3.867 \(\int \frac{1}{x \left (a+b x^n\right )^2 \left (c+d x^n\right )} \, dx\)

Optimal. Leaf size=101 \[ -\frac{b (b c-2 a d) \log \left (a+b x^n\right )}{a^2 n (b c-a d)^2}+\frac{\log (x)}{a^2 c}-\frac{d^2 \log \left (c+d x^n\right )}{c n (b c-a d)^2}+\frac{b}{a n (b c-a d) \left (a+b x^n\right )} \]

[Out]

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

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Rubi [A]  time = 0.280936, antiderivative size = 101, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ -\frac{b (b c-2 a d) \log \left (a+b x^n\right )}{a^2 n (b c-a d)^2}+\frac{\log (x)}{a^2 c}-\frac{d^2 \log \left (c+d x^n\right )}{c n (b c-a d)^2}+\frac{b}{a n (b c-a d) \left (a+b x^n\right )} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 41.3055, size = 85, normalized size = 0.84 \[ - \frac{d^{2} \log{\left (c + d x^{n} \right )}}{c n \left (a d - b c\right )^{2}} - \frac{b}{a n \left (a + b x^{n}\right ) \left (a d - b c\right )} + \frac{b \left (2 a d - b c\right ) \log{\left (a + b x^{n} \right )}}{a^{2} n \left (a d - b c\right )^{2}} + \frac{\log{\left (x^{n} \right )}}{a^{2} c n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.454578, size = 96, normalized size = 0.95 \[ \frac{\frac{b c (2 a d-b c) \left (a+b x^n\right ) \log \left (a+b x^n\right )-a \left (a d^2 \left (a+b x^n\right ) \log \left (c+d x^n\right )+b c (a d-b c)\right )}{n (b c-a d)^2 \left (a+b x^n\right )}+\log (x)}{a^2 c} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.018, size = 131, normalized size = 1.3 \[{\frac{\ln \left ({x}^{n} \right ) }{{a}^{2}nc}}-{\frac{b}{an \left ( ad-bc \right ) \left ( a+b{x}^{n} \right ) }}+2\,{\frac{b\ln \left ( a+b{x}^{n} \right ) d}{n \left ( ad-bc \right ) ^{2}a}}-{\frac{{b}^{2}\ln \left ( a+b{x}^{n} \right ) c}{n \left ( ad-bc \right ) ^{2}{a}^{2}}}-{\frac{{d}^{2}\ln \left ( c+d{x}^{n} \right ) }{nc \left ( ad-bc \right ) ^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.3978, size = 204, normalized size = 2.02 \[ -\frac{d^{2} \log \left (\frac{d x^{n} + c}{d}\right )}{b^{2} c^{3} n - 2 \, a b c^{2} d n + a^{2} c d^{2} n} - \frac{{\left (b^{2} c - 2 \, a b d\right )} \log \left (\frac{b x^{n} + a}{b}\right )}{a^{2} b^{2} c^{2} n - 2 \, a^{3} b c d n + a^{4} d^{2} n} + \frac{b}{a^{2} b c n - a^{3} d n +{\left (a b^{2} c n - a^{2} b d n\right )} x^{n}} + \frac{\log \left (x\right )}{a^{2} c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.251201, size = 302, normalized size = 2.99 \[ \frac{a b^{2} c^{2} - a^{2} b c d +{\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} n x^{n} \log \left (x\right ) +{\left (a b^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{2}\right )} n \log \left (x\right ) -{\left (a b^{2} c^{2} - 2 \, a^{2} b c d +{\left (b^{3} c^{2} - 2 \, a b^{2} c d\right )} x^{n}\right )} \log \left (b x^{n} + a\right ) -{\left (a^{2} b d^{2} x^{n} + a^{3} d^{2}\right )} \log \left (d x^{n} + c\right )}{{\left (a^{2} b^{3} c^{3} - 2 \, a^{3} b^{2} c^{2} d + a^{4} b c d^{2}\right )} n x^{n} +{\left (a^{3} b^{2} c^{3} - 2 \, a^{4} b c^{2} d + a^{5} c d^{2}\right )} n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{n} + a\right )}^{2}{\left (d x^{n} + c\right )} x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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