\(\int (d x)^m \sqrt {a+b x^2+c x^4} \, dx\) [1089]

Optimal result

Integrand size = 22, antiderivative size = 157 \[ \int (d x)^m \sqrt {a+b x^2+c x^4} \, dx=\frac {(d x)^{1+m} \sqrt {a+b x^2+c x^4} \operatorname {AppellF1}\left (\frac {1+m}{2},-\frac {1}{2},-\frac {1}{2},\frac {3+m}{2},-\frac {2 c x^2}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}\right )}{d (1+m) \sqrt {1+\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}}} \] Output:

(d*x)^(1+m)*(c*x^4+b*x^2+a)^(1/2)*AppellF1(1/2+1/2*m,-1/2,-1/2,3/2+1/2*m,- 
2*c*x^2/(b-(-4*a*c+b^2)^(1/2)),-2*c*x^2/(b+(-4*a*c+b^2)^(1/2)))/d/(1+m)/(1 
+2*c*x^2/(b-(-4*a*c+b^2)^(1/2)))^(1/2)/(1+2*c*x^2/(b+(-4*a*c+b^2)^(1/2)))^ 
(1/2)
 

Mathematica [A] (warning: unable to verify)

Time = 1.56 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.15 \[ \int (d x)^m \sqrt {a+b x^2+c x^4} \, dx=\frac {x (d x)^m \sqrt {a+b x^2+c x^4} \operatorname {AppellF1}\left (\frac {1+m}{2},-\frac {1}{2},-\frac {1}{2},\frac {3+m}{2},-\frac {2 c x^2}{b+\sqrt {b^2-4 a c}},\frac {2 c x^2}{-b+\sqrt {b^2-4 a c}}\right )}{(1+m) \sqrt {\frac {b-\sqrt {b^2-4 a c}+2 c x^2}{b-\sqrt {b^2-4 a c}}} \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^2}{b+\sqrt {b^2-4 a c}}}} \] Input:

Integrate[(d*x)^m*Sqrt[a + b*x^2 + c*x^4],x]
 

Output:

(x*(d*x)^m*Sqrt[a + b*x^2 + c*x^4]*AppellF1[(1 + m)/2, -1/2, -1/2, (3 + m) 
/2, (-2*c*x^2)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^2)/(-b + Sqrt[b^2 - 4*a*c]) 
])/((1 + m)*Sqrt[(b - Sqrt[b^2 - 4*a*c] + 2*c*x^2)/(b - Sqrt[b^2 - 4*a*c]) 
]*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)/(b + Sqrt[b^2 - 4*a*c])])
 

Rubi [A] (verified)

Time = 0.48 (sec) , antiderivative size = 157, 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 = {1461, 394}

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.

\(\displaystyle \int (d x)^m \sqrt {a+b x^2+c x^4} \, dx\)

\(\Big \downarrow \) 1461

\(\displaystyle \frac {\sqrt {a+b x^2+c x^4} \int (d x)^m \sqrt {\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}+1} \sqrt {\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}+1}dx}{\sqrt {\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}+1} \sqrt {\frac {2 c x^2}{\sqrt {b^2-4 a c}+b}+1}}\)

\(\Big \downarrow \) 394

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

Input:

Int[(d*x)^m*Sqrt[a + b*x^2 + c*x^4],x]
 

Output:

((d*x)^(1 + m)*Sqrt[a + b*x^2 + c*x^4]*AppellF1[(1 + m)/2, -1/2, -1/2, (3 
+ m)/2, (-2*c*x^2)/(b - Sqrt[b^2 - 4*a*c]), (-2*c*x^2)/(b + Sqrt[b^2 - 4*a 
*c])])/(d*(1 + m)*Sqrt[1 + (2*c*x^2)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[1 + (2* 
c*x^2)/(b + Sqrt[b^2 - 4*a*c])])
 

Defintions of rubi rules used

rule 394

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_ 
), x_Symbol] :> Simp[a^p*c^q*((e*x)^(m + 1)/(e*(m + 1)))*AppellF1[(m + 1)/2 
, -p, -q, 1 + (m + 1)/2, (-b)*(x^2/a), (-d)*(x^2/c)], x] /; FreeQ[{a, b, c, 
 d, e, m, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, 1] && (Int 
egerQ[p] || GtQ[a, 0]) && (IntegerQ[q] || GtQ[c, 0])
 

rule 1461

Int[((d_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] 
:> Simp[a^IntPart[p]*((a + b*x^2 + c*x^4)^FracPart[p]/((1 + 2*c*(x^2/(b + R 
t[b^2 - 4*a*c, 2])))^FracPart[p]*(1 + 2*c*(x^2/(b - Rt[b^2 - 4*a*c, 2])))^F 
racPart[p]))   Int[(d*x)^m*(1 + 2*c*(x^2/(b + Sqrt[b^2 - 4*a*c])))^p*(1 + 2 
*c*(x^2/(b - Sqrt[b^2 - 4*a*c])))^p, x], x] /; FreeQ[{a, b, c, d, m, p}, x]
 

Maple [F]

\[\int \left (d x \right )^{m} \sqrt {c \,x^{4}+b \,x^{2}+a}d x\]

Input:

int((d*x)^m*(c*x^4+b*x^2+a)^(1/2),x)
 

Output:

int((d*x)^m*(c*x^4+b*x^2+a)^(1/2),x)
 

Fricas [F]

\[ \int (d x)^m \sqrt {a+b x^2+c x^4} \, dx=\int { \sqrt {c x^{4} + b x^{2} + a} \left (d x\right )^{m} \,d x } \] Input:

integrate((d*x)^m*(c*x^4+b*x^2+a)^(1/2),x, algorithm="fricas")
 

Output:

integral(sqrt(c*x^4 + b*x^2 + a)*(d*x)^m, x)
 

Sympy [F]

\[ \int (d x)^m \sqrt {a+b x^2+c x^4} \, dx=\int \left (d x\right )^{m} \sqrt {a + b x^{2} + c x^{4}}\, dx \] Input:

integrate((d*x)**m*(c*x**4+b*x**2+a)**(1/2),x)
 

Output:

Integral((d*x)**m*sqrt(a + b*x**2 + c*x**4), x)
 

Maxima [F]

\[ \int (d x)^m \sqrt {a+b x^2+c x^4} \, dx=\int { \sqrt {c x^{4} + b x^{2} + a} \left (d x\right )^{m} \,d x } \] Input:

integrate((d*x)^m*(c*x^4+b*x^2+a)^(1/2),x, algorithm="maxima")
 

Output:

integrate(sqrt(c*x^4 + b*x^2 + a)*(d*x)^m, x)
 

Giac [F]

\[ \int (d x)^m \sqrt {a+b x^2+c x^4} \, dx=\int { \sqrt {c x^{4} + b x^{2} + a} \left (d x\right )^{m} \,d x } \] Input:

integrate((d*x)^m*(c*x^4+b*x^2+a)^(1/2),x, algorithm="giac")
 

Output:

integrate(sqrt(c*x^4 + b*x^2 + a)*(d*x)^m, x)
 

Mupad [F(-1)]

Timed out. \[ \int (d x)^m \sqrt {a+b x^2+c x^4} \, dx=\int {\left (d\,x\right )}^m\,\sqrt {c\,x^4+b\,x^2+a} \,d x \] Input:

int((d*x)^m*(a + b*x^2 + c*x^4)^(1/2),x)
 

Output:

int((d*x)^m*(a + b*x^2 + c*x^4)^(1/2), x)
 

Reduce [F]

\[ \int (d x)^m \sqrt {a+b x^2+c x^4} \, dx=\frac {d^{m} \left (x^{m} \sqrt {c \,x^{4}+b \,x^{2}+a}\, x +\left (\int \frac {x^{m} \sqrt {c \,x^{4}+b \,x^{2}+a}\, x^{2}}{c m \,x^{4}+3 c \,x^{4}+b m \,x^{2}+3 b \,x^{2}+a m +3 a}d x \right ) b m +3 \left (\int \frac {x^{m} \sqrt {c \,x^{4}+b \,x^{2}+a}\, x^{2}}{c m \,x^{4}+3 c \,x^{4}+b m \,x^{2}+3 b \,x^{2}+a m +3 a}d x \right ) b +2 \left (\int \frac {x^{m} \sqrt {c \,x^{4}+b \,x^{2}+a}}{c m \,x^{4}+3 c \,x^{4}+b m \,x^{2}+3 b \,x^{2}+a m +3 a}d x \right ) a m +6 \left (\int \frac {x^{m} \sqrt {c \,x^{4}+b \,x^{2}+a}}{c m \,x^{4}+3 c \,x^{4}+b m \,x^{2}+3 b \,x^{2}+a m +3 a}d x \right ) a \right )}{m +3} \] Input:

int((d*x)^m*(c*x^4+b*x^2+a)^(1/2),x)
 

Output:

(d**m*(x**m*sqrt(a + b*x**2 + c*x**4)*x + int((x**m*sqrt(a + b*x**2 + c*x* 
*4)*x**2)/(a*m + 3*a + b*m*x**2 + 3*b*x**2 + c*m*x**4 + 3*c*x**4),x)*b*m + 
 3*int((x**m*sqrt(a + b*x**2 + c*x**4)*x**2)/(a*m + 3*a + b*m*x**2 + 3*b*x 
**2 + c*m*x**4 + 3*c*x**4),x)*b + 2*int((x**m*sqrt(a + b*x**2 + c*x**4))/( 
a*m + 3*a + b*m*x**2 + 3*b*x**2 + c*m*x**4 + 3*c*x**4),x)*a*m + 6*int((x** 
m*sqrt(a + b*x**2 + c*x**4))/(a*m + 3*a + b*m*x**2 + 3*b*x**2 + c*m*x**4 + 
 3*c*x**4),x)*a))/(m + 3)