Integrand size = 22, antiderivative size = 152 \[ \int \frac {1}{x^2 \left (a+b x^n+c x^{2 n}\right )^{3/2}} \, dx=-\frac {\sqrt {1+\frac {2 c x^n}{b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}} \operatorname {AppellF1}\left (-\frac {1}{n},\frac {3}{2},\frac {3}{2},-\frac {1-n}{n},-\frac {2 c x^n}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}\right )}{a x \sqrt {a+b x^n+c x^{2 n}}} \] Output:
-(1+2*c*x^n/(b-(-4*a*c+b^2)^(1/2)))^(1/2)*(1+2*c*x^n/(b+(-4*a*c+b^2)^(1/2) ))^(1/2)*AppellF1(-1/n,3/2,3/2,-(1-n)/n,-2*c*x^n/(b-(-4*a*c+b^2)^(1/2)),-2 *c*x^n/(b+(-4*a*c+b^2)^(1/2)))/a/x/(a+b*x^n+c*x^(2*n))^(1/2)
Leaf count is larger than twice the leaf count of optimal. \(395\) vs. \(2(152)=304\).
Time = 1.01 (sec) , antiderivative size = 395, normalized size of antiderivative = 2.60 \[ \int \frac {1}{x^2 \left (a+b x^n+c x^{2 n}\right )^{3/2}} \, dx=\frac {(-1+n) \left (-4 a c (1+n)+b^2 (2+n)\right ) \sqrt {\frac {b-\sqrt {b^2-4 a c}+2 c x^n}{b-\sqrt {b^2-4 a c}}} \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^n}{b+\sqrt {b^2-4 a c}}} \operatorname {AppellF1}\left (-\frac {1}{n},\frac {1}{2},\frac {1}{2},\frac {-1+n}{n},-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}},\frac {2 c x^n}{-b+\sqrt {b^2-4 a c}}\right )-2 \left ((-1+n) \left (b^2-2 a c+b c x^n\right )+b c x^n \sqrt {\frac {b-\sqrt {b^2-4 a c}+2 c x^n}{b-\sqrt {b^2-4 a c}}} \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^n}{b+\sqrt {b^2-4 a c}}} \operatorname {AppellF1}\left (\frac {-1+n}{n},\frac {1}{2},\frac {1}{2},2-\frac {1}{n},-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}},\frac {2 c x^n}{-b+\sqrt {b^2-4 a c}}\right )\right )}{a \left (-b^2+4 a c\right ) (-1+n) n x \sqrt {a+x^n \left (b+c x^n\right )}} \] Input:
Integrate[1/(x^2*(a + b*x^n + c*x^(2*n))^(3/2)),x]
Output:
((-1 + n)*(-4*a*c*(1 + n) + b^2*(2 + n))*Sqrt[(b - Sqrt[b^2 - 4*a*c] + 2*c *x^n)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])]*AppellF1[-n^(-1), 1/2, 1/2, (-1 + n)/n, (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])] - 2*((-1 + n)*( b^2 - 2*a*c + b*c*x^n) + b*c*x^n*Sqrt[(b - Sqrt[b^2 - 4*a*c] + 2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^n)/(b + Sqrt[b^ 2 - 4*a*c])]*AppellF1[(-1 + n)/n, 1/2, 1/2, 2 - n^(-1), (-2*c*x^n)/(b + Sq rt[b^2 - 4*a*c]), (2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])]))/(a*(-b^2 + 4*a*c)* (-1 + n)*n*x*Sqrt[a + x^n*(b + c*x^n)])
Time = 0.34 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {1721, 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.
\(\displaystyle \int \frac {1}{x^2 \left (a+b x^n+c x^{2 n}\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 1721 |
\(\displaystyle \frac {\sqrt {\frac {2 c x^n}{b-\sqrt {b^2-4 a c}}+1} \sqrt {\frac {2 c x^n}{\sqrt {b^2-4 a c}+b}+1} \int \frac {1}{x^2 \left (\frac {2 c x^n}{b-\sqrt {b^2-4 a c}}+1\right )^{3/2} \left (\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}+1\right )^{3/2}}dx}{a \sqrt {a+b x^n+c x^{2 n}}}\) |
\(\Big \downarrow \) 1012 |
\(\displaystyle -\frac {\sqrt {\frac {2 c x^n}{b-\sqrt {b^2-4 a c}}+1} \sqrt {\frac {2 c x^n}{\sqrt {b^2-4 a c}+b}+1} \operatorname {AppellF1}\left (-\frac {1}{n},\frac {3}{2},\frac {3}{2},-\frac {1-n}{n},-\frac {2 c x^n}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}\right )}{a x \sqrt {a+b x^n+c x^{2 n}}}\) |
Input:
Int[1/(x^2*(a + b*x^n + c*x^(2*n))^(3/2)),x]
Output:
-((Sqrt[1 + (2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[1 + (2*c*x^n)/(b + Sqr t[b^2 - 4*a*c])]*AppellF1[-n^(-1), 3/2, 3/2, -((1 - n)/n), (-2*c*x^n)/(b - Sqrt[b^2 - 4*a*c]), (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])])/(a*x*Sqrt[a + b* x^n + c*x^(2*n)]))
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])
Int[((d_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_), x _Symbol] :> Simp[a^IntPart[p]*((a + b*x^n + c*x^(2*n))^FracPart[p]/((1 + 2* c*(x^n/(b + Rt[b^2 - 4*a*c, 2])))^FracPart[p]*(1 + 2*c*(x^n/(b - Rt[b^2 - 4 *a*c, 2])))^FracPart[p])) Int[(d*x)^m*(1 + 2*c*(x^n/(b + Sqrt[b^2 - 4*a*c ])))^p*(1 + 2*c*(x^n/(b - Sqrt[b^2 - 4*a*c])))^p, x], x] /; FreeQ[{a, b, c, d, m, n, p}, x] && EqQ[n2, 2*n]
\[\int \frac {1}{x^{2} \left (a +b \,x^{n}+c \,x^{2 n}\right )^{\frac {3}{2}}}d x\]
Input:
int(1/x^2/(a+b*x^n+c*x^(2*n))^(3/2),x)
Output:
int(1/x^2/(a+b*x^n+c*x^(2*n))^(3/2),x)
Exception generated. \[ \int \frac {1}{x^2 \left (a+b x^n+c x^{2 n}\right )^{3/2}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(1/x^2/(a+b*x^n+c*x^(2*n))^(3/2),x, algorithm="fricas")
Output:
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
\[ \int \frac {1}{x^2 \left (a+b x^n+c x^{2 n}\right )^{3/2}} \, dx=\int \frac {1}{x^{2} \left (a + b x^{n} + c x^{2 n}\right )^{\frac {3}{2}}}\, dx \] Input:
integrate(1/x**2/(a+b*x**n+c*x**(2*n))**(3/2),x)
Output:
Integral(1/(x**2*(a + b*x**n + c*x**(2*n))**(3/2)), x)
\[ \int \frac {1}{x^2 \left (a+b x^n+c x^{2 n}\right )^{3/2}} \, dx=\int { \frac {1}{{\left (c x^{2 \, n} + b x^{n} + a\right )}^{\frac {3}{2}} x^{2}} \,d x } \] Input:
integrate(1/x^2/(a+b*x^n+c*x^(2*n))^(3/2),x, algorithm="maxima")
Output:
integrate(1/((c*x^(2*n) + b*x^n + a)^(3/2)*x^2), x)
\[ \int \frac {1}{x^2 \left (a+b x^n+c x^{2 n}\right )^{3/2}} \, dx=\int { \frac {1}{{\left (c x^{2 \, n} + b x^{n} + a\right )}^{\frac {3}{2}} x^{2}} \,d x } \] Input:
integrate(1/x^2/(a+b*x^n+c*x^(2*n))^(3/2),x, algorithm="giac")
Output:
integrate(1/((c*x^(2*n) + b*x^n + a)^(3/2)*x^2), x)
Timed out. \[ \int \frac {1}{x^2 \left (a+b x^n+c x^{2 n}\right )^{3/2}} \, dx=\int \frac {1}{x^2\,{\left (a+b\,x^n+c\,x^{2\,n}\right )}^{3/2}} \,d x \] Input:
int(1/(x^2*(a + b*x^n + c*x^(2*n))^(3/2)),x)
Output:
int(1/(x^2*(a + b*x^n + c*x^(2*n))^(3/2)), x)
\[ \int \frac {1}{x^2 \left (a+b x^n+c x^{2 n}\right )^{3/2}} \, dx=\int \frac {\sqrt {x^{2 n} c +x^{n} b +a}}{x^{4 n} c^{2} x^{2}+2 x^{3 n} b c \,x^{2}+2 x^{2 n} a c \,x^{2}+x^{2 n} b^{2} x^{2}+2 x^{n} a b \,x^{2}+a^{2} x^{2}}d x \] Input:
int(1/x^2/(a+b*x^n+c*x^(2*n))^(3/2),x)
Output:
int(sqrt(x**(2*n)*c + x**n*b + a)/(x**(4*n)*c**2*x**2 + 2*x**(3*n)*b*c*x** 2 + 2*x**(2*n)*a*c*x**2 + x**(2*n)*b**2*x**2 + 2*x**n*a*b*x**2 + a**2*x**2 ),x)