3.1036 \(\int \frac{1}{x (a+b x^n)^2 (c+d x^n)} \, dx\)

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 = 0.105012, 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, Rules used = {446, 72} \[ -\frac{b (b c-2 a d) \log \left (a+b x^n\right )}{a^2 n (b c-a d)^2}+\frac{\log (x)}{a^2 c}-\frac{d^2 \log \left (c+d x^n\right )}{c n (b c-a d)^2}+\frac{b}{a n (b c-a d) \left (a+b x^n\right )} \]

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)

Rule 446

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] &&
NeQ[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]

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

Mathematica [A]  time = 0.174486, size = 97, normalized size = 0.96 \[ \frac{-\frac{b (b c-2 a d) \log \left (a+b x^n\right )}{a^2 (b c-a d)^2}+\frac{n \log (x)}{a^2 c}-\frac{d^2 \log \left (c+d x^n\right )}{c (b c-a d)^2}+\frac{b}{a (b c-a d) \left (a+b x^n\right )}}{n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.012, size = 131, normalized size = 1.3 \begin{align*} -{\frac{b}{n \left ( ad-bc \right ) a \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}}}+{\frac{\ln \left ({x}^{n} \right ) }{n{a}^{2}c}} \end{align*}

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

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Maxima [A]  time = 0.94676, size = 204, normalized size = 2.02 \begin{align*} -\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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(a+b*x^n)^2/(c+d*x^n),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 [B]  time = 1.14898, size = 458, normalized size = 4.53 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(a+b*x^n)^2/(c+d*x^n),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 [A]  time = 12.6121, size = 2227, normalized size = 22.05 \begin{align*} \text{result too large to display} \end{align*}

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]

Piecewise(((log(x)/c - log(c/d + x**n)/(c*n))/a**2, Eq(b, 0)), ((-x**(-2*n)/(2*c*n) + d*x**(-n)/(c**2*n) + d**
2*log(x)/c**3 - d**2*log(c/d + x**n)/(c**3*n))/b**2, Eq(a, 0)), (2*c**2*d**2*n*log(x)/(2*b**2*c**5*n + 4*b**2*
c**4*d*n*x**n + 2*b**2*c**3*d**2*n*x**(2*n)) - 2*c**2*d**2*log(c/d + x**n)/(2*b**2*c**5*n + 4*b**2*c**4*d*n*x*
*n + 2*b**2*c**3*d**2*n*x**(2*n)) + 4*c*d**3*n*x**n*log(x)/(2*b**2*c**5*n + 4*b**2*c**4*d*n*x**n + 2*b**2*c**3
*d**2*n*x**(2*n)) - 4*c*d**3*x**n*log(c/d + x**n)/(2*b**2*c**5*n + 4*b**2*c**4*d*n*x**n + 2*b**2*c**3*d**2*n*x
**(2*n)) - 4*c*d**3*x**n/(2*b**2*c**5*n + 4*b**2*c**4*d*n*x**n + 2*b**2*c**3*d**2*n*x**(2*n)) + 2*d**4*n*x**(2
*n)*log(x)/(2*b**2*c**5*n + 4*b**2*c**4*d*n*x**n + 2*b**2*c**3*d**2*n*x**(2*n)) - 2*d**4*x**(2*n)*log(c/d + x*
*n)/(2*b**2*c**5*n + 4*b**2*c**4*d*n*x**n + 2*b**2*c**3*d**2*n*x**(2*n)) - 3*d**4*x**(2*n)/(2*b**2*c**5*n + 4*
b**2*c**4*d*n*x**n + 2*b**2*c**3*d**2*n*x**(2*n)), Eq(a, b*c/d)), ((-a**2/(a**4*n*x**n + a**3*b*n*x**(2*n)) +
2*a*b*x**n*log(x**(-n) + b/a)/(a**4*n*x**n + a**3*b*n*x**(2*n)) + 2*b**2*x**(2*n)*log(x**(-n) + b/a)/(a**4*n*x
**n + a**3*b*n*x**(2*n)) + 2*b**2*x**(2*n)/(a**4*n*x**n + a**3*b*n*x**(2*n)))/d, Eq(c, 0)), (log(x)/((a + b)**
2*(c + d)), Eq(n, 0)), ((a*n*log(x)/(a**3*n + a**2*b*n*x**n) - a*log(a/b + x**n)/(a**3*n + a**2*b*n*x**n) + b*
n*x**n*log(x)/(a**3*n + a**2*b*n*x**n) - b*x**n*log(a/b + x**n)/(a**3*n + a**2*b*n*x**n) - b*x**n/(a**3*n + a*
*2*b*n*x**n))/c, Eq(d, 0)), (a**3*d**2*n*log(x)/(a**5*c*d**2*n - 2*a**4*b*c**2*d*n + a**4*b*c*d**2*n*x**n + a*
*3*b**2*c**3*n - 2*a**3*b**2*c**2*d*n*x**n + a**2*b**3*c**3*n*x**n) - a**3*d**2*log(c/d + x**n)/(a**5*c*d**2*n
- 2*a**4*b*c**2*d*n + a**4*b*c*d**2*n*x**n + a**3*b**2*c**3*n - 2*a**3*b**2*c**2*d*n*x**n + a**2*b**3*c**3*n*
x**n) - 2*a**2*b*c*d*n*log(x)/(a**5*c*d**2*n - 2*a**4*b*c**2*d*n + a**4*b*c*d**2*n*x**n + a**3*b**2*c**3*n - 2
*a**3*b**2*c**2*d*n*x**n + a**2*b**3*c**3*n*x**n) + 2*a**2*b*c*d*log(a/b + x**n)/(a**5*c*d**2*n - 2*a**4*b*c**
2*d*n + a**4*b*c*d**2*n*x**n + a**3*b**2*c**3*n - 2*a**3*b**2*c**2*d*n*x**n + a**2*b**3*c**3*n*x**n) + a**2*b*
d**2*n*x**n*log(x)/(a**5*c*d**2*n - 2*a**4*b*c**2*d*n + a**4*b*c*d**2*n*x**n + a**3*b**2*c**3*n - 2*a**3*b**2*
c**2*d*n*x**n + a**2*b**3*c**3*n*x**n) - a**2*b*d**2*x**n*log(c/d + x**n)/(a**5*c*d**2*n - 2*a**4*b*c**2*d*n +
a**4*b*c*d**2*n*x**n + a**3*b**2*c**3*n - 2*a**3*b**2*c**2*d*n*x**n + a**2*b**3*c**3*n*x**n) + a*b**2*c**2*n*
log(x)/(a**5*c*d**2*n - 2*a**4*b*c**2*d*n + a**4*b*c*d**2*n*x**n + a**3*b**2*c**3*n - 2*a**3*b**2*c**2*d*n*x**
n + a**2*b**3*c**3*n*x**n) - a*b**2*c**2*log(a/b + x**n)/(a**5*c*d**2*n - 2*a**4*b*c**2*d*n + a**4*b*c*d**2*n*
x**n + a**3*b**2*c**3*n - 2*a**3*b**2*c**2*d*n*x**n + a**2*b**3*c**3*n*x**n) - 2*a*b**2*c*d*n*x**n*log(x)/(a**
5*c*d**2*n - 2*a**4*b*c**2*d*n + a**4*b*c*d**2*n*x**n + a**3*b**2*c**3*n - 2*a**3*b**2*c**2*d*n*x**n + a**2*b*
*3*c**3*n*x**n) + 2*a*b**2*c*d*x**n*log(a/b + x**n)/(a**5*c*d**2*n - 2*a**4*b*c**2*d*n + a**4*b*c*d**2*n*x**n
+ a**3*b**2*c**3*n - 2*a**3*b**2*c**2*d*n*x**n + a**2*b**3*c**3*n*x**n) + a*b**2*c*d*x**n/(a**5*c*d**2*n - 2*a
**4*b*c**2*d*n + a**4*b*c*d**2*n*x**n + a**3*b**2*c**3*n - 2*a**3*b**2*c**2*d*n*x**n + a**2*b**3*c**3*n*x**n)
+ b**3*c**2*n*x**n*log(x)/(a**5*c*d**2*n - 2*a**4*b*c**2*d*n + a**4*b*c*d**2*n*x**n + a**3*b**2*c**3*n - 2*a**
3*b**2*c**2*d*n*x**n + a**2*b**3*c**3*n*x**n) - b**3*c**2*x**n*log(a/b + x**n)/(a**5*c*d**2*n - 2*a**4*b*c**2*
d*n + a**4*b*c*d**2*n*x**n + a**3*b**2*c**3*n - 2*a**3*b**2*c**2*d*n*x**n + a**2*b**3*c**3*n*x**n) - b**3*c**2
*x**n/(a**5*c*d**2*n - 2*a**4*b*c**2*d*n + a**4*b*c*d**2*n*x**n + a**3*b**2*c**3*n - 2*a**3*b**2*c**2*d*n*x**n
+ a**2*b**3*c**3*n*x**n), True))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b x^{n} + a\right )}^{2}{\left (d x^{n} + c\right )} x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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