\(\int \frac {(d+e x)^5}{a^2+2 a b x+b^2 x^2} \, dx\) [108]

Optimal result

Integrand size = 26, antiderivative size = 131 \[ \int \frac {(d+e x)^5}{a^2+2 a b x+b^2 x^2} \, dx=\frac {10 e^2 (b d-a e)^3 x}{b^5}-\frac {(b d-a e)^5}{b^6 (a+b x)}+\frac {5 e^3 (b d-a e)^2 (a+b x)^2}{b^6}+\frac {5 e^4 (b d-a e) (a+b x)^3}{3 b^6}+\frac {e^5 (a+b x)^4}{4 b^6}+\frac {5 e (b d-a e)^4 \log (a+b x)}{b^6} \] Output:

10*e^2*(-a*e+b*d)^3*x/b^5-(-a*e+b*d)^5/b^6/(b*x+a)+5*e^3*(-a*e+b*d)^2*(b*x 
+a)^2/b^6+5/3*e^4*(-a*e+b*d)*(b*x+a)^3/b^6+1/4*e^5*(b*x+a)^4/b^6+5*e*(-a*e 
+b*d)^4*ln(b*x+a)/b^6
 

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.76 \[ \int \frac {(d+e x)^5}{a^2+2 a b x+b^2 x^2} \, dx=\frac {12 a^5 e^5-12 a^4 b e^4 (5 d+4 e x)+30 a^3 b^2 e^3 \left (4 d^2+6 d e x-e^2 x^2\right )+10 a^2 b^3 e^2 \left (-12 d^3-24 d^2 e x+12 d e^2 x^2+e^3 x^3\right )-5 a b^4 e \left (-12 d^4-24 d^3 e x+36 d^2 e^2 x^2+8 d e^3 x^3+e^4 x^4\right )+b^5 \left (-12 d^5+120 d^3 e^2 x^2+60 d^2 e^3 x^3+20 d e^4 x^4+3 e^5 x^5\right )+60 e (b d-a e)^4 (a+b x) \log (a+b x)}{12 b^6 (a+b x)} \] Input:

Integrate[(d + e*x)^5/(a^2 + 2*a*b*x + b^2*x^2),x]
 

Output:

(12*a^5*e^5 - 12*a^4*b*e^4*(5*d + 4*e*x) + 30*a^3*b^2*e^3*(4*d^2 + 6*d*e*x 
 - e^2*x^2) + 10*a^2*b^3*e^2*(-12*d^3 - 24*d^2*e*x + 12*d*e^2*x^2 + e^3*x^ 
3) - 5*a*b^4*e*(-12*d^4 - 24*d^3*e*x + 36*d^2*e^2*x^2 + 8*d*e^3*x^3 + e^4* 
x^4) + b^5*(-12*d^5 + 120*d^3*e^2*x^2 + 60*d^2*e^3*x^3 + 20*d*e^4*x^4 + 3* 
e^5*x^5) + 60*e*(b*d - a*e)^4*(a + b*x)*Log[a + b*x])/(12*b^6*(a + b*x))
 

Rubi [A] (verified)

Time = 0.62 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {1098, 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.

\(\displaystyle \int \frac {(d+e x)^5}{a^2+2 a b x+b^2 x^2} \, dx\)

\(\Big \downarrow \) 1098

\(\displaystyle b^2 \int \frac {(d+e x)^5}{b^2 (a+b x)^2}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \int \frac {(d+e x)^5}{(a+b x)^2}dx\)

\(\Big \downarrow \) 49

\(\displaystyle \int \left (\frac {5 e^4 (a+b x)^2 (b d-a e)}{b^5}+\frac {10 e^3 (a+b x) (b d-a e)^2}{b^5}+\frac {10 e^2 (b d-a e)^3}{b^5}+\frac {5 e (b d-a e)^4}{b^5 (a+b x)}+\frac {(b d-a e)^5}{b^5 (a+b x)^2}+\frac {e^5 (a+b x)^3}{b^5}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {5 e^4 (a+b x)^3 (b d-a e)}{3 b^6}+\frac {5 e^3 (a+b x)^2 (b d-a e)^2}{b^6}-\frac {(b d-a e)^5}{b^6 (a+b x)}+\frac {5 e (b d-a e)^4 \log (a+b x)}{b^6}+\frac {e^5 (a+b x)^4}{4 b^6}+\frac {10 e^2 x (b d-a e)^3}{b^5}\)

Input:

Int[(d + e*x)^5/(a^2 + 2*a*b*x + b^2*x^2),x]
 

Output:

(10*e^2*(b*d - a*e)^3*x)/b^5 - (b*d - a*e)^5/(b^6*(a + b*x)) + (5*e^3*(b*d 
 - a*e)^2*(a + b*x)^2)/b^6 + (5*e^4*(b*d - a*e)*(a + b*x)^3)/(3*b^6) + (e^ 
5*(a + b*x)^4)/(4*b^6) + (5*e*(b*d - a*e)^4*Log[a + b*x])/b^6
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 49

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]
 

rule 1098

Int[((d_.) + (e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_ 
Symbol] :> Simp[1/c^p   Int[(d + e*x)^m*(b/2 + c*x)^(2*p), x], x] /; FreeQ[ 
{a, b, c, d, e, m}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(255\) vs. \(2(127)=254\).

Time = 1.12 (sec) , antiderivative size = 256, normalized size of antiderivative = 1.95

method	result	size
norman	\(\frac {\frac {e^{5} x^{5}}{4 b}-\frac {5 e^{2} \left (e^{3} a^{3}-4 a^{2} b d \,e^{2}+6 a \,b^{2} d^{2} e -4 b^{3} d^{3}\right ) x^{2}}{2 b^{4}}+\frac {5 e^{3} \left (e^{2} a^{2}-4 a b d e +6 b^{2} d^{2}\right ) x^{3}}{6 b^{3}}-\frac {5 e^{4} \left (a e -4 b d \right ) x^{4}}{12 b^{2}}-\frac {\left (5 e^{5} a^{5}-20 a^{4} b d \,e^{4}+30 a^{3} b^{2} d^{2} e^{3}-20 a^{2} b^{3} d^{3} e^{2}+5 a \,b^{4} d^{4} e -b^{5} d^{5}\right ) x}{a \,b^{5}}}{b x +a}+\frac {5 e \left (a^{4} e^{4}-4 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right ) \ln \left (b x +a \right )}{b^{6}}\)	\(256\)
default	\(-\frac {e^{2} \left (-\frac {1}{4} b^{3} x^{4} e^{3}+\frac {2}{3} a \,b^{2} e^{3} x^{3}-\frac {5}{3} b^{3} d \,e^{2} x^{3}-\frac {3}{2} a^{2} b \,e^{3} x^{2}+5 a \,b^{2} d \,e^{2} x^{2}-5 b^{3} d^{2} e \,x^{2}+4 e^{3} a^{3} x -15 a^{2} b d \,e^{2} x +20 a \,b^{2} d^{2} e x -10 b^{3} d^{3} x \right )}{b^{5}}-\frac {-e^{5} a^{5}+5 a^{4} b d \,e^{4}-10 a^{3} b^{2} d^{2} e^{3}+10 a^{2} b^{3} d^{3} e^{2}-5 a \,b^{4} d^{4} e +b^{5} d^{5}}{b^{6} \left (b x +a \right )}+\frac {5 e \left (a^{4} e^{4}-4 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right ) \ln \left (b x +a \right )}{b^{6}}\)	\(260\)
risch	\(\frac {e^{5} x^{4}}{4 b^{2}}-\frac {2 e^{5} a \,x^{3}}{3 b^{3}}+\frac {5 e^{4} d \,x^{3}}{3 b^{2}}+\frac {3 e^{5} a^{2} x^{2}}{2 b^{4}}-\frac {5 e^{4} a d \,x^{2}}{b^{3}}+\frac {5 e^{3} d^{2} x^{2}}{b^{2}}-\frac {4 e^{5} a^{3} x}{b^{5}}+\frac {15 e^{4} a^{2} d x}{b^{4}}-\frac {20 e^{3} a \,d^{2} x}{b^{3}}+\frac {10 e^{2} d^{3} x}{b^{2}}+\frac {e^{5} a^{5}}{b^{6} \left (b x +a \right )}-\frac {5 a^{4} d \,e^{4}}{b^{5} \left (b x +a \right )}+\frac {10 a^{3} d^{2} e^{3}}{b^{4} \left (b x +a \right )}-\frac {10 a^{2} d^{3} e^{2}}{b^{3} \left (b x +a \right )}+\frac {5 a \,d^{4} e}{b^{2} \left (b x +a \right )}-\frac {d^{5}}{b \left (b x +a \right )}+\frac {5 e^{5} \ln \left (b x +a \right ) a^{4}}{b^{6}}-\frac {20 e^{4} \ln \left (b x +a \right ) a^{3} d}{b^{5}}+\frac {30 e^{3} \ln \left (b x +a \right ) a^{2} d^{2}}{b^{4}}-\frac {20 e^{2} \ln \left (b x +a \right ) a \,d^{3}}{b^{3}}+\frac {5 e \ln \left (b x +a \right ) d^{4}}{b^{2}}\)	\(326\)
parallelrisch	\(\frac {-240 \ln \left (b x +a \right ) x \,a^{3} b^{2} d \,e^{4}-240 \ln \left (b x +a \right ) x a \,b^{4} d^{3} e^{2}-12 b^{5} d^{5}+360 \ln \left (b x +a \right ) x \,a^{2} b^{3} d^{2} e^{3}-240 a^{4} b d \,e^{4}+360 a^{3} b^{2} d^{2} e^{3}-240 a^{2} b^{3} d^{3} e^{2}+60 a \,b^{4} d^{4} e +120 x^{2} a^{2} b^{3} d \,e^{4}-180 x^{2} a \,b^{4} d^{2} e^{3}-40 x^{3} a \,b^{4} d \,e^{4}+60 \ln \left (b x +a \right ) x \,a^{4} b \,e^{5}+60 \ln \left (b x +a \right ) x \,b^{5} d^{4} e -240 \ln \left (b x +a \right ) a^{4} b d \,e^{4}+360 \ln \left (b x +a \right ) a^{3} b^{2} d^{2} e^{3}-240 \ln \left (b x +a \right ) a^{2} b^{3} d^{3} e^{2}+60 \ln \left (b x +a \right ) a \,b^{4} d^{4} e -5 x^{4} a \,b^{4} e^{5}+20 x^{4} b^{5} d \,e^{4}+10 x^{3} a^{2} b^{3} e^{5}+60 x^{3} b^{5} d^{2} e^{3}-30 x^{2} a^{3} b^{2} e^{5}+120 x^{2} b^{5} d^{3} e^{2}+60 e^{5} a^{5}+3 x^{5} e^{5} b^{5}+60 \ln \left (b x +a \right ) a^{5} e^{5}}{12 b^{6} \left (b x +a \right )}\)	\(389\)

Input:

int((e*x+d)^5/(b^2*x^2+2*a*b*x+a^2),x,method=_RETURNVERBOSE)
 

Output:

(1/4*e^5/b*x^5-5/2*e^2*(a^3*e^3-4*a^2*b*d*e^2+6*a*b^2*d^2*e-4*b^3*d^3)/b^4 
*x^2+5/6*e^3*(a^2*e^2-4*a*b*d*e+6*b^2*d^2)/b^3*x^3-5/12*e^4*(a*e-4*b*d)/b^ 
2*x^4-(5*a^5*e^5-20*a^4*b*d*e^4+30*a^3*b^2*d^2*e^3-20*a^2*b^3*d^3*e^2+5*a* 
b^4*d^4*e-b^5*d^5)/a/b^5*x)/(b*x+a)+5/b^6*e*(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)*ln(b*x+a)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 375 vs. \(2 (127) = 254\).

Time = 0.09 (sec) , antiderivative size = 375, normalized size of antiderivative = 2.86 \[ \int \frac {(d+e x)^5}{a^2+2 a b x+b^2 x^2} \, dx=\frac {3 \, b^{5} e^{5} x^{5} - 12 \, b^{5} d^{5} + 60 \, a b^{4} d^{4} e - 120 \, a^{2} b^{3} d^{3} e^{2} + 120 \, a^{3} b^{2} d^{2} e^{3} - 60 \, a^{4} b d e^{4} + 12 \, a^{5} e^{5} + 5 \, {\left (4 \, b^{5} d e^{4} - a b^{4} e^{5}\right )} x^{4} + 10 \, {\left (6 \, b^{5} d^{2} e^{3} - 4 \, a b^{4} d e^{4} + a^{2} b^{3} e^{5}\right )} x^{3} + 30 \, {\left (4 \, b^{5} d^{3} e^{2} - 6 \, a b^{4} d^{2} e^{3} + 4 \, a^{2} b^{3} d e^{4} - a^{3} b^{2} e^{5}\right )} x^{2} + 12 \, {\left (10 \, a b^{4} d^{3} e^{2} - 20 \, a^{2} b^{3} d^{2} e^{3} + 15 \, a^{3} b^{2} d e^{4} - 4 \, a^{4} b e^{5}\right )} x + 60 \, {\left (a b^{4} d^{4} e - 4 \, a^{2} b^{3} d^{3} e^{2} + 6 \, a^{3} b^{2} d^{2} e^{3} - 4 \, a^{4} b d e^{4} + a^{5} e^{5} + {\left (b^{5} d^{4} e - 4 \, a b^{4} d^{3} e^{2} + 6 \, a^{2} b^{3} d^{2} e^{3} - 4 \, a^{3} b^{2} d e^{4} + a^{4} b e^{5}\right )} x\right )} \log \left (b x + a\right )}{12 \, {\left (b^{7} x + a b^{6}\right )}} \] Input:

integrate((e*x+d)^5/(b^2*x^2+2*a*b*x+a^2),x, algorithm="fricas")
 

Output:

1/12*(3*b^5*e^5*x^5 - 12*b^5*d^5 + 60*a*b^4*d^4*e - 120*a^2*b^3*d^3*e^2 + 
120*a^3*b^2*d^2*e^3 - 60*a^4*b*d*e^4 + 12*a^5*e^5 + 5*(4*b^5*d*e^4 - a*b^4 
*e^5)*x^4 + 10*(6*b^5*d^2*e^3 - 4*a*b^4*d*e^4 + a^2*b^3*e^5)*x^3 + 30*(4*b 
^5*d^3*e^2 - 6*a*b^4*d^2*e^3 + 4*a^2*b^3*d*e^4 - a^3*b^2*e^5)*x^2 + 12*(10 
*a*b^4*d^3*e^2 - 20*a^2*b^3*d^2*e^3 + 15*a^3*b^2*d*e^4 - 4*a^4*b*e^5)*x + 
60*(a*b^4*d^4*e - 4*a^2*b^3*d^3*e^2 + 6*a^3*b^2*d^2*e^3 - 4*a^4*b*d*e^4 + 
a^5*e^5 + (b^5*d^4*e - 4*a*b^4*d^3*e^2 + 6*a^2*b^3*d^2*e^3 - 4*a^3*b^2*d*e 
^4 + a^4*b*e^5)*x)*log(b*x + a))/(b^7*x + a*b^6)
 

Sympy [A] (verification not implemented)

Time = 0.48 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.76 \[ \int \frac {(d+e x)^5}{a^2+2 a b x+b^2 x^2} \, dx=x^{3} \left (- \frac {2 a e^{5}}{3 b^{3}} + \frac {5 d e^{4}}{3 b^{2}}\right ) + x^{2} \cdot \left (\frac {3 a^{2} e^{5}}{2 b^{4}} - \frac {5 a d e^{4}}{b^{3}} + \frac {5 d^{2} e^{3}}{b^{2}}\right ) + x \left (- \frac {4 a^{3} e^{5}}{b^{5}} + \frac {15 a^{2} d e^{4}}{b^{4}} - \frac {20 a d^{2} e^{3}}{b^{3}} + \frac {10 d^{3} e^{2}}{b^{2}}\right ) + \frac {a^{5} e^{5} - 5 a^{4} b d e^{4} + 10 a^{3} b^{2} d^{2} e^{3} - 10 a^{2} b^{3} d^{3} e^{2} + 5 a b^{4} d^{4} e - b^{5} d^{5}}{a b^{6} + b^{7} x} + \frac {e^{5} x^{4}}{4 b^{2}} + \frac {5 e \left (a e - b d\right )^{4} \log {\left (a + b x \right )}}{b^{6}} \] Input:

integrate((e*x+d)**5/(b**2*x**2+2*a*b*x+a**2),x)
 

Output:

x**3*(-2*a*e**5/(3*b**3) + 5*d*e**4/(3*b**2)) + x**2*(3*a**2*e**5/(2*b**4) 
 - 5*a*d*e**4/b**3 + 5*d**2*e**3/b**2) + x*(-4*a**3*e**5/b**5 + 15*a**2*d* 
e**4/b**4 - 20*a*d**2*e**3/b**3 + 10*d**3*e**2/b**2) + (a**5*e**5 - 5*a**4 
*b*d*e**4 + 10*a**3*b**2*d**2*e**3 - 10*a**2*b**3*d**3*e**2 + 5*a*b**4*d** 
4*e - b**5*d**5)/(a*b**6 + b**7*x) + e**5*x**4/(4*b**2) + 5*e*(a*e - b*d)* 
*4*log(a + b*x)/b**6
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 265 vs. \(2 (127) = 254\).

Time = 0.04 (sec) , antiderivative size = 265, normalized size of antiderivative = 2.02 \[ \int \frac {(d+e x)^5}{a^2+2 a b x+b^2 x^2} \, dx=-\frac {b^{5} d^{5} - 5 \, a b^{4} d^{4} e + 10 \, a^{2} b^{3} d^{3} e^{2} - 10 \, a^{3} b^{2} d^{2} e^{3} + 5 \, a^{4} b d e^{4} - a^{5} e^{5}}{b^{7} x + a b^{6}} + \frac {3 \, b^{3} e^{5} x^{4} + 4 \, {\left (5 \, b^{3} d e^{4} - 2 \, a b^{2} e^{5}\right )} x^{3} + 6 \, {\left (10 \, b^{3} d^{2} e^{3} - 10 \, a b^{2} d e^{4} + 3 \, a^{2} b e^{5}\right )} x^{2} + 12 \, {\left (10 \, b^{3} d^{3} e^{2} - 20 \, a b^{2} d^{2} e^{3} + 15 \, a^{2} b d e^{4} - 4 \, a^{3} e^{5}\right )} x}{12 \, b^{5}} + \frac {5 \, {\left (b^{4} d^{4} e - 4 \, a b^{3} d^{3} e^{2} + 6 \, a^{2} b^{2} d^{2} e^{3} - 4 \, a^{3} b d e^{4} + a^{4} e^{5}\right )} \log \left (b x + a\right )}{b^{6}} \] Input:

integrate((e*x+d)^5/(b^2*x^2+2*a*b*x+a^2),x, algorithm="maxima")
 

Output:

-(b^5*d^5 - 5*a*b^4*d^4*e + 10*a^2*b^3*d^3*e^2 - 10*a^3*b^2*d^2*e^3 + 5*a^ 
4*b*d*e^4 - a^5*e^5)/(b^7*x + a*b^6) + 1/12*(3*b^3*e^5*x^4 + 4*(5*b^3*d*e^ 
4 - 2*a*b^2*e^5)*x^3 + 6*(10*b^3*d^2*e^3 - 10*a*b^2*d*e^4 + 3*a^2*b*e^5)*x 
^2 + 12*(10*b^3*d^3*e^2 - 20*a*b^2*d^2*e^3 + 15*a^2*b*d*e^4 - 4*a^3*e^5)*x 
)/b^5 + 5*(b^4*d^4*e - 4*a*b^3*d^3*e^2 + 6*a^2*b^2*d^2*e^3 - 4*a^3*b*d*e^4 
 + a^4*e^5)*log(b*x + a)/b^6
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 273 vs. \(2 (127) = 254\).

Time = 0.22 (sec) , antiderivative size = 273, normalized size of antiderivative = 2.08 \[ \int \frac {(d+e x)^5}{a^2+2 a b x+b^2 x^2} \, dx=\frac {5 \, {\left (b^{4} d^{4} e - 4 \, a b^{3} d^{3} e^{2} + 6 \, a^{2} b^{2} d^{2} e^{3} - 4 \, a^{3} b d e^{4} + a^{4} e^{5}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{6}} - \frac {b^{5} d^{5} - 5 \, a b^{4} d^{4} e + 10 \, a^{2} b^{3} d^{3} e^{2} - 10 \, a^{3} b^{2} d^{2} e^{3} + 5 \, a^{4} b d e^{4} - a^{5} e^{5}}{{\left (b x + a\right )} b^{6}} + \frac {3 \, b^{6} e^{5} x^{4} + 20 \, b^{6} d e^{4} x^{3} - 8 \, a b^{5} e^{5} x^{3} + 60 \, b^{6} d^{2} e^{3} x^{2} - 60 \, a b^{5} d e^{4} x^{2} + 18 \, a^{2} b^{4} e^{5} x^{2} + 120 \, b^{6} d^{3} e^{2} x - 240 \, a b^{5} d^{2} e^{3} x + 180 \, a^{2} b^{4} d e^{4} x - 48 \, a^{3} b^{3} e^{5} x}{12 \, b^{8}} \] Input:

integrate((e*x+d)^5/(b^2*x^2+2*a*b*x+a^2),x, algorithm="giac")
 

Output:

5*(b^4*d^4*e - 4*a*b^3*d^3*e^2 + 6*a^2*b^2*d^2*e^3 - 4*a^3*b*d*e^4 + a^4*e 
^5)*log(abs(b*x + a))/b^6 - (b^5*d^5 - 5*a*b^4*d^4*e + 10*a^2*b^3*d^3*e^2 
- 10*a^3*b^2*d^2*e^3 + 5*a^4*b*d*e^4 - a^5*e^5)/((b*x + a)*b^6) + 1/12*(3* 
b^6*e^5*x^4 + 20*b^6*d*e^4*x^3 - 8*a*b^5*e^5*x^3 + 60*b^6*d^2*e^3*x^2 - 60 
*a*b^5*d*e^4*x^2 + 18*a^2*b^4*e^5*x^2 + 120*b^6*d^3*e^2*x - 240*a*b^5*d^2* 
e^3*x + 180*a^2*b^4*d*e^4*x - 48*a^3*b^3*e^5*x)/b^8
 

Mupad [B] (verification not implemented)

Time = 8.77 (sec) , antiderivative size = 326, normalized size of antiderivative = 2.49 \[ \int \frac {(d+e x)^5}{a^2+2 a b x+b^2 x^2} \, dx=x\,\left (\frac {10\,d^3\,e^2}{b^2}-\frac {2\,a\,\left (\frac {2\,a\,\left (\frac {2\,a\,e^5}{b^3}-\frac {5\,d\,e^4}{b^2}\right )}{b}-\frac {a^2\,e^5}{b^4}+\frac {10\,d^2\,e^3}{b^2}\right )}{b}+\frac {a^2\,\left (\frac {2\,a\,e^5}{b^3}-\frac {5\,d\,e^4}{b^2}\right )}{b^2}\right )-x^3\,\left (\frac {2\,a\,e^5}{3\,b^3}-\frac {5\,d\,e^4}{3\,b^2}\right )+x^2\,\left (\frac {a\,\left (\frac {2\,a\,e^5}{b^3}-\frac {5\,d\,e^4}{b^2}\right )}{b}-\frac {a^2\,e^5}{2\,b^4}+\frac {5\,d^2\,e^3}{b^2}\right )+\frac {\ln \left (a+b\,x\right )\,\left (5\,a^4\,e^5-20\,a^3\,b\,d\,e^4+30\,a^2\,b^2\,d^2\,e^3-20\,a\,b^3\,d^3\,e^2+5\,b^4\,d^4\,e\right )}{b^6}+\frac {a^5\,e^5-5\,a^4\,b\,d\,e^4+10\,a^3\,b^2\,d^2\,e^3-10\,a^2\,b^3\,d^3\,e^2+5\,a\,b^4\,d^4\,e-b^5\,d^5}{b\,\left (x\,b^6+a\,b^5\right )}+\frac {e^5\,x^4}{4\,b^2} \] Input:

int((d + e*x)^5/(a^2 + b^2*x^2 + 2*a*b*x),x)
 

Output:

x*((10*d^3*e^2)/b^2 - (2*a*((2*a*((2*a*e^5)/b^3 - (5*d*e^4)/b^2))/b - (a^2 
*e^5)/b^4 + (10*d^2*e^3)/b^2))/b + (a^2*((2*a*e^5)/b^3 - (5*d*e^4)/b^2))/b 
^2) - x^3*((2*a*e^5)/(3*b^3) - (5*d*e^4)/(3*b^2)) + x^2*((a*((2*a*e^5)/b^3 
 - (5*d*e^4)/b^2))/b - (a^2*e^5)/(2*b^4) + (5*d^2*e^3)/b^2) + (log(a + b*x 
)*(5*a^4*e^5 + 5*b^4*d^4*e - 20*a*b^3*d^3*e^2 + 30*a^2*b^2*d^2*e^3 - 20*a^ 
3*b*d*e^4))/b^6 + (a^5*e^5 - b^5*d^5 - 10*a^2*b^3*d^3*e^2 + 10*a^3*b^2*d^2 
*e^3 + 5*a*b^4*d^4*e - 5*a^4*b*d*e^4)/(b*(a*b^5 + b^6*x)) + (e^5*x^4)/(4*b 
^2)
 

Reduce [B] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 415, normalized size of antiderivative = 3.17 \[ \int \frac {(d+e x)^5}{a^2+2 a b x+b^2 x^2} \, dx=\frac {-240 \,\mathrm {log}\left (b x +a \right ) a^{5} b d \,e^{4}+60 \,\mathrm {log}\left (b x +a \right ) a^{5} b \,e^{5} x +360 \,\mathrm {log}\left (b x +a \right ) a^{4} b^{2} d^{2} e^{3}-240 \,\mathrm {log}\left (b x +a \right ) a^{3} b^{3} d^{3} e^{2}-240 \,\mathrm {log}\left (b x +a \right ) a^{4} b^{2} d \,e^{4} x +360 \,\mathrm {log}\left (b x +a \right ) a^{3} b^{3} d^{2} e^{3} x -240 \,\mathrm {log}\left (b x +a \right ) a^{2} b^{4} d^{3} e^{2} x +60 \,\mathrm {log}\left (b x +a \right ) a \,b^{5} d^{4} e x -60 a^{5} b \,e^{5} x -30 a^{4} b^{2} e^{5} x^{2}+10 a^{3} b^{3} e^{5} x^{3}-5 a^{2} b^{4} e^{5} x^{4}+3 a \,b^{5} e^{5} x^{5}+60 \,\mathrm {log}\left (b x +a \right ) a^{6} e^{5}+12 b^{6} d^{5} x +60 \,\mathrm {log}\left (b x +a \right ) a^{2} b^{4} d^{4} e +240 a^{4} b^{2} d \,e^{4} x -360 a^{3} b^{3} d^{2} e^{3} x +120 a^{3} b^{3} d \,e^{4} x^{2}+240 a^{2} b^{4} d^{3} e^{2} x -180 a^{2} b^{4} d^{2} e^{3} x^{2}-40 a^{2} b^{4} d \,e^{4} x^{3}-60 a \,b^{5} d^{4} e x +120 a \,b^{5} d^{3} e^{2} x^{2}+60 a \,b^{5} d^{2} e^{3} x^{3}+20 a \,b^{5} d \,e^{4} x^{4}}{12 a \,b^{6} \left (b x +a \right )} \] Input:

int((e*x+d)^5/(b^2*x^2+2*a*b*x+a^2),x)
 

Output:

(60*log(a + b*x)*a**6*e**5 - 240*log(a + b*x)*a**5*b*d*e**4 + 60*log(a + b 
*x)*a**5*b*e**5*x + 360*log(a + b*x)*a**4*b**2*d**2*e**3 - 240*log(a + b*x 
)*a**4*b**2*d*e**4*x - 240*log(a + b*x)*a**3*b**3*d**3*e**2 + 360*log(a + 
b*x)*a**3*b**3*d**2*e**3*x + 60*log(a + b*x)*a**2*b**4*d**4*e - 240*log(a 
+ b*x)*a**2*b**4*d**3*e**2*x + 60*log(a + b*x)*a*b**5*d**4*e*x - 60*a**5*b 
*e**5*x + 240*a**4*b**2*d*e**4*x - 30*a**4*b**2*e**5*x**2 - 360*a**3*b**3* 
d**2*e**3*x + 120*a**3*b**3*d*e**4*x**2 + 10*a**3*b**3*e**5*x**3 + 240*a** 
2*b**4*d**3*e**2*x - 180*a**2*b**4*d**2*e**3*x**2 - 40*a**2*b**4*d*e**4*x* 
*3 - 5*a**2*b**4*e**5*x**4 - 60*a*b**5*d**4*e*x + 120*a*b**5*d**3*e**2*x** 
2 + 60*a*b**5*d**2*e**3*x**3 + 20*a*b**5*d*e**4*x**4 + 3*a*b**5*e**5*x**5 
+ 12*b**6*d**5*x)/(12*a*b**6*(a + b*x))