3.268 \(\int \frac{1}{x^2 (a+b x)^2 (c+d x)^3} \, dx\)

Optimal. Leaf size=195 \[ \frac{b^4 (2 b c-5 a d) \log (a+b x)}{a^3 (b c-a d)^4}-\frac{\log (x) (3 a d+2 b c)}{a^3 c^4}-\frac{b^4}{a^2 (a+b x) (b c-a d)^3}+\frac{d^3 \left (3 a^2 d^2-10 a b c d+10 b^2 c^2\right ) \log (c+d x)}{c^4 (b c-a d)^4}-\frac{1}{a^2 c^3 x}-\frac{2 d^3 (2 b c-a d)}{c^3 (c+d x) (b c-a d)^3}-\frac{d^3}{2 c^2 (c+d x)^2 (b c-a d)^2} \]

[Out]

-(1/(a^2*c^3*x)) - b^4/(a^2*(b*c - a*d)^3*(a + b*x)) - d^3/(2*c^2*(b*c - a*d)^2*
(c + d*x)^2) - (2*d^3*(2*b*c - a*d))/(c^3*(b*c - a*d)^3*(c + d*x)) - ((2*b*c + 3
*a*d)*Log[x])/(a^3*c^4) + (b^4*(2*b*c - 5*a*d)*Log[a + b*x])/(a^3*(b*c - a*d)^4)
+ (d^3*(10*b^2*c^2 - 10*a*b*c*d + 3*a^2*d^2)*Log[c + d*x])/(c^4*(b*c - a*d)^4)

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Rubi [A]  time = 0.476372, antiderivative size = 195, 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 \[ \frac{b^4 (2 b c-5 a d) \log (a+b x)}{a^3 (b c-a d)^4}-\frac{\log (x) (3 a d+2 b c)}{a^3 c^4}-\frac{b^4}{a^2 (a+b x) (b c-a d)^3}+\frac{d^3 \left (3 a^2 d^2-10 a b c d+10 b^2 c^2\right ) \log (c+d x)}{c^4 (b c-a d)^4}-\frac{1}{a^2 c^3 x}-\frac{2 d^3 (2 b c-a d)}{c^3 (c+d x) (b c-a d)^3}-\frac{d^3}{2 c^2 (c+d x)^2 (b c-a d)^2} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^2*(a + b*x)^2*(c + d*x)^3),x]

[Out]

-(1/(a^2*c^3*x)) - b^4/(a^2*(b*c - a*d)^3*(a + b*x)) - d^3/(2*c^2*(b*c - a*d)^2*
(c + d*x)^2) - (2*d^3*(2*b*c - a*d))/(c^3*(b*c - a*d)^3*(c + d*x)) - ((2*b*c + 3
*a*d)*Log[x])/(a^3*c^4) + (b^4*(2*b*c - 5*a*d)*Log[a + b*x])/(a^3*(b*c - a*d)^4)
+ (d^3*(10*b^2*c^2 - 10*a*b*c*d + 3*a^2*d^2)*Log[c + d*x])/(c^4*(b*c - a*d)^4)

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Rubi in Sympy [A]  time = 107.153, size = 184, normalized size = 0.94 \[ - \frac{d^{3}}{2 c^{2} \left (c + d x\right )^{2} \left (a d - b c\right )^{2}} - \frac{2 d^{3} \left (a d - 2 b c\right )}{c^{3} \left (c + d x\right ) \left (a d - b c\right )^{3}} + \frac{d^{3} \left (3 a^{2} d^{2} - 10 a b c d + 10 b^{2} c^{2}\right ) \log{\left (c + d x \right )}}{c^{4} \left (a d - b c\right )^{4}} + \frac{b^{4}}{a^{2} \left (a + b x\right ) \left (a d - b c\right )^{3}} - \frac{1}{a^{2} c^{3} x} - \frac{b^{4} \left (5 a d - 2 b c\right ) \log{\left (a + b x \right )}}{a^{3} \left (a d - b c\right )^{4}} - \frac{\left (3 a d + 2 b c\right ) \log{\left (x \right )}}{a^{3} c^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**2/(b*x+a)**2/(d*x+c)**3,x)

[Out]

-d**3/(2*c**2*(c + d*x)**2*(a*d - b*c)**2) - 2*d**3*(a*d - 2*b*c)/(c**3*(c + d*x
)*(a*d - b*c)**3) + d**3*(3*a**2*d**2 - 10*a*b*c*d + 10*b**2*c**2)*log(c + d*x)/
(c**4*(a*d - b*c)**4) + b**4/(a**2*(a + b*x)*(a*d - b*c)**3) - 1/(a**2*c**3*x) -
b**4*(5*a*d - 2*b*c)*log(a + b*x)/(a**3*(a*d - b*c)**4) - (3*a*d + 2*b*c)*log(x
)/(a**3*c**4)

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Mathematica [A]  time = 0.469531, size = 193, normalized size = 0.99 \[ \frac{b^4 (2 b c-5 a d) \log (a+b x)}{a^3 (b c-a d)^4}-\frac{\log (x) (3 a d+2 b c)}{a^3 c^4}+\frac{b^4}{a^2 (a+b x) (a d-b c)^3}+\frac{d^3 \left (3 a^2 d^2-10 a b c d+10 b^2 c^2\right ) \log (c+d x)}{c^4 (b c-a d)^4}-\frac{1}{a^2 c^3 x}+\frac{2 d^3 (a d-2 b c)}{c^3 (c+d x) (b c-a d)^3}-\frac{d^3}{2 c^2 (c+d x)^2 (b c-a d)^2} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^2*(a + b*x)^2*(c + d*x)^3),x]

[Out]

-(1/(a^2*c^3*x)) + b^4/(a^2*(-(b*c) + a*d)^3*(a + b*x)) - d^3/(2*c^2*(b*c - a*d)
^2*(c + d*x)^2) + (2*d^3*(-2*b*c + a*d))/(c^3*(b*c - a*d)^3*(c + d*x)) - ((2*b*c
+ 3*a*d)*Log[x])/(a^3*c^4) + (b^4*(2*b*c - 5*a*d)*Log[a + b*x])/(a^3*(b*c - a*d
)^4) + (d^3*(10*b^2*c^2 - 10*a*b*c*d + 3*a^2*d^2)*Log[c + d*x])/(c^4*(b*c - a*d)
^4)

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Maple [A]  time = 0.027, size = 266, normalized size = 1.4 \[ -{\frac{{d}^{3}}{2\,{c}^{2} \left ( ad-bc \right ) ^{2} \left ( dx+c \right ) ^{2}}}+3\,{\frac{{d}^{5}\ln \left ( dx+c \right ){a}^{2}}{{c}^{4} \left ( ad-bc \right ) ^{4}}}-10\,{\frac{{d}^{4}\ln \left ( dx+c \right ) ab}{{c}^{3} \left ( ad-bc \right ) ^{4}}}+10\,{\frac{{d}^{3}\ln \left ( dx+c \right ){b}^{2}}{{c}^{2} \left ( ad-bc \right ) ^{4}}}-2\,{\frac{{d}^{4}a}{{c}^{3} \left ( ad-bc \right ) ^{3} \left ( dx+c \right ) }}+4\,{\frac{{d}^{3}b}{{c}^{2} \left ( ad-bc \right ) ^{3} \left ( dx+c \right ) }}-{\frac{1}{{a}^{2}{c}^{3}x}}-3\,{\frac{\ln \left ( x \right ) d}{{a}^{2}{c}^{4}}}-2\,{\frac{b\ln \left ( x \right ) }{{a}^{3}{c}^{3}}}+{\frac{{b}^{4}}{ \left ( ad-bc \right ) ^{3}{a}^{2} \left ( bx+a \right ) }}-5\,{\frac{{b}^{4}\ln \left ( bx+a \right ) d}{ \left ( ad-bc \right ) ^{4}{a}^{2}}}+2\,{\frac{{b}^{5}\ln \left ( bx+a \right ) c}{ \left ( ad-bc \right ) ^{4}{a}^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^2/(b*x+a)^2/(d*x+c)^3,x)

[Out]

-1/2*d^3/c^2/(a*d-b*c)^2/(d*x+c)^2+3*d^5/c^4/(a*d-b*c)^4*ln(d*x+c)*a^2-10*d^4/c^
3/(a*d-b*c)^4*ln(d*x+c)*a*b+10*d^3/c^2/(a*d-b*c)^4*ln(d*x+c)*b^2-2*d^4/c^3/(a*d-
b*c)^3/(d*x+c)*a+4*d^3/c^2/(a*d-b*c)^3/(d*x+c)*b-1/a^2/c^3/x-3/a^2/c^4*ln(x)*d-2
/a^3/c^3*ln(x)*b+b^4/(a*d-b*c)^3/a^2/(b*x+a)-5*b^4/(a*d-b*c)^4/a^2*ln(b*x+a)*d+2
*b^5/(a*d-b*c)^4/a^3*ln(b*x+a)*c

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Maxima [A]  time = 1.41735, size = 863, normalized size = 4.43 \[ \frac{{\left (2 \, b^{5} c - 5 \, a b^{4} d\right )} \log \left (b x + a\right )}{a^{3} b^{4} c^{4} - 4 \, a^{4} b^{3} c^{3} d + 6 \, a^{5} b^{2} c^{2} d^{2} - 4 \, a^{6} b c d^{3} + a^{7} d^{4}} + \frac{{\left (10 \, b^{2} c^{2} d^{3} - 10 \, a b c d^{4} + 3 \, a^{2} d^{5}\right )} \log \left (d x + c\right )}{b^{4} c^{8} - 4 \, a b^{3} c^{7} d + 6 \, a^{2} b^{2} c^{6} d^{2} - 4 \, a^{3} b c^{5} d^{3} + a^{4} c^{4} d^{4}} - \frac{2 \, a b^{3} c^{5} - 6 \, a^{2} b^{2} c^{4} d + 6 \, a^{3} b c^{3} d^{2} - 2 \, a^{4} c^{2} d^{3} + 2 \,{\left (2 \, b^{4} c^{3} d^{2} - 3 \, a b^{3} c^{2} d^{3} + 7 \, a^{2} b^{2} c d^{4} - 3 \, a^{3} b d^{5}\right )} x^{3} +{\left (8 \, b^{4} c^{4} d - 10 \, a b^{3} c^{3} d^{2} + 15 \, a^{2} b^{2} c^{2} d^{3} + 5 \, a^{3} b c d^{4} - 6 \, a^{4} d^{5}\right )} x^{2} +{\left (4 \, b^{4} c^{5} - 2 \, a b^{3} c^{4} d - 6 \, a^{2} b^{2} c^{3} d^{2} + 19 \, a^{3} b c^{2} d^{3} - 9 \, a^{4} c d^{4}\right )} x}{2 \,{\left ({\left (a^{2} b^{4} c^{6} d^{2} - 3 \, a^{3} b^{3} c^{5} d^{3} + 3 \, a^{4} b^{2} c^{4} d^{4} - a^{5} b c^{3} d^{5}\right )} x^{4} +{\left (2 \, a^{2} b^{4} c^{7} d - 5 \, a^{3} b^{3} c^{6} d^{2} + 3 \, a^{4} b^{2} c^{5} d^{3} + a^{5} b c^{4} d^{4} - a^{6} c^{3} d^{5}\right )} x^{3} +{\left (a^{2} b^{4} c^{8} - a^{3} b^{3} c^{7} d - 3 \, a^{4} b^{2} c^{6} d^{2} + 5 \, a^{5} b c^{5} d^{3} - 2 \, a^{6} c^{4} d^{4}\right )} x^{2} +{\left (a^{3} b^{3} c^{8} - 3 \, a^{4} b^{2} c^{7} d + 3 \, a^{5} b c^{6} d^{2} - a^{6} c^{5} d^{3}\right )} x\right )}} - \frac{{\left (2 \, b c + 3 \, a d\right )} \log \left (x\right )}{a^{3} c^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^2*(d*x + c)^3*x^2),x, algorithm="maxima")

[Out]

(2*b^5*c - 5*a*b^4*d)*log(b*x + a)/(a^3*b^4*c^4 - 4*a^4*b^3*c^3*d + 6*a^5*b^2*c^
2*d^2 - 4*a^6*b*c*d^3 + a^7*d^4) + (10*b^2*c^2*d^3 - 10*a*b*c*d^4 + 3*a^2*d^5)*l
og(d*x + c)/(b^4*c^8 - 4*a*b^3*c^7*d + 6*a^2*b^2*c^6*d^2 - 4*a^3*b*c^5*d^3 + a^4
*c^4*d^4) - 1/2*(2*a*b^3*c^5 - 6*a^2*b^2*c^4*d + 6*a^3*b*c^3*d^2 - 2*a^4*c^2*d^3
+ 2*(2*b^4*c^3*d^2 - 3*a*b^3*c^2*d^3 + 7*a^2*b^2*c*d^4 - 3*a^3*b*d^5)*x^3 + (8*
b^4*c^4*d - 10*a*b^3*c^3*d^2 + 15*a^2*b^2*c^2*d^3 + 5*a^3*b*c*d^4 - 6*a^4*d^5)*x
^2 + (4*b^4*c^5 - 2*a*b^3*c^4*d - 6*a^2*b^2*c^3*d^2 + 19*a^3*b*c^2*d^3 - 9*a^4*c
*d^4)*x)/((a^2*b^4*c^6*d^2 - 3*a^3*b^3*c^5*d^3 + 3*a^4*b^2*c^4*d^4 - a^5*b*c^3*d
^5)*x^4 + (2*a^2*b^4*c^7*d - 5*a^3*b^3*c^6*d^2 + 3*a^4*b^2*c^5*d^3 + a^5*b*c^4*d
^4 - a^6*c^3*d^5)*x^3 + (a^2*b^4*c^8 - a^3*b^3*c^7*d - 3*a^4*b^2*c^6*d^2 + 5*a^5
*b*c^5*d^3 - 2*a^6*c^4*d^4)*x^2 + (a^3*b^3*c^8 - 3*a^4*b^2*c^7*d + 3*a^5*b*c^6*d
^2 - a^6*c^5*d^3)*x) - (2*b*c + 3*a*d)*log(x)/(a^3*c^4)

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Fricas [A]  time = 54.9218, size = 1638, normalized size = 8.4 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^2*(d*x + c)^3*x^2),x, algorithm="fricas")

[Out]

-1/2*(2*a^2*b^4*c^7 - 8*a^3*b^3*c^6*d + 12*a^4*b^2*c^5*d^2 - 8*a^5*b*c^4*d^3 + 2
*a^6*c^3*d^4 + 2*(2*a*b^5*c^5*d^2 - 5*a^2*b^4*c^4*d^3 + 10*a^3*b^3*c^3*d^4 - 10*
a^4*b^2*c^2*d^5 + 3*a^5*b*c*d^6)*x^3 + (8*a*b^5*c^6*d - 18*a^2*b^4*c^5*d^2 + 25*
a^3*b^3*c^4*d^3 - 10*a^4*b^2*c^3*d^4 - 11*a^5*b*c^2*d^5 + 6*a^6*c*d^6)*x^2 + (4*
a*b^5*c^7 - 6*a^2*b^4*c^6*d - 4*a^3*b^3*c^5*d^2 + 25*a^4*b^2*c^4*d^3 - 28*a^5*b*
c^3*d^4 + 9*a^6*c^2*d^5)*x - 2*((2*b^6*c^5*d^2 - 5*a*b^5*c^4*d^3)*x^4 + (4*b^6*c
^6*d - 8*a*b^5*c^5*d^2 - 5*a^2*b^4*c^4*d^3)*x^3 + (2*b^6*c^7 - a*b^5*c^6*d - 10*
a^2*b^4*c^5*d^2)*x^2 + (2*a*b^5*c^7 - 5*a^2*b^4*c^6*d)*x)*log(b*x + a) - 2*((10*
a^3*b^3*c^2*d^5 - 10*a^4*b^2*c*d^6 + 3*a^5*b*d^7)*x^4 + (20*a^3*b^3*c^3*d^4 - 10
*a^4*b^2*c^2*d^5 - 4*a^5*b*c*d^6 + 3*a^6*d^7)*x^3 + (10*a^3*b^3*c^4*d^3 + 10*a^4
*b^2*c^3*d^4 - 17*a^5*b*c^2*d^5 + 6*a^6*c*d^6)*x^2 + (10*a^4*b^2*c^4*d^3 - 10*a^
5*b*c^3*d^4 + 3*a^6*c^2*d^5)*x)*log(d*x + c) + 2*((2*b^6*c^5*d^2 - 5*a*b^5*c^4*d
^3 + 10*a^3*b^3*c^2*d^5 - 10*a^4*b^2*c*d^6 + 3*a^5*b*d^7)*x^4 + (4*b^6*c^6*d - 8
*a*b^5*c^5*d^2 - 5*a^2*b^4*c^4*d^3 + 20*a^3*b^3*c^3*d^4 - 10*a^4*b^2*c^2*d^5 - 4
*a^5*b*c*d^6 + 3*a^6*d^7)*x^3 + (2*b^6*c^7 - a*b^5*c^6*d - 10*a^2*b^4*c^5*d^2 +
10*a^3*b^3*c^4*d^3 + 10*a^4*b^2*c^3*d^4 - 17*a^5*b*c^2*d^5 + 6*a^6*c*d^6)*x^2 +
(2*a*b^5*c^7 - 5*a^2*b^4*c^6*d + 10*a^4*b^2*c^4*d^3 - 10*a^5*b*c^3*d^4 + 3*a^6*c
^2*d^5)*x)*log(x))/((a^3*b^5*c^8*d^2 - 4*a^4*b^4*c^7*d^3 + 6*a^5*b^3*c^6*d^4 - 4
*a^6*b^2*c^5*d^5 + a^7*b*c^4*d^6)*x^4 + (2*a^3*b^5*c^9*d - 7*a^4*b^4*c^8*d^2 + 8
*a^5*b^3*c^7*d^3 - 2*a^6*b^2*c^6*d^4 - 2*a^7*b*c^5*d^5 + a^8*c^4*d^6)*x^3 + (a^3
*b^5*c^10 - 2*a^4*b^4*c^9*d - 2*a^5*b^3*c^8*d^2 + 8*a^6*b^2*c^7*d^3 - 7*a^7*b*c^
6*d^4 + 2*a^8*c^5*d^5)*x^2 + (a^4*b^4*c^10 - 4*a^5*b^3*c^9*d + 6*a^6*b^2*c^8*d^2
- 4*a^7*b*c^7*d^3 + a^8*c^6*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**2/(b*x+a)**2/(d*x+c)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.362477, size = 698, normalized size = 3.58 \[ -\frac{b^{9}}{{\left (a^{2} b^{8} c^{3} - 3 \, a^{3} b^{7} c^{2} d + 3 \, a^{4} b^{6} c d^{2} - a^{5} b^{5} d^{3}\right )}{\left (b x + a\right )}} + \frac{{\left (10 \, b^{3} c^{2} d^{3} - 10 \, a b^{2} c d^{4} + 3 \, a^{2} b d^{5}\right )}{\rm ln}\left ({\left | \frac{b c}{b x + a} - \frac{a d}{b x + a} + d \right |}\right )}{b^{5} c^{8} - 4 \, a b^{4} c^{7} d + 6 \, a^{2} b^{3} c^{6} d^{2} - 4 \, a^{3} b^{2} c^{5} d^{3} + a^{4} b c^{4} d^{4}} - \frac{{\left (2 \, b^{2} c + 3 \, a b d\right )}{\rm ln}\left ({\left | -\frac{a}{b x + a} + 1 \right |}\right )}{a^{3} b c^{4}} + \frac{2 \, b^{5} c^{5} d^{2} - 8 \, a b^{4} c^{4} d^{3} + 12 \, a^{2} b^{3} c^{3} d^{4} - 17 \, a^{3} b^{2} c^{2} d^{5} + 6 \, a^{4} b c d^{6} + \frac{4 \, b^{7} c^{6} d - 20 \, a b^{6} c^{5} d^{2} + 40 \, a^{2} b^{5} c^{4} d^{3} - 50 \, a^{3} b^{4} c^{3} d^{4} + 43 \, a^{4} b^{3} c^{2} d^{5} - 12 \, a^{5} b^{2} c d^{6}}{{\left (b x + a\right )} b} + \frac{2 \,{\left (b^{9} c^{7} - 6 \, a b^{8} c^{6} d + 15 \, a^{2} b^{7} c^{5} d^{2} - 20 \, a^{3} b^{6} c^{4} d^{3} + 20 \, a^{4} b^{5} c^{3} d^{4} - 13 \, a^{5} b^{4} c^{2} d^{5} + 3 \, a^{6} b^{3} c d^{6}\right )}}{{\left (b x + a\right )}^{2} b^{2}}}{2 \,{\left (b c - a d\right )}^{4} a^{3}{\left (\frac{b c}{b x + a} - \frac{a d}{b x + a} + d\right )}^{2} c^{4}{\left (\frac{a}{b x + a} - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^2*(d*x + c)^3*x^2),x, algorithm="giac")

[Out]

-b^9/((a^2*b^8*c^3 - 3*a^3*b^7*c^2*d + 3*a^4*b^6*c*d^2 - a^5*b^5*d^3)*(b*x + a))
+ (10*b^3*c^2*d^3 - 10*a*b^2*c*d^4 + 3*a^2*b*d^5)*ln(abs(b*c/(b*x + a) - a*d/(b
*x + a) + d))/(b^5*c^8 - 4*a*b^4*c^7*d + 6*a^2*b^3*c^6*d^2 - 4*a^3*b^2*c^5*d^3 +
a^4*b*c^4*d^4) - (2*b^2*c + 3*a*b*d)*ln(abs(-a/(b*x + a) + 1))/(a^3*b*c^4) + 1/
2*(2*b^5*c^5*d^2 - 8*a*b^4*c^4*d^3 + 12*a^2*b^3*c^3*d^4 - 17*a^3*b^2*c^2*d^5 + 6
*a^4*b*c*d^6 + (4*b^7*c^6*d - 20*a*b^6*c^5*d^2 + 40*a^2*b^5*c^4*d^3 - 50*a^3*b^4
*c^3*d^4 + 43*a^4*b^3*c^2*d^5 - 12*a^5*b^2*c*d^6)/((b*x + a)*b) + 2*(b^9*c^7 - 6
*a*b^8*c^6*d + 15*a^2*b^7*c^5*d^2 - 20*a^3*b^6*c^4*d^3 + 20*a^4*b^5*c^3*d^4 - 13
*a^5*b^4*c^2*d^5 + 3*a^6*b^3*c*d^6)/((b*x + a)^2*b^2))/((b*c - a*d)^4*a^3*(b*c/(
b*x + a) - a*d/(b*x + a) + d)^2*c^4*(a/(b*x + a) - 1))