3.1048 \(\int \frac{x^{-1+3 n}}{(a+b x^n) (c+d x^n)} \, dx\)

Optimal. Leaf size=71 \[ \frac{a^2 \log \left (a+b x^n\right )}{b^2 n (b c-a d)}-\frac{c^2 \log \left (c+d x^n\right )}{d^2 n (b c-a d)}+\frac{x^n}{b d n} \]

[Out]

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

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Rubi [A]  time = 0.0691811, antiderivative size = 71, 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} \[ \frac{a^2 \log \left (a+b x^n\right )}{b^2 n (b c-a d)}-\frac{c^2 \log \left (c+d x^n\right )}{d^2 n (b c-a d)}+\frac{x^n}{b d n} \]

Antiderivative was successfully verified.

[In]

Int[x^(-1 + 3*n)/((a + b*x^n)*(c + d*x^n)),x]

[Out]

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

Mathematica [A]  time = 0.0665049, size = 66, normalized size = 0.93 \[ \frac{\frac{a^2 \log \left (a+b x^n\right )}{b^2 (b c-a d)}-\frac{c^2 \log \left (c+d x^n\right )}{d^2 (b c-a d)}+\frac{x^n}{b d}}{n} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(-1 + 3*n)/((a + b*x^n)*(c + d*x^n)),x]

[Out]

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

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Maple [A]  time = 0.026, size = 78, normalized size = 1.1 \begin{align*}{\frac{{{\rm e}^{n\ln \left ( x \right ) }}}{bdn}}+{\frac{{c}^{2}\ln \left ( c+d{{\rm e}^{n\ln \left ( x \right ) }} \right ) }{{d}^{2}n \left ( ad-bc \right ) }}-{\frac{{a}^{2}\ln \left ( a+b{{\rm e}^{n\ln \left ( x \right ) }} \right ) }{ \left ( ad-bc \right ){b}^{2}n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(-1+3*n)/(a+b*x^n)/(c+d*x^n),x)

[Out]

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

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Maxima [A]  time = 0.937863, size = 109, normalized size = 1.54 \begin{align*} \frac{a^{2} \log \left (\frac{b x^{n} + a}{b}\right )}{b^{3} c n - a b^{2} d n} - \frac{c^{2} \log \left (\frac{d x^{n} + c}{d}\right )}{b c d^{2} n - a d^{3} n} + \frac{x^{n}}{b d n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1+3*n)/(a+b*x^n)/(c+d*x^n),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.09306, size = 142, normalized size = 2. \begin{align*} \frac{a^{2} d^{2} \log \left (b x^{n} + a\right ) - b^{2} c^{2} \log \left (d x^{n} + c\right ) +{\left (b^{2} c d - a b d^{2}\right )} x^{n}}{{\left (b^{3} c d^{2} - a b^{2} d^{3}\right )} n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1+3*n)/(a+b*x^n)/(c+d*x^n),x, algorithm="fricas")

[Out]

(a^2*d^2*log(b*x^n + a) - b^2*c^2*log(d*x^n + c) + (b^2*c*d - a*b*d^2)*x^n)/((b^3*c*d^2 - a*b^2*d^3)*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+3*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^{3 \, 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+3*n)/(a+b*x^n)/(c+d*x^n),x, algorithm="giac")

[Out]

integrate(x^(3*n - 1)/((b*x^n + a)*(d*x^n + c)), x)