Integrand size = 30, antiderivative size = 314 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{5/2}} \, dx=\frac {2 (b d-a e)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^6 (a+b x) (d+e x)^{3/2}}-\frac {10 b (b d-a e)^4 \sqrt {a^2+2 a b x+b^2 x^2}}{e^6 (a+b x) \sqrt {d+e x}}-\frac {20 b^2 (b d-a e)^3 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}{e^6 (a+b x)}+\frac {20 b^3 (b d-a e)^2 (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^6 (a+b x)}-\frac {2 b^4 (b d-a e) (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}{e^6 (a+b x)}+\frac {2 b^5 (d+e x)^{7/2} \sqrt {a^2+2 a b x+b^2 x^2}}{7 e^6 (a+b x)} \] Output:
2/3*(-a*e+b*d)^5*((b*x+a)^2)^(1/2)/e^6/(b*x+a)/(e*x+d)^(3/2)-10*b*(-a*e+b* d)^4*((b*x+a)^2)^(1/2)/e^6/(b*x+a)/(e*x+d)^(1/2)-20*b^2*(-a*e+b*d)^3*(e*x+ d)^(1/2)*((b*x+a)^2)^(1/2)/e^6/(b*x+a)+20/3*b^3*(-a*e+b*d)^2*(e*x+d)^(3/2) *((b*x+a)^2)^(1/2)/e^6/(b*x+a)-2*b^4*(-a*e+b*d)*(e*x+d)^(5/2)*((b*x+a)^2)^ (1/2)/e^6/(b*x+a)+2/7*b^5*(e*x+d)^(7/2)*((b*x+a)^2)^(1/2)/e^6/(b*x+a)
Time = 0.14 (sec) , antiderivative size = 235, normalized size of antiderivative = 0.75 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{5/2}} \, dx=-\frac {2 \sqrt {(a+b x)^2} \left (7 a^5 e^5+35 a^4 b e^4 (2 d+3 e x)-70 a^3 b^2 e^3 \left (8 d^2+12 d e x+3 e^2 x^2\right )+70 a^2 b^3 e^2 \left (16 d^3+24 d^2 e x+6 d e^2 x^2-e^3 x^3\right )-7 a b^4 e \left (128 d^4+192 d^3 e x+48 d^2 e^2 x^2-8 d e^3 x^3+3 e^4 x^4\right )+b^5 \left (256 d^5+384 d^4 e x+96 d^3 e^2 x^2-16 d^2 e^3 x^3+6 d e^4 x^4-3 e^5 x^5\right )\right )}{21 e^6 (a+b x) (d+e x)^{3/2}} \] Input:
Integrate[(a^2 + 2*a*b*x + b^2*x^2)^(5/2)/(d + e*x)^(5/2),x]
Output:
(-2*Sqrt[(a + b*x)^2]*(7*a^5*e^5 + 35*a^4*b*e^4*(2*d + 3*e*x) - 70*a^3*b^2 *e^3*(8*d^2 + 12*d*e*x + 3*e^2*x^2) + 70*a^2*b^3*e^2*(16*d^3 + 24*d^2*e*x + 6*d*e^2*x^2 - e^3*x^3) - 7*a*b^4*e*(128*d^4 + 192*d^3*e*x + 48*d^2*e^2*x ^2 - 8*d*e^3*x^3 + 3*e^4*x^4) + b^5*(256*d^5 + 384*d^4*e*x + 96*d^3*e^2*x^ 2 - 16*d^2*e^3*x^3 + 6*d*e^4*x^4 - 3*e^5*x^5)))/(21*e^6*(a + b*x)*(d + e*x )^(3/2))
Time = 0.53 (sec) , antiderivative size = 180, normalized size of antiderivative = 0.57, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {1102, 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.
| \(\displaystyle \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{5/2}} \, dx\) |
| \(\Big \downarrow \) 1102 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {b^5 (a+b x)^5}{(d+e x)^{5/2}}dx}{b^5 (a+b x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {(a+b x)^5}{(d+e x)^{5/2}}dx}{a+b x}\) |
| \(\Big \downarrow \) 53 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (\frac {(d+e x)^{5/2} b^5}{e^5}-\frac {5 (b d-a e) (d+e x)^{3/2} b^4}{e^5}+\frac {10 (b d-a e)^2 \sqrt {d+e x} b^3}{e^5}-\frac {10 (b d-a e)^3 b^2}{e^5 \sqrt {d+e x}}+\frac {5 (b d-a e)^4 b}{e^5 (d+e x)^{3/2}}+\frac {(a e-b d)^5}{e^5 (d+e x)^{5/2}}\right )dx}{a+b x}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \left (-\frac {2 b^4 (d+e x)^{5/2} (b d-a e)}{e^6}+\frac {20 b^3 (d+e x)^{3/2} (b d-a e)^2}{3 e^6}-\frac {20 b^2 \sqrt {d+e x} (b d-a e)^3}{e^6}-\frac {10 b (b d-a e)^4}{e^6 \sqrt {d+e x}}+\frac {2 (b d-a e)^5}{3 e^6 (d+e x)^{3/2}}+\frac {2 b^5 (d+e x)^{7/2}}{7 e^6}\right )}{a+b x}\) |
Input:
Int[(a^2 + 2*a*b*x + b^2*x^2)^(5/2)/(d + e*x)^(5/2),x]
Output:
(Sqrt[a^2 + 2*a*b*x + b^2*x^2]*((2*(b*d - a*e)^5)/(3*e^6*(d + e*x)^(3/2)) - (10*b*(b*d - a*e)^4)/(e^6*Sqrt[d + e*x]) - (20*b^2*(b*d - a*e)^3*Sqrt[d + e*x])/e^6 + (20*b^3*(b*d - a*e)^2*(d + e*x)^(3/2))/(3*e^6) - (2*b^4*(b*d - a*e)*(d + e*x)^(5/2))/e^6 + (2*b^5*(d + e*x)^(7/2))/(7*e^6)))/(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_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> Simp[(a + b*x + c*x^2)^FracPart[p]/(c^IntPart[p]*(b/2 + c*x)^(2*F racPart[p])) Int[(d + e*x)^m*(b/2 + c*x)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && EqQ[b^2 - 4*a*c, 0]
Time = 1.21 (sec) , antiderivative size = 226, normalized size of antiderivative = 0.72
| method | result | size |
| risch | \(\frac {2 b^{2} \left (3 e^{3} x^{3} b^{3}+21 x^{2} a \,b^{2} e^{3}-12 x^{2} b^{3} d \,e^{2}+70 a^{2} b \,e^{3} x -98 x a \,b^{2} d \,e^{2}+37 b^{3} d^{2} e x +210 e^{3} a^{3}-560 a^{2} b d \,e^{2}+511 a \,b^{2} d^{2} e -158 b^{3} d^{3}\right ) \sqrt {e x +d}\, \sqrt {\left (b x +a \right )^{2}}}{21 e^{6} \left (b x +a \right )}-\frac {2 \left (15 b e x +a e +14 b d \right ) \left (a^{4} e^{4}-4 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right ) \sqrt {\left (b x +a \right )^{2}}}{3 e^{6} \left (e x +d \right )^{\frac {3}{2}} \left (b x +a \right )}\) | \(226\) |
| gosper | \(-\frac {2 \left (-3 x^{5} e^{5} b^{5}-21 x^{4} a \,b^{4} e^{5}+6 x^{4} b^{5} d \,e^{4}-70 x^{3} a^{2} b^{3} e^{5}+56 x^{3} a \,b^{4} d \,e^{4}-16 x^{3} b^{5} d^{2} e^{3}-210 x^{2} a^{3} b^{2} e^{5}+420 x^{2} a^{2} b^{3} d \,e^{4}-336 x^{2} a \,b^{4} d^{2} e^{3}+96 x^{2} b^{5} d^{3} e^{2}+105 a^{4} b \,e^{5} x -840 a^{3} b^{2} d \,e^{4} x +1680 x \,a^{2} b^{3} d^{2} e^{3}-1344 x a \,b^{4} d^{3} e^{2}+384 b^{5} d^{4} e x +7 e^{5} a^{5}+70 a^{4} b d \,e^{4}-560 a^{3} b^{2} d^{2} e^{3}+1120 a^{2} b^{3} d^{3} e^{2}-896 a \,b^{4} d^{4} e +256 b^{5} d^{5}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{21 \left (e x +d \right )^{\frac {3}{2}} e^{6} \left (b x +a \right )^{5}}\) | \(289\) |
| default | \(-\frac {2 \left (-3 x^{5} e^{5} b^{5}-21 x^{4} a \,b^{4} e^{5}+6 x^{4} b^{5} d \,e^{4}-70 x^{3} a^{2} b^{3} e^{5}+56 x^{3} a \,b^{4} d \,e^{4}-16 x^{3} b^{5} d^{2} e^{3}-210 x^{2} a^{3} b^{2} e^{5}+420 x^{2} a^{2} b^{3} d \,e^{4}-336 x^{2} a \,b^{4} d^{2} e^{3}+96 x^{2} b^{5} d^{3} e^{2}+105 a^{4} b \,e^{5} x -840 a^{3} b^{2} d \,e^{4} x +1680 x \,a^{2} b^{3} d^{2} e^{3}-1344 x a \,b^{4} d^{3} e^{2}+384 b^{5} d^{4} e x +7 e^{5} a^{5}+70 a^{4} b d \,e^{4}-560 a^{3} b^{2} d^{2} e^{3}+1120 a^{2} b^{3} d^{3} e^{2}-896 a \,b^{4} d^{4} e +256 b^{5} d^{5}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{21 \left (e x +d \right )^{\frac {3}{2}} e^{6} \left (b x +a \right )^{5}}\) | \(289\) |
| orering | \(-\frac {2 \left (-3 x^{5} e^{5} b^{5}-21 x^{4} a \,b^{4} e^{5}+6 x^{4} b^{5} d \,e^{4}-70 x^{3} a^{2} b^{3} e^{5}+56 x^{3} a \,b^{4} d \,e^{4}-16 x^{3} b^{5} d^{2} e^{3}-210 x^{2} a^{3} b^{2} e^{5}+420 x^{2} a^{2} b^{3} d \,e^{4}-336 x^{2} a \,b^{4} d^{2} e^{3}+96 x^{2} b^{5} d^{3} e^{2}+105 a^{4} b \,e^{5} x -840 a^{3} b^{2} d \,e^{4} x +1680 x \,a^{2} b^{3} d^{2} e^{3}-1344 x a \,b^{4} d^{3} e^{2}+384 b^{5} d^{4} e x +7 e^{5} a^{5}+70 a^{4} b d \,e^{4}-560 a^{3} b^{2} d^{2} e^{3}+1120 a^{2} b^{3} d^{3} e^{2}-896 a \,b^{4} d^{4} e +256 b^{5} d^{5}\right ) \left (b^{2} x^{2}+2 a b x +a^{2}\right )^{\frac {5}{2}}}{21 e^{6} \left (b x +a \right )^{5} \left (e x +d \right )^{\frac {3}{2}}}\) | \(298\) |
Input:
int((b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^(5/2),x,method=_RETURNVERBOSE)
Output:
2/21*b^2*(3*b^3*e^3*x^3+21*a*b^2*e^3*x^2-12*b^3*d*e^2*x^2+70*a^2*b*e^3*x-9 8*a*b^2*d*e^2*x+37*b^3*d^2*e*x+210*a^3*e^3-560*a^2*b*d*e^2+511*a*b^2*d^2*e -158*b^3*d^3)*(e*x+d)^(1/2)/e^6*((b*x+a)^2)^(1/2)/(b*x+a)-2/3*(15*b*e*x+a* e+14*b*d)*(a^4*e^4-4*a^3*b*d*e^3+6*a^2*b^2*d^2*e^2-4*a*b^3*d^3*e+b^4*d^4)/ e^6/(e*x+d)^(3/2)*((b*x+a)^2)^(1/2)/(b*x+a)
Time = 0.08 (sec) , antiderivative size = 283, normalized size of antiderivative = 0.90 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{5/2}} \, dx=\frac {2 \, {\left (3 \, b^{5} e^{5} x^{5} - 256 \, b^{5} d^{5} + 896 \, a b^{4} d^{4} e - 1120 \, a^{2} b^{3} d^{3} e^{2} + 560 \, a^{3} b^{2} d^{2} e^{3} - 70 \, a^{4} b d e^{4} - 7 \, a^{5} e^{5} - 3 \, {\left (2 \, b^{5} d e^{4} - 7 \, a b^{4} e^{5}\right )} x^{4} + 2 \, {\left (8 \, b^{5} d^{2} e^{3} - 28 \, a b^{4} d e^{4} + 35 \, a^{2} b^{3} e^{5}\right )} x^{3} - 6 \, {\left (16 \, b^{5} d^{3} e^{2} - 56 \, a b^{4} d^{2} e^{3} + 70 \, a^{2} b^{3} d e^{4} - 35 \, a^{3} b^{2} e^{5}\right )} x^{2} - 3 \, {\left (128 \, b^{5} d^{4} e - 448 \, a b^{4} d^{3} e^{2} + 560 \, a^{2} b^{3} d^{2} e^{3} - 280 \, a^{3} b^{2} d e^{4} + 35 \, a^{4} b e^{5}\right )} x\right )} \sqrt {e x + d}}{21 \, {\left (e^{8} x^{2} + 2 \, d e^{7} x + d^{2} e^{6}\right )}} \] Input:
integrate((b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^(5/2),x, algorithm="fricas")
Output:
2/21*(3*b^5*e^5*x^5 - 256*b^5*d^5 + 896*a*b^4*d^4*e - 1120*a^2*b^3*d^3*e^2 + 560*a^3*b^2*d^2*e^3 - 70*a^4*b*d*e^4 - 7*a^5*e^5 - 3*(2*b^5*d*e^4 - 7*a *b^4*e^5)*x^4 + 2*(8*b^5*d^2*e^3 - 28*a*b^4*d*e^4 + 35*a^2*b^3*e^5)*x^3 - 6*(16*b^5*d^3*e^2 - 56*a*b^4*d^2*e^3 + 70*a^2*b^3*d*e^4 - 35*a^3*b^2*e^5)* x^2 - 3*(128*b^5*d^4*e - 448*a*b^4*d^3*e^2 + 560*a^2*b^3*d^2*e^3 - 280*a^3 *b^2*d*e^4 + 35*a^4*b*e^5)*x)*sqrt(e*x + d)/(e^8*x^2 + 2*d*e^7*x + d^2*e^6 )
\[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{5/2}} \, dx=\int \frac {\left (\left (a + b x\right )^{2}\right )^{\frac {5}{2}}}{\left (d + e x\right )^{\frac {5}{2}}}\, dx \] Input:
integrate((b**2*x**2+2*a*b*x+a**2)**(5/2)/(e*x+d)**(5/2),x)
Output:
Integral(((a + b*x)**2)**(5/2)/(d + e*x)**(5/2), x)
Time = 0.05 (sec) , antiderivative size = 272, normalized size of antiderivative = 0.87 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{5/2}} \, dx=\frac {2 \, {\left (3 \, b^{5} e^{5} x^{5} - 256 \, b^{5} d^{5} + 896 \, a b^{4} d^{4} e - 1120 \, a^{2} b^{3} d^{3} e^{2} + 560 \, a^{3} b^{2} d^{2} e^{3} - 70 \, a^{4} b d e^{4} - 7 \, a^{5} e^{5} - 3 \, {\left (2 \, b^{5} d e^{4} - 7 \, a b^{4} e^{5}\right )} x^{4} + 2 \, {\left (8 \, b^{5} d^{2} e^{3} - 28 \, a b^{4} d e^{4} + 35 \, a^{2} b^{3} e^{5}\right )} x^{3} - 6 \, {\left (16 \, b^{5} d^{3} e^{2} - 56 \, a b^{4} d^{2} e^{3} + 70 \, a^{2} b^{3} d e^{4} - 35 \, a^{3} b^{2} e^{5}\right )} x^{2} - 3 \, {\left (128 \, b^{5} d^{4} e - 448 \, a b^{4} d^{3} e^{2} + 560 \, a^{2} b^{3} d^{2} e^{3} - 280 \, a^{3} b^{2} d e^{4} + 35 \, a^{4} b e^{5}\right )} x\right )}}{21 \, {\left (e^{7} x + d e^{6}\right )} \sqrt {e x + d}} \] Input:
integrate((b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^(5/2),x, algorithm="maxima")
Output:
2/21*(3*b^5*e^5*x^5 - 256*b^5*d^5 + 896*a*b^4*d^4*e - 1120*a^2*b^3*d^3*e^2 + 560*a^3*b^2*d^2*e^3 - 70*a^4*b*d*e^4 - 7*a^5*e^5 - 3*(2*b^5*d*e^4 - 7*a *b^4*e^5)*x^4 + 2*(8*b^5*d^2*e^3 - 28*a*b^4*d*e^4 + 35*a^2*b^3*e^5)*x^3 - 6*(16*b^5*d^3*e^2 - 56*a*b^4*d^2*e^3 + 70*a^2*b^3*d*e^4 - 35*a^3*b^2*e^5)* x^2 - 3*(128*b^5*d^4*e - 448*a*b^4*d^3*e^2 + 560*a^2*b^3*d^2*e^3 - 280*a^3 *b^2*d*e^4 + 35*a^4*b*e^5)*x)/((e^7*x + d*e^6)*sqrt(e*x + d))
Leaf count of result is larger than twice the leaf count of optimal. 461 vs. \(2 (230) = 460\).
Time = 0.17 (sec) , antiderivative size = 461, normalized size of antiderivative = 1.47 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{5/2}} \, dx=-\frac {2 \, {\left (15 \, {\left (e x + d\right )} b^{5} d^{4} \mathrm {sgn}\left (b x + a\right ) - b^{5} d^{5} \mathrm {sgn}\left (b x + a\right ) - 60 \, {\left (e x + d\right )} a b^{4} d^{3} e \mathrm {sgn}\left (b x + a\right ) + 5 \, a b^{4} d^{4} e \mathrm {sgn}\left (b x + a\right ) + 90 \, {\left (e x + d\right )} a^{2} b^{3} d^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) - 10 \, a^{2} b^{3} d^{3} e^{2} \mathrm {sgn}\left (b x + a\right ) - 60 \, {\left (e x + d\right )} a^{3} b^{2} d e^{3} \mathrm {sgn}\left (b x + a\right ) + 10 \, a^{3} b^{2} d^{2} e^{3} \mathrm {sgn}\left (b x + a\right ) + 15 \, {\left (e x + d\right )} a^{4} b e^{4} \mathrm {sgn}\left (b x + a\right ) - 5 \, a^{4} b d e^{4} \mathrm {sgn}\left (b x + a\right ) + a^{5} e^{5} \mathrm {sgn}\left (b x + a\right )\right )}}{3 \, {\left (e x + d\right )}^{\frac {3}{2}} e^{6}} + \frac {2 \, {\left (3 \, {\left (e x + d\right )}^{\frac {7}{2}} b^{5} e^{36} \mathrm {sgn}\left (b x + a\right ) - 21 \, {\left (e x + d\right )}^{\frac {5}{2}} b^{5} d e^{36} \mathrm {sgn}\left (b x + a\right ) + 70 \, {\left (e x + d\right )}^{\frac {3}{2}} b^{5} d^{2} e^{36} \mathrm {sgn}\left (b x + a\right ) - 210 \, \sqrt {e x + d} b^{5} d^{3} e^{36} \mathrm {sgn}\left (b x + a\right ) + 21 \, {\left (e x + d\right )}^{\frac {5}{2}} a b^{4} e^{37} \mathrm {sgn}\left (b x + a\right ) - 140 \, {\left (e x + d\right )}^{\frac {3}{2}} a b^{4} d e^{37} \mathrm {sgn}\left (b x + a\right ) + 630 \, \sqrt {e x + d} a b^{4} d^{2} e^{37} \mathrm {sgn}\left (b x + a\right ) + 70 \, {\left (e x + d\right )}^{\frac {3}{2}} a^{2} b^{3} e^{38} \mathrm {sgn}\left (b x + a\right ) - 630 \, \sqrt {e x + d} a^{2} b^{3} d e^{38} \mathrm {sgn}\left (b x + a\right ) + 210 \, \sqrt {e x + d} a^{3} b^{2} e^{39} \mathrm {sgn}\left (b x + a\right )\right )}}{21 \, e^{42}} \] Input:
integrate((b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^(5/2),x, algorithm="giac")
Output:
-2/3*(15*(e*x + d)*b^5*d^4*sgn(b*x + a) - b^5*d^5*sgn(b*x + a) - 60*(e*x + d)*a*b^4*d^3*e*sgn(b*x + a) + 5*a*b^4*d^4*e*sgn(b*x + a) + 90*(e*x + d)*a ^2*b^3*d^2*e^2*sgn(b*x + a) - 10*a^2*b^3*d^3*e^2*sgn(b*x + a) - 60*(e*x + d)*a^3*b^2*d*e^3*sgn(b*x + a) + 10*a^3*b^2*d^2*e^3*sgn(b*x + a) + 15*(e*x + d)*a^4*b*e^4*sgn(b*x + a) - 5*a^4*b*d*e^4*sgn(b*x + a) + a^5*e^5*sgn(b*x + a))/((e*x + d)^(3/2)*e^6) + 2/21*(3*(e*x + d)^(7/2)*b^5*e^36*sgn(b*x + a) - 21*(e*x + d)^(5/2)*b^5*d*e^36*sgn(b*x + a) + 70*(e*x + d)^(3/2)*b^5*d ^2*e^36*sgn(b*x + a) - 210*sqrt(e*x + d)*b^5*d^3*e^36*sgn(b*x + a) + 21*(e *x + d)^(5/2)*a*b^4*e^37*sgn(b*x + a) - 140*(e*x + d)^(3/2)*a*b^4*d*e^37*s gn(b*x + a) + 630*sqrt(e*x + d)*a*b^4*d^2*e^37*sgn(b*x + a) + 70*(e*x + d) ^(3/2)*a^2*b^3*e^38*sgn(b*x + a) - 630*sqrt(e*x + d)*a^2*b^3*d*e^38*sgn(b* x + a) + 210*sqrt(e*x + d)*a^3*b^2*e^39*sgn(b*x + a))/e^42
Time = 6.52 (sec) , antiderivative size = 330, normalized size of antiderivative = 1.05 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{5/2}} \, dx=\frac {\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}\,\left (\frac {2\,b^4\,x^5}{7\,e^2}-\frac {14\,a^5\,e^5+140\,a^4\,b\,d\,e^4-1120\,a^3\,b^2\,d^2\,e^3+2240\,a^2\,b^3\,d^3\,e^2-1792\,a\,b^4\,d^4\,e+512\,b^5\,d^5}{21\,b\,e^7}+\frac {2\,b^3\,x^4\,\left (7\,a\,e-2\,b\,d\right )}{7\,e^3}-\frac {x\,\left (210\,a^4\,b\,e^5-1680\,a^3\,b^2\,d\,e^4+3360\,a^2\,b^3\,d^2\,e^3-2688\,a\,b^4\,d^3\,e^2+768\,b^5\,d^4\,e\right )}{21\,b\,e^7}+\frac {4\,b^2\,x^3\,\left (35\,a^2\,e^2-28\,a\,b\,d\,e+8\,b^2\,d^2\right )}{21\,e^4}+\frac {4\,b\,x^2\,\left (35\,a^3\,e^3-70\,a^2\,b\,d\,e^2+56\,a\,b^2\,d^2\,e-16\,b^3\,d^3\right )}{7\,e^5}\right )}{x^2\,\sqrt {d+e\,x}+\frac {a\,d\,\sqrt {d+e\,x}}{b\,e}+\frac {x\,\left (21\,a\,e^7+21\,b\,d\,e^6\right )\,\sqrt {d+e\,x}}{21\,b\,e^7}} \] Input:
int((a^2 + b^2*x^2 + 2*a*b*x)^(5/2)/(d + e*x)^(5/2),x)
Output:
((a^2 + b^2*x^2 + 2*a*b*x)^(1/2)*((2*b^4*x^5)/(7*e^2) - (14*a^5*e^5 + 512* b^5*d^5 + 2240*a^2*b^3*d^3*e^2 - 1120*a^3*b^2*d^2*e^3 - 1792*a*b^4*d^4*e + 140*a^4*b*d*e^4)/(21*b*e^7) + (2*b^3*x^4*(7*a*e - 2*b*d))/(7*e^3) - (x*(2 10*a^4*b*e^5 + 768*b^5*d^4*e - 2688*a*b^4*d^3*e^2 - 1680*a^3*b^2*d*e^4 + 3 360*a^2*b^3*d^2*e^3))/(21*b*e^7) + (4*b^2*x^3*(35*a^2*e^2 + 8*b^2*d^2 - 28 *a*b*d*e))/(21*e^4) + (4*b*x^2*(35*a^3*e^3 - 16*b^3*d^3 + 56*a*b^2*d^2*e - 70*a^2*b*d*e^2))/(7*e^5)))/(x^2*(d + e*x)^(1/2) + (a*d*(d + e*x)^(1/2))/( b*e) + (x*(21*a*e^7 + 21*b*d*e^6)*(d + e*x)^(1/2))/(21*b*e^7))
Time = 0.27 (sec) , antiderivative size = 280, normalized size of antiderivative = 0.89 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{5/2}} \, dx=\frac {\frac {2}{7} b^{5} e^{5} x^{5}+2 a \,b^{4} e^{5} x^{4}-\frac {4}{7} b^{5} d \,e^{4} x^{4}+\frac {20}{3} a^{2} b^{3} e^{5} x^{3}-\frac {16}{3} a \,b^{4} d \,e^{4} x^{3}+\frac {32}{21} b^{5} d^{2} e^{3} x^{3}+20 a^{3} b^{2} e^{5} x^{2}-40 a^{2} b^{3} d \,e^{4} x^{2}+32 a \,b^{4} d^{2} e^{3} x^{2}-\frac {64}{7} b^{5} d^{3} e^{2} x^{2}-10 a^{4} b \,e^{5} x +80 a^{3} b^{2} d \,e^{4} x -160 a^{2} b^{3} d^{2} e^{3} x +128 a \,b^{4} d^{3} e^{2} x -\frac {256}{7} b^{5} d^{4} e x -\frac {2}{3} a^{5} e^{5}-\frac {20}{3} a^{4} b d \,e^{4}+\frac {160}{3} a^{3} b^{2} d^{2} e^{3}-\frac {320}{3} a^{2} b^{3} d^{3} e^{2}+\frac {256}{3} a \,b^{4} d^{4} e -\frac {512}{21} b^{5} d^{5}}{\sqrt {e x +d}\, e^{6} \left (e x +d \right )} \] Input:
int((b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^(5/2),x)
Output:
(2*( - 7*a**5*e**5 - 70*a**4*b*d*e**4 - 105*a**4*b*e**5*x + 560*a**3*b**2* d**2*e**3 + 840*a**3*b**2*d*e**4*x + 210*a**3*b**2*e**5*x**2 - 1120*a**2*b **3*d**3*e**2 - 1680*a**2*b**3*d**2*e**3*x - 420*a**2*b**3*d*e**4*x**2 + 7 0*a**2*b**3*e**5*x**3 + 896*a*b**4*d**4*e + 1344*a*b**4*d**3*e**2*x + 336* a*b**4*d**2*e**3*x**2 - 56*a*b**4*d*e**4*x**3 + 21*a*b**4*e**5*x**4 - 256* b**5*d**5 - 384*b**5*d**4*e*x - 96*b**5*d**3*e**2*x**2 + 16*b**5*d**2*e**3 *x**3 - 6*b**5*d*e**4*x**4 + 3*b**5*e**5*x**5))/(21*sqrt(d + e*x)*e**6*(d + e*x))