\(\int (d+e x^n) \sqrt {a+b x^n+c x^{2 n}} \, dx\) [83]

Optimal result

Integrand size = 26, antiderivative size = 292 \[ \int \left (d+e x^n\right ) \sqrt {a+b x^n+c x^{2 n}} \, dx=\frac {e x^{1+n} \sqrt {a+b x^n+c x^{2 n}} \operatorname {AppellF1}\left (1+\frac {1}{n},-\frac {1}{2},-\frac {1}{2},2+\frac {1}{n},-\frac {2 c x^n}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}\right )}{(1+n) \sqrt {1+\frac {2 c x^n}{b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}}}+\frac {d x \sqrt {a+b x^n+c x^{2 n}} \operatorname {AppellF1}\left (\frac {1}{n},-\frac {1}{2},-\frac {1}{2},1+\frac {1}{n},-\frac {2 c x^n}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}\right )}{\sqrt {1+\frac {2 c x^n}{b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}}} \] Output:

e*x^(1+n)*(a+b*x^n+c*x^(2*n))^(1/2)*AppellF1(1+1/n,-1/2,-1/2,2+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)))/(1+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)+d* 
x*(a+b*x^n+c*x^(2*n))^(1/2)*AppellF1(1/n,-1/2,-1/2,1+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)))/(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 [A] (warning: unable to verify)

Time = 1.89 (sec) , antiderivative size = 424, normalized size of antiderivative = 1.45 \[ \int \left (d+e x^n\right ) \sqrt {a+b x^n+c x^{2 n}} \, dx=\frac {x \left (-n \left (-4 a c e (1+n)+b^2 e (2+n)-2 b c d (1+2 n)\right ) x^n \sqrt {\frac {b-\sqrt {b^2-4 a c}+2 c x^n}{b-\sqrt {b^2-4 a c}}} \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^n}{b+\sqrt {b^2-4 a c}}} \operatorname {AppellF1}\left (1+\frac {1}{n},\frac {1}{2},\frac {1}{2},2+\frac {1}{n},-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}},\frac {2 c x^n}{-b+\sqrt {b^2-4 a c}}\right )+2 (1+n) \left (\left (a+x^n \left (b+c x^n\right )\right ) \left (b e n+2 c \left (d+2 d n+e (1+n) x^n\right )\right )+a n (-b e+2 c (d+2 d n)) \sqrt {\frac {b-\sqrt {b^2-4 a c}+2 c x^n}{b-\sqrt {b^2-4 a c}}} \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^n}{b+\sqrt {b^2-4 a c}}} \operatorname {AppellF1}\left (\frac {1}{n},\frac {1}{2},\frac {1}{2},1+\frac {1}{n},-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}},\frac {2 c x^n}{-b+\sqrt {b^2-4 a c}}\right )\right )\right )}{4 (1+n)^2 (c+2 c n) \sqrt {a+x^n \left (b+c x^n\right )}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.51 (sec) , antiderivative size = 292, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {1762, 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.

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

\(\Big \downarrow \) 1762

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

(e*x^(1 + n)*Sqrt[a + b*x^n + c*x^(2*n)]*AppellF1[1 + n^(-1), -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])])/((1 + n)*Sqrt[1 + (2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[1 + ( 
2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])]) + (d*x*Sqrt[a + b*x^n + c*x^(2*n)]*Appe 
llF1[n^(-1), -1/2, -1/2, 1 + n^(-1), (-2*c*x^n)/(b - Sqrt[b^2 - 4*a*c]), ( 
-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])])/(Sqrt[1 + (2*c*x^n)/(b - Sqrt[b^2 - 4* 
a*c])]*Sqrt[1 + (2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])])
 

Defintions of rubi rules used

rule 1762

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

rule 2009

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

Maple [F]

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

Input:

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

Output:

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

Fricas [F(-2)]

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

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

Output:

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

Sympy [F]

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

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

Output:

Integral((d + e*x**n)*sqrt(a + b*x**n + c*x**(2*n)), x)
 

Maxima [F]

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

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

Output:

integrate(sqrt(c*x^(2*n) + b*x^n + a)*(e*x^n + d), x)
 

Giac [F]

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

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

Output:

integrate(sqrt(c*x^(2*n) + b*x^n + a)*(e*x^n + d), x)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(2*x**n*sqrt(x**(2*n)*c + x**n*b + a)*b*e*n*x + 4*x**n*sqrt(x**(2*n)*c + x 
**n*b + a)*b*e*x + 4*sqrt(x**(2*n)*c + x**n*b + a)*a*e*n*x + 8*sqrt(x**(2* 
n)*c + x**n*b + a)*b*d*n*x + 4*sqrt(x**(2*n)*c + x**n*b + a)*b*d*x - 8*int 
(sqrt(x**(2*n)*c + x**n*b + a)/(2*x**(2*n)*c*n**2 + 5*x**(2*n)*c*n + 2*x** 
(2*n)*c + 2*x**n*b*n**2 + 5*x**n*b*n + 2*x**n*b + 2*a*n**2 + 5*a*n + 2*a), 
x)*a**2*e*n**3 - 20*int(sqrt(x**(2*n)*c + x**n*b + a)/(2*x**(2*n)*c*n**2 + 
 5*x**(2*n)*c*n + 2*x**(2*n)*c + 2*x**n*b*n**2 + 5*x**n*b*n + 2*x**n*b + 2 
*a*n**2 + 5*a*n + 2*a),x)*a**2*e*n**2 - 8*int(sqrt(x**(2*n)*c + x**n*b + a 
)/(2*x**(2*n)*c*n**2 + 5*x**(2*n)*c*n + 2*x**(2*n)*c + 2*x**n*b*n**2 + 5*x 
**n*b*n + 2*x**n*b + 2*a*n**2 + 5*a*n + 2*a),x)*a**2*e*n + 8*int(sqrt(x**( 
2*n)*c + x**n*b + a)/(2*x**(2*n)*c*n**2 + 5*x**(2*n)*c*n + 2*x**(2*n)*c + 
2*x**n*b*n**2 + 5*x**n*b*n + 2*x**n*b + 2*a*n**2 + 5*a*n + 2*a),x)*a*b*d*n 
**4 + 24*int(sqrt(x**(2*n)*c + x**n*b + a)/(2*x**(2*n)*c*n**2 + 5*x**(2*n) 
*c*n + 2*x**(2*n)*c + 2*x**n*b*n**2 + 5*x**n*b*n + 2*x**n*b + 2*a*n**2 + 5 
*a*n + 2*a),x)*a*b*d*n**3 + 18*int(sqrt(x**(2*n)*c + x**n*b + a)/(2*x**(2* 
n)*c*n**2 + 5*x**(2*n)*c*n + 2*x**(2*n)*c + 2*x**n*b*n**2 + 5*x**n*b*n + 2 
*x**n*b + 2*a*n**2 + 5*a*n + 2*a),x)*a*b*d*n**2 + 4*int(sqrt(x**(2*n)*c + 
x**n*b + a)/(2*x**(2*n)*c*n**2 + 5*x**(2*n)*c*n + 2*x**(2*n)*c + 2*x**n*b* 
n**2 + 5*x**n*b*n + 2*x**n*b + 2*a*n**2 + 5*a*n + 2*a),x)*a*b*d*n - 8*int( 
(x**(2*n)*sqrt(x**(2*n)*c + x**n*b + a))/(2*x**(2*n)*c*n**2 + 5*x**(2*n...