Integrand size = 30, antiderivative size = 405 \[ \int \frac {\left (c+d x+e x^2+f x^3\right ) \left (a+b x^4\right )^{3/2}}{x^{10}} \, dx=-\frac {b \left (\frac {224 c}{x^5}+\frac {315 d}{x^4}+\frac {480 e}{x^3}+\frac {840 f}{x^2}\right ) \sqrt {a+b x^4}}{1680}-\frac {4 b^2 c \sqrt {a+b x^4}}{15 a x}+\frac {4 b^{5/2} c x \sqrt {a+b x^4}}{15 a \left (\sqrt {a}+\sqrt {b} x^2\right )}-\frac {1}{504} \left (\frac {56 c}{x^9}+\frac {63 d}{x^8}+\frac {72 e}{x^7}+\frac {84 f}{x^6}\right ) \left (a+b x^4\right )^{3/2}+\frac {1}{2} b^{3/2} f \text {arctanh}\left (\frac {\sqrt {b} x^2}{\sqrt {a+b x^4}}\right )-\frac {3 b^2 d \text {arctanh}\left (\frac {\sqrt {a+b x^4}}{\sqrt {a}}\right )}{16 \sqrt {a}}-\frac {4 b^{9/4} c \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{15 a^{3/4} \sqrt {a+b x^4}}+\frac {2 b^{7/4} \left (7 \sqrt {b} c+15 \sqrt {a} e\right ) \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{105 a^{3/4} \sqrt {a+b x^4}} \]
-1/504*(56*c/x^9+63*d/x^8+72*e/x^7+84*f/x^6)*(b*x^4+a)^(3/2)+1/2*b^(3/2)*f *arctanh(x^2*b^(1/2)/(b*x^4+a)^(1/2))-3/16*b^2*d*arctanh((b*x^4+a)^(1/2)/a ^(1/2))/a^(1/2)-1/1680*b*(224*c/x^5+315*d/x^4+480*e/x^3+840*f/x^2)*(b*x^4+ a)^(1/2)-4/15*b^2*c*(b*x^4+a)^(1/2)/a/x+4/15*b^(5/2)*c*x*(b*x^4+a)^(1/2)/a /(a^(1/2)+x^2*b^(1/2))-4/15*b^(9/4)*c*(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2) ^(1/2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticE(sin(2*arctan(b^(1/4)*x/a ^(1/4))),1/2*2^(1/2))*(a^(1/2)+x^2*b^(1/2))*((b*x^4+a)/(a^(1/2)+x^2*b^(1/2 ))^2)^(1/2)/a^(3/4)/(b*x^4+a)^(1/2)+2/105*b^(7/4)*(cos(2*arctan(b^(1/4)*x/ a^(1/4)))^2)^(1/2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan (b^(1/4)*x/a^(1/4))),1/2*2^(1/2))*(15*e*a^(1/2)+7*c*b^(1/2))*(a^(1/2)+x^2* b^(1/2))*((b*x^4+a)/(a^(1/2)+x^2*b^(1/2))^2)^(1/2)/a^(3/4)/(b*x^4+a)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.32 (sec) , antiderivative size = 174, normalized size of antiderivative = 0.43 \[ \int \frac {\left (c+d x+e x^2+f x^3\right ) \left (a+b x^4\right )^{3/2}}{x^{10}} \, dx=-\frac {\sqrt {a+b x^4} \left (112 a^2 c \operatorname {Hypergeometric2F1}\left (-\frac {9}{4},-\frac {3}{2},-\frac {5}{4},-\frac {b x^4}{a}\right )+3 x \left (48 a^2 e x \operatorname {Hypergeometric2F1}\left (-\frac {7}{4},-\frac {3}{2},-\frac {3}{4},-\frac {b x^4}{a}\right )+7 \left (3 a d \left (2 a+5 b x^4\right ) \sqrt {1+\frac {b x^4}{a}}+9 b^2 d x^8 \text {arctanh}\left (\sqrt {1+\frac {b x^4}{a}}\right )+8 a^2 f x^2 \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},-\frac {3}{2},-\frac {1}{2},-\frac {b x^4}{a}\right )\right )\right )\right )}{1008 a x^9 \sqrt {1+\frac {b x^4}{a}}} \]
-1/1008*(Sqrt[a + b*x^4]*(112*a^2*c*Hypergeometric2F1[-9/4, -3/2, -5/4, -( (b*x^4)/a)] + 3*x*(48*a^2*e*x*Hypergeometric2F1[-7/4, -3/2, -3/4, -((b*x^4 )/a)] + 7*(3*a*d*(2*a + 5*b*x^4)*Sqrt[1 + (b*x^4)/a] + 9*b^2*d*x^8*ArcTanh [Sqrt[1 + (b*x^4)/a]] + 8*a^2*f*x^2*Hypergeometric2F1[-3/2, -3/2, -1/2, -( (b*x^4)/a)]))))/(a*x^9*Sqrt[1 + (b*x^4)/a])
Time = 0.78 (sec) , antiderivative size = 400, normalized size of antiderivative = 0.99, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2364, 27, 2364, 27, 2372, 2009}
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.
| \(\displaystyle \int \frac {\left (a+b x^4\right )^{3/2} \left (c+d x+e x^2+f x^3\right )}{x^{10}} \, dx\) |
| \(\Big \downarrow \) 2364 |
| \(\displaystyle -6 b \int -\frac {\left (84 f x^3+72 e x^2+63 d x+56 c\right ) \sqrt {b x^4+a}}{504 x^6}dx-\frac {1}{504} \left (a+b x^4\right )^{3/2} \left (\frac {56 c}{x^9}+\frac {63 d}{x^8}+\frac {72 e}{x^7}+\frac {84 f}{x^6}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{84} b \int \frac {\left (84 f x^3+72 e x^2+63 d x+56 c\right ) \sqrt {b x^4+a}}{x^6}dx-\frac {1}{504} \left (a+b x^4\right )^{3/2} \left (\frac {56 c}{x^9}+\frac {63 d}{x^8}+\frac {72 e}{x^7}+\frac {84 f}{x^6}\right )\) |
| \(\Big \downarrow \) 2364 |
| \(\displaystyle \frac {1}{84} b \left (-2 b \int -\frac {840 f x^3+480 e x^2+315 d x+224 c}{20 x^2 \sqrt {b x^4+a}}dx-\frac {1}{20} \sqrt {a+b x^4} \left (\frac {224 c}{x^5}+\frac {315 d}{x^4}+\frac {480 e}{x^3}+\frac {840 f}{x^2}\right )\right )-\frac {1}{504} \left (a+b x^4\right )^{3/2} \left (\frac {56 c}{x^9}+\frac {63 d}{x^8}+\frac {72 e}{x^7}+\frac {84 f}{x^6}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{84} b \left (\frac {1}{10} b \int \frac {840 f x^3+480 e x^2+315 d x+224 c}{x^2 \sqrt {b x^4+a}}dx-\frac {1}{20} \sqrt {a+b x^4} \left (\frac {224 c}{x^5}+\frac {315 d}{x^4}+\frac {480 e}{x^3}+\frac {840 f}{x^2}\right )\right )-\frac {1}{504} \left (a+b x^4\right )^{3/2} \left (\frac {56 c}{x^9}+\frac {63 d}{x^8}+\frac {72 e}{x^7}+\frac {84 f}{x^6}\right )\) |
| \(\Big \downarrow \) 2372 |
| \(\displaystyle \frac {1}{84} b \left (\frac {1}{10} b \int \left (\frac {480 e x^2+224 c}{x^2 \sqrt {b x^4+a}}+\frac {840 f x^2+315 d}{x \sqrt {b x^4+a}}\right )dx-\frac {1}{20} \sqrt {a+b x^4} \left (\frac {224 c}{x^5}+\frac {315 d}{x^4}+\frac {480 e}{x^3}+\frac {840 f}{x^2}\right )\right )-\frac {1}{504} \left (a+b x^4\right )^{3/2} \left (\frac {56 c}{x^9}+\frac {63 d}{x^8}+\frac {72 e}{x^7}+\frac {84 f}{x^6}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{84} b \left (\frac {1}{10} b \left (\frac {16 \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \left (15 \sqrt {a} e+7 \sqrt {b} c\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{a^{3/4} \sqrt [4]{b} \sqrt {a+b x^4}}-\frac {224 \sqrt [4]{b} c \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{a^{3/4} \sqrt {a+b x^4}}-\frac {315 d \text {arctanh}\left (\frac {\sqrt {a+b x^4}}{\sqrt {a}}\right )}{2 \sqrt {a}}+\frac {420 f \text {arctanh}\left (\frac {\sqrt {b} x^2}{\sqrt {a+b x^4}}\right )}{\sqrt {b}}-\frac {224 c \sqrt {a+b x^4}}{a x}+\frac {224 \sqrt {b} c x \sqrt {a+b x^4}}{a \left (\sqrt {a}+\sqrt {b} x^2\right )}\right )-\frac {1}{20} \sqrt {a+b x^4} \left (\frac {224 c}{x^5}+\frac {315 d}{x^4}+\frac {480 e}{x^3}+\frac {840 f}{x^2}\right )\right )-\frac {1}{504} \left (a+b x^4\right )^{3/2} \left (\frac {56 c}{x^9}+\frac {63 d}{x^8}+\frac {72 e}{x^7}+\frac {84 f}{x^6}\right )\) |
-1/504*(((56*c)/x^9 + (63*d)/x^8 + (72*e)/x^7 + (84*f)/x^6)*(a + b*x^4)^(3 /2)) + (b*(-1/20*(((224*c)/x^5 + (315*d)/x^4 + (480*e)/x^3 + (840*f)/x^2)* Sqrt[a + b*x^4]) + (b*((-224*c*Sqrt[a + b*x^4])/(a*x) + (224*Sqrt[b]*c*x*S qrt[a + b*x^4])/(a*(Sqrt[a] + Sqrt[b]*x^2)) + (420*f*ArcTanh[(Sqrt[b]*x^2) /Sqrt[a + b*x^4]])/Sqrt[b] - (315*d*ArcTanh[Sqrt[a + b*x^4]/Sqrt[a]])/(2*S qrt[a]) - (224*b^(1/4)*c*(Sqrt[a] + Sqrt[b]*x^2)*Sqrt[(a + b*x^4)/(Sqrt[a] + Sqrt[b]*x^2)^2]*EllipticE[2*ArcTan[(b^(1/4)*x)/a^(1/4)], 1/2])/(a^(3/4) *Sqrt[a + b*x^4]) + (16*(7*Sqrt[b]*c + 15*Sqrt[a]*e)*(Sqrt[a] + Sqrt[b]*x^ 2)*Sqrt[(a + b*x^4)/(Sqrt[a] + Sqrt[b]*x^2)^2]*EllipticF[2*ArcTan[(b^(1/4) *x)/a^(1/4)], 1/2])/(a^(3/4)*b^(1/4)*Sqrt[a + b*x^4])))/10))/84
3.6.24.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(Pq_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Module[{u = IntHide[x^m*Pq, x]}, Simp[u*(a + b*x^n)^p, x] - Simp[b*n*p Int[x^(m + n)*(a + b*x^n)^(p - 1)*ExpandToSum[u/x^(m + 1), x], x], x]] /; FreeQ[{a, b} , x] && PolyQ[Pq, x] && IGtQ[n, 0] && GtQ[p, 0] && LtQ[m + Expon[Pq, x] + 1 , 0]
Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Mo dule[{q = Expon[Pq, x], j, k}, Int[Sum[((c*x)^(m + j)/c^j)*Sum[Coeff[Pq, x, j + k*(n/2)]*x^(k*(n/2)), {k, 0, 2*((q - j)/n) + 1}]*(a + b*x^n)^p, {j, 0, n/2 - 1}], x]] /; FreeQ[{a, b, c, m, p}, x] && PolyQ[Pq, x] && IGtQ[n/2, 0 ] && !PolyQ[Pq, x^(n/2)]
Result contains complex when optimal does not.
Time = 2.40 (sec) , antiderivative size = 324, normalized size of antiderivative = 0.80
| method | result | size |
| risch | \(-\frac {\sqrt {b \,x^{4}+a}\, \left (1344 b^{2} c \,x^{8}+3360 a b f \,x^{7}+2160 a e b \,x^{6}+1575 x^{5} d b a +1232 a b c \,x^{4}+840 a^{2} f \,x^{3}+720 a^{2} e \,x^{2}+630 a^{2} d x +560 a^{2} c \right )}{5040 x^{9} a}+\frac {b^{2} \left (\frac {480 a e \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, F\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}+\frac {224 i \sqrt {b}\, c \sqrt {a}\, \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (F\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-E\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}+\frac {420 a f \ln \left (x^{2} \sqrt {b}+\sqrt {b \,x^{4}+a}\right )}{\sqrt {b}}-\frac {315 \sqrt {a}\, d \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{4}+a}}{x^{2}}\right )}{2}\right )}{840 a}\) | \(324\) |
| elliptic | \(-\frac {a c \sqrt {b \,x^{4}+a}}{9 x^{9}}-\frac {a d \sqrt {b \,x^{4}+a}}{8 x^{8}}-\frac {a e \sqrt {b \,x^{4}+a}}{7 x^{7}}-\frac {a f \sqrt {b \,x^{4}+a}}{6 x^{6}}-\frac {11 b c \sqrt {b \,x^{4}+a}}{45 x^{5}}-\frac {5 b d \sqrt {b \,x^{4}+a}}{16 x^{4}}-\frac {3 b e \sqrt {b \,x^{4}+a}}{7 x^{3}}-\frac {2 b f \sqrt {b \,x^{4}+a}}{3 x^{2}}-\frac {4 b^{2} c \sqrt {b \,x^{4}+a}}{15 a x}+\frac {4 b^{2} e \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, F\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{7 \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}+\frac {b^{\frac {3}{2}} f \ln \left (2 x^{2} \sqrt {b}+2 \sqrt {b \,x^{4}+a}\right )}{2}+\frac {4 i b^{\frac {5}{2}} c \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (F\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-E\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{15 \sqrt {a}\, \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}-\frac {3 b^{2} d \,\operatorname {arctanh}\left (\frac {\sqrt {a}}{\sqrt {b \,x^{4}+a}}\right )}{16 \sqrt {a}}\) | \(372\) |
| default | \(f \left (\frac {b^{\frac {3}{2}} \ln \left (x^{2} \sqrt {b}+\sqrt {b \,x^{4}+a}\right )}{2}-\frac {a \sqrt {b \,x^{4}+a}}{6 x^{6}}-\frac {2 b \sqrt {b \,x^{4}+a}}{3 x^{2}}\right )+e \left (-\frac {a \sqrt {b \,x^{4}+a}}{7 x^{7}}-\frac {3 b \sqrt {b \,x^{4}+a}}{7 x^{3}}+\frac {4 b^{2} \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, F\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{7 \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\right )+d \left (-\frac {3 b^{2} \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{4}+a}}{x^{2}}\right )}{16 \sqrt {a}}-\frac {a \sqrt {b \,x^{4}+a}}{8 x^{8}}-\frac {5 b \sqrt {b \,x^{4}+a}}{16 x^{4}}\right )+c \left (-\frac {a \sqrt {b \,x^{4}+a}}{9 x^{9}}-\frac {11 b \sqrt {b \,x^{4}+a}}{45 x^{5}}-\frac {4 b^{2} \sqrt {b \,x^{4}+a}}{15 a x}+\frac {4 i b^{\frac {5}{2}} \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (F\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-E\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{15 \sqrt {a}\, \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\right )\) | \(377\) |
-1/5040*(b*x^4+a)^(1/2)*(1344*b^2*c*x^8+3360*a*b*f*x^7+2160*a*b*e*x^6+1575 *a*b*d*x^5+1232*a*b*c*x^4+840*a^2*f*x^3+720*a^2*e*x^2+630*a^2*d*x+560*a^2* c)/x^9/a+1/840/a*b^2*(480*a*e/(I/a^(1/2)*b^(1/2))^(1/2)*(1-I/a^(1/2)*b^(1/ 2)*x^2)^(1/2)*(1+I/a^(1/2)*b^(1/2)*x^2)^(1/2)/(b*x^4+a)^(1/2)*EllipticF(x* (I/a^(1/2)*b^(1/2))^(1/2),I)+224*I*b^(1/2)*c*a^(1/2)/(I/a^(1/2)*b^(1/2))^( 1/2)*(1-I/a^(1/2)*b^(1/2)*x^2)^(1/2)*(1+I/a^(1/2)*b^(1/2)*x^2)^(1/2)/(b*x^ 4+a)^(1/2)*(EllipticF(x*(I/a^(1/2)*b^(1/2))^(1/2),I)-EllipticE(x*(I/a^(1/2 )*b^(1/2))^(1/2),I))+420*a*f*ln(x^2*b^(1/2)+(b*x^4+a)^(1/2))/b^(1/2)-315/2 *a^(1/2)*d*ln((2*a+2*a^(1/2)*(b*x^4+a)^(1/2))/x^2))
\[ \int \frac {\left (c+d x+e x^2+f x^3\right ) \left (a+b x^4\right )^{3/2}}{x^{10}} \, dx=\int { \frac {{\left (b x^{4} + a\right )}^{\frac {3}{2}} {\left (f x^{3} + e x^{2} + d x + c\right )}}{x^{10}} \,d x } \]
integral((b*f*x^7 + b*e*x^6 + b*d*x^5 + b*c*x^4 + a*f*x^3 + a*e*x^2 + a*d* x + a*c)*sqrt(b*x^4 + a)/x^10, x)
Result contains complex when optimal does not.
Time = 6.79 (sec) , antiderivative size = 449, normalized size of antiderivative = 1.11 \[ \int \frac {\left (c+d x+e x^2+f x^3\right ) \left (a+b x^4\right )^{3/2}}{x^{10}} \, dx=\frac {a^{\frac {3}{2}} c \Gamma \left (- \frac {9}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {9}{4}, - \frac {1}{2} \\ - \frac {5}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 x^{9} \Gamma \left (- \frac {5}{4}\right )} + \frac {a^{\frac {3}{2}} e \Gamma \left (- \frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {7}{4}, - \frac {1}{2} \\ - \frac {3}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 x^{7} \Gamma \left (- \frac {3}{4}\right )} + \frac {\sqrt {a} b c \Gamma \left (- \frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {5}{4}, - \frac {1}{2} \\ - \frac {1}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 x^{5} \Gamma \left (- \frac {1}{4}\right )} + \frac {\sqrt {a} b e \Gamma \left (- \frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, - \frac {1}{2} \\ \frac {1}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 x^{3} \Gamma \left (\frac {1}{4}\right )} - \frac {\sqrt {a} b f}{2 x^{2} \sqrt {1 + \frac {b x^{4}}{a}}} - \frac {a^{2} d}{8 \sqrt {b} x^{10} \sqrt {\frac {a}{b x^{4}} + 1}} - \frac {3 a \sqrt {b} d}{16 x^{6} \sqrt {\frac {a}{b x^{4}} + 1}} - \frac {a \sqrt {b} f \sqrt {\frac {a}{b x^{4}} + 1}}{6 x^{4}} - \frac {b^{\frac {3}{2}} d \sqrt {\frac {a}{b x^{4}} + 1}}{4 x^{2}} - \frac {b^{\frac {3}{2}} d}{16 x^{2} \sqrt {\frac {a}{b x^{4}} + 1}} - \frac {b^{\frac {3}{2}} f \sqrt {\frac {a}{b x^{4}} + 1}}{6} + \frac {b^{\frac {3}{2}} f \operatorname {asinh}{\left (\frac {\sqrt {b} x^{2}}{\sqrt {a}} \right )}}{2} - \frac {3 b^{2} d \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x^{2}} \right )}}{16 \sqrt {a}} - \frac {b^{2} f x^{2}}{2 \sqrt {a} \sqrt {1 + \frac {b x^{4}}{a}}} \]
a**(3/2)*c*gamma(-9/4)*hyper((-9/4, -1/2), (-5/4,), b*x**4*exp_polar(I*pi) /a)/(4*x**9*gamma(-5/4)) + a**(3/2)*e*gamma(-7/4)*hyper((-7/4, -1/2), (-3/ 4,), b*x**4*exp_polar(I*pi)/a)/(4*x**7*gamma(-3/4)) + sqrt(a)*b*c*gamma(-5 /4)*hyper((-5/4, -1/2), (-1/4,), b*x**4*exp_polar(I*pi)/a)/(4*x**5*gamma(- 1/4)) + sqrt(a)*b*e*gamma(-3/4)*hyper((-3/4, -1/2), (1/4,), b*x**4*exp_pol ar(I*pi)/a)/(4*x**3*gamma(1/4)) - sqrt(a)*b*f/(2*x**2*sqrt(1 + b*x**4/a)) - a**2*d/(8*sqrt(b)*x**10*sqrt(a/(b*x**4) + 1)) - 3*a*sqrt(b)*d/(16*x**6*s qrt(a/(b*x**4) + 1)) - a*sqrt(b)*f*sqrt(a/(b*x**4) + 1)/(6*x**4) - b**(3/2 )*d*sqrt(a/(b*x**4) + 1)/(4*x**2) - b**(3/2)*d/(16*x**2*sqrt(a/(b*x**4) + 1)) - b**(3/2)*f*sqrt(a/(b*x**4) + 1)/6 + b**(3/2)*f*asinh(sqrt(b)*x**2/sq rt(a))/2 - 3*b**2*d*asinh(sqrt(a)/(sqrt(b)*x**2))/(16*sqrt(a)) - b**2*f*x* *2/(2*sqrt(a)*sqrt(1 + b*x**4/a))
\[ \int \frac {\left (c+d x+e x^2+f x^3\right ) \left (a+b x^4\right )^{3/2}}{x^{10}} \, dx=\int { \frac {{\left (b x^{4} + a\right )}^{\frac {3}{2}} {\left (f x^{3} + e x^{2} + d x + c\right )}}{x^{10}} \,d x } \]
\[ \int \frac {\left (c+d x+e x^2+f x^3\right ) \left (a+b x^4\right )^{3/2}}{x^{10}} \, dx=\int { \frac {{\left (b x^{4} + a\right )}^{\frac {3}{2}} {\left (f x^{3} + e x^{2} + d x + c\right )}}{x^{10}} \,d x } \]
Timed out. \[ \int \frac {\left (c+d x+e x^2+f x^3\right ) \left (a+b x^4\right )^{3/2}}{x^{10}} \, dx=\int \frac {{\left (b\,x^4+a\right )}^{3/2}\,\left (f\,x^3+e\,x^2+d\,x+c\right )}{x^{10}} \,d x \]