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

Optimal. Leaf size=107 \[ \frac{x \left (c d^2-a e^2\right ) \, _2F_1\left (1,\frac{1}{2 n};\frac{1}{2} \left (2+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )}{a c}+\frac{2 d e x^{n+1} \, _2F_1\left (1,\frac{n+1}{2 n};\frac{1}{2} \left (3+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )}{a (n+1)}+\frac{e^2 x}{c} \]

[Out]

(e^2*x)/c + ((c*d^2 - a*e^2)*x*Hypergeometric2F1[1, 1/(2*n), (2 + n^(-1))/2, -((c*x^(2*n))/a)])/(a*c) + (2*d*e
*x^(1 + n)*Hypergeometric2F1[1, (1 + n)/(2*n), (3 + n^(-1))/2, -((c*x^(2*n))/a)])/(a*(1 + n))

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Rubi [A]  time = 0.097461, antiderivative size = 107, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {1425, 1418, 245, 364} \[ \frac{x \left (c d^2-a e^2\right ) \, _2F_1\left (1,\frac{1}{2 n};\frac{1}{2} \left (2+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )}{a c}+\frac{2 d e x^{n+1} \, _2F_1\left (1,\frac{n+1}{2 n};\frac{1}{2} \left (3+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )}{a (n+1)}+\frac{e^2 x}{c} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x^n)^2/(a + c*x^(2*n)),x]

[Out]

(e^2*x)/c + ((c*d^2 - a*e^2)*x*Hypergeometric2F1[1, 1/(2*n), (2 + n^(-1))/2, -((c*x^(2*n))/a)])/(a*c) + (2*d*e
*x^(1 + n)*Hypergeometric2F1[1, (1 + n)/(2*n), (3 + n^(-1))/2, -((c*x^(2*n))/a)])/(a*(1 + n))

Rule 1425

Int[((d_) + (e_.)*(x_)^(n_))^(q_)/((a_) + (c_.)*(x_)^(n2_)), x_Symbol] :> Int[ExpandIntegrand[(d + e*x^n)^q/(a
+ c*x^(2*n)), x], x] /; FreeQ[{a, c, d, e, n}, x] && EqQ[n2, 2*n] && NeQ[c*d^2 + a*e^2, 0] && IntegerQ[q]

Rule 1418

Int[((d_) + (e_.)*(x_)^(n_))/((a_) + (c_.)*(x_)^(n2_)), x_Symbol] :> Dist[d, Int[1/(a + c*x^(2*n)), x], x] + D
ist[e, Int[x^n/(a + c*x^(2*n)), x], x] /; FreeQ[{a, c, d, e, n}, x] && EqQ[n2, 2*n] && NeQ[c*d^2 + a*e^2, 0] &
& (PosQ[a*c] ||  !IntegerQ[n])

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])

Rule 364

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

Rubi steps

\begin{align*} \int \frac{\left (d+e x^n\right )^2}{a+c x^{2 n}} \, dx &=\int \left (\frac{e^2}{c}+\frac{c d^2-a e^2+2 c d e x^n}{c \left (a+c x^{2 n}\right )}\right ) \, dx\\ &=\frac{e^2 x}{c}+\frac{\int \frac{c d^2-a e^2+2 c d e x^n}{a+c x^{2 n}} \, dx}{c}\\ &=\frac{e^2 x}{c}+(2 d e) \int \frac{x^n}{a+c x^{2 n}} \, dx+\frac{\left (c d^2-a e^2\right ) \int \frac{1}{a+c x^{2 n}} \, dx}{c}\\ &=\frac{e^2 x}{c}+\frac{\left (c d^2-a e^2\right ) x \, _2F_1\left (1,\frac{1}{2 n};\frac{1}{2} \left (2+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )}{a c}+\frac{2 d e x^{1+n} \, _2F_1\left (1,\frac{1+n}{2 n};\frac{1}{2} \left (3+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )}{a (1+n)}\\ \end{align*}

Mathematica [A]  time = 0.150999, size = 107, normalized size = 1. \[ \frac{x \left (c d^2-a e^2\right ) \, _2F_1\left (1,\frac{1}{2 n};\frac{1}{2} \left (2+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )}{a c}+\frac{2 d e x^{n+1} \, _2F_1\left (1,\frac{n+1}{2 n};\frac{1}{2} \left (3+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )}{a (n+1)}+\frac{e^2 x}{c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(e^2*x)/c + ((c*d^2 - a*e^2)*x*Hypergeometric2F1[1, 1/(2*n), (2 + n^(-1))/2, -((c*x^(2*n))/a)])/(a*c) + (2*d*e
*x^(1 + n)*Hypergeometric2F1[1, (1 + n)/(2*n), (3 + n^(-1))/2, -((c*x^(2*n))/a)])/(a*(1 + n))

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Maple [F]  time = 0.062, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( d+e{x}^{n} \right ) ^{2}}{a+c{x}^{2\,n}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [C]  time = 7.57686, size = 207, normalized size = 1.93 \begin{align*} - \frac{e^{2} x \Phi \left (\frac{a x^{- 2 n} e^{i \pi }}{c}, 1, \frac{e^{i \pi }}{2 n}\right ) \Gamma \left (\frac{1}{2 n}\right )}{4 c n^{2} \Gamma \left (1 + \frac{1}{2 n}\right )} + \frac{d^{2} x \Phi \left (\frac{c x^{2 n} e^{i \pi }}{a}, 1, \frac{1}{2 n}\right ) \Gamma \left (\frac{1}{2 n}\right )}{4 a n^{2} \Gamma \left (1 + \frac{1}{2 n}\right )} + \frac{d e x x^{n} \Phi \left (\frac{c x^{2 n} e^{i \pi }}{a}, 1, \frac{1}{2} + \frac{1}{2 n}\right ) \Gamma \left (\frac{1}{2} + \frac{1}{2 n}\right )}{2 a n \Gamma \left (\frac{3}{2} + \frac{1}{2 n}\right )} + \frac{d e x x^{n} \Phi \left (\frac{c x^{2 n} e^{i \pi }}{a}, 1, \frac{1}{2} + \frac{1}{2 n}\right ) \Gamma \left (\frac{1}{2} + \frac{1}{2 n}\right )}{2 a n^{2} \Gamma \left (\frac{3}{2} + \frac{1}{2 n}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-e**2*x*lerchphi(a*x**(-2*n)*exp_polar(I*pi)/c, 1, exp_polar(I*pi)/(2*n))*gamma(1/(2*n))/(4*c*n**2*gamma(1 + 1
/(2*n))) + d**2*x*lerchphi(c*x**(2*n)*exp_polar(I*pi)/a, 1, 1/(2*n))*gamma(1/(2*n))/(4*a*n**2*gamma(1 + 1/(2*n
))) + d*e*x*x**n*lerchphi(c*x**(2*n)*exp_polar(I*pi)/a, 1, 1/2 + 1/(2*n))*gamma(1/2 + 1/(2*n))/(2*a*n*gamma(3/
2 + 1/(2*n))) + d*e*x*x**n*lerchphi(c*x**(2*n)*exp_polar(I*pi)/a, 1, 1/2 + 1/(2*n))*gamma(1/2 + 1/(2*n))/(2*a*
n**2*gamma(3/2 + 1/(2*n)))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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