3.296 \(\int \frac{x}{(a+b x)^2 (c+d x)^3} \, dx\)

Optimal. Leaf size=121 \[ \frac{a b}{(a+b x) (b c-a d)^3}+\frac{a d+b c}{(c+d x) (b c-a d)^3}+\frac{c}{2 (c+d x)^2 (b c-a d)^2}+\frac{b (2 a d+b c) \log (a+b x)}{(b c-a d)^4}-\frac{b (2 a d+b c) \log (c+d x)}{(b c-a d)^4} \]

[Out]

(a*b)/((b*c - a*d)^3*(a + b*x)) + c/(2*(b*c - a*d)^2*(c + d*x)^2) + (b*c + a*d)/((b*c - a*d)^3*(c + d*x)) + (b
*(b*c + 2*a*d)*Log[a + b*x])/(b*c - a*d)^4 - (b*(b*c + 2*a*d)*Log[c + d*x])/(b*c - a*d)^4

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Rubi [A]  time = 0.101683, antiderivative size = 121, 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} \[ \frac{a b}{(a+b x) (b c-a d)^3}+\frac{a d+b c}{(c+d x) (b c-a d)^3}+\frac{c}{2 (c+d x)^2 (b c-a d)^2}+\frac{b (2 a d+b c) \log (a+b x)}{(b c-a d)^4}-\frac{b (2 a d+b c) \log (c+d x)}{(b c-a d)^4} \]

Antiderivative was successfully verified.

[In]

Int[x/((a + b*x)^2*(c + d*x)^3),x]

[Out]

(a*b)/((b*c - a*d)^3*(a + b*x)) + c/(2*(b*c - a*d)^2*(c + d*x)^2) + (b*c + a*d)/((b*c - a*d)^3*(c + d*x)) + (b
*(b*c + 2*a*d)*Log[a + b*x])/(b*c - a*d)^4 - (b*(b*c + 2*a*d)*Log[c + d*x])/(b*c - a*d)^4

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)^2 (c+d x)^3} \, dx &=\int \left (-\frac{a b^2}{(b c-a d)^3 (a+b x)^2}+\frac{b^2 (b c+2 a d)}{(b c-a d)^4 (a+b x)}-\frac{c d}{(b c-a d)^2 (c+d x)^3}-\frac{d (b c+a d)}{(b c-a d)^3 (c+d x)^2}-\frac{b d (b c+2 a d)}{(b c-a d)^4 (c+d x)}\right ) \, dx\\ &=\frac{a b}{(b c-a d)^3 (a+b x)}+\frac{c}{2 (b c-a d)^2 (c+d x)^2}+\frac{b c+a d}{(b c-a d)^3 (c+d x)}+\frac{b (b c+2 a d) \log (a+b x)}{(b c-a d)^4}-\frac{b (b c+2 a d) \log (c+d x)}{(b c-a d)^4}\\ \end{align*}

Mathematica [A]  time = 0.103097, size = 111, normalized size = 0.92 \[ \frac{\frac{c (b c-a d)^2}{(c+d x)^2}+\frac{2 a b (b c-a d)}{a+b x}+\frac{2 (a d+b c) (b c-a d)}{c+d x}+2 b (2 a d+b c) \log (a+b x)-2 b (2 a d+b c) \log (c+d x)}{2 (b c-a d)^4} \]

Antiderivative was successfully verified.

[In]

Integrate[x/((a + b*x)^2*(c + d*x)^3),x]

[Out]

((2*a*b*(b*c - a*d))/(a + b*x) + (c*(b*c - a*d)^2)/(c + d*x)^2 + (2*(b*c - a*d)*(b*c + a*d))/(c + d*x) + 2*b*(
b*c + 2*a*d)*Log[a + b*x] - 2*b*(b*c + 2*a*d)*Log[c + d*x])/(2*(b*c - a*d)^4)

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Maple [A]  time = 0.011, size = 170, normalized size = 1.4 \begin{align*} -{\frac{ad}{ \left ( ad-bc \right ) ^{3} \left ( dx+c \right ) }}-{\frac{bc}{ \left ( ad-bc \right ) ^{3} \left ( dx+c \right ) }}+{\frac{c}{2\, \left ( ad-bc \right ) ^{2} \left ( dx+c \right ) ^{2}}}-2\,{\frac{b\ln \left ( dx+c \right ) ad}{ \left ( ad-bc \right ) ^{4}}}-{\frac{{b}^{2}\ln \left ( dx+c \right ) c}{ \left ( ad-bc \right ) ^{4}}}-{\frac{ab}{ \left ( ad-bc \right ) ^{3} \left ( bx+a \right ) }}+2\,{\frac{b\ln \left ( bx+a \right ) ad}{ \left ( ad-bc \right ) ^{4}}}+{\frac{{b}^{2}\ln \left ( bx+a \right ) c}{ \left ( ad-bc \right ) ^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(b*x+a)^2/(d*x+c)^3,x)

[Out]

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

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Maxima [B]  time = 1.23528, size = 548, normalized size = 4.53 \begin{align*} \frac{{\left (b^{2} c + 2 \, a b d\right )} \log \left (b x + a\right )}{b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}} - \frac{{\left (b^{2} c + 2 \, a b d\right )} \log \left (d x + c\right )}{b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}} + \frac{5 \, a b c^{2} + a^{2} c d + 2 \,{\left (b^{2} c d + 2 \, a b d^{2}\right )} x^{2} +{\left (3 \, b^{2} c^{2} + 7 \, a b c d + 2 \, a^{2} d^{2}\right )} x}{2 \,{\left (a b^{3} c^{5} - 3 \, a^{2} b^{2} c^{4} d + 3 \, a^{3} b c^{3} d^{2} - a^{4} c^{2} d^{3} +{\left (b^{4} c^{3} d^{2} - 3 \, a b^{3} c^{2} d^{3} + 3 \, a^{2} b^{2} c d^{4} - a^{3} b d^{5}\right )} x^{3} +{\left (2 \, b^{4} c^{4} d - 5 \, a b^{3} c^{3} d^{2} + 3 \, a^{2} b^{2} c^{2} d^{3} + a^{3} b c d^{4} - a^{4} d^{5}\right )} x^{2} +{\left (b^{4} c^{5} - a b^{3} c^{4} d - 3 \, a^{2} b^{2} c^{3} d^{2} + 5 \, a^{3} b c^{2} d^{3} - 2 \, a^{4} c d^{4}\right )} x\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(b*x+a)^2/(d*x+c)^3,x, algorithm="maxima")

[Out]

(b^2*c + 2*a*b*d)*log(b*x + a)/(b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4) - (b^2*
c + 2*a*b*d)*log(d*x + c)/(b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4) + 1/2*(5*a*b
*c^2 + a^2*c*d + 2*(b^2*c*d + 2*a*b*d^2)*x^2 + (3*b^2*c^2 + 7*a*b*c*d + 2*a^2*d^2)*x)/(a*b^3*c^5 - 3*a^2*b^2*c
^4*d + 3*a^3*b*c^3*d^2 - a^4*c^2*d^3 + (b^4*c^3*d^2 - 3*a*b^3*c^2*d^3 + 3*a^2*b^2*c*d^4 - a^3*b*d^5)*x^3 + (2*
b^4*c^4*d - 5*a*b^3*c^3*d^2 + 3*a^2*b^2*c^2*d^3 + a^3*b*c*d^4 - a^4*d^5)*x^2 + (b^4*c^5 - a*b^3*c^4*d - 3*a^2*
b^2*c^3*d^2 + 5*a^3*b*c^2*d^3 - 2*a^4*c*d^4)*x)

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Fricas [B]  time = 3.00991, size = 1184, normalized size = 9.79 \begin{align*} \frac{5 \, a b^{2} c^{3} - 4 \, a^{2} b c^{2} d - a^{3} c d^{2} + 2 \,{\left (b^{3} c^{2} d + a b^{2} c d^{2} - 2 \, a^{2} b d^{3}\right )} x^{2} +{\left (3 \, b^{3} c^{3} + 4 \, a b^{2} c^{2} d - 5 \, a^{2} b c d^{2} - 2 \, a^{3} d^{3}\right )} x + 2 \,{\left (a b^{2} c^{3} + 2 \, a^{2} b c^{2} d +{\left (b^{3} c d^{2} + 2 \, a b^{2} d^{3}\right )} x^{3} +{\left (2 \, b^{3} c^{2} d + 5 \, a b^{2} c d^{2} + 2 \, a^{2} b d^{3}\right )} x^{2} +{\left (b^{3} c^{3} + 4 \, a b^{2} c^{2} d + 4 \, a^{2} b c d^{2}\right )} x\right )} \log \left (b x + a\right ) - 2 \,{\left (a b^{2} c^{3} + 2 \, a^{2} b c^{2} d +{\left (b^{3} c d^{2} + 2 \, a b^{2} d^{3}\right )} x^{3} +{\left (2 \, b^{3} c^{2} d + 5 \, a b^{2} c d^{2} + 2 \, a^{2} b d^{3}\right )} x^{2} +{\left (b^{3} c^{3} + 4 \, a b^{2} c^{2} d + 4 \, a^{2} b c d^{2}\right )} x\right )} \log \left (d x + c\right )}{2 \,{\left (a b^{4} c^{6} - 4 \, a^{2} b^{3} c^{5} d + 6 \, a^{3} b^{2} c^{4} d^{2} - 4 \, a^{4} b c^{3} d^{3} + a^{5} c^{2} d^{4} +{\left (b^{5} c^{4} d^{2} - 4 \, a b^{4} c^{3} d^{3} + 6 \, a^{2} b^{3} c^{2} d^{4} - 4 \, a^{3} b^{2} c d^{5} + a^{4} b d^{6}\right )} x^{3} +{\left (2 \, b^{5} c^{5} d - 7 \, a b^{4} c^{4} d^{2} + 8 \, a^{2} b^{3} c^{3} d^{3} - 2 \, a^{3} b^{2} c^{2} d^{4} - 2 \, a^{4} b c d^{5} + a^{5} d^{6}\right )} x^{2} +{\left (b^{5} c^{6} - 2 \, a b^{4} c^{5} d - 2 \, a^{2} b^{3} c^{4} d^{2} + 8 \, a^{3} b^{2} c^{3} d^{3} - 7 \, a^{4} b c^{2} d^{4} + 2 \, a^{5} c d^{5}\right )} x\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(b*x+a)^2/(d*x+c)^3,x, algorithm="fricas")

[Out]

1/2*(5*a*b^2*c^3 - 4*a^2*b*c^2*d - a^3*c*d^2 + 2*(b^3*c^2*d + a*b^2*c*d^2 - 2*a^2*b*d^3)*x^2 + (3*b^3*c^3 + 4*
a*b^2*c^2*d - 5*a^2*b*c*d^2 - 2*a^3*d^3)*x + 2*(a*b^2*c^3 + 2*a^2*b*c^2*d + (b^3*c*d^2 + 2*a*b^2*d^3)*x^3 + (2
*b^3*c^2*d + 5*a*b^2*c*d^2 + 2*a^2*b*d^3)*x^2 + (b^3*c^3 + 4*a*b^2*c^2*d + 4*a^2*b*c*d^2)*x)*log(b*x + a) - 2*
(a*b^2*c^3 + 2*a^2*b*c^2*d + (b^3*c*d^2 + 2*a*b^2*d^3)*x^3 + (2*b^3*c^2*d + 5*a*b^2*c*d^2 + 2*a^2*b*d^3)*x^2 +
(b^3*c^3 + 4*a*b^2*c^2*d + 4*a^2*b*c*d^2)*x)*log(d*x + c))/(a*b^4*c^6 - 4*a^2*b^3*c^5*d + 6*a^3*b^2*c^4*d^2 -
4*a^4*b*c^3*d^3 + a^5*c^2*d^4 + (b^5*c^4*d^2 - 4*a*b^4*c^3*d^3 + 6*a^2*b^3*c^2*d^4 - 4*a^3*b^2*c*d^5 + a^4*b*
d^6)*x^3 + (2*b^5*c^5*d - 7*a*b^4*c^4*d^2 + 8*a^2*b^3*c^3*d^3 - 2*a^3*b^2*c^2*d^4 - 2*a^4*b*c*d^5 + a^5*d^6)*x
^2 + (b^5*c^6 - 2*a*b^4*c^5*d - 2*a^2*b^3*c^4*d^2 + 8*a^3*b^2*c^3*d^3 - 7*a^4*b*c^2*d^4 + 2*a^5*c*d^5)*x)

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Sympy [B]  time = 2.77603, size = 770, normalized size = 6.36 \begin{align*} - \frac{b \left (2 a d + b c\right ) \log{\left (x + \frac{- \frac{a^{5} b d^{5} \left (2 a d + b c\right )}{\left (a d - b c\right )^{4}} + \frac{5 a^{4} b^{2} c d^{4} \left (2 a d + b c\right )}{\left (a d - b c\right )^{4}} - \frac{10 a^{3} b^{3} c^{2} d^{3} \left (2 a d + b c\right )}{\left (a d - b c\right )^{4}} + \frac{10 a^{2} b^{4} c^{3} d^{2} \left (2 a d + b c\right )}{\left (a d - b c\right )^{4}} + 2 a^{2} b d^{2} - \frac{5 a b^{5} c^{4} d \left (2 a d + b c\right )}{\left (a d - b c\right )^{4}} + 3 a b^{2} c d + \frac{b^{6} c^{5} \left (2 a d + b c\right )}{\left (a d - b c\right )^{4}} + b^{3} c^{2}}{4 a b^{2} d^{2} + 2 b^{3} c d} \right )}}{\left (a d - b c\right )^{4}} + \frac{b \left (2 a d + b c\right ) \log{\left (x + \frac{\frac{a^{5} b d^{5} \left (2 a d + b c\right )}{\left (a d - b c\right )^{4}} - \frac{5 a^{4} b^{2} c d^{4} \left (2 a d + b c\right )}{\left (a d - b c\right )^{4}} + \frac{10 a^{3} b^{3} c^{2} d^{3} \left (2 a d + b c\right )}{\left (a d - b c\right )^{4}} - \frac{10 a^{2} b^{4} c^{3} d^{2} \left (2 a d + b c\right )}{\left (a d - b c\right )^{4}} + 2 a^{2} b d^{2} + \frac{5 a b^{5} c^{4} d \left (2 a d + b c\right )}{\left (a d - b c\right )^{4}} + 3 a b^{2} c d - \frac{b^{6} c^{5} \left (2 a d + b c\right )}{\left (a d - b c\right )^{4}} + b^{3} c^{2}}{4 a b^{2} d^{2} + 2 b^{3} c d} \right )}}{\left (a d - b c\right )^{4}} - \frac{a^{2} c d + 5 a b c^{2} + x^{2} \left (4 a b d^{2} + 2 b^{2} c d\right ) + x \left (2 a^{2} d^{2} + 7 a b c d + 3 b^{2} c^{2}\right )}{2 a^{4} c^{2} d^{3} - 6 a^{3} b c^{3} d^{2} + 6 a^{2} b^{2} c^{4} d - 2 a b^{3} c^{5} + x^{3} \left (2 a^{3} b d^{5} - 6 a^{2} b^{2} c d^{4} + 6 a b^{3} c^{2} d^{3} - 2 b^{4} c^{3} d^{2}\right ) + x^{2} \left (2 a^{4} d^{5} - 2 a^{3} b c d^{4} - 6 a^{2} b^{2} c^{2} d^{3} + 10 a b^{3} c^{3} d^{2} - 4 b^{4} c^{4} d\right ) + x \left (4 a^{4} c d^{4} - 10 a^{3} b c^{2} d^{3} + 6 a^{2} b^{2} c^{3} d^{2} + 2 a b^{3} c^{4} d - 2 b^{4} c^{5}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(b*x+a)**2/(d*x+c)**3,x)

[Out]

-b*(2*a*d + b*c)*log(x + (-a**5*b*d**5*(2*a*d + b*c)/(a*d - b*c)**4 + 5*a**4*b**2*c*d**4*(2*a*d + b*c)/(a*d -
b*c)**4 - 10*a**3*b**3*c**2*d**3*(2*a*d + b*c)/(a*d - b*c)**4 + 10*a**2*b**4*c**3*d**2*(2*a*d + b*c)/(a*d - b*
c)**4 + 2*a**2*b*d**2 - 5*a*b**5*c**4*d*(2*a*d + b*c)/(a*d - b*c)**4 + 3*a*b**2*c*d + b**6*c**5*(2*a*d + b*c)/
(a*d - b*c)**4 + b**3*c**2)/(4*a*b**2*d**2 + 2*b**3*c*d))/(a*d - b*c)**4 + b*(2*a*d + b*c)*log(x + (a**5*b*d**
5*(2*a*d + b*c)/(a*d - b*c)**4 - 5*a**4*b**2*c*d**4*(2*a*d + b*c)/(a*d - b*c)**4 + 10*a**3*b**3*c**2*d**3*(2*a
*d + b*c)/(a*d - b*c)**4 - 10*a**2*b**4*c**3*d**2*(2*a*d + b*c)/(a*d - b*c)**4 + 2*a**2*b*d**2 + 5*a*b**5*c**4
*d*(2*a*d + b*c)/(a*d - b*c)**4 + 3*a*b**2*c*d - b**6*c**5*(2*a*d + b*c)/(a*d - b*c)**4 + b**3*c**2)/(4*a*b**2
*d**2 + 2*b**3*c*d))/(a*d - b*c)**4 - (a**2*c*d + 5*a*b*c**2 + x**2*(4*a*b*d**2 + 2*b**2*c*d) + x*(2*a**2*d**2
+ 7*a*b*c*d + 3*b**2*c**2))/(2*a**4*c**2*d**3 - 6*a**3*b*c**3*d**2 + 6*a**2*b**2*c**4*d - 2*a*b**3*c**5 + x**
3*(2*a**3*b*d**5 - 6*a**2*b**2*c*d**4 + 6*a*b**3*c**2*d**3 - 2*b**4*c**3*d**2) + x**2*(2*a**4*d**5 - 2*a**3*b*
c*d**4 - 6*a**2*b**2*c**2*d**3 + 10*a*b**3*c**3*d**2 - 4*b**4*c**4*d) + x*(4*a**4*c*d**4 - 10*a**3*b*c**2*d**3
+ 6*a**2*b**2*c**3*d**2 + 2*a*b**3*c**4*d - 2*b**4*c**5))

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Giac [B]  time = 1.19555, size = 344, normalized size = 2.84 \begin{align*} \frac{\frac{2 \, a b^{5}}{{\left (b^{6} c^{3} - 3 \, a b^{5} c^{2} d + 3 \, a^{2} b^{4} c d^{2} - a^{3} b^{3} d^{3}\right )}{\left (b x + a\right )}} - \frac{2 \,{\left (b^{4} c + 2 \, a b^{3} d\right )} \log \left ({\left | \frac{b c}{b x + a} - \frac{a d}{b x + a} + d \right |}\right )}{b^{5} c^{4} - 4 \, a b^{4} c^{3} d + 6 \, a^{2} b^{3} c^{2} d^{2} - 4 \, a^{3} b^{2} c d^{3} + a^{4} b d^{4}} - \frac{3 \, b^{3} c d^{2} + 2 \, a b^{2} d^{3} + \frac{2 \,{\left (2 \, b^{5} c^{2} d - a b^{4} c d^{2} - a^{2} b^{3} d^{3}\right )}}{{\left (b x + a\right )} b}}{{\left (b c - a d\right )}^{4}{\left (\frac{b c}{b x + a} - \frac{a d}{b x + a} + d\right )}^{2}}}{2 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(b*x+a)^2/(d*x+c)^3,x, algorithm="giac")

[Out]

1/2*(2*a*b^5/((b^6*c^3 - 3*a*b^5*c^2*d + 3*a^2*b^4*c*d^2 - a^3*b^3*d^3)*(b*x + a)) - 2*(b^4*c + 2*a*b^3*d)*log
(abs(b*c/(b*x + a) - a*d/(b*x + a) + d))/(b^5*c^4 - 4*a*b^4*c^3*d + 6*a^2*b^3*c^2*d^2 - 4*a^3*b^2*c*d^3 + a^4*
b*d^4) - (3*b^3*c*d^2 + 2*a*b^2*d^3 + 2*(2*b^5*c^2*d - a*b^4*c*d^2 - a^2*b^3*d^3)/((b*x + a)*b))/((b*c - a*d)^
4*(b*c/(b*x + a) - a*d/(b*x + a) + d)^2))/b