Integrand size = 24, antiderivative size = 586 \[ \int \frac {1}{x^6 \sqrt [4]{a+b x^2} \left (c+d x^2\right )^2} \, dx=\frac {b \left (7 b^3 c^3+13 a b^2 c^2 d+40 a^2 b c d^2-70 a^3 d^3\right ) x}{20 a^3 c^4 (b c-a d) \sqrt [4]{a+b x^2}}-\frac {(2 b c-7 a d) \left (a+b x^2\right )^{3/4}}{10 a c^2 (b c-a d) x^5}+\frac {\left (7 b^2 c^2+13 a b c d-35 a^2 d^2\right ) \left (a+b x^2\right )^{3/4}}{30 a^2 c^3 (b c-a d) x^3}-\frac {\left (7 b^3 c^3+13 a b^2 c^2 d+40 a^2 b c d^2-70 a^3 d^3\right ) \left (a+b x^2\right )^{3/4}}{20 a^3 c^4 (b c-a d) x}-\frac {d \left (a+b x^2\right )^{3/4}}{2 c (b c-a d) x^5 \left (c+d x^2\right )}-\frac {\sqrt {b} \left (7 b^3 c^3+13 a b^2 c^2 d+40 a^2 b c d^2-70 a^3 d^3\right ) \sqrt [4]{1+\frac {b x^2}{a}} E\left (\left .\frac {1}{2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{20 a^{5/2} c^4 (b c-a d) \sqrt [4]{a+b x^2}}+\frac {\sqrt [4]{a} d^{5/2} (15 b c-14 a d) \sqrt {-\frac {b x^2}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} \sqrt {d}}{\sqrt {-b c+a d}},\arcsin \left (\frac {\sqrt [4]{a+b x^2}}{\sqrt [4]{a}}\right ),-1\right )}{4 c^4 (-b c+a d)^{3/2} x}-\frac {\sqrt [4]{a} d^{5/2} (15 b c-14 a d) \sqrt {-\frac {b x^2}{a}} \operatorname {EllipticPi}\left (\frac {\sqrt {a} \sqrt {d}}{\sqrt {-b c+a d}},\arcsin \left (\frac {\sqrt [4]{a+b x^2}}{\sqrt [4]{a}}\right ),-1\right )}{4 c^4 (-b c+a d)^{3/2} x} \] Output:
1/20*b*(-70*a^3*d^3+40*a^2*b*c*d^2+13*a*b^2*c^2*d+7*b^3*c^3)*x/a^3/c^4/(-a *d+b*c)/(b*x^2+a)^(1/4)-1/10*(-7*a*d+2*b*c)*(b*x^2+a)^(3/4)/a/c^2/(-a*d+b* c)/x^5+1/30*(-35*a^2*d^2+13*a*b*c*d+7*b^2*c^2)*(b*x^2+a)^(3/4)/a^2/c^3/(-a *d+b*c)/x^3-1/20*(-70*a^3*d^3+40*a^2*b*c*d^2+13*a*b^2*c^2*d+7*b^3*c^3)*(b* x^2+a)^(3/4)/a^3/c^4/(-a*d+b*c)/x-1/2*d*(b*x^2+a)^(3/4)/c/(-a*d+b*c)/x^5/( d*x^2+c)-1/20*b^(1/2)*(-70*a^3*d^3+40*a^2*b*c*d^2+13*a*b^2*c^2*d+7*b^3*c^3 )*(1+b*x^2/a)^(1/4)*EllipticE(sin(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)^(1/4)+1/4*a^(1/4)*d^(5/2)*(-14*a*d+15*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)^(3/2)/x-1/4*a^(1/4)*d^(5/2)*(-14*a*d+15*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)^(3/2)/x
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 11.19 (sec) , antiderivative size = 491, normalized size of antiderivative = 0.84 \[ \int \frac {1}{x^6 \sqrt [4]{a+b x^2} \left (c+d x^2\right )^2} \, dx=\frac {\frac {b d \left (-7 b^3 c^3-13 a b^2 c^2 d-40 a^2 b c d^2+70 a^3 d^3\right ) x^8 \sqrt [4]{1+\frac {b x^2}{a}} \operatorname {AppellF1}\left (\frac {3}{2},\frac {1}{4},1,\frac {5}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )}{a^3 (-b c+a d)}+2 c \left (-12 c^2+\frac {2 b c^2 x^2}{a}+40 c d x^2-\frac {7 b^2 c^2 x^4}{a^2}-\frac {20 b c d x^4}{a}-180 d^2 x^4-\frac {21 b^3 c^2 x^6}{a^3}-\frac {60 b^2 c d x^6}{a^2}-\frac {180 b d^2 x^6}{a}+\frac {30 a d^4 x^6}{(b c-a d) \left (c+d x^2\right )}+\frac {30 b d^4 x^8}{(b c-a d) \left (c+d x^2\right )}-\frac {9 c \left (7 b^4 c^4+13 a b^3 c^3 d+40 a^2 b^2 c^2 d^2-220 a^3 b c d^3+140 a^4 d^4\right ) x^6 \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{4},1,\frac {3}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )}{a^2 (b c-a d) \left (c+d x^2\right ) \left (-6 a c \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{4},1,\frac {3}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )+x^2 \left (4 a d \operatorname {AppellF1}\left (\frac {3}{2},\frac {1}{4},2,\frac {5}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )+b c \operatorname {AppellF1}\left (\frac {3}{2},\frac {5}{4},1,\frac {5}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )\right )\right )}\right )}{120 c^5 x^5 \sqrt [4]{a+b x^2}} \] Input:
Integrate[1/(x^6*(a + b*x^2)^(1/4)*(c + d*x^2)^2),x]
Output:
((b*d*(-7*b^3*c^3 - 13*a*b^2*c^2*d - 40*a^2*b*c*d^2 + 70*a^3*d^3)*x^8*(1 + (b*x^2)/a)^(1/4)*AppellF1[3/2, 1/4, 1, 5/2, -((b*x^2)/a), -((d*x^2)/c)])/ (a^3*(-(b*c) + a*d)) + 2*c*(-12*c^2 + (2*b*c^2*x^2)/a + 40*c*d*x^2 - (7*b^ 2*c^2*x^4)/a^2 - (20*b*c*d*x^4)/a - 180*d^2*x^4 - (21*b^3*c^2*x^6)/a^3 - ( 60*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)) - (9*c*(7*b^4*c^4 + 13 *a*b^3*c^3*d + 40*a^2*b^2*c^2*d^2 - 220*a^3*b*c*d^3 + 140*a^4*d^4)*x^6*App ellF1[1/2, 1/4, 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, 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*App ellF1[3/2, 5/4, 1, 5/2, -((b*x^2)/a), -((d*x^2)/c)])))))/(120*c^5*x^5*(a + b*x^2)^(1/4))
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 0.19 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.11, 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.
| \(\displaystyle \int \frac {1}{x^6 \sqrt [4]{a+b x^2} \left (c+d x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 395 |
| \(\displaystyle \frac {\sqrt [4]{\frac {b x^2}{a}+1} \int \frac {1}{x^6 \sqrt [4]{\frac {b x^2}{a}+1} \left (d x^2+c\right )^2}dx}{\sqrt [4]{a+b x^2}}\) |
| \(\Big \downarrow \) 394 |
| \(\displaystyle -\frac {\sqrt [4]{\frac {b x^2}{a}+1} \operatorname {AppellF1}\left (-\frac {5}{2},\frac {1}{4},2,-\frac {3}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )}{5 c^2 x^5 \sqrt [4]{a+b x^2}}\) |
Input:
Int[1/(x^6*(a + b*x^2)^(1/4)*(c + d*x^2)^2),x]
Output:
-1/5*((1 + (b*x^2)/a)^(1/4)*AppellF1[-5/2, 1/4, 2, -3/2, -((b*x^2)/a), -(( d*x^2)/c)])/(c^2*x^5*(a + b*x^2)^(1/4))
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])
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])
\[\int \frac {1}{x^{6} \left (b \,x^{2}+a \right )^{\frac {1}{4}} \left (x^{2} d +c \right )^{2}}d x\]
Input:
int(1/x^6/(b*x^2+a)^(1/4)/(d*x^2+c)^2,x)
Output:
int(1/x^6/(b*x^2+a)^(1/4)/(d*x^2+c)^2,x)
Timed out. \[ \int \frac {1}{x^6 \sqrt [4]{a+b x^2} \left (c+d x^2\right )^2} \, dx=\text {Timed out} \] Input:
integrate(1/x^6/(b*x^2+a)^(1/4)/(d*x^2+c)^2,x, algorithm="fricas")
Output:
Timed out
\[ \int \frac {1}{x^6 \sqrt [4]{a+b x^2} \left (c+d x^2\right )^2} \, dx=\int \frac {1}{x^{6} \sqrt [4]{a + b x^{2}} \left (c + d x^{2}\right )^{2}}\, dx \] Input:
integrate(1/x**6/(b*x**2+a)**(1/4)/(d*x**2+c)**2,x)
Output:
Integral(1/(x**6*(a + b*x**2)**(1/4)*(c + d*x**2)**2), x)
\[ \int \frac {1}{x^6 \sqrt [4]{a+b x^2} \left (c+d x^2\right )^2} \, dx=\int { \frac {1}{{\left (b x^{2} + a\right )}^{\frac {1}{4}} {\left (d x^{2} + c\right )}^{2} x^{6}} \,d x } \] Input:
integrate(1/x^6/(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^6), x)
\[ \int \frac {1}{x^6 \sqrt [4]{a+b x^2} \left (c+d x^2\right )^2} \, dx=\int { \frac {1}{{\left (b x^{2} + a\right )}^{\frac {1}{4}} {\left (d x^{2} + c\right )}^{2} x^{6}} \,d x } \] Input:
integrate(1/x^6/(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^6), x)
Timed out. \[ \int \frac {1}{x^6 \sqrt [4]{a+b x^2} \left (c+d x^2\right )^2} \, dx=\int \frac {1}{x^6\,{\left (b\,x^2+a\right )}^{1/4}\,{\left (d\,x^2+c\right )}^2} \,d x \] Input:
int(1/(x^6*(a + b*x^2)^(1/4)*(c + d*x^2)^2),x)
Output:
int(1/(x^6*(a + b*x^2)^(1/4)*(c + d*x^2)^2), x)
\[ \int \frac {1}{x^6 \sqrt [4]{a+b x^2} \left (c+d x^2\right )^2} \, dx=\int \frac {1}{\left (b \,x^{2}+a \right )^{\frac {1}{4}} c^{2} x^{6}+2 \left (b \,x^{2}+a \right )^{\frac {1}{4}} c d \,x^{8}+\left (b \,x^{2}+a \right )^{\frac {1}{4}} d^{2} x^{10}}d x \] Input:
int(1/x^6/(b*x^2+a)^(1/4)/(d*x^2+c)^2,x)
Output:
int(1/((a + b*x**2)**(1/4)*c**2*x**6 + 2*(a + b*x**2)**(1/4)*c*d*x**8 + (a + b*x**2)**(1/4)*d**2*x**10),x)