\(\int \frac {1}{x^3 (a+\frac {b}{c+d x^n})^{3/2}} \, dx\) [333]

Optimal result

Integrand size = 21, antiderivative size = 110 \[ \int \frac {1}{x^3 \left (a+\frac {b}{c+d x^n}\right )^{3/2}} \, dx=-\frac {c \sqrt {1+\frac {a d x^n}{b+a c}} \operatorname {AppellF1}\left (-\frac {2}{n},-\frac {3}{2},\frac {3}{2},-\frac {2-n}{n},-\frac {d x^n}{c},-\frac {a d x^n}{b+a c}\right )}{2 (b+a c) x^2 \sqrt {1+\frac {d x^n}{c}} \sqrt {a+\frac {b}{c+d x^n}}} \] Output:

-1/2*c*(1+a*d*x^n/(a*c+b))^(1/2)*AppellF1(-2/n,-3/2,3/2,-(2-n)/n,-d*x^n/c, 
-a*d*x^n/(a*c+b))/(a*c+b)/x^2/(1+d*x^n/c)^(1/2)/(a+b/(c+d*x^n))^(1/2)
                                                                                    
                                                                                    
 

Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(229\) vs. \(2(110)=220\).

Time = 2.54 (sec) , antiderivative size = 229, normalized size of antiderivative = 2.08 \[ \int \frac {1}{x^3 \left (a+\frac {b}{c+d x^n}\right )^{3/2}} \, dx=\frac {\sqrt {\frac {b+a c+a d x^n}{c+d x^n}} \left (-4 b \left (c+d x^n\right )+\frac {\sqrt {\frac {b+a c+a d x^n}{b+a c}} \sqrt {1+\frac {d x^n}{c}} \left (-c (-2+n) (-4 b+a c n) \operatorname {AppellF1}\left (-\frac {2}{n},\frac {1}{2},\frac {1}{2},\frac {-2+n}{n},-\frac {d x^n}{c},-\frac {a d x^n}{b+a c}\right )+2 d (b (-4+n)+a c n) x^n \operatorname {AppellF1}\left (\frac {-2+n}{n},\frac {1}{2},\frac {1}{2},2-\frac {2}{n},-\frac {d x^n}{c},-\frac {a d x^n}{b+a c}\right )\right )}{-2+n}\right )}{2 a (b+a c) n x^2 \left (b+a \left (c+d x^n\right )\right )} \] Input:

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

Output:

(Sqrt[(b + a*c + a*d*x^n)/(c + d*x^n)]*(-4*b*(c + d*x^n) + (Sqrt[(b + a*c 
+ a*d*x^n)/(b + a*c)]*Sqrt[1 + (d*x^n)/c]*(-(c*(-2 + n)*(-4*b + a*c*n)*App 
ellF1[-2/n, 1/2, 1/2, (-2 + n)/n, -((d*x^n)/c), -((a*d*x^n)/(b + a*c))]) + 
 2*d*(b*(-4 + n) + a*c*n)*x^n*AppellF1[(-2 + n)/n, 1/2, 1/2, 2 - 2/n, -((d 
*x^n)/c), -((a*d*x^n)/(b + a*c))]))/(-2 + n)))/(2*a*(b + a*c)*n*x^2*(b + a 
*(c + d*x^n)))
 

Rubi [A] (verified)

Time = 0.71 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.07, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {2057, 2058, 1013, 1013, 1012}

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.

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

\(\Big \downarrow \) 2057

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

\(\Big \downarrow \) 2058

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

\(\Big \downarrow \) 1013

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

\(\Big \downarrow \) 1013

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

\(\Big \downarrow \) 1012

\(\displaystyle -\frac {c \sqrt {\frac {a d x^n}{a c+b}+1} \operatorname {AppellF1}\left (-\frac {2}{n},-\frac {3}{2},\frac {3}{2},-\frac {2-n}{n},-\frac {d x^n}{c},-\frac {a d x^n}{b+a c}\right )}{2 x^2 (a c+b) \sqrt {\frac {d x^n}{c}+1} \sqrt {\frac {a c+a d x^n+b}{c+d x^n}}}\)

Input:

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

Output:

-1/2*(c*Sqrt[1 + (a*d*x^n)/(b + a*c)]*AppellF1[-2/n, -3/2, 3/2, -((2 - n)/ 
n), -((d*x^n)/c), -((a*d*x^n)/(b + a*c))])/((b + a*c)*x^2*Sqrt[(b + a*c + 
a*d*x^n)/(c + d*x^n)]*Sqrt[1 + (d*x^n)/c])
 

Defintions of rubi rules used

rule 1012

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_ 
))^(q_), x_Symbol] :> Simp[a^p*c^q*((e*x)^(m + 1)/(e*(m + 1)))*AppellF1[(m 
+ 1)/n, -p, -q, 1 + (m + 1)/n, (-b)*(x^n/a), (-d)*(x^n/c)], x] /; FreeQ[{a, 
 b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, n 
 - 1] && (IntegerQ[p] || GtQ[a, 0]) && (IntegerQ[q] || GtQ[c, 0])
 

rule 1013

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_ 
))^(q_), x_Symbol] :> Simp[a^IntPart[p]*((a + b*x^n)^FracPart[p]/(1 + b*(x^ 
n/a))^FracPart[p])   Int[(e*x)^m*(1 + b*(x^n/a))^p*(c + d*x^n)^q, x], x] /; 
 FreeQ[{a, b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] & 
& NeQ[m, n - 1] &&  !(IntegerQ[p] || GtQ[a, 0])
 

rule 2057

Int[(u_.)*((a_) + (b_.)/((c_) + (d_.)*(x_)^(n_)))^(p_), x_Symbol] :> Int[u* 
((b + a*c + a*d*x^n)/(c + d*x^n))^p, x] /; FreeQ[{a, b, c, d, n, p}, x]
 

rule 2058

Int[(u_.)*((e_.)*((a_.) + (b_.)*(x_)^(n_.))^(q_.)*((c_) + (d_.)*(x_)^(n_))^ 
(r_.))^(p_), x_Symbol] :> Simp[Simp[(e*(a + b*x^n)^q*(c + d*x^n)^r)^p/((a + 
 b*x^n)^(p*q)*(c + d*x^n)^(p*r))]   Int[u*(a + b*x^n)^(p*q)*(c + d*x^n)^(p* 
r), x], x] /; FreeQ[{a, b, c, d, e, n, p, q, r}, x]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F(-2)]

Exception generated. \[ \int \frac {1}{x^3 \left (a+\frac {b}{c+d x^n}\right )^{3/2}} \, dx=\text {Exception raised: TypeError} \] Input:

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

Output:

Exception raised: TypeError >>  Error detected within library code:   inte 
grate: implementation incomplete (constant residues)
 

Sympy [F]

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

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

Output:

Integral(1/(x**3*((a*c + a*d*x**n + b)/(c + d*x**n))**(3/2)), x)
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {1}{x^3 \left (a+\frac {b}{c+d x^n}\right )^{3/2}} \, dx=\text {too large to display} \] Input:

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

Output:

( - 4*sqrt(x**n*d + c)*sqrt(x**n*a*d + a*c + b)*c - 8*x**n*int((x**(2*n)*s 
qrt(x**n*d + c)*sqrt(x**n*a*d + a*c + b))/(8*x**(3*n)*a**3*c*d**3*x**3 - x 
**(3*n)*a**2*b*d**3*n*x**3 + 4*x**(3*n)*a**2*b*d**3*x**3 + 24*x**(2*n)*a** 
3*c**2*d**2*x**3 - 3*x**(2*n)*a**2*b*c*d**2*n*x**3 + 28*x**(2*n)*a**2*b*c* 
d**2*x**3 - 2*x**(2*n)*a*b**2*d**2*n*x**3 + 8*x**(2*n)*a*b**2*d**2*x**3 + 
24*x**n*a**3*c**3*d*x**3 - 3*x**n*a**2*b*c**2*d*n*x**3 + 44*x**n*a**2*b*c* 
*2*d*x**3 - 4*x**n*a*b**2*c*d*n*x**3 + 24*x**n*a*b**2*c*d*x**3 - x**n*b**3 
*d*n*x**3 + 4*x**n*b**3*d*x**3 + 8*a**3*c**4*x**3 - a**2*b*c**3*n*x**3 + 2 
0*a**2*b*c**3*x**3 - 2*a*b**2*c**2*n*x**3 + 16*a*b**2*c**2*x**3 - b**3*c*n 
*x**3 + 4*b**3*c*x**3),x)*a**2*b*c*d**3*n*x**2 + 32*x**n*int((x**(2*n)*sqr 
t(x**n*d + c)*sqrt(x**n*a*d + a*c + b))/(8*x**(3*n)*a**3*c*d**3*x**3 - x** 
(3*n)*a**2*b*d**3*n*x**3 + 4*x**(3*n)*a**2*b*d**3*x**3 + 24*x**(2*n)*a**3* 
c**2*d**2*x**3 - 3*x**(2*n)*a**2*b*c*d**2*n*x**3 + 28*x**(2*n)*a**2*b*c*d* 
*2*x**3 - 2*x**(2*n)*a*b**2*d**2*n*x**3 + 8*x**(2*n)*a*b**2*d**2*x**3 + 24 
*x**n*a**3*c**3*d*x**3 - 3*x**n*a**2*b*c**2*d*n*x**3 + 44*x**n*a**2*b*c**2 
*d*x**3 - 4*x**n*a*b**2*c*d*n*x**3 + 24*x**n*a*b**2*c*d*x**3 - x**n*b**3*d 
*n*x**3 + 4*x**n*b**3*d*x**3 + 8*a**3*c**4*x**3 - a**2*b*c**3*n*x**3 + 20* 
a**2*b*c**3*x**3 - 2*a*b**2*c**2*n*x**3 + 16*a*b**2*c**2*x**3 - b**3*c*n*x 
**3 + 4*b**3*c*x**3),x)*a**2*b*c*d**3*x**2 + x**n*int((x**(2*n)*sqrt(x**n* 
d + c)*sqrt(x**n*a*d + a*c + b))/(8*x**(3*n)*a**3*c*d**3*x**3 - x**(3*n...