\(\int \frac {x^2}{(a+b x) (c+d x)} \, dx\) [339]

Optimal result

Integrand size = 18, antiderivative size = 56 \[ \int \frac {x^2}{(a+b x) (c+d x)} \, dx=\frac {x}{b d}+\frac {a^2 \log (a+b x)}{b^2 (b c-a d)}-\frac {c^2 \log (c+d x)}{d^2 (b c-a d)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 56, 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} \[ \int \frac {x^2}{(a+b x) (c+d x)} \, dx=\frac {a^2 \log (a+b x)}{b^2 (b c-a d)}-\frac {c^2 \log (c+d x)}{d^2 (b c-a d)}+\frac {x}{b d} \]

[In]

Int[x^2/((a + b*x)*(c + d*x)),x]

[Out]

x/(b*d) + (a^2*Log[a + b*x])/(b^2*(b*c - a*d)) - (c^2*Log[c + d*x])/(d^2*(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 {1}{b d}+\frac {a^2}{b (b c-a d) (a+b x)}+\frac {c^2}{d (-b c+a d) (c+d x)}\right ) \, dx \\ & = \frac {x}{b d}+\frac {a^2 \log (a+b x)}{b^2 (b c-a d)}-\frac {c^2 \log (c+d x)}{d^2 (b c-a d)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.00 \[ \int \frac {x^2}{(a+b x) (c+d x)} \, dx=\frac {x}{b d}+\frac {a^2 \log (a+b x)}{b^2 (b c-a d)}-\frac {c^2 \log (c+d x)}{d^2 (b c-a d)} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.81 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.02

method	result	size
default	\(\frac {x}{b d}+\frac {c^{2} \ln \left (d x +c \right )}{d^{2} \left (d a -b c \right )}-\frac {a^{2} \ln \left (b x +a \right )}{b^{2} \left (d a -b c \right )}\)	\(57\)
norman	\(\frac {x}{b d}+\frac {c^{2} \ln \left (d x +c \right )}{d^{2} \left (d a -b c \right )}-\frac {a^{2} \ln \left (b x +a \right )}{b^{2} \left (d a -b c \right )}\)	\(57\)
risch	\(\frac {x}{b d}+\frac {c^{2} \ln \left (-d x -c \right )}{d^{2} \left (d a -b c \right )}-\frac {a^{2} \ln \left (b x +a \right )}{b^{2} \left (d a -b c \right )}\)	\(60\)
parallelrisch	\(-\frac {a^{2} \ln \left (b x +a \right ) d^{2}-c^{2} \ln \left (d x +c \right ) b^{2}-a b \,d^{2} x +x \,b^{2} c d}{b^{2} d^{2} \left (d a -b c \right )}\)	\(62\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.16 \[ \int \frac {x^2}{(a+b x) (c+d x)} \, dx=\frac {a^{2} d^{2} \log \left (b x + a\right ) - b^{2} c^{2} \log \left (d x + c\right ) + {\left (b^{2} c d - a b d^{2}\right )} x}{b^{3} c d^{2} - a b^{2} d^{3}} \]

[In]

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

[Out]

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 190 vs. \(2 (44) = 88\).

Time = 0.61 (sec) , antiderivative size = 190, normalized size of antiderivative = 3.39 \[ \int \frac {x^2}{(a+b x) (c+d x)} \, dx=- \frac {a^{2} \log {\left (x + \frac {\frac {a^{4} d^{3}}{b \left (a d - b c\right )} - \frac {2 a^{3} c d^{2}}{a d - b c} + \frac {a^{2} b c^{2} d}{a d - b c} + a^{2} c d + a b c^{2}}{a^{2} d^{2} + b^{2} c^{2}} \right )}}{b^{2} \left (a d - b c\right )} + \frac {c^{2} \log {\left (x + \frac {- \frac {a^{2} b c^{2} d}{a d - b c} + a^{2} c d + \frac {2 a b^{2} c^{3}}{a d - b c} + a b c^{2} - \frac {b^{3} c^{4}}{d \left (a d - b c\right )}}{a^{2} d^{2} + b^{2} c^{2}} \right )}}{d^{2} \left (a d - b c\right )} + \frac {x}{b d} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.07 \[ \int \frac {x^2}{(a+b x) (c+d x)} \, dx=\frac {a^{2} \log \left (b x + a\right )}{b^{3} c - a b^{2} d} - \frac {c^{2} \log \left (d x + c\right )}{b c d^{2} - a d^{3}} + \frac {x}{b d} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.11 \[ \int \frac {x^2}{(a+b x) (c+d x)} \, dx=\frac {a^{2} \log \left ({\left | b x + a \right |}\right )}{b^{3} c - a b^{2} d} - \frac {c^{2} \log \left ({\left | d x + c \right |}\right )}{b c d^{2} - a d^{3}} + \frac {x}{b d} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.09 \[ \int \frac {x^2}{(a+b x) (c+d x)} \, dx=-\frac {a^2\,d^2\,\ln \left (a+b\,x\right )-b^2\,c^2\,\ln \left (c+d\,x\right )-a\,b\,d^2\,x+b^2\,c\,d\,x}{b^2\,d^2\,\left (a\,d-b\,c\right )} \]

[In]

int(x^2/((a + b*x)*(c + d*x)),x)

[Out]

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