\(\int \frac {(c+d x^n)^{-1/n}}{(a+b x^n)^2} \, dx\) [334]

Optimal result

Integrand size = 23, antiderivative size = 127 \[ \int \frac {\left (c+d x^n\right )^{-1/n}}{\left (a+b x^n\right )^2} \, dx=\frac {b x \left (c+d x^n\right )^{-\frac {1-n}{n}}}{a (b c-a d) n \left (a+b x^n\right )}-\frac {(b c (1-n)+a d n) x \left (c+d x^n\right )^{-1/n} \operatorname {Hypergeometric2F1}\left (1,\frac {1}{n},1+\frac {1}{n},-\frac {(b c-a d) x^n}{a \left (c+d x^n\right )}\right )}{a^2 (b c-a d) n} \]

[Out]

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

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {390, 387} \[ \int \frac {\left (c+d x^n\right )^{-1/n}}{\left (a+b x^n\right )^2} \, dx=\frac {b x \left (c+d x^n\right )^{-\frac {1-n}{n}}}{a n (b c-a d) \left (a+b x^n\right )}-\frac {x \left (c+d x^n\right )^{-1/n} (a d n+b c (1-n)) \operatorname {Hypergeometric2F1}\left (1,\frac {1}{n},1+\frac {1}{n},-\frac {(b c-a d) x^n}{a \left (d x^n+c\right )}\right )}{a^2 n (b c-a d)} \]

[In]

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

[Out]

(b*x)/(a*(b*c - a*d)*n*(a + b*x^n)*(c + d*x^n)^((1 - n)/n)) - ((b*c*(1 - n) + a*d*n)*x*Hypergeometric2F1[1, n^
(-1), 1 + n^(-1), -(((b*c - a*d)*x^n)/(a*(c + d*x^n)))])/(a^2*(b*c - a*d)*n*(c + d*x^n)^n^(-1))

Rule 387

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[a^p*(x/(c^(p + 1)*(c + d*x^
n)^(1/n)))*Hypergeometric2F1[1/n, -p, 1 + 1/n, (-(b*c - a*d))*(x^n/(a*(c + d*x^n)))], x] /; FreeQ[{a, b, c, d,
n, q}, x] && NeQ[b*c - a*d, 0] && EqQ[n*(p + q + 1) + 1, 0] && ILtQ[p, 0]

Rule 390

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(-b)*x*(a + b*x^n)^(p + 1)*
((c + d*x^n)^(q + 1)/(a*n*(p + 1)*(b*c - a*d))), x] + Dist[(b*c + n*(p + 1)*(b*c - a*d))/(a*n*(p + 1)*(b*c - a
*d)), Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, n, q}, x] && NeQ[b*c - a*d, 0] && Eq
Q[n*(p + q + 2) + 1, 0] && (LtQ[p, -1] ||  !LtQ[q, -1]) && NeQ[p, -1]

Rubi steps \begin{align*} \text {integral}& = \frac {b x \left (c+d x^n\right )^{-\frac {1-n}{n}}}{a (b c-a d) n \left (a+b x^n\right )}-\frac {(b c-(b c-a d) n) \int \frac {\left (c+d x^n\right )^{-1/n}}{a+b x^n} \, dx}{a (b c-a d) n} \\ & = \frac {b x \left (c+d x^n\right )^{-\frac {1-n}{n}}}{a (b c-a d) n \left (a+b x^n\right )}-\frac {(b c (1-n)+a d n) x \left (c+d x^n\right )^{-1/n} \, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};-\frac {(b c-a d) x^n}{a \left (c+d x^n\right )}\right )}{a^2 (b c-a d) n} \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 5 in optimal.

Time = 8.56 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.67 \[ \int \frac {\left (c+d x^n\right )^{-1/n}}{\left (a+b x^n\right )^2} \, dx=\frac {x \left (c+d x^n\right )^{\frac {-1+n}{n}} \left (b x^n \Phi \left (\frac {(-b c+a d) x^n}{a \left (c+d x^n\right )},1,1+\frac {1}{n}\right )+\left (a n+b (-1+n) x^n\right ) \Phi \left (\frac {(-b c+a d) x^n}{a \left (c+d x^n\right )},1,\frac {1}{n}\right )\right )}{n \left (a+b x^n\right ) \left (-b (b c-a d) x^{2 n} \Phi \left (\frac {(-b c+a d) x^n}{a \left (c+d x^n\right )},1,1+\frac {1}{n}\right )+a \left (c+d x^n\right ) \left (n \left (a+b x^n\right )-b x^n \Phi \left (\frac {(-b c+a d) x^n}{a \left (c+d x^n\right )},1,\frac {1}{n}\right )\right )\right )} \]

[In]

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

[Out]

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

Maple [F]

\[\int \frac {\left (c +d \,x^{n}\right )^{-\frac {1}{n}}}{\left (a +b \,x^{n}\right )^{2}}d x\]

[In]

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

[Out]

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

Fricas [F]

\[ \int \frac {\left (c+d x^n\right )^{-1/n}}{\left (a+b x^n\right )^2} \, dx=\int { \frac {1}{{\left (b x^{n} + a\right )}^{2} {\left (d x^{n} + c\right )}^{\left (\frac {1}{n}\right )}} \,d x } \]

[In]

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

[Out]

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

Sympy [F(-2)]

Exception generated. \[ \int \frac {\left (c+d x^n\right )^{-1/n}}{\left (a+b x^n\right )^2} \, dx=\text {Exception raised: HeuristicGCDFailed} \]

[In]

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

[Out]

Exception raised: HeuristicGCDFailed >> no luck

Maxima [F]

\[ \int \frac {\left (c+d x^n\right )^{-1/n}}{\left (a+b x^n\right )^2} \, dx=\int { \frac {1}{{\left (b x^{n} + a\right )}^{2} {\left (d x^{n} + c\right )}^{\left (\frac {1}{n}\right )}} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int \frac {\left (c+d x^n\right )^{-1/n}}{\left (a+b x^n\right )^2} \, dx=\int { \frac {1}{{\left (b x^{n} + a\right )}^{2} {\left (d x^{n} + c\right )}^{\left (\frac {1}{n}\right )}} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (c+d x^n\right )^{-1/n}}{\left (a+b x^n\right )^2} \, dx=\int \frac {1}{{\left (a+b\,x^n\right )}^2\,{\left (c+d\,x^n\right )}^{1/n}} \,d x \]

[In]

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

[Out]

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