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

Optimal. Leaf size=107 \[ \frac{\log (x) \left (a^2 d^2+a b c d+b^2 c^2\right )}{a^3 c^3}-\frac{b^3 \log (a+b x)}{a^3 (b c-a d)}+\frac{a d+b c}{a^2 c^2 x}+\frac{d^3 \log (c+d x)}{c^3 (b c-a d)}-\frac{1}{2 a c x^2} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0801743, antiderivative size = 107, 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} \[ \frac{\log (x) \left (a^2 d^2+a b c d+b^2 c^2\right )}{a^3 c^3}-\frac{b^3 \log (a+b x)}{a^3 (b c-a d)}+\frac{a d+b c}{a^2 c^2 x}+\frac{d^3 \log (c+d x)}{c^3 (b c-a d)}-\frac{1}{2 a c x^2} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^3*(a + b*x)*(c + d*x)),x]

[Out]

-1/(2*a*c*x^2) + (b*c + a*d)/(a^2*c^2*x) + ((b^2*c^2 + a*b*c*d + a^2*d^2)*Log[x])/(a^3*c^3) - (b^3*Log[a + b*x
])/(a^3*(b*c - a*d)) + (d^3*Log[c + d*x])/(c^3*(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^3 (a+b x) (c+d x)} \, dx &=\int \left (\frac{1}{a c x^3}+\frac{-b c-a d}{a^2 c^2 x^2}+\frac{b^2 c^2+a b c d+a^2 d^2}{a^3 c^3 x}+\frac{b^4}{a^3 (-b c+a d) (a+b x)}+\frac{d^4}{c^3 (b c-a d) (c+d x)}\right ) \, dx\\ &=-\frac{1}{2 a c x^2}+\frac{b c+a d}{a^2 c^2 x}+\frac{\left (b^2 c^2+a b c d+a^2 d^2\right ) \log (x)}{a^3 c^3}-\frac{b^3 \log (a+b x)}{a^3 (b c-a d)}+\frac{d^3 \log (c+d x)}{c^3 (b c-a d)}\\ \end{align*}

Mathematica [A]  time = 0.0489417, size = 106, normalized size = 0.99 \[ \frac{\log (x) \left (a^2 d^2+a b c d+b^2 c^2\right )}{a^3 c^3}+\frac{b^3 \log (a+b x)}{a^3 (a d-b c)}+\frac{a d+b c}{a^2 c^2 x}+\frac{d^3 \log (c+d x)}{c^3 (b c-a d)}-\frac{1}{2 a c x^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.008, size = 117, normalized size = 1.1 \begin{align*} -{\frac{{d}^{3}\ln \left ( dx+c \right ) }{{c}^{3} \left ( ad-bc \right ) }}-{\frac{1}{2\,ac{x}^{2}}}+{\frac{d}{a{c}^{2}x}}+{\frac{b}{{a}^{2}cx}}+{\frac{\ln \left ( x \right ){d}^{2}}{a{c}^{3}}}+{\frac{b\ln \left ( x \right ) d}{{a}^{2}{c}^{2}}}+{\frac{\ln \left ( x \right ){b}^{2}}{{a}^{3}c}}+{\frac{{b}^{3}\ln \left ( bx+a \right ) }{ \left ( ad-bc \right ){a}^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^3/(b*x+a)/(d*x+c),x)

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.03271, size = 143, normalized size = 1.34 \begin{align*} -\frac{b^{3} \log \left (b x + a\right )}{a^{3} b c - a^{4} d} + \frac{d^{3} \log \left (d x + c\right )}{b c^{4} - a c^{3} d} + \frac{{\left (b^{2} c^{2} + a b c d + a^{2} d^{2}\right )} \log \left (x\right )}{a^{3} c^{3}} - \frac{a c - 2 \,{\left (b c + a d\right )} x}{2 \, a^{2} c^{2} x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 21.3442, size = 247, normalized size = 2.31 \begin{align*} -\frac{2 \, b^{3} c^{3} x^{2} \log \left (b x + a\right ) - 2 \, a^{3} d^{3} x^{2} \log \left (d x + c\right ) + a^{2} b c^{3} - a^{3} c^{2} d - 2 \,{\left (b^{3} c^{3} - a^{3} d^{3}\right )} x^{2} \log \left (x\right ) - 2 \,{\left (a b^{2} c^{3} - a^{3} c d^{2}\right )} x}{2 \,{\left (a^{3} b c^{4} - a^{4} c^{3} d\right )} x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

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

[Out]

Exception raised: NotImplementedError