Integrand size = 30, antiderivative size = 474 \[ \int \frac {\left (c+d x+e x^2+f x^3\right ) \left (a+b x^4\right )^{3/2}}{x^{14}} \, dx=-\frac {b \left (\frac {12320 c}{x^9}+\frac {15015 d}{x^8}+\frac {18720 e}{x^7}+\frac {24024 f}{x^6}\right ) \sqrt {a+b x^4}}{240240}-\frac {4 b^2 c \sqrt {a+b x^4}}{195 a x^5}-\frac {b^2 d \sqrt {a+b x^4}}{32 a x^4}-\frac {4 b^2 e \sqrt {a+b x^4}}{77 a x^3}-\frac {b^2 f \sqrt {a+b x^4}}{10 a x^2}+\frac {4 b^3 c \sqrt {a+b x^4}}{65 a^2 x}-\frac {4 b^{7/2} c x \sqrt {a+b x^4}}{65 a^2 \left (\sqrt {a}+\sqrt {b} x^2\right )}-\frac {\left (\frac {660 c}{x^{13}}+\frac {715 d}{x^{12}}+\frac {780 e}{x^{11}}+\frac {858 f}{x^{10}}\right ) \left (a+b x^4\right )^{3/2}}{8580}+\frac {b^3 d \text {arctanh}\left (\frac {\sqrt {a+b x^4}}{\sqrt {a}}\right )}{32 a^{3/2}}+\frac {4 b^{13/4} c \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{65 a^{7/4} \sqrt {a+b x^4}}-\frac {2 b^{11/4} \left (77 \sqrt {b} c+65 \sqrt {a} e\right ) \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{5005 a^{7/4} \sqrt {a+b x^4}} \]
-1/8580*(660*c/x^13+715*d/x^12+780*e/x^11+858*f/x^10)*(b*x^4+a)^(3/2)+1/32 *b^3*d*arctanh((b*x^4+a)^(1/2)/a^(1/2))/a^(3/2)-1/240240*b*(12320*c/x^9+15 015*d/x^8+18720*e/x^7+24024*f/x^6)*(b*x^4+a)^(1/2)-4/195*b^2*c*(b*x^4+a)^( 1/2)/a/x^5-1/32*b^2*d*(b*x^4+a)^(1/2)/a/x^4-4/77*b^2*e*(b*x^4+a)^(1/2)/a/x ^3-1/10*b^2*f*(b*x^4+a)^(1/2)/a/x^2+4/65*b^3*c*(b*x^4+a)^(1/2)/a^2/x-4/65* b^(7/2)*c*x*(b*x^4+a)^(1/2)/a^2/(a^(1/2)+x^2*b^(1/2))+4/65*b^(13/4)*c*(cos (2*arctan(b^(1/4)*x/a^(1/4)))^2)^(1/2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*El lipticE(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^(7/4)/(b*x^4+a)^(1/2)-2/5005 *b^(11/4)*(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))*(65*e* a^(1/2)+77*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^(7/4)/(b*x^4+a)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.21 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.32 \[ \int \frac {\left (c+d x+e x^2+f x^3\right ) \left (a+b x^4\right )^{3/2}}{x^{14}} \, dx=-\frac {\sqrt {a+b x^4} \left (110 a^5 c \operatorname {Hypergeometric2F1}\left (-\frac {13}{4},-\frac {3}{2},-\frac {9}{4},-\frac {b x^4}{a}\right )+13 x^2 \left (10 a^5 e \operatorname {Hypergeometric2F1}\left (-\frac {11}{4},-\frac {3}{2},-\frac {7}{4},-\frac {b x^4}{a}\right )+11 x \left (a+b x^4\right )^2 \sqrt {1+\frac {b x^4}{a}} \left (a^3 f-b^3 d x^{10} \operatorname {Hypergeometric2F1}\left (\frac {5}{2},4,\frac {7}{2},1+\frac {b x^4}{a}\right )\right )\right )\right )}{1430 a^4 x^{13} \sqrt {1+\frac {b x^4}{a}}} \]
-1/1430*(Sqrt[a + b*x^4]*(110*a^5*c*Hypergeometric2F1[-13/4, -3/2, -9/4, - ((b*x^4)/a)] + 13*x^2*(10*a^5*e*Hypergeometric2F1[-11/4, -3/2, -7/4, -((b* x^4)/a)] + 11*x*(a + b*x^4)^2*Sqrt[1 + (b*x^4)/a]*(a^3*f - b^3*d*x^10*Hype rgeometric2F1[5/2, 4, 7/2, 1 + (b*x^4)/a]))))/(a^4*x^13*Sqrt[1 + (b*x^4)/a ])
Time = 0.93 (sec) , antiderivative size = 455, normalized size of antiderivative = 0.96, 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.
| \(\displaystyle \int \frac {\left (a+b x^4\right )^{3/2} \left (c+d x+e x^2+f x^3\right )}{x^{14}} \, dx\) |
| \(\Big \downarrow \) 2364 |
| \(\displaystyle -6 b \int -\frac {\left (858 f x^3+780 e x^2+715 d x+660 c\right ) \sqrt {b x^4+a}}{8580 x^{10}}dx-\frac {\left (a+b x^4\right )^{3/2} \left (\frac {660 c}{x^{13}}+\frac {715 d}{x^{12}}+\frac {780 e}{x^{11}}+\frac {858 f}{x^{10}}\right )}{8580}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {b \int \frac {\left (858 f x^3+780 e x^2+715 d x+660 c\right ) \sqrt {b x^4+a}}{x^{10}}dx}{1430}-\frac {\left (a+b x^4\right )^{3/2} \left (\frac {660 c}{x^{13}}+\frac {715 d}{x^{12}}+\frac {780 e}{x^{11}}+\frac {858 f}{x^{10}}\right )}{8580}\) |
| \(\Big \downarrow \) 2364 |
| \(\displaystyle \frac {b \left (-2 b \int -\frac {24024 f x^3+18720 e x^2+15015 d x+12320 c}{168 x^6 \sqrt {b x^4+a}}dx-\frac {1}{168} \sqrt {a+b x^4} \left (\frac {12320 c}{x^9}+\frac {15015 d}{x^8}+\frac {18720 e}{x^7}+\frac {24024 f}{x^6}\right )\right )}{1430}-\frac {\left (a+b x^4\right )^{3/2} \left (\frac {660 c}{x^{13}}+\frac {715 d}{x^{12}}+\frac {780 e}{x^{11}}+\frac {858 f}{x^{10}}\right )}{8580}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {b \left (\frac {1}{84} b \int \frac {24024 f x^3+18720 e x^2+15015 d x+12320 c}{x^6 \sqrt {b x^4+a}}dx-\frac {1}{168} \sqrt {a+b x^4} \left (\frac {12320 c}{x^9}+\frac {15015 d}{x^8}+\frac {18720 e}{x^7}+\frac {24024 f}{x^6}\right )\right )}{1430}-\frac {\left (a+b x^4\right )^{3/2} \left (\frac {660 c}{x^{13}}+\frac {715 d}{x^{12}}+\frac {780 e}{x^{11}}+\frac {858 f}{x^{10}}\right )}{8580}\) |
| \(\Big \downarrow \) 2372 |
| \(\displaystyle \frac {b \left (\frac {1}{84} b \int \left (\frac {18720 e x^2+12320 c}{x^6 \sqrt {b x^4+a}}+\frac {24024 f x^2+15015 d}{x^5 \sqrt {b x^4+a}}\right )dx-\frac {1}{168} \sqrt {a+b x^4} \left (\frac {12320 c}{x^9}+\frac {15015 d}{x^8}+\frac {18720 e}{x^7}+\frac {24024 f}{x^6}\right )\right )}{1430}-\frac {\left (a+b x^4\right )^{3/2} \left (\frac {660 c}{x^{13}}+\frac {715 d}{x^{12}}+\frac {780 e}{x^{11}}+\frac {858 f}{x^{10}}\right )}{8580}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {b \left (\frac {1}{84} b \left (-\frac {48 b^{3/4} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \left (65 \sqrt {a} e+77 \sqrt {b} c\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{a^{7/4} \sqrt {a+b x^4}}+\frac {7392 b^{5/4} c \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{a^{7/4} \sqrt {a+b x^4}}+\frac {15015 b d \text {arctanh}\left (\frac {\sqrt {a+b x^4}}{\sqrt {a}}\right )}{4 a^{3/2}}-\frac {7392 b^{3/2} c x \sqrt {a+b x^4}}{a^2 \left (\sqrt {a}+\sqrt {b} x^2\right )}+\frac {7392 b c \sqrt {a+b x^4}}{a^2 x}-\frac {2464 c \sqrt {a+b x^4}}{a x^5}-\frac {15015 d \sqrt {a+b x^4}}{4 a x^4}-\frac {6240 e \sqrt {a+b x^4}}{a x^3}-\frac {12012 f \sqrt {a+b x^4}}{a x^2}\right )-\frac {1}{168} \sqrt {a+b x^4} \left (\frac {12320 c}{x^9}+\frac {15015 d}{x^8}+\frac {18720 e}{x^7}+\frac {24024 f}{x^6}\right )\right )}{1430}-\frac {\left (a+b x^4\right )^{3/2} \left (\frac {660 c}{x^{13}}+\frac {715 d}{x^{12}}+\frac {780 e}{x^{11}}+\frac {858 f}{x^{10}}\right )}{8580}\) |
-1/8580*(((660*c)/x^13 + (715*d)/x^12 + (780*e)/x^11 + (858*f)/x^10)*(a + b*x^4)^(3/2)) + (b*(-1/168*(((12320*c)/x^9 + (15015*d)/x^8 + (18720*e)/x^7 + (24024*f)/x^6)*Sqrt[a + b*x^4]) + (b*((-2464*c*Sqrt[a + b*x^4])/(a*x^5) - (15015*d*Sqrt[a + b*x^4])/(4*a*x^4) - (6240*e*Sqrt[a + b*x^4])/(a*x^3) - (12012*f*Sqrt[a + b*x^4])/(a*x^2) + (7392*b*c*Sqrt[a + b*x^4])/(a^2*x) - (7392*b^(3/2)*c*x*Sqrt[a + b*x^4])/(a^2*(Sqrt[a] + Sqrt[b]*x^2)) + (15015 *b*d*ArcTanh[Sqrt[a + b*x^4]/Sqrt[a]])/(4*a^(3/2)) + (7392*b^(5/4)*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^(7/4)*Sqrt[a + b*x^4]) - (48*b^(3/4 )*(77*Sqrt[b]*c + 65*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^(7/4)*Sqrt[a + b*x^4])))/84))/1430
3.6.28.3.1 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 = 3.65 (sec) , antiderivative size = 347, normalized size of antiderivative = 0.73
| method | result | size |
| risch | \(-\frac {\sqrt {b \,x^{4}+a}\, \left (-29568 b^{3} c \,x^{12}+48048 x^{11} a \,b^{2} f +24960 a \,b^{2} e \,x^{10}+15015 x^{9} a \,b^{2} d +9856 a \,b^{2} c \,x^{8}+96096 a^{2} b f \,x^{7}+81120 a^{2} b e \,x^{6}+70070 x^{5} b d \,a^{2}+61600 a^{2} b c \,x^{4}+48048 f \,a^{3} x^{3}+43680 a^{3} e \,x^{2}+40040 a^{3} d x +36960 c \,a^{3}\right )}{480480 x^{13} a^{2}}-\frac {b^{3} \left (\frac {4160 a e \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, F\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}+\frac {4928 i \sqrt {b}\, c \sqrt {a}\, \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (F\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-E\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}-\frac {5005 \sqrt {a}\, d \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{4}+a}}{x^{2}}\right )}{2}\right )}{80080 a^{2}}\) | \(347\) |
| default | \(d \left (-\frac {a \sqrt {b \,x^{4}+a}}{12 x^{12}}-\frac {7 b \sqrt {b \,x^{4}+a}}{48 x^{8}}-\frac {b^{2} \sqrt {b \,x^{4}+a}}{32 a \,x^{4}}+\frac {b^{3} \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{4}+a}}{x^{2}}\right )}{32 a^{\frac {3}{2}}}\right )-\frac {f \left (b^{2} x^{8}+2 a b \,x^{4}+a^{2}\right ) \sqrt {b \,x^{4}+a}}{10 x^{10} a}+e \left (-\frac {a \sqrt {b \,x^{4}+a}}{11 x^{11}}-\frac {13 b \sqrt {b \,x^{4}+a}}{77 x^{7}}-\frac {4 b^{2} \sqrt {b \,x^{4}+a}}{77 a \,x^{3}}-\frac {4 b^{3} \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, F\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{77 a \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\right )+c \left (-\frac {a \sqrt {b \,x^{4}+a}}{13 x^{13}}-\frac {5 b \sqrt {b \,x^{4}+a}}{39 x^{9}}-\frac {4 b^{2} \sqrt {b \,x^{4}+a}}{195 a \,x^{5}}+\frac {4 b^{3} \sqrt {b \,x^{4}+a}}{65 a^{2} x}-\frac {4 i b^{\frac {7}{2}} \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (F\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-E\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{65 a^{\frac {3}{2}} \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\right )\) | \(420\) |
| elliptic | \(-\frac {a c \sqrt {b \,x^{4}+a}}{13 x^{13}}-\frac {a d \sqrt {b \,x^{4}+a}}{12 x^{12}}-\frac {a e \sqrt {b \,x^{4}+a}}{11 x^{11}}-\frac {a f \sqrt {b \,x^{4}+a}}{10 x^{10}}-\frac {5 b c \sqrt {b \,x^{4}+a}}{39 x^{9}}-\frac {7 b d \sqrt {b \,x^{4}+a}}{48 x^{8}}-\frac {13 b e \sqrt {b \,x^{4}+a}}{77 x^{7}}-\frac {b f \sqrt {b \,x^{4}+a}}{5 x^{6}}-\frac {4 b^{2} c \sqrt {b \,x^{4}+a}}{195 a \,x^{5}}-\frac {b^{2} d \sqrt {b \,x^{4}+a}}{32 a \,x^{4}}-\frac {4 b^{2} e \sqrt {b \,x^{4}+a}}{77 a \,x^{3}}-\frac {b^{2} f \sqrt {b \,x^{4}+a}}{10 a \,x^{2}}+\frac {4 b^{3} c \sqrt {b \,x^{4}+a}}{65 a^{2} x}-\frac {4 b^{3} e \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, F\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{77 a \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}-\frac {4 i b^{\frac {7}{2}} c \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (F\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-E\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{65 a^{\frac {3}{2}} \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}+\frac {d \,b^{3} \operatorname {arctanh}\left (\frac {\sqrt {a}}{\sqrt {b \,x^{4}+a}}\right )}{32 a^{\frac {3}{2}}}\) | \(432\) |
-1/480480*(b*x^4+a)^(1/2)*(-29568*b^3*c*x^12+48048*a*b^2*f*x^11+24960*a*b^ 2*e*x^10+15015*a*b^2*d*x^9+9856*a*b^2*c*x^8+96096*a^2*b*f*x^7+81120*a^2*b* e*x^6+70070*a^2*b*d*x^5+61600*a^2*b*c*x^4+48048*a^3*f*x^3+43680*a^3*e*x^2+ 40040*a^3*d*x+36960*a^3*c)/x^13/a^2-1/80080/a^2*b^3*(4160*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)+4928*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))-5005/2*a^(1/2)*d*ln( (2*a+2*a^(1/2)*(b*x^4+a)^(1/2))/x^2))
Time = 0.13 (sec) , antiderivative size = 258, normalized size of antiderivative = 0.54 \[ \int \frac {\left (c+d x+e x^2+f x^3\right ) \left (a+b x^4\right )^{3/2}}{x^{14}} \, dx=\frac {59136 \, \sqrt {a} b^{3} c x^{13} \left (-\frac {b}{a}\right )^{\frac {3}{4}} E(\arcsin \left (x \left (-\frac {b}{a}\right )^{\frac {1}{4}}\right )\,|\,-1) + 15015 \, \sqrt {a} b^{3} d x^{13} \log \left (-\frac {b x^{4} + 2 \, \sqrt {b x^{4} + a} \sqrt {a} + 2 \, a}{x^{4}}\right ) - 768 \, {\left (77 \, b^{3} c - 65 \, a b^{2} e\right )} \sqrt {a} x^{13} \left (-\frac {b}{a}\right )^{\frac {3}{4}} F(\arcsin \left (x \left (-\frac {b}{a}\right )^{\frac {1}{4}}\right )\,|\,-1) + 2 \, {\left (29568 \, b^{3} c x^{12} - 48048 \, a b^{2} f x^{11} - 24960 \, a b^{2} e x^{10} - 15015 \, a b^{2} d x^{9} - 9856 \, a b^{2} c x^{8} - 96096 \, a^{2} b f x^{7} - 81120 \, a^{2} b e x^{6} - 70070 \, a^{2} b d x^{5} - 61600 \, a^{2} b c x^{4} - 48048 \, a^{3} f x^{3} - 43680 \, a^{3} e x^{2} - 40040 \, a^{3} d x - 36960 \, a^{3} c\right )} \sqrt {b x^{4} + a}}{960960 \, a^{2} x^{13}} \]
1/960960*(59136*sqrt(a)*b^3*c*x^13*(-b/a)^(3/4)*elliptic_e(arcsin(x*(-b/a) ^(1/4)), -1) + 15015*sqrt(a)*b^3*d*x^13*log(-(b*x^4 + 2*sqrt(b*x^4 + a)*sq rt(a) + 2*a)/x^4) - 768*(77*b^3*c - 65*a*b^2*e)*sqrt(a)*x^13*(-b/a)^(3/4)* elliptic_f(arcsin(x*(-b/a)^(1/4)), -1) + 2*(29568*b^3*c*x^12 - 48048*a*b^2 *f*x^11 - 24960*a*b^2*e*x^10 - 15015*a*b^2*d*x^9 - 9856*a*b^2*c*x^8 - 9609 6*a^2*b*f*x^7 - 81120*a^2*b*e*x^6 - 70070*a^2*b*d*x^5 - 61600*a^2*b*c*x^4 - 48048*a^3*f*x^3 - 43680*a^3*e*x^2 - 40040*a^3*d*x - 36960*a^3*c)*sqrt(b* x^4 + a))/(a^2*x^13)
Result contains complex when optimal does not.
Time = 11.57 (sec) , antiderivative size = 403, normalized size of antiderivative = 0.85 \[ \int \frac {\left (c+d x+e x^2+f x^3\right ) \left (a+b x^4\right )^{3/2}}{x^{14}} \, dx=\frac {a^{\frac {3}{2}} c \Gamma \left (- \frac {13}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {13}{4}, - \frac {1}{2} \\ - \frac {9}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 x^{13} \Gamma \left (- \frac {9}{4}\right )} + \frac {a^{\frac {3}{2}} e \Gamma \left (- \frac {11}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {11}{4}, - \frac {1}{2} \\ - \frac {7}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 x^{11} \Gamma \left (- \frac {7}{4}\right )} + \frac {\sqrt {a} b c \Gamma \left (- \frac {9}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {9}{4}, - \frac {1}{2} \\ - \frac {5}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 x^{9} \Gamma \left (- \frac {5}{4}\right )} + \frac {\sqrt {a} b e \Gamma \left (- \frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {7}{4}, - \frac {1}{2} \\ - \frac {3}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 x^{7} \Gamma \left (- \frac {3}{4}\right )} - \frac {a^{2} d}{12 \sqrt {b} x^{14} \sqrt {\frac {a}{b x^{4}} + 1}} - \frac {11 a \sqrt {b} d}{48 x^{10} \sqrt {\frac {a}{b x^{4}} + 1}} - \frac {a \sqrt {b} f \sqrt {\frac {a}{b x^{4}} + 1}}{10 x^{8}} - \frac {17 b^{\frac {3}{2}} d}{96 x^{6} \sqrt {\frac {a}{b x^{4}} + 1}} - \frac {b^{\frac {3}{2}} f \sqrt {\frac {a}{b x^{4}} + 1}}{5 x^{4}} - \frac {b^{\frac {5}{2}} d}{32 a x^{2} \sqrt {\frac {a}{b x^{4}} + 1}} - \frac {b^{\frac {5}{2}} f \sqrt {\frac {a}{b x^{4}} + 1}}{10 a} + \frac {b^{3} d \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x^{2}} \right )}}{32 a^{\frac {3}{2}}} \]
a**(3/2)*c*gamma(-13/4)*hyper((-13/4, -1/2), (-9/4,), b*x**4*exp_polar(I*p i)/a)/(4*x**13*gamma(-9/4)) + a**(3/2)*e*gamma(-11/4)*hyper((-11/4, -1/2), (-7/4,), b*x**4*exp_polar(I*pi)/a)/(4*x**11*gamma(-7/4)) + sqrt(a)*b*c*ga mma(-9/4)*hyper((-9/4, -1/2), (-5/4,), b*x**4*exp_polar(I*pi)/a)/(4*x**9*g amma(-5/4)) + sqrt(a)*b*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)) - a**2*d/(12*sqrt(b)*x**14*sqrt(a/ (b*x**4) + 1)) - 11*a*sqrt(b)*d/(48*x**10*sqrt(a/(b*x**4) + 1)) - a*sqrt(b )*f*sqrt(a/(b*x**4) + 1)/(10*x**8) - 17*b**(3/2)*d/(96*x**6*sqrt(a/(b*x**4 ) + 1)) - b**(3/2)*f*sqrt(a/(b*x**4) + 1)/(5*x**4) - b**(5/2)*d/(32*a*x**2 *sqrt(a/(b*x**4) + 1)) - b**(5/2)*f*sqrt(a/(b*x**4) + 1)/(10*a) + b**3*d*a sinh(sqrt(a)/(sqrt(b)*x**2))/(32*a**(3/2))
\[ \int \frac {\left (c+d x+e x^2+f x^3\right ) \left (a+b x^4\right )^{3/2}}{x^{14}} \, dx=\int { \frac {{\left (b x^{4} + a\right )}^{\frac {3}{2}} {\left (f x^{3} + e x^{2} + d x + c\right )}}{x^{14}} \,d x } \]
\[ \int \frac {\left (c+d x+e x^2+f x^3\right ) \left (a+b x^4\right )^{3/2}}{x^{14}} \, dx=\int { \frac {{\left (b x^{4} + a\right )}^{\frac {3}{2}} {\left (f x^{3} + e x^{2} + d x + c\right )}}{x^{14}} \,d x } \]
Timed out. \[ \int \frac {\left (c+d x+e x^2+f x^3\right ) \left (a+b x^4\right )^{3/2}}{x^{14}} \, dx=\int \frac {{\left (b\,x^4+a\right )}^{3/2}\,\left (f\,x^3+e\,x^2+d\,x+c\right )}{x^{14}} \,d x \]