\(\int (a+b x^n+c x^{2 n})^{3/2} (A+B x^n+C x^{2 n}) \, dx\) [23]

Optimal result

Integrand size = 33, antiderivative size = 378 \[ \int \left (a+b x^n+c x^{2 n}\right )^{3/2} \left (A+B x^n+C x^{2 n}\right ) \, dx=\frac {C x \left (a+b x^n+c x^{2 n}\right )^{5/2}}{c (1+5 n)}-\frac {a (b C (2+5 n)-2 B (c+5 c n)) x^{1+n} \sqrt {a+b x^n+c x^{2 n}} \operatorname {AppellF1}\left (1+\frac {1}{n},-\frac {3}{2},-\frac {3}{2},2+\frac {1}{n},-\frac {2 c x^n}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}\right )}{2 c (1+n) (1+5 n) \sqrt {1+\frac {2 c x^n}{b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}}}-\frac {a (a C-A (c+5 c n)) x \sqrt {a+b x^n+c x^{2 n}} \operatorname {AppellF1}\left (\frac {1}{n},-\frac {3}{2},-\frac {3}{2},1+\frac {1}{n},-\frac {2 c x^n}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}\right )}{c (1+5 n) \sqrt {1+\frac {2 c x^n}{b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}}} \] Output:

C*x*(a+b*x^n+c*x^(2*n))^(5/2)/c/(1+5*n)-1/2*a*(b*C*(2+5*n)-2*B*(5*c*n+c))* 
x^(1+n)*(a+b*x^n+c*x^(2*n))^(1/2)*AppellF1(1+1/n,-3/2,-3/2,2+1/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)))/c/(1+n)/(1+5*n)/(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)-a*(C*a-A*(5*c*n+c))*x*(a+b*x^n+c*x^(2*n))^(1/2)*AppellF1(1/n,-3/2,-3 
/2,1+1/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)))/ 
c/(1+5*n)/(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. \(19261\) vs. \(2(378)=756\).

Time = 9.09 (sec) , antiderivative size = 19261, normalized size of antiderivative = 50.96 \[ \int \left (a+b x^n+c x^{2 n}\right )^{3/2} \left (A+B x^n+C x^{2 n}\right ) \, dx=\text {Result too large to show} \] Input:

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

Output:

Result too large to show
 

Rubi [F]

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.

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

\(\Big \downarrow \) 2329

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

Input:

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

Output:

$Aborted
 

Defintions of rubi rules used

rule 2329

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.))^(p_.), x_Symbol] :> 
 Unintegrable[Pq*(a + b*x^n + c*x^(2*n))^p, x] /; FreeQ[{a, b, c, n, p}, x] 
 && EqQ[n2, 2*n] && (PolyQ[Pq, x] || PolyQ[Pq, x^n])
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F(-2)]

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

integrate((a+b*x^n+c*x^(2*n))^(3/2)*(A+B*x^n+C*x^(2*n)),x, algorithm="fric 
as")
 

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

integrate((a+b*x^n+c*x^(2*n))^(3/2)*(A+B*x^n+C*x^(2*n)),x, algorithm="maxi 
ma")
 

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(384*x**(4*n)*sqrt(x**(2*n)*c + x**n*b + a)*c**3*n**4*x + 1184*x**(4*n)*sq 
rt(x**(2*n)*c + x**n*b + a)*c**3*n**3*x + 976*x**(4*n)*sqrt(x**(2*n)*c + x 
**n*b + a)*c**3*n**2*x + 304*x**(4*n)*sqrt(x**(2*n)*c + x**n*b + a)*c**3*n 
*x + 32*x**(4*n)*sqrt(x**(2*n)*c + x**n*b + a)*c**3*x + 1008*x**(3*n)*sqrt 
(x**(2*n)*c + x**n*b + a)*b*c**2*n**4*x + 3048*x**(3*n)*sqrt(x**(2*n)*c + 
x**n*b + a)*b*c**2*n**3*x + 2392*x**(3*n)*sqrt(x**(2*n)*c + x**n*b + a)*b* 
c**2*n**2*x + 688*x**(3*n)*sqrt(x**(2*n)*c + x**n*b + a)*b*c**2*n*x + 64*x 
**(3*n)*sqrt(x**(2*n)*c + x**n*b + a)*b*c**2*x + 1408*x**(2*n)*sqrt(x**(2* 
n)*c + x**n*b + a)*a*c**2*n**4*x + 4128*x**(2*n)*sqrt(x**(2*n)*c + x**n*b 
+ a)*a*c**2*n**3*x + 2992*x**(2*n)*sqrt(x**(2*n)*c + x**n*b + a)*a*c**2*n* 
*2*x + 768*x**(2*n)*sqrt(x**(2*n)*c + x**n*b + a)*a*c**2*n*x + 64*x**(2*n) 
*sqrt(x**(2*n)*c + x**n*b + a)*a*c**2*x + 744*x**(2*n)*sqrt(x**(2*n)*c + x 
**n*b + a)*b**2*c*n**4*x + 2164*x**(2*n)*sqrt(x**(2*n)*c + x**n*b + a)*b** 
2*c*n**3*x + 1536*x**(2*n)*sqrt(x**(2*n)*c + x**n*b + a)*b**2*c*n**2*x + 3 
84*x**(2*n)*sqrt(x**(2*n)*c + x**n*b + a)*b**2*c*n*x + 32*x**(2*n)*sqrt(x* 
*(2*n)*c + x**n*b + a)*b**2*c*x + 2488*x**n*sqrt(x**(2*n)*c + x**n*b + a)* 
a*b*c*n**4*x + 6728*x**n*sqrt(x**(2*n)*c + x**n*b + a)*a*b*c*n**3*x + 3912 
*x**n*sqrt(x**(2*n)*c + x**n*b + a)*a*b*c*n**2*x + 848*x**n*sqrt(x**(2*n)* 
c + x**n*b + a)*a*b*c*n*x + 64*x**n*sqrt(x**(2*n)*c + x**n*b + a)*a*b*c*x 
+ 30*x**n*sqrt(x**(2*n)*c + x**n*b + a)*b**3*n**4*x + 60*x**n*sqrt(x**(...