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

Optimal. Leaf size=76 \[ \frac{b^2 \log (a+b x)}{a^2 (b c-a d)}-\frac{\log (x) (a d+b c)}{a^2 c^2}-\frac{d^2 \log (c+d x)}{c^2 (b c-a d)}-\frac{1}{a c x} \]

[Out]

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

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Rubi [A]  time = 0.135863, antiderivative size = 76, 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 \[ \frac{b^2 \log (a+b x)}{a^2 (b c-a d)}-\frac{\log (x) (a d+b c)}{a^2 c^2}-\frac{d^2 \log (c+d x)}{c^2 (b c-a d)}-\frac{1}{a c x} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 32.8926, size = 63, normalized size = 0.83 \[ \frac{d^{2} \log{\left (c + d x \right )}}{c^{2} \left (a d - b c\right )} - \frac{1}{a c x} - \frac{b^{2} \log{\left (a + b x \right )}}{a^{2} \left (a d - b c\right )} - \frac{\left (a d + b c\right ) \log{\left (x \right )}}{a^{2} c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

d**2*log(c + d*x)/(c**2*(a*d - b*c)) - 1/(a*c*x) - b**2*log(a + b*x)/(a**2*(a*d
- b*c)) - (a*d + b*c)*log(x)/(a**2*c**2)

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Mathematica [A]  time = 0.0621593, size = 78, normalized size = 1.03 \[ -\frac{b^2 \log (a+b x)}{a^2 (a d-b c)}+\frac{\log (x) (-a d-b c)}{a^2 c^2}-\frac{d^2 \log (c+d x)}{c^2 (b c-a d)}-\frac{1}{a c x} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.019, size = 82, normalized size = 1.1 \[{\frac{{d}^{2}\ln \left ( dx+c \right ) }{{c}^{2} \left ( ad-bc \right ) }}-{\frac{1}{acx}}-{\frac{\ln \left ( x \right ) d}{a{c}^{2}}}-{\frac{b\ln \left ( x \right ) }{{a}^{2}c}}-{\frac{{b}^{2}\ln \left ( bx+a \right ) }{ \left ( ad-bc \right ){a}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^2/(b*x+a)/(d*x+c),x)

[Out]

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

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Maxima [A]  time = 1.35481, size = 108, normalized size = 1.42 \[ \frac{b^{2} \log \left (b x + a\right )}{a^{2} b c - a^{3} d} - \frac{d^{2} \log \left (d x + c\right )}{b c^{3} - a c^{2} d} - \frac{{\left (b c + a d\right )} \log \left (x\right )}{a^{2} c^{2}} - \frac{1}{a c x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.349368, size = 119, normalized size = 1.57 \[ \frac{b^{2} c^{2} x \log \left (b x + a\right ) - a^{2} d^{2} x \log \left (d x + c\right ) - a b c^{2} + a^{2} c d -{\left (b^{2} c^{2} - a^{2} d^{2}\right )} x \log \left (x\right )}{{\left (a^{2} b c^{3} - a^{3} c^{2} d\right )} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: NotImplementedError