\(\int \frac {1}{(d+e x^n)^{3/2} (a+c x^{2 n})} \, dx\) [45]

Optimal result

Integrand size = 23, antiderivative size = 150 \[ \int \frac {1}{\left (d+e x^n\right )^{3/2} \left (a+c x^{2 n}\right )} \, dx=\frac {x \sqrt {1+\frac {e x^n}{d}} \operatorname {AppellF1}\left (\frac {1}{n},\frac {3}{2},1,1+\frac {1}{n},-\frac {e x^n}{d},-\frac {\sqrt {c} x^n}{\sqrt {-a}}\right )}{2 a d \sqrt {d+e x^n}}+\frac {x \sqrt {1+\frac {e x^n}{d}} \operatorname {AppellF1}\left (\frac {1}{n},\frac {3}{2},1,1+\frac {1}{n},-\frac {e x^n}{d},\frac {\sqrt {c} x^n}{\sqrt {-a}}\right )}{2 a d \sqrt {d+e x^n}} \] Output:

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

Mathematica [F]

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

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

Output:

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

Rubi [A] (verified)

Time = 0.98 (sec) , antiderivative size = 260, normalized size of antiderivative = 1.73, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {1757, 779, 778, 7293, 2009}

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.

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

\(\Big \downarrow \) 1757

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

\(\Big \downarrow \) 779

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

\(\Big \downarrow \) 778

\(\displaystyle \frac {c \int \frac {d-e x^n}{\sqrt {e x^n+d} \left (c x^{2 n}+a\right )}dx}{a e^2+c d^2}+\frac {e^2 x \sqrt {\frac {e x^n}{d}+1} \operatorname {Hypergeometric2F1}\left (\frac {3}{2},\frac {1}{n},1+\frac {1}{n},-\frac {e x^n}{d}\right )}{d \left (a e^2+c d^2\right ) \sqrt {d+e x^n}}\)

\(\Big \downarrow \) 7293

\(\displaystyle \frac {c \int \left (\frac {-d-\frac {\sqrt {-a} e}{\sqrt {c}}}{2 \sqrt {-a} \sqrt {c} \left (x^n+\frac {\sqrt {-a}}{\sqrt {c}}\right ) \sqrt {e x^n+d}}-\frac {d-\frac {\sqrt {-a} e}{\sqrt {c}}}{2 \sqrt {-a} \sqrt {c} \left (\frac {\sqrt {-a}}{\sqrt {c}}-x^n\right ) \sqrt {e x^n+d}}\right )dx}{a e^2+c d^2}+\frac {e^2 x \sqrt {\frac {e x^n}{d}+1} \operatorname {Hypergeometric2F1}\left (\frac {3}{2},\frac {1}{n},1+\frac {1}{n},-\frac {e x^n}{d}\right )}{d \left (a e^2+c d^2\right ) \sqrt {d+e x^n}}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {c \left (\frac {x \left (\frac {\sqrt {-a} e}{\sqrt {c}}+d\right ) \sqrt {\frac {e x^n}{d}+1} \operatorname {AppellF1}\left (\frac {1}{n},1,\frac {1}{2},1+\frac {1}{n},-\frac {\sqrt {c} x^n}{\sqrt {-a}},-\frac {e x^n}{d}\right )}{2 a \sqrt {d+e x^n}}+\frac {x \left (d-\frac {\sqrt {-a} e}{\sqrt {c}}\right ) \sqrt {\frac {e x^n}{d}+1} \operatorname {AppellF1}\left (\frac {1}{n},1,\frac {1}{2},1+\frac {1}{n},\frac {\sqrt {c} x^n}{\sqrt {-a}},-\frac {e x^n}{d}\right )}{2 a \sqrt {d+e x^n}}\right )}{a e^2+c d^2}+\frac {e^2 x \sqrt {\frac {e x^n}{d}+1} \operatorname {Hypergeometric2F1}\left (\frac {3}{2},\frac {1}{n},1+\frac {1}{n},-\frac {e x^n}{d}\right )}{d \left (a e^2+c d^2\right ) \sqrt {d+e x^n}}\)

Input:

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

Output:

(c*(((d + (Sqrt[-a]*e)/Sqrt[c])*x*Sqrt[1 + (e*x^n)/d]*AppellF1[n^(-1), 1, 
1/2, 1 + n^(-1), -((Sqrt[c]*x^n)/Sqrt[-a]), -((e*x^n)/d)])/(2*a*Sqrt[d + e 
*x^n]) + ((d - (Sqrt[-a]*e)/Sqrt[c])*x*Sqrt[1 + (e*x^n)/d]*AppellF1[n^(-1) 
, 1, 1/2, 1 + n^(-1), (Sqrt[c]*x^n)/Sqrt[-a], -((e*x^n)/d)])/(2*a*Sqrt[d + 
 e*x^n])))/(c*d^2 + a*e^2) + (e^2*x*Sqrt[1 + (e*x^n)/d]*Hypergeometric2F1[ 
3/2, n^(-1), 1 + n^(-1), -((e*x^n)/d)])/(d*(c*d^2 + a*e^2)*Sqrt[d + e*x^n] 
)
 

Defintions of rubi rules used

rule 778

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*x*Hypergeometric2F 
1[-p, 1/n, 1/n + 1, (-b)*(x^n/a)], x] /; FreeQ[{a, b, n, p}, x] &&  !IGtQ[p 
, 0] &&  !IntegerQ[1/n] &&  !ILtQ[Simplify[1/n + p], 0] && (IntegerQ[p] || 
GtQ[a, 0])
 

rule 779

Int[((a_) + (b_.)*(x_)^(n_))^(p_), 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, x], x 
] /; FreeQ[{a, b, n, p}, x] &&  !IGtQ[p, 0] &&  !IntegerQ[1/n] &&  !ILtQ[Si 
mplify[1/n + p], 0] &&  !(IntegerQ[p] || GtQ[a, 0])
 

rule 1757

Int[((d_) + (e_.)*(x_)^(n_))^(q_)/((a_) + (c_.)*(x_)^(n2_)), x_Symbol] :> S 
imp[e^2/(c*d^2 + a*e^2)   Int[(d + e*x^n)^q, x], x] + Simp[c/(c*d^2 + a*e^2 
)   Int[(d + e*x^n)^(q + 1)*((d - e*x^n)/(a + c*x^(2*n))), x], x] /; FreeQ[ 
{a, c, d, e, n}, x] && EqQ[n2, 2*n] && NeQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[ 
q] && LtQ[q, -1]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 7293

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v] 
]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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