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

Optimal. Leaf size=56 \[ \frac{1}{(c+d x) (b c-a d)}+\frac{b \log (a+b x)}{(b c-a d)^2}-\frac{b \log (c+d x)}{(b c-a d)^2} \]

[Out]

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

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Rubi [A]  time = 0.0675686, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067 \[ \frac{1}{(c+d x) (b c-a d)}+\frac{b \log (a+b x)}{(b c-a d)^2}-\frac{b \log (c+d x)}{(b c-a d)^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 23.5796, size = 46, normalized size = 0.82 \[ \frac{b \log{\left (a + b x \right )}}{\left (a d - b c\right )^{2}} - \frac{b \log{\left (c + d x \right )}}{\left (a d - b c\right )^{2}} - \frac{1}{\left (c + d x\right ) \left (a d - b c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.038372, size = 53, normalized size = 0.95 \[ \frac{b (c+d x) \log (a+b x)-a d-b (c+d x) \log (c+d x)+b c}{(c+d x) (b c-a d)^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0., size = 58, normalized size = 1. \[ -{\frac{1}{ \left ( ad-bc \right ) \left ( dx+c \right ) }}-{\frac{b\ln \left ( dx+c \right ) }{ \left ( ad-bc \right ) ^{2}}}+{\frac{b\ln \left ( bx+a \right ) }{ \left ( ad-bc \right ) ^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b*x+a)/(d*x+c)^2,x)

[Out]

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

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Maxima [A]  time = 1.35174, size = 122, normalized size = 2.18 \[ \frac{b \log \left (b x + a\right )}{b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}} - \frac{b \log \left (d x + c\right )}{b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}} + \frac{1}{b c^{2} - a c d +{\left (b c d - a d^{2}\right )} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.216132, size = 124, normalized size = 2.21 \[ \frac{b c - a d +{\left (b d x + b c\right )} \log \left (b x + a\right ) -{\left (b d x + b c\right )} \log \left (d x + c\right )}{b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2} +{\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 4.95236, size = 233, normalized size = 4.16 \[ - \frac{b \log{\left (x + \frac{- \frac{a^{3} b d^{3}}{\left (a d - b c\right )^{2}} + \frac{3 a^{2} b^{2} c d^{2}}{\left (a d - b c\right )^{2}} - \frac{3 a b^{3} c^{2} d}{\left (a d - b c\right )^{2}} + a b d + \frac{b^{4} c^{3}}{\left (a d - b c\right )^{2}} + b^{2} c}{2 b^{2} d} \right )}}{\left (a d - b c\right )^{2}} + \frac{b \log{\left (x + \frac{\frac{a^{3} b d^{3}}{\left (a d - b c\right )^{2}} - \frac{3 a^{2} b^{2} c d^{2}}{\left (a d - b c\right )^{2}} + \frac{3 a b^{3} c^{2} d}{\left (a d - b c\right )^{2}} + a b d - \frac{b^{4} c^{3}}{\left (a d - b c\right )^{2}} + b^{2} c}{2 b^{2} d} \right )}}{\left (a d - b c\right )^{2}} - \frac{1}{a c d - b c^{2} + x \left (a d^{2} - b c d\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.264592, size = 104, normalized size = 1.86 \[ \frac{b d{\rm ln}\left ({\left | b - \frac{b c}{d x + c} + \frac{a d}{d x + c} \right |}\right )}{b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}} + \frac{d}{{\left (b c d - a d^{2}\right )}{\left (d x + c\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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