Integrand size = 26, antiderivative size = 390 \[ \int \frac {1}{x^6 \left (a+b x^2\right )^{5/4} \left (c+d x^2\right )^{5/4}} \, dx=-\frac {1}{5 a c x^5 \sqrt [4]{a+b x^2} \sqrt [4]{c+d x^2}}-\frac {b (11 b c-a d)}{5 a^2 c (b c-a d) x^3 \sqrt [4]{a+b x^2} \sqrt [4]{c+d x^2}}-\frac {d \left (11 b^2 c^2-2 a b c d+11 a^2 d^2\right ) \left (a+b x^2\right )^{3/4}}{5 a^2 c^2 (b c-a d)^2 x^3 \sqrt [4]{c+d x^2}}+\frac {(b c+a d) \left (77 b^2 c^2-94 a b c d+77 a^2 d^2\right ) \left (a+b x^2\right )^{3/4} \left (c+d x^2\right )^{3/4}}{30 a^3 c^3 (b c-a d)^2 x^3}-\frac {\left (77 b^4 c^4-28 a b^3 c^3 d-18 a^2 b^2 c^2 d^2-28 a^3 b c d^3+77 a^4 d^4\right ) \left (a+b x^2\right )^{3/4} \sqrt [4]{\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {1}{4},\frac {1}{2},\frac {(b c-a d) x^2}{c \left (a+b x^2\right )}\right )}{20 a^4 c^3 (b c-a d)^2 x \sqrt [4]{c+d x^2}} \] Output:
-1/5/a/c/x^5/(b*x^2+a)^(1/4)/(d*x^2+c)^(1/4)-1/5*b*(-a*d+11*b*c)/a^2/c/(-a *d+b*c)/x^3/(b*x^2+a)^(1/4)/(d*x^2+c)^(1/4)-1/5*d*(11*a^2*d^2-2*a*b*c*d+11 *b^2*c^2)*(b*x^2+a)^(3/4)/a^2/c^2/(-a*d+b*c)^2/x^3/(d*x^2+c)^(1/4)+1/30*(a *d+b*c)*(77*a^2*d^2-94*a*b*c*d+77*b^2*c^2)*(b*x^2+a)^(3/4)*(d*x^2+c)^(3/4) /a^3/c^3/(-a*d+b*c)^2/x^3-1/20*(77*a^4*d^4-28*a^3*b*c*d^3-18*a^2*b^2*c^2*d ^2-28*a*b^3*c^3*d+77*b^4*c^4)*(b*x^2+a)^(3/4)*(a*(d*x^2+c)/c/(b*x^2+a))^(1 /4)*hypergeom([-1/2, 1/4],[1/2],(-a*d+b*c)*x^2/c/(b*x^2+a))/a^4/c^3/(-a*d+ b*c)^2/x/(d*x^2+c)^(1/4)
Result contains higher order function than in optimal. Order 6 vs. order 5 in optimal.
Time = 11.50 (sec) , antiderivative size = 691, normalized size of antiderivative = 1.77 \[ \int \frac {1}{x^6 \left (a+b x^2\right )^{5/4} \left (c+d x^2\right )^{5/4}} \, dx=\frac {-12 a^3 b^2 c^5+24 a^4 b c^4 d-12 a^5 c^3 d^2+22 a^2 b^3 c^5 x^2-22 a^3 b^2 c^4 d x^2-22 a^4 b c^3 d^2 x^2+22 a^5 c^2 d^3 x^2-77 a b^4 c^5 x^4+72 a^2 b^3 c^4 d x^4+10 a^3 b^2 c^3 d^2 x^4+72 a^4 b c^2 d^3 x^4-77 a^5 c d^4 x^4-231 b^5 c^5 x^6+7 a b^4 c^4 d x^6+104 a^2 b^3 c^3 d^2 x^6+104 a^3 b^2 c^2 d^3 x^6+7 a^4 b c d^4 x^6-231 a^5 d^5 x^6-231 b^5 c^4 d x^8+84 a b^4 c^3 d^2 x^8+54 a^2 b^3 c^2 d^3 x^8+84 a^3 b^2 c d^4 x^8-231 a^4 b d^5 x^8+2 b d \left (77 b^4 c^4-28 a b^3 c^3 d-18 a^2 b^2 c^2 d^2-28 a^3 b c d^3+77 a^4 d^4\right ) x^8 \sqrt [4]{1+\frac {b x^2}{a}} \sqrt [4]{1+\frac {d x^2}{c}} \operatorname {AppellF1}\left (\frac {3}{2},\frac {1}{4},\frac {1}{4},\frac {5}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )-\frac {9 a c \left (77 b^5 c^5+49 a b^4 c^4 d-46 a^2 b^3 c^3 d^2-46 a^3 b^2 c^2 d^3+49 a^4 b c d^4+77 a^5 d^5\right ) x^6 \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{4},\frac {1}{4},\frac {3}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )}{-6 a c \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{4},\frac {1}{4},\frac {3}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )+x^2 \left (a d \operatorname {AppellF1}\left (\frac {3}{2},\frac {1}{4},\frac {5}{4},\frac {5}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )+b c \operatorname {AppellF1}\left (\frac {3}{2},\frac {5}{4},\frac {1}{4},\frac {5}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )\right )}}{60 a^4 c^4 (b c-a d)^2 x^5 \sqrt [4]{a+b x^2} \sqrt [4]{c+d x^2}} \] Input:
Integrate[1/(x^6*(a + b*x^2)^(5/4)*(c + d*x^2)^(5/4)),x]
Output:
(-12*a^3*b^2*c^5 + 24*a^4*b*c^4*d - 12*a^5*c^3*d^2 + 22*a^2*b^3*c^5*x^2 - 22*a^3*b^2*c^4*d*x^2 - 22*a^4*b*c^3*d^2*x^2 + 22*a^5*c^2*d^3*x^2 - 77*a*b^ 4*c^5*x^4 + 72*a^2*b^3*c^4*d*x^4 + 10*a^3*b^2*c^3*d^2*x^4 + 72*a^4*b*c^2*d ^3*x^4 - 77*a^5*c*d^4*x^4 - 231*b^5*c^5*x^6 + 7*a*b^4*c^4*d*x^6 + 104*a^2* b^3*c^3*d^2*x^6 + 104*a^3*b^2*c^2*d^3*x^6 + 7*a^4*b*c*d^4*x^6 - 231*a^5*d^ 5*x^6 - 231*b^5*c^4*d*x^8 + 84*a*b^4*c^3*d^2*x^8 + 54*a^2*b^3*c^2*d^3*x^8 + 84*a^3*b^2*c*d^4*x^8 - 231*a^4*b*d^5*x^8 + 2*b*d*(77*b^4*c^4 - 28*a*b^3* c^3*d - 18*a^2*b^2*c^2*d^2 - 28*a^3*b*c*d^3 + 77*a^4*d^4)*x^8*(1 + (b*x^2) /a)^(1/4)*(1 + (d*x^2)/c)^(1/4)*AppellF1[3/2, 1/4, 1/4, 5/2, -((b*x^2)/a), -((d*x^2)/c)] - (9*a*c*(77*b^5*c^5 + 49*a*b^4*c^4*d - 46*a^2*b^3*c^3*d^2 - 46*a^3*b^2*c^2*d^3 + 49*a^4*b*c*d^4 + 77*a^5*d^5)*x^6*AppellF1[1/2, 1/4, 1/4, 3/2, -((b*x^2)/a), -((d*x^2)/c)])/(-6*a*c*AppellF1[1/2, 1/4, 1/4, 3/ 2, -((b*x^2)/a), -((d*x^2)/c)] + x^2*(a*d*AppellF1[3/2, 1/4, 5/4, 5/2, -(( b*x^2)/a), -((d*x^2)/c)] + b*c*AppellF1[3/2, 5/4, 1/4, 5/2, -((b*x^2)/a), -((d*x^2)/c)])))/(60*a^4*c^4*(b*c - a*d)^2*x^5*(a + b*x^2)^(1/4)*(c + d*x^ 2)^(1/4))
Leaf count is larger than twice the leaf count of optimal. \(1468\) vs. \(2(390)=780\).
Time = 7.18 (sec) , antiderivative size = 1468, normalized size of antiderivative = 3.76, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {395, 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.
| \(\displaystyle \int \frac {1}{x^6 \left (a+b x^2\right )^{5/4} \left (c+d x^2\right )^{5/4}} \, dx\) |
| \(\Big \downarrow \) 395 |
| \(\displaystyle \frac {\sqrt [4]{\frac {b x^2}{a}+1} \int \frac {1}{x^6 \left (\frac {b x^2}{a}+1\right )^{5/4} \left (d x^2+c\right )^{5/4}}dx}{a \sqrt [4]{a+b x^2}}\) |
| \(\Big \downarrow \) 395 |
| \(\displaystyle \frac {\sqrt [4]{\frac {b x^2}{a}+1} \sqrt [4]{\frac {d x^2}{c}+1} \int \frac {1}{x^6 \left (\frac {b x^2}{a}+1\right )^{5/4} \left (\frac {d x^2}{c}+1\right )^{5/4}}dx}{a c \sqrt [4]{a+b x^2} \sqrt [4]{c+d x^2}}\) |
| \(\Big \downarrow \) 394 |
| \(\displaystyle -\frac {\operatorname {Gamma}\left (\frac {1}{4}\right ) \left (384 b c d^4 \operatorname {Hypergeometric2F1}\left (1,\frac {5}{4},\frac {5}{2},\frac {(b c-a d) x^2}{c \left (b x^2+a\right )}\right ) x^{10}-400 a d^5 \operatorname {Hypergeometric2F1}\left (2,\frac {9}{4},\frac {7}{2},\frac {(b c-a d) x^2}{c \left (b x^2+a\right )}\right ) x^{10}+400 b c d^4 \operatorname {Hypergeometric2F1}\left (2,\frac {9}{4},\frac {7}{2},\frac {(b c-a d) x^2}{c \left (b x^2+a\right )}\right ) x^{10}-8 a d^5 \, _5F_4\left (2,2,2,2,\frac {9}{4};1,1,1,\frac {7}{2};\frac {(b c-a d) x^2}{c \left (b x^2+a\right )}\right ) x^{10}+8 b c d^4 \, _5F_4\left (2,2,2,2,\frac {9}{4};1,1,1,\frac {7}{2};\frac {(b c-a d) x^2}{c \left (b x^2+a\right )}\right ) x^{10}+384 a c d^4 \operatorname {Hypergeometric2F1}\left (1,\frac {5}{4},\frac {5}{2},\frac {(b c-a d) x^2}{c \left (b x^2+a\right )}\right ) x^8+576 b c^2 d^3 \operatorname {Hypergeometric2F1}\left (1,\frac {5}{4},\frac {5}{2},\frac {(b c-a d) x^2}{c \left (b x^2+a\right )}\right ) x^8-720 a c d^4 \operatorname {Hypergeometric2F1}\left (2,\frac {9}{4},\frac {7}{2},\frac {(b c-a d) x^2}{c \left (b x^2+a\right )}\right ) x^8+720 b c^2 d^3 \operatorname {Hypergeometric2F1}\left (2,\frac {9}{4},\frac {7}{2},\frac {(b c-a d) x^2}{c \left (b x^2+a\right )}\right ) x^8-32 a c d^4 \, _5F_4\left (2,2,2,2,\frac {9}{4};1,1,1,\frac {7}{2};\frac {(b c-a d) x^2}{c \left (b x^2+a\right )}\right ) x^8+32 b c^2 d^3 \, _5F_4\left (2,2,2,2,\frac {9}{4};1,1,1,\frac {7}{2};\frac {(b c-a d) x^2}{c \left (b x^2+a\right )}\right ) x^8+576 a c^2 d^3 \operatorname {Hypergeometric2F1}\left (1,\frac {5}{4},\frac {5}{2},\frac {(b c-a d) x^2}{c \left (b x^2+a\right )}\right ) x^6+144 b c^3 d^2 \operatorname {Hypergeometric2F1}\left (1,\frac {5}{4},\frac {5}{2},\frac {(b c-a d) x^2}{c \left (b x^2+a\right )}\right ) x^6-300 a c^2 d^3 \operatorname {Hypergeometric2F1}\left (2,\frac {9}{4},\frac {7}{2},\frac {(b c-a d) x^2}{c \left (b x^2+a\right )}\right ) x^6+300 b c^3 d^2 \operatorname {Hypergeometric2F1}\left (2,\frac {9}{4},\frac {7}{2},\frac {(b c-a d) x^2}{c \left (b x^2+a\right )}\right ) x^6-48 a c^2 d^3 \, _5F_4\left (2,2,2,2,\frac {9}{4};1,1,1,\frac {7}{2};\frac {(b c-a d) x^2}{c \left (b x^2+a\right )}\right ) x^6+48 b c^3 d^2 \, _5F_4\left (2,2,2,2,\frac {9}{4};1,1,1,\frac {7}{2};\frac {(b c-a d) x^2}{c \left (b x^2+a\right )}\right ) x^6+144 a c^3 d^2 \operatorname {Hypergeometric2F1}\left (1,\frac {5}{4},\frac {5}{2},\frac {(b c-a d) x^2}{c \left (b x^2+a\right )}\right ) x^4-24 b c^4 d \operatorname {Hypergeometric2F1}\left (1,\frac {5}{4},\frac {5}{2},\frac {(b c-a d) x^2}{c \left (b x^2+a\right )}\right ) x^4+20 a c^3 d^2 \operatorname {Hypergeometric2F1}\left (2,\frac {9}{4},\frac {7}{2},\frac {(b c-a d) x^2}{c \left (b x^2+a\right )}\right ) x^4-20 b c^4 d \operatorname {Hypergeometric2F1}\left (2,\frac {9}{4},\frac {7}{2},\frac {(b c-a d) x^2}{c \left (b x^2+a\right )}\right ) x^4+80 d (b c-a d) \left (d x^2+c\right )^3 \, _4F_3\left (2,2,2,\frac {9}{4};1,1,\frac {7}{2};\frac {(b c-a d) x^2}{c \left (b x^2+a\right )}\right ) x^4-32 a c^3 d^2 \, _5F_4\left (2,2,2,2,\frac {9}{4};1,1,1,\frac {7}{2};\frac {(b c-a d) x^2}{c \left (b x^2+a\right )}\right ) x^4+32 b c^4 d \, _5F_4\left (2,2,2,2,\frac {9}{4};1,1,1,\frac {7}{2};\frac {(b c-a d) x^2}{c \left (b x^2+a\right )}\right ) x^4+9 b c^5 \operatorname {Hypergeometric2F1}\left (1,\frac {5}{4},\frac {5}{2},\frac {(b c-a d) x^2}{c \left (b x^2+a\right )}\right ) x^2-24 a c^4 d \operatorname {Hypergeometric2F1}\left (1,\frac {5}{4},\frac {5}{2},\frac {(b c-a d) x^2}{c \left (b x^2+a\right )}\right ) x^2-20 (b c-a d) \left (d x^2+c\right )^2 \left (-14 d^2 x^4-4 c d x^2+c^2\right ) \, _3F_2\left (2,2,\frac {9}{4};1,\frac {7}{2};\frac {(b c-a d) x^2}{c \left (b x^2+a\right )}\right ) x^2+8 b c^5 \, _5F_4\left (2,2,2,2,\frac {9}{4};1,1,1,\frac {7}{2};\frac {(b c-a d) x^2}{c \left (b x^2+a\right )}\right ) x^2-8 a c^4 d \, _5F_4\left (2,2,2,2,\frac {9}{4};1,1,1,\frac {7}{2};\frac {(b c-a d) x^2}{c \left (b x^2+a\right )}\right ) x^2+9 a c^5 \operatorname {Hypergeometric2F1}\left (1,\frac {5}{4},\frac {5}{2},\frac {(b c-a d) x^2}{c \left (b x^2+a\right )}\right )\right )}{180 a c^6 x^5 \left (b x^2+a\right )^{5/4} \left (\frac {b x^2}{a}+1\right ) \sqrt [4]{d x^2+c} \operatorname {Gamma}\left (\frac {5}{4}\right )}\) |
Input:
Int[1/(x^6*(a + b*x^2)^(5/4)*(c + d*x^2)^(5/4)),x]
Output:
-1/180*(Gamma[1/4]*(9*a*c^5*Hypergeometric2F1[1, 5/4, 5/2, ((b*c - a*d)*x^ 2)/(c*(a + b*x^2))] + 9*b*c^5*x^2*Hypergeometric2F1[1, 5/4, 5/2, ((b*c - a *d)*x^2)/(c*(a + b*x^2))] - 24*a*c^4*d*x^2*Hypergeometric2F1[1, 5/4, 5/2, ((b*c - a*d)*x^2)/(c*(a + b*x^2))] - 24*b*c^4*d*x^4*Hypergeometric2F1[1, 5 /4, 5/2, ((b*c - a*d)*x^2)/(c*(a + b*x^2))] + 144*a*c^3*d^2*x^4*Hypergeome tric2F1[1, 5/4, 5/2, ((b*c - a*d)*x^2)/(c*(a + b*x^2))] + 144*b*c^3*d^2*x^ 6*Hypergeometric2F1[1, 5/4, 5/2, ((b*c - a*d)*x^2)/(c*(a + b*x^2))] + 576* a*c^2*d^3*x^6*Hypergeometric2F1[1, 5/4, 5/2, ((b*c - a*d)*x^2)/(c*(a + b*x ^2))] + 576*b*c^2*d^3*x^8*Hypergeometric2F1[1, 5/4, 5/2, ((b*c - a*d)*x^2) /(c*(a + b*x^2))] + 384*a*c*d^4*x^8*Hypergeometric2F1[1, 5/4, 5/2, ((b*c - a*d)*x^2)/(c*(a + b*x^2))] + 384*b*c*d^4*x^10*Hypergeometric2F1[1, 5/4, 5 /2, ((b*c - a*d)*x^2)/(c*(a + b*x^2))] - 20*b*c^4*d*x^4*Hypergeometric2F1[ 2, 9/4, 7/2, ((b*c - a*d)*x^2)/(c*(a + b*x^2))] + 20*a*c^3*d^2*x^4*Hyperge ometric2F1[2, 9/4, 7/2, ((b*c - a*d)*x^2)/(c*(a + b*x^2))] + 300*b*c^3*d^2 *x^6*Hypergeometric2F1[2, 9/4, 7/2, ((b*c - a*d)*x^2)/(c*(a + b*x^2))] - 3 00*a*c^2*d^3*x^6*Hypergeometric2F1[2, 9/4, 7/2, ((b*c - a*d)*x^2)/(c*(a + b*x^2))] + 720*b*c^2*d^3*x^8*Hypergeometric2F1[2, 9/4, 7/2, ((b*c - a*d)*x ^2)/(c*(a + b*x^2))] - 720*a*c*d^4*x^8*Hypergeometric2F1[2, 9/4, 7/2, ((b* c - a*d)*x^2)/(c*(a + b*x^2))] + 400*b*c*d^4*x^10*Hypergeometric2F1[2, 9/4 , 7/2, ((b*c - a*d)*x^2)/(c*(a + b*x^2))] - 400*a*d^5*x^10*Hypergeometr...
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])
\[\int \frac {1}{x^{6} \left (b \,x^{2}+a \right )^{\frac {5}{4}} \left (x^{2} d +c \right )^{\frac {5}{4}}}d x\]
Input:
int(1/x^6/(b*x^2+a)^(5/4)/(d*x^2+c)^(5/4),x)
Output:
int(1/x^6/(b*x^2+a)^(5/4)/(d*x^2+c)^(5/4),x)
\[ \int \frac {1}{x^6 \left (a+b x^2\right )^{5/4} \left (c+d x^2\right )^{5/4}} \, dx=\int { \frac {1}{{\left (b x^{2} + a\right )}^{\frac {5}{4}} {\left (d x^{2} + c\right )}^{\frac {5}{4}} x^{6}} \,d x } \] Input:
integrate(1/x^6/(b*x^2+a)^(5/4)/(d*x^2+c)^(5/4),x, algorithm="fricas")
Output:
integral((b*x^2 + a)^(3/4)*(d*x^2 + c)^(3/4)/(b^2*d^2*x^14 + 2*(b^2*c*d + a*b*d^2)*x^12 + (b^2*c^2 + 4*a*b*c*d + a^2*d^2)*x^10 + a^2*c^2*x^6 + 2*(a* b*c^2 + a^2*c*d)*x^8), x)
\[ \int \frac {1}{x^6 \left (a+b x^2\right )^{5/4} \left (c+d x^2\right )^{5/4}} \, dx=\int \frac {1}{x^{6} \left (a + b x^{2}\right )^{\frac {5}{4}} \left (c + d x^{2}\right )^{\frac {5}{4}}}\, dx \] Input:
integrate(1/x**6/(b*x**2+a)**(5/4)/(d*x**2+c)**(5/4),x)
Output:
Integral(1/(x**6*(a + b*x**2)**(5/4)*(c + d*x**2)**(5/4)), x)
\[ \int \frac {1}{x^6 \left (a+b x^2\right )^{5/4} \left (c+d x^2\right )^{5/4}} \, dx=\int { \frac {1}{{\left (b x^{2} + a\right )}^{\frac {5}{4}} {\left (d x^{2} + c\right )}^{\frac {5}{4}} x^{6}} \,d x } \] Input:
integrate(1/x^6/(b*x^2+a)^(5/4)/(d*x^2+c)^(5/4),x, algorithm="maxima")
Output:
integrate(1/((b*x^2 + a)^(5/4)*(d*x^2 + c)^(5/4)*x^6), x)
\[ \int \frac {1}{x^6 \left (a+b x^2\right )^{5/4} \left (c+d x^2\right )^{5/4}} \, dx=\int { \frac {1}{{\left (b x^{2} + a\right )}^{\frac {5}{4}} {\left (d x^{2} + c\right )}^{\frac {5}{4}} x^{6}} \,d x } \] Input:
integrate(1/x^6/(b*x^2+a)^(5/4)/(d*x^2+c)^(5/4),x, algorithm="giac")
Output:
integrate(1/((b*x^2 + a)^(5/4)*(d*x^2 + c)^(5/4)*x^6), x)
Timed out. \[ \int \frac {1}{x^6 \left (a+b x^2\right )^{5/4} \left (c+d x^2\right )^{5/4}} \, dx=\int \frac {1}{x^6\,{\left (b\,x^2+a\right )}^{5/4}\,{\left (d\,x^2+c\right )}^{5/4}} \,d x \] Input:
int(1/(x^6*(a + b*x^2)^(5/4)*(c + d*x^2)^(5/4)),x)
Output:
int(1/(x^6*(a + b*x^2)^(5/4)*(c + d*x^2)^(5/4)), x)
\[ \int \frac {1}{x^6 \left (a+b x^2\right )^{5/4} \left (c+d x^2\right )^{5/4}} \, dx=\int \frac {1}{\left (d \,x^{2}+c \right )^{\frac {1}{4}} \left (b \,x^{2}+a \right )^{\frac {1}{4}} a c \,x^{6}+\left (d \,x^{2}+c \right )^{\frac {1}{4}} \left (b \,x^{2}+a \right )^{\frac {1}{4}} a d \,x^{8}+\left (d \,x^{2}+c \right )^{\frac {1}{4}} \left (b \,x^{2}+a \right )^{\frac {1}{4}} b c \,x^{8}+\left (d \,x^{2}+c \right )^{\frac {1}{4}} \left (b \,x^{2}+a \right )^{\frac {1}{4}} b d \,x^{10}}d x \] Input:
int(1/x^6/(b*x^2+a)^(5/4)/(d*x^2+c)^(5/4),x)
Output:
int(1/((c + d*x**2)**(1/4)*(a + b*x**2)**(1/4)*a*c*x**6 + (c + d*x**2)**(1 /4)*(a + b*x**2)**(1/4)*a*d*x**8 + (c + d*x**2)**(1/4)*(a + b*x**2)**(1/4) *b*c*x**8 + (c + d*x**2)**(1/4)*(a + b*x**2)**(1/4)*b*d*x**10),x)