\(\int \frac {1}{(a+b x^n)^{3/2} (c+d x^n)} \, dx\) [105]

Optimal result

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

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

Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(382\) vs. \(2(64)=128\).

Time = 0.80 (sec) , antiderivative size = 382, normalized size of antiderivative = 5.97 \[ \int \frac {1}{\left (a+b x^n\right )^{3/2} \left (c+d x^n\right )} \, dx=-\frac {x \left (\frac {b d (-2+n) x^n \sqrt {1+\frac {b x^n}{a}} \operatorname {AppellF1}\left (1+\frac {1}{n},\frac {1}{2},1,2+\frac {1}{n},-\frac {b x^n}{a},-\frac {d x^n}{c}\right )}{c (1+n)}+\frac {2 \left (2 a b d n x^n \left (c+d x^n\right ) \operatorname {AppellF1}\left (1+\frac {1}{n},\frac {1}{2},2,2+\frac {1}{n},-\frac {b x^n}{a},-\frac {d x^n}{c}\right )+b^2 c n x^n \left (c+d x^n\right ) \operatorname {AppellF1}\left (1+\frac {1}{n},\frac {3}{2},1,2+\frac {1}{n},-\frac {b x^n}{a},-\frac {d x^n}{c}\right )+a c (1+n) \left (a d n-b \left (c n+2 d x^n\right )\right ) \operatorname {AppellF1}\left (\frac {1}{n},\frac {1}{2},1,1+\frac {1}{n},-\frac {b x^n}{a},-\frac {d x^n}{c}\right )\right )}{\left (c+d x^n\right ) \left (2 a d n x^n \operatorname {AppellF1}\left (1+\frac {1}{n},\frac {1}{2},2,2+\frac {1}{n},-\frac {b x^n}{a},-\frac {d x^n}{c}\right )+b c n x^n \operatorname {AppellF1}\left (1+\frac {1}{n},\frac {3}{2},1,2+\frac {1}{n},-\frac {b x^n}{a},-\frac {d x^n}{c}\right )-2 a c (1+n) \operatorname {AppellF1}\left (\frac {1}{n},\frac {1}{2},1,1+\frac {1}{n},-\frac {b x^n}{a},-\frac {d x^n}{c}\right )\right )}\right )}{a (-b c+a d) n \sqrt {a+b x^n}} \] Input:

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

Output:

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

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

\(\Big \downarrow \) 937

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

\(\Big \downarrow \) 936

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

Input:

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

Output:

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

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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