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

Optimal. Leaf size=61 \[ -\frac{c}{d (c+d x) (b c-a d)}-\frac{a \log (a+b x)}{(b c-a d)^2}+\frac{a \log (c+d x)}{(b c-a d)^2} \]

[Out]

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

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Rubi [A]  time = 0.0872763, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062 \[ -\frac{c}{d (c+d x) (b c-a d)}-\frac{a \log (a+b x)}{(b c-a d)^2}+\frac{a \log (c+d x)}{(b c-a d)^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 25.2652, size = 48, normalized size = 0.79 \[ - \frac{a \log{\left (a + b x \right )}}{\left (a d - b c\right )^{2}} + \frac{a \log{\left (c + d x \right )}}{\left (a d - b c\right )^{2}} + \frac{c}{d \left (c + d x\right ) \left (a d - b c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0515013, size = 60, normalized size = 0.98 \[ \frac{c}{d (c+d x) (a d-b c)}-\frac{a \log (a+b x)}{(b c-a d)^2}+\frac{a \log (c+d x)}{(b c-a d)^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.013, size = 61, normalized size = 1. \[{\frac{a\ln \left ( dx+c \right ) }{ \left ( ad-bc \right ) ^{2}}}+{\frac{c}{d \left ( ad-bc \right ) \left ( dx+c \right ) }}-{\frac{a\ln \left ( bx+a \right ) }{ \left ( ad-bc \right ) ^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(b*x+a)/(d*x+c)^2,x)

[Out]

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

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Maxima [A]  time = 1.34932, size = 132, normalized size = 2.16 \[ -\frac{a \log \left (b x + a\right )}{b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}} + \frac{a \log \left (d x + c\right )}{b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}} - \frac{c}{b c^{2} d - a c d^{2} +{\left (b c d^{2} - a d^{3}\right )} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.224019, size = 144, normalized size = 2.36 \[ -\frac{b c^{2} - a c d +{\left (a d^{2} x + a c d\right )} \log \left (b x + a\right ) -{\left (a d^{2} x + a c d\right )} \log \left (d x + c\right )}{b^{2} c^{3} d - 2 \, a b c^{2} d^{2} + a^{2} c d^{3} +{\left (b^{2} c^{2} d^{2} - 2 \, a b c d^{3} + a^{2} d^{4}\right )} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 5.06708, size = 238, normalized size = 3.9 \[ \frac{a \log{\left (x + \frac{- \frac{a^{4} d^{3}}{\left (a d - b c\right )^{2}} + \frac{3 a^{3} b c d^{2}}{\left (a d - b c\right )^{2}} - \frac{3 a^{2} b^{2} c^{2} d}{\left (a d - b c\right )^{2}} + a^{2} d + \frac{a b^{3} c^{3}}{\left (a d - b c\right )^{2}} + a b c}{2 a b d} \right )}}{\left (a d - b c\right )^{2}} - \frac{a \log{\left (x + \frac{\frac{a^{4} d^{3}}{\left (a d - b c\right )^{2}} - \frac{3 a^{3} b c d^{2}}{\left (a d - b c\right )^{2}} + \frac{3 a^{2} b^{2} c^{2} d}{\left (a d - b c\right )^{2}} + a^{2} d - \frac{a b^{3} c^{3}}{\left (a d - b c\right )^{2}} + a b c}{2 a b d} \right )}}{\left (a d - b c\right )^{2}} + \frac{c}{a c d^{2} - b c^{2} d + x \left (a d^{3} - b c d^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.30503, size = 115, normalized size = 1.89 \[ -\frac{\frac{a d^{2}{\rm ln}\left ({\left | b - \frac{b c}{d x + c} + \frac{a d}{d x + c} \right |}\right )}{b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}} + \frac{c d}{{\left (b c d - a d^{2}\right )}{\left (d x + c\right )}}}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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