Integrand size = 33, antiderivative size = 210 \[ \int \frac {(a+b x) (d+e x)^4}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=\frac {4 e^2 (b d-a e)^2 x (a+b x)}{b^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 e (b d-a e) (a+b x) (d+e x)^2}{b^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {4 e (a+b x) (d+e x)^3}{3 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(d+e x)^4}{b \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {4 e (b d-a e)^3 (a+b x) \log (a+b x)}{b^5 \sqrt {a^2+2 a b x+b^2 x^2}} \]
4*e^2*(-a*e+b*d)^2*x*(b*x+a)/b^4/((b*x+a)^2)^(1/2)+2*e*(-a*e+b*d)*(b*x+a)* (e*x+d)^2/b^3/((b*x+a)^2)^(1/2)+4/3*e*(b*x+a)*(e*x+d)^3/b^2/((b*x+a)^2)^(1 /2)-(e*x+d)^4/b/((b*x+a)^2)^(1/2)+4*e*(-a*e+b*d)^3*(b*x+a)*ln(b*x+a)/b^5/( (b*x+a)^2)^(1/2)
Time = 1.06 (sec) , antiderivative size = 170, normalized size of antiderivative = 0.81 \[ \int \frac {(a+b x) (d+e x)^4}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=\frac {-3 a^4 e^4+3 a^3 b e^3 (4 d+3 e x)+6 a^2 b^2 e^2 \left (-3 d^2-4 d e x+e^2 x^2\right )-2 a b^3 e \left (-6 d^3-9 d^2 e x+9 d e^2 x^2+e^3 x^3\right )+b^4 \left (-3 d^4+18 d^2 e^2 x^2+6 d e^3 x^3+e^4 x^4\right )-12 e (-b d+a e)^3 (a+b x) \log (a+b x)}{3 b^5 \sqrt {(a+b x)^2}} \]
(-3*a^4*e^4 + 3*a^3*b*e^3*(4*d + 3*e*x) + 6*a^2*b^2*e^2*(-3*d^2 - 4*d*e*x + e^2*x^2) - 2*a*b^3*e*(-6*d^3 - 9*d^2*e*x + 9*d*e^2*x^2 + e^3*x^3) + b^4* (-3*d^4 + 18*d^2*e^2*x^2 + 6*d*e^3*x^3 + e^4*x^4) - 12*e*(-(b*d) + a*e)^3* (a + b*x)*Log[a + b*x])/(3*b^5*Sqrt[(a + b*x)^2])
Time = 0.33 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.62, 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.
| \(\displaystyle \int \frac {(a+b x) (d+e x)^4}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 1187 |
| \(\displaystyle \frac {b^3 (a+b x) \int \frac {(d+e x)^4}{b^3 (a+b x)^2}dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {(a+b x) \int \frac {(d+e x)^4}{(a+b x)^2}dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
| \(\Big \downarrow \) 49 |
| \(\displaystyle \frac {(a+b x) \int \left (\frac {(a+b x)^2 e^4}{b^4}+\frac {4 (b d-a e) (a+b x) e^3}{b^4}+\frac {6 (b d-a e)^2 e^2}{b^4}+\frac {4 (b d-a e)^3 e}{b^4 (a+b x)}+\frac {(b d-a e)^4}{b^4 (a+b x)^2}\right )dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {(a+b x) \left (\frac {2 e^3 (a+b x)^2 (b d-a e)}{b^5}-\frac {(b d-a e)^4}{b^5 (a+b x)}+\frac {4 e (b d-a e)^3 \log (a+b x)}{b^5}+\frac {e^4 (a+b x)^3}{3 b^5}+\frac {6 e^2 x (b d-a e)^2}{b^4}\right )}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
((a + b*x)*((6*e^2*(b*d - a*e)^2*x)/b^4 - (b*d - a*e)^4/(b^5*(a + b*x)) + (2*e^3*(b*d - a*e)*(a + b*x)^2)/b^5 + (e^4*(a + b*x)^3)/(3*b^5) + (4*e*(b* d - a*e)^3*Log[a + b*x])/b^5))/Sqrt[a^2 + 2*a*b*x + b^2*x^2]
3.21.25.3.1 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 = 0.66 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.03
| method | result | size |
| risch | \(\frac {\sqrt {\left (b x +a \right )^{2}}\, e^{2} \left (\frac {1}{3} b^{2} e^{2} x^{3}-a b \,e^{2} x^{2}+2 b^{2} d e \,x^{2}+3 e^{2} a^{2} x -8 a b d e x +6 b^{2} d^{2} x \right )}{\left (b x +a \right ) b^{4}}-\frac {4 \sqrt {\left (b x +a \right )^{2}}\, e \left (a^{3} e^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right ) \ln \left (b x +a \right )}{\left (b x +a \right ) b^{5}}-\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (e^{4} a^{4}-4 b d \,e^{3} a^{3}+6 b^{2} d^{2} e^{2} a^{2}-4 b^{3} d^{3} e a +b^{4} d^{4}\right )}{\left (b x +a \right )^{2} b^{5}}\) | \(216\) |
| default | \(-\frac {\left (-e^{4} x^{4} b^{4}+2 x^{3} a \,b^{3} e^{4}-6 x^{3} b^{4} d \,e^{3}+12 \ln \left (b x +a \right ) x \,a^{3} b \,e^{4}-36 \ln \left (b x +a \right ) x \,a^{2} b^{2} d \,e^{3}+36 \ln \left (b x +a \right ) x a \,b^{3} d^{2} e^{2}-12 \ln \left (b x +a \right ) b^{4} d^{3} e x -6 x^{2} a^{2} b^{2} e^{4}+18 x^{2} a \,b^{3} d \,e^{3}-18 x^{2} b^{4} d^{2} e^{2}+12 \ln \left (b x +a \right ) a^{4} e^{4}-36 \ln \left (b x +a \right ) a^{3} b d \,e^{3}+36 \ln \left (b x +a \right ) a^{2} b^{2} d^{2} e^{2}-12 \ln \left (b x +a \right ) a \,b^{3} d^{3} e -9 x \,a^{3} b \,e^{4}+24 x \,a^{2} b^{2} d \,e^{3}-18 x a \,b^{3} d^{2} e^{2}+3 e^{4} a^{4}-12 b d \,e^{3} a^{3}+18 b^{2} d^{2} e^{2} a^{2}-12 b^{3} d^{3} e a +3 b^{4} d^{4}\right ) \left (b x +a \right )^{2}}{3 b^{5} \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}\) | \(321\) |
((b*x+a)^2)^(1/2)/(b*x+a)*e^2/b^4*(1/3*b^2*e^2*x^3-a*b*e^2*x^2+2*b^2*d*e*x ^2+3*e^2*a^2*x-8*a*b*d*e*x+6*b^2*d^2*x)-4*((b*x+a)^2)^(1/2)/(b*x+a)/b^5*e* (a^3*e^3-3*a^2*b*d*e^2+3*a*b^2*d^2*e-b^3*d^3)*ln(b*x+a)-((b*x+a)^2)^(1/2)/ (b*x+a)^2*(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)/ b^5
Time = 0.35 (sec) , antiderivative size = 268, normalized size of antiderivative = 1.28 \[ \int \frac {(a+b x) (d+e x)^4}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=\frac {b^{4} e^{4} x^{4} - 3 \, b^{4} d^{4} + 12 \, a b^{3} d^{3} e - 18 \, a^{2} b^{2} d^{2} e^{2} + 12 \, a^{3} b d e^{3} - 3 \, a^{4} e^{4} + 2 \, {\left (3 \, b^{4} d e^{3} - a b^{3} e^{4}\right )} x^{3} + 6 \, {\left (3 \, b^{4} d^{2} e^{2} - 3 \, a b^{3} d e^{3} + a^{2} b^{2} e^{4}\right )} x^{2} + 3 \, {\left (6 \, a b^{3} d^{2} e^{2} - 8 \, a^{2} b^{2} d e^{3} + 3 \, a^{3} b e^{4}\right )} x + 12 \, {\left (a b^{3} d^{3} e - 3 \, a^{2} b^{2} d^{2} e^{2} + 3 \, a^{3} b d e^{3} - a^{4} e^{4} + {\left (b^{4} d^{3} e - 3 \, a b^{3} d^{2} e^{2} + 3 \, a^{2} b^{2} d e^{3} - a^{3} b e^{4}\right )} x\right )} \log \left (b x + a\right )}{3 \, {\left (b^{6} x + a b^{5}\right )}} \]
1/3*(b^4*e^4*x^4 - 3*b^4*d^4 + 12*a*b^3*d^3*e - 18*a^2*b^2*d^2*e^2 + 12*a^ 3*b*d*e^3 - 3*a^4*e^4 + 2*(3*b^4*d*e^3 - a*b^3*e^4)*x^3 + 6*(3*b^4*d^2*e^2 - 3*a*b^3*d*e^3 + a^2*b^2*e^4)*x^2 + 3*(6*a*b^3*d^2*e^2 - 8*a^2*b^2*d*e^3 + 3*a^3*b*e^4)*x + 12*(a*b^3*d^3*e - 3*a^2*b^2*d^2*e^2 + 3*a^3*b*d*e^3 - a^4*e^4 + (b^4*d^3*e - 3*a*b^3*d^2*e^2 + 3*a^2*b^2*d*e^3 - a^3*b*e^4)*x)*l og(b*x + a))/(b^6*x + a*b^5)
\[ \int \frac {(a+b x) (d+e x)^4}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=\int \frac {\left (a + b x\right ) \left (d + e x\right )^{4}}{\left (\left (a + b x\right )^{2}\right )^{\frac {3}{2}}}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 754 vs. \(2 (153) = 306\).
Time = 0.20 (sec) , antiderivative size = 754, normalized size of antiderivative = 3.59 \[ \int \frac {(a+b x) (d+e x)^4}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=\frac {e^{4} x^{4}}{3 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} b} - \frac {7 \, a e^{4} x^{3}}{6 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} b^{2}} + \frac {9 \, a^{2} e^{4} x^{2}}{2 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} b^{3}} - \frac {10 \, a^{3} e^{4} \log \left (x + \frac {a}{b}\right )}{b^{5}} + \frac {9 \, a^{4} e^{4}}{\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} b^{5}} + \frac {{\left (4 \, b d e^{3} + a e^{4}\right )} x^{3}}{2 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} b^{2}} - \frac {20 \, a^{4} e^{4} x}{b^{6} {\left (x + \frac {a}{b}\right )}^{2}} - \frac {5 \, {\left (4 \, b d e^{3} + a e^{4}\right )} a x^{2}}{2 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} b^{3}} + \frac {2 \, {\left (3 \, b d^{2} e^{2} + 2 \, a d e^{3}\right )} x^{2}}{\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} b^{2}} - \frac {a d^{4}}{2 \, b^{3} {\left (x + \frac {a}{b}\right )}^{2}} - \frac {39 \, a^{5} e^{4}}{2 \, b^{7} {\left (x + \frac {a}{b}\right )}^{2}} + \frac {6 \, {\left (4 \, b d e^{3} + a e^{4}\right )} a^{2} \log \left (x + \frac {a}{b}\right )}{b^{5}} - \frac {6 \, {\left (3 \, b d^{2} e^{2} + 2 \, a d e^{3}\right )} a \log \left (x + \frac {a}{b}\right )}{b^{4}} + \frac {2 \, {\left (2 \, b d^{3} e + 3 \, a d^{2} e^{2}\right )} \log \left (x + \frac {a}{b}\right )}{b^{3}} - \frac {5 \, {\left (4 \, b d e^{3} + a e^{4}\right )} a^{3}}{\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} b^{5}} + \frac {4 \, {\left (3 \, b d^{2} e^{2} + 2 \, a d e^{3}\right )} a^{2}}{\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} b^{4}} - \frac {b d^{4} + 4 \, a d^{3} e}{\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} b^{2}} + \frac {12 \, {\left (4 \, b d e^{3} + a e^{4}\right )} a^{3} x}{b^{6} {\left (x + \frac {a}{b}\right )}^{2}} - \frac {12 \, {\left (3 \, b d^{2} e^{2} + 2 \, a d e^{3}\right )} a^{2} x}{b^{5} {\left (x + \frac {a}{b}\right )}^{2}} + \frac {4 \, {\left (2 \, b d^{3} e + 3 \, a d^{2} e^{2}\right )} a x}{b^{4} {\left (x + \frac {a}{b}\right )}^{2}} + \frac {23 \, {\left (4 \, b d e^{3} + a e^{4}\right )} a^{4}}{2 \, b^{7} {\left (x + \frac {a}{b}\right )}^{2}} - \frac {11 \, {\left (3 \, b d^{2} e^{2} + 2 \, a d e^{3}\right )} a^{3}}{b^{6} {\left (x + \frac {a}{b}\right )}^{2}} + \frac {3 \, {\left (2 \, b d^{3} e + 3 \, a d^{2} e^{2}\right )} a^{2}}{b^{5} {\left (x + \frac {a}{b}\right )}^{2}} + \frac {{\left (b d^{4} + 4 \, a d^{3} e\right )} a}{2 \, b^{4} {\left (x + \frac {a}{b}\right )}^{2}} \]
1/3*e^4*x^4/(sqrt(b^2*x^2 + 2*a*b*x + a^2)*b) - 7/6*a*e^4*x^3/(sqrt(b^2*x^ 2 + 2*a*b*x + a^2)*b^2) + 9/2*a^2*e^4*x^2/(sqrt(b^2*x^2 + 2*a*b*x + a^2)*b ^3) - 10*a^3*e^4*log(x + a/b)/b^5 + 9*a^4*e^4/(sqrt(b^2*x^2 + 2*a*b*x + a^ 2)*b^5) + 1/2*(4*b*d*e^3 + a*e^4)*x^3/(sqrt(b^2*x^2 + 2*a*b*x + a^2)*b^2) - 20*a^4*e^4*x/(b^6*(x + a/b)^2) - 5/2*(4*b*d*e^3 + a*e^4)*a*x^2/(sqrt(b^2 *x^2 + 2*a*b*x + a^2)*b^3) + 2*(3*b*d^2*e^2 + 2*a*d*e^3)*x^2/(sqrt(b^2*x^2 + 2*a*b*x + a^2)*b^2) - 1/2*a*d^4/(b^3*(x + a/b)^2) - 39/2*a^5*e^4/(b^7*( x + a/b)^2) + 6*(4*b*d*e^3 + a*e^4)*a^2*log(x + a/b)/b^5 - 6*(3*b*d^2*e^2 + 2*a*d*e^3)*a*log(x + a/b)/b^4 + 2*(2*b*d^3*e + 3*a*d^2*e^2)*log(x + a/b) /b^3 - 5*(4*b*d*e^3 + a*e^4)*a^3/(sqrt(b^2*x^2 + 2*a*b*x + a^2)*b^5) + 4*( 3*b*d^2*e^2 + 2*a*d*e^3)*a^2/(sqrt(b^2*x^2 + 2*a*b*x + a^2)*b^4) - (b*d^4 + 4*a*d^3*e)/(sqrt(b^2*x^2 + 2*a*b*x + a^2)*b^2) + 12*(4*b*d*e^3 + a*e^4)* a^3*x/(b^6*(x + a/b)^2) - 12*(3*b*d^2*e^2 + 2*a*d*e^3)*a^2*x/(b^5*(x + a/b )^2) + 4*(2*b*d^3*e + 3*a*d^2*e^2)*a*x/(b^4*(x + a/b)^2) + 23/2*(4*b*d*e^3 + a*e^4)*a^4/(b^7*(x + a/b)^2) - 11*(3*b*d^2*e^2 + 2*a*d*e^3)*a^3/(b^6*(x + a/b)^2) + 3*(2*b*d^3*e + 3*a*d^2*e^2)*a^2/(b^5*(x + a/b)^2) + 1/2*(b*d^ 4 + 4*a*d^3*e)*a/(b^4*(x + a/b)^2)
Time = 0.27 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.01 \[ \int \frac {(a+b x) (d+e x)^4}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=\frac {4 \, {\left (b^{3} d^{3} e - 3 \, a b^{2} d^{2} e^{2} + 3 \, a^{2} b d e^{3} - a^{3} e^{4}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{5} \mathrm {sgn}\left (b x + a\right )} + \frac {b^{4} e^{4} x^{3} + 6 \, b^{4} d e^{3} x^{2} - 3 \, a b^{3} e^{4} x^{2} + 18 \, b^{4} d^{2} e^{2} x - 24 \, a b^{3} d e^{3} x + 9 \, a^{2} b^{2} e^{4} x}{3 \, b^{6} \mathrm {sgn}\left (b x + a\right )} - \frac {b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4}}{{\left (b x + a\right )} b^{5} \mathrm {sgn}\left (b x + a\right )} \]
4*(b^3*d^3*e - 3*a*b^2*d^2*e^2 + 3*a^2*b*d*e^3 - a^3*e^4)*log(abs(b*x + a) )/(b^5*sgn(b*x + a)) + 1/3*(b^4*e^4*x^3 + 6*b^4*d*e^3*x^2 - 3*a*b^3*e^4*x^ 2 + 18*b^4*d^2*e^2*x - 24*a*b^3*d*e^3*x + 9*a^2*b^2*e^4*x)/(b^6*sgn(b*x + a)) - (b^4*d^4 - 4*a*b^3*d^3*e + 6*a^2*b^2*d^2*e^2 - 4*a^3*b*d*e^3 + a^4*e ^4)/((b*x + a)*b^5*sgn(b*x + a))
Timed out. \[ \int \frac {(a+b x) (d+e x)^4}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=\int \frac {\left (a+b\,x\right )\,{\left (d+e\,x\right )}^4}{{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2}} \,d x \]