Integrand size = 30, antiderivative size = 424 \[ \int \frac {\left (c+d x+e x^2+f x^3\right ) \left (a+b x^4\right )^{3/2}}{x^{12}} \, dx=-\frac {b \left (\frac {1440 c}{x^7}+\frac {1848 d}{x^6}+\frac {2464 e}{x^5}+\frac {3465 f}{x^4}\right ) \sqrt {a+b x^4}}{18480}-\frac {4 b^2 c \sqrt {a+b x^4}}{77 a x^3}-\frac {b^2 d \sqrt {a+b x^4}}{10 a x^2}-\frac {4 b^2 e \sqrt {a+b x^4}}{15 a x}+\frac {4 b^{5/2} e x \sqrt {a+b x^4}}{15 a \left (\sqrt {a}+\sqrt {b} x^2\right )}-\frac {\left (\frac {360 c}{x^{11}}+\frac {396 d}{x^{10}}+\frac {440 e}{x^9}+\frac {495 f}{x^8}\right ) \left (a+b x^4\right )^{3/2}}{3960}-\frac {3 b^2 f \text {arctanh}\left (\frac {\sqrt {a+b x^4}}{\sqrt {a}}\right )}{16 \sqrt {a}}-\frac {4 b^{9/4} e \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{15 a^{3/4} \sqrt {a+b x^4}}-\frac {2 b^{9/4} \left (15 \sqrt {b} c-77 \sqrt {a} e\right ) \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{1155 a^{5/4} \sqrt {a+b x^4}} \]
-1/3960*(360*c/x^11+396*d/x^10+440*e/x^9+495*f/x^8)*(b*x^4+a)^(3/2)-3/16*b ^2*f*arctanh((b*x^4+a)^(1/2)/a^(1/2))/a^(1/2)-1/18480*b*(1440*c/x^7+1848*d /x^6+2464*e/x^5+3465*f/x^4)*(b*x^4+a)^(1/2)-4/77*b^2*c*(b*x^4+a)^(1/2)/a/x ^3-1/10*b^2*d*(b*x^4+a)^(1/2)/a/x^2-4/15*b^2*e*(b*x^4+a)^(1/2)/a/x+4/15*b^ (5/2)*e*x*(b*x^4+a)^(1/2)/a/(a^(1/2)+x^2*b^(1/2))-4/15*b^(9/4)*e*(cos(2*ar ctan(b^(1/4)*x/a^(1/4)))^2)^(1/2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*Ellipti cE(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)-2/1155*b^(9 /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))*(-77*e*a^(1/ 2)+15*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^(5/4)/(b*x^4+a)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.36 (sec) , antiderivative size = 172, normalized size of antiderivative = 0.41 \[ \int \frac {\left (c+d x+e x^2+f x^3\right ) \left (a+b x^4\right )^{3/2}}{x^{12}} \, dx=-\frac {\sqrt {a+b x^4} \left (720 a^2 c \operatorname {Hypergeometric2F1}\left (-\frac {11}{4},-\frac {3}{2},-\frac {7}{4},-\frac {b x^4}{a}\right )+11 x \left (9 \sqrt {1+\frac {b x^4}{a}} \left (8 b^2 d x^8+2 a^2 \left (4 d+5 f x^2\right )+a b x^4 \left (16 d+25 f x^2\right )\right )+135 b^2 f x^{10} \text {arctanh}\left (\sqrt {1+\frac {b x^4}{a}}\right )+80 a^2 e x \operatorname {Hypergeometric2F1}\left (-\frac {9}{4},-\frac {3}{2},-\frac {5}{4},-\frac {b x^4}{a}\right )\right )\right )}{7920 a x^{11} \sqrt {1+\frac {b x^4}{a}}} \]
-1/7920*(Sqrt[a + b*x^4]*(720*a^2*c*Hypergeometric2F1[-11/4, -3/2, -7/4, - ((b*x^4)/a)] + 11*x*(9*Sqrt[1 + (b*x^4)/a]*(8*b^2*d*x^8 + 2*a^2*(4*d + 5*f *x^2) + a*b*x^4*(16*d + 25*f*x^2)) + 135*b^2*f*x^10*ArcTanh[Sqrt[1 + (b*x^ 4)/a]] + 80*a^2*e*x*Hypergeometric2F1[-9/4, -3/2, -5/4, -((b*x^4)/a)])))/( a*x^11*Sqrt[1 + (b*x^4)/a])
Time = 0.84 (sec) , antiderivative size = 411, normalized size of antiderivative = 0.97, 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.
| \(\displaystyle \int \frac {\left (a+b x^4\right )^{3/2} \left (c+d x+e x^2+f x^3\right )}{x^{12}} \, dx\) |
| \(\Big \downarrow \) 2364 |
| \(\displaystyle -6 b \int -\frac {\left (495 f x^3+440 e x^2+396 d x+360 c\right ) \sqrt {b x^4+a}}{3960 x^8}dx-\frac {\left (a+b x^4\right )^{3/2} \left (\frac {360 c}{x^{11}}+\frac {396 d}{x^{10}}+\frac {440 e}{x^9}+\frac {495 f}{x^8}\right )}{3960}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{660} b \int \frac {\left (495 f x^3+440 e x^2+396 d x+360 c\right ) \sqrt {b x^4+a}}{x^8}dx-\frac {\left (a+b x^4\right )^{3/2} \left (\frac {360 c}{x^{11}}+\frac {396 d}{x^{10}}+\frac {440 e}{x^9}+\frac {495 f}{x^8}\right )}{3960}\) |
| \(\Big \downarrow \) 2364 |
| \(\displaystyle \frac {1}{660} b \left (-2 b \int -\frac {3465 f x^3+2464 e x^2+1848 d x+1440 c}{28 x^4 \sqrt {b x^4+a}}dx-\frac {1}{28} \sqrt {a+b x^4} \left (\frac {1440 c}{x^7}+\frac {1848 d}{x^6}+\frac {2464 e}{x^5}+\frac {3465 f}{x^4}\right )\right )-\frac {\left (a+b x^4\right )^{3/2} \left (\frac {360 c}{x^{11}}+\frac {396 d}{x^{10}}+\frac {440 e}{x^9}+\frac {495 f}{x^8}\right )}{3960}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{660} b \left (\frac {1}{14} b \int \frac {3465 f x^3+2464 e x^2+1848 d x+1440 c}{x^4 \sqrt {b x^4+a}}dx-\frac {1}{28} \sqrt {a+b x^4} \left (\frac {1440 c}{x^7}+\frac {1848 d}{x^6}+\frac {2464 e}{x^5}+\frac {3465 f}{x^4}\right )\right )-\frac {\left (a+b x^4\right )^{3/2} \left (\frac {360 c}{x^{11}}+\frac {396 d}{x^{10}}+\frac {440 e}{x^9}+\frac {495 f}{x^8}\right )}{3960}\) |
| \(\Big \downarrow \) 2372 |
| \(\displaystyle \frac {1}{660} b \left (\frac {1}{14} b \int \left (\frac {2464 e x^2+1440 c}{x^4 \sqrt {b x^4+a}}+\frac {3465 f x^2+1848 d}{x^3 \sqrt {b x^4+a}}\right )dx-\frac {1}{28} \sqrt {a+b x^4} \left (\frac {1440 c}{x^7}+\frac {1848 d}{x^6}+\frac {2464 e}{x^5}+\frac {3465 f}{x^4}\right )\right )-\frac {\left (a+b x^4\right )^{3/2} \left (\frac {360 c}{x^{11}}+\frac {396 d}{x^{10}}+\frac {440 e}{x^9}+\frac {495 f}{x^8}\right )}{3960}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{660} b \left (\frac {1}{14} b \left (-\frac {16 \sqrt [4]{b} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \left (15 \sqrt {b} c-77 \sqrt {a} e\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{a^{5/4} \sqrt {a+b x^4}}-\frac {2464 \sqrt [4]{b} e \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{a^{3/4} \sqrt {a+b x^4}}-\frac {3465 f \text {arctanh}\left (\frac {\sqrt {a+b x^4}}{\sqrt {a}}\right )}{2 \sqrt {a}}-\frac {480 c \sqrt {a+b x^4}}{a x^3}-\frac {924 d \sqrt {a+b x^4}}{a x^2}-\frac {2464 e \sqrt {a+b x^4}}{a x}+\frac {2464 \sqrt {b} e x \sqrt {a+b x^4}}{a \left (\sqrt {a}+\sqrt {b} x^2\right )}\right )-\frac {1}{28} \sqrt {a+b x^4} \left (\frac {1440 c}{x^7}+\frac {1848 d}{x^6}+\frac {2464 e}{x^5}+\frac {3465 f}{x^4}\right )\right )-\frac {\left (a+b x^4\right )^{3/2} \left (\frac {360 c}{x^{11}}+\frac {396 d}{x^{10}}+\frac {440 e}{x^9}+\frac {495 f}{x^8}\right )}{3960}\) |
-1/3960*(((360*c)/x^11 + (396*d)/x^10 + (440*e)/x^9 + (495*f)/x^8)*(a + b* x^4)^(3/2)) + (b*(-1/28*(((1440*c)/x^7 + (1848*d)/x^6 + (2464*e)/x^5 + (34 65*f)/x^4)*Sqrt[a + b*x^4]) + (b*((-480*c*Sqrt[a + b*x^4])/(a*x^3) - (924* d*Sqrt[a + b*x^4])/(a*x^2) - (2464*e*Sqrt[a + b*x^4])/(a*x) + (2464*Sqrt[b ]*e*x*Sqrt[a + b*x^4])/(a*(Sqrt[a] + Sqrt[b]*x^2)) - (3465*f*ArcTanh[Sqrt[ a + b*x^4]/Sqrt[a]])/(2*Sqrt[a]) - (2464*b^(1/4)*e*(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*b^(1/4)*(15*Sqrt[b]*c - 77*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^(5/4)*Sqrt[a + b *x^4])))/14))/660
3.6.26.3.1 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 = 3.20 (sec) , antiderivative size = 317, normalized size of antiderivative = 0.75
| method | result | size |
| risch | \(-\frac {\sqrt {b \,x^{4}+a}\, \left (14784 b^{2} e \,x^{10}+5544 b^{2} d \,x^{9}+2880 b^{2} c \,x^{8}+17325 a b f \,x^{7}+13552 a e b \,x^{6}+11088 x^{5} d b a +9360 a b c \,x^{4}+6930 a^{2} f \,x^{3}+6160 a^{2} e \,x^{2}+5544 a^{2} d x +5040 a^{2} c \right )}{55440 x^{11} a}+\frac {b^{2} \left (-\frac {480 b c \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, F\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}+\frac {2464 i \sqrt {b}\, e \sqrt {a}\, \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (F\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-E\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}-\frac {3465 \sqrt {a}\, f \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{4}+a}}{x^{2}}\right )}{2}\right )}{9240 a}\) | \(317\) |
| default | \(f \left (-\frac {3 b^{2} \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{4}+a}}{x^{2}}\right )}{16 \sqrt {a}}-\frac {a \sqrt {b \,x^{4}+a}}{8 x^{8}}-\frac {5 b \sqrt {b \,x^{4}+a}}{16 x^{4}}\right )-\frac {d \left (b^{2} x^{8}+2 a b \,x^{4}+a^{2}\right ) \sqrt {b \,x^{4}+a}}{10 x^{10} a}+c \left (-\frac {a \sqrt {b \,x^{4}+a}}{11 x^{11}}-\frac {13 b \sqrt {b \,x^{4}+a}}{77 x^{7}}-\frac {4 b^{2} \sqrt {b \,x^{4}+a}}{77 a \,x^{3}}-\frac {4 b^{3} \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, F\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{77 a \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\right )+e \left (-\frac {a \sqrt {b \,x^{4}+a}}{9 x^{9}}-\frac {11 b \sqrt {b \,x^{4}+a}}{45 x^{5}}-\frac {4 b^{2} \sqrt {b \,x^{4}+a}}{15 a x}+\frac {4 i b^{\frac {5}{2}} \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (F\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-E\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{15 \sqrt {a}\, \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\right )\) | \(380\) |
| elliptic | \(-\frac {a c \sqrt {b \,x^{4}+a}}{11 x^{11}}-\frac {a d \sqrt {b \,x^{4}+a}}{10 x^{10}}-\frac {a e \sqrt {b \,x^{4}+a}}{9 x^{9}}-\frac {a f \sqrt {b \,x^{4}+a}}{8 x^{8}}-\frac {13 b c \sqrt {b \,x^{4}+a}}{77 x^{7}}-\frac {b d \sqrt {b \,x^{4}+a}}{5 x^{6}}-\frac {11 b e \sqrt {b \,x^{4}+a}}{45 x^{5}}-\frac {5 b f \sqrt {b \,x^{4}+a}}{16 x^{4}}-\frac {4 b^{2} c \sqrt {b \,x^{4}+a}}{77 a \,x^{3}}-\frac {b^{2} d \sqrt {b \,x^{4}+a}}{10 a \,x^{2}}-\frac {4 b^{2} e \sqrt {b \,x^{4}+a}}{15 a x}-\frac {4 b^{3} c \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, F\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{77 a \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}+\frac {4 i b^{\frac {5}{2}} e \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (F\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-E\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{15 \sqrt {a}\, \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}-\frac {3 b^{2} f \,\operatorname {arctanh}\left (\frac {\sqrt {a}}{\sqrt {b \,x^{4}+a}}\right )}{16 \sqrt {a}}\) | \(390\) |
-1/55440*(b*x^4+a)^(1/2)*(14784*b^2*e*x^10+5544*b^2*d*x^9+2880*b^2*c*x^8+1 7325*a*b*f*x^7+13552*a*b*e*x^6+11088*a*b*d*x^5+9360*a*b*c*x^4+6930*a^2*f*x ^3+6160*a^2*e*x^2+5544*a^2*d*x+5040*a^2*c)/x^11/a+1/9240/a*b^2*(-480*b*c/( 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)+24 64*I*b^(1/2)*e*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))-3465/2*a^ (1/2)*f*ln((2*a+2*a^(1/2)*(b*x^4+a)^(1/2))/x^2))
Time = 0.35 (sec) , antiderivative size = 227, normalized size of antiderivative = 0.54 \[ \int \frac {\left (c+d x+e x^2+f x^3\right ) \left (a+b x^4\right )^{3/2}}{x^{12}} \, dx=-\frac {29568 \, \sqrt {a} b^{2} e x^{11} \left (-\frac {b}{a}\right )^{\frac {3}{4}} E(\arcsin \left (x \left (-\frac {b}{a}\right )^{\frac {1}{4}}\right )\,|\,-1) - 10395 \, \sqrt {a} b^{2} f x^{11} \log \left (-\frac {b x^{4} - 2 \, \sqrt {b x^{4} + a} \sqrt {a} + 2 \, a}{x^{4}}\right ) - 384 \, {\left (15 \, b^{2} c + 77 \, b^{2} e\right )} \sqrt {a} x^{11} \left (-\frac {b}{a}\right )^{\frac {3}{4}} F(\arcsin \left (x \left (-\frac {b}{a}\right )^{\frac {1}{4}}\right )\,|\,-1) + 2 \, {\left (14784 \, b^{2} e x^{10} + 5544 \, b^{2} d x^{9} + 2880 \, b^{2} c x^{8} + 17325 \, a b f x^{7} + 13552 \, a b e x^{6} + 11088 \, a b d x^{5} + 9360 \, a b c x^{4} + 6930 \, a^{2} f x^{3} + 6160 \, a^{2} e x^{2} + 5544 \, a^{2} d x + 5040 \, a^{2} c\right )} \sqrt {b x^{4} + a}}{110880 \, a x^{11}} \]
-1/110880*(29568*sqrt(a)*b^2*e*x^11*(-b/a)^(3/4)*elliptic_e(arcsin(x*(-b/a )^(1/4)), -1) - 10395*sqrt(a)*b^2*f*x^11*log(-(b*x^4 - 2*sqrt(b*x^4 + a)*s qrt(a) + 2*a)/x^4) - 384*(15*b^2*c + 77*b^2*e)*sqrt(a)*x^11*(-b/a)^(3/4)*e lliptic_f(arcsin(x*(-b/a)^(1/4)), -1) + 2*(14784*b^2*e*x^10 + 5544*b^2*d*x ^9 + 2880*b^2*c*x^8 + 17325*a*b*f*x^7 + 13552*a*b*e*x^6 + 11088*a*b*d*x^5 + 9360*a*b*c*x^4 + 6930*a^2*f*x^3 + 6160*a^2*e*x^2 + 5544*a^2*d*x + 5040*a ^2*c)*sqrt(b*x^4 + a))/(a*x^11)
Result contains complex when optimal does not.
Time = 6.99 (sec) , antiderivative size = 401, normalized size of antiderivative = 0.95 \[ \int \frac {\left (c+d x+e x^2+f x^3\right ) \left (a+b x^4\right )^{3/2}}{x^{12}} \, dx=\frac {a^{\frac {3}{2}} c \Gamma \left (- \frac {11}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {11}{4}, - \frac {1}{2} \\ - \frac {7}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 x^{11} \Gamma \left (- \frac {7}{4}\right )} + \frac {a^{\frac {3}{2}} e \Gamma \left (- \frac {9}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {9}{4}, - \frac {1}{2} \\ - \frac {5}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 x^{9} \Gamma \left (- \frac {5}{4}\right )} + \frac {\sqrt {a} b c \Gamma \left (- \frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {7}{4}, - \frac {1}{2} \\ - \frac {3}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 x^{7} \Gamma \left (- \frac {3}{4}\right )} + \frac {\sqrt {a} b e \Gamma \left (- \frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {5}{4}, - \frac {1}{2} \\ - \frac {1}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 x^{5} \Gamma \left (- \frac {1}{4}\right )} - \frac {a^{2} f}{8 \sqrt {b} x^{10} \sqrt {\frac {a}{b x^{4}} + 1}} - \frac {a \sqrt {b} d \sqrt {\frac {a}{b x^{4}} + 1}}{10 x^{8}} - \frac {3 a \sqrt {b} f}{16 x^{6} \sqrt {\frac {a}{b x^{4}} + 1}} - \frac {b^{\frac {3}{2}} d \sqrt {\frac {a}{b x^{4}} + 1}}{5 x^{4}} - \frac {b^{\frac {3}{2}} f \sqrt {\frac {a}{b x^{4}} + 1}}{4 x^{2}} - \frac {b^{\frac {3}{2}} f}{16 x^{2} \sqrt {\frac {a}{b x^{4}} + 1}} - \frac {b^{\frac {5}{2}} d \sqrt {\frac {a}{b x^{4}} + 1}}{10 a} - \frac {3 b^{2} f \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x^{2}} \right )}}{16 \sqrt {a}} \]
a**(3/2)*c*gamma(-11/4)*hyper((-11/4, -1/2), (-7/4,), b*x**4*exp_polar(I*p i)/a)/(4*x**11*gamma(-7/4)) + a**(3/2)*e*gamma(-9/4)*hyper((-9/4, -1/2), ( -5/4,), b*x**4*exp_polar(I*pi)/a)/(4*x**9*gamma(-5/4)) + sqrt(a)*b*c*gamma (-7/4)*hyper((-7/4, -1/2), (-3/4,), b*x**4*exp_polar(I*pi)/a)/(4*x**7*gamm a(-3/4)) + sqrt(a)*b*e*gamma(-5/4)*hyper((-5/4, -1/2), (-1/4,), b*x**4*exp _polar(I*pi)/a)/(4*x**5*gamma(-1/4)) - a**2*f/(8*sqrt(b)*x**10*sqrt(a/(b*x **4) + 1)) - a*sqrt(b)*d*sqrt(a/(b*x**4) + 1)/(10*x**8) - 3*a*sqrt(b)*f/(1 6*x**6*sqrt(a/(b*x**4) + 1)) - b**(3/2)*d*sqrt(a/(b*x**4) + 1)/(5*x**4) - b**(3/2)*f*sqrt(a/(b*x**4) + 1)/(4*x**2) - b**(3/2)*f/(16*x**2*sqrt(a/(b*x **4) + 1)) - b**(5/2)*d*sqrt(a/(b*x**4) + 1)/(10*a) - 3*b**2*f*asinh(sqrt( a)/(sqrt(b)*x**2))/(16*sqrt(a))
\[ \int \frac {\left (c+d x+e x^2+f x^3\right ) \left (a+b x^4\right )^{3/2}}{x^{12}} \, dx=\int { \frac {{\left (b x^{4} + a\right )}^{\frac {3}{2}} {\left (f x^{3} + e x^{2} + d x + c\right )}}{x^{12}} \,d x } \]
\[ \int \frac {\left (c+d x+e x^2+f x^3\right ) \left (a+b x^4\right )^{3/2}}{x^{12}} \, dx=\int { \frac {{\left (b x^{4} + a\right )}^{\frac {3}{2}} {\left (f x^{3} + e x^{2} + d x + c\right )}}{x^{12}} \,d x } \]
Timed out. \[ \int \frac {\left (c+d x+e x^2+f x^3\right ) \left (a+b x^4\right )^{3/2}}{x^{12}} \, dx=\int \frac {{\left (b\,x^4+a\right )}^{3/2}\,\left (f\,x^3+e\,x^2+d\,x+c\right )}{x^{12}} \,d x \]