Integrand size = 33, antiderivative size = 356 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^7} \, dx=-\frac {(b d-a e)^6 \sqrt {a^2+2 a b x+b^2 x^2}}{6 e^7 (a+b x) (d+e x)^6}+\frac {6 b (b d-a e)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{5 e^7 (a+b x) (d+e x)^5}-\frac {15 b^2 (b d-a e)^4 \sqrt {a^2+2 a b x+b^2 x^2}}{4 e^7 (a+b x) (d+e x)^4}+\frac {20 b^3 (b d-a e)^3 \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^7 (a+b x) (d+e x)^3}-\frac {15 b^4 (b d-a e)^2 \sqrt {a^2+2 a b x+b^2 x^2}}{2 e^7 (a+b x) (d+e x)^2}+\frac {6 b^5 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}{e^7 (a+b x) (d+e x)}+\frac {b^6 \sqrt {a^2+2 a b x+b^2 x^2} \log (d+e x)}{e^7 (a+b x)} \]
-1/6*(-a*e+b*d)^6*((b*x+a)^2)^(1/2)/e^7/(b*x+a)/(e*x+d)^6+6/5*b*(-a*e+b*d) ^5*((b*x+a)^2)^(1/2)/e^7/(b*x+a)/(e*x+d)^5-15/4*b^2*(-a*e+b*d)^4*((b*x+a)^ 2)^(1/2)/e^7/(b*x+a)/(e*x+d)^4+20/3*b^3*(-a*e+b*d)^3*((b*x+a)^2)^(1/2)/e^7 /(b*x+a)/(e*x+d)^3-15/2*b^4*(-a*e+b*d)^2*((b*x+a)^2)^(1/2)/e^7/(b*x+a)/(e* x+d)^2+6*b^5*(-a*e+b*d)*((b*x+a)^2)^(1/2)/e^7/(b*x+a)/(e*x+d)+b^6*ln(e*x+d )*((b*x+a)^2)^(1/2)/e^7/(b*x+a)
Time = 1.12 (sec) , antiderivative size = 258, normalized size of antiderivative = 0.72 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^7} \, dx=\frac {\sqrt {(a+b x)^2} \left ((b d-a e) \left (10 a^5 e^5+2 a^4 b e^4 (11 d+36 e x)+a^3 b^2 e^3 \left (37 d^2+162 d e x+225 e^2 x^2\right )+a^2 b^3 e^2 \left (57 d^3+282 d^2 e x+525 d e^2 x^2+400 e^3 x^3\right )+a b^4 e \left (87 d^4+462 d^3 e x+975 d^2 e^2 x^2+1000 d e^3 x^3+450 e^4 x^4\right )+b^5 \left (147 d^5+822 d^4 e x+1875 d^3 e^2 x^2+2200 d^2 e^3 x^3+1350 d e^4 x^4+360 e^5 x^5\right )\right )+60 b^6 (d+e x)^6 \log (d+e x)\right )}{60 e^7 (a+b x) (d+e x)^6} \]
(Sqrt[(a + b*x)^2]*((b*d - a*e)*(10*a^5*e^5 + 2*a^4*b*e^4*(11*d + 36*e*x) + a^3*b^2*e^3*(37*d^2 + 162*d*e*x + 225*e^2*x^2) + a^2*b^3*e^2*(57*d^3 + 2 82*d^2*e*x + 525*d*e^2*x^2 + 400*e^3*x^3) + a*b^4*e*(87*d^4 + 462*d^3*e*x + 975*d^2*e^2*x^2 + 1000*d*e^3*x^3 + 450*e^4*x^4) + b^5*(147*d^5 + 822*d^4 *e*x + 1875*d^3*e^2*x^2 + 2200*d^2*e^3*x^3 + 1350*d*e^4*x^4 + 360*e^5*x^5) ) + 60*b^6*(d + e*x)^6*Log[d + e*x]))/(60*e^7*(a + b*x)*(d + e*x)^6)
Time = 0.40 (sec) , antiderivative size = 195, normalized size of antiderivative = 0.55, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.121, Rules used = {1187, 27, 49, 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.
| \(\displaystyle \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^7} \, dx\) |
| \(\Big \downarrow \) 1187 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {b^5 (a+b x)^6}{(d+e x)^7}dx}{b^5 (a+b x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {(a+b x)^6}{(d+e x)^7}dx}{a+b x}\) |
| \(\Big \downarrow \) 49 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (\frac {b^6}{e^6 (d+e x)}-\frac {6 (b d-a e) b^5}{e^6 (d+e x)^2}+\frac {15 (b d-a e)^2 b^4}{e^6 (d+e x)^3}-\frac {20 (b d-a e)^3 b^3}{e^6 (d+e x)^4}+\frac {15 (b d-a e)^4 b^2}{e^6 (d+e x)^5}-\frac {6 (b d-a e)^5 b}{e^6 (d+e x)^6}+\frac {(a e-b d)^6}{e^6 (d+e x)^7}\right )dx}{a+b x}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \left (\frac {6 b^5 (b d-a e)}{e^7 (d+e x)}-\frac {15 b^4 (b d-a e)^2}{2 e^7 (d+e x)^2}+\frac {20 b^3 (b d-a e)^3}{3 e^7 (d+e x)^3}-\frac {15 b^2 (b d-a e)^4}{4 e^7 (d+e x)^4}+\frac {6 b (b d-a e)^5}{5 e^7 (d+e x)^5}-\frac {(b d-a e)^6}{6 e^7 (d+e x)^6}+\frac {b^6 \log (d+e x)}{e^7}\right )}{a+b x}\) |
(Sqrt[a^2 + 2*a*b*x + b^2*x^2]*(-1/6*(b*d - a*e)^6/(e^7*(d + e*x)^6) + (6* b*(b*d - a*e)^5)/(5*e^7*(d + e*x)^5) - (15*b^2*(b*d - a*e)^4)/(4*e^7*(d + e*x)^4) + (20*b^3*(b*d - a*e)^3)/(3*e^7*(d + e*x)^3) - (15*b^4*(b*d - a*e) ^2)/(2*e^7*(d + e*x)^2) + (6*b^5*(b*d - a*e))/(e^7*(d + e*x)) + (b^6*Log[d + e*x])/e^7))/(a + b*x)
3.21.4.3.1 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}, x] && IGtQ[m, 0] && IGtQ[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 = 1.37 (sec) , antiderivative size = 374, normalized size of antiderivative = 1.05
| method | result | size |
| risch | \(\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (-\frac {6 b^{5} \left (a e -b d \right ) x^{5}}{e^{2}}-\frac {15 b^{4} \left (e^{2} a^{2}+2 a b d e -3 b^{2} d^{2}\right ) x^{4}}{2 e^{3}}-\frac {10 b^{3} \left (2 a^{3} e^{3}+3 a^{2} b d \,e^{2}+6 a \,b^{2} d^{2} e -11 b^{3} d^{3}\right ) x^{3}}{3 e^{4}}-\frac {5 b^{2} \left (3 e^{4} a^{4}+4 b d \,e^{3} a^{3}+6 b^{2} d^{2} e^{2} a^{2}+12 b^{3} d^{3} e a -25 b^{4} d^{4}\right ) x^{2}}{4 e^{5}}-\frac {b \left (12 e^{5} a^{5}+15 b d \,e^{4} a^{4}+20 b^{2} d^{2} e^{3} a^{3}+30 b^{3} d^{3} e^{2} a^{2}+60 b^{4} d^{4} e a -137 b^{5} d^{5}\right ) x}{10 e^{6}}-\frac {10 e^{6} a^{6}+12 b d \,e^{5} a^{5}+15 b^{2} d^{2} e^{4} a^{4}+20 b^{3} d^{3} e^{3} a^{3}+30 b^{4} d^{4} e^{2} a^{2}+60 b^{5} d^{5} e a -147 b^{6} d^{6}}{60 e^{7}}\right )}{\left (b x +a \right ) \left (e x +d \right )^{6}}+\frac {b^{6} \ln \left (e x +d \right ) \sqrt {\left (b x +a \right )^{2}}}{e^{7} \left (b x +a \right )}\) | \(374\) |
| default | \(\frac {\left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}} \left (-15 b^{2} d^{2} e^{4} a^{4}-20 b^{3} d^{3} e^{3} a^{3}-30 b^{4} d^{4} e^{2} a^{2}-60 b^{5} d^{5} e a -12 b d \,e^{5} a^{5}-72 a^{5} b \,e^{6} x +822 b^{6} d^{5} e x -360 a \,b^{5} e^{6} x^{5}+360 b^{6} d \,e^{5} x^{5}-450 a^{2} b^{4} e^{6} x^{4}+1350 b^{6} d^{2} e^{4} x^{4}-400 a^{3} b^{3} e^{6} x^{3}+2200 b^{6} d^{3} e^{3} x^{3}-225 a^{4} b^{2} e^{6} x^{2}+1875 b^{6} d^{4} e^{2} x^{2}+60 \ln \left (e x +d \right ) b^{6} e^{6} x^{6}-900 a \,b^{5} d \,e^{5} x^{4}-600 a^{2} b^{4} d \,e^{5} x^{3}-1200 a \,b^{5} d^{2} e^{4} x^{3}-300 a^{3} b^{3} d \,e^{5} x^{2}-450 a^{2} b^{4} d^{2} e^{4} x^{2}-900 a \,b^{5} d^{3} e^{3} x^{2}-90 a^{4} b^{2} d \,e^{5} x +900 \ln \left (e x +d \right ) b^{6} d^{2} e^{4} x^{4}+900 \ln \left (e x +d \right ) b^{6} d^{4} e^{2} x^{2}+1200 \ln \left (e x +d \right ) b^{6} d^{3} e^{3} x^{3}-10 e^{6} a^{6}+147 b^{6} d^{6}+360 \ln \left (e x +d \right ) b^{6} d \,e^{5} x^{5}-120 a^{3} b^{3} d^{2} e^{4} x -180 a^{2} b^{4} d^{3} e^{3} x -360 a \,b^{5} d^{4} e^{2} x +360 \ln \left (e x +d \right ) b^{6} d^{5} e x +60 \ln \left (e x +d \right ) b^{6} d^{6}\right )}{60 \left (b x +a \right )^{5} e^{7} \left (e x +d \right )^{6}}\) | \(507\) |
((b*x+a)^2)^(1/2)/(b*x+a)*(-6*b^5*(a*e-b*d)/e^2*x^5-15/2*b^4*(a^2*e^2+2*a* b*d*e-3*b^2*d^2)/e^3*x^4-10/3*b^3*(2*a^3*e^3+3*a^2*b*d*e^2+6*a*b^2*d^2*e-1 1*b^3*d^3)/e^4*x^3-5/4*b^2*(3*a^4*e^4+4*a^3*b*d*e^3+6*a^2*b^2*d^2*e^2+12*a *b^3*d^3*e-25*b^4*d^4)/e^5*x^2-1/10*b*(12*a^5*e^5+15*a^4*b*d*e^4+20*a^3*b^ 2*d^2*e^3+30*a^2*b^3*d^3*e^2+60*a*b^4*d^4*e-137*b^5*d^5)/e^6*x-1/60*(10*a^ 6*e^6+12*a^5*b*d*e^5+15*a^4*b^2*d^2*e^4+20*a^3*b^3*d^3*e^3+30*a^2*b^4*d^4* e^2+60*a*b^5*d^5*e-147*b^6*d^6)/e^7)/(e*x+d)^6+b^6*ln(e*x+d)*((b*x+a)^2)^( 1/2)/e^7/(b*x+a)
Time = 0.37 (sec) , antiderivative size = 492, normalized size of antiderivative = 1.38 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^7} \, dx=\frac {147 \, b^{6} d^{6} - 60 \, a b^{5} d^{5} e - 30 \, a^{2} b^{4} d^{4} e^{2} - 20 \, a^{3} b^{3} d^{3} e^{3} - 15 \, a^{4} b^{2} d^{2} e^{4} - 12 \, a^{5} b d e^{5} - 10 \, a^{6} e^{6} + 360 \, {\left (b^{6} d e^{5} - a b^{5} e^{6}\right )} x^{5} + 450 \, {\left (3 \, b^{6} d^{2} e^{4} - 2 \, a b^{5} d e^{5} - a^{2} b^{4} e^{6}\right )} x^{4} + 200 \, {\left (11 \, b^{6} d^{3} e^{3} - 6 \, a b^{5} d^{2} e^{4} - 3 \, a^{2} b^{4} d e^{5} - 2 \, a^{3} b^{3} e^{6}\right )} x^{3} + 75 \, {\left (25 \, b^{6} d^{4} e^{2} - 12 \, a b^{5} d^{3} e^{3} - 6 \, a^{2} b^{4} d^{2} e^{4} - 4 \, a^{3} b^{3} d e^{5} - 3 \, a^{4} b^{2} e^{6}\right )} x^{2} + 6 \, {\left (137 \, b^{6} d^{5} e - 60 \, a b^{5} d^{4} e^{2} - 30 \, a^{2} b^{4} d^{3} e^{3} - 20 \, a^{3} b^{3} d^{2} e^{4} - 15 \, a^{4} b^{2} d e^{5} - 12 \, a^{5} b e^{6}\right )} x + 60 \, {\left (b^{6} e^{6} x^{6} + 6 \, b^{6} d e^{5} x^{5} + 15 \, b^{6} d^{2} e^{4} x^{4} + 20 \, b^{6} d^{3} e^{3} x^{3} + 15 \, b^{6} d^{4} e^{2} x^{2} + 6 \, b^{6} d^{5} e x + b^{6} d^{6}\right )} \log \left (e x + d\right )}{60 \, {\left (e^{13} x^{6} + 6 \, d e^{12} x^{5} + 15 \, d^{2} e^{11} x^{4} + 20 \, d^{3} e^{10} x^{3} + 15 \, d^{4} e^{9} x^{2} + 6 \, d^{5} e^{8} x + d^{6} e^{7}\right )}} \]
1/60*(147*b^6*d^6 - 60*a*b^5*d^5*e - 30*a^2*b^4*d^4*e^2 - 20*a^3*b^3*d^3*e ^3 - 15*a^4*b^2*d^2*e^4 - 12*a^5*b*d*e^5 - 10*a^6*e^6 + 360*(b^6*d*e^5 - a *b^5*e^6)*x^5 + 450*(3*b^6*d^2*e^4 - 2*a*b^5*d*e^5 - a^2*b^4*e^6)*x^4 + 20 0*(11*b^6*d^3*e^3 - 6*a*b^5*d^2*e^4 - 3*a^2*b^4*d*e^5 - 2*a^3*b^3*e^6)*x^3 + 75*(25*b^6*d^4*e^2 - 12*a*b^5*d^3*e^3 - 6*a^2*b^4*d^2*e^4 - 4*a^3*b^3*d *e^5 - 3*a^4*b^2*e^6)*x^2 + 6*(137*b^6*d^5*e - 60*a*b^5*d^4*e^2 - 30*a^2*b ^4*d^3*e^3 - 20*a^3*b^3*d^2*e^4 - 15*a^4*b^2*d*e^5 - 12*a^5*b*e^6)*x + 60* (b^6*e^6*x^6 + 6*b^6*d*e^5*x^5 + 15*b^6*d^2*e^4*x^4 + 20*b^6*d^3*e^3*x^3 + 15*b^6*d^4*e^2*x^2 + 6*b^6*d^5*e*x + b^6*d^6)*log(e*x + d))/(e^13*x^6 + 6 *d*e^12*x^5 + 15*d^2*e^11*x^4 + 20*d^3*e^10*x^3 + 15*d^4*e^9*x^2 + 6*d^5*e ^8*x + d^6*e^7)
Timed out. \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^7} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^7} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(a*e-b*d>0)', see `assume?` for m ore detail
Time = 0.27 (sec) , antiderivative size = 527, normalized size of antiderivative = 1.48 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^7} \, dx=\frac {b^{6} \log \left ({\left | e x + d \right |}\right ) \mathrm {sgn}\left (b x + a\right )}{e^{7}} + \frac {360 \, {\left (b^{6} d e^{4} \mathrm {sgn}\left (b x + a\right ) - a b^{5} e^{5} \mathrm {sgn}\left (b x + a\right )\right )} x^{5} + 450 \, {\left (3 \, b^{6} d^{2} e^{3} \mathrm {sgn}\left (b x + a\right ) - 2 \, a b^{5} d e^{4} \mathrm {sgn}\left (b x + a\right ) - a^{2} b^{4} e^{5} \mathrm {sgn}\left (b x + a\right )\right )} x^{4} + 200 \, {\left (11 \, b^{6} d^{3} e^{2} \mathrm {sgn}\left (b x + a\right ) - 6 \, a b^{5} d^{2} e^{3} \mathrm {sgn}\left (b x + a\right ) - 3 \, a^{2} b^{4} d e^{4} \mathrm {sgn}\left (b x + a\right ) - 2 \, a^{3} b^{3} e^{5} \mathrm {sgn}\left (b x + a\right )\right )} x^{3} + 75 \, {\left (25 \, b^{6} d^{4} e \mathrm {sgn}\left (b x + a\right ) - 12 \, a b^{5} d^{3} e^{2} \mathrm {sgn}\left (b x + a\right ) - 6 \, a^{2} b^{4} d^{2} e^{3} \mathrm {sgn}\left (b x + a\right ) - 4 \, a^{3} b^{3} d e^{4} \mathrm {sgn}\left (b x + a\right ) - 3 \, a^{4} b^{2} e^{5} \mathrm {sgn}\left (b x + a\right )\right )} x^{2} + 6 \, {\left (137 \, b^{6} d^{5} \mathrm {sgn}\left (b x + a\right ) - 60 \, a b^{5} d^{4} e \mathrm {sgn}\left (b x + a\right ) - 30 \, a^{2} b^{4} d^{3} e^{2} \mathrm {sgn}\left (b x + a\right ) - 20 \, a^{3} b^{3} d^{2} e^{3} \mathrm {sgn}\left (b x + a\right ) - 15 \, a^{4} b^{2} d e^{4} \mathrm {sgn}\left (b x + a\right ) - 12 \, a^{5} b e^{5} \mathrm {sgn}\left (b x + a\right )\right )} x + \frac {147 \, b^{6} d^{6} \mathrm {sgn}\left (b x + a\right ) - 60 \, a b^{5} d^{5} e \mathrm {sgn}\left (b x + a\right ) - 30 \, a^{2} b^{4} d^{4} e^{2} \mathrm {sgn}\left (b x + a\right ) - 20 \, a^{3} b^{3} d^{3} e^{3} \mathrm {sgn}\left (b x + a\right ) - 15 \, a^{4} b^{2} d^{2} e^{4} \mathrm {sgn}\left (b x + a\right ) - 12 \, a^{5} b d e^{5} \mathrm {sgn}\left (b x + a\right ) - 10 \, a^{6} e^{6} \mathrm {sgn}\left (b x + a\right )}{e}}{60 \, {\left (e x + d\right )}^{6} e^{6}} \]
b^6*log(abs(e*x + d))*sgn(b*x + a)/e^7 + 1/60*(360*(b^6*d*e^4*sgn(b*x + a) - a*b^5*e^5*sgn(b*x + a))*x^5 + 450*(3*b^6*d^2*e^3*sgn(b*x + a) - 2*a*b^5 *d*e^4*sgn(b*x + a) - a^2*b^4*e^5*sgn(b*x + a))*x^4 + 200*(11*b^6*d^3*e^2* sgn(b*x + a) - 6*a*b^5*d^2*e^3*sgn(b*x + a) - 3*a^2*b^4*d*e^4*sgn(b*x + a) - 2*a^3*b^3*e^5*sgn(b*x + a))*x^3 + 75*(25*b^6*d^4*e*sgn(b*x + a) - 12*a* b^5*d^3*e^2*sgn(b*x + a) - 6*a^2*b^4*d^2*e^3*sgn(b*x + a) - 4*a^3*b^3*d*e^ 4*sgn(b*x + a) - 3*a^4*b^2*e^5*sgn(b*x + a))*x^2 + 6*(137*b^6*d^5*sgn(b*x + a) - 60*a*b^5*d^4*e*sgn(b*x + a) - 30*a^2*b^4*d^3*e^2*sgn(b*x + a) - 20* a^3*b^3*d^2*e^3*sgn(b*x + a) - 15*a^4*b^2*d*e^4*sgn(b*x + a) - 12*a^5*b*e^ 5*sgn(b*x + a))*x + (147*b^6*d^6*sgn(b*x + a) - 60*a*b^5*d^5*e*sgn(b*x + a ) - 30*a^2*b^4*d^4*e^2*sgn(b*x + a) - 20*a^3*b^3*d^3*e^3*sgn(b*x + a) - 15 *a^4*b^2*d^2*e^4*sgn(b*x + a) - 12*a^5*b*d*e^5*sgn(b*x + a) - 10*a^6*e^6*s gn(b*x + a))/e)/((e*x + d)^6*e^6)
Timed out. \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^7} \, dx=\int \frac {\left (a+b\,x\right )\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2}}{{\left (d+e\,x\right )}^7} \,d x \]