\(\int \frac {1}{(d+e x^n)^{3/2} (a+b x^n+c x^{2 n})} \, dx\) [82]

Optimal result

Integrand size = 28, antiderivative size = 204 \[ \int \frac {1}{\left (d+e x^n\right )^{3/2} \left (a+b x^n+c x^{2 n}\right )} \, dx=-\frac {2 c x \sqrt {1+\frac {e x^n}{d}} \operatorname {AppellF1}\left (\frac {1}{n},\frac {3}{2},1,1+\frac {1}{n},-\frac {e x^n}{d},-\frac {2 c x^n}{b-\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c-b \sqrt {b^2-4 a c}\right ) d \sqrt {d+e x^n}}-\frac {2 c x \sqrt {1+\frac {e x^n}{d}} \operatorname {AppellF1}\left (\frac {1}{n},\frac {3}{2},1,1+\frac {1}{n},-\frac {e x^n}{d},-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c+b \sqrt {b^2-4 a c}\right ) d \sqrt {d+e x^n}} \] Output:

-2*c*x*(1+e*x^n/d)^(1/2)*AppellF1(1/n,1,3/2,1+1/n,-2*c*x^n/(b-(-4*a*c+b^2) 
^(1/2)),-e*x^n/d)/(b^2-4*a*c-b*(-4*a*c+b^2)^(1/2))/d/(d+e*x^n)^(1/2)-2*c*x 
*(1+e*x^n/d)^(1/2)*AppellF1(1/n,1,3/2,1+1/n,-2*c*x^n/(b+(-4*a*c+b^2)^(1/2) 
),-e*x^n/d)/(b*(-4*a*c+b^2)^(1/2)-4*a*c+b^2)/d/(d+e*x^n)^(1/2)
 

Mathematica [F]

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

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

Output:

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

Rubi [A] (warning: unable to verify)

Time = 1.31 (sec) , antiderivative size = 324, normalized size of antiderivative = 1.59, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {1756, 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.

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

\(\Big \downarrow \) 1756

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

\(\Big \downarrow \) 779

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

\(\Big \downarrow \) 778

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

\(\Big \downarrow \) 7293

\(\displaystyle \frac {\int \left (\frac {-c e-\frac {c (b e-2 c d)}{\sqrt {b^2-4 a c}}}{\left (2 c x^n+b-\sqrt {b^2-4 a c}\right ) \sqrt {e x^n+d}}+\frac {\frac {c (b e-2 c d)}{\sqrt {b^2-4 a c}}-c e}{\left (2 c x^n+b+\sqrt {b^2-4 a c}\right ) \sqrt {e x^n+d}}\right )dx}{a e^2-b d e+c d^2}+\frac {e^2 x \sqrt {\frac {e x^n}{d}+1} \operatorname {Hypergeometric2F1}\left (\frac {3}{2},\frac {1}{n},1+\frac {1}{n},-\frac {e x^n}{d}\right )}{d \sqrt {d+e x^n} \left (a e^2-b d e+c d^2\right )}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {-\frac {c x \sqrt {\frac {e x^n}{d}+1} \left (e-\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) \operatorname {AppellF1}\left (\frac {1}{n},1,\frac {1}{2},1+\frac {1}{n},-\frac {2 c x^n}{b-\sqrt {b^2-4 a c}},-\frac {e x^n}{d}\right )}{\left (b-\sqrt {b^2-4 a c}\right ) \sqrt {d+e x^n}}-\frac {c x \sqrt {\frac {e x^n}{d}+1} \left (\frac {2 c d-b e}{\sqrt {b^2-4 a c}}+e\right ) \operatorname {AppellF1}\left (\frac {1}{n},1,\frac {1}{2},1+\frac {1}{n},-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}},-\frac {e x^n}{d}\right )}{\left (\sqrt {b^2-4 a c}+b\right ) \sqrt {d+e x^n}}}{a e^2-b d e+c d^2}+\frac {e^2 x \sqrt {\frac {e x^n}{d}+1} \operatorname {Hypergeometric2F1}\left (\frac {3}{2},\frac {1}{n},1+\frac {1}{n},-\frac {e x^n}{d}\right )}{d \sqrt {d+e x^n} \left (a e^2-b d e+c d^2\right )}\)

Input:

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

Output:

(-((c*(e - (2*c*d - b*e)/Sqrt[b^2 - 4*a*c])*x*Sqrt[1 + (e*x^n)/d]*AppellF1 
[n^(-1), 1, 1/2, 1 + n^(-1), (-2*c*x^n)/(b - Sqrt[b^2 - 4*a*c]), -((e*x^n) 
/d)])/((b - Sqrt[b^2 - 4*a*c])*Sqrt[d + e*x^n])) - (c*(e + (2*c*d - b*e)/S 
qrt[b^2 - 4*a*c])*x*Sqrt[1 + (e*x^n)/d]*AppellF1[n^(-1), 1, 1/2, 1 + n^(-1 
), (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]), -((e*x^n)/d)])/((b + Sqrt[b^2 - 4*a 
*c])*Sqrt[d + e*x^n]))/(c*d^2 - b*d*e + 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 - b* 
d*e + 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 1756

Int[((d_) + (e_.)*(x_)^(n_))^(q_)/((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_ 
)), x_Symbol] :> Simp[e^2/(c*d^2 - b*d*e + a*e^2)   Int[(d + e*x^n)^q, x], 
x] + Simp[1/(c*d^2 - b*d*e + a*e^2)   Int[(d + e*x^n)^(q + 1)*((c*d - b*e - 
 c*e*x^n)/(a + b*x^n + c*x^(2*n))), x], x] /; FreeQ[{a, b, c, d, e, n}, x] 
&& EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + 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]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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