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

Optimal. Leaf size=127 \[ \frac{b x \left (c+d x^n\right )^{-\frac{1-n}{n}}}{a n (b c-a d) \left (a+b x^n\right )}-\frac{x \left (c+d x^n\right )^{-1/n} (a d n+b c (1-n)) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{(b c-a d) x^n}{a \left (d x^n+c\right )}\right )}{a^2 n (b c-a d)} \]

[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))

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Rubi [A]  time = 0.0522504, antiderivative size = 127, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {382, 379} \[ \frac{b x \left (c+d x^n\right )^{-\frac{1-n}{n}}}{a n (b c-a d) \left (a+b x^n\right )}-\frac{x \left (c+d x^n\right )^{-1/n} (a d n+b c (1-n)) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{(b c-a d) x^n}{a \left (d x^n+c\right )}\right )}{a^2 n (b c-a d)} \]

Antiderivative was successfully verified.

[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 382

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] && EqQ[
n*(p + q + 2) + 1, 0] && (LtQ[p, -1] ||  !LtQ[q, -1]) && NeQ[p, -1]

Rule 379

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(a^p*x*Hypergeometric2F1[1/
n, -p, 1 + 1/n, -(((b*c - a*d)*x^n)/(a*(c + d*x^n)))])/(c^(p + 1)*(c + d*x^n)^(1/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]

Rubi steps

\begin{align*} \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-(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 [B]  time = 49.8391, size = 1070, normalized size = 8.43 \[ \frac{c^2 (2 n+1) (3 n+1) x \left (b x^n+a\right ) \left (d x^n+c\right )^{-1/n} \left (\frac{d x^n}{c}+1\right ) \text{Gamma}\left (2+\frac{1}{n}\right ) \text{Gamma}\left (3+\frac{1}{n}\right ) \left (\frac{2 (b c-a d) n \left (d x^n+c\right ) \, _2F_1\left (2,3;3+\frac{1}{n};\frac{(b c-a d) x^n}{c \left (b x^n+a\right )}\right ) x^n}{\left (b x^n+a\right ) \text{Gamma}\left (3+\frac{1}{n}\right )}+\frac{c \left (d n x^n+c+c n\right ) \, _2F_1\left (1,2;2+\frac{1}{n};\frac{(b c-a d) x^n}{c \left (b x^n+a\right )}\right )}{\text{Gamma}\left (2+\frac{1}{n}\right )}\right )}{-c d (1-n) (2 n+1) (3 n+1) \left (b x^n+a\right )^2 \left (2 (b c-a d) n \left (d x^n+c\right ) \text{Gamma}\left (2+\frac{1}{n}\right ) \, _2F_1\left (2,3;3+\frac{1}{n};\frac{(b c-a d) x^n}{c \left (b x^n+a\right )}\right ) x^n+c \left (b x^n+a\right ) \left (d n x^n+c+c n\right ) \text{Gamma}\left (3+\frac{1}{n}\right ) \, _2F_1\left (1,2;2+\frac{1}{n};\frac{(b c-a d) x^n}{c \left (b x^n+a\right )}\right )\right ) x^n-2 b c n (2 n+1) (3 n+1) \left (b x^n+a\right ) \left (d x^n+c\right ) \left (2 (b c-a d) n \left (d x^n+c\right ) \text{Gamma}\left (2+\frac{1}{n}\right ) \, _2F_1\left (2,3;3+\frac{1}{n};\frac{(b c-a d) x^n}{c \left (b x^n+a\right )}\right ) x^n+c \left (b x^n+a\right ) \left (d n x^n+c+c n\right ) \text{Gamma}\left (3+\frac{1}{n}\right ) \, _2F_1\left (1,2;2+\frac{1}{n};\frac{(b c-a d) x^n}{c \left (b x^n+a\right )}\right )\right ) x^n+n^2 \left (d x^n+c\right ) \left (2 c d (b c-a d) (2 n+1) (3 n+1) \left (b x^n+a\right )^2 \text{Gamma}\left (2+\frac{1}{n}\right ) \, _2F_1\left (2,3;3+\frac{1}{n};\frac{(b c-a d) x^n}{c \left (b x^n+a\right )}\right ) x^n-2 b c (b c-a d) (2 n+1) (3 n+1) \left (b x^n+a\right ) \left (d x^n+c\right ) \text{Gamma}\left (2+\frac{1}{n}\right ) \, _2F_1\left (2,3;3+\frac{1}{n};\frac{(b c-a d) x^n}{c \left (b x^n+a\right )}\right ) x^n+12 a (b c-a d)^2 n (2 n+1) \left (d x^n+c\right ) \text{Gamma}\left (2+\frac{1}{n}\right ) \, _2F_1\left (3,4;4+\frac{1}{n};\frac{(b c-a d) x^n}{c \left (b x^n+a\right )}\right ) x^n+c^2 d (2 n+1) (3 n+1) \left (b x^n+a\right )^3 \text{Gamma}\left (3+\frac{1}{n}\right ) \, _2F_1\left (1,2;2+\frac{1}{n};\frac{(b c-a d) x^n}{c \left (b x^n+a\right )}\right )+2 c (b c-a d) (2 n+1) (3 n+1) \left (b x^n+a\right )^2 \left (d x^n+c\right ) \text{Gamma}\left (2+\frac{1}{n}\right ) \, _2F_1\left (2,3;3+\frac{1}{n};\frac{(b c-a d) x^n}{c \left (b x^n+a\right )}\right )+2 a c (b c-a d) (3 n+1) \left (b x^n+a\right ) \left (d n x^n+c+c n\right ) \text{Gamma}\left (3+\frac{1}{n}\right ) \, _2F_1\left (2,3;3+\frac{1}{n};\frac{(b c-a d) x^n}{c \left (b x^n+a\right )}\right )\right ) x^n+c (2 n+1) (3 n+1) \left (b x^n+a\right )^2 \left (d x^n+c\right ) \left (2 (b c-a d) n \left (d x^n+c\right ) \text{Gamma}\left (2+\frac{1}{n}\right ) \, _2F_1\left (2,3;3+\frac{1}{n};\frac{(b c-a d) x^n}{c \left (b x^n+a\right )}\right ) x^n+c \left (b x^n+a\right ) \left (d n x^n+c+c n\right ) \text{Gamma}\left (3+\frac{1}{n}\right ) \, _2F_1\left (1,2;2+\frac{1}{n};\frac{(b c-a d) x^n}{c \left (b x^n+a\right )}\right )\right )} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(c^2*(1 + 2*n)*(1 + 3*n)*x*(a + b*x^n)*(1 + (d*x^n)/c)*Gamma[2 + n^(-1)]*Gamma[3 + n^(-1)]*((c*(c + c*n + d*n*
x^n)*Hypergeometric2F1[1, 2, 2 + n^(-1), ((b*c - a*d)*x^n)/(c*(a + b*x^n))])/Gamma[2 + n^(-1)] + (2*(b*c - a*d
)*n*x^n*(c + d*x^n)*Hypergeometric2F1[2, 3, 3 + n^(-1), ((b*c - a*d)*x^n)/(c*(a + b*x^n))])/((a + b*x^n)*Gamma
[3 + n^(-1)])))/((c + d*x^n)^n^(-1)*(-(c*d*(1 - n)*(1 + 2*n)*(1 + 3*n)*x^n*(a + b*x^n)^2*(c*(a + b*x^n)*(c + c
*n + d*n*x^n)*Gamma[3 + n^(-1)]*Hypergeometric2F1[1, 2, 2 + n^(-1), ((b*c - a*d)*x^n)/(c*(a + b*x^n))] + 2*(b*
c - a*d)*n*x^n*(c + d*x^n)*Gamma[2 + n^(-1)]*Hypergeometric2F1[2, 3, 3 + n^(-1), ((b*c - a*d)*x^n)/(c*(a + b*x
^n))])) - 2*b*c*n*(1 + 2*n)*(1 + 3*n)*x^n*(a + b*x^n)*(c + d*x^n)*(c*(a + b*x^n)*(c + c*n + d*n*x^n)*Gamma[3 +
n^(-1)]*Hypergeometric2F1[1, 2, 2 + n^(-1), ((b*c - a*d)*x^n)/(c*(a + b*x^n))] + 2*(b*c - a*d)*n*x^n*(c + d*x
^n)*Gamma[2 + n^(-1)]*Hypergeometric2F1[2, 3, 3 + n^(-1), ((b*c - a*d)*x^n)/(c*(a + b*x^n))]) + c*(1 + 2*n)*(1
+ 3*n)*(a + b*x^n)^2*(c + d*x^n)*(c*(a + b*x^n)*(c + c*n + d*n*x^n)*Gamma[3 + n^(-1)]*Hypergeometric2F1[1, 2,
2 + n^(-1), ((b*c - a*d)*x^n)/(c*(a + b*x^n))] + 2*(b*c - a*d)*n*x^n*(c + d*x^n)*Gamma[2 + n^(-1)]*Hypergeome
tric2F1[2, 3, 3 + n^(-1), ((b*c - a*d)*x^n)/(c*(a + b*x^n))]) + n^2*x^n*(c + d*x^n)*(c^2*d*(1 + 2*n)*(1 + 3*n)
*(a + b*x^n)^3*Gamma[3 + n^(-1)]*Hypergeometric2F1[1, 2, 2 + n^(-1), ((b*c - a*d)*x^n)/(c*(a + b*x^n))] + 2*c*
d*(b*c - a*d)*(1 + 2*n)*(1 + 3*n)*x^n*(a + b*x^n)^2*Gamma[2 + n^(-1)]*Hypergeometric2F1[2, 3, 3 + n^(-1), ((b*
c - a*d)*x^n)/(c*(a + b*x^n))] - 2*b*c*(b*c - a*d)*(1 + 2*n)*(1 + 3*n)*x^n*(a + b*x^n)*(c + d*x^n)*Gamma[2 + n
^(-1)]*Hypergeometric2F1[2, 3, 3 + n^(-1), ((b*c - a*d)*x^n)/(c*(a + b*x^n))] + 2*c*(b*c - a*d)*(1 + 2*n)*(1 +
3*n)*(a + b*x^n)^2*(c + d*x^n)*Gamma[2 + n^(-1)]*Hypergeometric2F1[2, 3, 3 + n^(-1), ((b*c - a*d)*x^n)/(c*(a
+ b*x^n))] + 2*a*c*(b*c - a*d)*(1 + 3*n)*(a + b*x^n)*(c + c*n + d*n*x^n)*Gamma[3 + n^(-1)]*Hypergeometric2F1[2
, 3, 3 + n^(-1), ((b*c - a*d)*x^n)/(c*(a + b*x^n))] + 12*a*(b*c - a*d)^2*n*(1 + 2*n)*x^n*(c + d*x^n)*Gamma[2 +
n^(-1)]*Hypergeometric2F1[3, 4, 4 + n^(-1), ((b*c - a*d)*x^n)/(c*(a + b*x^n))])))

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Maple [F]  time = 0.682, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( a+b{x}^{n} \right ) ^{2}\sqrt [n]{c+d{x}^{n}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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

Verification of antiderivative is not currently implemented for this CAS.

[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)

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

Verification of antiderivative is not currently implemented for this CAS.

[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)

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[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)