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

Optimal. Leaf size=147 \[ -\frac {a^5 \log (a+b x)}{b^4 (b c-a d)^2}+\frac {x \left (a^2 d^2+2 a b c d+3 b^2 c^2\right )}{b^3 d^4}-\frac {x^2 (a d+2 b c)}{2 b^2 d^3}-\frac {c^5}{d^5 (c+d x) (b c-a d)}-\frac {c^4 (4 b c-5 a d) \log (c+d x)}{d^5 (b c-a d)^2}+\frac {x^3}{3 b d^2} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.16, antiderivative size = 147, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {88} \[ \frac {x \left (a^2 d^2+2 a b c d+3 b^2 c^2\right )}{b^3 d^4}-\frac {a^5 \log (a+b x)}{b^4 (b c-a d)^2}-\frac {x^2 (a d+2 b c)}{2 b^2 d^3}-\frac {c^5}{d^5 (c+d x) (b c-a d)}-\frac {c^4 (4 b c-5 a d) \log (c+d x)}{d^5 (b c-a d)^2}+\frac {x^3}{3 b d^2} \]

Antiderivative was successfully verified.

[In]

Int[x^5/((a + b*x)*(c + d*x)^2),x]

[Out]

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

Rule 88

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 0.19, size = 147, normalized size = 1.00 \[ -\frac {a^5 \log (a+b x)}{b^4 (b c-a d)^2}+\frac {x \left (a^2 d^2+2 a b c d+3 b^2 c^2\right )}{b^3 d^4}-\frac {x^2 (a d+2 b c)}{2 b^2 d^3}+\frac {c^5}{d^5 (c+d x) (a d-b c)}+\frac {\left (5 a c^4 d-4 b c^5\right ) \log (c+d x)}{d^5 (b c-a d)^2}+\frac {x^3}{3 b d^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [B]  time = 0.93, size = 337, normalized size = 2.29 \[ -\frac {6 \, b^{5} c^{6} - 6 \, a b^{4} c^{5} d - 2 \, {\left (b^{5} c^{2} d^{4} - 2 \, a b^{4} c d^{5} + a^{2} b^{3} d^{6}\right )} x^{4} + {\left (4 \, b^{5} c^{3} d^{3} - 5 \, a b^{4} c^{2} d^{4} - 2 \, a^{2} b^{3} c d^{5} + 3 \, a^{3} b^{2} d^{6}\right )} x^{3} - 3 \, {\left (4 \, b^{5} c^{4} d^{2} - 5 \, a b^{4} c^{3} d^{3} - a^{3} b^{2} c d^{5} + 2 \, a^{4} b d^{6}\right )} x^{2} - 6 \, {\left (3 \, b^{5} c^{5} d - 4 \, a b^{4} c^{4} d^{2} + a^{4} b c d^{5}\right )} x + 6 \, {\left (a^{5} d^{6} x + a^{5} c d^{5}\right )} \log \left (b x + a\right ) + 6 \, {\left (4 \, b^{5} c^{6} - 5 \, a b^{4} c^{5} d + {\left (4 \, b^{5} c^{5} d - 5 \, a b^{4} c^{4} d^{2}\right )} x\right )} \log \left (d x + c\right )}{6 \, {\left (b^{6} c^{3} d^{5} - 2 \, a b^{5} c^{2} d^{6} + a^{2} b^{4} c d^{7} + {\left (b^{6} c^{2} d^{6} - 2 \, a b^{5} c d^{7} + a^{2} b^{4} d^{8}\right )} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*(6*b^5*c^6 - 6*a*b^4*c^5*d - 2*(b^5*c^2*d^4 - 2*a*b^4*c*d^5 + a^2*b^3*d^6)*x^4 + (4*b^5*c^3*d^3 - 5*a*b^4
*c^2*d^4 - 2*a^2*b^3*c*d^5 + 3*a^3*b^2*d^6)*x^3 - 3*(4*b^5*c^4*d^2 - 5*a*b^4*c^3*d^3 - a^3*b^2*c*d^5 + 2*a^4*b
*d^6)*x^2 - 6*(3*b^5*c^5*d - 4*a*b^4*c^4*d^2 + a^4*b*c*d^5)*x + 6*(a^5*d^6*x + a^5*c*d^5)*log(b*x + a) + 6*(4*
b^5*c^6 - 5*a*b^4*c^5*d + (4*b^5*c^5*d - 5*a*b^4*c^4*d^2)*x)*log(d*x + c))/(b^6*c^3*d^5 - 2*a*b^5*c^2*d^6 + a^
2*b^4*c*d^7 + (b^6*c^2*d^6 - 2*a*b^5*c*d^7 + a^2*b^4*d^8)*x)

________________________________________________________________________________________

giac [A]  time = 0.81, size = 246, normalized size = 1.67 \[ -\frac {c^{5} d^{4}}{{\left (b c d^{9} - a d^{10}\right )} {\left (d x + c\right )}} - \frac {a^{5} d \log \left ({\left | b - \frac {b c}{d x + c} + \frac {a d}{d x + c} \right |}\right )}{b^{6} c^{2} d - 2 \, a b^{5} c d^{2} + a^{2} b^{4} d^{3}} + \frac {{\left (2 \, b^{3} - \frac {3 \, {\left (4 \, b^{3} c d + a b^{2} d^{2}\right )}}{{\left (d x + c\right )} d} + \frac {6 \, {\left (6 \, b^{3} c^{2} d^{2} + 3 \, a b^{2} c d^{3} + a^{2} b d^{4}\right )}}{{\left (d x + c\right )}^{2} d^{2}}\right )} {\left (d x + c\right )}^{3}}{6 \, b^{4} d^{5}} + \frac {{\left (4 \, b^{3} c^{3} + 3 \, a b^{2} c^{2} d + 2 \, a^{2} b c d^{2} + a^{3} d^{3}\right )} \log \left (\frac {{\left | d x + c \right |}}{{\left (d x + c\right )}^{2} {\left | d \right |}}\right )}{b^{4} d^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-c^5*d^4/((b*c*d^9 - a*d^10)*(d*x + c)) - a^5*d*log(abs(b - b*c/(d*x + c) + a*d/(d*x + c)))/(b^6*c^2*d - 2*a*b
^5*c*d^2 + a^2*b^4*d^3) + 1/6*(2*b^3 - 3*(4*b^3*c*d + a*b^2*d^2)/((d*x + c)*d) + 6*(6*b^3*c^2*d^2 + 3*a*b^2*c*
d^3 + a^2*b*d^4)/((d*x + c)^2*d^2))*(d*x + c)^3/(b^4*d^5) + (4*b^3*c^3 + 3*a*b^2*c^2*d + 2*a^2*b*c*d^2 + a^3*d
^3)*log(abs(d*x + c)/((d*x + c)^2*abs(d)))/(b^4*d^5)

________________________________________________________________________________________

maple [A]  time = 0.01, size = 169, normalized size = 1.15 \[ -\frac {a^{5} \ln \left (b x +a \right )}{\left (a d -b c \right )^{2} b^{4}}+\frac {5 a \,c^{4} \ln \left (d x +c \right )}{\left (a d -b c \right )^{2} d^{4}}-\frac {4 b \,c^{5} \ln \left (d x +c \right )}{\left (a d -b c \right )^{2} d^{5}}+\frac {x^{3}}{3 b \,d^{2}}+\frac {c^{5}}{\left (a d -b c \right ) \left (d x +c \right ) d^{5}}-\frac {a \,x^{2}}{2 b^{2} d^{2}}-\frac {c \,x^{2}}{b \,d^{3}}+\frac {a^{2} x}{b^{3} d^{2}}+\frac {2 a c x}{b^{2} d^{3}}+\frac {3 c^{2} x}{b \,d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 1.08, size = 192, normalized size = 1.31 \[ -\frac {a^{5} \log \left (b x + a\right )}{b^{6} c^{2} - 2 \, a b^{5} c d + a^{2} b^{4} d^{2}} - \frac {c^{5}}{b c^{2} d^{5} - a c d^{6} + {\left (b c d^{6} - a d^{7}\right )} x} - \frac {{\left (4 \, b c^{5} - 5 \, a c^{4} d\right )} \log \left (d x + c\right )}{b^{2} c^{2} d^{5} - 2 \, a b c d^{6} + a^{2} d^{7}} + \frac {2 \, b^{2} d^{2} x^{3} - 3 \, {\left (2 \, b^{2} c d + a b d^{2}\right )} x^{2} + 6 \, {\left (3 \, b^{2} c^{2} + 2 \, a b c d + a^{2} d^{2}\right )} x}{6 \, b^{3} d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a^5*log(b*x + a)/(b^6*c^2 - 2*a*b^5*c*d + a^2*b^4*d^2) - c^5/(b*c^2*d^5 - a*c*d^6 + (b*c*d^6 - a*d^7)*x) - (4
*b*c^5 - 5*a*c^4*d)*log(d*x + c)/(b^2*c^2*d^5 - 2*a*b*c*d^6 + a^2*d^7) + 1/6*(2*b^2*d^2*x^3 - 3*(2*b^2*c*d + a
*b*d^2)*x^2 + 6*(3*b^2*c^2 + 2*a*b*c*d + a^2*d^2)*x)/(b^3*d^4)

________________________________________________________________________________________

mupad [B]  time = 0.61, size = 205, normalized size = 1.39 \[ \frac {x^3}{3\,b\,d^2}-\frac {a^5\,\ln \left (a+b\,x\right )}{a^2\,b^4\,d^2-2\,a\,b^5\,c\,d+b^6\,c^2}-\frac {\ln \left (c+d\,x\right )\,\left (4\,b\,c^5-5\,a\,c^4\,d\right )}{a^2\,d^7-2\,a\,b\,c\,d^6+b^2\,c^2\,d^5}-x\,\left (\frac {b\,c^2+2\,a\,d\,c}{b^2\,d^4}-\frac {{\left (a\,d^2+2\,b\,c\,d\right )}^2}{b^3\,d^6}\right )-\frac {x^2\,\left (a\,d^2+2\,b\,c\,d\right )}{2\,b^2\,d^4}+\frac {b^3\,c^5}{d\,\left (x\,b^3\,d^5+c\,b^3\,d^4\right )\,\left (a\,d-b\,c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5/((a + b*x)*(c + d*x)^2),x)

[Out]

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

________________________________________________________________________________________

sympy [B]  time = 6.00, size = 464, normalized size = 3.16 \[ - \frac {a^{5} \log {\left (x + \frac {\frac {a^{8} d^{7}}{b \left (a d - b c\right )^{2}} - \frac {3 a^{7} c d^{6}}{\left (a d - b c\right )^{2}} + \frac {3 a^{6} b c^{2} d^{5}}{\left (a d - b c\right )^{2}} - \frac {a^{5} b^{2} c^{3} d^{4}}{\left (a d - b c\right )^{2}} + a^{5} c d^{4} + 5 a^{2} b^{3} c^{4} d - 4 a b^{4} c^{5}}{a^{5} d^{5} + 5 a b^{4} c^{4} d - 4 b^{5} c^{5}} \right )}}{b^{4} \left (a d - b c\right )^{2}} + \frac {c^{5}}{a c d^{6} - b c^{2} d^{5} + x \left (a d^{7} - b c d^{6}\right )} + \frac {c^{4} \left (5 a d - 4 b c\right ) \log {\left (x + \frac {a^{5} c d^{4} - \frac {a^{3} b^{3} c^{4} d^{2} \left (5 a d - 4 b c\right )}{\left (a d - b c\right )^{2}} + \frac {3 a^{2} b^{4} c^{5} d \left (5 a d - 4 b c\right )}{\left (a d - b c\right )^{2}} + 5 a^{2} b^{3} c^{4} d - \frac {3 a b^{5} c^{6} \left (5 a d - 4 b c\right )}{\left (a d - b c\right )^{2}} - 4 a b^{4} c^{5} + \frac {b^{6} c^{7} \left (5 a d - 4 b c\right )}{d \left (a d - b c\right )^{2}}}{a^{5} d^{5} + 5 a b^{4} c^{4} d - 4 b^{5} c^{5}} \right )}}{d^{5} \left (a d - b c\right )^{2}} + x^{2} \left (- \frac {a}{2 b^{2} d^{2}} - \frac {c}{b d^{3}}\right ) + x \left (\frac {a^{2}}{b^{3} d^{2}} + \frac {2 a c}{b^{2} d^{3}} + \frac {3 c^{2}}{b d^{4}}\right ) + \frac {x^{3}}{3 b d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a**5*log(x + (a**8*d**7/(b*(a*d - b*c)**2) - 3*a**7*c*d**6/(a*d - b*c)**2 + 3*a**6*b*c**2*d**5/(a*d - b*c)**2
- a**5*b**2*c**3*d**4/(a*d - b*c)**2 + a**5*c*d**4 + 5*a**2*b**3*c**4*d - 4*a*b**4*c**5)/(a**5*d**5 + 5*a*b**
4*c**4*d - 4*b**5*c**5))/(b**4*(a*d - b*c)**2) + c**5/(a*c*d**6 - b*c**2*d**5 + x*(a*d**7 - b*c*d**6)) + c**4*
(5*a*d - 4*b*c)*log(x + (a**5*c*d**4 - a**3*b**3*c**4*d**2*(5*a*d - 4*b*c)/(a*d - b*c)**2 + 3*a**2*b**4*c**5*d
*(5*a*d - 4*b*c)/(a*d - b*c)**2 + 5*a**2*b**3*c**4*d - 3*a*b**5*c**6*(5*a*d - 4*b*c)/(a*d - b*c)**2 - 4*a*b**4
*c**5 + b**6*c**7*(5*a*d - 4*b*c)/(d*(a*d - b*c)**2))/(a**5*d**5 + 5*a*b**4*c**4*d - 4*b**5*c**5))/(d**5*(a*d
- b*c)**2) + x**2*(-a/(2*b**2*d**2) - c/(b*d**3)) + x*(a**2/(b**3*d**2) + 2*a*c/(b**2*d**3) + 3*c**2/(b*d**4))
+ x**3/(3*b*d**2)

________________________________________________________________________________________