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

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

[Out]

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

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Rubi [A]  time = 0.12, antiderivative size = 144, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.08, size = 139, normalized size = 0.97 \[ \frac {6 x^3 \log (x) \left (b^4 c^4-a^4 d^4\right )+a \left (a^3 c d \left (-2 c^2+3 c d x-6 d^2 x^2\right )+6 a^3 d^4 x^3 \log (c+d x)+2 a^2 b c^4-3 a b^2 c^4 x+6 b^3 c^4 x^2\right )-6 b^4 c^4 x^3 \log (a+b x)}{6 a^4 c^4 x^3 (a d-b c)} \]

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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fricas [A]  time = 7.76, size = 149, normalized size = 1.03 \[ \frac {6 \, b^{4} c^{4} x^{3} \log \left (b x + a\right ) - 6 \, a^{4} d^{4} x^{3} \log \left (d x + c\right ) - 2 \, a^{3} b c^{4} + 2 \, a^{4} c^{3} d - 6 \, {\left (b^{4} c^{4} - a^{4} d^{4}\right )} x^{3} \log \relax (x) - 6 \, {\left (a b^{3} c^{4} - a^{4} c d^{3}\right )} x^{2} + 3 \, {\left (a^{2} b^{2} c^{4} - a^{4} c^{2} d^{2}\right )} x}{6 \, {\left (a^{4} b c^{5} - a^{5} c^{4} d\right )} x^{3}} \]

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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giac [A]  time = 0.99, size = 175, normalized size = 1.22 \[ \frac {b^{5} \log \left ({\left | b x + a \right |}\right )}{a^{4} b^{2} c - a^{5} b d} - \frac {d^{5} \log \left ({\left | d x + c \right |}\right )}{b c^{5} d - a c^{4} d^{2}} - \frac {{\left (b^{3} c^{3} + a b^{2} c^{2} d + a^{2} b c d^{2} + a^{3} d^{3}\right )} \log \left ({\left | x \right |}\right )}{a^{4} c^{4}} - \frac {2 \, a^{3} c^{3} + 6 \, {\left (a b^{2} c^{3} + a^{2} b c^{2} d + a^{3} c d^{2}\right )} x^{2} - 3 \, {\left (a^{2} b c^{3} + a^{3} c^{2} d\right )} x}{6 \, a^{4} c^{4} x^{3}} \]

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]

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

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maple [A]  time = 0.01, size = 179, normalized size = 1.24 \[ -\frac {b^{4} \ln \left (b x +a \right )}{\left (a d -b c \right ) a^{4}}+\frac {d^{4} \ln \left (d x +c \right )}{\left (a d -b c \right ) c^{4}}-\frac {d^{3} \ln \relax (x )}{a \,c^{4}}-\frac {b \,d^{2} \ln \relax (x )}{a^{2} c^{3}}-\frac {b^{2} d \ln \relax (x )}{a^{3} c^{2}}-\frac {b^{3} \ln \relax (x )}{a^{4} c}-\frac {d^{2}}{a \,c^{3} x}-\frac {b d}{a^{2} c^{2} x}-\frac {b^{2}}{a^{3} c x}+\frac {d}{2 a \,c^{2} x^{2}}+\frac {b}{2 a^{2} c \,x^{2}}-\frac {1}{3 a c \,x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.13, size = 156, normalized size = 1.08 \[ \frac {b^{4} \log \left (b x + a\right )}{a^{4} b c - a^{5} d} - \frac {d^{4} \log \left (d x + c\right )}{b c^{5} - a c^{4} d} - \frac {{\left (b^{3} c^{3} + a b^{2} c^{2} d + a^{2} b c d^{2} + a^{3} d^{3}\right )} \log \relax (x)}{a^{4} c^{4}} - \frac {2 \, a^{2} c^{2} + 6 \, {\left (b^{2} c^{2} + a b c d + a^{2} d^{2}\right )} x^{2} - 3 \, {\left (a b c^{2} + a^{2} c d\right )} x}{6 \, a^{3} c^{3} x^{3}} \]

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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mupad [B]  time = 0.55, size = 153, normalized size = 1.06 \[ \frac {d^4\,\ln \left (c+d\,x\right )}{c^4\,\left (a\,d-b\,c\right )}-\frac {b^4\,\ln \left (a+b\,x\right )}{a^5\,d-a^4\,b\,c}-\frac {\ln \relax (x)\,\left (a^3\,d^3+a^2\,b\,c\,d^2+a\,b^2\,c^2\,d+b^3\,c^3\right )}{a^4\,c^4}-\frac {\frac {1}{3\,a\,c}-\frac {x\,\left (a\,d+b\,c\right )}{2\,a^2\,c^2}+\frac {x^2\,\left (a^2\,d^2+a\,b\,c\,d+b^2\,c^2\right )}{a^3\,c^3}}{x^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^4*(a + b*x)*(c + d*x)),x)

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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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