\(\int \frac {1}{(a+b x^n)^{3/2} (c+d x^n)^2} \, dx\) [106]

Optimal result

Integrand size = 21, antiderivative size = 64 \[ \int \frac {1}{\left (a+b x^n\right )^{3/2} \left (c+d x^n\right )^2} \, dx=\frac {x \sqrt {1+\frac {b x^n}{a}} \operatorname {AppellF1}\left (\frac {1}{n},\frac {3}{2},2,1+\frac {1}{n},-\frac {b x^n}{a},-\frac {d x^n}{c}\right )}{a c^2 \sqrt {a+b x^n}} \] Output:

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

Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(1294\) vs. \(2(64)=128\).

Time = 1.22 (sec) , antiderivative size = 1294, normalized size of antiderivative = 20.22 \[ \int \frac {1}{\left (a+b x^n\right )^{3/2} \left (c+d x^n\right )^2} \, dx =\text {Too large to display} \] Input:

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

Output:

(x*(8*a*b^2*c^4*(1 + n)^2*AppellF1[n^(-1), 1/2, 1, 1 + n^(-1), -((b*x^n)/a 
), -((d*x^n)/c)] + 4*a^3*c^2*d^2*(1 + n)^2*AppellF1[n^(-1), 1/2, 1, 1 + n^ 
(-1), -((b*x^n)/a), -((d*x^n)/c)] - 4*a*b^2*c^4*n*(1 + n)^2*AppellF1[n^(-1 
), 1/2, 1, 1 + n^(-1), -((b*x^n)/a), -((d*x^n)/c)] + 8*a^2*b*c^3*d*n*(1 + 
n)^2*AppellF1[n^(-1), 1/2, 1, 1 + n^(-1), -((b*x^n)/a), -((d*x^n)/c)] - 4* 
a^3*c^2*d^2*n*(1 + n)^2*AppellF1[n^(-1), 1/2, 1, 1 + n^(-1), -((b*x^n)/a), 
 -((d*x^n)/c)] + 2*c*(1 + n)*(a*d^2*(a + b*x^n) + 2*b^2*c*(c + d*x^n))*(2* 
a*d*n*x^n*AppellF1[1 + n^(-1), 1/2, 2, 2 + n^(-1), -((b*x^n)/a), -((d*x^n) 
/c)] + b*c*n*x^n*AppellF1[1 + n^(-1), 3/2, 1, 2 + n^(-1), -((b*x^n)/a), -( 
(d*x^n)/c)] - 2*a*c*(1 + n)*AppellF1[n^(-1), 1/2, 1, 1 + n^(-1), -((b*x^n) 
/a), -((d*x^n)/c)]) - 4*b^2*c*d*x^n*Sqrt[1 + (b*x^n)/a]*(c + d*x^n)*Appell 
F1[1 + n^(-1), 1/2, 1, 2 + n^(-1), -((b*x^n)/a), -((d*x^n)/c)]*(2*a*d*n*x^ 
n*AppellF1[1 + n^(-1), 1/2, 2, 2 + n^(-1), -((b*x^n)/a), -((d*x^n)/c)] + b 
*c*n*x^n*AppellF1[1 + n^(-1), 3/2, 1, 2 + n^(-1), -((b*x^n)/a), -((d*x^n)/ 
c)] - 2*a*c*(1 + n)*AppellF1[n^(-1), 1/2, 1, 1 + n^(-1), -((b*x^n)/a), -(( 
d*x^n)/c)]) - 2*a*b*d^2*x^n*Sqrt[1 + (b*x^n)/a]*(c + d*x^n)*AppellF1[1 + n 
^(-1), 1/2, 1, 2 + n^(-1), -((b*x^n)/a), -((d*x^n)/c)]*(2*a*d*n*x^n*Appell 
F1[1 + n^(-1), 1/2, 2, 2 + n^(-1), -((b*x^n)/a), -((d*x^n)/c)] + b*c*n*x^n 
*AppellF1[1 + n^(-1), 3/2, 1, 2 + n^(-1), -((b*x^n)/a), -((d*x^n)/c)] - 2* 
a*c*(1 + n)*AppellF1[n^(-1), 1/2, 1, 1 + n^(-1), -((b*x^n)/a), -((d*x^n...
 

Rubi [A] (verified)

Time = 0.33 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {937, 936}

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.

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

\(\Big \downarrow \) 937

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

\(\Big \downarrow \) 936

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 936

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

rule 937

Int[((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[(1 + b*(x^n/a))^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, n, p, q 
}, x] && NeQ[b*c - a*d, 0] && NeQ[n, -1] &&  !(IntegerQ[p] || GtQ[a, 0])
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

integral(sqrt(b*x^n + a)/(b^2*d^2*x^(4*n) + a^2*c^2 + 2*(b^2*c*d + a*b*d^2 
)*x^(3*n) + (b^2*c^2 + 4*a*b*c*d + a^2*d^2)*x^(2*n) + 2*(a*b*c^2 + a^2*c*d 
)*x^n), x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {1}{\left (a+b x^n\right )^{3/2} \left (c+d x^n\right )^2} \, dx=\int \frac {\sqrt {x^{n} b +a}}{x^{4 n} b^{2} d^{2}+2 x^{3 n} a b \,d^{2}+2 x^{3 n} b^{2} c d +x^{2 n} a^{2} d^{2}+4 x^{2 n} a b c d +x^{2 n} b^{2} c^{2}+2 x^{n} a^{2} c d +2 x^{n} a b \,c^{2}+a^{2} c^{2}}d x \] Input:

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

Output:

int(sqrt(x**n*b + a)/(x**(4*n)*b**2*d**2 + 2*x**(3*n)*a*b*d**2 + 2*x**(3*n 
)*b**2*c*d + x**(2*n)*a**2*d**2 + 4*x**(2*n)*a*b*c*d + x**(2*n)*b**2*c**2 
+ 2*x**n*a**2*c*d + 2*x**n*a*b*c**2 + a**2*c**2),x)