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

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

________________________________________________________________________________________

Rubi [A]  time = 0.11, antiderivative size = 101, normalized size of antiderivative = 1.00, 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} \begin {gather*} -\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 )} \end {gather*}

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 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]

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]]

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 = 0.33, size = 97, normalized size = 0.96 \begin {gather*} \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} \end {gather*}

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

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.13, size = 109, normalized size = 1.08 \begin {gather*} \frac {\left (2 a b d-b^2 c\right ) \log \left (a+b x^n\right )}{a^2 n (a d-b c)^2}+\frac {\log \left (x^n\right )}{a^2 c n}-\frac {d^2 \log \left (c+d x^n\right )}{c n (b c-a d)^2}-\frac {b}{a n (a d-b c) \left (a+b x^n\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[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^n]/(a^2*c*n) + ((-(b^2*c) + 2*a*b*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)

________________________________________________________________________________________

fricas [B]  time = 0.47, size = 224, normalized size = 2.22 \begin {gather*} \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 \relax (x) + {\left (a b^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{2}\right )} n \log \relax (x) - {\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 {gather*}

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)

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{{\left (b x^{n} + a\right )}^{2} {\left (d x^{n} + c\right )} x}\,{d x} \end {gather*}

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)

________________________________________________________________________________________

maple [A]  time = 0.10, size = 131, normalized size = 1.30 \begin {gather*} \frac {2 b d \ln \left (b \,x^{n}+a \right )}{\left (a d -b c \right )^{2} a n}-\frac {b^{2} c \ln \left (b \,x^{n}+a \right )}{\left (a d -b c \right )^{2} a^{2} n}-\frac {d^{2} \ln \left (d \,x^{n}+c \right )}{\left (a d -b c \right )^{2} c n}-\frac {b}{\left (a d -b c \right ) \left (b \,x^{n}+a \right ) a n}+\frac {\ln \left (x^{n}\right )}{a^{2} c n} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 0.67, size = 151, normalized size = 1.50 \begin {gather*} -\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 \relax (x)}{a^{2} c} \end {gather*}

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
)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{x\,{\left (a+b\,x^n\right )}^2\,\left (c+d\,x^n\right )} \,d x \end {gather*}

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]

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

________________________________________________________________________________________

sympy [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: HeuristicGCDFailed} \end {gather*}

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]

Exception raised: HeuristicGCDFailed

________________________________________________________________________________________