3.1881 \(\int \frac{(c+d x)^{1+2 n-2 (1+n)}}{(a+b x)^2} \, dx\)

Optimal. Leaf size=57 \[ -\frac{1}{(a+b x) (b c-a d)}-\frac{d \log (a+b x)}{(b c-a d)^2}+\frac{d \log (c+d x)}{(b c-a d)^2} \]

[Out]

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

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Rubi [A]  time = 0.0709204, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ -\frac{1}{(a+b x) (b c-a d)}-\frac{d \log (a+b x)}{(b c-a d)^2}+\frac{d \log (c+d x)}{(b c-a d)^2} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x)^(1 + 2*n - 2*(1 + n))/(a + b*x)^2,x]

[Out]

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

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Rubi in Sympy [A]  time = 12.8051, size = 46, normalized size = 0.81 \[ - \frac{d \log{\left (a + b x \right )}}{\left (a d - b c\right )^{2}} + \frac{d \log{\left (c + d x \right )}}{\left (a d - b c\right )^{2}} + \frac{1}{\left (a + b x\right ) \left (a d - b c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(b*x+a)**2/(d*x+c),x)

[Out]

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

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Mathematica [A]  time = 0.0410186, size = 53, normalized size = 0.93 \[ \frac{d (a+b x) \log (c+d x)-d (a+b x) \log (a+b x)+a d-b c}{(a+b x) (b c-a d)^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(c + d*x)^(1 + 2*n - 2*(1 + n))/(a + b*x)^2,x]

[Out]

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

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Maple [A]  time = 0.003, size = 57, normalized size = 1. \[{\frac{d\ln \left ( dx+c \right ) }{ \left ( ad-bc \right ) ^{2}}}+{\frac{1}{ \left ( ad-bc \right ) \left ( bx+a \right ) }}-{\frac{d\ln \left ( bx+a \right ) }{ \left ( ad-bc \right ) ^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b*x+a)^2/(d*x+c),x)

[Out]

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

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Maxima [A]  time = 1.34407, size = 124, normalized size = 2.18 \[ -\frac{d \log \left (b x + a\right )}{b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}} + \frac{d \log \left (d x + c\right )}{b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}} - \frac{1}{a b c - a^{2} d +{\left (b^{2} c - a b d\right )} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^2*(d*x + c)),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.215933, size = 126, normalized size = 2.21 \[ -\frac{b c - a d +{\left (b d x + a d\right )} \log \left (b x + a\right ) -{\left (b d x + a d\right )} \log \left (d x + c\right )}{a b^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{2} +{\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^2*(d*x + c)),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 1.50607, size = 233, normalized size = 4.09 \[ \frac{d \log{\left (x + \frac{- \frac{a^{3} d^{4}}{\left (a d - b c\right )^{2}} + \frac{3 a^{2} b c d^{3}}{\left (a d - b c\right )^{2}} - \frac{3 a b^{2} c^{2} d^{2}}{\left (a d - b c\right )^{2}} + a d^{2} + \frac{b^{3} c^{3} d}{\left (a d - b c\right )^{2}} + b c d}{2 b d^{2}} \right )}}{\left (a d - b c\right )^{2}} - \frac{d \log{\left (x + \frac{\frac{a^{3} d^{4}}{\left (a d - b c\right )^{2}} - \frac{3 a^{2} b c d^{3}}{\left (a d - b c\right )^{2}} + \frac{3 a b^{2} c^{2} d^{2}}{\left (a d - b c\right )^{2}} + a d^{2} - \frac{b^{3} c^{3} d}{\left (a d - b c\right )^{2}} + b c d}{2 b d^{2}} \right )}}{\left (a d - b c\right )^{2}} + \frac{1}{a^{2} d - a b c + x \left (a b d - b^{2} c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(b*x+a)**2/(d*x+c),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.272628, size = 105, normalized size = 1.84 \[ \frac{b d{\rm ln}\left ({\left | \frac{b c}{b x + a} - \frac{a d}{b x + a} + d \right |}\right )}{b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}} - \frac{b}{{\left (b^{2} c - a b d\right )}{\left (b x + a\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^2*(d*x + c)),x, algorithm="giac")

[Out]

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