\(\int \frac {(d+e x)^5}{(a^2+2 a b x+b^2 x^2)^3} \, dx\) [1528]

Optimal result

Integrand size = 26, antiderivative size = 138 \[ \int \frac {(d+e x)^5}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=-\frac {(b d-a e)^5}{5 b^6 (a+b x)^5}-\frac {5 e (b d-a e)^4}{4 b^6 (a+b x)^4}-\frac {10 e^2 (b d-a e)^3}{3 b^6 (a+b x)^3}-\frac {5 e^3 (b d-a e)^2}{b^6 (a+b x)^2}-\frac {5 e^4 (b d-a e)}{b^6 (a+b x)}+\frac {e^5 \log (a+b x)}{b^6} \]

[Out]

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

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {27, 45} \[ \int \frac {(d+e x)^5}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=-\frac {5 e^4 (b d-a e)}{b^6 (a+b x)}-\frac {5 e^3 (b d-a e)^2}{b^6 (a+b x)^2}-\frac {10 e^2 (b d-a e)^3}{3 b^6 (a+b x)^3}-\frac {5 e (b d-a e)^4}{4 b^6 (a+b x)^4}-\frac {(b d-a e)^5}{5 b^6 (a+b x)^5}+\frac {e^5 \log (a+b x)}{b^6} \]

[In]

Int[(d + e*x)^5/(a^2 + 2*a*b*x + b^2*x^2)^3,x]

[Out]

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

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 45

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] && NeQ[b*c - a*d, 0] && 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])

Rubi steps \begin{align*} \text {integral}& = \int \frac {(d+e x)^5}{(a+b x)^6} \, dx \\ & = \int \left (\frac {(b d-a e)^5}{b^5 (a+b x)^6}+\frac {5 e (b d-a e)^4}{b^5 (a+b x)^5}+\frac {10 e^2 (b d-a e)^3}{b^5 (a+b x)^4}+\frac {10 e^3 (b d-a e)^2}{b^5 (a+b x)^3}+\frac {5 e^4 (b d-a e)}{b^5 (a+b x)^2}+\frac {e^5}{b^5 (a+b x)}\right ) \, dx \\ & = -\frac {(b d-a e)^5}{5 b^6 (a+b x)^5}-\frac {5 e (b d-a e)^4}{4 b^6 (a+b x)^4}-\frac {10 e^2 (b d-a e)^3}{3 b^6 (a+b x)^3}-\frac {5 e^3 (b d-a e)^2}{b^6 (a+b x)^2}-\frac {5 e^4 (b d-a e)}{b^6 (a+b x)}+\frac {e^5 \log (a+b x)}{b^6} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.24 \[ \int \frac {(d+e x)^5}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=-\frac {(b d-a e) \left (137 a^4 e^4+a^3 b e^3 (77 d+625 e x)+a^2 b^2 e^2 \left (47 d^2+325 d e x+1100 e^2 x^2\right )+a b^3 e \left (27 d^3+175 d^2 e x+500 d e^2 x^2+900 e^3 x^3\right )+b^4 \left (12 d^4+75 d^3 e x+200 d^2 e^2 x^2+300 d e^3 x^3+300 e^4 x^4\right )\right )}{60 b^6 (a+b x)^5}+\frac {e^5 \log (a+b x)}{b^6} \]

[In]

Integrate[(d + e*x)^5/(a^2 + 2*a*b*x + b^2*x^2)^3,x]

[Out]

-1/60*((b*d - a*e)*(137*a^4*e^4 + a^3*b*e^3*(77*d + 625*e*x) + a^2*b^2*e^2*(47*d^2 + 325*d*e*x + 1100*e^2*x^2)
+ a*b^3*e*(27*d^3 + 175*d^2*e*x + 500*d*e^2*x^2 + 900*e^3*x^3) + b^4*(12*d^4 + 75*d^3*e*x + 200*d^2*e^2*x^2 +
300*d*e^3*x^3 + 300*e^4*x^4)))/(b^6*(a + b*x)^5) + (e^5*Log[a + b*x])/b^6

Maple [A] (verified)

Time = 2.27 (sec) , antiderivative size = 261, normalized size of antiderivative = 1.89

method	result	size
norman	\(\frac {\frac {137 a^{5} e^{5}-60 a^{4} b d \,e^{4}-30 a^{3} b^{2} d^{2} e^{3}-20 a^{2} b^{3} d^{3} e^{2}-15 a \,b^{4} d^{4} e -12 b^{5} d^{5}}{60 b^{6}}+\frac {5 \left (a \,e^{5}-b d \,e^{4}\right ) x^{4}}{b^{2}}+\frac {5 \left (3 a^{2} e^{5}-2 a d \,e^{4} b -d^{2} e^{3} b^{2}\right ) x^{3}}{b^{3}}+\frac {5 \left (11 a^{3} e^{5}-6 a^{2} d \,e^{4} b -3 d^{2} e^{3} a \,b^{2}-2 b^{3} d^{3} e^{2}\right ) x^{2}}{3 b^{4}}+\frac {5 \left (25 e^{5} a^{4}-12 a^{3} b d \,e^{4}-6 a^{2} b^{2} d^{2} e^{3}-4 a \,b^{3} d^{3} e^{2}-3 b^{4} d^{4} e \right ) x}{12 b^{5}}}{\left (b x +a \right )^{5}}+\frac {e^{5} \ln \left (b x +a \right )}{b^{6}}\)	\(261\)
default	\(\frac {10 e^{2} \left (a^{3} e^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right )}{3 b^{6} \left (b x +a \right )^{3}}-\frac {-a^{5} e^{5}+5 a^{4} b d \,e^{4}-10 a^{3} b^{2} d^{2} e^{3}+10 a^{2} b^{3} d^{3} e^{2}-5 a \,b^{4} d^{4} e +b^{5} d^{5}}{5 b^{6} \left (b x +a \right )^{5}}+\frac {e^{5} \ln \left (b x +a \right )}{b^{6}}-\frac {5 e \left (e^{4} a^{4}-4 b \,e^{3} d \,a^{3}+6 b^{2} e^{2} d^{2} a^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right )}{4 b^{6} \left (b x +a \right )^{4}}-\frac {5 e^{3} \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )}{b^{6} \left (b x +a \right )^{2}}+\frac {5 e^{4} \left (a e -b d \right )}{b^{6} \left (b x +a \right )}\)	\(263\)
risch	\(\frac {\frac {5 e^{4} \left (a e -b d \right ) x^{4}}{b^{2}}+\frac {5 e^{3} \left (3 a^{2} e^{2}-2 a b d e -b^{2} d^{2}\right ) x^{3}}{b^{3}}+\frac {5 e^{2} \left (11 a^{3} e^{3}-6 a^{2} b d \,e^{2}-3 a \,b^{2} d^{2} e -2 b^{3} d^{3}\right ) x^{2}}{3 b^{4}}+\frac {5 e \left (25 e^{4} a^{4}-12 b \,e^{3} d \,a^{3}-6 b^{2} e^{2} d^{2} a^{2}-4 a \,b^{3} d^{3} e -3 b^{4} d^{4}\right ) x}{12 b^{5}}+\frac {137 a^{5} e^{5}-60 a^{4} b d \,e^{4}-30 a^{3} b^{2} d^{2} e^{3}-20 a^{2} b^{3} d^{3} e^{2}-15 a \,b^{4} d^{4} e -12 b^{5} d^{5}}{60 b^{6}}}{\left (b x +a \right ) \left (b^{2} x^{2}+2 a b x +a^{2}\right )^{2}}+\frac {e^{5} \ln \left (b x +a \right )}{b^{6}}\)	\(271\)
parallelrisch	\(\frac {-12 b^{5} d^{5}+137 a^{5} e^{5}-60 a^{4} b d \,e^{4}-30 a^{3} b^{2} d^{2} e^{3}-20 a^{2} b^{3} d^{3} e^{2}-15 a \,b^{4} d^{4} e -300 a^{3} b^{2} d \,e^{4} x +625 a^{4} b \,e^{5} x -600 x^{3} a \,b^{4} d \,e^{4}-600 x^{2} a^{2} b^{3} d \,e^{4}-300 x^{2} a \,b^{4} d^{2} e^{3}+300 \ln \left (b x +a \right ) x \,a^{4} b \,e^{5}-300 x^{4} b^{5} d \,e^{4}+900 x^{3} a^{2} b^{3} e^{5}-300 x^{3} b^{5} d^{2} e^{3}+300 \ln \left (b x +a \right ) x^{4} a \,b^{4} e^{5}-75 b^{5} d^{4} e x +60 \ln \left (b x +a \right ) a^{5} e^{5}+300 x^{4} a \,b^{4} e^{5}+600 \ln \left (b x +a \right ) x^{3} a^{2} b^{3} e^{5}+600 \ln \left (b x +a \right ) x^{2} a^{3} b^{2} e^{5}+1100 x^{2} a^{3} b^{2} e^{5}-200 x^{2} b^{5} d^{3} e^{2}+60 \ln \left (b x +a \right ) x^{5} b^{5} e^{5}-150 x \,a^{2} b^{3} d^{2} e^{3}-100 x a \,b^{4} d^{3} e^{2}}{60 b^{6} \left (b^{2} x^{2}+2 a b x +a^{2}\right )^{2} \left (b x +a \right )}\)	\(385\)

[In]

int((e*x+d)^5/(b^2*x^2+2*a*b*x+a^2)^3,x,method=_RETURNVERBOSE)

[Out]

(1/60*(137*a^5*e^5-60*a^4*b*d*e^4-30*a^3*b^2*d^2*e^3-20*a^2*b^3*d^3*e^2-15*a*b^4*d^4*e-12*b^5*d^5)/b^6+5*(a*e^
5-b*d*e^4)/b^2*x^4+5*(3*a^2*e^5-2*a*b*d*e^4-b^2*d^2*e^3)/b^3*x^3+5/3*(11*a^3*e^5-6*a^2*b*d*e^4-3*a*b^2*d^2*e^3
-2*b^3*d^3*e^2)/b^4*x^2+5/12*(25*a^4*e^5-12*a^3*b*d*e^4-6*a^2*b^2*d^2*e^3-4*a*b^3*d^3*e^2-3*b^4*d^4*e)/b^5*x)/
(b*x+a)^5+e^5*ln(b*x+a)/b^6

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 372 vs. \(2 (132) = 264\).

Time = 0.31 (sec) , antiderivative size = 372, normalized size of antiderivative = 2.70 \[ \int \frac {(d+e x)^5}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=-\frac {12 \, b^{5} d^{5} + 15 \, a b^{4} d^{4} e + 20 \, a^{2} b^{3} d^{3} e^{2} + 30 \, a^{3} b^{2} d^{2} e^{3} + 60 \, a^{4} b d e^{4} - 137 \, a^{5} e^{5} + 300 \, {\left (b^{5} d e^{4} - a b^{4} e^{5}\right )} x^{4} + 300 \, {\left (b^{5} d^{2} e^{3} + 2 \, a b^{4} d e^{4} - 3 \, a^{2} b^{3} e^{5}\right )} x^{3} + 100 \, {\left (2 \, b^{5} d^{3} e^{2} + 3 \, a b^{4} d^{2} e^{3} + 6 \, a^{2} b^{3} d e^{4} - 11 \, a^{3} b^{2} e^{5}\right )} x^{2} + 25 \, {\left (3 \, b^{5} d^{4} e + 4 \, a b^{4} d^{3} e^{2} + 6 \, a^{2} b^{3} d^{2} e^{3} + 12 \, a^{3} b^{2} d e^{4} - 25 \, a^{4} b e^{5}\right )} x - 60 \, {\left (b^{5} e^{5} x^{5} + 5 \, a b^{4} e^{5} x^{4} + 10 \, a^{2} b^{3} e^{5} x^{3} + 10 \, a^{3} b^{2} e^{5} x^{2} + 5 \, a^{4} b e^{5} x + a^{5} e^{5}\right )} \log \left (b x + a\right )}{60 \, {\left (b^{11} x^{5} + 5 \, a b^{10} x^{4} + 10 \, a^{2} b^{9} x^{3} + 10 \, a^{3} b^{8} x^{2} + 5 \, a^{4} b^{7} x + a^{5} b^{6}\right )}} \]

[In]

integrate((e*x+d)^5/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="fricas")

[Out]

-1/60*(12*b^5*d^5 + 15*a*b^4*d^4*e + 20*a^2*b^3*d^3*e^2 + 30*a^3*b^2*d^2*e^3 + 60*a^4*b*d*e^4 - 137*a^5*e^5 +
300*(b^5*d*e^4 - a*b^4*e^5)*x^4 + 300*(b^5*d^2*e^3 + 2*a*b^4*d*e^4 - 3*a^2*b^3*e^5)*x^3 + 100*(2*b^5*d^3*e^2 +
3*a*b^4*d^2*e^3 + 6*a^2*b^3*d*e^4 - 11*a^3*b^2*e^5)*x^2 + 25*(3*b^5*d^4*e + 4*a*b^4*d^3*e^2 + 6*a^2*b^3*d^2*e
^3 + 12*a^3*b^2*d*e^4 - 25*a^4*b*e^5)*x - 60*(b^5*e^5*x^5 + 5*a*b^4*e^5*x^4 + 10*a^2*b^3*e^5*x^3 + 10*a^3*b^2*
e^5*x^2 + 5*a^4*b*e^5*x + a^5*e^5)*log(b*x + a))/(b^11*x^5 + 5*a*b^10*x^4 + 10*a^2*b^9*x^3 + 10*a^3*b^8*x^2 +
5*a^4*b^7*x + a^5*b^6)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 326 vs. \(2 (126) = 252\).

Time = 106.25 (sec) , antiderivative size = 326, normalized size of antiderivative = 2.36 \[ \int \frac {(d+e x)^5}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {137 a^{5} e^{5} - 60 a^{4} b d e^{4} - 30 a^{3} b^{2} d^{2} e^{3} - 20 a^{2} b^{3} d^{3} e^{2} - 15 a b^{4} d^{4} e - 12 b^{5} d^{5} + x^{4} \cdot \left (300 a b^{4} e^{5} - 300 b^{5} d e^{4}\right ) + x^{3} \cdot \left (900 a^{2} b^{3} e^{5} - 600 a b^{4} d e^{4} - 300 b^{5} d^{2} e^{3}\right ) + x^{2} \cdot \left (1100 a^{3} b^{2} e^{5} - 600 a^{2} b^{3} d e^{4} - 300 a b^{4} d^{2} e^{3} - 200 b^{5} d^{3} e^{2}\right ) + x \left (625 a^{4} b e^{5} - 300 a^{3} b^{2} d e^{4} - 150 a^{2} b^{3} d^{2} e^{3} - 100 a b^{4} d^{3} e^{2} - 75 b^{5} d^{4} e\right )}{60 a^{5} b^{6} + 300 a^{4} b^{7} x + 600 a^{3} b^{8} x^{2} + 600 a^{2} b^{9} x^{3} + 300 a b^{10} x^{4} + 60 b^{11} x^{5}} + \frac {e^{5} \log {\left (a + b x \right )}}{b^{6}} \]

[In]

integrate((e*x+d)**5/(b**2*x**2+2*a*b*x+a**2)**3,x)

[Out]

(137*a**5*e**5 - 60*a**4*b*d*e**4 - 30*a**3*b**2*d**2*e**3 - 20*a**2*b**3*d**3*e**2 - 15*a*b**4*d**4*e - 12*b*
*5*d**5 + x**4*(300*a*b**4*e**5 - 300*b**5*d*e**4) + x**3*(900*a**2*b**3*e**5 - 600*a*b**4*d*e**4 - 300*b**5*d
**2*e**3) + x**2*(1100*a**3*b**2*e**5 - 600*a**2*b**3*d*e**4 - 300*a*b**4*d**2*e**3 - 200*b**5*d**3*e**2) + x*
(625*a**4*b*e**5 - 300*a**3*b**2*d*e**4 - 150*a**2*b**3*d**2*e**3 - 100*a*b**4*d**3*e**2 - 75*b**5*d**4*e))/(6
0*a**5*b**6 + 300*a**4*b**7*x + 600*a**3*b**8*x**2 + 600*a**2*b**9*x**3 + 300*a*b**10*x**4 + 60*b**11*x**5) +
e**5*log(a + b*x)/b**6

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 310 vs. \(2 (132) = 264\).

Time = 0.21 (sec) , antiderivative size = 310, normalized size of antiderivative = 2.25 \[ \int \frac {(d+e x)^5}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=-\frac {12 \, b^{5} d^{5} + 15 \, a b^{4} d^{4} e + 20 \, a^{2} b^{3} d^{3} e^{2} + 30 \, a^{3} b^{2} d^{2} e^{3} + 60 \, a^{4} b d e^{4} - 137 \, a^{5} e^{5} + 300 \, {\left (b^{5} d e^{4} - a b^{4} e^{5}\right )} x^{4} + 300 \, {\left (b^{5} d^{2} e^{3} + 2 \, a b^{4} d e^{4} - 3 \, a^{2} b^{3} e^{5}\right )} x^{3} + 100 \, {\left (2 \, b^{5} d^{3} e^{2} + 3 \, a b^{4} d^{2} e^{3} + 6 \, a^{2} b^{3} d e^{4} - 11 \, a^{3} b^{2} e^{5}\right )} x^{2} + 25 \, {\left (3 \, b^{5} d^{4} e + 4 \, a b^{4} d^{3} e^{2} + 6 \, a^{2} b^{3} d^{2} e^{3} + 12 \, a^{3} b^{2} d e^{4} - 25 \, a^{4} b e^{5}\right )} x}{60 \, {\left (b^{11} x^{5} + 5 \, a b^{10} x^{4} + 10 \, a^{2} b^{9} x^{3} + 10 \, a^{3} b^{8} x^{2} + 5 \, a^{4} b^{7} x + a^{5} b^{6}\right )}} + \frac {e^{5} \log \left (b x + a\right )}{b^{6}} \]

[In]

integrate((e*x+d)^5/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="maxima")

[Out]

-1/60*(12*b^5*d^5 + 15*a*b^4*d^4*e + 20*a^2*b^3*d^3*e^2 + 30*a^3*b^2*d^2*e^3 + 60*a^4*b*d*e^4 - 137*a^5*e^5 +
300*(b^5*d*e^4 - a*b^4*e^5)*x^4 + 300*(b^5*d^2*e^3 + 2*a*b^4*d*e^4 - 3*a^2*b^3*e^5)*x^3 + 100*(2*b^5*d^3*e^2 +
3*a*b^4*d^2*e^3 + 6*a^2*b^3*d*e^4 - 11*a^3*b^2*e^5)*x^2 + 25*(3*b^5*d^4*e + 4*a*b^4*d^3*e^2 + 6*a^2*b^3*d^2*e
^3 + 12*a^3*b^2*d*e^4 - 25*a^4*b*e^5)*x)/(b^11*x^5 + 5*a*b^10*x^4 + 10*a^2*b^9*x^3 + 10*a^3*b^8*x^2 + 5*a^4*b^
7*x + a^5*b^6) + e^5*log(b*x + a)/b^6

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 264, normalized size of antiderivative = 1.91 \[ \int \frac {(d+e x)^5}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {e^{5} \log \left ({\left | b x + a \right |}\right )}{b^{6}} - \frac {300 \, {\left (b^{4} d e^{4} - a b^{3} e^{5}\right )} x^{4} + 300 \, {\left (b^{4} d^{2} e^{3} + 2 \, a b^{3} d e^{4} - 3 \, a^{2} b^{2} e^{5}\right )} x^{3} + 100 \, {\left (2 \, b^{4} d^{3} e^{2} + 3 \, a b^{3} d^{2} e^{3} + 6 \, a^{2} b^{2} d e^{4} - 11 \, a^{3} b e^{5}\right )} x^{2} + 25 \, {\left (3 \, b^{4} d^{4} e + 4 \, a b^{3} d^{3} e^{2} + 6 \, a^{2} b^{2} d^{2} e^{3} + 12 \, a^{3} b d e^{4} - 25 \, a^{4} e^{5}\right )} x + \frac {12 \, b^{5} d^{5} + 15 \, a b^{4} d^{4} e + 20 \, a^{2} b^{3} d^{3} e^{2} + 30 \, a^{3} b^{2} d^{2} e^{3} + 60 \, a^{4} b d e^{4} - 137 \, a^{5} e^{5}}{b}}{60 \, {\left (b x + a\right )}^{5} b^{5}} \]

[In]

integrate((e*x+d)^5/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="giac")

[Out]

e^5*log(abs(b*x + a))/b^6 - 1/60*(300*(b^4*d*e^4 - a*b^3*e^5)*x^4 + 300*(b^4*d^2*e^3 + 2*a*b^3*d*e^4 - 3*a^2*b
^2*e^5)*x^3 + 100*(2*b^4*d^3*e^2 + 3*a*b^3*d^2*e^3 + 6*a^2*b^2*d*e^4 - 11*a^3*b*e^5)*x^2 + 25*(3*b^4*d^4*e + 4
*a*b^3*d^3*e^2 + 6*a^2*b^2*d^2*e^3 + 12*a^3*b*d*e^4 - 25*a^4*e^5)*x + (12*b^5*d^5 + 15*a*b^4*d^4*e + 20*a^2*b^
3*d^3*e^2 + 30*a^3*b^2*d^2*e^3 + 60*a^4*b*d*e^4 - 137*a^5*e^5)/b)/((b*x + a)^5*b^5)

Mupad [B] (verification not implemented)

Time = 9.69 (sec) , antiderivative size = 261, normalized size of antiderivative = 1.89 \[ \int \frac {(d+e x)^5}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {e^5\,\ln \left (a+b\,x\right )}{b^6}-\frac {x\,\left (-\frac {125\,a^4\,b\,e^5}{12}+5\,a^3\,b^2\,d\,e^4+\frac {5\,a^2\,b^3\,d^2\,e^3}{2}+\frac {5\,a\,b^4\,d^3\,e^2}{3}+\frac {5\,b^5\,d^4\,e}{4}\right )-x^4\,\left (5\,a\,b^4\,e^5-5\,b^5\,d\,e^4\right )+x^3\,\left (-15\,a^2\,b^3\,e^5+10\,a\,b^4\,d\,e^4+5\,b^5\,d^2\,e^3\right )-\frac {137\,a^5\,e^5}{60}+\frac {b^5\,d^5}{5}+x^2\,\left (-\frac {55\,a^3\,b^2\,e^5}{3}+10\,a^2\,b^3\,d\,e^4+5\,a\,b^4\,d^2\,e^3+\frac {10\,b^5\,d^3\,e^2}{3}\right )+\frac {a^2\,b^3\,d^3\,e^2}{3}+\frac {a^3\,b^2\,d^2\,e^3}{2}+\frac {a\,b^4\,d^4\,e}{4}+a^4\,b\,d\,e^4}{b^6\,{\left (a+b\,x\right )}^5} \]

[In]

int((d + e*x)^5/(a^2 + b^2*x^2 + 2*a*b*x)^3,x)

[Out]

(e^5*log(a + b*x))/b^6 - (x*((5*b^5*d^4*e)/4 - (125*a^4*b*e^5)/12 + (5*a*b^4*d^3*e^2)/3 + 5*a^3*b^2*d*e^4 + (5
*a^2*b^3*d^2*e^3)/2) - x^4*(5*a*b^4*e^5 - 5*b^5*d*e^4) + x^3*(5*b^5*d^2*e^3 - 15*a^2*b^3*e^5 + 10*a*b^4*d*e^4)
- (137*a^5*e^5)/60 + (b^5*d^5)/5 + x^2*((10*b^5*d^3*e^2)/3 - (55*a^3*b^2*e^5)/3 + 5*a*b^4*d^2*e^3 + 10*a^2*b^
3*d*e^4) + (a^2*b^3*d^3*e^2)/3 + (a^3*b^2*d^2*e^3)/2 + (a*b^4*d^4*e)/4 + a^4*b*d*e^4)/(b^6*(a + b*x)^5)