\(\int \frac {x^3}{(a+b x) (c+d x)^2} \, dx\) [244]

Optimal result

Integrand size = 18, antiderivative size = 91 \[ \int \frac {x^3}{(a+b x) (c+d x)^2} \, dx=\frac {x}{b d^2}-\frac {c^3}{d^3 (b c-a d) (c+d x)}-\frac {a^3 \log (a+b x)}{b^2 (b c-a d)^2}-\frac {c^2 (2 b c-3 a d) \log (c+d x)}{d^3 (b c-a d)^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 91, 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 = {90} \[ \int \frac {x^3}{(a+b x) (c+d x)^2} \, dx=-\frac {a^3 \log (a+b x)}{b^2 (b c-a d)^2}-\frac {c^3}{d^3 (c+d x) (b c-a d)}-\frac {c^2 (2 b c-3 a d) \log (c+d x)}{d^3 (b c-a d)^2}+\frac {x}{b d^2} \]

[In]

Int[x^3/((a + b*x)*(c + d*x)^2),x]

[Out]

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

Rule 90

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*} \text {integral}& = \int \left (\frac {1}{b d^2}-\frac {a^3}{b (b c-a d)^2 (a+b x)}-\frac {c^3}{d^2 (-b c+a d) (c+d x)^2}-\frac {c^2 (2 b c-3 a d)}{d^2 (-b c+a d)^2 (c+d x)}\right ) \, dx \\ & = \frac {x}{b d^2}-\frac {c^3}{d^3 (b c-a d) (c+d x)}-\frac {a^3 \log (a+b x)}{b^2 (b c-a d)^2}-\frac {c^2 (2 b c-3 a d) \log (c+d x)}{d^3 (b c-a d)^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.96 \[ \int \frac {x^3}{(a+b x) (c+d x)^2} \, dx=-\frac {a^3 \log (a+b x)}{b^2 (b c-a d)^2}+\frac {\frac {d x}{b}+\frac {c^3}{(-b c+a d) (c+d x)}-\frac {c^2 (2 b c-3 a d) \log (c+d x)}{(b c-a d)^2}}{d^3} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 1.26 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.99

method	result	size
default	\(\frac {x}{b \,d^{2}}+\frac {c^{2} \left (3 a d -2 b c \right ) \ln \left (d x +c \right )}{d^{3} \left (a d -b c \right )^{2}}+\frac {c^{3}}{d^{3} \left (a d -b c \right ) \left (d x +c \right )}-\frac {a^{3} \ln \left (b x +a \right )}{b^{2} \left (a d -b c \right )^{2}}\)	\(90\)
norman	\(\frac {\frac {x^{2}}{b d}+\frac {\left (-a d c +2 b \,c^{2}\right ) c}{d^{3} b \left (a d -b c \right )}}{d x +c}+\frac {c^{2} \left (3 a d -2 b c \right ) \ln \left (d x +c \right )}{d^{3} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}-\frac {a^{3} \ln \left (b x +a \right )}{\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) b^{2}}\)	\(133\)
risch	\(\frac {x}{b \,d^{2}}+\frac {c^{3}}{d^{3} \left (a d -b c \right ) \left (d x +c \right )}-\frac {a^{3} \ln \left (b x +a \right )}{\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) b^{2}}+\frac {3 c^{2} \ln \left (-d x -c \right ) a}{d^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}-\frac {2 c^{3} \ln \left (-d x -c \right ) b}{d^{3} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}\)	\(153\)
parallelrisch	\(-\frac {\ln \left (b x +a \right ) x \,a^{3} d^{4}-3 \ln \left (d x +c \right ) x a \,b^{2} c^{2} d^{2}+2 \ln \left (d x +c \right ) x \,b^{3} c^{3} d -x^{2} a^{2} b \,d^{4}+2 x^{2} a \,b^{2} c \,d^{3}-x^{2} b^{3} c^{2} d^{2}+\ln \left (b x +a \right ) a^{3} c \,d^{3}-3 \ln \left (d x +c \right ) a \,b^{2} c^{3} d +2 \ln \left (d x +c \right ) b^{3} c^{4}+a^{2} d^{2} b \,c^{2}-3 a \,b^{2} c^{3} d +2 c^{4} b^{3}}{\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \left (d x +c \right ) b^{2} d^{3}}\)	\(201\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 230 vs. \(2 (91) = 182\).

Time = 0.23 (sec) , antiderivative size = 230, normalized size of antiderivative = 2.53 \[ \int \frac {x^3}{(a+b x) (c+d x)^2} \, dx=-\frac {b^{3} c^{4} - a b^{2} c^{3} d - {\left (b^{3} c^{2} d^{2} - 2 \, a b^{2} c d^{3} + a^{2} b d^{4}\right )} x^{2} - {\left (b^{3} c^{3} d - 2 \, a b^{2} c^{2} d^{2} + a^{2} b c d^{3}\right )} x + {\left (a^{3} d^{4} x + a^{3} c d^{3}\right )} \log \left (b x + a\right ) + {\left (2 \, b^{3} c^{4} - 3 \, a b^{2} c^{3} d + {\left (2 \, b^{3} c^{3} d - 3 \, a b^{2} c^{2} d^{2}\right )} x\right )} \log \left (d x + c\right )}{b^{4} c^{3} d^{3} - 2 \, a b^{3} c^{2} d^{4} + a^{2} b^{2} c d^{5} + {\left (b^{4} c^{2} d^{4} - 2 \, a b^{3} c d^{5} + a^{2} b^{2} d^{6}\right )} x} \]

[In]

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

[Out]

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 400 vs. \(2 (78) = 156\).

Time = 1.95 (sec) , antiderivative size = 400, normalized size of antiderivative = 4.40 \[ \int \frac {x^3}{(a+b x) (c+d x)^2} \, dx=- \frac {a^{3} \log {\left (x + \frac {\frac {a^{6} d^{5}}{b \left (a d - b c\right )^{2}} - \frac {3 a^{5} c d^{4}}{\left (a d - b c\right )^{2}} + \frac {3 a^{4} b c^{2} d^{3}}{\left (a d - b c\right )^{2}} - \frac {a^{3} b^{2} c^{3} d^{2}}{\left (a d - b c\right )^{2}} + a^{3} c d^{2} + 3 a^{2} b c^{2} d - 2 a b^{2} c^{3}}{a^{3} d^{3} + 3 a b^{2} c^{2} d - 2 b^{3} c^{3}} \right )}}{b^{2} \left (a d - b c\right )^{2}} + \frac {c^{3}}{a c d^{4} - b c^{2} d^{3} + x \left (a d^{5} - b c d^{4}\right )} + \frac {c^{2} \cdot \left (3 a d - 2 b c\right ) \log {\left (x + \frac {- \frac {a^{3} b c^{2} d^{2} \cdot \left (3 a d - 2 b c\right )}{\left (a d - b c\right )^{2}} + a^{3} c d^{2} + \frac {3 a^{2} b^{2} c^{3} d \left (3 a d - 2 b c\right )}{\left (a d - b c\right )^{2}} + 3 a^{2} b c^{2} d - \frac {3 a b^{3} c^{4} \cdot \left (3 a d - 2 b c\right )}{\left (a d - b c\right )^{2}} - 2 a b^{2} c^{3} + \frac {b^{4} c^{5} \cdot \left (3 a d - 2 b c\right )}{d \left (a d - b c\right )^{2}}}{a^{3} d^{3} + 3 a b^{2} c^{2} d - 2 b^{3} c^{3}} \right )}}{d^{3} \left (a d - b c\right )^{2}} + \frac {x}{b d^{2}} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.49 \[ \int \frac {x^3}{(a+b x) (c+d x)^2} \, dx=-\frac {a^{3} \log \left (b x + a\right )}{b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}} - \frac {c^{3}}{b c^{2} d^{3} - a c d^{4} + {\left (b c d^{4} - a d^{5}\right )} x} - \frac {{\left (2 \, b c^{3} - 3 \, a c^{2} d\right )} \log \left (d x + c\right )}{b^{2} c^{2} d^{3} - 2 \, a b c d^{4} + a^{2} d^{5}} + \frac {x}{b d^{2}} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.53 \[ \int \frac {x^3}{(a+b x) (c+d x)^2} \, dx=-\frac {a^{3} d \log \left ({\left | b - \frac {b c}{d x + c} + \frac {a d}{d x + c} \right |}\right )}{b^{4} c^{2} d - 2 \, a b^{3} c d^{2} + a^{2} b^{2} d^{3}} - \frac {c^{3} d^{2}}{{\left (b c d^{5} - a d^{6}\right )} {\left (d x + c\right )}} + \frac {d x + c}{b d^{3}} + \frac {{\left (2 \, b c + a d\right )} \log \left (\frac {{\left | d x + c \right |}}{{\left (d x + c\right )}^{2} {\left | d \right |}}\right )}{b^{2} d^{3}} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.51 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.44 \[ \int \frac {x^3}{(a+b x) (c+d x)^2} \, dx=\frac {x}{b\,d^2}-\frac {\ln \left (c+d\,x\right )\,\left (2\,b\,c^3-3\,a\,c^2\,d\right )}{a^2\,d^5-2\,a\,b\,c\,d^4+b^2\,c^2\,d^3}-\frac {a^3\,\ln \left (a+b\,x\right )}{a^2\,b^2\,d^2-2\,a\,b^3\,c\,d+b^4\,c^2}+\frac {b\,c^3}{d\,\left (b\,x\,d^3+b\,c\,d^2\right )\,\left (a\,d-b\,c\right )} \]

[In]

int(x^3/((a + b*x)*(c + d*x)^2),x)

[Out]

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