*d*e^2))/(3*e^8) + (2*b*x^2*(3*a^4*e^4 + 128*b ^4*d^4 + 48*a^2*b^2*d^2*e^2 - 192*a*b^3*d^3*e + 8*a^3*b*d*e^3))/e^9 - (4*b ^4*x^5*(3*a*e - 2*b*d))/e^6 + (10*b^3*x^4*(3*a^2*e^2 + 8*b^2*d^2 - 12*a*b* d*e))/e^7))/(x^5*(d + e*x)^(1/2) + (a*d^4*(d + e*x)^(1/2))/(b*e^4) + (x^4* (63*a*e^11 + 252*b*d*e^10)*(d + e*x)^(1/2))/(63*b*e^11) + (2*d*x^3*(2*a*e + 3*b*d)*(d + e*x)^(1/2))/(b*e^2) + (d^3*x*(4*a*e + b*d)*(d + e*x)^(1/2))/ (b*e^4) + (2*d^2*x^2*(3*a*e + 2*b*d)*(d + e*x)^(1/2))/(b*e^3))
Time = 0.25 (sec) , antiderivative size = 417, normalized size of antiderivative = 1.13 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{11/2}} \, dx=\frac {-\frac {8}{21} a^{5} b d \,e^{5}-\frac {12}{7} a^{5} b \,e^{6} x -\frac {16}{21} a^{4} b^{2} d^{2} e^{4}-6 a^{4} b^{2} e^{6} x^{2}-\frac {128}{63} a^{3} b^{3} d^{3} e^{3}-\frac {40}{3} a^{3} b^{3} e^{6} x^{3}-\frac {256}{21} a^{2} b^{4} d^{4} e^{2}+\frac {1024}{21} a \,b^{5} d^{5} e +12 a \,b^{5} e^{6} x^{5}-\frac {1024}{7} b^{6} d^{5} e x -256 b^{6} d^{4} e^{2} x^{2}-\frac {640}{3} b^{6} d^{3} e^{3} x^{3}-8 b^{6} d \,e^{5} x^{5}+\frac {2}{3} b^{6} e^{6} x^{6}-\frac {24}{7} a^{4} b^{2} d \,e^{5} x -\frac {64}{7} a^{3} b^{3} d^{2} e^{4} x -16 a^{3} b^{3} d \,e^{5} x^{2}-\frac {384}{7} a^{2} b^{4} d^{3} e^{3} x -96 a^{2} b^{4} d^{2} e^{4} x^{2}-80 a^{2} b^{4} d \,e^{5} x^{3}+\frac {1536}{7} a \,b^{5} d^{4} e^{2} x +384 a \,b^{5} d^{3} e^{3} x^{2}+320 a \,b^{5} d^{2} e^{4} x^{3}-30 a^{2} b^{4} e^{6} x^{4}-80 b^{6} d^{2} e^{4} x^{4}+120 a \,b^{5} d \,e^{5} x^{4}-\frac {2}{9} a^{6} e^{6}-\frac {2048}{63} b^{6} d^{6}}{\sqrt {e x +d}\, e^{7} \left (e^{4} x^{4}+4 d \,e^{3} x^{3}+6 d^{2} e^{2} x^{2}+4 d^{3} e x +d^{4}\right )} \] Input:
int((b*x+a)*(b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^(11/2),x)
Output:
(2*( - 7*a**6*e**6 - 12*a**5*b*d*e**5 - 54*a**5*b*e**6*x - 24*a**4*b**2*d* *2*e**4 - 108*a**4*b**2*d*e**5*x - 189*a**4*b**2*e**6*x**2 - 64*a**3*b**3* d**3*e**3 - 288*a**3*b**3*d**2*e**4*x - 504*a**3*b**3*d*e**5*x**2 - 420*a* *3*b**3*e**6*x**3 - 384*a**2*b**4*d**4*e**2 - 1728*a**2*b**4*d**3*e**3*x - 3024*a**2*b**4*d**2*e**4*x**2 - 2520*a**2*b**4*d*e**5*x**3 - 945*a**2*b** 4*e**6*x**4 + 1536*a*b**5*d**5*e + 6912*a*b**5*d**4*e**2*x + 12096*a*b**5* d**3*e**3*x**2 + 10080*a*b**5*d**2*e**4*x**3 + 3780*a*b**5*d*e**5*x**4 + 3 78*a*b**5*e**6*x**5 - 1024*b**6*d**6 - 4608*b**6*d**5*e*x - 8064*b**6*d**4 *e**2*x**2 - 6720*b**6*d**3*e**3*x**3 - 2520*b**6*d**2*e**4*x**4 - 252*b** 6*d*e**5*x**5 + 21*b**6*e**6*x**6))/(63*sqrt(d + e*x)*e**7*(d**4 + 4*d**3* e*x + 6*d**2*e**2*x**2 + 4*d*e**3*x**3 + e**4*x**4))