\(\int \frac {1}{(d x)^{3/2} (a+b x^2+c x^4)^{3/2}} \, dx\) [1082]

Optimal result

Integrand size = 24, antiderivative size = 148 \[ \int \frac {1}{(d x)^{3/2} \left (a+b x^2+c x^4\right )^{3/2}} \, dx=-\frac {2 \sqrt {1+\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}} \operatorname {AppellF1}\left (-\frac {1}{4},\frac {3}{2},\frac {3}{2},\frac {3}{4},-\frac {2 c x^2}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}\right )}{a d \sqrt {d x} \sqrt {a+b x^2+c x^4}} \] Output:

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

Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(409\) vs. \(2(148)=296\).

Time = 11.54 (sec) , antiderivative size = 409, normalized size of antiderivative = 2.76 \[ \int \frac {1}{(d x)^{3/2} \left (a+b x^2+c x^4\right )^{3/2}} \, dx=-\frac {x \left (-7 \left (8 a^2 c-3 b^2 x^2 \left (b+c x^2\right )+a \left (-2 b^2+11 b c x^2+10 c^2 x^4\right )\right )-7 b \left (b^2-3 a c\right ) x^2 \sqrt {\frac {b-\sqrt {b^2-4 a c}+2 c x^2}{b-\sqrt {b^2-4 a c}}} \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^2}{b+\sqrt {b^2-4 a c}}} \operatorname {AppellF1}\left (\frac {3}{4},\frac {1}{2},\frac {1}{2},\frac {7}{4},-\frac {2 c x^2}{b+\sqrt {b^2-4 a c}},\frac {2 c x^2}{-b+\sqrt {b^2-4 a c}}\right )+3 c \left (-3 b^2+10 a c\right ) x^4 \sqrt {\frac {b-\sqrt {b^2-4 a c}+2 c x^2}{b-\sqrt {b^2-4 a c}}} \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^2}{b+\sqrt {b^2-4 a c}}} \operatorname {AppellF1}\left (\frac {7}{4},\frac {1}{2},\frac {1}{2},\frac {11}{4},-\frac {2 c x^2}{b+\sqrt {b^2-4 a c}},\frac {2 c x^2}{-b+\sqrt {b^2-4 a c}}\right )\right )}{7 a^2 \left (b^2-4 a c\right ) (d x)^{3/2} \sqrt {a+b x^2+c x^4}} \] Input:

Integrate[1/((d*x)^(3/2)*(a + b*x^2 + c*x^4)^(3/2)),x]
 

Output:

-1/7*(x*(-7*(8*a^2*c - 3*b^2*x^2*(b + c*x^2) + a*(-2*b^2 + 11*b*c*x^2 + 10 
*c^2*x^4)) - 7*b*(b^2 - 3*a*c)*x^2*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])]*AppellF1[3/4, 1/2, 1/2, 7/4, (-2*c*x^2)/(b + Sqrt[b^2 - 4*a 
*c]), (2*c*x^2)/(-b + Sqrt[b^2 - 4*a*c])] + 3*c*(-3*b^2 + 10*a*c)*x^4*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])]*AppellF1[7/4, 1/2, 1/2, 
11/4, (-2*c*x^2)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^2)/(-b + Sqrt[b^2 - 4*a*c 
])]))/(a^2*(b^2 - 4*a*c)*(d*x)^(3/2)*Sqrt[a + b*x^2 + c*x^4])
 

Rubi [A] (verified)

Time = 0.48 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, 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.

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

\(\Big \downarrow \) 1461

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

\(\Big \downarrow \) 394

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

Input:

Int[1/((d*x)^(3/2)*(a + b*x^2 + c*x^4)^(3/2)),x]
 

Output:

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

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]

\[\int \frac {1}{\left (d x \right )^{\frac {3}{2}} \left (c \,x^{4}+b \,x^{2}+a \right )^{\frac {3}{2}}}d x\]

Input:

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

Output:

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

Fricas [F]

\[ \int \frac {1}{(d x)^{3/2} \left (a+b x^2+c x^4\right )^{3/2}} \, dx=\int { \frac {1}{{\left (c x^{4} + b x^{2} + a\right )}^{\frac {3}{2}} \left (d x\right )^{\frac {3}{2}}} \,d x } \] Input:

integrate(1/(d*x)^(3/2)/(c*x^4+b*x^2+a)^(3/2),x, algorithm="fricas")
 

Output:

integral(sqrt(c*x^4 + b*x^2 + a)*sqrt(d*x)/(c^2*d^2*x^10 + 2*b*c*d^2*x^8 + 
 (b^2 + 2*a*c)*d^2*x^6 + 2*a*b*d^2*x^4 + a^2*d^2*x^2), x)
 

Sympy [F]

\[ \int \frac {1}{(d x)^{3/2} \left (a+b x^2+c x^4\right )^{3/2}} \, dx=\int \frac {1}{\left (d x\right )^{\frac {3}{2}} \left (a + b x^{2} + c x^{4}\right )^{\frac {3}{2}}}\, dx \] Input:

integrate(1/(d*x)**(3/2)/(c*x**4+b*x**2+a)**(3/2),x)
 

Output:

Integral(1/((d*x)**(3/2)*(a + b*x**2 + c*x**4)**(3/2)), x)
 

Maxima [F]

\[ \int \frac {1}{(d x)^{3/2} \left (a+b x^2+c x^4\right )^{3/2}} \, dx=\int { \frac {1}{{\left (c x^{4} + b x^{2} + a\right )}^{\frac {3}{2}} \left (d x\right )^{\frac {3}{2}}} \,d x } \] Input:

integrate(1/(d*x)^(3/2)/(c*x^4+b*x^2+a)^(3/2),x, algorithm="maxima")
 

Output:

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

Giac [F]

\[ \int \frac {1}{(d x)^{3/2} \left (a+b x^2+c x^4\right )^{3/2}} \, dx=\int { \frac {1}{{\left (c x^{4} + b x^{2} + a\right )}^{\frac {3}{2}} \left (d x\right )^{\frac {3}{2}}} \,d x } \] Input:

integrate(1/(d*x)^(3/2)/(c*x^4+b*x^2+a)^(3/2),x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(d x)^{3/2} \left (a+b x^2+c x^4\right )^{3/2}} \, dx=\int \frac {1}{{\left (d\,x\right )}^{3/2}\,{\left (c\,x^4+b\,x^2+a\right )}^{3/2}} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {1}{(d x)^{3/2} \left (a+b x^2+c x^4\right )^{3/2}} \, dx=\frac {\sqrt {d}\, \left (-2 \sqrt {c \,x^{4}+b \,x^{2}+a}-5 \sqrt {x}\, \left (\int \frac {\sqrt {x}\, \sqrt {c \,x^{4}+b \,x^{2}+a}\, x^{2}}{c^{2} x^{8}+2 b c \,x^{6}+2 a c \,x^{4}+b^{2} x^{4}+2 a b \,x^{2}+a^{2}}d x \right ) a c -5 \sqrt {x}\, \left (\int \frac {\sqrt {x}\, \sqrt {c \,x^{4}+b \,x^{2}+a}\, x^{2}}{c^{2} x^{8}+2 b c \,x^{6}+2 a c \,x^{4}+b^{2} x^{4}+2 a b \,x^{2}+a^{2}}d x \right ) b c \,x^{2}-5 \sqrt {x}\, \left (\int \frac {\sqrt {x}\, \sqrt {c \,x^{4}+b \,x^{2}+a}\, x^{2}}{c^{2} x^{8}+2 b c \,x^{6}+2 a c \,x^{4}+b^{2} x^{4}+2 a b \,x^{2}+a^{2}}d x \right ) c^{2} x^{4}-3 \sqrt {x}\, \left (\int \frac {\sqrt {x}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{c^{2} x^{8}+2 b c \,x^{6}+2 a c \,x^{4}+b^{2} x^{4}+2 a b \,x^{2}+a^{2}}d x \right ) a b -3 \sqrt {x}\, \left (\int \frac {\sqrt {x}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{c^{2} x^{8}+2 b c \,x^{6}+2 a c \,x^{4}+b^{2} x^{4}+2 a b \,x^{2}+a^{2}}d x \right ) b^{2} x^{2}-3 \sqrt {x}\, \left (\int \frac {\sqrt {x}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{c^{2} x^{8}+2 b c \,x^{6}+2 a c \,x^{4}+b^{2} x^{4}+2 a b \,x^{2}+a^{2}}d x \right ) b c \,x^{4}\right )}{\sqrt {x}\, a \,d^{2} \left (c \,x^{4}+b \,x^{2}+a \right )} \] Input:

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

Output:

(sqrt(d)*( - 2*sqrt(a + b*x**2 + c*x**4) - 5*sqrt(x)*int((sqrt(x)*sqrt(a + 
 b*x**2 + c*x**4)*x**2)/(a**2 + 2*a*b*x**2 + 2*a*c*x**4 + b**2*x**4 + 2*b* 
c*x**6 + c**2*x**8),x)*a*c - 5*sqrt(x)*int((sqrt(x)*sqrt(a + b*x**2 + c*x* 
*4)*x**2)/(a**2 + 2*a*b*x**2 + 2*a*c*x**4 + b**2*x**4 + 2*b*c*x**6 + c**2* 
x**8),x)*b*c*x**2 - 5*sqrt(x)*int((sqrt(x)*sqrt(a + b*x**2 + c*x**4)*x**2) 
/(a**2 + 2*a*b*x**2 + 2*a*c*x**4 + b**2*x**4 + 2*b*c*x**6 + c**2*x**8),x)* 
c**2*x**4 - 3*sqrt(x)*int((sqrt(x)*sqrt(a + b*x**2 + c*x**4))/(a**2 + 2*a* 
b*x**2 + 2*a*c*x**4 + b**2*x**4 + 2*b*c*x**6 + c**2*x**8),x)*a*b - 3*sqrt( 
x)*int((sqrt(x)*sqrt(a + b*x**2 + c*x**4))/(a**2 + 2*a*b*x**2 + 2*a*c*x**4 
 + b**2*x**4 + 2*b*c*x**6 + c**2*x**8),x)*b**2*x**2 - 3*sqrt(x)*int((sqrt( 
x)*sqrt(a + b*x**2 + c*x**4))/(a**2 + 2*a*b*x**2 + 2*a*c*x**4 + b**2*x**4 
+ 2*b*c*x**6 + c**2*x**8),x)*b*c*x**4))/(sqrt(x)*a*d**2*(a + b*x**2 + c*x* 
*4))