\(\int \frac {1}{x^6 (a+b x^2)^{3/4} (c+d x^2)^2} \, dx\) [1549]

Optimal result

Integrand size = 24, antiderivative size = 508 \[ \int \frac {1}{x^6 \left (a+b x^2\right )^{3/4} \left (c+d x^2\right )^2} \, dx=-\frac {(2 b c-7 a d) \sqrt [4]{a+b x^2}}{10 a c^2 (b c-a d) x^5}+\frac {\left (9 b^2 c^2+11 a b c d-35 a^2 d^2\right ) \sqrt [4]{a+b x^2}}{30 a^2 c^3 (b c-a d) x^3}-\frac {\left (9 b^3 c^3+11 a b^2 c^2 d+16 a^2 b c d^2-42 a^3 d^3\right ) \sqrt [4]{a+b x^2}}{12 a^3 c^4 (b c-a d) x}-\frac {d \sqrt [4]{a+b x^2}}{2 c (b c-a d) x^5 \left (c+d x^2\right )}-\frac {\sqrt {b} \left (9 b^3 c^3+11 a b^2 c^2 d+16 a^2 b c d^2-42 a^3 d^3\right ) \left (1+\frac {b x^2}{a}\right )^{3/4} \operatorname {EllipticF}\left (\frac {1}{2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),2\right )}{12 a^{5/2} c^4 (b c-a d) \left (a+b x^2\right )^{3/4}}-\frac {\sqrt [4]{a} d^3 (17 b c-14 a d) \sqrt {-\frac {b x^2}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} \sqrt {d}}{\sqrt {-b c+a d}},\arcsin \left (\frac {\sqrt [4]{a+b x^2}}{\sqrt [4]{a}}\right ),-1\right )}{4 c^4 (b c-a d)^2 x}-\frac {\sqrt [4]{a} d^3 (17 b c-14 a d) \sqrt {-\frac {b x^2}{a}} \operatorname {EllipticPi}\left (\frac {\sqrt {a} \sqrt {d}}{\sqrt {-b c+a d}},\arcsin \left (\frac {\sqrt [4]{a+b x^2}}{\sqrt [4]{a}}\right ),-1\right )}{4 c^4 (b c-a d)^2 x} \] Output:

-1/10*(-7*a*d+2*b*c)*(b*x^2+a)^(1/4)/a/c^2/(-a*d+b*c)/x^5+1/30*(-35*a^2*d^ 
2+11*a*b*c*d+9*b^2*c^2)*(b*x^2+a)^(1/4)/a^2/c^3/(-a*d+b*c)/x^3-1/12*(-42*a 
^3*d^3+16*a^2*b*c*d^2+11*a*b^2*c^2*d+9*b^3*c^3)*(b*x^2+a)^(1/4)/a^3/c^4/(- 
a*d+b*c)/x-1/2*d*(b*x^2+a)^(1/4)/c/(-a*d+b*c)/x^5/(d*x^2+c)-1/12*b^(1/2)*( 
-42*a^3*d^3+16*a^2*b*c*d^2+11*a*b^2*c^2*d+9*b^3*c^3)*(1+b*x^2/a)^(3/4)*Inv 
erseJacobiAM(1/2*arctan(b^(1/2)*x/a^(1/2)),2^(1/2))/a^(5/2)/c^4/(-a*d+b*c) 
/(b*x^2+a)^(3/4)-1/4*a^(1/4)*d^3*(-14*a*d+17*b*c)*(-b*x^2/a)^(1/2)*Ellipti 
cPi((b*x^2+a)^(1/4)/a^(1/4),-a^(1/2)*d^(1/2)/(a*d-b*c)^(1/2),I)/c^4/(-a*d+ 
b*c)^2/x-1/4*a^(1/4)*d^3*(-14*a*d+17*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^4/(-a*d+b*c)^2/x
 

Mathematica [C] (warning: unable to verify)

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

Time = 11.08 (sec) , antiderivative size = 496, normalized size of antiderivative = 0.98 \[ \int \frac {1}{x^6 \left (a+b x^2\right )^{3/4} \left (c+d x^2\right )^2} \, dx=\frac {\frac {b d \left (9 b^3 c^3+11 a b^2 c^2 d+16 a^2 b c d^2-42 a^3 d^3\right ) x^3 \left (1+\frac {b x^2}{a}\right )^{3/4} \operatorname {AppellF1}\left (\frac {3}{2},\frac {3}{4},1,\frac {5}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )}{a^3 c^5 (-b c+a d)}+\frac {6 \left (-12 c^2+\frac {6 b c^2 x^2}{a}+40 c d x^2-\frac {27 b^2 c^2 x^4}{a^2}-\frac {60 b c d x^4}{a}-180 d^2 x^4-\frac {45 b^3 c^2 x^6}{a^3}-\frac {100 b^2 c d x^6}{a^2}-\frac {180 b d^2 x^6}{a}+\frac {30 a d^4 x^6}{(b c-a d) \left (c+d x^2\right )}+\frac {30 b d^4 x^8}{(b c-a d) \left (c+d x^2\right )}+\frac {15 c \left (9 b^4 c^4+11 a b^3 c^3 d+16 a^2 b^2 c^2 d^2+60 a^3 b c d^3-84 a^4 d^4\right ) x^6 \operatorname {AppellF1}\left (\frac {1}{2},\frac {3}{4},1,\frac {3}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )}{a^2 (b c-a d) \left (c+d x^2\right ) \left (-6 a c \operatorname {AppellF1}\left (\frac {1}{2},\frac {3}{4},1,\frac {3}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )+x^2 \left (4 a d \operatorname {AppellF1}\left (\frac {3}{2},\frac {3}{4},2,\frac {5}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )+3 b c \operatorname {AppellF1}\left (\frac {3}{2},\frac {7}{4},1,\frac {5}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )\right )\right )}\right )}{5 c^4 x^5}}{72 \left (a+b x^2\right )^{3/4}} \] Input:

Integrate[1/(x^6*(a + b*x^2)^(3/4)*(c + d*x^2)^2),x]
 

Output:

((b*d*(9*b^3*c^3 + 11*a*b^2*c^2*d + 16*a^2*b*c*d^2 - 42*a^3*d^3)*x^3*(1 + 
(b*x^2)/a)^(3/4)*AppellF1[3/2, 3/4, 1, 5/2, -((b*x^2)/a), -((d*x^2)/c)])/( 
a^3*c^5*(-(b*c) + a*d)) + (6*(-12*c^2 + (6*b*c^2*x^2)/a + 40*c*d*x^2 - (27 
*b^2*c^2*x^4)/a^2 - (60*b*c*d*x^4)/a - 180*d^2*x^4 - (45*b^3*c^2*x^6)/a^3 
- (100*b^2*c*d*x^6)/a^2 - (180*b*d^2*x^6)/a + (30*a*d^4*x^6)/((b*c - a*d)* 
(c + d*x^2)) + (30*b*d^4*x^8)/((b*c - a*d)*(c + d*x^2)) + (15*c*(9*b^4*c^4 
 + 11*a*b^3*c^3*d + 16*a^2*b^2*c^2*d^2 + 60*a^3*b*c*d^3 - 84*a^4*d^4)*x^6* 
AppellF1[1/2, 3/4, 1, 3/2, -((b*x^2)/a), -((d*x^2)/c)])/(a^2*(b*c - a*d)*( 
c + d*x^2)*(-6*a*c*AppellF1[1/2, 3/4, 1, 3/2, -((b*x^2)/a), -((d*x^2)/c)] 
+ x^2*(4*a*d*AppellF1[3/2, 3/4, 2, 5/2, -((b*x^2)/a), -((d*x^2)/c)] + 3*b* 
c*AppellF1[3/2, 7/4, 1, 5/2, -((b*x^2)/a), -((d*x^2)/c)])))))/(5*c^4*x^5)) 
/(72*(a + b*x^2)^(3/4))
 

Rubi [C] (verified)

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

Time = 0.20 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.13, 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.

\(\displaystyle \int \frac {1}{x^6 \left (a+b x^2\right )^{3/4} \left (c+d x^2\right )^2} \, dx\)

\(\Big \downarrow \) 395

\(\displaystyle \frac {\left (\frac {b x^2}{a}+1\right )^{3/4} \int \frac {1}{x^6 \left (\frac {b x^2}{a}+1\right )^{3/4} \left (d x^2+c\right )^2}dx}{\left (a+b x^2\right )^{3/4}}\)

\(\Big \downarrow \) 394

\(\displaystyle -\frac {\left (\frac {b x^2}{a}+1\right )^{3/4} \operatorname {AppellF1}\left (-\frac {5}{2},\frac {3}{4},2,-\frac {3}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )}{5 c^2 x^5 \left (a+b x^2\right )^{3/4}}\)

Input:

Int[1/(x^6*(a + b*x^2)^(3/4)*(c + d*x^2)^2),x]
 

Output:

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

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]

\[\int \frac {1}{x^{6} \left (b \,x^{2}+a \right )^{\frac {3}{4}} \left (x^{2} d +c \right )^{2}}d x\]

Input:

int(1/x^6/(b*x^2+a)^(3/4)/(d*x^2+c)^2,x)
 

Output:

int(1/x^6/(b*x^2+a)^(3/4)/(d*x^2+c)^2,x)
 

Fricas [F(-1)]

Timed out. \[ \int \frac {1}{x^6 \left (a+b x^2\right )^{3/4} \left (c+d x^2\right )^2} \, dx=\text {Timed out} \] Input:

integrate(1/x^6/(b*x^2+a)^(3/4)/(d*x^2+c)^2,x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F]

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

integrate(1/x**6/(b*x**2+a)**(3/4)/(d*x**2+c)**2,x)
 

Output:

Integral(1/(x**6*(a + b*x**2)**(3/4)*(c + d*x**2)**2), x)
 

Maxima [F]

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

integrate(1/x^6/(b*x^2+a)^(3/4)/(d*x^2+c)^2,x, algorithm="maxima")
 

Output:

integrate(1/((b*x^2 + a)^(3/4)*(d*x^2 + c)^2*x^6), x)
 

Giac [F]

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

integrate(1/x^6/(b*x^2+a)^(3/4)/(d*x^2+c)^2,x, algorithm="giac")
 

Output:

integrate(1/((b*x^2 + a)^(3/4)*(d*x^2 + c)^2*x^6), x)
 

Mupad [F(-1)]

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

int(1/(x^6*(a + b*x^2)^(3/4)*(c + d*x^2)^2),x)
 

Output:

int(1/(x^6*(a + b*x^2)^(3/4)*(c + d*x^2)^2), x)
 

Reduce [F]

\[ \int \frac {1}{x^6 \left (a+b x^2\right )^{3/4} \left (c+d x^2\right )^2} \, dx=\int \frac {1}{\left (b \,x^{2}+a \right )^{\frac {3}{4}} c^{2} x^{6}+2 \left (b \,x^{2}+a \right )^{\frac {3}{4}} c d \,x^{8}+\left (b \,x^{2}+a \right )^{\frac {3}{4}} d^{2} x^{10}}d x \] Input:

int(1/x^6/(b*x^2+a)^(3/4)/(d*x^2+c)^2,x)
 

Output:

int(1/((a + b*x**2)**(3/4)*c**2*x**6 + 2*(a + b*x**2)**(3/4)*c*d*x**8 + (a 
 + b*x**2)**(3/4)*d**2*x**10),x)