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

Optimal. Leaf size=91 \[ -\frac {a^2}{b (a+b x) (b c-a d)^2}-\frac {c^2}{d (c+d x) (b c-a d)^2}-\frac {2 a c \log (a+b x)}{(b c-a d)^3}+\frac {2 a c \log (c+d x)}{(b c-a d)^3} \]

[Out]

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

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Rubi [A]  time = 0.06, antiderivative size = 91, 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} \[ -\frac {a^2}{b (a+b x) (b c-a d)^2}-\frac {c^2}{d (c+d x) (b c-a d)^2}-\frac {2 a c \log (a+b x)}{(b c-a d)^3}+\frac {2 a c \log (c+d x)}{(b c-a d)^3} \]

Antiderivative was successfully verified.

[In]

Int[x^2/((a + b*x)^2*(c + d*x)^2),x]

[Out]

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

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

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Mathematica [A]  time = 0.18, size = 71, normalized size = 0.78 \[ \frac {-\left ((b c-a d) \left (\frac {a^2}{b (a+b x)}+\frac {c^2}{d (c+d x)}\right )\right )-2 a c \log (a+b x)+2 a c \log (c+d x)}{(b c-a d)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 0.90, size = 303, normalized size = 3.33 \[ -\frac {a b^{2} c^{3} - a^{3} c d^{2} + {\left (b^{3} c^{3} - a b^{2} c^{2} d + a^{2} b c d^{2} - a^{3} d^{3}\right )} x + 2 \, {\left (a b^{2} c d^{2} x^{2} + a^{2} b c^{2} d + {\left (a b^{2} c^{2} d + a^{2} b c d^{2}\right )} x\right )} \log \left (b x + a\right ) - 2 \, {\left (a b^{2} c d^{2} x^{2} + a^{2} b c^{2} d + {\left (a b^{2} c^{2} d + a^{2} b c d^{2}\right )} x\right )} \log \left (d x + c\right )}{a b^{4} c^{4} d - 3 \, a^{2} b^{3} c^{3} d^{2} + 3 \, a^{3} b^{2} c^{2} d^{3} - a^{4} b c d^{4} + {\left (b^{5} c^{3} d^{2} - 3 \, a b^{4} c^{2} d^{3} + 3 \, a^{2} b^{3} c d^{4} - a^{3} b^{2} d^{5}\right )} x^{2} + {\left (b^{5} c^{4} d - 2 \, a b^{4} c^{3} d^{2} + 2 \, a^{3} b^{2} c d^{4} - a^{4} b d^{5}\right )} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.98, size = 153, normalized size = 1.68 \[ \frac {2 \, a b c \log \left ({\left | \frac {b c}{b x + a} - \frac {a d}{b x + a} + d \right |}\right )}{b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}} - \frac {a^{2} b}{{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} {\left (b x + a\right )}} + \frac {b c^{2}}{{\left (b c - a d\right )}^{3} {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.01, size = 92, normalized size = 1.01 \[ \frac {2 a c \ln \left (b x +a \right )}{\left (a d -b c \right )^{3}}-\frac {2 a c \ln \left (d x +c \right )}{\left (a d -b c \right )^{3}}-\frac {a^{2}}{\left (a d -b c \right )^{2} \left (b x +a \right ) b}-\frac {c^{2}}{\left (a d -b c \right )^{2} \left (d x +c \right ) d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 1.15, size = 242, normalized size = 2.66 \[ -\frac {2 \, a c \log \left (b x + a\right )}{b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}} + \frac {2 \, a c \log \left (d x + c\right )}{b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}} - \frac {a b c^{2} + a^{2} c d + {\left (b^{2} c^{2} + a^{2} d^{2}\right )} x}{a b^{3} c^{3} d - 2 \, a^{2} b^{2} c^{2} d^{2} + a^{3} b c d^{3} + {\left (b^{4} c^{2} d^{2} - 2 \, a b^{3} c d^{3} + a^{2} b^{2} d^{4}\right )} x^{2} + {\left (b^{4} c^{3} d - a b^{3} c^{2} d^{2} - a^{2} b^{2} c d^{3} + a^{3} b d^{4}\right )} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.45, size = 208, normalized size = 2.29 \[ \frac {4\,a\,c\,\mathrm {atanh}\left (\frac {a^3\,d^3-a^2\,b\,c\,d^2-a\,b^2\,c^2\,d+b^3\,c^3}{{\left (a\,d-b\,c\right )}^3}+\frac {2\,b\,d\,x\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}{{\left (a\,d-b\,c\right )}^3}\right )}{{\left (a\,d-b\,c\right )}^3}-\frac {\frac {x\,\left (a^2\,d^2+b^2\,c^2\right )}{b\,d\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}+\frac {a\,c\,\left (a\,d+b\,c\right )}{b\,d\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}}{b\,d\,x^2+\left (a\,d+b\,c\right )\,x+a\,c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/((a + b*x)^2*(c + d*x)^2),x)

[Out]

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

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sympy [B]  time = 1.39, size = 439, normalized size = 4.82 \[ - \frac {2 a c \log {\left (x + \frac {- \frac {2 a^{5} c d^{4}}{\left (a d - b c\right )^{3}} + \frac {8 a^{4} b c^{2} d^{3}}{\left (a d - b c\right )^{3}} - \frac {12 a^{3} b^{2} c^{3} d^{2}}{\left (a d - b c\right )^{3}} + \frac {8 a^{2} b^{3} c^{4} d}{\left (a d - b c\right )^{3}} + 2 a^{2} c d - \frac {2 a b^{4} c^{5}}{\left (a d - b c\right )^{3}} + 2 a b c^{2}}{4 a b c d} \right )}}{\left (a d - b c\right )^{3}} + \frac {2 a c \log {\left (x + \frac {\frac {2 a^{5} c d^{4}}{\left (a d - b c\right )^{3}} - \frac {8 a^{4} b c^{2} d^{3}}{\left (a d - b c\right )^{3}} + \frac {12 a^{3} b^{2} c^{3} d^{2}}{\left (a d - b c\right )^{3}} - \frac {8 a^{2} b^{3} c^{4} d}{\left (a d - b c\right )^{3}} + 2 a^{2} c d + \frac {2 a b^{4} c^{5}}{\left (a d - b c\right )^{3}} + 2 a b c^{2}}{4 a b c d} \right )}}{\left (a d - b c\right )^{3}} + \frac {- a^{2} c d - a b c^{2} + x \left (- a^{2} d^{2} - b^{2} c^{2}\right )}{a^{3} b c d^{3} - 2 a^{2} b^{2} c^{2} d^{2} + a b^{3} c^{3} d + x^{2} \left (a^{2} b^{2} d^{4} - 2 a b^{3} c d^{3} + b^{4} c^{2} d^{2}\right ) + x \left (a^{3} b d^{4} - a^{2} b^{2} c d^{3} - a b^{3} c^{2} d^{2} + b^{4} c^{3} d\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*a*c*log(x + (-2*a**5*c*d**4/(a*d - b*c)**3 + 8*a**4*b*c**2*d**3/(a*d - b*c)**3 - 12*a**3*b**2*c**3*d**2/(a*
d - b*c)**3 + 8*a**2*b**3*c**4*d/(a*d - b*c)**3 + 2*a**2*c*d - 2*a*b**4*c**5/(a*d - b*c)**3 + 2*a*b*c**2)/(4*a
*b*c*d))/(a*d - b*c)**3 + 2*a*c*log(x + (2*a**5*c*d**4/(a*d - b*c)**3 - 8*a**4*b*c**2*d**3/(a*d - b*c)**3 + 12
*a**3*b**2*c**3*d**2/(a*d - b*c)**3 - 8*a**2*b**3*c**4*d/(a*d - b*c)**3 + 2*a**2*c*d + 2*a*b**4*c**5/(a*d - b*
c)**3 + 2*a*b*c**2)/(4*a*b*c*d))/(a*d - b*c)**3 + (-a**2*c*d - a*b*c**2 + x*(-a**2*d**2 - b**2*c**2))/(a**3*b*
c*d**3 - 2*a**2*b**2*c**2*d**2 + a*b**3*c**3*d + x**2*(a**2*b**2*d**4 - 2*a*b**3*c*d**3 + b**4*c**2*d**2) + x*
(a**3*b*d**4 - a**2*b**2*c*d**3 - a*b**3*c**2*d**2 + b**4*c**3*d))

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