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

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 = 0.0355882, 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, Rules used = {77} \[ -\frac{c}{d (c+d x) (b c-a d)}-\frac{a \log (a+b x)}{(b c-a d)^2}+\frac{a \log (c+d x)}{(b c-a d)^2} \]

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

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

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

Mathematica [A]  time = 0.0295518, 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 = 0.007, size = 61, normalized size = 1. \begin{align*}{\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}}} \end{align*}

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 = 1.23646, size = 132, normalized size = 2.16 \begin{align*} -\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} \end{align*}

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 = 2.31251, size = 224, normalized size = 3.67 \begin{align*} -\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} \end{align*}

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 [B]  time = 1.17505, size = 238, normalized size = 3.9 \begin{align*} \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 )} \end{align*}

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 [A]  time = 1.19232, size = 115, normalized size = 1.89 \begin{align*} -\frac{\frac{a d^{2} \log \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} \end{align*}

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*log(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