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

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 = 0.054235, 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, Rules used = {72} \[ \frac{b^2 \log (a+b x)}{a^2 (b c-a d)}-\frac{\log (x) (a d+b c)}{a^2 c^2}-\frac{d^2 \log (c+d x)}{c^2 (b c-a d)}-\frac{1}{a c x} \]

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))

Rule 72

Int[((e_.) + (f_.)*(x_))^(p_.)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Int[ExpandIntegrand[(
e + f*x)^p/((a + b*x)*(c + d*x)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IntegerQ[p]

Rubi steps

\begin{align*} \int \frac{1}{x^2 (a+b x) (c+d x)} \, dx &=\int \left (\frac{1}{a c x^2}+\frac{-b c-a d}{a^2 c^2 x}-\frac{b^3}{a^2 (-b c+a d) (a+b x)}-\frac{d^3}{c^2 (b c-a d) (c+d x)}\right ) \, dx\\ &=-\frac{1}{a c x}-\frac{(b c+a d) \log (x)}{a^2 c^2}+\frac{b^2 \log (a+b x)}{a^2 (b c-a d)}-\frac{d^2 \log (c+d x)}{c^2 (b c-a d)}\\ \end{align*}

Mathematica [A]  time = 0.0362894, 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 = 0.01, size = 82, normalized size = 1.1 \begin{align*}{\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}}} \end{align*}

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 = 1.977, size = 108, normalized size = 1.42 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(b*x+a)/(d*x+c),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 = 3.48285, size = 176, normalized size = 2.32 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(b*x+a)/(d*x+c),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 [B]  time = 66.8629, size = 1119, normalized size = 14.72 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**2/(b*x+a)/(d*x+c),x)

[Out]

d**2*log(x + (-2*a**9*d**9/(a*d - b*c)**2 + 5*a**8*b*c*d**8/(a*d - b*c)**2 - 3*a**7*b**2*c**2*d**7/(a*d - b*c)
**2 + a**7*b*c*d**7/(a*d - b*c) + 2*a**7*d**7 - 3*a**6*b**3*c**3*d**6/(a*d - b*c)**2 + a**6*b**2*c**2*d**6/(a*
d - b*c) + 5*a**5*b**4*c**4*d**5/(a*d - b*c)**2 - 4*a**5*b**3*c**3*d**5/(a*d - b*c) - 3*a**5*b**2*c**2*d**5 -
2*a**4*b**5*c**5*d**4/(a*d - b*c)**2 + a**4*b**4*c**4*d**4/(a*d - b*c) + a**4*b**3*c**3*d**4 + a**3*b**5*c**5*
d**3/(a*d - b*c) + a**3*b**4*c**4*d**3 - 3*a**2*b**5*c**5*d**2 + 2*b**7*c**7)/(2*a**6*b*d**7 - 3*a**4*b**3*c**
2*d**5 - 3*a**2*b**5*c**4*d**3 + 2*b**7*c**6*d))/(c**2*(a*d - b*c)) - 1/(a*c*x) - b**2*log(x + (2*a**7*d**7 -
2*a**5*b**4*c**4*d**5/(a*d - b*c)**2 - a**5*b**3*c**3*d**5/(a*d - b*c) - 3*a**5*b**2*c**2*d**5 + 5*a**4*b**5*c
**5*d**4/(a*d - b*c)**2 - a**4*b**4*c**4*d**4/(a*d - b*c) + a**4*b**3*c**3*d**4 - 3*a**3*b**6*c**6*d**3/(a*d -
b*c)**2 + 4*a**3*b**5*c**5*d**3/(a*d - b*c) + a**3*b**4*c**4*d**3 - 3*a**2*b**7*c**7*d**2/(a*d - b*c)**2 - a*
*2*b**6*c**6*d**2/(a*d - b*c) - 3*a**2*b**5*c**5*d**2 + 5*a*b**8*c**8*d/(a*d - b*c)**2 - a*b**7*c**7*d/(a*d -
b*c) - 2*b**9*c**9/(a*d - b*c)**2 + 2*b**7*c**7)/(2*a**6*b*d**7 - 3*a**4*b**3*c**2*d**5 - 3*a**2*b**5*c**4*d**
3 + 2*b**7*c**6*d))/(a**2*(a*d - b*c)) - (a*d + b*c)*log(x + (2*a**7*d**7 - 3*a**5*b**2*c**2*d**5 - a**5*b*c*d
**5*(a*d + b*c) - 2*a**5*d**5*(a*d + b*c)**2 + a**4*b**3*c**3*d**4 - a**4*b**2*c**2*d**4*(a*d + b*c) + 5*a**4*
b*c*d**4*(a*d + b*c)**2 + a**3*b**4*c**4*d**3 + 4*a**3*b**3*c**3*d**3*(a*d + b*c) - 3*a**3*b**2*c**2*d**3*(a*d
+ b*c)**2 - 3*a**2*b**5*c**5*d**2 - a**2*b**4*c**4*d**2*(a*d + b*c) - 3*a**2*b**3*c**3*d**2*(a*d + b*c)**2 -
a*b**5*c**5*d*(a*d + b*c) + 5*a*b**4*c**4*d*(a*d + b*c)**2 + 2*b**7*c**7 - 2*b**5*c**5*(a*d + b*c)**2)/(2*a**6
*b*d**7 - 3*a**4*b**3*c**2*d**5 - 3*a**2*b**5*c**4*d**3 + 2*b**7*c**6*d))/(a**2*c**2)

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: NotImplementedError