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

Optimal. Leaf size=85 \[ -\frac {a}{(c+d x) (b c-a d)^2}-\frac {c}{2 d (c+d x)^2 (b c-a d)}-\frac {a b \log (a+b x)}{(b c-a d)^3}+\frac {a b \log (c+d x)}{(b c-a d)^3} \]

[Out]

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

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Rubi [A]  time = 0.05, antiderivative size = 85, normalized size of antiderivative = 1.00, 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} \[ -\frac {a}{(c+d x) (b c-a d)^2}-\frac {c}{2 d (c+d x)^2 (b c-a d)}-\frac {a b \log (a+b x)}{(b c-a d)^3}+\frac {a b \log (c+d x)}{(b c-a d)^3} \]

Antiderivative was successfully verified.

[In]

Int[x/((a + b*x)*(c + d*x)^3),x]

[Out]

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

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

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Mathematica [A]  time = 0.06, size = 85, normalized size = 1.00 \[ -\frac {a}{(c+d x) (b c-a d)^2}+\frac {c}{2 d (c+d x)^2 (a d-b c)}-\frac {a b \log (a+b x)}{(b c-a d)^3}+\frac {a b \log (c+d x)}{(b c-a d)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 0.60, size = 244, normalized size = 2.87 \[ -\frac {b^{2} c^{3} - a^{2} c d^{2} + 2 \, {\left (a b c d^{2} - a^{2} d^{3}\right )} x + 2 \, {\left (a b d^{3} x^{2} + 2 \, a b c d^{2} x + a b c^{2} d\right )} \log \left (b x + a\right ) - 2 \, {\left (a b d^{3} x^{2} + 2 \, a b c d^{2} x + a b c^{2} d\right )} \log \left (d x + c\right )}{2 \, {\left (b^{3} c^{5} d - 3 \, a b^{2} c^{4} d^{2} + 3 \, a^{2} b c^{3} d^{3} - a^{3} c^{2} d^{4} + {\left (b^{3} c^{3} d^{3} - 3 \, a b^{2} c^{2} d^{4} + 3 \, a^{2} b c d^{5} - a^{3} d^{6}\right )} x^{2} + 2 \, {\left (b^{3} c^{4} d^{2} - 3 \, a b^{2} c^{3} d^{3} + 3 \, a^{2} b c^{2} d^{4} - a^{3} c d^{5}\right )} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 1.07, size = 165, normalized size = 1.94 \[ -\frac {a b^{2} \log \left ({\left | b x + a \right |}\right )}{b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}} + \frac {a b d \log \left ({\left | d x + c \right |}\right )}{b^{3} c^{3} d - 3 \, a b^{2} c^{2} d^{2} + 3 \, a^{2} b c d^{3} - a^{3} d^{4}} - \frac {b^{2} c^{3} - a^{2} c d^{2} + 2 \, {\left (a b c d^{2} - a^{2} d^{3}\right )} x}{2 \, {\left (b c - a d\right )}^{3} {\left (d x + c\right )}^{2} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.01, size = 84, normalized size = 0.99 \[ \frac {a b \ln \left (b x +a \right )}{\left (a d -b c \right )^{3}}-\frac {a b \ln \left (d x +c \right )}{\left (a d -b c \right )^{3}}-\frac {a}{\left (a d -b c \right )^{2} \left (d x +c \right )}+\frac {c}{2 \left (a d -b c \right ) \left (d x +c \right )^{2} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 1.11, size = 208, normalized size = 2.45 \[ -\frac {a b \log \left (b x + a\right )}{b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}} + \frac {a b \log \left (d x + c\right )}{b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}} - \frac {2 \, a d^{2} x + b c^{2} + a c d}{2 \, {\left (b^{2} c^{4} d - 2 \, a b c^{3} d^{2} + a^{2} c^{2} d^{3} + {\left (b^{2} c^{2} d^{3} - 2 \, a b c d^{4} + a^{2} d^{5}\right )} x^{2} + 2 \, {\left (b^{2} c^{3} d^{2} - 2 \, a b c^{2} d^{3} + a^{2} c d^{4}\right )} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.16, size = 185, normalized size = 2.18 \[ \frac {2\,a\,b\,\mathrm {atanh}\left (\frac {a^3\,d^3-a^2\,b\,c\,d^2-a\,b^2\,c^2\,d+b^3\,c^3}{{\left (a\,d-b\,c\right )}^3}+\frac {2\,b\,d\,x\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}{{\left (a\,d-b\,c\right )}^3}\right )}{{\left (a\,d-b\,c\right )}^3}-\frac {\frac {b\,c^2+a\,d\,c}{2\,d\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}+\frac {a\,d\,x}{a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2}}{c^2+2\,c\,d\,x+d^2\,x^2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/((a + b*x)*(c + d*x)^3),x)

[Out]

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

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sympy [B]  time = 1.88, size = 401, normalized size = 4.72 \[ - \frac {a b \log {\left (x + \frac {- \frac {a^{5} b d^{4}}{\left (a d - b c\right )^{3}} + \frac {4 a^{4} b^{2} c d^{3}}{\left (a d - b c\right )^{3}} - \frac {6 a^{3} b^{3} c^{2} d^{2}}{\left (a d - b c\right )^{3}} + \frac {4 a^{2} b^{4} c^{3} d}{\left (a d - b c\right )^{3}} + a^{2} b d - \frac {a b^{5} c^{4}}{\left (a d - b c\right )^{3}} + a b^{2} c}{2 a b^{2} d} \right )}}{\left (a d - b c\right )^{3}} + \frac {a b \log {\left (x + \frac {\frac {a^{5} b d^{4}}{\left (a d - b c\right )^{3}} - \frac {4 a^{4} b^{2} c d^{3}}{\left (a d - b c\right )^{3}} + \frac {6 a^{3} b^{3} c^{2} d^{2}}{\left (a d - b c\right )^{3}} - \frac {4 a^{2} b^{4} c^{3} d}{\left (a d - b c\right )^{3}} + a^{2} b d + \frac {a b^{5} c^{4}}{\left (a d - b c\right )^{3}} + a b^{2} c}{2 a b^{2} d} \right )}}{\left (a d - b c\right )^{3}} + \frac {- a c d - 2 a d^{2} x - b c^{2}}{2 a^{2} c^{2} d^{3} - 4 a b c^{3} d^{2} + 2 b^{2} c^{4} d + x^{2} \left (2 a^{2} d^{5} - 4 a b c d^{4} + 2 b^{2} c^{2} d^{3}\right ) + x \left (4 a^{2} c d^{4} - 8 a b c^{2} d^{3} + 4 b^{2} c^{3} d^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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