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

Optimal. Leaf size=73 \[ \frac{x (b c-a d (1-n)) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{d x^n}{c}\right )}{c^2 d n}-\frac{x (b c-a d)}{c d n \left (c+d x^n\right )} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0308483, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {385, 245} \[ \frac{x (b c-a d (1-n)) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{d x^n}{c}\right )}{c^2 d n}-\frac{x (b c-a d)}{c d n \left (c+d x^n\right )} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^n)/(c + d*x^n)^2,x]

[Out]

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

Rule 385

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

Rule 245

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*x*Hypergeometric2F1[-p, 1/n, 1/n + 1, -((b*x^n)/a)],
x] /; FreeQ[{a, b, n, p}, x] &&  !IGtQ[p, 0] &&  !IntegerQ[1/n] &&  !ILtQ[Simplify[1/n + p], 0] && (IntegerQ[p
] || GtQ[a, 0])

Rubi steps

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

Mathematica [A]  time = 0.0366805, size = 56, normalized size = 0.77 \[ \frac{x \left (\frac{b}{c+d x^n}-\frac{(a d (n-1)+b c) \, _2F_1\left (2,\frac{1}{n};1+\frac{1}{n};-\frac{d x^n}{c}\right )}{c^2}\right )}{d-d n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [F]  time = 0.36, size = 0, normalized size = 0. \begin{align*} \int{\frac{a+b{x}^{n}}{ \left ( c+d{x}^{n} \right ) ^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [C]  time = 4.6078, size = 592, normalized size = 8.11 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*(n*x*lerchphi(d*x**n*exp_polar(I*pi)/c, 1, 1/n)*gamma(1/n)/(c*(c*n**3*gamma(1 + 1/n) + d*n**3*x**n*gamma(1 +
1/n))) + n*x*gamma(1/n)/(c*(c*n**3*gamma(1 + 1/n) + d*n**3*x**n*gamma(1 + 1/n))) - x*lerchphi(d*x**n*exp_pola
r(I*pi)/c, 1, 1/n)*gamma(1/n)/(c*(c*n**3*gamma(1 + 1/n) + d*n**3*x**n*gamma(1 + 1/n))) + d*n*x*x**n*lerchphi(d
*x**n*exp_polar(I*pi)/c, 1, 1/n)*gamma(1/n)/(c**2*(c*n**3*gamma(1 + 1/n) + d*n**3*x**n*gamma(1 + 1/n))) - d*x*
x**n*lerchphi(d*x**n*exp_polar(I*pi)/c, 1, 1/n)*gamma(1/n)/(c**2*(c*n**3*gamma(1 + 1/n) + d*n**3*x**n*gamma(1
+ 1/n)))) + b*(n**2*x*x**n*gamma(1 + 1/n)/(c*(c*n**3*gamma(2 + 1/n) + d*n**3*x**n*gamma(2 + 1/n))) - n*x*x**n*
lerchphi(d*x**n*exp_polar(I*pi)/c, 1, 1 + 1/n)*gamma(1 + 1/n)/(c*(c*n**3*gamma(2 + 1/n) + d*n**3*x**n*gamma(2
+ 1/n))) + n*x*x**n*gamma(1 + 1/n)/(c*(c*n**3*gamma(2 + 1/n) + d*n**3*x**n*gamma(2 + 1/n))) - x*x**n*lerchphi(
d*x**n*exp_polar(I*pi)/c, 1, 1 + 1/n)*gamma(1 + 1/n)/(c*(c*n**3*gamma(2 + 1/n) + d*n**3*x**n*gamma(2 + 1/n)))
- d*n*x*x**(2*n)*lerchphi(d*x**n*exp_polar(I*pi)/c, 1, 1 + 1/n)*gamma(1 + 1/n)/(c**2*(c*n**3*gamma(2 + 1/n) +
d*n**3*x**n*gamma(2 + 1/n))) - d*x*x**(2*n)*lerchphi(d*x**n*exp_polar(I*pi)/c, 1, 1 + 1/n)*gamma(1 + 1/n)/(c**
2*(c*n**3*gamma(2 + 1/n) + d*n**3*x**n*gamma(2 + 1/n))))

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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