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

Optimal. Leaf size=72 \[ \frac{x (a d-b c (1-n)) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right )}{a^2 b n}+\frac{x (b c-a d)}{a b n \left (a+b x^n\right )} \]

[Out]

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

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Rubi [A]  time = 0.0307196, antiderivative size = 72, 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} \[ \frac{x (a d-b c (1-n)) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right )}{a^2 b n}+\frac{x (b c-a d)}{a b n \left (a+b x^n\right )} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x^n)/(a + b*x^n)^2,x]

[Out]

((b*c - a*d)*x)/(a*b*n*(a + b*x^n)) + ((a*d - b*c*(1 - n))*x*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), -((b*x^n
)/a)])/(a^2*b*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{c+d x^n}{\left (a+b x^n\right )^2} \, dx &=\frac{(b c-a d) x}{a b n \left (a+b x^n\right )}+\frac{(a d-b c (1-n)) \int \frac{1}{a+b x^n} \, dx}{a b n}\\ &=\frac{(b c-a d) x}{a b n \left (a+b x^n\right )}+\frac{(a d-b c (1-n)) x \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right )}{a^2 b n}\\ \end{align*}

Mathematica [A]  time = 0.0389247, size = 56, normalized size = 0.78 \[ \frac{x \left (\frac{d}{a+b x^n}-\frac{(a d+b c (n-1)) \, _2F_1\left (2,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right )}{a^2}\right )}{b-b n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.356, size = 0, normalized size = 0. \begin{align*} \int{\frac{c+d{x}^{n}}{ \left ( a+b{x}^{n} \right ) ^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [C]  time = 4.51472, size = 592, normalized size = 8.22 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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