3.226 \(\int \frac{1}{x (a+b x) (c+d x)^3} \, dx\)

Optimal. Leaf size=134 \[ \frac{d \left (a^2 d^2-3 a b c d+3 b^2 c^2\right ) \log (c+d x)}{c^3 (b c-a d)^3}-\frac{b^3 \log (a+b x)}{a (b c-a d)^3}-\frac{d (2 b c-a d)}{c^2 (c+d x) (b c-a d)^2}-\frac{d}{2 c (c+d x)^2 (b c-a d)}+\frac{\log (x)}{a c^3} \]

[Out]

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

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Rubi [A]  time = 0.240977, antiderivative size = 134, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056 \[ \frac{d \left (a^2 d^2-3 a b c d+3 b^2 c^2\right ) \log (c+d x)}{c^3 (b c-a d)^3}-\frac{b^3 \log (a+b x)}{a (b c-a d)^3}-\frac{d (2 b c-a d)}{c^2 (c+d x) (b c-a d)^2}-\frac{d}{2 c (c+d x)^2 (b c-a d)}+\frac{\log (x)}{a c^3} \]

Antiderivative was successfully verified.

[In]  Int[1/(x*(a + b*x)*(c + d*x)^3),x]

[Out]

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

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Rubi in Sympy [A]  time = 45.609, size = 117, normalized size = 0.87 \[ \frac{d}{2 c \left (c + d x\right )^{2} \left (a d - b c\right )} + \frac{d \left (a d - 2 b c\right )}{c^{2} \left (c + d x\right ) \left (a d - b c\right )^{2}} - \frac{d \left (a^{2} d^{2} - 3 a b c d + 3 b^{2} c^{2}\right ) \log{\left (c + d x \right )}}{c^{3} \left (a d - b c\right )^{3}} + \frac{b^{3} \log{\left (a + b x \right )}}{a \left (a d - b c\right )^{3}} + \frac{\log{\left (x \right )}}{a c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x/(b*x+a)/(d*x+c)**3,x)

[Out]

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

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Mathematica [A]  time = 0.467649, size = 116, normalized size = 0.87 \[ \frac{\frac{d \left (\frac{c (b c-a d) (b c (5 c+4 d x)-a d (3 c+2 d x))}{(c+d x)^2}-2 \left (a^2 d^2-3 a b c d+3 b^2 c^2\right ) \log (c+d x)\right )}{c^3}+\frac{2 b^3 \log (a+b x)}{a}}{2 (a d-b c)^3}+\frac{\log (x)}{a c^3} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x*(a + b*x)*(c + d*x)^3),x]

[Out]

Log[x]/(a*c^3) + ((2*b^3*Log[a + b*x])/a + (d*((c*(b*c - a*d)*(-(a*d*(3*c + 2*d*
x)) + b*c*(5*c + 4*d*x)))/(c + d*x)^2 - 2*(3*b^2*c^2 - 3*a*b*c*d + a^2*d^2)*Log[
c + d*x]))/c^3)/(2*(-(b*c) + a*d)^3)

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Maple [A]  time = 0.018, size = 184, normalized size = 1.4 \[{\frac{d}{2\,c \left ( ad-bc \right ) \left ( dx+c \right ) ^{2}}}+{\frac{{d}^{2}a}{{c}^{2} \left ( ad-bc \right ) ^{2} \left ( dx+c \right ) }}-2\,{\frac{bd}{c \left ( ad-bc \right ) ^{2} \left ( dx+c \right ) }}-{\frac{{d}^{3}\ln \left ( dx+c \right ){a}^{2}}{{c}^{3} \left ( ad-bc \right ) ^{3}}}+3\,{\frac{{d}^{2}\ln \left ( dx+c \right ) ab}{{c}^{2} \left ( ad-bc \right ) ^{3}}}-3\,{\frac{d\ln \left ( dx+c \right ){b}^{2}}{c \left ( ad-bc \right ) ^{3}}}+{\frac{\ln \left ( x \right ) }{a{c}^{3}}}+{\frac{{b}^{3}\ln \left ( bx+a \right ) }{ \left ( ad-bc \right ) ^{3}a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(b*x+a)/(d*x+c)^3,x)

[Out]

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

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Maxima [A]  time = 1.36192, size = 359, normalized size = 2.68 \[ -\frac{b^{3} \log \left (b x + a\right )}{a b^{3} c^{3} - 3 \, a^{2} b^{2} c^{2} d + 3 \, a^{3} b c d^{2} - a^{4} d^{3}} + \frac{{\left (3 \, b^{2} c^{2} d - 3 \, a b c d^{2} + a^{2} d^{3}\right )} \log \left (d x + c\right )}{b^{3} c^{6} - 3 \, a b^{2} c^{5} d + 3 \, a^{2} b c^{4} d^{2} - a^{3} c^{3} d^{3}} - \frac{5 \, b c^{2} d - 3 \, a c d^{2} + 2 \,{\left (2 \, b c d^{2} - a d^{3}\right )} x}{2 \,{\left (b^{2} c^{6} - 2 \, a b c^{5} d + a^{2} c^{4} d^{2} +{\left (b^{2} c^{4} d^{2} - 2 \, a b c^{3} d^{3} + a^{2} c^{2} d^{4}\right )} x^{2} + 2 \,{\left (b^{2} c^{5} d - 2 \, a b c^{4} d^{2} + a^{2} c^{3} d^{3}\right )} x\right )}} + \frac{\log \left (x\right )}{a c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)*(d*x + c)^3*x),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 7.67893, size = 683, normalized size = 5.1 \[ -\frac{5 \, a b^{2} c^{4} d - 8 \, a^{2} b c^{3} d^{2} + 3 \, a^{3} c^{2} d^{3} + 2 \,{\left (2 \, a b^{2} c^{3} d^{2} - 3 \, a^{2} b c^{2} d^{3} + a^{3} c d^{4}\right )} x + 2 \,{\left (b^{3} c^{3} d^{2} x^{2} + 2 \, b^{3} c^{4} d x + b^{3} c^{5}\right )} \log \left (b x + a\right ) - 2 \,{\left (3 \, a b^{2} c^{4} d - 3 \, a^{2} b c^{3} d^{2} + a^{3} c^{2} d^{3} +{\left (3 \, a b^{2} c^{2} d^{3} - 3 \, a^{2} b c d^{4} + a^{3} d^{5}\right )} x^{2} + 2 \,{\left (3 \, a b^{2} c^{3} d^{2} - 3 \, a^{2} b c^{2} d^{3} + a^{3} c d^{4}\right )} x\right )} \log \left (d x + c\right ) - 2 \,{\left (b^{3} c^{5} - 3 \, a b^{2} c^{4} d + 3 \, a^{2} b c^{3} d^{2} - a^{3} c^{2} d^{3} +{\left (b^{3} c^{3} d^{2} - 3 \, a b^{2} c^{2} d^{3} + 3 \, a^{2} b c d^{4} - a^{3} d^{5}\right )} x^{2} + 2 \,{\left (b^{3} c^{4} d - 3 \, a b^{2} c^{3} d^{2} + 3 \, a^{2} b c^{2} d^{3} - a^{3} c d^{4}\right )} x\right )} \log \left (x\right )}{2 \,{\left (a b^{3} c^{8} - 3 \, a^{2} b^{2} c^{7} d + 3 \, a^{3} b c^{6} d^{2} - a^{4} c^{5} d^{3} +{\left (a b^{3} c^{6} d^{2} - 3 \, a^{2} b^{2} c^{5} d^{3} + 3 \, a^{3} b c^{4} d^{4} - a^{4} c^{3} d^{5}\right )} x^{2} + 2 \,{\left (a b^{3} c^{7} d - 3 \, a^{2} b^{2} c^{6} d^{2} + 3 \, a^{3} b c^{5} d^{3} - a^{4} c^{4} d^{4}\right )} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)*(d*x + c)^3*x),x, algorithm="fricas")

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x/(b*x+a)/(d*x+c)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.278585, size = 316, normalized size = 2.36 \[ -\frac{b^{4}{\rm ln}\left ({\left | b x + a \right |}\right )}{a b^{4} c^{3} - 3 \, a^{2} b^{3} c^{2} d + 3 \, a^{3} b^{2} c d^{2} - a^{4} b d^{3}} + \frac{{\left (3 \, b^{2} c^{2} d^{2} - 3 \, a b c d^{3} + a^{2} d^{4}\right )}{\rm ln}\left ({\left | d x + c \right |}\right )}{b^{3} c^{6} d - 3 \, a b^{2} c^{5} d^{2} + 3 \, a^{2} b c^{4} d^{3} - a^{3} c^{3} d^{4}} + \frac{{\rm ln}\left ({\left | x \right |}\right )}{a c^{3}} - \frac{5 \, b^{2} c^{4} d - 8 \, a b c^{3} d^{2} + 3 \, a^{2} c^{2} d^{3} + 2 \,{\left (2 \, b^{2} c^{3} d^{2} - 3 \, a b c^{2} d^{3} + a^{2} c d^{4}\right )} x}{2 \,{\left (b c - a d\right )}^{3}{\left (d x + c\right )}^{2} c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)*(d*x + c)^3*x),x, algorithm="giac")

[Out]

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