3.1042 \(\int \frac{x^{-1+2 n}}{(a+b x^n) (c+d x^n)} \, dx\)

Optimal. Leaf size=54 \[ \frac{c \log \left (c+d x^n\right )}{d n (b c-a d)}-\frac{a \log \left (a+b x^n\right )}{b n (b c-a d)} \]

[Out]

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

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Rubi [A]  time = 0.0543653, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {446, 72} \[ \frac{c \log \left (c+d x^n\right )}{d n (b c-a d)}-\frac{a \log \left (a+b x^n\right )}{b n (b c-a d)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0350836, size = 44, normalized size = 0.81 \[ -\frac{a d \log \left (a+b x^n\right )-b c \log \left (c+d x^n\right )}{b^2 c d n-a b d^2 n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.02, size = 59, normalized size = 1.1 \begin{align*}{\frac{a\ln \left ( a+b{{\rm e}^{n\ln \left ( x \right ) }} \right ) }{ \left ( ad-bc \right ) bn}}-{\frac{c\ln \left ( c+d{{\rm e}^{n\ln \left ( x \right ) }} \right ) }{dn \left ( ad-bc \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a/(a*d-b*c)/b/n*ln(a+b*exp(n*ln(x)))-c/d/n/(a*d-b*c)*ln(c+d*exp(n*ln(x)))

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Maxima [A]  time = 0.945042, size = 81, normalized size = 1.5 \begin{align*} -\frac{a \log \left (\frac{b x^{n} + a}{b}\right )}{b^{2} c n - a b d n} + \frac{c \log \left (\frac{d x^{n} + c}{d}\right )}{b c d n - a d^{2} n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.07062, size = 92, normalized size = 1.7 \begin{align*} -\frac{a d \log \left (b x^{n} + a\right ) - b c \log \left (d x^{n} + c\right )}{{\left (b^{2} c d - a b d^{2}\right )} n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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