\(\int \frac {(c x)^{-1-\frac {3 n}{2}}}{a+b x^n} \, dx\) [665]

Optimal result

Integrand size = 21, antiderivative size = 100 \[ \int \frac {(c x)^{-1-\frac {3 n}{2}}}{a+b x^n} \, dx=-\frac {2 (c x)^{-3 n/2}}{3 a c n}+\frac {2 b x^n (c x)^{-3 n/2}}{a^2 c n}-\frac {2 b^{3/2} x^{3 n/2} (c x)^{-3 n/2} \arctan \left (\frac {\sqrt {a} x^{-n/2}}{\sqrt {b}}\right )}{a^{5/2} c n} \] Output:

-2/3/a/c/n/((c*x)^(3/2*n))+2*b*x^n/a^2/c/n/((c*x)^(3/2*n))-2*b^(3/2)*x^(3/ 
2*n)*arctan(a^(1/2)/b^(1/2)/(x^(1/2*n)))/a^(5/2)/c/n/((c*x)^(3/2*n))
 

Mathematica [C] (verified)

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

Time = 0.01 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.39 \[ \int \frac {(c x)^{-1-\frac {3 n}{2}}}{a+b x^n} \, dx=-\frac {2 x (c x)^{-1-\frac {3 n}{2}} \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},1,-\frac {1}{2},-\frac {b x^n}{a}\right )}{3 a n} \] Input:

Integrate[(c*x)^(-1 - (3*n)/2)/(a + b*x^n),x]
 

Output:

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

Rubi [A] (verified)

Time = 0.36 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.90, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {887, 886, 868, 772, 262, 218}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {(c x)^{-\frac {3 n}{2}-1}}{a+b x^n} \, dx\)

\(\Big \downarrow \) 887

\(\displaystyle \frac {x^{3 n/2} (c x)^{-3 n/2} \int \frac {x^{-\frac {3 n}{2}-1}}{b x^n+a}dx}{c}\)

\(\Big \downarrow \) 886

\(\displaystyle \frac {x^{3 n/2} (c x)^{-3 n/2} \left (-\frac {b \int \frac {x^{-\frac {n}{2}-1}}{b x^n+a}dx}{a}-\frac {2 x^{-3 n/2}}{3 a n}\right )}{c}\)

\(\Big \downarrow \) 868

\(\displaystyle \frac {x^{3 n/2} (c x)^{-3 n/2} \left (\frac {2 b \int \frac {1}{b x^n+a}dx^{-n/2}}{a n}-\frac {2 x^{-3 n/2}}{3 a n}\right )}{c}\)

\(\Big \downarrow \) 772

\(\displaystyle \frac {x^{3 n/2} (c x)^{-3 n/2} \left (\frac {2 b \int \frac {x^{-n}}{a x^{-n}+b}dx^{-n/2}}{a n}-\frac {2 x^{-3 n/2}}{3 a n}\right )}{c}\)

\(\Big \downarrow \) 262

\(\displaystyle \frac {x^{3 n/2} (c x)^{-3 n/2} \left (\frac {2 b \left (\frac {x^{-n/2}}{a}-\frac {b \int \frac {1}{a x^{-n}+b}dx^{-n/2}}{a}\right )}{a n}-\frac {2 x^{-3 n/2}}{3 a n}\right )}{c}\)

\(\Big \downarrow \) 218

\(\displaystyle \frac {x^{3 n/2} (c x)^{-3 n/2} \left (\frac {2 b \left (\frac {x^{-n/2}}{a}-\frac {\sqrt {b} \arctan \left (\frac {\sqrt {a} x^{-n/2}}{\sqrt {b}}\right )}{a^{3/2}}\right )}{a n}-\frac {2 x^{-3 n/2}}{3 a n}\right )}{c}\)

Input:

Int[(c*x)^(-1 - (3*n)/2)/(a + b*x^n),x]
 

Output:

(x^((3*n)/2)*(-2/(3*a*n*x^((3*n)/2)) + (2*b*(1/(a*x^(n/2)) - (Sqrt[b]*ArcT 
an[Sqrt[a]/(Sqrt[b]*x^(n/2))])/a^(3/2)))/(a*n)))/(c*(c*x)^((3*n)/2))
 

Defintions of rubi rules used

rule 218

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R 
t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
 

rule 262

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x) 
^(m - 1)*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] - Simp[a*c^2*((m - 1)/ 
(b*(m + 2*p + 1)))   Int[(c*x)^(m - 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b 
, c, p}, x] && GtQ[m, 2 - 1] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[a, b, c 
, 2, m, p, x]
 

rule 772

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

rule 868

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[1/(m + 1) 
 Subst[Int[(a + b*x^Simplify[n/(m + 1)])^p, x], x, x^(m + 1)], x] /; FreeQ[ 
{a, b, m, n, p}, x] && IntegerQ[Simplify[n/(m + 1)]] &&  !IntegerQ[n]
 

rule 886

Int[(x_)^(m_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[x^(m + 1)/(a*(m + 
 1)), x] - Simp[b/a   Int[x^Simplify[m + n]/(a + b*x^n), x], x] /; FreeQ[{a 
, b, m, n}, x] && FractionQ[Simplify[(m + 1)/n]] && SumSimplerQ[m, n]
 

rule 887

Int[((c_)*(x_))^(m_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[c^IntPart[ 
m]*((c*x)^FracPart[m]/x^FracPart[m])   Int[x^m/(a + b*x^n), x], x] /; FreeQ 
[{a, b, c, m, n}, x] && FractionQ[Simplify[(m + 1)/n]] && (SumSimplerQ[m, n 
] || SumSimplerQ[m, -n])
 

Maple [F]

\[\int \frac {\left (c x \right )^{-1-\frac {3 n}{2}}}{a +b \,x^{n}}d x\]

Input:

int((c*x)^(-1-3/2*n)/(a+b*x^n),x)
 

Output:

int((c*x)^(-1-3/2*n)/(a+b*x^n),x)
 

Fricas [A] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 504, normalized size of antiderivative = 5.04 \[ \int \frac {(c x)^{-1-\frac {3 n}{2}}}{a+b x^n} \, dx =\text {Too large to display} \] Input:

integrate((c*x)^(-1-3/2*n)/(a+b*x^n),x, algorithm="fricas")
 

Output:

[1/3*(3*b*c^(-n - 2/3)*sqrt(-b*c^(-n - 2/3)/a)*log(-(2*a^2*b*c^(-n - 2/3)* 
x^(4/3)*e^(-2/3*(3*n + 2)*log(c) - 2/3*(3*n + 2)*log(x)) - a^3*x^2*e^(-(3* 
n + 2)*log(c) - (3*n + 2)*log(x)) - 2*a*b^2*c^(-2*n - 4/3)*x^(2/3)*e^(-1/3 
*(3*n + 2)*log(c) - 1/3*(3*n + 2)*log(x)) + b^3*c^(-3*n - 2) - 2*(a^2*b*c^ 
(-n - 2/3)*x*e^(-1/2*(3*n + 2)*log(c) - 1/2*(3*n + 2)*log(x)) - a^3*x^(5/3 
)*e^(-5/6*(3*n + 2)*log(c) - 5/6*(3*n + 2)*log(x)) - a*b^2*c^(-2*n - 4/3)* 
x^(1/3)*e^(-1/6*(3*n + 2)*log(c) - 1/6*(3*n + 2)*log(x)))*sqrt(-b*c^(-n - 
2/3)/a))/(a^3*x^2*e^(-(3*n + 2)*log(c) - (3*n + 2)*log(x)) + b^3*c^(-3*n - 
 2))) + 6*b*c^(-n - 2/3)*x^(1/3)*e^(-1/6*(3*n + 2)*log(c) - 1/6*(3*n + 2)* 
log(x)) - 2*a*x*e^(-1/2*(3*n + 2)*log(c) - 1/2*(3*n + 2)*log(x)))/(a^2*n), 
 -2/3*(3*b*c^(-n - 2/3)*sqrt(b*c^(-n - 2/3)/a)*arctan(a*sqrt(b*c^(-n - 2/3 
)/a)*x^(1/3)*e^(-1/6*(3*n + 2)*log(c) - 1/6*(3*n + 2)*log(x))/(b*c^(-n - 2 
/3))) - 3*b*c^(-n - 2/3)*x^(1/3)*e^(-1/6*(3*n + 2)*log(c) - 1/6*(3*n + 2)* 
log(x)) + a*x*e^(-1/2*(3*n + 2)*log(c) - 1/2*(3*n + 2)*log(x)))/(a^2*n)]
 

Sympy [A] (verification not implemented)

Time = 1.33 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.87 \[ \int \frac {(c x)^{-1-\frac {3 n}{2}}}{a+b x^n} \, dx=- \frac {2 c^{- \frac {3 n}{2} - 1} x^{- \frac {3 n}{2}}}{3 a n} + \frac {2 b c^{- \frac {3 n}{2} - 1} x^{- \frac {n}{2}}}{a^{2} n} + \frac {2 b^{\frac {3}{2}} c^{- \frac {3 n}{2} - 1} \operatorname {atan}{\left (\frac {\sqrt {b} x^{\frac {n}{2}}}{\sqrt {a}} \right )}}{a^{\frac {5}{2}} n} \] Input:

integrate((c*x)**(-1-3/2*n)/(a+b*x**n),x)
 

Output:

-2*c**(-3*n/2 - 1)/(3*a*n*x**(3*n/2)) + 2*b*c**(-3*n/2 - 1)/(a**2*n*x**(n/ 
2)) + 2*b**(3/2)*c**(-3*n/2 - 1)*atan(sqrt(b)*x**(n/2)/sqrt(a))/(a**(5/2)* 
n)
 

Maxima [F]

\[ \int \frac {(c x)^{-1-\frac {3 n}{2}}}{a+b x^n} \, dx=\int { \frac {\left (c x\right )^{-\frac {3}{2} \, n - 1}}{b x^{n} + a} \,d x } \] Input:

integrate((c*x)^(-1-3/2*n)/(a+b*x^n),x, algorithm="maxima")
 

Output:

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

Giac [F]

\[ \int \frac {(c x)^{-1-\frac {3 n}{2}}}{a+b x^n} \, dx=\int { \frac {\left (c x\right )^{-\frac {3}{2} \, n - 1}}{b x^{n} + a} \,d x } \] Input:

integrate((c*x)^(-1-3/2*n)/(a+b*x^n),x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {(c x)^{-1-\frac {3 n}{2}}}{a+b x^n} \, dx=\int \frac {1}{{\left (c\,x\right )}^{\frac {3\,n}{2}+1}\,\left (a+b\,x^n\right )} \,d x \] Input:

int(1/((c*x)^((3*n)/2 + 1)*(a + b*x^n)),x)
 

Output:

int(1/((c*x)^((3*n)/2 + 1)*(a + b*x^n)), x)
 

Reduce [F]

\[ \int \frac {(c x)^{-1-\frac {3 n}{2}}}{a+b x^n} \, dx=\frac {\int \frac {1}{x^{\frac {5 n}{2}} b x +x^{\frac {3 n}{2}} a x}d x}{c^{\frac {3 n}{2}} c} \] Input:

int((c*x)^(-1-3/2*n)/(a+b*x^n),x)
                                                                                    
                                                                                    
 

Output:

int(1/(x**((5*n)/2)*b*x + x**((3*n)/2)*a*x),x)/(c**((3*n)/2)*c)