\(\int \frac {(d+e x^n)^3}{a+c x^{2 n}} \, dx\) [42]

Optimal result

Integrand size = 21, antiderivative size = 141 \[ \int \frac {\left (d+e x^n\right )^3}{a+c x^{2 n}} \, dx=\frac {3 d e^2 x}{c}+\frac {e^3 x^{1+n}}{c (1+n)}+\frac {d \left (c d^2-3 a e^2\right ) x \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2 n},\frac {1}{2} \left (2+\frac {1}{n}\right ),-\frac {c x^{2 n}}{a}\right )}{a c}+\frac {e \left (3 c d^2-a e^2\right ) x^{1+n} \operatorname {Hypergeometric2F1}\left (1,\frac {1+n}{2 n},\frac {1}{2} \left (3+\frac {1}{n}\right ),-\frac {c x^{2 n}}{a}\right )}{a c (1+n)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.10 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {1439, 1432, 251, 371} \[ \int \frac {\left (d+e x^n\right )^3}{a+c x^{2 n}} \, dx=\frac {e x^{n+1} \left (3 c d^2-a e^2\right ) \operatorname {Hypergeometric2F1}\left (1,\frac {n+1}{2 n},\frac {1}{2} \left (3+\frac {1}{n}\right ),-\frac {c x^{2 n}}{a}\right )}{a c (n+1)}+\frac {d x \left (c d^2-3 a e^2\right ) \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2 n},\frac {1}{2} \left (2+\frac {1}{n}\right ),-\frac {c x^{2 n}}{a}\right )}{a c}+\frac {3 d e^2 x}{c}+\frac {e^3 x^{n+1}}{c (n+1)} \]

[In]

Int[(d + e*x^n)^3/(a + c*x^(2*n)),x]

[Out]

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

Rule 251

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 371

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*((c*x)^(m + 1)/(c*(m + 1)))*Hyperg
eometric2F1[-p, (m + 1)/n, (m + 1)/n + 1, (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rule 1432

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 1439

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]

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

Mathematica [A] (verified)

Time = 0.56 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.90 \[ \int \frac {\left (d+e x^n\right )^3}{a+c x^{2 n}} \, dx=\frac {x \left (d \left (c d^2-3 a e^2\right ) (1+n) \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2 n},\frac {1}{2} \left (2+\frac {1}{n}\right ),-\frac {c x^{2 n}}{a}\right )+e \left (a e \left (3 d (1+n)+e x^n\right )+\left (3 c d^2-a e^2\right ) x^n \operatorname {Hypergeometric2F1}\left (1,\frac {1+n}{2 n},\frac {1}{2} \left (3+\frac {1}{n}\right ),-\frac {c x^{2 n}}{a}\right )\right )\right )}{a c (1+n)} \]

[In]

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

[Out]

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

Maple [F]

\[\int \frac {\left (d +e \,x^{n}\right )^{3}}{a +c \,x^{2 n}}d x\]

[In]

int((d+e*x^n)^3/(a+c*x^(2*n)),x)

[Out]

int((d+e*x^n)^3/(a+c*x^(2*n)),x)

Fricas [F]

\[ \int \frac {\left (d+e x^n\right )^3}{a+c x^{2 n}} \, dx=\int { \frac {{\left (e x^{n} + d\right )}^{3}}{c x^{2 \, n} + a} \,d x } \]

[In]

integrate((d+e*x^n)^3/(a+c*x^(2*n)),x, algorithm="fricas")

[Out]

integral((e^3*x^(3*n) + 3*d*e^2*x^(2*n) + 3*d^2*e*x^n + d^3)/(c*x^(2*n) + a), x)

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 4.01 (sec) , antiderivative size = 466, normalized size of antiderivative = 3.30 \[ \int \frac {\left (d+e x^n\right )^3}{a+c x^{2 n}} \, dx=\frac {a^{\frac {1}{2 n}} a^{-1 - \frac {1}{2 n}} d^{3} x \Phi \left (\frac {c x^{2 n} e^{i \pi }}{a}, 1, \frac {1}{2 n}\right ) \Gamma \left (\frac {1}{2 n}\right )}{4 n^{2} \Gamma \left (1 + \frac {1}{2 n}\right )} + \frac {3 a^{- \frac {5}{2} - \frac {1}{2 n}} a^{\frac {3}{2} + \frac {1}{2 n}} e^{3} x^{3 n + 1} \Phi \left (\frac {c x^{2 n} e^{i \pi }}{a}, 1, \frac {3}{2} + \frac {1}{2 n}\right ) \Gamma \left (\frac {3}{2} + \frac {1}{2 n}\right )}{4 n \Gamma \left (\frac {5}{2} + \frac {1}{2 n}\right )} + \frac {a^{- \frac {5}{2} - \frac {1}{2 n}} a^{\frac {3}{2} + \frac {1}{2 n}} e^{3} x^{3 n + 1} \Phi \left (\frac {c x^{2 n} e^{i \pi }}{a}, 1, \frac {3}{2} + \frac {1}{2 n}\right ) \Gamma \left (\frac {3}{2} + \frac {1}{2 n}\right )}{4 n^{2} \Gamma \left (\frac {5}{2} + \frac {1}{2 n}\right )} + \frac {3 a^{- \frac {3}{2} - \frac {1}{2 n}} a^{\frac {1}{2} + \frac {1}{2 n}} d^{2} e x^{n + 1} \Phi \left (\frac {c x^{2 n} e^{i \pi }}{a}, 1, \frac {1}{2} + \frac {1}{2 n}\right ) \Gamma \left (\frac {1}{2} + \frac {1}{2 n}\right )}{4 n \Gamma \left (\frac {3}{2} + \frac {1}{2 n}\right )} + \frac {3 a^{- \frac {3}{2} - \frac {1}{2 n}} a^{\frac {1}{2} + \frac {1}{2 n}} d^{2} e x^{n + 1} \Phi \left (\frac {c x^{2 n} e^{i \pi }}{a}, 1, \frac {1}{2} + \frac {1}{2 n}\right ) \Gamma \left (\frac {1}{2} + \frac {1}{2 n}\right )}{4 n^{2} \Gamma \left (\frac {3}{2} + \frac {1}{2 n}\right )} - \frac {3 a^{- \frac {1}{2 n}} a^{1 + \frac {1}{2 n}} c^{\frac {1}{2 n}} c^{-1 - \frac {1}{2 n}} d e^{2} x \Phi \left (\frac {a x^{- 2 n} e^{i \pi }}{c}, 1, \frac {e^{i \pi }}{2 n}\right ) \Gamma \left (\frac {1}{2 n}\right )}{4 a n^{2} \Gamma \left (1 + \frac {1}{2 n}\right )} \]

[In]

integrate((d+e*x**n)**3/(a+c*x**(2*n)),x)

[Out]

a**(1/(2*n))*a**(-1 - 1/(2*n))*d**3*x*lerchphi(c*x**(2*n)*exp_polar(I*pi)/a, 1, 1/(2*n))*gamma(1/(2*n))/(4*n**
2*gamma(1 + 1/(2*n))) + 3*a**(-5/2 - 1/(2*n))*a**(3/2 + 1/(2*n))*e**3*x**(3*n + 1)*lerchphi(c*x**(2*n)*exp_pol
ar(I*pi)/a, 1, 3/2 + 1/(2*n))*gamma(3/2 + 1/(2*n))/(4*n*gamma(5/2 + 1/(2*n))) + a**(-5/2 - 1/(2*n))*a**(3/2 +
1/(2*n))*e**3*x**(3*n + 1)*lerchphi(c*x**(2*n)*exp_polar(I*pi)/a, 1, 3/2 + 1/(2*n))*gamma(3/2 + 1/(2*n))/(4*n*
*2*gamma(5/2 + 1/(2*n))) + 3*a**(-3/2 - 1/(2*n))*a**(1/2 + 1/(2*n))*d**2*e*x**(n + 1)*lerchphi(c*x**(2*n)*exp_
polar(I*pi)/a, 1, 1/2 + 1/(2*n))*gamma(1/2 + 1/(2*n))/(4*n*gamma(3/2 + 1/(2*n))) + 3*a**(-3/2 - 1/(2*n))*a**(1
/2 + 1/(2*n))*d**2*e*x**(n + 1)*lerchphi(c*x**(2*n)*exp_polar(I*pi)/a, 1, 1/2 + 1/(2*n))*gamma(1/2 + 1/(2*n))/
(4*n**2*gamma(3/2 + 1/(2*n))) - 3*a**(1 + 1/(2*n))*c**(1/(2*n))*c**(-1 - 1/(2*n))*d*e**2*x*lerchphi(a*exp_pola
r(I*pi)/(c*x**(2*n)), 1, exp_polar(I*pi)/(2*n))*gamma(1/(2*n))/(4*a*a**(1/(2*n))*n**2*gamma(1 + 1/(2*n)))

Maxima [F]

\[ \int \frac {\left (d+e x^n\right )^3}{a+c x^{2 n}} \, dx=\int { \frac {{\left (e x^{n} + d\right )}^{3}}{c x^{2 \, n} + a} \,d x } \]

[In]

integrate((d+e*x^n)^3/(a+c*x^(2*n)),x, algorithm="maxima")

[Out]

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

Giac [F]

\[ \int \frac {\left (d+e x^n\right )^3}{a+c x^{2 n}} \, dx=\int { \frac {{\left (e x^{n} + d\right )}^{3}}{c x^{2 \, n} + a} \,d x } \]

[In]

integrate((d+e*x^n)^3/(a+c*x^(2*n)),x, algorithm="giac")

[Out]

integrate((e*x^n + d)^3/(c*x^(2*n) + a), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (d+e x^n\right )^3}{a+c x^{2 n}} \, dx=\int \frac {{\left (d+e\,x^n\right )}^3}{a+c\,x^{2\,n}} \,d x \]

[In]

int((d + e*x^n)^3/(a + c*x^(2*n)),x)

[Out]

int((d + e*x^n)^3/(a + c*x^(2*n)), x)