Integrand size = 30, antiderivative size = 399 \[ \int \frac {\left (c+d x+e x^2+f x^3\right ) \left (a+b x^4\right )^{3/2}}{x^{11}} \, dx=-\frac {b \left (\frac {168 c}{x^6}+\frac {224 d}{x^5}+\frac {315 e}{x^4}+\frac {480 f}{x^3}\right ) \sqrt {a+b x^4}}{1680}-\frac {b^2 c \sqrt {a+b x^4}}{10 a x^2}-\frac {4 b^2 d \sqrt {a+b x^4}}{15 a x}+\frac {4 b^{5/2} d x \sqrt {a+b x^4}}{15 a \left (\sqrt {a}+\sqrt {b} x^2\right )}-\frac {\left (\frac {252 c}{x^{10}}+\frac {280 d}{x^9}+\frac {315 e}{x^8}+\frac {360 f}{x^7}\right ) \left (a+b x^4\right )^{3/2}}{2520}-\frac {3 b^2 e \text {arctanh}\left (\frac {\sqrt {a+b x^4}}{\sqrt {a}}\right )}{16 \sqrt {a}}-\frac {4 b^{9/4} d \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{15 a^{3/4} \sqrt {a+b x^4}}+\frac {2 b^{7/4} \left (7 \sqrt {b} d+15 \sqrt {a} f\right ) \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{105 a^{3/4} \sqrt {a+b x^4}} \]
-1/2520*(252*c/x^10+280*d/x^9+315*e/x^8+360*f/x^7)*(b*x^4+a)^(3/2)-3/16*b^ 2*e*arctanh((b*x^4+a)^(1/2)/a^(1/2))/a^(1/2)-1/1680*b*(168*c/x^6+224*d/x^5 +315*e/x^4+480*f/x^3)*(b*x^4+a)^(1/2)-1/10*b^2*c*(b*x^4+a)^(1/2)/a/x^2-4/1 5*b^2*d*(b*x^4+a)^(1/2)/a/x+4/15*b^(5/2)*d*x*(b*x^4+a)^(1/2)/a/(a^(1/2)+x^ 2*b^(1/2))-4/15*b^(9/4)*d*(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*f*a^(1/2)+7*d*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 = 10.34 (sec) , antiderivative size = 171, normalized size of antiderivative = 0.43 \[ \int \frac {\left (c+d x+e x^2+f x^3\right ) \left (a+b x^4\right )^{3/2}}{x^{11}} \, dx=-\frac {\sqrt {a+b x^4} \left (63 \sqrt {1+\frac {b x^4}{a}} \left (8 b^2 c x^8+2 a^2 \left (4 c+5 e x^2\right )+a b x^4 \left (16 c+25 e x^2\right )\right )+945 b^2 e x^{10} \text {arctanh}\left (\sqrt {1+\frac {b x^4}{a}}\right )+560 a^2 d x \operatorname {Hypergeometric2F1}\left (-\frac {9}{4},-\frac {3}{2},-\frac {5}{4},-\frac {b x^4}{a}\right )+720 a^2 f x^3 \operatorname {Hypergeometric2F1}\left (-\frac {7}{4},-\frac {3}{2},-\frac {3}{4},-\frac {b x^4}{a}\right )\right )}{5040 a x^{10} \sqrt {1+\frac {b x^4}{a}}} \]
-1/5040*(Sqrt[a + b*x^4]*(63*Sqrt[1 + (b*x^4)/a]*(8*b^2*c*x^8 + 2*a^2*(4*c + 5*e*x^2) + a*b*x^4*(16*c + 25*e*x^2)) + 945*b^2*e*x^10*ArcTanh[Sqrt[1 + (b*x^4)/a]] + 560*a^2*d*x*Hypergeometric2F1[-9/4, -3/2, -5/4, -((b*x^4)/a )] + 720*a^2*f*x^3*Hypergeometric2F1[-7/4, -3/2, -3/4, -((b*x^4)/a)]))/(a* x^10*Sqrt[1 + (b*x^4)/a])
Time = 0.79 (sec) , antiderivative size = 391, normalized size of antiderivative = 0.98, 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.
| \(\displaystyle \int \frac {\left (a+b x^4\right )^{3/2} \left (c+d x+e x^2+f x^3\right )}{x^{11}} \, dx\) |
| \(\Big \downarrow \) 2364 |
| \(\displaystyle -6 b \int -\frac {\left (360 f x^3+315 e x^2+280 d x+252 c\right ) \sqrt {b x^4+a}}{2520 x^7}dx-\frac {\left (a+b x^4\right )^{3/2} \left (\frac {252 c}{x^{10}}+\frac {280 d}{x^9}+\frac {315 e}{x^8}+\frac {360 f}{x^7}\right )}{2520}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{420} b \int \frac {\left (360 f x^3+315 e x^2+280 d x+252 c\right ) \sqrt {b x^4+a}}{x^7}dx-\frac {\left (a+b x^4\right )^{3/2} \left (\frac {252 c}{x^{10}}+\frac {280 d}{x^9}+\frac {315 e}{x^8}+\frac {360 f}{x^7}\right )}{2520}\) |
| \(\Big \downarrow \) 2364 |
| \(\displaystyle \frac {1}{420} b \left (-2 b \int -\frac {480 f x^3+315 e x^2+224 d x+168 c}{4 x^3 \sqrt {b x^4+a}}dx-\frac {1}{4} \sqrt {a+b x^4} \left (\frac {168 c}{x^6}+\frac {224 d}{x^5}+\frac {315 e}{x^4}+\frac {480 f}{x^3}\right )\right )-\frac {\left (a+b x^4\right )^{3/2} \left (\frac {252 c}{x^{10}}+\frac {280 d}{x^9}+\frac {315 e}{x^8}+\frac {360 f}{x^7}\right )}{2520}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{420} b \left (\frac {1}{2} b \int \frac {480 f x^3+315 e x^2+224 d x+168 c}{x^3 \sqrt {b x^4+a}}dx-\frac {1}{4} \sqrt {a+b x^4} \left (\frac {168 c}{x^6}+\frac {224 d}{x^5}+\frac {315 e}{x^4}+\frac {480 f}{x^3}\right )\right )-\frac {\left (a+b x^4\right )^{3/2} \left (\frac {252 c}{x^{10}}+\frac {280 d}{x^9}+\frac {315 e}{x^8}+\frac {360 f}{x^7}\right )}{2520}\) |
| \(\Big \downarrow \) 2372 |
| \(\displaystyle \frac {1}{420} b \left (\frac {1}{2} b \int \left (\frac {315 e x^2+168 c}{x^3 \sqrt {b x^4+a}}+\frac {480 f x^2+224 d}{x^2 \sqrt {b x^4+a}}\right )dx-\frac {1}{4} \sqrt {a+b x^4} \left (\frac {168 c}{x^6}+\frac {224 d}{x^5}+\frac {315 e}{x^4}+\frac {480 f}{x^3}\right )\right )-\frac {\left (a+b x^4\right )^{3/2} \left (\frac {252 c}{x^{10}}+\frac {280 d}{x^9}+\frac {315 e}{x^8}+\frac {360 f}{x^7}\right )}{2520}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{420} b \left (\frac {1}{2} b \left (\frac {16 \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \left (15 \sqrt {a} f+7 \sqrt {b} d\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{a^{3/4} \sqrt [4]{b} \sqrt {a+b x^4}}-\frac {224 \sqrt [4]{b} d \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{a^{3/4} \sqrt {a+b x^4}}-\frac {315 e \text {arctanh}\left (\frac {\sqrt {a+b x^4}}{\sqrt {a}}\right )}{2 \sqrt {a}}-\frac {84 c \sqrt {a+b x^4}}{a x^2}-\frac {224 d \sqrt {a+b x^4}}{a x}+\frac {224 \sqrt {b} d x \sqrt {a+b x^4}}{a \left (\sqrt {a}+\sqrt {b} x^2\right )}\right )-\frac {1}{4} \sqrt {a+b x^4} \left (\frac {168 c}{x^6}+\frac {224 d}{x^5}+\frac {315 e}{x^4}+\frac {480 f}{x^3}\right )\right )-\frac {\left (a+b x^4\right )^{3/2} \left (\frac {252 c}{x^{10}}+\frac {280 d}{x^9}+\frac {315 e}{x^8}+\frac {360 f}{x^7}\right )}{2520}\) |
-1/2520*(((252*c)/x^10 + (280*d)/x^9 + (315*e)/x^8 + (360*f)/x^7)*(a + b*x ^4)^(3/2)) + (b*(-1/4*(((168*c)/x^6 + (224*d)/x^5 + (315*e)/x^4 + (480*f)/ x^3)*Sqrt[a + b*x^4]) + (b*((-84*c*Sqrt[a + b*x^4])/(a*x^2) - (224*d*Sqrt[ a + b*x^4])/(a*x) + (224*Sqrt[b]*d*x*Sqrt[a + b*x^4])/(a*(Sqrt[a] + Sqrt[b ]*x^2)) - (315*e*ArcTanh[Sqrt[a + b*x^4]/Sqrt[a]])/(2*Sqrt[a]) - (224*b^(1 /4)*d*(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]*d + 15*Sqrt[a]*f)*(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])))/2))/420
3.6.25.3.1 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 = 2.83 (sec) , antiderivative size = 308, normalized size of antiderivative = 0.77
| method | result | size |
| risch | \(-\frac {\sqrt {b \,x^{4}+a}\, \left (1344 b^{2} d \,x^{9}+504 b^{2} c \,x^{8}+2160 a b f \,x^{7}+1575 a e b \,x^{6}+1232 x^{5} d b a +1008 a b c \,x^{4}+720 a^{2} f \,x^{3}+630 a^{2} e \,x^{2}+560 a^{2} d x +504 a^{2} c \right )}{5040 x^{10} a}+\frac {b^{2} \left (\frac {480 a f \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, F\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}+\frac {224 i \sqrt {b}\, d \sqrt {a}\, \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (F\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-E\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}-\frac {315 \sqrt {a}\, e \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{4}+a}}{x^{2}}\right )}{2}\right )}{840 a}\) | \(308\) |
| default | \(f \left (-\frac {a \sqrt {b \,x^{4}+a}}{7 x^{7}}-\frac {3 b \sqrt {b \,x^{4}+a}}{7 x^{3}}+\frac {4 b^{2} \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, F\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{7 \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\right )+e \left (-\frac {3 b^{2} \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{4}+a}}{x^{2}}\right )}{16 \sqrt {a}}-\frac {a \sqrt {b \,x^{4}+a}}{8 x^{8}}-\frac {5 b \sqrt {b \,x^{4}+a}}{16 x^{4}}\right )-\frac {c \left (b^{2} x^{8}+2 a b \,x^{4}+a^{2}\right ) \sqrt {b \,x^{4}+a}}{10 x^{10} a}+d \left (-\frac {a \sqrt {b \,x^{4}+a}}{9 x^{9}}-\frac {11 b \sqrt {b \,x^{4}+a}}{45 x^{5}}-\frac {4 b^{2} \sqrt {b \,x^{4}+a}}{15 a x}+\frac {4 i b^{\frac {5}{2}} \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (F\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-E\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{15 \sqrt {a}\, \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\right )\) | \(357\) |
| elliptic | \(-\frac {a c \sqrt {b \,x^{4}+a}}{10 x^{10}}-\frac {a d \sqrt {b \,x^{4}+a}}{9 x^{9}}-\frac {a e \sqrt {b \,x^{4}+a}}{8 x^{8}}-\frac {a f \sqrt {b \,x^{4}+a}}{7 x^{7}}-\frac {b c \sqrt {b \,x^{4}+a}}{5 x^{6}}-\frac {11 b d \sqrt {b \,x^{4}+a}}{45 x^{5}}-\frac {5 b e \sqrt {b \,x^{4}+a}}{16 x^{4}}-\frac {3 b f \sqrt {b \,x^{4}+a}}{7 x^{3}}-\frac {b^{2} c \sqrt {b \,x^{4}+a}}{10 a \,x^{2}}-\frac {4 b^{2} d \sqrt {b \,x^{4}+a}}{15 a x}+\frac {4 b^{2} f \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, F\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{7 \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}+\frac {4 i d \,b^{\frac {5}{2}} \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (F\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-E\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{15 \sqrt {a}\, \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}-\frac {3 b^{2} e \,\operatorname {arctanh}\left (\frac {\sqrt {a}}{\sqrt {b \,x^{4}+a}}\right )}{16 \sqrt {a}}\) | \(366\) |
-1/5040*(b*x^4+a)^(1/2)*(1344*b^2*d*x^9+504*b^2*c*x^8+2160*a*b*f*x^7+1575* a*b*e*x^6+1232*a*b*d*x^5+1008*a*b*c*x^4+720*a^2*f*x^3+630*a^2*e*x^2+560*a^ 2*d*x+504*a^2*c)/x^10/a+1/840/a*b^2*(480*a*f/(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)*d*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)-Ellipt icE(x*(I/a^(1/2)*b^(1/2))^(1/2),I))-315/2*a^(1/2)*e*ln((2*a+2*a^(1/2)*(b*x ^4+a)^(1/2))/x^2))
Time = 0.21 (sec) , antiderivative size = 217, normalized size of antiderivative = 0.54 \[ \int \frac {\left (c+d x+e x^2+f x^3\right ) \left (a+b x^4\right )^{3/2}}{x^{11}} \, dx=-\frac {2688 \, \sqrt {a} b^{2} d x^{10} \left (-\frac {b}{a}\right )^{\frac {3}{4}} E(\arcsin \left (x \left (-\frac {b}{a}\right )^{\frac {1}{4}}\right )\,|\,-1) - 945 \, \sqrt {a} b^{2} e x^{10} \log \left (-\frac {b x^{4} - 2 \, \sqrt {b x^{4} + a} \sqrt {a} + 2 \, a}{x^{4}}\right ) - 384 \, {\left (7 \, b^{2} d - 15 \, a b f\right )} \sqrt {a} x^{10} \left (-\frac {b}{a}\right )^{\frac {3}{4}} F(\arcsin \left (x \left (-\frac {b}{a}\right )^{\frac {1}{4}}\right )\,|\,-1) + 2 \, {\left (1344 \, b^{2} d x^{9} + 504 \, b^{2} c x^{8} + 2160 \, a b f x^{7} + 1575 \, a b e x^{6} + 1232 \, a b d x^{5} + 1008 \, a b c x^{4} + 720 \, a^{2} f x^{3} + 630 \, a^{2} e x^{2} + 560 \, a^{2} d x + 504 \, a^{2} c\right )} \sqrt {b x^{4} + a}}{10080 \, a x^{10}} \]
-1/10080*(2688*sqrt(a)*b^2*d*x^10*(-b/a)^(3/4)*elliptic_e(arcsin(x*(-b/a)^ (1/4)), -1) - 945*sqrt(a)*b^2*e*x^10*log(-(b*x^4 - 2*sqrt(b*x^4 + a)*sqrt( a) + 2*a)/x^4) - 384*(7*b^2*d - 15*a*b*f)*sqrt(a)*x^10*(-b/a)^(3/4)*ellipt ic_f(arcsin(x*(-b/a)^(1/4)), -1) + 2*(1344*b^2*d*x^9 + 504*b^2*c*x^8 + 216 0*a*b*f*x^7 + 1575*a*b*e*x^6 + 1232*a*b*d*x^5 + 1008*a*b*c*x^4 + 720*a^2*f *x^3 + 630*a^2*e*x^2 + 560*a^2*d*x + 504*a^2*c)*sqrt(b*x^4 + a))/(a*x^10)
Result contains complex when optimal does not.
Time = 6.67 (sec) , antiderivative size = 398, normalized size of antiderivative = 1.00 \[ \int \frac {\left (c+d x+e x^2+f x^3\right ) \left (a+b x^4\right )^{3/2}}{x^{11}} \, dx=\frac {a^{\frac {3}{2}} d \Gamma \left (- \frac {9}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {9}{4}, - \frac {1}{2} \\ - \frac {5}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 x^{9} \Gamma \left (- \frac {5}{4}\right )} + \frac {a^{\frac {3}{2}} f \Gamma \left (- \frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {7}{4}, - \frac {1}{2} \\ - \frac {3}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 x^{7} \Gamma \left (- \frac {3}{4}\right )} + \frac {\sqrt {a} b d \Gamma \left (- \frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {5}{4}, - \frac {1}{2} \\ - \frac {1}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 x^{5} \Gamma \left (- \frac {1}{4}\right )} + \frac {\sqrt {a} b f \Gamma \left (- \frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, - \frac {1}{2} \\ \frac {1}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 x^{3} \Gamma \left (\frac {1}{4}\right )} - \frac {a^{2} e}{8 \sqrt {b} x^{10} \sqrt {\frac {a}{b x^{4}} + 1}} - \frac {a \sqrt {b} c \sqrt {\frac {a}{b x^{4}} + 1}}{10 x^{8}} - \frac {3 a \sqrt {b} e}{16 x^{6} \sqrt {\frac {a}{b x^{4}} + 1}} - \frac {b^{\frac {3}{2}} c \sqrt {\frac {a}{b x^{4}} + 1}}{5 x^{4}} - \frac {b^{\frac {3}{2}} e \sqrt {\frac {a}{b x^{4}} + 1}}{4 x^{2}} - \frac {b^{\frac {3}{2}} e}{16 x^{2} \sqrt {\frac {a}{b x^{4}} + 1}} - \frac {b^{\frac {5}{2}} c \sqrt {\frac {a}{b x^{4}} + 1}}{10 a} - \frac {3 b^{2} e \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x^{2}} \right )}}{16 \sqrt {a}} \]
a**(3/2)*d*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)*f*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*d*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*f*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)) - a**2*e/(8*sqrt(b)*x**10*sqrt(a/(b*x**4) + 1)) - a*sqrt(b)*c*sqrt(a/(b*x**4) + 1)/(10*x**8) - 3*a*sqrt(b)*e/(16*x** 6*sqrt(a/(b*x**4) + 1)) - b**(3/2)*c*sqrt(a/(b*x**4) + 1)/(5*x**4) - b**(3 /2)*e*sqrt(a/(b*x**4) + 1)/(4*x**2) - b**(3/2)*e/(16*x**2*sqrt(a/(b*x**4) + 1)) - b**(5/2)*c*sqrt(a/(b*x**4) + 1)/(10*a) - 3*b**2*e*asinh(sqrt(a)/(s qrt(b)*x**2))/(16*sqrt(a))
\[ \int \frac {\left (c+d x+e x^2+f x^3\right ) \left (a+b x^4\right )^{3/2}}{x^{11}} \, dx=\int { \frac {{\left (b x^{4} + a\right )}^{\frac {3}{2}} {\left (f x^{3} + e x^{2} + d x + c\right )}}{x^{11}} \,d x } \]
-1/10*(b*x^4 + a)^(5/2)*c/(a*x^10) + integrate((b*f*x^6 + b*e*x^5 + b*d*x^ 4 + a*f*x^2 + a*e*x + a*d)*sqrt(b*x^4 + a)/x^10, x)
\[ \int \frac {\left (c+d x+e x^2+f x^3\right ) \left (a+b x^4\right )^{3/2}}{x^{11}} \, dx=\int { \frac {{\left (b x^{4} + a\right )}^{\frac {3}{2}} {\left (f x^{3} + e x^{2} + d x + c\right )}}{x^{11}} \,d x } \]
Timed out. \[ \int \frac {\left (c+d x+e x^2+f x^3\right ) \left (a+b x^4\right )^{3/2}}{x^{11}} \, dx=\int \frac {{\left (b\,x^4+a\right )}^{3/2}\,\left (f\,x^3+e\,x^2+d\,x+c\right )}{x^{11}} \,d x \]