Integrand size = 15, antiderivative size = 130 \[ \int \frac {(a+b x)^5}{(c+d x)^2} \, dx=-\frac {10 b^2 (b c-a d)^3 x}{d^5}+\frac {(b c-a d)^5}{d^6 (c+d x)}+\frac {5 b^3 (b c-a d)^2 (c+d x)^2}{d^6}-\frac {5 b^4 (b c-a d) (c+d x)^3}{3 d^6}+\frac {b^5 (c+d x)^4}{4 d^6}+\frac {5 b (b c-a d)^4 \log (c+d x)}{d^6} \] Output:
-10*b^2*(-a*d+b*c)^3*x/d^5+(-a*d+b*c)^5/d^6/(d*x+c)+5*b^3*(-a*d+b*c)^2*(d* x+c)^2/d^6-5/3*b^4*(-a*d+b*c)*(d*x+c)^3/d^6+1/4*b^5*(d*x+c)^4/d^6+5*b*(-a* d+b*c)^4*ln(d*x+c)/d^6
Time = 0.04 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.75 \[ \int \frac {(a+b x)^5}{(c+d x)^2} \, dx=\frac {60 a^4 b c d^4-12 a^5 d^5+120 a^3 b^2 d^3 \left (-c^2+c d x+d^2 x^2\right )+60 a^2 b^3 d^2 \left (2 c^3-4 c^2 d x-3 c d^2 x^2+d^3 x^3\right )+20 a b^4 d \left (-3 c^4+9 c^3 d x+6 c^2 d^2 x^2-2 c d^3 x^3+d^4 x^4\right )+b^5 \left (12 c^5-48 c^4 d x-30 c^3 d^2 x^2+10 c^2 d^3 x^3-5 c d^4 x^4+3 d^5 x^5\right )+60 b (b c-a d)^4 (c+d x) \log (c+d x)}{12 d^6 (c+d x)} \] Input:
Integrate[(a + b*x)^5/(c + d*x)^2,x]
Output:
(60*a^4*b*c*d^4 - 12*a^5*d^5 + 120*a^3*b^2*d^3*(-c^2 + c*d*x + d^2*x^2) + 60*a^2*b^3*d^2*(2*c^3 - 4*c^2*d*x - 3*c*d^2*x^2 + d^3*x^3) + 20*a*b^4*d*(- 3*c^4 + 9*c^3*d*x + 6*c^2*d^2*x^2 - 2*c*d^3*x^3 + d^4*x^4) + b^5*(12*c^5 - 48*c^4*d*x - 30*c^3*d^2*x^2 + 10*c^2*d^3*x^3 - 5*c*d^4*x^4 + 3*d^5*x^5) + 60*b*(b*c - a*d)^4*(c + d*x)*Log[c + d*x])/(12*d^6*(c + d*x))
Time = 0.31 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {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.
\(\displaystyle \int \frac {(a+b x)^5}{(c+d x)^2} \, dx\) |
\(\Big \downarrow \) 49 |
\(\displaystyle \int \left (-\frac {5 b^4 (c+d x)^2 (b c-a d)}{d^5}+\frac {10 b^3 (c+d x) (b c-a d)^2}{d^5}-\frac {10 b^2 (b c-a d)^3}{d^5}+\frac {5 b (b c-a d)^4}{d^5 (c+d x)}+\frac {(a d-b c)^5}{d^5 (c+d x)^2}+\frac {b^5 (c+d x)^3}{d^5}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {5 b^4 (c+d x)^3 (b c-a d)}{3 d^6}+\frac {5 b^3 (c+d x)^2 (b c-a d)^2}{d^6}-\frac {10 b^2 x (b c-a d)^3}{d^5}+\frac {(b c-a d)^5}{d^6 (c+d x)}+\frac {5 b (b c-a d)^4 \log (c+d x)}{d^6}+\frac {b^5 (c+d x)^4}{4 d^6}\) |
Input:
Int[(a + b*x)^5/(c + d*x)^2,x]
Output:
(-10*b^2*(b*c - a*d)^3*x)/d^5 + (b*c - a*d)^5/(d^6*(c + d*x)) + (5*b^3*(b* c - a*d)^2*(c + d*x)^2)/d^6 - (5*b^4*(b*c - a*d)*(c + d*x)^3)/(3*d^6) + (b ^5*(c + d*x)^4)/(4*d^6) + (5*b*(b*c - a*d)^4*Log[c + d*x])/d^6
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]
Leaf count of result is larger than twice the leaf count of optimal. \(255\) vs. \(2(126)=252\).
Time = 0.10 (sec) , antiderivative size = 256, normalized size of antiderivative = 1.97
method | result | size |
norman | \(\frac {\frac {\left (a^{5} d^{5}-5 a^{4} b c \,d^{4}+20 a^{3} b^{2} c^{2} d^{3}-30 a^{2} b^{3} c^{3} d^{2}+20 a \,b^{4} c^{4} d -5 c^{5} b^{5}\right ) x}{d^{5} c}+\frac {b^{5} x^{5}}{4 d}+\frac {5 b^{2} \left (4 a^{3} d^{3}-6 a^{2} b c \,d^{2}+4 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) x^{2}}{2 d^{4}}+\frac {5 b^{3} \left (6 a^{2} d^{2}-4 a b c d +b^{2} c^{2}\right ) x^{3}}{6 d^{3}}+\frac {5 b^{4} \left (4 a d -b c \right ) x^{4}}{12 d^{2}}}{x d +c}+\frac {5 b \left (d^{4} a^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +c^{4} b^{4}\right ) \ln \left (x d +c \right )}{d^{6}}\) | \(256\) |
default | \(\frac {b^{2} \left (\frac {1}{4} d^{3} x^{4} b^{3}+\frac {5}{3} x^{3} a \,b^{2} d^{3}-\frac {2}{3} x^{3} b^{3} c \,d^{2}+5 x^{2} a^{2} b \,d^{3}-5 x^{2} a \,b^{2} c \,d^{2}+\frac {3}{2} x^{2} b^{3} c^{2} d +10 a^{3} d^{3} x -20 a^{2} b c \,d^{2} x +15 a \,b^{2} c^{2} d x -4 b^{3} c^{3} x \right )}{d^{5}}-\frac {a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -c^{5} b^{5}}{d^{6} \left (x d +c \right )}+\frac {5 b \left (d^{4} a^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +c^{4} b^{4}\right ) \ln \left (x d +c \right )}{d^{6}}\) | \(259\) |
risch | \(\frac {b^{5} x^{4}}{4 d^{2}}+\frac {5 b^{4} x^{3} a}{3 d^{2}}-\frac {2 b^{5} x^{3} c}{3 d^{3}}+\frac {5 b^{3} x^{2} a^{2}}{d^{2}}-\frac {5 b^{4} x^{2} a c}{d^{3}}+\frac {3 b^{5} x^{2} c^{2}}{2 d^{4}}+\frac {10 b^{2} a^{3} x}{d^{2}}-\frac {20 b^{3} a^{2} c x}{d^{3}}+\frac {15 b^{4} a \,c^{2} x}{d^{4}}-\frac {4 b^{5} c^{3} x}{d^{5}}-\frac {a^{5}}{d \left (x d +c \right )}+\frac {5 a^{4} b c}{d^{2} \left (x d +c \right )}-\frac {10 a^{3} b^{2} c^{2}}{d^{3} \left (x d +c \right )}+\frac {10 a^{2} b^{3} c^{3}}{d^{4} \left (x d +c \right )}-\frac {5 a \,b^{4} c^{4}}{d^{5} \left (x d +c \right )}+\frac {c^{5} b^{5}}{d^{6} \left (x d +c \right )}+\frac {5 b \ln \left (x d +c \right ) a^{4}}{d^{2}}-\frac {20 b^{2} \ln \left (x d +c \right ) a^{3} c}{d^{3}}+\frac {30 b^{3} \ln \left (x d +c \right ) a^{2} c^{2}}{d^{4}}-\frac {20 b^{4} \ln \left (x d +c \right ) a \,c^{3}}{d^{5}}+\frac {5 b^{5} \ln \left (x d +c \right ) c^{4}}{d^{6}}\) | \(326\) |
parallelrisch | \(\frac {60 \ln \left (x d +c \right ) x \,a^{4} b \,d^{5}+60 \ln \left (x d +c \right ) x \,b^{5} c^{4} d +60 c^{5} b^{5}+60 x^{3} a^{2} b^{3} d^{5}+10 x^{3} b^{5} c^{2} d^{3}+120 x^{2} a^{3} b^{2} d^{5}-30 x^{2} b^{5} c^{3} d^{2}+20 x^{4} a \,b^{4} d^{5}-5 x^{4} b^{5} c \,d^{4}-12 a^{5} d^{5}-40 x^{3} a \,b^{4} c \,d^{4}-180 x^{2} a^{2} b^{3} c \,d^{4}+120 x^{2} a \,b^{4} c^{2} d^{3}+60 \ln \left (x d +c \right ) a^{4} b c \,d^{4}+3 x^{5} b^{5} d^{5}+360 \ln \left (x d +c \right ) x \,a^{2} b^{3} c^{2} d^{3}-240 \ln \left (x d +c \right ) x a \,b^{4} c^{3} d^{2}+60 a^{4} b c \,d^{4}-240 a^{3} b^{2} c^{2} d^{3}+360 a^{2} b^{3} c^{3} d^{2}-240 a \,b^{4} c^{4} d +60 \ln \left (x d +c \right ) b^{5} c^{5}-240 \ln \left (x d +c \right ) a^{3} b^{2} c^{2} d^{3}+360 \ln \left (x d +c \right ) a^{2} b^{3} c^{3} d^{2}-240 \ln \left (x d +c \right ) a \,b^{4} c^{4} d -240 \ln \left (x d +c \right ) x \,a^{3} b^{2} c \,d^{4}}{12 d^{6} \left (x d +c \right )}\) | \(389\) |
Input:
int((b*x+a)^5/(d*x+c)^2,x,method=_RETURNVERBOSE)
Output:
((a^5*d^5-5*a^4*b*c*d^4+20*a^3*b^2*c^2*d^3-30*a^2*b^3*c^3*d^2+20*a*b^4*c^4 *d-5*b^5*c^5)/d^5/c*x+1/4*b^5/d*x^5+5/2*b^2*(4*a^3*d^3-6*a^2*b*c*d^2+4*a*b ^2*c^2*d-b^3*c^3)/d^4*x^2+5/6*b^3*(6*a^2*d^2-4*a*b*c*d+b^2*c^2)/d^3*x^3+5/ 12*b^4*(4*a*d-b*c)/d^2*x^4)/(d*x+c)+5/d^6*b*(a^4*d^4-4*a^3*b*c*d^3+6*a^2*b ^2*c^2*d^2-4*a*b^3*c^3*d+b^4*c^4)*ln(d*x+c)
Leaf count of result is larger than twice the leaf count of optimal. 373 vs. \(2 (126) = 252\).
Time = 0.08 (sec) , antiderivative size = 373, normalized size of antiderivative = 2.87 \[ \int \frac {(a+b x)^5}{(c+d x)^2} \, dx=\frac {3 \, b^{5} d^{5} x^{5} + 12 \, b^{5} c^{5} - 60 \, a b^{4} c^{4} d + 120 \, a^{2} b^{3} c^{3} d^{2} - 120 \, a^{3} b^{2} c^{2} d^{3} + 60 \, a^{4} b c d^{4} - 12 \, a^{5} d^{5} - 5 \, {\left (b^{5} c d^{4} - 4 \, a b^{4} d^{5}\right )} x^{4} + 10 \, {\left (b^{5} c^{2} d^{3} - 4 \, a b^{4} c d^{4} + 6 \, a^{2} b^{3} d^{5}\right )} x^{3} - 30 \, {\left (b^{5} c^{3} d^{2} - 4 \, a b^{4} c^{2} d^{3} + 6 \, a^{2} b^{3} c d^{4} - 4 \, a^{3} b^{2} d^{5}\right )} x^{2} - 12 \, {\left (4 \, b^{5} c^{4} d - 15 \, a b^{4} c^{3} d^{2} + 20 \, a^{2} b^{3} c^{2} d^{3} - 10 \, a^{3} b^{2} c d^{4}\right )} x + 60 \, {\left (b^{5} c^{5} - 4 \, a b^{4} c^{4} d + 6 \, a^{2} b^{3} c^{3} d^{2} - 4 \, a^{3} b^{2} c^{2} d^{3} + a^{4} b c d^{4} + {\left (b^{5} c^{4} d - 4 \, a b^{4} c^{3} d^{2} + 6 \, a^{2} b^{3} c^{2} d^{3} - 4 \, a^{3} b^{2} c d^{4} + a^{4} b d^{5}\right )} x\right )} \log \left (d x + c\right )}{12 \, {\left (d^{7} x + c d^{6}\right )}} \] Input:
integrate((b*x+a)^5/(d*x+c)^2,x, algorithm="fricas")
Output:
1/12*(3*b^5*d^5*x^5 + 12*b^5*c^5 - 60*a*b^4*c^4*d + 120*a^2*b^3*c^3*d^2 - 120*a^3*b^2*c^2*d^3 + 60*a^4*b*c*d^4 - 12*a^5*d^5 - 5*(b^5*c*d^4 - 4*a*b^4 *d^5)*x^4 + 10*(b^5*c^2*d^3 - 4*a*b^4*c*d^4 + 6*a^2*b^3*d^5)*x^3 - 30*(b^5 *c^3*d^2 - 4*a*b^4*c^2*d^3 + 6*a^2*b^3*c*d^4 - 4*a^3*b^2*d^5)*x^2 - 12*(4* b^5*c^4*d - 15*a*b^4*c^3*d^2 + 20*a^2*b^3*c^2*d^3 - 10*a^3*b^2*c*d^4)*x + 60*(b^5*c^5 - 4*a*b^4*c^4*d + 6*a^2*b^3*c^3*d^2 - 4*a^3*b^2*c^2*d^3 + a^4* b*c*d^4 + (b^5*c^4*d - 4*a*b^4*c^3*d^2 + 6*a^2*b^3*c^2*d^3 - 4*a^3*b^2*c*d ^4 + a^4*b*d^5)*x)*log(d*x + c))/(d^7*x + c*d^6)
Time = 0.43 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.78 \[ \int \frac {(a+b x)^5}{(c+d x)^2} \, dx=\frac {b^{5} x^{4}}{4 d^{2}} + \frac {5 b \left (a d - b c\right )^{4} \log {\left (c + d x \right )}}{d^{6}} + x^{3} \cdot \left (\frac {5 a b^{4}}{3 d^{2}} - \frac {2 b^{5} c}{3 d^{3}}\right ) + x^{2} \cdot \left (\frac {5 a^{2} b^{3}}{d^{2}} - \frac {5 a b^{4} c}{d^{3}} + \frac {3 b^{5} c^{2}}{2 d^{4}}\right ) + x \left (\frac {10 a^{3} b^{2}}{d^{2}} - \frac {20 a^{2} b^{3} c}{d^{3}} + \frac {15 a b^{4} c^{2}}{d^{4}} - \frac {4 b^{5} c^{3}}{d^{5}}\right ) + \frac {- a^{5} d^{5} + 5 a^{4} b c d^{4} - 10 a^{3} b^{2} c^{2} d^{3} + 10 a^{2} b^{3} c^{3} d^{2} - 5 a b^{4} c^{4} d + b^{5} c^{5}}{c d^{6} + d^{7} x} \] Input:
integrate((b*x+a)**5/(d*x+c)**2,x)
Output:
b**5*x**4/(4*d**2) + 5*b*(a*d - b*c)**4*log(c + d*x)/d**6 + x**3*(5*a*b**4 /(3*d**2) - 2*b**5*c/(3*d**3)) + x**2*(5*a**2*b**3/d**2 - 5*a*b**4*c/d**3 + 3*b**5*c**2/(2*d**4)) + x*(10*a**3*b**2/d**2 - 20*a**2*b**3*c/d**3 + 15* a*b**4*c**2/d**4 - 4*b**5*c**3/d**5) + (-a**5*d**5 + 5*a**4*b*c*d**4 - 10* a**3*b**2*c**2*d**3 + 10*a**2*b**3*c**3*d**2 - 5*a*b**4*c**4*d + b**5*c**5 )/(c*d**6 + d**7*x)
Leaf count of result is larger than twice the leaf count of optimal. 264 vs. \(2 (126) = 252\).
Time = 0.03 (sec) , antiderivative size = 264, normalized size of antiderivative = 2.03 \[ \int \frac {(a+b x)^5}{(c+d x)^2} \, dx=\frac {b^{5} c^{5} - 5 \, a b^{4} c^{4} d + 10 \, a^{2} b^{3} c^{3} d^{2} - 10 \, a^{3} b^{2} c^{2} d^{3} + 5 \, a^{4} b c d^{4} - a^{5} d^{5}}{d^{7} x + c d^{6}} + \frac {3 \, b^{5} d^{3} x^{4} - 4 \, {\left (2 \, b^{5} c d^{2} - 5 \, a b^{4} d^{3}\right )} x^{3} + 6 \, {\left (3 \, b^{5} c^{2} d - 10 \, a b^{4} c d^{2} + 10 \, a^{2} b^{3} d^{3}\right )} x^{2} - 12 \, {\left (4 \, b^{5} c^{3} - 15 \, a b^{4} c^{2} d + 20 \, a^{2} b^{3} c d^{2} - 10 \, a^{3} b^{2} d^{3}\right )} x}{12 \, d^{5}} + \frac {5 \, {\left (b^{5} c^{4} - 4 \, a b^{4} c^{3} d + 6 \, a^{2} b^{3} c^{2} d^{2} - 4 \, a^{3} b^{2} c d^{3} + a^{4} b d^{4}\right )} \log \left (d x + c\right )}{d^{6}} \] Input:
integrate((b*x+a)^5/(d*x+c)^2,x, algorithm="maxima")
Output:
(b^5*c^5 - 5*a*b^4*c^4*d + 10*a^2*b^3*c^3*d^2 - 10*a^3*b^2*c^2*d^3 + 5*a^4 *b*c*d^4 - a^5*d^5)/(d^7*x + c*d^6) + 1/12*(3*b^5*d^3*x^4 - 4*(2*b^5*c*d^2 - 5*a*b^4*d^3)*x^3 + 6*(3*b^5*c^2*d - 10*a*b^4*c*d^2 + 10*a^2*b^3*d^3)*x^ 2 - 12*(4*b^5*c^3 - 15*a*b^4*c^2*d + 20*a^2*b^3*c*d^2 - 10*a^3*b^2*d^3)*x) /d^5 + 5*(b^5*c^4 - 4*a*b^4*c^3*d + 6*a^2*b^3*c^2*d^2 - 4*a^3*b^2*c*d^3 + a^4*b*d^4)*log(d*x + c)/d^6
Leaf count of result is larger than twice the leaf count of optimal. 339 vs. \(2 (126) = 252\).
Time = 0.13 (sec) , antiderivative size = 339, normalized size of antiderivative = 2.61 \[ \int \frac {(a+b x)^5}{(c+d x)^2} \, dx=\frac {{\left (3 \, b^{5} - \frac {20 \, {\left (b^{5} c d - a b^{4} d^{2}\right )}}{{\left (d x + c\right )} d} + \frac {60 \, {\left (b^{5} c^{2} d^{2} - 2 \, a b^{4} c d^{3} + a^{2} b^{3} d^{4}\right )}}{{\left (d x + c\right )}^{2} d^{2}} - \frac {120 \, {\left (b^{5} c^{3} d^{3} - 3 \, a b^{4} c^{2} d^{4} + 3 \, a^{2} b^{3} c d^{5} - a^{3} b^{2} d^{6}\right )}}{{\left (d x + c\right )}^{3} d^{3}}\right )} {\left (d x + c\right )}^{4}}{12 \, d^{6}} - \frac {5 \, {\left (b^{5} c^{4} - 4 \, a b^{4} c^{3} d + 6 \, a^{2} b^{3} c^{2} d^{2} - 4 \, a^{3} b^{2} c d^{3} + a^{4} b d^{4}\right )} \log \left (\frac {{\left | d x + c \right |}}{{\left (d x + c\right )}^{2} {\left | d \right |}}\right )}{d^{6}} + \frac {\frac {b^{5} c^{5} d^{4}}{d x + c} - \frac {5 \, a b^{4} c^{4} d^{5}}{d x + c} + \frac {10 \, a^{2} b^{3} c^{3} d^{6}}{d x + c} - \frac {10 \, a^{3} b^{2} c^{2} d^{7}}{d x + c} + \frac {5 \, a^{4} b c d^{8}}{d x + c} - \frac {a^{5} d^{9}}{d x + c}}{d^{10}} \] Input:
integrate((b*x+a)^5/(d*x+c)^2,x, algorithm="giac")
Output:
1/12*(3*b^5 - 20*(b^5*c*d - a*b^4*d^2)/((d*x + c)*d) + 60*(b^5*c^2*d^2 - 2 *a*b^4*c*d^3 + a^2*b^3*d^4)/((d*x + c)^2*d^2) - 120*(b^5*c^3*d^3 - 3*a*b^4 *c^2*d^4 + 3*a^2*b^3*c*d^5 - a^3*b^2*d^6)/((d*x + c)^3*d^3))*(d*x + c)^4/d ^6 - 5*(b^5*c^4 - 4*a*b^4*c^3*d + 6*a^2*b^3*c^2*d^2 - 4*a^3*b^2*c*d^3 + a^ 4*b*d^4)*log(abs(d*x + c)/((d*x + c)^2*abs(d)))/d^6 + (b^5*c^5*d^4/(d*x + c) - 5*a*b^4*c^4*d^5/(d*x + c) + 10*a^2*b^3*c^3*d^6/(d*x + c) - 10*a^3*b^2 *c^2*d^7/(d*x + c) + 5*a^4*b*c*d^8/(d*x + c) - a^5*d^9/(d*x + c))/d^10
Time = 0.05 (sec) , antiderivative size = 327, normalized size of antiderivative = 2.52 \[ \int \frac {(a+b x)^5}{(c+d x)^2} \, dx=x^3\,\left (\frac {5\,a\,b^4}{3\,d^2}-\frac {2\,b^5\,c}{3\,d^3}\right )+x\,\left (\frac {2\,c\,\left (\frac {2\,c\,\left (\frac {5\,a\,b^4}{d^2}-\frac {2\,b^5\,c}{d^3}\right )}{d}-\frac {10\,a^2\,b^3}{d^2}+\frac {b^5\,c^2}{d^4}\right )}{d}+\frac {10\,a^3\,b^2}{d^2}-\frac {c^2\,\left (\frac {5\,a\,b^4}{d^2}-\frac {2\,b^5\,c}{d^3}\right )}{d^2}\right )-x^2\,\left (\frac {c\,\left (\frac {5\,a\,b^4}{d^2}-\frac {2\,b^5\,c}{d^3}\right )}{d}-\frac {5\,a^2\,b^3}{d^2}+\frac {b^5\,c^2}{2\,d^4}\right )+\frac {\ln \left (c+d\,x\right )\,\left (5\,a^4\,b\,d^4-20\,a^3\,b^2\,c\,d^3+30\,a^2\,b^3\,c^2\,d^2-20\,a\,b^4\,c^3\,d+5\,b^5\,c^4\right )}{d^6}-\frac {a^5\,d^5-5\,a^4\,b\,c\,d^4+10\,a^3\,b^2\,c^2\,d^3-10\,a^2\,b^3\,c^3\,d^2+5\,a\,b^4\,c^4\,d-b^5\,c^5}{d\,\left (x\,d^6+c\,d^5\right )}+\frac {b^5\,x^4}{4\,d^2} \] Input:
int((a + b*x)^5/(c + d*x)^2,x)
Output:
x^3*((5*a*b^4)/(3*d^2) - (2*b^5*c)/(3*d^3)) + x*((2*c*((2*c*((5*a*b^4)/d^2 - (2*b^5*c)/d^3))/d - (10*a^2*b^3)/d^2 + (b^5*c^2)/d^4))/d + (10*a^3*b^2) /d^2 - (c^2*((5*a*b^4)/d^2 - (2*b^5*c)/d^3))/d^2) - x^2*((c*((5*a*b^4)/d^2 - (2*b^5*c)/d^3))/d - (5*a^2*b^3)/d^2 + (b^5*c^2)/(2*d^4)) + (log(c + d*x )*(5*b^5*c^4 + 5*a^4*b*d^4 - 20*a^3*b^2*c*d^3 + 30*a^2*b^3*c^2*d^2 - 20*a* b^4*c^3*d))/d^6 - (a^5*d^5 - b^5*c^5 - 10*a^2*b^3*c^3*d^2 + 10*a^3*b^2*c^2 *d^3 + 5*a*b^4*c^4*d - 5*a^4*b*c*d^4)/(d*(c*d^5 + d^6*x)) + (b^5*x^4)/(4*d ^2)
Time = 0.17 (sec) , antiderivative size = 415, normalized size of antiderivative = 3.19 \[ \int \frac {(a+b x)^5}{(c+d x)^2} \, dx=\frac {60 \,\mathrm {log}\left (d x +c \right ) a^{4} b \,c^{2} d^{4}-240 \,\mathrm {log}\left (d x +c \right ) a^{3} b^{2} c^{3} d^{3}+360 \,\mathrm {log}\left (d x +c \right ) a^{2} b^{3} c^{4} d^{2}-240 \,\mathrm {log}\left (d x +c \right ) a \,b^{4} c^{5} d +60 \,\mathrm {log}\left (d x +c \right ) b^{5} c^{5} d x -60 a^{4} b c \,d^{5} x +240 a^{3} b^{2} c^{2} d^{4} x +120 a^{3} b^{2} c \,d^{5} x^{2}-360 a^{2} b^{3} c^{3} d^{3} x -180 a^{2} b^{3} c^{2} d^{4} x^{2}+60 a^{2} b^{3} c \,d^{5} x^{3}+240 a \,b^{4} c^{4} d^{2} x +120 a \,b^{4} c^{3} d^{3} x^{2}-40 a \,b^{4} c^{2} d^{4} x^{3}+20 a \,b^{4} c \,d^{5} x^{4}-240 \,\mathrm {log}\left (d x +c \right ) a \,b^{4} c^{4} d^{2} x -60 b^{5} c^{5} d x -30 b^{5} c^{4} d^{2} x^{2}+10 b^{5} c^{3} d^{3} x^{3}-5 b^{5} c^{2} d^{4} x^{4}+3 b^{5} c \,d^{5} x^{5}+60 \,\mathrm {log}\left (d x +c \right ) a^{4} b c \,d^{5} x -240 \,\mathrm {log}\left (d x +c \right ) a^{3} b^{2} c^{2} d^{4} x +360 \,\mathrm {log}\left (d x +c \right ) a^{2} b^{3} c^{3} d^{3} x +60 \,\mathrm {log}\left (d x +c \right ) b^{5} c^{6}+12 a^{5} d^{6} x}{12 c \,d^{6} \left (d x +c \right )} \] Input:
int((b*x+a)^5/(d*x+c)^2,x)
Output:
(60*log(c + d*x)*a**4*b*c**2*d**4 + 60*log(c + d*x)*a**4*b*c*d**5*x - 240* log(c + d*x)*a**3*b**2*c**3*d**3 - 240*log(c + d*x)*a**3*b**2*c**2*d**4*x + 360*log(c + d*x)*a**2*b**3*c**4*d**2 + 360*log(c + d*x)*a**2*b**3*c**3*d **3*x - 240*log(c + d*x)*a*b**4*c**5*d - 240*log(c + d*x)*a*b**4*c**4*d**2 *x + 60*log(c + d*x)*b**5*c**6 + 60*log(c + d*x)*b**5*c**5*d*x + 12*a**5*d **6*x - 60*a**4*b*c*d**5*x + 240*a**3*b**2*c**2*d**4*x + 120*a**3*b**2*c*d **5*x**2 - 360*a**2*b**3*c**3*d**3*x - 180*a**2*b**3*c**2*d**4*x**2 + 60*a **2*b**3*c*d**5*x**3 + 240*a*b**4*c**4*d**2*x + 120*a*b**4*c**3*d**3*x**2 - 40*a*b**4*c**2*d**4*x**3 + 20*a*b**4*c*d**5*x**4 - 60*b**5*c**5*d*x - 30 *b**5*c**4*d**2*x**2 + 10*b**5*c**3*d**3*x**3 - 5*b**5*c**2*d**4*x**4 + 3* b**5*c*d**5*x**5)/(12*c*d**6*(c + d*x))