Integrand size = 21, antiderivative size = 357 \[ \int \frac {1}{\left (a+b x^4\right )^{11/4} \left (c+d x^4\right )} \, dx=\frac {b x}{7 a (b c-a d) \left (a+b x^4\right )^{7/4}}+\frac {b (6 b c-13 a d) x}{21 a^2 (b c-a d)^2 \left (a+b x^4\right )^{3/4}}-\frac {b^{3/2} \left (12 b^2 c^2-38 a b c d+47 a^2 d^2\right ) \left (1+\frac {a}{b x^4}\right )^{3/4} x^3 \operatorname {EllipticF}\left (\frac {1}{2} \cot ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right ),2\right )}{21 a^{5/2} (b c-a d)^3 \left (a+b x^4\right )^{3/4}}-\frac {d^3 \sqrt {\frac {a}{a+b x^4}} \sqrt {a+b x^4} \operatorname {EllipticPi}\left (-\frac {\sqrt {b c-a d}}{\sqrt {b} \sqrt {c}},\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right ),-1\right )}{2 \sqrt [4]{b} c (b c-a d)^3}-\frac {d^3 \sqrt {\frac {a}{a+b x^4}} \sqrt {a+b x^4} \operatorname {EllipticPi}\left (\frac {\sqrt {b c-a d}}{\sqrt {b} \sqrt {c}},\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right ),-1\right )}{2 \sqrt [4]{b} c (b c-a d)^3} \] Output:
1/7*b*x/a/(-a*d+b*c)/(b*x^4+a)^(7/4)+1/21*b*(-13*a*d+6*b*c)*x/a^2/(-a*d+b* c)^2/(b*x^4+a)^(3/4)-1/21*b^(3/2)*(47*a^2*d^2-38*a*b*c*d+12*b^2*c^2)*(1+a/ b/x^4)^(3/4)*x^3*InverseJacobiAM(1/2*arccot(b^(1/2)*x^2/a^(1/2)),2^(1/2))/ a^(5/2)/(-a*d+b*c)^3/(b*x^4+a)^(3/4)-1/2*d^3*(a/(b*x^4+a))^(1/2)*(b*x^4+a) ^(1/2)*EllipticPi(b^(1/4)*x/(b*x^4+a)^(1/4),-(-a*d+b*c)^(1/2)/b^(1/2)/c^(1 /2),I)/b^(1/4)/c/(-a*d+b*c)^3-1/2*d^3*(a/(b*x^4+a))^(1/2)*(b*x^4+a)^(1/2)* EllipticPi(b^(1/4)*x/(b*x^4+a)^(1/4),(-a*d+b*c)^(1/2)/b^(1/2)/c^(1/2),I)/b ^(1/4)/c/(-a*d+b*c)^3
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 10.94 (sec) , antiderivative size = 430, normalized size of antiderivative = 1.20 \[ \int \frac {1}{\left (a+b x^4\right )^{11/4} \left (c+d x^4\right )} \, dx=\frac {x \left (-\frac {2 b d (-6 b c+13 a d) x^4 \left (1+\frac {b x^4}{a}\right )^{3/4} \operatorname {AppellF1}\left (\frac {5}{4},\frac {3}{4},1,\frac {9}{4},-\frac {b x^4}{a},-\frac {d x^4}{c}\right )}{c}-\frac {5 \left (5 a c \left (21 a^3 d^2+6 b^3 c x^4 \left (3 c+d x^4\right )+a^2 b d \left (-42 c+5 d x^4\right )+a b^2 \left (21 c^2-30 c d x^4-13 d^2 x^8\right )\right ) \operatorname {AppellF1}\left (\frac {1}{4},\frac {3}{4},1,\frac {5}{4},-\frac {b x^4}{a},-\frac {d x^4}{c}\right )+b x^4 \left (c+d x^4\right ) \left (16 a^2 d-6 b^2 c x^4+a b \left (-9 c+13 d x^4\right )\right ) \left (4 a d \operatorname {AppellF1}\left (\frac {5}{4},\frac {3}{4},2,\frac {9}{4},-\frac {b x^4}{a},-\frac {d x^4}{c}\right )+3 b c \operatorname {AppellF1}\left (\frac {5}{4},\frac {7}{4},1,\frac {9}{4},-\frac {b x^4}{a},-\frac {d x^4}{c}\right )\right )\right )}{\left (a+b x^4\right ) \left (c+d x^4\right ) \left (-5 a c \operatorname {AppellF1}\left (\frac {1}{4},\frac {3}{4},1,\frac {5}{4},-\frac {b x^4}{a},-\frac {d x^4}{c}\right )+x^4 \left (4 a d \operatorname {AppellF1}\left (\frac {5}{4},\frac {3}{4},2,\frac {9}{4},-\frac {b x^4}{a},-\frac {d x^4}{c}\right )+3 b c \operatorname {AppellF1}\left (\frac {5}{4},\frac {7}{4},1,\frac {9}{4},-\frac {b x^4}{a},-\frac {d x^4}{c}\right )\right )\right )}\right )}{105 a^2 (b c-a d)^2 \left (a+b x^4\right )^{3/4}} \] Input:
Integrate[1/((a + b*x^4)^(11/4)*(c + d*x^4)),x]
Output:
(x*((-2*b*d*(-6*b*c + 13*a*d)*x^4*(1 + (b*x^4)/a)^(3/4)*AppellF1[5/4, 3/4, 1, 9/4, -((b*x^4)/a), -((d*x^4)/c)])/c - (5*(5*a*c*(21*a^3*d^2 + 6*b^3*c* x^4*(3*c + d*x^4) + a^2*b*d*(-42*c + 5*d*x^4) + a*b^2*(21*c^2 - 30*c*d*x^4 - 13*d^2*x^8))*AppellF1[1/4, 3/4, 1, 5/4, -((b*x^4)/a), -((d*x^4)/c)] + b *x^4*(c + d*x^4)*(16*a^2*d - 6*b^2*c*x^4 + a*b*(-9*c + 13*d*x^4))*(4*a*d*A ppellF1[5/4, 3/4, 2, 9/4, -((b*x^4)/a), -((d*x^4)/c)] + 3*b*c*AppellF1[5/4 , 7/4, 1, 9/4, -((b*x^4)/a), -((d*x^4)/c)])))/((a + b*x^4)*(c + d*x^4)*(-5 *a*c*AppellF1[1/4, 3/4, 1, 5/4, -((b*x^4)/a), -((d*x^4)/c)] + x^4*(4*a*d*A ppellF1[5/4, 3/4, 2, 9/4, -((b*x^4)/a), -((d*x^4)/c)] + 3*b*c*AppellF1[5/4 , 7/4, 1, 9/4, -((b*x^4)/a), -((d*x^4)/c)])))))/(105*a^2*(b*c - a*d)^2*(a + b*x^4)^(3/4))
Time = 1.21 (sec) , antiderivative size = 358, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.619, Rules used = {931, 25, 1024, 25, 404, 768, 858, 807, 229, 923, 925, 27, 1542}
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.
| \(\displaystyle \int \frac {1}{\left (a+b x^4\right )^{11/4} \left (c+d x^4\right )} \, dx\) |
| \(\Big \downarrow \) 931 |
| \(\displaystyle \frac {b x}{7 a \left (a+b x^4\right )^{7/4} (b c-a d)}-\frac {\int -\frac {6 b d x^4+6 b c-7 a d}{\left (b x^4+a\right )^{7/4} \left (d x^4+c\right )}dx}{7 a (b c-a d)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {6 b d x^4+6 b c-7 a d}{\left (b x^4+a\right )^{7/4} \left (d x^4+c\right )}dx}{7 a (b c-a d)}+\frac {b x}{7 a \left (a+b x^4\right )^{7/4} (b c-a d)}\) |
| \(\Big \downarrow \) 1024 |
| \(\displaystyle \frac {\frac {b x (6 b c-13 a d)}{3 a \left (a+b x^4\right )^{3/4} (b c-a d)}-\frac {\int -\frac {2 b d (6 b c-13 a d) x^4+12 b^2 c^2+21 a^2 d^2-26 a b c d}{\left (b x^4+a\right )^{3/4} \left (d x^4+c\right )}dx}{3 a (b c-a d)}}{7 a (b c-a d)}+\frac {b x}{7 a \left (a+b x^4\right )^{7/4} (b c-a d)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int \frac {2 b d (6 b c-13 a d) x^4+12 b^2 c^2+21 a^2 d^2-26 a b c d}{\left (b x^4+a\right )^{3/4} \left (d x^4+c\right )}dx}{3 a (b c-a d)}+\frac {b x (6 b c-13 a d)}{3 a \left (a+b x^4\right )^{3/4} (b c-a d)}}{7 a (b c-a d)}+\frac {b x}{7 a \left (a+b x^4\right )^{7/4} (b c-a d)}\) |
| \(\Big \downarrow \) 404 |
| \(\displaystyle \frac {\frac {\frac {b \left (47 a^2 d^2-38 a b c d+12 b^2 c^2\right ) \int \frac {1}{\left (b x^4+a\right )^{3/4}}dx}{b c-a d}-\frac {21 a^2 d^3 \int \frac {\sqrt [4]{b x^4+a}}{d x^4+c}dx}{b c-a d}}{3 a (b c-a d)}+\frac {b x (6 b c-13 a d)}{3 a \left (a+b x^4\right )^{3/4} (b c-a d)}}{7 a (b c-a d)}+\frac {b x}{7 a \left (a+b x^4\right )^{7/4} (b c-a d)}\) |
| \(\Big \downarrow \) 768 |
| \(\displaystyle \frac {\frac {\frac {b x^3 \left (\frac {a}{b x^4}+1\right )^{3/4} \left (47 a^2 d^2-38 a b c d+12 b^2 c^2\right ) \int \frac {1}{\left (\frac {a}{b x^4}+1\right )^{3/4} x^3}dx}{\left (a+b x^4\right )^{3/4} (b c-a d)}-\frac {21 a^2 d^3 \int \frac {\sqrt [4]{b x^4+a}}{d x^4+c}dx}{b c-a d}}{3 a (b c-a d)}+\frac {b x (6 b c-13 a d)}{3 a \left (a+b x^4\right )^{3/4} (b c-a d)}}{7 a (b c-a d)}+\frac {b x}{7 a \left (a+b x^4\right )^{7/4} (b c-a d)}\) |
| \(\Big \downarrow \) 858 |
| \(\displaystyle \frac {\frac {-\frac {b x^3 \left (\frac {a}{b x^4}+1\right )^{3/4} \left (47 a^2 d^2-38 a b c d+12 b^2 c^2\right ) \int \frac {1}{\left (\frac {a}{b x^4}+1\right )^{3/4} x}d\frac {1}{x}}{\left (a+b x^4\right )^{3/4} (b c-a d)}-\frac {21 a^2 d^3 \int \frac {\sqrt [4]{b x^4+a}}{d x^4+c}dx}{b c-a d}}{3 a (b c-a d)}+\frac {b x (6 b c-13 a d)}{3 a \left (a+b x^4\right )^{3/4} (b c-a d)}}{7 a (b c-a d)}+\frac {b x}{7 a \left (a+b x^4\right )^{7/4} (b c-a d)}\) |
| \(\Big \downarrow \) 807 |
| \(\displaystyle \frac {\frac {-\frac {b x^3 \left (\frac {a}{b x^4}+1\right )^{3/4} \left (47 a^2 d^2-38 a b c d+12 b^2 c^2\right ) \int \frac {1}{\left (\frac {a}{b x^2}+1\right )^{3/4}}d\frac {1}{x^2}}{2 \left (a+b x^4\right )^{3/4} (b c-a d)}-\frac {21 a^2 d^3 \int \frac {\sqrt [4]{b x^4+a}}{d x^4+c}dx}{b c-a d}}{3 a (b c-a d)}+\frac {b x (6 b c-13 a d)}{3 a \left (a+b x^4\right )^{3/4} (b c-a d)}}{7 a (b c-a d)}+\frac {b x}{7 a \left (a+b x^4\right )^{7/4} (b c-a d)}\) |
| \(\Big \downarrow \) 229 |
| \(\displaystyle \frac {\frac {-\frac {21 a^2 d^3 \int \frac {\sqrt [4]{b x^4+a}}{d x^4+c}dx}{b c-a d}-\frac {b^{3/2} x^3 \left (\frac {a}{b x^4}+1\right )^{3/4} \left (47 a^2 d^2-38 a b c d+12 b^2 c^2\right ) \operatorname {EllipticF}\left (\frac {1}{2} \arctan \left (\frac {\sqrt {a}}{\sqrt {b} x^2}\right ),2\right )}{\sqrt {a} \left (a+b x^4\right )^{3/4} (b c-a d)}}{3 a (b c-a d)}+\frac {b x (6 b c-13 a d)}{3 a \left (a+b x^4\right )^{3/4} (b c-a d)}}{7 a (b c-a d)}+\frac {b x}{7 a \left (a+b x^4\right )^{7/4} (b c-a d)}\) |
| \(\Big \downarrow \) 923 |
| \(\displaystyle \frac {\frac {-\frac {21 a^2 d^3 \sqrt {\frac {a}{a+b x^4}} \sqrt {a+b x^4} \int \frac {1}{\sqrt {1-\frac {b x^4}{b x^4+a}} \left (c-\frac {(b c-a d) x^4}{b x^4+a}\right )}d\frac {x}{\sqrt [4]{b x^4+a}}}{b c-a d}-\frac {b^{3/2} x^3 \left (\frac {a}{b x^4}+1\right )^{3/4} \left (47 a^2 d^2-38 a b c d+12 b^2 c^2\right ) \operatorname {EllipticF}\left (\frac {1}{2} \arctan \left (\frac {\sqrt {a}}{\sqrt {b} x^2}\right ),2\right )}{\sqrt {a} \left (a+b x^4\right )^{3/4} (b c-a d)}}{3 a (b c-a d)}+\frac {b x (6 b c-13 a d)}{3 a \left (a+b x^4\right )^{3/4} (b c-a d)}}{7 a (b c-a d)}+\frac {b x}{7 a \left (a+b x^4\right )^{7/4} (b c-a d)}\) |
| \(\Big \downarrow \) 925 |
| \(\displaystyle \frac {\frac {-\frac {21 a^2 d^3 \sqrt {\frac {a}{a+b x^4}} \sqrt {a+b x^4} \left (\frac {\int \frac {\sqrt {c}}{\sqrt {1-\frac {b x^4}{b x^4+a}} \left (\sqrt {c}-\frac {\sqrt {b c-a d} x^2}{\sqrt {b x^4+a}}\right )}d\frac {x}{\sqrt [4]{b x^4+a}}}{2 c}+\frac {\int \frac {\sqrt {c}}{\sqrt {1-\frac {b x^4}{b x^4+a}} \left (\frac {\sqrt {b c-a d} x^2}{\sqrt {b x^4+a}}+\sqrt {c}\right )}d\frac {x}{\sqrt [4]{b x^4+a}}}{2 c}\right )}{b c-a d}-\frac {b^{3/2} x^3 \left (\frac {a}{b x^4}+1\right )^{3/4} \left (47 a^2 d^2-38 a b c d+12 b^2 c^2\right ) \operatorname {EllipticF}\left (\frac {1}{2} \arctan \left (\frac {\sqrt {a}}{\sqrt {b} x^2}\right ),2\right )}{\sqrt {a} \left (a+b x^4\right )^{3/4} (b c-a d)}}{3 a (b c-a d)}+\frac {b x (6 b c-13 a d)}{3 a \left (a+b x^4\right )^{3/4} (b c-a d)}}{7 a (b c-a d)}+\frac {b x}{7 a \left (a+b x^4\right )^{7/4} (b c-a d)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {-\frac {21 a^2 d^3 \sqrt {\frac {a}{a+b x^4}} \sqrt {a+b x^4} \left (\frac {\int \frac {1}{\sqrt {1-\frac {b x^4}{b x^4+a}} \left (\sqrt {c}-\frac {\sqrt {b c-a d} x^2}{\sqrt {b x^4+a}}\right )}d\frac {x}{\sqrt [4]{b x^4+a}}}{2 \sqrt {c}}+\frac {\int \frac {1}{\sqrt {1-\frac {b x^4}{b x^4+a}} \left (\frac {\sqrt {b c-a d} x^2}{\sqrt {b x^4+a}}+\sqrt {c}\right )}d\frac {x}{\sqrt [4]{b x^4+a}}}{2 \sqrt {c}}\right )}{b c-a d}-\frac {b^{3/2} x^3 \left (\frac {a}{b x^4}+1\right )^{3/4} \left (47 a^2 d^2-38 a b c d+12 b^2 c^2\right ) \operatorname {EllipticF}\left (\frac {1}{2} \arctan \left (\frac {\sqrt {a}}{\sqrt {b} x^2}\right ),2\right )}{\sqrt {a} \left (a+b x^4\right )^{3/4} (b c-a d)}}{3 a (b c-a d)}+\frac {b x (6 b c-13 a d)}{3 a \left (a+b x^4\right )^{3/4} (b c-a d)}}{7 a (b c-a d)}+\frac {b x}{7 a \left (a+b x^4\right )^{7/4} (b c-a d)}\) |
| \(\Big \downarrow \) 1542 |
| \(\displaystyle \frac {\frac {-\frac {21 a^2 d^3 \sqrt {\frac {a}{a+b x^4}} \sqrt {a+b x^4} \left (\frac {\operatorname {EllipticPi}\left (-\frac {\sqrt {b c-a d}}{\sqrt {b} \sqrt {c}},\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{b x^4+a}}\right ),-1\right )}{2 \sqrt [4]{b} c}+\frac {\operatorname {EllipticPi}\left (\frac {\sqrt {b c-a d}}{\sqrt {b} \sqrt {c}},\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{b x^4+a}}\right ),-1\right )}{2 \sqrt [4]{b} c}\right )}{b c-a d}-\frac {b^{3/2} x^3 \left (\frac {a}{b x^4}+1\right )^{3/4} \left (47 a^2 d^2-38 a b c d+12 b^2 c^2\right ) \operatorname {EllipticF}\left (\frac {1}{2} \arctan \left (\frac {\sqrt {a}}{\sqrt {b} x^2}\right ),2\right )}{\sqrt {a} \left (a+b x^4\right )^{3/4} (b c-a d)}}{3 a (b c-a d)}+\frac {b x (6 b c-13 a d)}{3 a \left (a+b x^4\right )^{3/4} (b c-a d)}}{7 a (b c-a d)}+\frac {b x}{7 a \left (a+b x^4\right )^{7/4} (b c-a d)}\) |
Input:
Int[1/((a + b*x^4)^(11/4)*(c + d*x^4)),x]
Output:
(b*x)/(7*a*(b*c - a*d)*(a + b*x^4)^(7/4)) + ((b*(6*b*c - 13*a*d)*x)/(3*a*( b*c - a*d)*(a + b*x^4)^(3/4)) + (-((b^(3/2)*(12*b^2*c^2 - 38*a*b*c*d + 47* a^2*d^2)*(1 + a/(b*x^4))^(3/4)*x^3*EllipticF[ArcTan[Sqrt[a]/(Sqrt[b]*x^2)] /2, 2])/(Sqrt[a]*(b*c - a*d)*(a + b*x^4)^(3/4))) - (21*a^2*d^3*Sqrt[a/(a + b*x^4)]*Sqrt[a + b*x^4]*(EllipticPi[-(Sqrt[b*c - a*d]/(Sqrt[b]*Sqrt[c])), ArcSin[(b^(1/4)*x)/(a + b*x^4)^(1/4)], -1]/(2*b^(1/4)*c) + EllipticPi[Sqr t[b*c - a*d]/(Sqrt[b]*Sqrt[c]), ArcSin[(b^(1/4)*x)/(a + b*x^4)^(1/4)], -1] /(2*b^(1/4)*c)))/(b*c - a*d))/(3*a*(b*c - a*d)))/(7*a*(b*c - a*d))
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-3/4), x_Symbol] :> Simp[(2/(a^(3/4)*Rt[b/a, 2]) )*EllipticF[(1/2)*ArcTan[Rt[b/a, 2]*x], 2], x] /; FreeQ[{a, b}, x] && GtQ[a , 0] && PosQ[b/a]
Int[((e_) + (f_.)*(x_)^4)/(((a_) + (b_.)*(x_)^4)^(3/4)*((c_) + (d_.)*(x_)^4 )), x_Symbol] :> Simp[(b*e - a*f)/(b*c - a*d) Int[1/(a + b*x^4)^(3/4), x] , x] - Simp[(d*e - c*f)/(b*c - a*d) Int[(a + b*x^4)^(1/4)/(c + d*x^4), x] , x] /; FreeQ[{a, b, c, d, e, f}, x]
Int[((a_) + (b_.)*(x_)^4)^(-3/4), x_Symbol] :> Simp[x^3*((1 + a/(b*x^4))^(3 /4)/(a + b*x^4)^(3/4)) Int[1/(x^3*(1 + a/(b*x^4))^(3/4)), x], x] /; FreeQ [{a, b}, x]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Simp[1/k Subst[Int[x^((m + 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Subst[Int[(a + b/x^n)^p/x^(m + 2), x], x, 1/x] /; FreeQ[{a, b, p}, x] && ILtQ[n, 0] && Int egerQ[m]
Int[((a_) + (b_.)*(x_)^4)^(1/4)/((c_) + (d_.)*(x_)^4), x_Symbol] :> Simp[Sq rt[a + b*x^4]*Sqrt[a/(a + b*x^4)] Subst[Int[1/(Sqrt[1 - b*x^4]*(c - (b*c - a*d)*x^4)), x], x, x/(a + b*x^4)^(1/4)], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]
Int[1/(Sqrt[(a_) + (b_.)*(x_)^4]*((c_) + (d_.)*(x_)^4)), x_Symbol] :> Simp[ 1/(2*c) Int[1/(Sqrt[a + b*x^4]*(1 - Rt[-d/c, 2]*x^2)), x], x] + Simp[1/(2 *c) Int[1/(Sqrt[a + b*x^4]*(1 + Rt[-d/c, 2]*x^2)), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]
Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(-b)*x*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*n*(p + 1)*(b*c - a*d))), x] + Simp[1/(a*n*(p + 1)*(b*c - a*d)) Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[b*c + n*(p + 1)*(b*c - a*d) + d*b*(n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, n, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && !( !IntegerQ[p] && IntegerQ[q] && LtQ[q, -1]) && IntBinomialQ[a, b, c, d, n, p, q, x]
Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f _.)*(x_)^(n_)), x_Symbol] :> Simp[(-(b*e - a*f))*x*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*n*(b*c - a*d)*(p + 1))), x] + Simp[1/(a*n*(b*c - a*d)*( p + 1)) Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(b*e - a*f) + e*n*(b *c - a*d)*(p + 1) + d*(b*e - a*f)*(n*(p + q + 2) + 1)*x^n, x], x], x] /; Fr eeQ[{a, b, c, d, e, f, n, q}, x] && LtQ[p, -1]
Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> With[ {q = Rt[-c/a, 4]}, Simp[(1/(d*Sqrt[a]*q))*EllipticPi[-e/(d*q^2), ArcSin[q*x ], -1], x]] /; FreeQ[{a, c, d, e}, x] && NegQ[c/a] && GtQ[a, 0]
\[\int \frac {1}{\left (b \,x^{4}+a \right )^{\frac {11}{4}} \left (d \,x^{4}+c \right )}d x\]
Input:
int(1/(b*x^4+a)^(11/4)/(d*x^4+c),x)
Output:
int(1/(b*x^4+a)^(11/4)/(d*x^4+c),x)
Timed out. \[ \int \frac {1}{\left (a+b x^4\right )^{11/4} \left (c+d x^4\right )} \, dx=\text {Timed out} \] Input:
integrate(1/(b*x^4+a)^(11/4)/(d*x^4+c),x, algorithm="fricas")
Output:
Timed out
\[ \int \frac {1}{\left (a+b x^4\right )^{11/4} \left (c+d x^4\right )} \, dx=\int \frac {1}{\left (a + b x^{4}\right )^{\frac {11}{4}} \left (c + d x^{4}\right )}\, dx \] Input:
integrate(1/(b*x**4+a)**(11/4)/(d*x**4+c),x)
Output:
Integral(1/((a + b*x**4)**(11/4)*(c + d*x**4)), x)
\[ \int \frac {1}{\left (a+b x^4\right )^{11/4} \left (c+d x^4\right )} \, dx=\int { \frac {1}{{\left (b x^{4} + a\right )}^{\frac {11}{4}} {\left (d x^{4} + c\right )}} \,d x } \] Input:
integrate(1/(b*x^4+a)^(11/4)/(d*x^4+c),x, algorithm="maxima")
Output:
integrate(1/((b*x^4 + a)^(11/4)*(d*x^4 + c)), x)
\[ \int \frac {1}{\left (a+b x^4\right )^{11/4} \left (c+d x^4\right )} \, dx=\int { \frac {1}{{\left (b x^{4} + a\right )}^{\frac {11}{4}} {\left (d x^{4} + c\right )}} \,d x } \] Input:
integrate(1/(b*x^4+a)^(11/4)/(d*x^4+c),x, algorithm="giac")
Output:
integrate(1/((b*x^4 + a)^(11/4)*(d*x^4 + c)), x)
Timed out. \[ \int \frac {1}{\left (a+b x^4\right )^{11/4} \left (c+d x^4\right )} \, dx=\int \frac {1}{{\left (b\,x^4+a\right )}^{11/4}\,\left (d\,x^4+c\right )} \,d x \] Input:
int(1/((a + b*x^4)^(11/4)*(c + d*x^4)),x)
Output:
int(1/((a + b*x^4)^(11/4)*(c + d*x^4)), x)
\[ \int \frac {1}{\left (a+b x^4\right )^{11/4} \left (c+d x^4\right )} \, dx=\int \frac {1}{\left (b \,x^{4}+a \right )^{\frac {3}{4}} a^{2} c +\left (b \,x^{4}+a \right )^{\frac {3}{4}} a^{2} d \,x^{4}+2 \left (b \,x^{4}+a \right )^{\frac {3}{4}} a b c \,x^{4}+2 \left (b \,x^{4}+a \right )^{\frac {3}{4}} a b d \,x^{8}+\left (b \,x^{4}+a \right )^{\frac {3}{4}} b^{2} c \,x^{8}+\left (b \,x^{4}+a \right )^{\frac {3}{4}} b^{2} d \,x^{12}}d x \] Input:
int(1/(b*x^4+a)^(11/4)/(d*x^4+c),x)
Output:
int(1/((a + b*x**4)**(3/4)*a**2*c + (a + b*x**4)**(3/4)*a**2*d*x**4 + 2*(a + b*x**4)**(3/4)*a*b*c*x**4 + 2*(a + b*x**4)**(3/4)*a*b*d*x**8 + (a + b*x **4)**(3/4)*b**2*c*x**8 + (a + b*x**4)**(3/4)*b**2*d*x**12),x)