\(\int \frac {(a+\frac {b}{c+d x^n})^{3/2}}{x^3} \, dx\) [323]

Optimal result

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

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

Mathematica [A] (warning: unable to verify)

Time = 2.60 (sec) , antiderivative size = 220, normalized size of antiderivative = 2.00 \[ \int \frac {\left (a+\frac {b}{c+d x^n}\right )^{3/2}}{x^3} \, dx=\frac {\sqrt {\frac {b+a c+a d x^n}{c+d x^n}} \left (4 b+\frac {\sqrt {\frac {b+a c+a d x^n}{b+a c}} \sqrt {1+\frac {d x^n}{c}} \left (-\left ((b+a c) (-2+n) (a c n+b (4+n)) \operatorname {AppellF1}\left (-\frac {2}{n},\frac {1}{2},\frac {1}{2},\frac {-2+n}{n},-\frac {d x^n}{c},-\frac {a d x^n}{b+a c}\right )\right )+2 a d (4 b+a c n) x^n \operatorname {AppellF1}\left (\frac {-2+n}{n},\frac {1}{2},\frac {1}{2},2-\frac {2}{n},-\frac {d x^n}{c},-\frac {a d x^n}{b+a c}\right )\right )}{(-2+n) \left (b+a \left (c+d x^n\right )\right )}\right )}{2 c n x^2} \] Input:

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

Output:

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

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.

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

\(\Big \downarrow \) 2057

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

\(\Big \downarrow \) 2058

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

\(\Big \downarrow \) 1013

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

\(\Big \downarrow \) 1013

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

\(\Big \downarrow \) 1012

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

Input:

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

Output:

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

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

Input:

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

Output:

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

Fricas [F(-2)]

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

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

( - 4*sqrt(x**n*d + c)*sqrt(x**n*a*d + a*c + b)*a**2*c - 4*sqrt(x**n*d + c 
)*sqrt(x**n*a*d + a*c + b)*a*b + 8*x**n*int((x**(2*n)*sqrt(x**n*d + c)*sqr 
t(x**n*a*d + a*c + b))/(8*x**(3*n)*a**2*c*d**3*x**3 + x**(3*n)*a*b*d**3*n* 
x**3 + 4*x**(3*n)*a*b*d**3*x**3 + 24*x**(2*n)*a**2*c**2*d**2*x**3 + 3*x**( 
2*n)*a*b*c*d**2*n*x**3 + 20*x**(2*n)*a*b*c*d**2*x**3 + x**(2*n)*b**2*d**2* 
n*x**3 + 4*x**(2*n)*b**2*d**2*x**3 + 24*x**n*a**2*c**3*d*x**3 + 3*x**n*a*b 
*c**2*d*n*x**3 + 28*x**n*a*b*c**2*d*x**3 + 2*x**n*b**2*c*d*n*x**3 + 8*x**n 
*b**2*c*d*x**3 + 8*a**2*c**4*x**3 + a*b*c**3*n*x**3 + 12*a*b*c**3*x**3 + b 
**2*c**2*n*x**3 + 4*b**2*c**2*x**3),x)*a**3*b*c*d**3*n*x**2 - 32*x**n*int( 
(x**(2*n)*sqrt(x**n*d + c)*sqrt(x**n*a*d + a*c + b))/(8*x**(3*n)*a**2*c*d* 
*3*x**3 + x**(3*n)*a*b*d**3*n*x**3 + 4*x**(3*n)*a*b*d**3*x**3 + 24*x**(2*n 
)*a**2*c**2*d**2*x**3 + 3*x**(2*n)*a*b*c*d**2*n*x**3 + 20*x**(2*n)*a*b*c*d 
**2*x**3 + x**(2*n)*b**2*d**2*n*x**3 + 4*x**(2*n)*b**2*d**2*x**3 + 24*x**n 
*a**2*c**3*d*x**3 + 3*x**n*a*b*c**2*d*n*x**3 + 28*x**n*a*b*c**2*d*x**3 + 2 
*x**n*b**2*c*d*n*x**3 + 8*x**n*b**2*c*d*x**3 + 8*a**2*c**4*x**3 + a*b*c**3 
*n*x**3 + 12*a*b*c**3*x**3 + b**2*c**2*n*x**3 + 4*b**2*c**2*x**3),x)*a**3* 
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**2*c*d**3*x**3 + x**(3*n)*a*b*d**3*n*x**3 + 4*x**(3*n)*a 
*b*d**3*x**3 + 24*x**(2*n)*a**2*c**2*d**2*x**3 + 3*x**(2*n)*a*b*c*d**2*n*x 
**3 + 20*x**(2*n)*a*b*c*d**2*x**3 + x**(2*n)*b**2*d**2*n*x**3 + 4*x**(2...