\(\int \frac {x^5}{(a+b x) (c+d x)} \, dx\) [232]

Optimal result

Integrand size = 18, antiderivative size = 145 \[ \int \frac {x^5}{(a+b x) (c+d x)} \, dx=-\frac {(b c+a d) \left (b^2 c^2+a^2 d^2\right ) x}{b^4 d^4}+\frac {\left (b^2 c^2+a b c d+a^2 d^2\right ) x^2}{2 b^3 d^3}-\frac {(b c+a d) x^3}{3 b^2 d^2}+\frac {x^4}{4 b d}-\frac {a^5 \log (a+b x)}{b^5 (b c-a d)}+\frac {c^5 \log (c+d x)}{d^5 (b c-a d)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.10 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {84} \[ \int \frac {x^5}{(a+b x) (c+d x)} \, dx=-\frac {a^5 \log (a+b x)}{b^5 (b c-a d)}-\frac {x (a d+b c) \left (a^2 d^2+b^2 c^2\right )}{b^4 d^4}+\frac {x^2 \left (a^2 d^2+a b c d+b^2 c^2\right )}{2 b^3 d^3}-\frac {x^3 (a d+b c)}{3 b^2 d^2}+\frac {c^5 \log (c+d x)}{d^5 (b c-a d)}+\frac {x^4}{4 b d} \]

[In]

Int[x^5/((a + b*x)*(c + d*x)),x]

[Out]

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

Rule 84

Int[((e_.) + (f_.)*(x_))^(p_.)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Int[ExpandIntegrand[(
e + f*x)^p/((a + b*x)*(c + d*x)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IntegerQ[p]

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

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.92 \[ \int \frac {x^5}{(a+b x) (c+d x)} \, dx=\frac {b d x \left (12 a^4 d^4-6 a^3 b d^4 x+4 a^2 b^2 d^4 x^2-3 a b^3 d^4 x^3+b^4 c \left (-12 c^3+6 c^2 d x-4 c d^2 x^2+3 d^3 x^3\right )\right )-12 a^5 d^5 \log (a+b x)+12 b^5 c^5 \log (c+d x)}{12 b^5 d^5 (b c-a d)} \]

[In]

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

[Out]

(b*d*x*(12*a^4*d^4 - 6*a^3*b*d^4*x + 4*a^2*b^2*d^4*x^2 - 3*a*b^3*d^4*x^3 + b^4*c*(-12*c^3 + 6*c^2*d*x - 4*c*d^
2*x^2 + 3*d^3*x^3)) - 12*a^5*d^5*Log[a + b*x] + 12*b^5*c^5*Log[c + d*x])/(12*b^5*d^5*(b*c - a*d))

Maple [A] (verified)

Time = 0.47 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.02

method	result	size
parallelrisch	\(\frac {3 x^{4} a \,b^{4} d^{5}-3 x^{4} b^{5} c \,d^{4}-4 x^{3} a^{2} b^{3} d^{5}+4 x^{3} b^{5} c^{2} d^{3}+6 x^{2} a^{3} b^{2} d^{5}-6 x^{2} b^{5} c^{3} d^{2}+12 a^{5} \ln \left (b x +a \right ) d^{5}-12 c^{5} \ln \left (d x +c \right ) b^{5}-12 x \,a^{4} b \,d^{5}+12 x \,b^{5} c^{4} d}{12 b^{5} d^{5} \left (a d -b c \right )}\)	\(148\)
norman	\(\frac {x^{4}}{4 b d}-\frac {\left (a d +b c \right ) x^{3}}{3 b^{2} d^{2}}+\frac {\left (a^{2} d^{2}+a b c d +b^{2} c^{2}\right ) x^{2}}{2 b^{3} d^{3}}-\frac {\left (a^{3} d^{3}+a^{2} b c \,d^{2}+a \,b^{2} c^{2} d +b^{3} c^{3}\right ) x}{b^{4} d^{4}}+\frac {a^{5} \ln \left (b x +a \right )}{b^{5} \left (a d -b c \right )}-\frac {c^{5} \ln \left (d x +c \right )}{d^{5} \left (a d -b c \right )}\)	\(151\)
default	\(-\frac {-\frac {1}{4} d^{3} x^{4} b^{3}+\frac {1}{3} x^{3} a \,b^{2} d^{3}+\frac {1}{3} x^{3} b^{3} c \,d^{2}-\frac {1}{2} x^{2} a^{2} b \,d^{3}-\frac {1}{2} x^{2} a \,b^{2} c \,d^{2}-\frac {1}{2} x^{2} b^{3} c^{2} d +a^{3} d^{3} x +a^{2} b c \,d^{2} x +a \,b^{2} c^{2} d x +b^{3} c^{3} x}{b^{4} d^{4}}-\frac {c^{5} \ln \left (d x +c \right )}{d^{5} \left (a d -b c \right )}+\frac {a^{5} \ln \left (b x +a \right )}{b^{5} \left (a d -b c \right )}\)	\(166\)
risch	\(\frac {x^{4}}{4 b d}-\frac {x^{3} a}{3 b^{2} d}-\frac {x^{3} c}{3 b \,d^{2}}+\frac {x^{2} a^{2}}{2 b^{3} d}+\frac {x^{2} a c}{2 b^{2} d^{2}}+\frac {x^{2} c^{2}}{2 b \,d^{3}}-\frac {a^{3} x}{b^{4} d}-\frac {a^{2} c x}{b^{3} d^{2}}-\frac {a \,c^{2} x}{b^{2} d^{3}}-\frac {c^{3} x}{b \,d^{4}}+\frac {a^{5} \ln \left (-b x -a \right )}{b^{5} \left (a d -b c \right )}-\frac {c^{5} \ln \left (d x +c \right )}{d^{5} \left (a d -b c \right )}\)	\(178\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.03 \[ \int \frac {x^5}{(a+b x) (c+d x)} \, dx=-\frac {12 \, a^{5} d^{5} \log \left (b x + a\right ) - 12 \, b^{5} c^{5} \log \left (d x + c\right ) - 3 \, {\left (b^{5} c d^{4} - a b^{4} d^{5}\right )} x^{4} + 4 \, {\left (b^{5} c^{2} d^{3} - a^{2} b^{3} d^{5}\right )} x^{3} - 6 \, {\left (b^{5} c^{3} d^{2} - a^{3} b^{2} d^{5}\right )} x^{2} + 12 \, {\left (b^{5} c^{4} d - a^{4} b d^{5}\right )} x}{12 \, {\left (b^{6} c d^{5} - a b^{5} d^{6}\right )}} \]

[In]

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

[Out]

-1/12*(12*a^5*d^5*log(b*x + a) - 12*b^5*c^5*log(d*x + c) - 3*(b^5*c*d^4 - a*b^4*d^5)*x^4 + 4*(b^5*c^2*d^3 - a^
2*b^3*d^5)*x^3 - 6*(b^5*c^3*d^2 - a^3*b^2*d^5)*x^2 + 12*(b^5*c^4*d - a^4*b*d^5)*x)/(b^6*c*d^5 - a*b^5*d^6)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 306 vs. \(2 (134) = 268\).

Time = 1.35 (sec) , antiderivative size = 306, normalized size of antiderivative = 2.11 \[ \int \frac {x^5}{(a+b x) (c+d x)} \, dx=\frac {a^{5} \log {\left (x + \frac {\frac {a^{7} d^{6}}{b \left (a d - b c\right )} - \frac {2 a^{6} c d^{5}}{a d - b c} + \frac {a^{5} b c^{2} d^{4}}{a d - b c} + a^{5} c d^{4} + a b^{4} c^{5}}{a^{5} d^{5} + b^{5} c^{5}} \right )}}{b^{5} \left (a d - b c\right )} - \frac {c^{5} \log {\left (x + \frac {a^{5} c d^{4} - \frac {a^{2} b^{4} c^{5} d}{a d - b c} + \frac {2 a b^{5} c^{6}}{a d - b c} + a b^{4} c^{5} - \frac {b^{6} c^{7}}{d \left (a d - b c\right )}}{a^{5} d^{5} + b^{5} c^{5}} \right )}}{d^{5} \left (a d - b c\right )} + x^{3} \left (- \frac {a}{3 b^{2} d} - \frac {c}{3 b d^{2}}\right ) + x^{2} \left (\frac {a^{2}}{2 b^{3} d} + \frac {a c}{2 b^{2} d^{2}} + \frac {c^{2}}{2 b d^{3}}\right ) + x \left (- \frac {a^{3}}{b^{4} d} - \frac {a^{2} c}{b^{3} d^{2}} - \frac {a c^{2}}{b^{2} d^{3}} - \frac {c^{3}}{b d^{4}}\right ) + \frac {x^{4}}{4 b d} \]

[In]

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

[Out]

a**5*log(x + (a**7*d**6/(b*(a*d - b*c)) - 2*a**6*c*d**5/(a*d - b*c) + a**5*b*c**2*d**4/(a*d - b*c) + a**5*c*d*
*4 + a*b**4*c**5)/(a**5*d**5 + b**5*c**5))/(b**5*(a*d - b*c)) - c**5*log(x + (a**5*c*d**4 - a**2*b**4*c**5*d/(
a*d - b*c) + 2*a*b**5*c**6/(a*d - b*c) + a*b**4*c**5 - b**6*c**7/(d*(a*d - b*c)))/(a**5*d**5 + b**5*c**5))/(d*
*5*(a*d - b*c)) + x**3*(-a/(3*b**2*d) - c/(3*b*d**2)) + x**2*(a**2/(2*b**3*d) + a*c/(2*b**2*d**2) + c**2/(2*b*
d**3)) + x*(-a**3/(b**4*d) - a**2*c/(b**3*d**2) - a*c**2/(b**2*d**3) - c**3/(b*d**4)) + x**4/(4*b*d)

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.11 \[ \int \frac {x^5}{(a+b x) (c+d x)} \, dx=-\frac {a^{5} \log \left (b x + a\right )}{b^{6} c - a b^{5} d} + \frac {c^{5} \log \left (d x + c\right )}{b c d^{5} - a d^{6}} + \frac {3 \, b^{3} d^{3} x^{4} - 4 \, {\left (b^{3} c d^{2} + a b^{2} d^{3}\right )} x^{3} + 6 \, {\left (b^{3} c^{2} d + a b^{2} c d^{2} + a^{2} b d^{3}\right )} x^{2} - 12 \, {\left (b^{3} c^{3} + a b^{2} c^{2} d + a^{2} b c d^{2} + a^{3} d^{3}\right )} x}{12 \, b^{4} d^{4}} \]

[In]

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

[Out]

-a^5*log(b*x + a)/(b^6*c - a*b^5*d) + c^5*log(d*x + c)/(b*c*d^5 - a*d^6) + 1/12*(3*b^3*d^3*x^4 - 4*(b^3*c*d^2
+ a*b^2*d^3)*x^3 + 6*(b^3*c^2*d + a*b^2*c*d^2 + a^2*b*d^3)*x^2 - 12*(b^3*c^3 + a*b^2*c^2*d + a^2*b*c*d^2 + a^3
*d^3)*x)/(b^4*d^4)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.21 \[ \int \frac {x^5}{(a+b x) (c+d x)} \, dx=-\frac {a^{5} \log \left ({\left | b x + a \right |}\right )}{b^{6} c - a b^{5} d} + \frac {c^{5} \log \left ({\left | d x + c \right |}\right )}{b c d^{5} - a d^{6}} + \frac {3 \, b^{3} d^{3} x^{4} - 4 \, b^{3} c d^{2} x^{3} - 4 \, a b^{2} d^{3} x^{3} + 6 \, b^{3} c^{2} d x^{2} + 6 \, a b^{2} c d^{2} x^{2} + 6 \, a^{2} b d^{3} x^{2} - 12 \, b^{3} c^{3} x - 12 \, a b^{2} c^{2} d x - 12 \, a^{2} b c d^{2} x - 12 \, a^{3} d^{3} x}{12 \, b^{4} d^{4}} \]

[In]

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

[Out]

-a^5*log(abs(b*x + a))/(b^6*c - a*b^5*d) + c^5*log(abs(d*x + c))/(b*c*d^5 - a*d^6) + 1/12*(3*b^3*d^3*x^4 - 4*b
^3*c*d^2*x^3 - 4*a*b^2*d^3*x^3 + 6*b^3*c^2*d*x^2 + 6*a*b^2*c*d^2*x^2 + 6*a^2*b*d^3*x^2 - 12*b^3*c^3*x - 12*a*b
^2*c^2*d*x - 12*a^2*b*c*d^2*x - 12*a^3*d^3*x)/(b^4*d^4)

Mupad [B] (verification not implemented)

Time = 0.56 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.20 \[ \int \frac {x^5}{(a+b x) (c+d x)} \, dx=x^2\,\left (\frac {{\left (a\,d+b\,c\right )}^2}{2\,b^3\,d^3}-\frac {a\,c}{2\,b^2\,d^2}\right )-x\,\left (\frac {\left (a\,d+b\,c\right )\,\left (\frac {{\left (a\,d+b\,c\right )}^2}{b^3\,d^3}-\frac {a\,c}{b^2\,d^2}\right )}{b\,d}-\frac {a\,c\,\left (a\,d+b\,c\right )}{b^3\,d^3}\right )-\frac {a^5\,\ln \left (a+b\,x\right )}{b^6\,c-a\,b^5\,d}+\frac {x^4}{4\,b\,d}-\frac {c^5\,\ln \left (c+d\,x\right )}{d^5\,\left (a\,d-b\,c\right )}-\frac {x^3\,\left (a\,d+b\,c\right )}{3\,b^2\,d^2} \]

[In]

int(x^5/((a + b*x)*(c + d*x)),x)

[Out]

x^2*((a*d + b*c)^2/(2*b^3*d^3) - (a*c)/(2*b^2*d^2)) - x*(((a*d + b*c)*((a*d + b*c)^2/(b^3*d^3) - (a*c)/(b^2*d^
2)))/(b*d) - (a*c*(a*d + b*c))/(b^3*d^3)) - (a^5*log(a + b*x))/(b^6*c - a*b^5*d) + x^4/(4*b*d) - (c^5*log(c +
d*x))/(d^5*(a*d - b*c)) - (x^3*(a*d + b*c))/(3*b^2*d^2)