3.241 \(\int \frac{1}{x^4 (a+b x) (c+d x)} \, dx\)

Optimal. Leaf size=144 \[ -\frac{a^2 d^2+a b c d+b^2 c^2}{a^3 c^3 x}-\frac{\log (x) (a d+b c) \left (a^2 d^2+b^2 c^2\right )}{a^4 c^4}+\frac{b^4 \log (a+b x)}{a^4 (b c-a d)}+\frac{a d+b c}{2 a^2 c^2 x^2}-\frac{d^4 \log (c+d x)}{c^4 (b c-a d)}-\frac{1}{3 a c x^3} \]

[Out]

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

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Rubi [A]  time = 0.11801, antiderivative size = 144, 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} \[ -\frac{a^2 d^2+a b c d+b^2 c^2}{a^3 c^3 x}-\frac{\log (x) (a d+b c) \left (a^2 d^2+b^2 c^2\right )}{a^4 c^4}+\frac{b^4 \log (a+b x)}{a^4 (b c-a d)}+\frac{a d+b c}{2 a^2 c^2 x^2}-\frac{d^4 \log (c+d x)}{c^4 (b c-a d)}-\frac{1}{3 a c x^3} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^4*(a + b*x)*(c + d*x)),x]

[Out]

-1/(3*a*c*x^3) + (b*c + a*d)/(2*a^2*c^2*x^2) - (b^2*c^2 + a*b*c*d + a^2*d^2)/(a^3*c^3*x) - ((b*c + a*d)*(b^2*c
^2 + a^2*d^2)*Log[x])/(a^4*c^4) + (b^4*Log[a + b*x])/(a^4*(b*c - a*d)) - (d^4*Log[c + d*x])/(c^4*(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^4 (a+b x) (c+d x)} \, dx &=\int \left (\frac{1}{a c x^4}+\frac{-b c-a d}{a^2 c^2 x^3}+\frac{b^2 c^2+a b c d+a^2 d^2}{a^3 c^3 x^2}-\frac{(b c+a d) \left (b^2 c^2+a^2 d^2\right )}{a^4 c^4 x}-\frac{b^5}{a^4 (-b c+a d) (a+b x)}-\frac{d^5}{c^4 (b c-a d) (c+d x)}\right ) \, dx\\ &=-\frac{1}{3 a c x^3}+\frac{b c+a d}{2 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 c+a d) \left (b^2 c^2+a^2 d^2\right ) \log (x)}{a^4 c^4}+\frac{b^4 \log (a+b x)}{a^4 (b c-a d)}-\frac{d^4 \log (c+d x)}{c^4 (b c-a d)}\\ \end{align*}

Mathematica [A]  time = 0.0622287, size = 139, normalized size = 0.97 \[ \frac{6 x^3 \log (x) \left (b^4 c^4-a^4 d^4\right )+a \left (2 a^2 b c^4+a^3 c d \left (-2 c^2+3 c d x-6 d^2 x^2\right )+6 a^3 d^4 x^3 \log (c+d x)-3 a b^2 c^4 x+6 b^3 c^4 x^2\right )-6 b^4 c^4 x^3 \log (a+b x)}{6 a^4 c^4 x^3 (a d-b c)} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^4*(a + b*x)*(c + d*x)),x]

[Out]

(6*(b^4*c^4 - a^4*d^4)*x^3*Log[x] - 6*b^4*c^4*x^3*Log[a + b*x] + a*(2*a^2*b*c^4 - 3*a*b^2*c^4*x + 6*b^3*c^4*x^
2 + a^3*c*d*(-2*c^2 + 3*c*d*x - 6*d^2*x^2) + 6*a^3*d^4*x^3*Log[c + d*x]))/(6*a^4*c^4*(-(b*c) + a*d)*x^3)

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Maple [A]  time = 0.011, size = 179, normalized size = 1.2 \begin{align*}{\frac{{d}^{4}\ln \left ( dx+c \right ) }{{c}^{4} \left ( ad-bc \right ) }}-{\frac{1}{3\,ac{x}^{3}}}+{\frac{d}{2\,a{c}^{2}{x}^{2}}}+{\frac{b}{2\,{a}^{2}c{x}^{2}}}-{\frac{{d}^{2}}{a{c}^{3}x}}-{\frac{bd}{{a}^{2}{c}^{2}x}}-{\frac{{b}^{2}}{{a}^{3}cx}}-{\frac{\ln \left ( x \right ){d}^{3}}{a{c}^{4}}}-{\frac{b\ln \left ( x \right ){d}^{2}}{{a}^{2}{c}^{3}}}-{\frac{\ln \left ( x \right ){b}^{2}d}{{a}^{3}{c}^{2}}}-{\frac{\ln \left ( x \right ){b}^{3}}{{a}^{4}c}}-{\frac{{b}^{4}\ln \left ( bx+a \right ) }{ \left ( ad-bc \right ){a}^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^4/(b*x+a)/(d*x+c),x)

[Out]

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

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Maxima [A]  time = 1.05948, size = 211, normalized size = 1.47 \begin{align*} \frac{b^{4} \log \left (b x + a\right )}{a^{4} b c - a^{5} d} - \frac{d^{4} \log \left (d x + c\right )}{b c^{5} - a c^{4} d} - \frac{{\left (b^{3} c^{3} + a b^{2} c^{2} d + a^{2} b c d^{2} + a^{3} d^{3}\right )} \log \left (x\right )}{a^{4} c^{4}} - \frac{2 \, a^{2} c^{2} + 6 \,{\left (b^{2} c^{2} + a b c d + a^{2} d^{2}\right )} x^{2} - 3 \,{\left (a b c^{2} + a^{2} c d\right )} x}{6 \, a^{3} c^{3} x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(b*x+a)/(d*x+c),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 31.1353, size = 300, normalized size = 2.08 \begin{align*} \frac{6 \, b^{4} c^{4} x^{3} \log \left (b x + a\right ) - 6 \, a^{4} d^{4} x^{3} \log \left (d x + c\right ) - 2 \, a^{3} b c^{4} + 2 \, a^{4} c^{3} d - 6 \,{\left (b^{4} c^{4} - a^{4} d^{4}\right )} x^{3} \log \left (x\right ) - 6 \,{\left (a b^{3} c^{4} - a^{4} c d^{3}\right )} x^{2} + 3 \,{\left (a^{2} b^{2} c^{4} - a^{4} c^{2} d^{2}\right )} x}{6 \,{\left (a^{4} b c^{5} - a^{5} c^{4} d\right )} x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(b*x+a)/(d*x+c),x, algorithm="fricas")

[Out]

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

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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**4/(b*x+a)/(d*x+c),x)

[Out]

Timed out

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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^4/(b*x+a)/(d*x+c),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError