Integrand size = 33, antiderivative size = 298 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{d+e x} \, dx=-\frac {b (b d-a e)^5 x \sqrt {a^2+2 a b x+b^2 x^2}}{e^6 (a+b x)}+\frac {(b d-a e)^4 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}{2 e^5}-\frac {(b d-a e)^3 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^4}+\frac {(b d-a e)^2 (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}{4 e^3}-\frac {(b d-a e) (a+b x)^4 \sqrt {a^2+2 a b x+b^2 x^2}}{5 e^2}+\frac {(a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{6 e}+\frac {(b d-a e)^6 \sqrt {a^2+2 a b x+b^2 x^2} \log (d+e x)}{e^7 (a+b x)} \]
-b*(-a*e+b*d)^5*x*((b*x+a)^2)^(1/2)/e^6/(b*x+a)+1/2*(-a*e+b*d)^4*(b*x+a)*( (b*x+a)^2)^(1/2)/e^5-1/3*(-a*e+b*d)^3*(b*x+a)^2*((b*x+a)^2)^(1/2)/e^4+1/4* (-a*e+b*d)^2*(b*x+a)^3*((b*x+a)^2)^(1/2)/e^3-1/5*(-a*e+b*d)*(b*x+a)^4*((b* x+a)^2)^(1/2)/e^2+1/6*(b*x+a)^5*((b*x+a)^2)^(1/2)/e+(-a*e+b*d)^6*ln(e*x+d) *((b*x+a)^2)^(1/2)/e^7/(b*x+a)
Time = 1.09 (sec) , antiderivative size = 248, normalized size of antiderivative = 0.83 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{d+e x} \, dx=\frac {\sqrt {(a+b x)^2} \left (b e x \left (360 a^5 e^5+450 a^4 b e^4 (-2 d+e x)+200 a^3 b^2 e^3 \left (6 d^2-3 d e x+2 e^2 x^2\right )+75 a^2 b^3 e^2 \left (-12 d^3+6 d^2 e x-4 d e^2 x^2+3 e^3 x^3\right )+6 a b^4 e \left (60 d^4-30 d^3 e x+20 d^2 e^2 x^2-15 d e^3 x^3+12 e^4 x^4\right )+b^5 \left (-60 d^5+30 d^4 e x-20 d^3 e^2 x^2+15 d^2 e^3 x^3-12 d e^4 x^4+10 e^5 x^5\right )\right )+60 (b d-a e)^6 \log (d+e x)\right )}{60 e^7 (a+b x)} \]
(Sqrt[(a + b*x)^2]*(b*e*x*(360*a^5*e^5 + 450*a^4*b*e^4*(-2*d + e*x) + 200* a^3*b^2*e^3*(6*d^2 - 3*d*e*x + 2*e^2*x^2) + 75*a^2*b^3*e^2*(-12*d^3 + 6*d^ 2*e*x - 4*d*e^2*x^2 + 3*e^3*x^3) + 6*a*b^4*e*(60*d^4 - 30*d^3*e*x + 20*d^2 *e^2*x^2 - 15*d*e^3*x^3 + 12*e^4*x^4) + b^5*(-60*d^5 + 30*d^4*e*x - 20*d^3 *e^2*x^2 + 15*d^2*e^3*x^3 - 12*d*e^4*x^4 + 10*e^5*x^5)) + 60*(b*d - a*e)^6 *Log[d + e*x]))/(60*e^7*(a + b*x))
Time = 0.34 (sec) , antiderivative size = 174, normalized size of antiderivative = 0.58, 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.
| \(\displaystyle \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{d+e x} \, dx\) |
| \(\Big \downarrow \) 1187 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {b^5 (a+b x)^6}{d+e x}dx}{b^5 (a+b x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {(a+b x)^6}{d+e x}dx}{a+b x}\) |
| \(\Big \downarrow \) 49 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (\frac {(a e-b d)^6}{e^6 (d+e x)}-\frac {b (b d-a e)^5}{e^6}+\frac {b (a+b x)^5}{e}-\frac {b (b d-a e) (a+b x)^4}{e^2}+\frac {b (b d-a e)^2 (a+b x)^3}{e^3}-\frac {b (b d-a e)^3 (a+b x)^2}{e^4}+\frac {b (b d-a e)^4 (a+b x)}{e^5}\right )dx}{a+b x}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \left (\frac {(b d-a e)^6 \log (d+e x)}{e^7}-\frac {b x (b d-a e)^5}{e^6}+\frac {(a+b x)^2 (b d-a e)^4}{2 e^5}-\frac {(a+b x)^3 (b d-a e)^3}{3 e^4}+\frac {(a+b x)^4 (b d-a e)^2}{4 e^3}-\frac {(a+b x)^5 (b d-a e)}{5 e^2}+\frac {(a+b x)^6}{6 e}\right )}{a+b x}\) |
(Sqrt[a^2 + 2*a*b*x + b^2*x^2]*(-((b*(b*d - a*e)^5*x)/e^6) + ((b*d - a*e)^ 4*(a + b*x)^2)/(2*e^5) - ((b*d - a*e)^3*(a + b*x)^3)/(3*e^4) + ((b*d - a*e )^2*(a + b*x)^4)/(4*e^3) - ((b*d - a*e)*(a + b*x)^5)/(5*e^2) + (a + b*x)^6 /(6*e) + ((b*d - a*e)^6*Log[d + e*x])/e^7))/(a + b*x)
3.20.98.3.1 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]
Leaf count of result is larger than twice the leaf count of optimal. \(427\) vs. \(2(211)=422\).
Time = 0.35 (sec) , antiderivative size = 428, normalized size of antiderivative = 1.44
| method | result | size |
| default | \(\frac {\left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}} \left (360 a^{5} b \,e^{6} x -60 b^{6} d^{5} e x +72 a \,b^{5} e^{6} x^{5}-12 b^{6} d \,e^{5} x^{5}+225 a^{2} b^{4} e^{6} x^{4}+15 b^{6} d^{2} e^{4} x^{4}+400 a^{3} b^{3} e^{6} x^{3}-20 b^{6} d^{3} e^{3} x^{3}+450 a^{4} b^{2} e^{6} x^{2}+30 b^{6} d^{4} e^{2} x^{2}-90 a \,b^{5} d \,e^{5} x^{4}-300 a^{2} b^{4} d \,e^{5} x^{3}+120 a \,b^{5} d^{2} e^{4} x^{3}-600 a^{3} b^{3} d \,e^{5} x^{2}+450 a^{2} b^{4} d^{2} e^{4} x^{2}-180 a \,b^{5} d^{3} e^{3} x^{2}-900 a^{4} b^{2} d \,e^{5} x -1200 \ln \left (e x +d \right ) a^{3} b^{3} d^{3} e^{3}+900 \ln \left (e x +d \right ) a^{2} b^{4} d^{4} e^{2}-360 \ln \left (e x +d \right ) a \,b^{5} d^{5} e -360 \ln \left (e x +d \right ) a^{5} b d \,e^{5}+900 \ln \left (e x +d \right ) a^{4} b^{2} d^{2} e^{4}+1200 a^{3} b^{3} d^{2} e^{4} x -900 a^{2} b^{4} d^{3} e^{3} x +360 a \,b^{5} d^{4} e^{2} x +10 b^{6} e^{6} x^{6}+60 \ln \left (e x +d \right ) a^{6} e^{6}+60 \ln \left (e x +d \right ) b^{6} d^{6}\right )}{60 \left (b x +a \right )^{5} e^{7}}\) | \(428\) |
| risch | \(\frac {\sqrt {\left (b x +a \right )^{2}}\, b \left (\frac {x^{6} b^{5} e^{5}}{6}+\frac {\left (\left (\left (2 a e -b d \right ) b^{2} e^{2}+b e \left (a b \,e^{2}+b^{2} d e \right )\right ) b^{2} e^{2}+b^{3} e^{3} \left (3 a b \,e^{2}-b^{2} d e \right )\right ) x^{5}}{5}+\frac {\left (\left (\left (2 a e -b d \right ) \left (a b \,e^{2}+b^{2} d e \right )+b e \left (e^{2} a^{2}-a b d e +b^{2} d^{2}\right )\right ) b^{2} e^{2}+\left (\left (2 a e -b d \right ) b^{2} e^{2}+b e \left (a b \,e^{2}+b^{2} d e \right )\right ) \left (3 a b \,e^{2}-b^{2} d e \right )+b^{3} e^{3} \left (3 e^{2} a^{2}-3 a b d e +b^{2} d^{2}\right )\right ) x^{4}}{4}+\frac {\left (\left (2 a e -b d \right ) \left (e^{2} a^{2}-a b d e +b^{2} d^{2}\right ) b^{2} e^{2}+\left (\left (2 a e -b d \right ) \left (a b \,e^{2}+b^{2} d e \right )+b e \left (e^{2} a^{2}-a b d e +b^{2} d^{2}\right )\right ) \left (3 a b \,e^{2}-b^{2} d e \right )+\left (\left (2 a e -b d \right ) b^{2} e^{2}+b e \left (a b \,e^{2}+b^{2} d e \right )\right ) \left (3 e^{2} a^{2}-3 a b d e +b^{2} d^{2}\right )\right ) x^{3}}{3}+\frac {\left (\left (2 a e -b d \right ) \left (e^{2} a^{2}-a b d e +b^{2} d^{2}\right ) \left (3 a b \,e^{2}-b^{2} d e \right )+\left (\left (2 a e -b d \right ) \left (a b \,e^{2}+b^{2} d e \right )+b e \left (e^{2} a^{2}-a b d e +b^{2} d^{2}\right )\right ) \left (3 e^{2} a^{2}-3 a b d e +b^{2} d^{2}\right )\right ) x^{2}}{2}+\left (2 a e -b d \right ) \left (e^{2} a^{2}-a b d e +b^{2} d^{2}\right ) \left (3 e^{2} a^{2}-3 a b d e +b^{2} d^{2}\right ) x \right )}{\left (b x +a \right ) e^{6}}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (e^{6} a^{6}-6 b d \,e^{5} a^{5}+15 b^{2} d^{2} e^{4} a^{4}-20 b^{3} d^{3} e^{3} a^{3}+15 b^{4} d^{4} e^{2} a^{2}-6 b^{5} d^{5} e a +b^{6} d^{6}\right ) \ln \left (e x +d \right )}{\left (b x +a \right ) e^{7}}\) | \(685\) |
1/60*((b*x+a)^2)^(5/2)*(360*a^5*b*e^6*x-60*b^6*d^5*e*x+72*a*b^5*e^6*x^5-12 *b^6*d*e^5*x^5+225*a^2*b^4*e^6*x^4+15*b^6*d^2*e^4*x^4+400*a^3*b^3*e^6*x^3- 20*b^6*d^3*e^3*x^3+450*a^4*b^2*e^6*x^2+30*b^6*d^4*e^2*x^2-90*a*b^5*d*e^5*x ^4-300*a^2*b^4*d*e^5*x^3+120*a*b^5*d^2*e^4*x^3-600*a^3*b^3*d*e^5*x^2+450*a ^2*b^4*d^2*e^4*x^2-180*a*b^5*d^3*e^3*x^2-900*a^4*b^2*d*e^5*x-1200*ln(e*x+d )*a^3*b^3*d^3*e^3+900*ln(e*x+d)*a^2*b^4*d^4*e^2-360*ln(e*x+d)*a*b^5*d^5*e- 360*ln(e*x+d)*a^5*b*d*e^5+900*ln(e*x+d)*a^4*b^2*d^2*e^4+1200*a^3*b^3*d^2*e ^4*x-900*a^2*b^4*d^3*e^3*x+360*a*b^5*d^4*e^2*x+10*b^6*e^6*x^6+60*ln(e*x+d) *a^6*e^6+60*ln(e*x+d)*b^6*d^6)/(b*x+a)^5/e^7
Time = 0.29 (sec) , antiderivative size = 351, normalized size of antiderivative = 1.18 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{d+e x} \, dx=\frac {10 \, b^{6} e^{6} x^{6} - 12 \, {\left (b^{6} d e^{5} - 6 \, a b^{5} e^{6}\right )} x^{5} + 15 \, {\left (b^{6} d^{2} e^{4} - 6 \, a b^{5} d e^{5} + 15 \, a^{2} b^{4} e^{6}\right )} x^{4} - 20 \, {\left (b^{6} d^{3} e^{3} - 6 \, a b^{5} d^{2} e^{4} + 15 \, a^{2} b^{4} d e^{5} - 20 \, a^{3} b^{3} e^{6}\right )} x^{3} + 30 \, {\left (b^{6} d^{4} e^{2} - 6 \, a b^{5} d^{3} e^{3} + 15 \, a^{2} b^{4} d^{2} e^{4} - 20 \, a^{3} b^{3} d e^{5} + 15 \, a^{4} b^{2} e^{6}\right )} x^{2} - 60 \, {\left (b^{6} d^{5} e - 6 \, a b^{5} d^{4} e^{2} + 15 \, a^{2} b^{4} d^{3} e^{3} - 20 \, a^{3} b^{3} d^{2} e^{4} + 15 \, a^{4} b^{2} d e^{5} - 6 \, a^{5} b e^{6}\right )} x + 60 \, {\left (b^{6} d^{6} - 6 \, a b^{5} d^{5} e + 15 \, a^{2} b^{4} d^{4} e^{2} - 20 \, a^{3} b^{3} d^{3} e^{3} + 15 \, a^{4} b^{2} d^{2} e^{4} - 6 \, a^{5} b d e^{5} + a^{6} e^{6}\right )} \log \left (e x + d\right )}{60 \, e^{7}} \]
1/60*(10*b^6*e^6*x^6 - 12*(b^6*d*e^5 - 6*a*b^5*e^6)*x^5 + 15*(b^6*d^2*e^4 - 6*a*b^5*d*e^5 + 15*a^2*b^4*e^6)*x^4 - 20*(b^6*d^3*e^3 - 6*a*b^5*d^2*e^4 + 15*a^2*b^4*d*e^5 - 20*a^3*b^3*e^6)*x^3 + 30*(b^6*d^4*e^2 - 6*a*b^5*d^3*e ^3 + 15*a^2*b^4*d^2*e^4 - 20*a^3*b^3*d*e^5 + 15*a^4*b^2*e^6)*x^2 - 60*(b^6 *d^5*e - 6*a*b^5*d^4*e^2 + 15*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 - 6*a^5*b*e^6)*x + 60*(b^6*d^6 - 6*a*b^5*d^5*e + 15*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 - 6*a^5*b*d*e^5 + a^6*e^6) *log(e*x + d))/e^7
\[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{d+e x} \, dx=\int \frac {\left (a + b x\right ) \left (\left (a + b x\right )^{2}\right )^{\frac {5}{2}}}{d + e x}\, dx \]
Exception generated. \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{d+e x} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more de tails)Is e
Leaf count of result is larger than twice the leaf count of optimal. 543 vs. \(2 (211) = 422\).
Time = 0.28 (sec) , antiderivative size = 543, normalized size of antiderivative = 1.82 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{d+e x} \, dx=\frac {10 \, b^{6} e^{5} x^{6} \mathrm {sgn}\left (b x + a\right ) - 12 \, b^{6} d e^{4} x^{5} \mathrm {sgn}\left (b x + a\right ) + 72 \, a b^{5} e^{5} x^{5} \mathrm {sgn}\left (b x + a\right ) + 15 \, b^{6} d^{2} e^{3} x^{4} \mathrm {sgn}\left (b x + a\right ) - 90 \, a b^{5} d e^{4} x^{4} \mathrm {sgn}\left (b x + a\right ) + 225 \, a^{2} b^{4} e^{5} x^{4} \mathrm {sgn}\left (b x + a\right ) - 20 \, b^{6} d^{3} e^{2} x^{3} \mathrm {sgn}\left (b x + a\right ) + 120 \, a b^{5} d^{2} e^{3} x^{3} \mathrm {sgn}\left (b x + a\right ) - 300 \, a^{2} b^{4} d e^{4} x^{3} \mathrm {sgn}\left (b x + a\right ) + 400 \, a^{3} b^{3} e^{5} x^{3} \mathrm {sgn}\left (b x + a\right ) + 30 \, b^{6} d^{4} e x^{2} \mathrm {sgn}\left (b x + a\right ) - 180 \, a b^{5} d^{3} e^{2} x^{2} \mathrm {sgn}\left (b x + a\right ) + 450 \, a^{2} b^{4} d^{2} e^{3} x^{2} \mathrm {sgn}\left (b x + a\right ) - 600 \, a^{3} b^{3} d e^{4} x^{2} \mathrm {sgn}\left (b x + a\right ) + 450 \, a^{4} b^{2} e^{5} x^{2} \mathrm {sgn}\left (b x + a\right ) - 60 \, b^{6} d^{5} x \mathrm {sgn}\left (b x + a\right ) + 360 \, a b^{5} d^{4} e x \mathrm {sgn}\left (b x + a\right ) - 900 \, a^{2} b^{4} d^{3} e^{2} x \mathrm {sgn}\left (b x + a\right ) + 1200 \, a^{3} b^{3} d^{2} e^{3} x \mathrm {sgn}\left (b x + a\right ) - 900 \, a^{4} b^{2} d e^{4} x \mathrm {sgn}\left (b x + a\right ) + 360 \, a^{5} b e^{5} x \mathrm {sgn}\left (b x + a\right )}{60 \, e^{6}} + \frac {{\left (b^{6} d^{6} \mathrm {sgn}\left (b x + a\right ) - 6 \, a b^{5} d^{5} e \mathrm {sgn}\left (b x + a\right ) + 15 \, 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 ) - 6 \, a^{5} b d e^{5} \mathrm {sgn}\left (b x + a\right ) + a^{6} e^{6} \mathrm {sgn}\left (b x + a\right )\right )} \log \left ({\left | e x + d \right |}\right )}{e^{7}} \]
1/60*(10*b^6*e^5*x^6*sgn(b*x + a) - 12*b^6*d*e^4*x^5*sgn(b*x + a) + 72*a*b ^5*e^5*x^5*sgn(b*x + a) + 15*b^6*d^2*e^3*x^4*sgn(b*x + a) - 90*a*b^5*d*e^4 *x^4*sgn(b*x + a) + 225*a^2*b^4*e^5*x^4*sgn(b*x + a) - 20*b^6*d^3*e^2*x^3* sgn(b*x + a) + 120*a*b^5*d^2*e^3*x^3*sgn(b*x + a) - 300*a^2*b^4*d*e^4*x^3* sgn(b*x + a) + 400*a^3*b^3*e^5*x^3*sgn(b*x + a) + 30*b^6*d^4*e*x^2*sgn(b*x + a) - 180*a*b^5*d^3*e^2*x^2*sgn(b*x + a) + 450*a^2*b^4*d^2*e^3*x^2*sgn(b *x + a) - 600*a^3*b^3*d*e^4*x^2*sgn(b*x + a) + 450*a^4*b^2*e^5*x^2*sgn(b*x + a) - 60*b^6*d^5*x*sgn(b*x + a) + 360*a*b^5*d^4*e*x*sgn(b*x + a) - 900*a ^2*b^4*d^3*e^2*x*sgn(b*x + a) + 1200*a^3*b^3*d^2*e^3*x*sgn(b*x + a) - 900* a^4*b^2*d*e^4*x*sgn(b*x + a) + 360*a^5*b*e^5*x*sgn(b*x + a))/e^6 + (b^6*d^ 6*sgn(b*x + a) - 6*a*b^5*d^5*e*sgn(b*x + a) + 15*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) - 6*a^5*b*d*e^5*sgn(b*x + a) + a^6*e^6*sgn(b*x + a))*log(abs(e*x + d))/e^7
Timed out. \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{d+e x} \, dx=\int \frac {\left (a+b\,x\right )\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2}}{d+e\,x} \,d x \]