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

Optimal. Leaf size=77 \[ \frac{a^2 \log (a+b x)}{b (b c-a d)^2}+\frac{c^2}{d^2 (c+d x) (b c-a d)}+\frac{c (b c-2 a d) \log (c+d x)}{d^2 (b c-a d)^2} \]

[Out]

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

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Rubi [A]  time = 0.141331, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056 \[ \frac{a^2 \log (a+b x)}{b (b c-a d)^2}+\frac{c^2}{d^2 (c+d x) (b c-a d)}+\frac{c (b c-2 a d) \log (c+d x)}{d^2 (b c-a d)^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 36.9121, size = 66, normalized size = 0.86 \[ \frac{a^{2} \log{\left (a + b x \right )}}{b \left (a d - b c\right )^{2}} - \frac{c^{2}}{d^{2} \left (c + d x\right ) \left (a d - b c\right )} - \frac{c \left (2 a d - b c\right ) \log{\left (c + d x \right )}}{d^{2} \left (a d - b c\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**2/(b*x+a)/(d*x+c)**2,x)

[Out]

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

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Mathematica [A]  time = 0.076785, size = 77, normalized size = 1. \[ \frac{a^2 d^2 (c+d x) \log (a+b x)+b c ((c+d x) (b c-2 a d) \log (c+d x)+c (b c-a d))}{b d^2 (c+d x) (b c-a d)^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.013, size = 97, normalized size = 1.3 \[ -{\frac{{c}^{2}}{{d}^{2} \left ( ad-bc \right ) \left ( dx+c \right ) }}-2\,{\frac{c\ln \left ( dx+c \right ) a}{d \left ( ad-bc \right ) ^{2}}}+{\frac{{c}^{2}\ln \left ( dx+c \right ) b}{ \left ( ad-bc \right ) ^{2}{d}^{2}}}+{\frac{{a}^{2}\ln \left ( bx+a \right ) }{ \left ( ad-bc \right ) ^{2}b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2/(b*x+a)/(d*x+c)^2,x)

[Out]

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

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Maxima [A]  time = 1.36686, size = 162, normalized size = 2.1 \[ \frac{a^{2} \log \left (b x + a\right )}{b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}} + \frac{c^{2}}{b c^{2} d^{2} - a c d^{3} +{\left (b c d^{3} - a d^{4}\right )} x} + \frac{{\left (b c^{2} - 2 \, a c d\right )} \log \left (d x + c\right )}{b^{2} c^{2} d^{2} - 2 \, a b c d^{3} + a^{2} d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.222827, size = 200, normalized size = 2.6 \[ \frac{b^{2} c^{3} - a b c^{2} d +{\left (a^{2} d^{3} x + a^{2} c d^{2}\right )} \log \left (b x + a\right ) +{\left (b^{2} c^{3} - 2 \, a b c^{2} d +{\left (b^{2} c^{2} d - 2 \, a b c d^{2}\right )} x\right )} \log \left (d x + c\right )}{b^{3} c^{3} d^{2} - 2 \, a b^{2} c^{2} d^{3} + a^{2} b c d^{4} +{\left (b^{3} c^{2} d^{3} - 2 \, a b^{2} c d^{4} + a^{2} b d^{5}\right )} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 8.92094, size = 333, normalized size = 4.32 \[ \frac{a^{2} \log{\left (x + \frac{\frac{a^{5} d^{4}}{b \left (a d - b c\right )^{2}} - \frac{3 a^{4} c d^{3}}{\left (a d - b c\right )^{2}} + \frac{3 a^{3} b c^{2} d^{2}}{\left (a d - b c\right )^{2}} - \frac{a^{2} b^{2} c^{3} d}{\left (a d - b c\right )^{2}} + 3 a^{2} c d - a b c^{2}}{a^{2} d^{2} + 2 a b c d - b^{2} c^{2}} \right )}}{b \left (a d - b c\right )^{2}} - \frac{c^{2}}{a c d^{3} - b c^{2} d^{2} + x \left (a d^{4} - b c d^{3}\right )} - \frac{c \left (2 a d - b c\right ) \log{\left (x + \frac{- \frac{a^{3} c d^{2} \left (2 a d - b c\right )}{\left (a d - b c\right )^{2}} + \frac{3 a^{2} b c^{2} d \left (2 a d - b c\right )}{\left (a d - b c\right )^{2}} + 3 a^{2} c d - \frac{3 a b^{2} c^{3} \left (2 a d - b c\right )}{\left (a d - b c\right )^{2}} - a b c^{2} + \frac{b^{3} c^{4} \left (2 a d - b c\right )}{d \left (a d - b c\right )^{2}}}{a^{2} d^{2} + 2 a b c d - b^{2} c^{2}} \right )}}{d^{2} \left (a d - b c\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.280791, size = 154, normalized size = 2. \[ \frac{a^{2} d{\rm ln}\left ({\left | b - \frac{b c}{d x + c} + \frac{a d}{d x + c} \right |}\right )}{b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}} + \frac{c^{2} d}{{\left (b c d^{3} - a d^{4}\right )}{\left (d x + c\right )}} - \frac{{\rm ln}\left (\frac{{\left | d x + c \right |}}{{\left (d x + c\right )}^{2}{\left | d \right |}}\right )}{b d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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