\(\int (a+b x^n) (c+d x^n)^{-2-\frac {1}{n}} \, dx\) [323]

Optimal result

Integrand size = 23, antiderivative size = 58 \[ \int \left (a+b x^n\right ) \left (c+d x^n\right )^{-2-\frac {1}{n}} \, dx=\frac {x \left (a+b x^n\right ) \left (c+d x^n\right )^{-1-\frac {1}{n}}}{c (1+n)}+\frac {a n x \left (c+d x^n\right )^{-1/n}}{c^2 (1+n)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {386, 197} \[ \int \left (a+b x^n\right ) \left (c+d x^n\right )^{-2-\frac {1}{n}} \, dx=\frac {x \left (a+b x^n\right ) \left (c+d x^n\right )^{-\frac {1}{n}-1}}{c (n+1)}+\frac {a n x \left (c+d x^n\right )^{-1/n}}{c^2 (n+1)} \]

[In]

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

[Out]

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

Rule 197

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x*((a + b*x^n)^(p + 1)/a), x] /; FreeQ[{a, b, n, p}, x] &
& EqQ[1/n + p + 1, 0]

Rule 386

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

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

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 0.17 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.41 \[ \int \left (a+b x^n\right ) \left (c+d x^n\right )^{-2-\frac {1}{n}} \, dx=\frac {x \left (c+d x^n\right )^{-\frac {1+n}{n}} \left (b c x^n+a (1+n) \left (c+d x^n\right ) \left (1+\frac {d x^n}{c}\right )^{\frac {1}{n}} \operatorname {Hypergeometric2F1}\left (2+\frac {1}{n},\frac {1}{n},1+\frac {1}{n},-\frac {d x^n}{c}\right )\right )}{c^2 (1+n)} \]

[In]

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

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(198\) vs. \(2(58)=116\).

Time = 4.21 (sec) , antiderivative size = 199, normalized size of antiderivative = 3.43

method	result	size
parallelrisch	\(\frac {x \,x^{2 n} \left (c +d \,x^{n}\right )^{-\frac {1+2 n}{n}} a \,d^{2} n +x \,x^{2 n} \left (c +d \,x^{n}\right )^{-\frac {1+2 n}{n}} b c d +2 x \,x^{n} \left (c +d \,x^{n}\right )^{-\frac {1+2 n}{n}} a c d n +x \,x^{n} \left (c +d \,x^{n}\right )^{-\frac {1+2 n}{n}} a c d +x \,x^{n} \left (c +d \,x^{n}\right )^{-\frac {1+2 n}{n}} b \,c^{2}+x \left (c +d \,x^{n}\right )^{-\frac {1+2 n}{n}} a \,c^{2} n +x \left (c +d \,x^{n}\right )^{-\frac {1+2 n}{n}} a \,c^{2}}{c^{2} \left (1+n \right )}\)	\(199\)

[In]

int((a+b*x^n)*(c+d*x^n)^(-2-1/n),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.47 \[ \int \left (a+b x^n\right ) \left (c+d x^n\right )^{-2-\frac {1}{n}} \, dx=\frac {{\left (a d^{2} n + b c d\right )} x x^{2 \, n} + {\left (2 \, a c d n + b c^{2} + a c d\right )} x x^{n} + {\left (a c^{2} n + a c^{2}\right )} x}{{\left (c^{2} n + c^{2}\right )} {\left (d x^{n} + c\right )}^{\frac {2 \, n + 1}{n}}} \]

[In]

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

[Out]

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 311 vs. \(2 (48) = 96\).

Time = 2.21 (sec) , antiderivative size = 311, normalized size of antiderivative = 5.36 \[ \int \left (a+b x^n\right ) \left (c+d x^n\right )^{-2-\frac {1}{n}} \, dx=\frac {a c c^{\frac {1}{n}} c^{-2 - \frac {1}{n}} n \Gamma \left (\frac {1}{n}\right )}{c d^{\frac {1}{n}} n^{2} \left (\frac {c x^{- n}}{d} + 1\right )^{\frac {1}{n}} \Gamma \left (2 + \frac {1}{n}\right ) + d d^{\frac {1}{n}} n^{2} x^{n} \left (\frac {c x^{- n}}{d} + 1\right )^{\frac {1}{n}} \Gamma \left (2 + \frac {1}{n}\right )} + \frac {a c c^{\frac {1}{n}} c^{-2 - \frac {1}{n}} \Gamma \left (\frac {1}{n}\right )}{c d^{\frac {1}{n}} n^{2} \left (\frac {c x^{- n}}{d} + 1\right )^{\frac {1}{n}} \Gamma \left (2 + \frac {1}{n}\right ) + d d^{\frac {1}{n}} n^{2} x^{n} \left (\frac {c x^{- n}}{d} + 1\right )^{\frac {1}{n}} \Gamma \left (2 + \frac {1}{n}\right )} + \frac {a c^{\frac {1}{n}} c^{-2 - \frac {1}{n}} d n x^{n} \Gamma \left (\frac {1}{n}\right )}{c d^{\frac {1}{n}} n^{2} \left (\frac {c x^{- n}}{d} + 1\right )^{\frac {1}{n}} \Gamma \left (2 + \frac {1}{n}\right ) + d d^{\frac {1}{n}} n^{2} x^{n} \left (\frac {c x^{- n}}{d} + 1\right )^{\frac {1}{n}} \Gamma \left (2 + \frac {1}{n}\right )} + \frac {b c^{-2 - \frac {1}{n}} c^{1 + \frac {1}{n}} d^{-1 - \frac {1}{n}} \left (\frac {c x^{- n}}{d} + 1\right )^{-1 - \frac {1}{n}} \Gamma \left (1 + \frac {1}{n}\right )}{n \Gamma \left (2 + \frac {1}{n}\right )} \]

[In]

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

[Out]

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

Maxima [F]

\[ \int \left (a+b x^n\right ) \left (c+d x^n\right )^{-2-\frac {1}{n}} \, dx=\int { {\left (b x^{n} + a\right )} {\left (d x^{n} + c\right )}^{-\frac {1}{n} - 2} \,d x } \]

[In]

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

[Out]

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

Giac [F(-2)]

Exception generated. \[ \int \left (a+b x^n\right ) \left (c+d x^n\right )^{-2-\frac {1}{n}} \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Unable to divide, perhaps due to rounding error%%%{1,[0,0,2,2,1,1,0,1]%%%}+%%%{1,[0,0,2,1,1,1,0,1]%%%}+%%%{
1,[0,0,2,1,

Mupad [F(-1)]

Timed out. \[ \int \left (a+b x^n\right ) \left (c+d x^n\right )^{-2-\frac {1}{n}} \, dx=\int \frac {a+b\,x^n}{{\left (c+d\,x^n\right )}^{\frac {1}{n}+2}} \,d x \]

[In]

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

[Out]

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