Integrand size = 21, antiderivative size = 230 \[ \int \frac {\left (a+b x^4\right )^{7/4}}{\left (c+d x^4\right )^2} \, dx=-\frac {(b c-a d) x \left (a+b x^4\right )^{3/4}}{4 c d \left (c+d x^4\right )}+\frac {b^{7/4} \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{2 d^2}-\frac {(b c-a d)^{3/4} (4 b c+3 a d) \arctan \left (\frac {\sqrt [4]{b c-a d} x}{\sqrt [4]{c} \sqrt [4]{a+b x^4}}\right )}{8 c^{7/4} d^2}+\frac {b^{7/4} \text {arctanh}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{2 d^2}-\frac {(b c-a d)^{3/4} (4 b c+3 a d) \text {arctanh}\left (\frac {\sqrt [4]{b c-a d} x}{\sqrt [4]{c} \sqrt [4]{a+b x^4}}\right )}{8 c^{7/4} d^2} \]
-1/4*(-a*d+b*c)*x*(b*x^4+a)^(3/4)/c/d/(d*x^4+c)+1/2*b^(7/4)*arctan(b^(1/4) *x/(b*x^4+a)^(1/4))/d^2-1/8*(-a*d+b*c)^(3/4)*(3*a*d+4*b*c)*arctan((-a*d+b* c)^(1/4)*x/c^(1/4)/(b*x^4+a)^(1/4))/c^(7/4)/d^2+1/2*b^(7/4)*arctanh(b^(1/4 )*x/(b*x^4+a)^(1/4))/d^2-1/8*(-a*d+b*c)^(3/4)*(3*a*d+4*b*c)*arctanh((-a*d+ b*c)^(1/4)*x/c^(1/4)/(b*x^4+a)^(1/4))/c^(7/4)/d^2
Result contains complex when optimal does not.
Time = 2.08 (sec) , antiderivative size = 341, normalized size of antiderivative = 1.48 \[ \int \frac {\left (a+b x^4\right )^{7/4}}{\left (c+d x^4\right )^2} \, dx=\frac {\left (\frac {1}{16}+\frac {i}{16}\right ) \left (-\frac {(2-2 i) d (b c-a d) x \left (a+b x^4\right )^{3/4}}{c \left (c+d x^4\right )}+(4-4 i) b^{7/4} \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )-\frac {\left (4 b^2 c^2-a b c d-3 a^2 d^2\right ) \arctan \left (\frac {\frac {(1-i) \sqrt [4]{b c-a d} x^2}{\sqrt [4]{c} \sqrt [4]{a+b x^4}}-\frac {(1+i) \sqrt [4]{c} \sqrt [4]{a+b x^4}}{\sqrt [4]{b c-a d}}}{2 x}\right )}{c^{7/4} \sqrt [4]{b c-a d}}+(4-4 i) b^{7/4} \text {arctanh}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )-\frac {\left (4 b^2 c^2-a b c d-3 a^2 d^2\right ) \text {arctanh}\left (\frac {\frac {(1-i) \sqrt [4]{b c-a d} x^2}{\sqrt [4]{c} \sqrt [4]{a+b x^4}}+\frac {(1+i) \sqrt [4]{c} \sqrt [4]{a+b x^4}}{\sqrt [4]{b c-a d}}}{2 x}\right )}{c^{7/4} \sqrt [4]{b c-a d}}\right )}{d^2} \]
((1/16 + I/16)*(((-2 + 2*I)*d*(b*c - a*d)*x*(a + b*x^4)^(3/4))/(c*(c + d*x ^4)) + (4 - 4*I)*b^(7/4)*ArcTan[(b^(1/4)*x)/(a + b*x^4)^(1/4)] - ((4*b^2*c ^2 - a*b*c*d - 3*a^2*d^2)*ArcTan[(((1 - I)*(b*c - a*d)^(1/4)*x^2)/(c^(1/4) *(a + b*x^4)^(1/4)) - ((1 + I)*c^(1/4)*(a + b*x^4)^(1/4))/(b*c - a*d)^(1/4 ))/(2*x)])/(c^(7/4)*(b*c - a*d)^(1/4)) + (4 - 4*I)*b^(7/4)*ArcTanh[(b^(1/4 )*x)/(a + b*x^4)^(1/4)] - ((4*b^2*c^2 - a*b*c*d - 3*a^2*d^2)*ArcTanh[(((1 - I)*(b*c - a*d)^(1/4)*x^2)/(c^(1/4)*(a + b*x^4)^(1/4)) + ((1 + I)*c^(1/4) *(a + b*x^4)^(1/4))/(b*c - a*d)^(1/4))/(2*x)])/(c^(7/4)*(b*c - a*d)^(1/4)) ))/d^2
Time = 0.41 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.06, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.476, Rules used = {930, 1026, 770, 756, 216, 219, 902, 756, 218, 221}
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.
| \(\displaystyle \int \frac {\left (a+b x^4\right )^{7/4}}{\left (c+d x^4\right )^2} \, dx\) |
| \(\Big \downarrow \) 930 |
| \(\displaystyle \frac {\int \frac {4 b^2 c x^4+a (b c+3 a d)}{\sqrt [4]{b x^4+a} \left (d x^4+c\right )}dx}{4 c d}-\frac {x \left (a+b x^4\right )^{3/4} (b c-a d)}{4 c d \left (c+d x^4\right )}\) |
| \(\Big \downarrow \) 1026 |
| \(\displaystyle \frac {\frac {4 b^2 c \int \frac {1}{\sqrt [4]{b x^4+a}}dx}{d}-\frac {(b c-a d) (3 a d+4 b c) \int \frac {1}{\sqrt [4]{b x^4+a} \left (d x^4+c\right )}dx}{d}}{4 c d}-\frac {x \left (a+b x^4\right )^{3/4} (b c-a d)}{4 c d \left (c+d x^4\right )}\) |
| \(\Big \downarrow \) 770 |
| \(\displaystyle \frac {\frac {4 b^2 c \int \frac {1}{1-\frac {b x^4}{b x^4+a}}d\frac {x}{\sqrt [4]{b x^4+a}}}{d}-\frac {(b c-a d) (3 a d+4 b c) \int \frac {1}{\sqrt [4]{b x^4+a} \left (d x^4+c\right )}dx}{d}}{4 c d}-\frac {x \left (a+b x^4\right )^{3/4} (b c-a d)}{4 c d \left (c+d x^4\right )}\) |
| \(\Big \downarrow \) 756 |
| \(\displaystyle \frac {\frac {4 b^2 c \left (\frac {1}{2} \int \frac {1}{1-\frac {\sqrt {b} x^2}{\sqrt {b x^4+a}}}d\frac {x}{\sqrt [4]{b x^4+a}}+\frac {1}{2} \int \frac {1}{\frac {\sqrt {b} x^2}{\sqrt {b x^4+a}}+1}d\frac {x}{\sqrt [4]{b x^4+a}}\right )}{d}-\frac {(b c-a d) (3 a d+4 b c) \int \frac {1}{\sqrt [4]{b x^4+a} \left (d x^4+c\right )}dx}{d}}{4 c d}-\frac {x \left (a+b x^4\right )^{3/4} (b c-a d)}{4 c d \left (c+d x^4\right )}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {\frac {4 b^2 c \left (\frac {1}{2} \int \frac {1}{1-\frac {\sqrt {b} x^2}{\sqrt {b x^4+a}}}d\frac {x}{\sqrt [4]{b x^4+a}}+\frac {\arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{2 \sqrt [4]{b}}\right )}{d}-\frac {(b c-a d) (3 a d+4 b c) \int \frac {1}{\sqrt [4]{b x^4+a} \left (d x^4+c\right )}dx}{d}}{4 c d}-\frac {x \left (a+b x^4\right )^{3/4} (b c-a d)}{4 c d \left (c+d x^4\right )}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {4 b^2 c \left (\frac {\arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{2 \sqrt [4]{b}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{2 \sqrt [4]{b}}\right )}{d}-\frac {(b c-a d) (3 a d+4 b c) \int \frac {1}{\sqrt [4]{b x^4+a} \left (d x^4+c\right )}dx}{d}}{4 c d}-\frac {x \left (a+b x^4\right )^{3/4} (b c-a d)}{4 c d \left (c+d x^4\right )}\) |
| \(\Big \downarrow \) 902 |
| \(\displaystyle \frac {\frac {4 b^2 c \left (\frac {\arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{2 \sqrt [4]{b}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{2 \sqrt [4]{b}}\right )}{d}-\frac {(b c-a d) (3 a d+4 b c) \int \frac {1}{c-\frac {(b c-a d) x^4}{b x^4+a}}d\frac {x}{\sqrt [4]{b x^4+a}}}{d}}{4 c d}-\frac {x \left (a+b x^4\right )^{3/4} (b c-a d)}{4 c d \left (c+d x^4\right )}\) |
| \(\Big \downarrow \) 756 |
| \(\displaystyle \frac {\frac {4 b^2 c \left (\frac {\arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{2 \sqrt [4]{b}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{2 \sqrt [4]{b}}\right )}{d}-\frac {(b c-a d) (3 a d+4 b c) \left (\frac {\int \frac {1}{\sqrt {c}-\frac {\sqrt {b c-a d} x^2}{\sqrt {b x^4+a}}}d\frac {x}{\sqrt [4]{b x^4+a}}}{2 \sqrt {c}}+\frac {\int \frac {1}{\frac {\sqrt {b c-a d} x^2}{\sqrt {b x^4+a}}+\sqrt {c}}d\frac {x}{\sqrt [4]{b x^4+a}}}{2 \sqrt {c}}\right )}{d}}{4 c d}-\frac {x \left (a+b x^4\right )^{3/4} (b c-a d)}{4 c d \left (c+d x^4\right )}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {\frac {4 b^2 c \left (\frac {\arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{2 \sqrt [4]{b}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{2 \sqrt [4]{b}}\right )}{d}-\frac {(b c-a d) (3 a d+4 b c) \left (\frac {\int \frac {1}{\sqrt {c}-\frac {\sqrt {b c-a d} x^2}{\sqrt {b x^4+a}}}d\frac {x}{\sqrt [4]{b x^4+a}}}{2 \sqrt {c}}+\frac {\arctan \left (\frac {x \sqrt [4]{b c-a d}}{\sqrt [4]{c} \sqrt [4]{a+b x^4}}\right )}{2 c^{3/4} \sqrt [4]{b c-a d}}\right )}{d}}{4 c d}-\frac {x \left (a+b x^4\right )^{3/4} (b c-a d)}{4 c d \left (c+d x^4\right )}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\frac {4 b^2 c \left (\frac {\arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{2 \sqrt [4]{b}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{2 \sqrt [4]{b}}\right )}{d}-\frac {(b c-a d) (3 a d+4 b c) \left (\frac {\arctan \left (\frac {x \sqrt [4]{b c-a d}}{\sqrt [4]{c} \sqrt [4]{a+b x^4}}\right )}{2 c^{3/4} \sqrt [4]{b c-a d}}+\frac {\text {arctanh}\left (\frac {x \sqrt [4]{b c-a d}}{\sqrt [4]{c} \sqrt [4]{a+b x^4}}\right )}{2 c^{3/4} \sqrt [4]{b c-a d}}\right )}{d}}{4 c d}-\frac {x \left (a+b x^4\right )^{3/4} (b c-a d)}{4 c d \left (c+d x^4\right )}\) |
-1/4*((b*c - a*d)*x*(a + b*x^4)^(3/4))/(c*d*(c + d*x^4)) + ((4*b^2*c*(ArcT an[(b^(1/4)*x)/(a + b*x^4)^(1/4)]/(2*b^(1/4)) + ArcTanh[(b^(1/4)*x)/(a + b *x^4)^(1/4)]/(2*b^(1/4))))/d - ((b*c - a*d)*(4*b*c + 3*a*d)*(ArcTan[((b*c - a*d)^(1/4)*x)/(c^(1/4)*(a + b*x^4)^(1/4))]/(2*c^(3/4)*(b*c - a*d)^(1/4)) + ArcTanh[((b*c - a*d)^(1/4)*x)/(c^(1/4)*(a + b*x^4)^(1/4))]/(2*c^(3/4)*( b*c - a*d)^(1/4))))/d)/(4*c*d)
3.3.6.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2 ]], s = Denominator[Rt[-a/b, 2]]}, Simp[r/(2*a) Int[1/(r - s*x^2), x], x] + Simp[r/(2*a) Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] && !GtQ[a /b, 0]
Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^(p + 1/n) Subst[In t[1/(1 - b*x^n)^(p + 1/n + 1), x], x, x/(a + b*x^n)^(1/n)], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[-1, p, 0] && NeQ[p, -2^(-1)] && IntegerQ[p + 1 /n]
Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Su bst[Int[1/(c - (b*c - a*d)*x^n), x], x, x/(a + b*x^n)^(1/n)] /; FreeQ[{a, b , c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[n*p + 1, 0] && IntegerQ[n]
Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(a*d - c*b)*x*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q - 1)/(a*b*n*(p + 1))), x] - Simp[1/(a*b*n*(p + 1)) Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^(q - 2)*Simp[c*(a*d - c*b*(n*(p + 1) + 1)) + d*(a*d*(n*(q - 1) + 1) - b*c*(n*( p + q) + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && GtQ[q, 1] && IntBinomialQ[a, b, c, d, n, p, q, x]
Int[(((a_) + (b_.)*(x_)^(n_))^(p_)*((e_) + (f_.)*(x_)^(n_)))/((c_) + (d_.)* (x_)^(n_)), x_Symbol] :> Simp[f/d Int[(a + b*x^n)^p, x], x] + Simp[(d*e - c*f)/d Int[(a + b*x^n)^p/(c + d*x^n), x], x] /; FreeQ[{a, b, c, d, e, f, p, n}, x]
Leaf count of result is larger than twice the leaf count of optimal. \(407\) vs. \(2(186)=372\).
Time = 4.51 (sec) , antiderivative size = 408, normalized size of antiderivative = 1.77
| method | result | size |
| pseudoelliptic | \(-\frac {-\left (b \,x^{4}+a \right )^{\frac {3}{4}} x \left (a d -b c \right ) d c \left (\frac {a d -b c}{c}\right )^{\frac {1}{4}}+\left (\left (\frac {a d -b c}{c}\right )^{\frac {1}{4}} c^{2} \left (2 \arctan \left (\frac {\left (b \,x^{4}+a \right )^{\frac {1}{4}}}{b^{\frac {1}{4}} x}\right )-\ln \left (\frac {-b^{\frac {1}{4}} x -\left (b \,x^{4}+a \right )^{\frac {1}{4}}}{b^{\frac {1}{4}} x -\left (b \,x^{4}+a \right )^{\frac {1}{4}}}\right )\right ) b^{\frac {7}{4}}+\frac {\sqrt {2}\, \left (3 a^{2} d^{2}+a b c d -4 b^{2} c^{2}\right ) \left (\ln \left (\frac {-\left (\frac {a d -b c}{c}\right )^{\frac {1}{4}} \left (b \,x^{4}+a \right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {\frac {a d -b c}{c}}\, x^{2}+\sqrt {b \,x^{4}+a}}{\left (\frac {a d -b c}{c}\right )^{\frac {1}{4}} \left (b \,x^{4}+a \right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {\frac {a d -b c}{c}}\, x^{2}+\sqrt {b \,x^{4}+a}}\right )-2 \arctan \left (\frac {\left (\frac {a d -b c}{c}\right )^{\frac {1}{4}} x -\sqrt {2}\, \left (b \,x^{4}+a \right )^{\frac {1}{4}}}{\left (\frac {a d -b c}{c}\right )^{\frac {1}{4}} x}\right )+2 \arctan \left (\frac {\left (\frac {a d -b c}{c}\right )^{\frac {1}{4}} x +\sqrt {2}\, \left (b \,x^{4}+a \right )^{\frac {1}{4}}}{\left (\frac {a d -b c}{c}\right )^{\frac {1}{4}} x}\right )\right )}{8}\right ) \left (d \,x^{4}+c \right )}{4 \left (\frac {a d -b c}{c}\right )^{\frac {1}{4}} d^{2} c^{2} \left (d \,x^{4}+c \right )}\) | \(408\) |
-1/4*(-(b*x^4+a)^(3/4)*x*(a*d-b*c)*d*c*((a*d-b*c)/c)^(1/4)+(((a*d-b*c)/c)^ (1/4)*c^2*(2*arctan(1/b^(1/4)/x*(b*x^4+a)^(1/4))-ln((-b^(1/4)*x-(b*x^4+a)^ (1/4))/(b^(1/4)*x-(b*x^4+a)^(1/4))))*b^(7/4)+1/8*2^(1/2)*(3*a^2*d^2+a*b*c* d-4*b^2*c^2)*(ln((-((a*d-b*c)/c)^(1/4)*(b*x^4+a)^(1/4)*2^(1/2)*x+((a*d-b*c )/c)^(1/2)*x^2+(b*x^4+a)^(1/2))/(((a*d-b*c)/c)^(1/4)*(b*x^4+a)^(1/4)*2^(1/ 2)*x+((a*d-b*c)/c)^(1/2)*x^2+(b*x^4+a)^(1/2)))-2*arctan((((a*d-b*c)/c)^(1/ 4)*x-2^(1/2)*(b*x^4+a)^(1/4))/((a*d-b*c)/c)^(1/4)/x)+2*arctan((((a*d-b*c)/ c)^(1/4)*x+2^(1/2)*(b*x^4+a)^(1/4))/((a*d-b*c)/c)^(1/4)/x)))*(d*x^4+c))/(( a*d-b*c)/c)^(1/4)/d^2/c^2/(d*x^4+c)
Result contains complex when optimal does not.
Time = 0.86 (sec) , antiderivative size = 1440, normalized size of antiderivative = 6.26 \[ \int \frac {\left (a+b x^4\right )^{7/4}}{\left (c+d x^4\right )^2} \, dx=\text {Too large to display} \]
-1/16*(4*(b*x^4 + a)^(3/4)*(b*c - a*d)*x + (c*d^2*x^4 + c^2*d)*((256*b^7*c ^7 - 672*a^2*b^5*c^5*d^2 - 112*a^3*b^4*c^4*d^3 + 609*a^4*b^3*c^3*d^4 + 189 *a^5*b^2*c^2*d^5 - 189*a^6*b*c*d^6 - 81*a^7*d^7)/(c^7*d^8))^(1/4)*log((c^5 *d^6*x*((256*b^7*c^7 - 672*a^2*b^5*c^5*d^2 - 112*a^3*b^4*c^4*d^3 + 609*a^4 *b^3*c^3*d^4 + 189*a^5*b^2*c^2*d^5 - 189*a^6*b*c*d^6 - 81*a^7*d^7)/(c^7*d^ 8))^(3/4) + (64*b^5*c^5 + 16*a*b^4*c^4*d - 116*a^2*b^3*c^3*d^2 - 45*a^3*b^ 2*c^2*d^3 + 54*a^4*b*c*d^4 + 27*a^5*d^5)*(b*x^4 + a)^(1/4))/x) - (c*d^2*x^ 4 + c^2*d)*((256*b^7*c^7 - 672*a^2*b^5*c^5*d^2 - 112*a^3*b^4*c^4*d^3 + 609 *a^4*b^3*c^3*d^4 + 189*a^5*b^2*c^2*d^5 - 189*a^6*b*c*d^6 - 81*a^7*d^7)/(c^ 7*d^8))^(1/4)*log(-(c^5*d^6*x*((256*b^7*c^7 - 672*a^2*b^5*c^5*d^2 - 112*a^ 3*b^4*c^4*d^3 + 609*a^4*b^3*c^3*d^4 + 189*a^5*b^2*c^2*d^5 - 189*a^6*b*c*d^ 6 - 81*a^7*d^7)/(c^7*d^8))^(3/4) - (64*b^5*c^5 + 16*a*b^4*c^4*d - 116*a^2* b^3*c^3*d^2 - 45*a^3*b^2*c^2*d^3 + 54*a^4*b*c*d^4 + 27*a^5*d^5)*(b*x^4 + a )^(1/4))/x) + (-I*c*d^2*x^4 - I*c^2*d)*((256*b^7*c^7 - 672*a^2*b^5*c^5*d^2 - 112*a^3*b^4*c^4*d^3 + 609*a^4*b^3*c^3*d^4 + 189*a^5*b^2*c^2*d^5 - 189*a ^6*b*c*d^6 - 81*a^7*d^7)/(c^7*d^8))^(1/4)*log((I*c^5*d^6*x*((256*b^7*c^7 - 672*a^2*b^5*c^5*d^2 - 112*a^3*b^4*c^4*d^3 + 609*a^4*b^3*c^3*d^4 + 189*a^5 *b^2*c^2*d^5 - 189*a^6*b*c*d^6 - 81*a^7*d^7)/(c^7*d^8))^(3/4) + (64*b^5*c^ 5 + 16*a*b^4*c^4*d - 116*a^2*b^3*c^3*d^2 - 45*a^3*b^2*c^2*d^3 + 54*a^4*b*c *d^4 + 27*a^5*d^5)*(b*x^4 + a)^(1/4))/x) + (I*c*d^2*x^4 + I*c^2*d)*((25...
\[ \int \frac {\left (a+b x^4\right )^{7/4}}{\left (c+d x^4\right )^2} \, dx=\int \frac {\left (a + b x^{4}\right )^{\frac {7}{4}}}{\left (c + d x^{4}\right )^{2}}\, dx \]
\[ \int \frac {\left (a+b x^4\right )^{7/4}}{\left (c+d x^4\right )^2} \, dx=\int { \frac {{\left (b x^{4} + a\right )}^{\frac {7}{4}}}{{\left (d x^{4} + c\right )}^{2}} \,d x } \]
\[ \int \frac {\left (a+b x^4\right )^{7/4}}{\left (c+d x^4\right )^2} \, dx=\int { \frac {{\left (b x^{4} + a\right )}^{\frac {7}{4}}}{{\left (d x^{4} + c\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {\left (a+b x^4\right )^{7/4}}{\left (c+d x^4\right )^2} \, dx=\int \frac {{\left (b\,x^4+a\right )}^{7/4}}{{\left (d\,x^4+c\right )}^2} \,d x \]