\(\int (a+b x^n+c x^{2 n})^{3/2} \, dx\) [101]

Optimal result

Integrand size = 18, antiderivative size = 140 \[ \int \left (a+b x^n+c x^{2 n}\right )^{3/2} \, dx=\frac {a x \sqrt {a+b x^n+c x^{2 n}} \operatorname {AppellF1}\left (\frac {1}{n},-\frac {3}{2},-\frac {3}{2},1+\frac {1}{n},-\frac {2 c x^n}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}\right )}{\sqrt {1+\frac {2 c x^n}{b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}}} \] Output:

a*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)))/(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. \(466\) vs. \(2(140)=280\).

Time = 1.72 (sec) , antiderivative size = 466, normalized size of antiderivative = 3.33 \[ \int \left (a+b x^n+c x^{2 n}\right )^{3/2} \, dx=\frac {x \left (-3 b n^2 \left (b^2 (2+n)-4 a c (2+3 n)\right ) x^n \sqrt {\frac {b-\sqrt {b^2-4 a c}+2 c x^n}{b-\sqrt {b^2-4 a c}}} \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^n}{b+\sqrt {b^2-4 a c}}} \operatorname {AppellF1}\left (1+\frac {1}{n},\frac {1}{2},\frac {1}{2},2+\frac {1}{n},-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}},\frac {2 c x^n}{-b+\sqrt {b^2-4 a c}}\right )+2 (1+n) \left (\left (3 b^2 n^2+4 a c \left (1+6 n+8 n^2\right )+2 b c \left (2+9 n+7 n^2\right ) x^n+4 c^2 \left (1+3 n+2 n^2\right ) x^{2 n}\right ) \left (a+x^n \left (b+c x^n\right )\right )-3 a n^2 \left (b^2-4 a c (1+2 n)\right ) \sqrt {\frac {b-\sqrt {b^2-4 a c}+2 c x^n}{b-\sqrt {b^2-4 a c}}} \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^n}{b+\sqrt {b^2-4 a c}}} \operatorname {AppellF1}\left (\frac {1}{n},\frac {1}{2},\frac {1}{2},1+\frac {1}{n},-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}},\frac {2 c x^n}{-b+\sqrt {b^2-4 a c}}\right )\right )\right )}{8 c (1+n)^2 (1+2 n) (1+3 n) \sqrt {a+x^n \left (b+c x^n\right )}} \] Input:

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

Output:

(x*(-3*b*n^2*(b^2*(2 + n) - 4*a*c*(2 + 3*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)*((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)) - 3*a*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])]*A 
ppellF1[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])])))/(8*c*(1 + n)^2*(1 + 2*n)*(1 + 3*n)* 
Sqrt[a + x^n*(b + c*x^n)])
 

Rubi [A] (verified)

Time = 0.29 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {1686, 936}

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.

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

\(\Big \downarrow \) 1686

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

\(\Big \downarrow \) 936

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

Input:

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

Output:

(a*x*Sqrt[a + b*x^n + c*x^(2*n)]*AppellF1[n^(-1), -3/2, -3/2, 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 936

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] 
:> Simp[a^p*c^q*x*AppellF1[1/n, -p, -q, 1 + 1/n, (-b)*(x^n/a), (-d)*(x^n/c) 
], x] /; FreeQ[{a, b, c, d, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[n, -1] 
 && (IntegerQ[p] || GtQ[a, 0]) && (IntegerQ[q] || GtQ[c, 0])
 

rule 1686

Int[((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])))^FracPar 
t[p]))   Int[(1 + 2*c*(x^n/(b + Sqrt[b^2 - 4*a*c])))^p*(1 + 2*c*(x^n/(b - S 
qrt[b^2 - 4*a*c])))^p, x], x] /; FreeQ[{a, b, c, n, p}, x] && EqQ[n2, 2*n] 
&& NeQ[b^2 - 4*a*c, 0] &&  !IntegerQ[p]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F(-2)]

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

integrate((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 (has polynomial part)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(8*x**(2*n)*sqrt(x**(2*n)*c + x**n*b + a)*c*n**2*x + 20*x**(2*n)*sqrt(x**( 
2*n)*c + x**n*b + a)*c*n*x + 8*x**(2*n)*sqrt(x**(2*n)*c + x**n*b + a)*c*x 
+ 14*x**n*sqrt(x**(2*n)*c + x**n*b + a)*b*n**2*x + 32*x**n*sqrt(x**(2*n)*c 
 + x**n*b + a)*b*n*x + 8*x**n*sqrt(x**(2*n)*c + x**n*b + a)*b*x + 68*sqrt( 
x**(2*n)*c + x**n*b + a)*a*n**2*x + 44*sqrt(x**(2*n)*c + x**n*b + a)*a*n*x 
 + 8*sqrt(x**(2*n)*c + x**n*b + a)*a*x + 144*int(sqrt(x**(2*n)*c + x**n*b 
+ a)/(6*x**(2*n)*c*n**3 + 17*x**(2*n)*c*n**2 + 11*x**(2*n)*c*n + 2*x**(2*n 
)*c + 6*x**n*b*n**3 + 17*x**n*b*n**2 + 11*x**n*b*n + 2*x**n*b + 6*a*n**3 + 
 17*a*n**2 + 11*a*n + 2*a),x)*a**2*n**6 + 408*int(sqrt(x**(2*n)*c + x**n*b 
 + a)/(6*x**(2*n)*c*n**3 + 17*x**(2*n)*c*n**2 + 11*x**(2*n)*c*n + 2*x**(2* 
n)*c + 6*x**n*b*n**3 + 17*x**n*b*n**2 + 11*x**n*b*n + 2*x**n*b + 6*a*n**3 
+ 17*a*n**2 + 11*a*n + 2*a),x)*a**2*n**5 + 264*int(sqrt(x**(2*n)*c + x**n* 
b + a)/(6*x**(2*n)*c*n**3 + 17*x**(2*n)*c*n**2 + 11*x**(2*n)*c*n + 2*x**(2 
*n)*c + 6*x**n*b*n**3 + 17*x**n*b*n**2 + 11*x**n*b*n + 2*x**n*b + 6*a*n**3 
 + 17*a*n**2 + 11*a*n + 2*a),x)*a**2*n**4 + 48*int(sqrt(x**(2*n)*c + x**n* 
b + a)/(6*x**(2*n)*c*n**3 + 17*x**(2*n)*c*n**2 + 11*x**(2*n)*c*n + 2*x**(2 
*n)*c + 6*x**n*b*n**3 + 17*x**n*b*n**2 + 11*x**n*b*n + 2*x**n*b + 6*a*n**3 
 + 17*a*n**2 + 11*a*n + 2*a),x)*a**2*n**3 - 216*int((x**(2*n)*sqrt(x**(2*n 
)*c + x**n*b + a))/(6*x**(2*n)*c*n**3 + 17*x**(2*n)*c*n**2 + 11*x**(2*n)*c 
*n + 2*x**(2*n)*c + 6*x**n*b*n**3 + 17*x**n*b*n**2 + 11*x**n*b*n + 2*x*...