\(\int \frac {(a+b x^n+c x^{2 n})^{3/2}}{x^2} \, dx\) [224]

Optimal result

Integrand size = 22, antiderivative size = 150 \[ \int \frac {\left (a+b x^n+c x^{2 n}\right )^{3/2}}{x^2} \, dx=-\frac {a \sqrt {a+b x^n+c x^{2 n}} \operatorname {AppellF1}\left (-\frac {1}{n},-\frac {3}{2},-\frac {3}{2},-\frac {1-n}{n},-\frac {2 c x^n}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}\right )}{x \sqrt {1+\frac {2 c x^n}{b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}}} \] Output:

-a*(a+b*x^n+c*x^(2*n))^(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)))/x/(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)
 

Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(477\) vs. \(2(150)=300\).

Time = 1.61 (sec) , antiderivative size = 477, normalized size of antiderivative = 3.18 \[ \int \frac {\left (a+b x^n+c x^{2 n}\right )^{3/2}}{x^2} \, dx=\frac {2 (-1+n) \left (3 b^2 n^2+4 a c \left (1-6 n+8 n^2\right )+2 b c \left (2-9 n+7 n^2\right ) x^n+4 c^2 \left (1-3 n+2 n^2\right ) x^{2 n}\right ) \left (a+x^n \left (b+c x^n\right )\right )-6 a (-1+n) n^2 \left (b^2+4 a c (-1+2 n)\right ) \sqrt {\frac {b-\sqrt {b^2-4 a c}+2 c x^n}{b-\sqrt {b^2-4 a c}}} \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^n}{b+\sqrt {b^2-4 a c}}} \operatorname {AppellF1}\left (-\frac {1}{n},\frac {1}{2},\frac {1}{2},\frac {-1+n}{n},-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}},\frac {2 c x^n}{-b+\sqrt {b^2-4 a c}}\right )-3 b \left (4 a c (2-3 n)+b^2 (-2+n)\right ) n^2 x^n \sqrt {\frac {b-\sqrt {b^2-4 a c}+2 c x^n}{b-\sqrt {b^2-4 a c}}} \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^n}{b+\sqrt {b^2-4 a c}}} \operatorname {AppellF1}\left (\frac {-1+n}{n},\frac {1}{2},\frac {1}{2},2-\frac {1}{n},-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}},\frac {2 c x^n}{-b+\sqrt {b^2-4 a c}}\right )}{8 c (-1+n)^2 (-1+2 n) (-1+3 n) x \sqrt {a+x^n \left (b+c x^n\right )}} \] Input:

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

Output:

(2*(-1 + n)*(3*b^2*n^2 + 4*a*c*(1 - 6*n + 8*n^2) + 2*b*c*(2 - 9*n + 7*n^2) 
*x^n + 4*c^2*(1 - 3*n + 2*n^2)*x^(2*n))*(a + x^n*(b + c*x^n)) - 6*a*(-1 + 
n)*n^2*(b^2 + 4*a*c*(-1 + 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])] - 3*b*(4*a*c*(2 - 3*n) + 
 b^2*(-2 + n))*n^2*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 + Sqrt[b^2 - 4 
*a*c]), (2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])])/(8*c*(-1 + n)^2*(-1 + 2*n)*(- 
1 + 3*n)*x*Sqrt[a + x^n*(b + c*x^n)])
 

Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 150, 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.

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

\(\Big \downarrow \) 1721

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

\(\Big \downarrow \) 1012

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 1012

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])
 

rule 1721

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]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F(-2)]

Exception generated. \[ \int \frac {\left (a+b x^n+c x^{2 n}\right )^{3/2}}{x^2} \, dx=\text {Exception raised: TypeError} \] Input:

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

Output:

Exception raised: TypeError >>  Error detected within library code:   inte 
grate: implementation incomplete (has polynomial part)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\left (a+b x^n+c x^{2 n}\right )^{3/2}}{x^2} \, dx=\text {too large to display} \] Input:

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

Output:

(8*x**(2*n)*sqrt(x**(2*n)*c + x**n*b + a)*c*n**2 - 20*x**(2*n)*sqrt(x**(2* 
n)*c + x**n*b + a)*c*n + 8*x**(2*n)*sqrt(x**(2*n)*c + x**n*b + a)*c + 14*x 
**n*sqrt(x**(2*n)*c + x**n*b + a)*b*n**2 - 32*x**n*sqrt(x**(2*n)*c + x**n* 
b + a)*b*n + 8*x**n*sqrt(x**(2*n)*c + x**n*b + a)*b + 68*sqrt(x**(2*n)*c + 
 x**n*b + a)*a*n**2 - 44*sqrt(x**(2*n)*c + x**n*b + a)*a*n + 8*sqrt(x**(2* 
n)*c + x**n*b + a)*a + 144*int(sqrt(x**(2*n)*c + x**n*b + a)/(6*x**(2*n)*c 
*n**3*x**2 - 17*x**(2*n)*c*n**2*x**2 + 11*x**(2*n)*c*n*x**2 - 2*x**(2*n)*c 
*x**2 + 6*x**n*b*n**3*x**2 - 17*x**n*b*n**2*x**2 + 11*x**n*b*n*x**2 - 2*x* 
*n*b*x**2 + 6*a*n**3*x**2 - 17*a*n**2*x**2 + 11*a*n*x**2 - 2*a*x**2),x)*a* 
*2*n**6*x - 408*int(sqrt(x**(2*n)*c + x**n*b + a)/(6*x**(2*n)*c*n**3*x**2 
- 17*x**(2*n)*c*n**2*x**2 + 11*x**(2*n)*c*n*x**2 - 2*x**(2*n)*c*x**2 + 6*x 
**n*b*n**3*x**2 - 17*x**n*b*n**2*x**2 + 11*x**n*b*n*x**2 - 2*x**n*b*x**2 + 
 6*a*n**3*x**2 - 17*a*n**2*x**2 + 11*a*n*x**2 - 2*a*x**2),x)*a**2*n**5*x + 
 264*int(sqrt(x**(2*n)*c + x**n*b + a)/(6*x**(2*n)*c*n**3*x**2 - 17*x**(2* 
n)*c*n**2*x**2 + 11*x**(2*n)*c*n*x**2 - 2*x**(2*n)*c*x**2 + 6*x**n*b*n**3* 
x**2 - 17*x**n*b*n**2*x**2 + 11*x**n*b*n*x**2 - 2*x**n*b*x**2 + 6*a*n**3*x 
**2 - 17*a*n**2*x**2 + 11*a*n*x**2 - 2*a*x**2),x)*a**2*n**4*x - 48*int(sqr 
t(x**(2*n)*c + x**n*b + a)/(6*x**(2*n)*c*n**3*x**2 - 17*x**(2*n)*c*n**2*x* 
*2 + 11*x**(2*n)*c*n*x**2 - 2*x**(2*n)*c*x**2 + 6*x**n*b*n**3*x**2 - 17*x* 
*n*b*n**2*x**2 + 11*x**n*b*n*x**2 - 2*x**n*b*x**2 + 6*a*n**3*x**2 - 17*...