3.1044 \(\int \frac{x^{-1+2 n}}{(a+b x^n)^3 (c+d x^n)} \, dx\)

Optimal. Leaf size=105 \[ \frac{a}{2 b n (b c-a d) \left (a+b x^n\right )^2}-\frac{c}{n (b c-a d)^2 \left (a+b x^n\right )}-\frac{c d \log \left (a+b x^n\right )}{n (b c-a d)^3}+\frac{c d \log \left (c+d x^n\right )}{n (b c-a d)^3} \]

[Out]

a/(2*b*(b*c - a*d)*n*(a + b*x^n)^2) - c/((b*c - a*d)^2*n*(a + b*x^n)) - (c*d*Log[a + b*x^n])/((b*c - a*d)^3*n)
+ (c*d*Log[c + d*x^n])/((b*c - a*d)^3*n)

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Rubi [A]  time = 0.0899089, antiderivative size = 105, 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, 77} \[ \frac{a}{2 b n (b c-a d) \left (a+b x^n\right )^2}-\frac{c}{n (b c-a d)^2 \left (a+b x^n\right )}-\frac{c d \log \left (a+b x^n\right )}{n (b c-a d)^3}+\frac{c d \log \left (c+d x^n\right )}{n (b c-a d)^3} \]

Antiderivative was successfully verified.

[In]

Int[x^(-1 + 2*n)/((a + b*x^n)^3*(c + d*x^n)),x]

[Out]

a/(2*b*(b*c - a*d)*n*(a + b*x^n)^2) - c/((b*c - a*d)^2*n*(a + b*x^n)) - (c*d*Log[a + b*x^n])/((b*c - a*d)^3*n)
+ (c*d*Log[c + d*x^n])/((b*c - a*d)^3*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 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

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

Mathematica [A]  time = 0.11534, size = 97, normalized size = 0.92 \[ \frac{\frac{a}{2 b (b c-a d) \left (a+b x^n\right )^2}-\frac{c}{(b c-a d)^2 \left (a+b x^n\right )}-\frac{c d \log \left (a+b x^n\right )}{(b c-a d)^3}+\frac{c d \log \left (c+d x^n\right )}{(b c-a d)^3}}{n} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(-1 + 2*n)/((a + b*x^n)^3*(c + d*x^n)),x]

[Out]

(a/(2*b*(b*c - a*d)*(a + b*x^n)^2) - c/((b*c - a*d)^2*(a + b*x^n)) - (c*d*Log[a + b*x^n])/(b*c - a*d)^3 + (c*d
*Log[c + d*x^n])/(b*c - a*d)^3)/n

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Maple [A]  time = 0.056, size = 203, normalized size = 1.9 \begin{align*}{\frac{1}{ \left ( a+b{{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}} \left ( -{\frac{bc{{\rm e}^{n\ln \left ( x \right ) }}}{ \left ({a}^{2}{d}^{2}-2\,abcd+{b}^{2}{c}^{2} \right ) n}}+{\frac{a \left ( -adb-{b}^{2}c \right ) }{ \left ( 2\,{a}^{2}{d}^{2}-4\,abcd+2\,{b}^{2}{c}^{2} \right ){b}^{2}n}} \right ) }+{\frac{cd\ln \left ( a+b{{\rm e}^{n\ln \left ( x \right ) }} \right ) }{n \left ({a}^{3}{d}^{3}-3\,{a}^{2}bc{d}^{2}+3\,a{b}^{2}{c}^{2}d-{b}^{3}{c}^{3} \right ) }}-{\frac{cd\ln \left ( c+d{{\rm e}^{n\ln \left ( x \right ) }} \right ) }{n \left ({a}^{3}{d}^{3}-3\,{a}^{2}bc{d}^{2}+3\,a{b}^{2}{c}^{2}d-{b}^{3}{c}^{3} \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(-1+2*n)/(a+b*x^n)^3/(c+d*x^n),x)

[Out]

(-b*c/(a^2*d^2-2*a*b*c*d+b^2*c^2)/n*exp(n*ln(x))+1/2*a*(-a*b*d-b^2*c)/(a^2*d^2-2*a*b*c*d+b^2*c^2)/b^2/n)/(a+b*
exp(n*ln(x)))^2+c*d/n/(a^3*d^3-3*a^2*b*c*d^2+3*a*b^2*c^2*d-b^3*c^3)*ln(a+b*exp(n*ln(x)))-c*d/n/(a^3*d^3-3*a^2*
b*c*d^2+3*a*b^2*c^2*d-b^3*c^3)*ln(c+d*exp(n*ln(x)))

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Maxima [B]  time = 0.96995, size = 328, normalized size = 3.12 \begin{align*} -\frac{c d \log \left (\frac{b x^{n} + a}{b}\right )}{b^{3} c^{3} n - 3 \, a b^{2} c^{2} d n + 3 \, a^{2} b c d^{2} n - a^{3} d^{3} n} + \frac{c d \log \left (\frac{d x^{n} + c}{d}\right )}{b^{3} c^{3} n - 3 \, a b^{2} c^{2} d n + 3 \, a^{2} b c d^{2} n - a^{3} d^{3} n} - \frac{2 \, b^{2} c x^{n} + a b c + a^{2} d}{2 \,{\left (a^{2} b^{3} c^{2} n - 2 \, a^{3} b^{2} c d n + a^{4} b d^{2} n +{\left (b^{5} c^{2} n - 2 \, a b^{4} c d n + a^{2} b^{3} d^{2} n\right )} x^{2 \, n} + 2 \,{\left (a b^{4} c^{2} n - 2 \, a^{2} b^{3} c d n + a^{3} b^{2} d^{2} n\right )} x^{n}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1+2*n)/(a+b*x^n)^3/(c+d*x^n),x, algorithm="maxima")

[Out]

-c*d*log((b*x^n + a)/b)/(b^3*c^3*n - 3*a*b^2*c^2*d*n + 3*a^2*b*c*d^2*n - a^3*d^3*n) + c*d*log((d*x^n + c)/d)/(
b^3*c^3*n - 3*a*b^2*c^2*d*n + 3*a^2*b*c*d^2*n - a^3*d^3*n) - 1/2*(2*b^2*c*x^n + a*b*c + a^2*d)/(a^2*b^3*c^2*n
- 2*a^3*b^2*c*d*n + a^4*b*d^2*n + (b^5*c^2*n - 2*a*b^4*c*d*n + a^2*b^3*d^2*n)*x^(2*n) + 2*(a*b^4*c^2*n - 2*a^2
*b^3*c*d*n + a^3*b^2*d^2*n)*x^n)

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Fricas [B]  time = 1.10648, size = 540, normalized size = 5.14 \begin{align*} -\frac{a b^{2} c^{2} - a^{3} d^{2} + 2 \,{\left (b^{3} c^{2} - a b^{2} c d\right )} x^{n} + 2 \,{\left (b^{3} c d x^{2 \, n} + 2 \, a b^{2} c d x^{n} + a^{2} b c d\right )} \log \left (b x^{n} + a\right ) - 2 \,{\left (b^{3} c d x^{2 \, n} + 2 \, a b^{2} c d x^{n} + a^{2} b c d\right )} \log \left (d x^{n} + c\right )}{2 \,{\left ({\left (b^{6} c^{3} - 3 \, a b^{5} c^{2} d + 3 \, a^{2} b^{4} c d^{2} - a^{3} b^{3} d^{3}\right )} n x^{2 \, n} + 2 \,{\left (a b^{5} c^{3} - 3 \, a^{2} b^{4} c^{2} d + 3 \, a^{3} b^{3} c d^{2} - a^{4} b^{2} d^{3}\right )} n x^{n} +{\left (a^{2} b^{4} c^{3} - 3 \, a^{3} b^{3} c^{2} d + 3 \, a^{4} b^{2} c d^{2} - a^{5} b d^{3}\right )} n\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1+2*n)/(a+b*x^n)^3/(c+d*x^n),x, algorithm="fricas")

[Out]

-1/2*(a*b^2*c^2 - a^3*d^2 + 2*(b^3*c^2 - a*b^2*c*d)*x^n + 2*(b^3*c*d*x^(2*n) + 2*a*b^2*c*d*x^n + a^2*b*c*d)*lo
g(b*x^n + a) - 2*(b^3*c*d*x^(2*n) + 2*a*b^2*c*d*x^n + a^2*b*c*d)*log(d*x^n + c))/((b^6*c^3 - 3*a*b^5*c^2*d + 3
*a^2*b^4*c*d^2 - a^3*b^3*d^3)*n*x^(2*n) + 2*(a*b^5*c^3 - 3*a^2*b^4*c^2*d + 3*a^3*b^3*c*d^2 - a^4*b^2*d^3)*n*x^
n + (a^2*b^4*c^3 - 3*a^3*b^3*c^2*d + 3*a^4*b^2*c*d^2 - a^5*b*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+2*n)/(a+b*x**n)**3/(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 )}^{3}{\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)^3/(c+d*x^n),x, algorithm="giac")

[Out]

integrate(x^(2*n - 1)/((b*x^n + a)^3*(d*x^n + c)), x)