\(\int \frac {1}{x^2 \sqrt [4]{a+b x^2} (c+d x^2)^2} \, dx\) [1539]

Optimal result

Integrand size = 24, antiderivative size = 333 \[ \int \frac {1}{x^2 \sqrt [4]{a+b x^2} \left (c+d x^2\right )^2} \, dx=-\frac {1}{c x \sqrt [4]{a+b x^2} \left (c+d x^2\right )}-\frac {3 d x}{2 c^2 \sqrt [4]{a+b x^2} \left (c+d x^2\right )}-\frac {\sqrt {b} (2 b c-3 a d) \sqrt [4]{1+\frac {b x^2}{a}} E\left (\left .\frac {1}{2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{2 \sqrt {a} c^2 (b c-a d) \sqrt [4]{a+b x^2}}+\frac {\sqrt [4]{a} \sqrt {d} (7 b c-6 a d) \sqrt {-\frac {b x^2}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} \sqrt {d}}{\sqrt {-b c+a d}},\arcsin \left (\frac {\sqrt [4]{a+b x^2}}{\sqrt [4]{a}}\right ),-1\right )}{4 c^2 (-b c+a d)^{3/2} x}-\frac {\sqrt [4]{a} \sqrt {d} (7 b c-6 a d) \sqrt {-\frac {b x^2}{a}} \operatorname {EllipticPi}\left (\frac {\sqrt {a} \sqrt {d}}{\sqrt {-b c+a d}},\arcsin \left (\frac {\sqrt [4]{a+b x^2}}{\sqrt [4]{a}}\right ),-1\right )}{4 c^2 (-b c+a d)^{3/2} x} \] Output:

-1/c/x/(b*x^2+a)^(1/4)/(d*x^2+c)-3/2*d*x/c^2/(b*x^2+a)^(1/4)/(d*x^2+c)-1/2 
*b^(1/2)*(-3*a*d+2*b*c)*(1+b*x^2/a)^(1/4)*EllipticE(sin(1/2*arctan(b^(1/2) 
*x/a^(1/2))),2^(1/2))/a^(1/2)/c^2/(-a*d+b*c)/(b*x^2+a)^(1/4)+1/4*a^(1/4)*d 
^(1/2)*(-6*a*d+7*b*c)*(-b*x^2/a)^(1/2)*EllipticPi((b*x^2+a)^(1/4)/a^(1/4), 
-a^(1/2)*d^(1/2)/(a*d-b*c)^(1/2),I)/c^2/(a*d-b*c)^(3/2)/x-1/4*a^(1/4)*d^(1 
/2)*(-6*a*d+7*b*c)*(-b*x^2/a)^(1/2)*EllipticPi((b*x^2+a)^(1/4)/a^(1/4),a^( 
1/2)*d^(1/2)/(a*d-b*c)^(1/2),I)/c^2/(a*d-b*c)^(3/2)/x
                                                                                    
                                                                                    
 

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.

Time = 10.57 (sec) , antiderivative size = 413, normalized size of antiderivative = 1.24 \[ \int \frac {1}{x^2 \sqrt [4]{a+b x^2} \left (c+d x^2\right )^2} \, dx=\frac {\frac {b d (-2 b c+3 a d) x^4 \sqrt [4]{1+\frac {b x^2}{a}} \operatorname {AppellF1}\left (\frac {3}{2},\frac {1}{4},1,\frac {5}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )}{-b c+a d}+\frac {6 c \left (-6 a c \left (-b^2 c x^2 \left (c+2 d x^2\right )+2 a^2 d \left (c+3 d x^2\right )+a b \left (-2 c^2-5 c d x^2+3 d^2 x^4\right )\right ) \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{4},1,\frac {3}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )+x^2 \left (a+b x^2\right ) \left (-2 b c \left (c+d x^2\right )+a d \left (2 c+3 d x^2\right )\right ) \left (4 a d \operatorname {AppellF1}\left (\frac {3}{2},\frac {1}{4},2,\frac {5}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )+b c \operatorname {AppellF1}\left (\frac {3}{2},\frac {5}{4},1,\frac {5}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )\right )\right )}{(b c-a d) \left (c+d x^2\right ) \left (-6 a c \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{4},1,\frac {3}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )+x^2 \left (4 a d \operatorname {AppellF1}\left (\frac {3}{2},\frac {1}{4},2,\frac {5}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )+b c \operatorname {AppellF1}\left (\frac {3}{2},\frac {5}{4},1,\frac {5}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )\right )\right )}}{12 a c^3 x \sqrt [4]{a+b x^2}} \] Input:

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

Output:

((b*d*(-2*b*c + 3*a*d)*x^4*(1 + (b*x^2)/a)^(1/4)*AppellF1[3/2, 1/4, 1, 5/2 
, -((b*x^2)/a), -((d*x^2)/c)])/(-(b*c) + a*d) + (6*c*(-6*a*c*(-(b^2*c*x^2* 
(c + 2*d*x^2)) + 2*a^2*d*(c + 3*d*x^2) + a*b*(-2*c^2 - 5*c*d*x^2 + 3*d^2*x 
^4))*AppellF1[1/2, 1/4, 1, 3/2, -((b*x^2)/a), -((d*x^2)/c)] + x^2*(a + b*x 
^2)*(-2*b*c*(c + d*x^2) + a*d*(2*c + 3*d*x^2))*(4*a*d*AppellF1[3/2, 1/4, 2 
, 5/2, -((b*x^2)/a), -((d*x^2)/c)] + b*c*AppellF1[3/2, 5/4, 1, 5/2, -((b*x 
^2)/a), -((d*x^2)/c)])))/((b*c - a*d)*(c + d*x^2)*(-6*a*c*AppellF1[1/2, 1/ 
4, 1, 3/2, -((b*x^2)/a), -((d*x^2)/c)] + x^2*(4*a*d*AppellF1[3/2, 1/4, 2, 
5/2, -((b*x^2)/a), -((d*x^2)/c)] + b*c*AppellF1[3/2, 5/4, 1, 5/2, -((b*x^2 
)/a), -((d*x^2)/c)]))))/(12*a*c^3*x*(a + b*x^2)^(1/4))
 

Rubi [C] (verified)

Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.

Time = 0.19 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.19, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {395, 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.

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

\(\Big \downarrow \) 395

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

\(\Big \downarrow \) 394

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

Input:

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

Output:

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

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 395

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

Maple [F]

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

Input:

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

Output:

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

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

int(1/((a + b*x**2)**(1/4)*c**2*x**2 + 2*(a + b*x**2)**(1/4)*c*d*x**4 + (a 
 + b*x**2)**(1/4)*d**2*x**6),x)