Integrand size = 30, antiderivative size = 480 \[ \int \frac {\left (c+d x+e x^2+f x^3\right ) \left (a+b x^4\right )^{3/2}}{x^{13}} \, dx=-\frac {a c \sqrt {a+b x^4}}{12 x^{12}}-\frac {7 b c \sqrt {a+b x^4}}{48 x^8}+\frac {12 b d \sqrt {a+b x^4}}{385 x^7}+\frac {4 b f \sqrt {a+b x^4}}{45 x^5}-\frac {b^2 c \sqrt {a+b x^4}}{32 a x^4}-\frac {4 b^2 d \sqrt {a+b x^4}}{77 a x^3}-\frac {4 b^2 f \sqrt {a+b x^4}}{15 \sqrt {a} x \left (\sqrt {a}+\sqrt {b} x^2\right )}+\frac {2 \left (55 a f-27 b d x^2\right ) \sqrt {a+b x^4}}{495 x^9}-\frac {\left (3 d+11 f x^2\right ) \left (a+b x^4\right )^{3/2}}{33 x^{11}}-\frac {e \left (a+b x^4\right )^{5/2}}{10 a x^{10}}+\frac {b^3 c \text {arctanh}\left (\frac {\sqrt {a+b x^4}}{\sqrt {a}}\right )}{32 a^{3/2}}-\frac {4 b^{9/4} f \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{15 a^{3/4} \sqrt {a+b x^4}}-\frac {2 b^{9/4} \left (15 \sqrt {b} d-77 \sqrt {a} f\right ) \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{1155 a^{5/4} \sqrt {a+b x^4}} \] Output:
-1/12*a*c*(b*x^4+a)^(1/2)/x^12-7/48*b*c*(b*x^4+a)^(1/2)/x^8+12/385*b*d*(b* x^4+a)^(1/2)/x^7+4/45*b*f*(b*x^4+a)^(1/2)/x^5-1/32*b^2*c*(b*x^4+a)^(1/2)/a /x^4-4/77*b^2*d*(b*x^4+a)^(1/2)/a/x^3-4/15*b^2*f*(b*x^4+a)^(1/2)/a^(1/2)/x /(a^(1/2)+b^(1/2)*x^2)+2/495*(-27*b*d*x^2+55*a*f)*(b*x^4+a)^(1/2)/x^9-1/33 *(11*f*x^2+3*d)*(b*x^4+a)^(3/2)/x^11-1/10*e*(b*x^4+a)^(5/2)/a/x^10+1/32*b^ 3*c*arctanh((b*x^4+a)^(1/2)/a^(1/2))/a^(3/2)-4/15*b^(9/4)*f*(a^(1/2)+b^(1/ 2)*x^2)*((b*x^4+a)/(a^(1/2)+b^(1/2)*x^2)^2)^(1/2)*EllipticE(sin(2*arctan(b ^(1/4)*x/a^(1/4))),1/2*2^(1/2))/a^(3/4)/(b*x^4+a)^(1/2)-2/1155*b^(9/4)*(15 *b^(1/2)*d-77*a^(1/2)*f)*(a^(1/2)+b^(1/2)*x^2)*((b*x^4+a)/(a^(1/2)+b^(1/2) *x^2)^2)^(1/2)*InverseJacobiAM(2*arctan(b^(1/4)*x/a^(1/4)),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 = 10.17 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.31 \[ \int \frac {\left (c+d x+e x^2+f x^3\right ) \left (a+b x^4\right )^{3/2}}{x^{13}} \, dx=-\frac {\sqrt {a+b x^4} \left (90 a^5 d \operatorname {Hypergeometric2F1}\left (-\frac {11}{4},-\frac {3}{2},-\frac {7}{4},-\frac {b x^4}{a}\right )+11 x \left (10 a^5 f x \operatorname {Hypergeometric2F1}\left (-\frac {9}{4},-\frac {3}{2},-\frac {5}{4},-\frac {b x^4}{a}\right )+9 \left (a+b x^4\right )^2 \sqrt {1+\frac {b x^4}{a}} \left (a^3 e-b^3 c x^{10} \operatorname {Hypergeometric2F1}\left (\frac {5}{2},4,\frac {7}{2},1+\frac {b x^4}{a}\right )\right )\right )\right )}{990 a^4 x^{11} \sqrt {1+\frac {b x^4}{a}}} \] Input:
Integrate[((c + d*x + e*x^2 + f*x^3)*(a + b*x^4)^(3/2))/x^13,x]
Output:
-1/990*(Sqrt[a + b*x^4]*(90*a^5*d*Hypergeometric2F1[-11/4, -3/2, -7/4, -(( b*x^4)/a)] + 11*x*(10*a^5*f*x*Hypergeometric2F1[-9/4, -3/2, -5/4, -((b*x^4 )/a)] + 9*(a + b*x^4)^2*Sqrt[1 + (b*x^4)/a]*(a^3*e - b^3*c*x^10*Hypergeome tric2F1[5/2, 4, 7/2, 1 + (b*x^4)/a]))))/(a^4*x^11*Sqrt[1 + (b*x^4)/a])
Time = 1.46 (sec) , antiderivative size = 434, normalized size of antiderivative = 0.90, 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.
| \(\displaystyle \int \frac {\left (a+b x^4\right )^{3/2} \left (c+d x+e x^2+f x^3\right )}{x^{13}} \, dx\) |
| \(\Big \downarrow \) 2364 |
| \(\displaystyle -6 b \int -\frac {\left (220 f x^3+198 e x^2+180 d x+165 c\right ) \sqrt {b x^4+a}}{1980 x^9}dx-\frac {\left (a+b x^4\right )^{3/2} \left (\frac {165 c}{x^{12}}+\frac {180 d}{x^{11}}+\frac {198 e}{x^{10}}+\frac {220 f}{x^9}\right )}{1980}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{330} b \int \frac {\left (220 f x^3+198 e x^2+180 d x+165 c\right ) \sqrt {b x^4+a}}{x^9}dx-\frac {\left (a+b x^4\right )^{3/2} \left (\frac {165 c}{x^{12}}+\frac {180 d}{x^{11}}+\frac {198 e}{x^{10}}+\frac {220 f}{x^9}\right )}{1980}\) |
| \(\Big \downarrow \) 2364 |
| \(\displaystyle \frac {1}{330} b \left (-2 b \int -\frac {2464 f x^3+1848 e x^2+1440 d x+1155 c}{56 x^5 \sqrt {b x^4+a}}dx-\frac {1}{56} \sqrt {a+b x^4} \left (\frac {1155 c}{x^8}+\frac {1440 d}{x^7}+\frac {1848 e}{x^6}+\frac {2464 f}{x^5}\right )\right )-\frac {\left (a+b x^4\right )^{3/2} \left (\frac {165 c}{x^{12}}+\frac {180 d}{x^{11}}+\frac {198 e}{x^{10}}+\frac {220 f}{x^9}\right )}{1980}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{330} b \left (\frac {1}{28} b \int \frac {2464 f x^3+1848 e x^2+1440 d x+1155 c}{x^5 \sqrt {b x^4+a}}dx-\frac {1}{56} \sqrt {a+b x^4} \left (\frac {1155 c}{x^8}+\frac {1440 d}{x^7}+\frac {1848 e}{x^6}+\frac {2464 f}{x^5}\right )\right )-\frac {\left (a+b x^4\right )^{3/2} \left (\frac {165 c}{x^{12}}+\frac {180 d}{x^{11}}+\frac {198 e}{x^{10}}+\frac {220 f}{x^9}\right )}{1980}\) |
| \(\Big \downarrow \) 2372 |
| \(\displaystyle \frac {1}{330} b \left (\frac {1}{28} b \int \left (\frac {1848 e x^2+1155 c}{x^5 \sqrt {b x^4+a}}+\frac {2464 f x^2+1440 d}{x^4 \sqrt {b x^4+a}}\right )dx-\frac {1}{56} \sqrt {a+b x^4} \left (\frac {1155 c}{x^8}+\frac {1440 d}{x^7}+\frac {1848 e}{x^6}+\frac {2464 f}{x^5}\right )\right )-\frac {\left (a+b x^4\right )^{3/2} \left (\frac {165 c}{x^{12}}+\frac {180 d}{x^{11}}+\frac {198 e}{x^{10}}+\frac {220 f}{x^9}\right )}{1980}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{330} b \left (\frac {1}{28} b \left (-\frac {16 \sqrt [4]{b} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \left (15 \sqrt {b} d-77 \sqrt {a} f\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{a^{5/4} \sqrt {a+b x^4}}-\frac {2464 \sqrt [4]{b} f \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{a^{3/4} \sqrt {a+b x^4}}+\frac {1155 b c \text {arctanh}\left (\frac {\sqrt {a+b x^4}}{\sqrt {a}}\right )}{4 a^{3/2}}-\frac {1155 c \sqrt {a+b x^4}}{4 a x^4}-\frac {480 d \sqrt {a+b x^4}}{a x^3}-\frac {924 e \sqrt {a+b x^4}}{a x^2}-\frac {2464 f \sqrt {a+b x^4}}{a x}+\frac {2464 \sqrt {b} f x \sqrt {a+b x^4}}{a \left (\sqrt {a}+\sqrt {b} x^2\right )}\right )-\frac {1}{56} \sqrt {a+b x^4} \left (\frac {1155 c}{x^8}+\frac {1440 d}{x^7}+\frac {1848 e}{x^6}+\frac {2464 f}{x^5}\right )\right )-\frac {\left (a+b x^4\right )^{3/2} \left (\frac {165 c}{x^{12}}+\frac {180 d}{x^{11}}+\frac {198 e}{x^{10}}+\frac {220 f}{x^9}\right )}{1980}\) |
Input:
Int[((c + d*x + e*x^2 + f*x^3)*(a + b*x^4)^(3/2))/x^13,x]
Output:
-1/1980*(((165*c)/x^12 + (180*d)/x^11 + (198*e)/x^10 + (220*f)/x^9)*(a + b *x^4)^(3/2)) + (b*(-1/56*(((1155*c)/x^8 + (1440*d)/x^7 + (1848*e)/x^6 + (2 464*f)/x^5)*Sqrt[a + b*x^4]) + (b*((-1155*c*Sqrt[a + b*x^4])/(4*a*x^4) - ( 480*d*Sqrt[a + b*x^4])/(a*x^3) - (924*e*Sqrt[a + b*x^4])/(a*x^2) - (2464*f *Sqrt[a + b*x^4])/(a*x) + (2464*Sqrt[b]*f*x*Sqrt[a + b*x^4])/(a*(Sqrt[a] + Sqrt[b]*x^2)) + (1155*b*c*ArcTanh[Sqrt[a + b*x^4]/Sqrt[a]])/(4*a^(3/2)) - (2464*b^(1/4)*f*(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]*d - 77*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^(5/4)*Sqrt[a + b*x^4])))/28))/330
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 = 5.10 (sec) , antiderivative size = 325, normalized size of antiderivative = 0.68
| method | result | size |
| risch | \(-\frac {\sqrt {b \,x^{4}+a}\, \left (29568 b^{2} f \,x^{11}+11088 e \,b^{2} x^{10}+5760 b^{2} d \,x^{9}+3465 b^{2} c \,x^{8}+27104 x^{7} a b f +22176 a b e \,x^{6}+18720 x^{5} a d b +16170 a b c \,x^{4}+12320 a^{2} f \,x^{3}+11088 a^{2} e \,x^{2}+10080 a^{2} d x +9240 a^{2} c \right )}{110880 x^{12} a}-\frac {b^{3} \left (\frac {960 d \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}-\frac {1155 c \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{4}+a}}{x^{2}}\right )}{2 \sqrt {a}}-\frac {4928 i f \sqrt {a}\, \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}\, \sqrt {b}}\right )}{18480 a}\) | \(325\) |
| default | \(c \left (-\frac {a \sqrt {b \,x^{4}+a}}{12 x^{12}}-\frac {7 b \sqrt {b \,x^{4}+a}}{48 x^{8}}-\frac {b^{2} \sqrt {b \,x^{4}+a}}{32 a \,x^{4}}+\frac {b^{3} \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{4}+a}}{x^{2}}\right )}{32 a^{\frac {3}{2}}}\right )+d \left (-\frac {a \sqrt {b \,x^{4}+a}}{11 x^{11}}-\frac {13 b \sqrt {b \,x^{4}+a}}{77 x^{7}}-\frac {4 b^{2} \sqrt {b \,x^{4}+a}}{77 a \,x^{3}}-\frac {4 b^{3} \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{77 a \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\right )-\frac {e \sqrt {b \,x^{4}+a}\, \left (b^{2} x^{8}+2 b \,x^{4} a +a^{2}\right )}{10 x^{10} a}+f \left (-\frac {a \sqrt {b \,x^{4}+a}}{9 x^{9}}-\frac {11 b \sqrt {b \,x^{4}+a}}{45 x^{5}}-\frac {4 b^{2} \sqrt {b \,x^{4}+a}}{15 a x}+\frac {4 i b^{\frac {5}{2}} \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{15 \sqrt {a}\, \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\right )\) | \(400\) |
| elliptic | \(-\frac {a c \sqrt {b \,x^{4}+a}}{12 x^{12}}-\frac {a d \sqrt {b \,x^{4}+a}}{11 x^{11}}-\frac {a e \sqrt {b \,x^{4}+a}}{10 x^{10}}-\frac {a f \sqrt {b \,x^{4}+a}}{9 x^{9}}-\frac {7 b c \sqrt {b \,x^{4}+a}}{48 x^{8}}-\frac {13 b d \sqrt {b \,x^{4}+a}}{77 x^{7}}-\frac {b e \sqrt {b \,x^{4}+a}}{5 x^{6}}-\frac {11 b f \sqrt {b \,x^{4}+a}}{45 x^{5}}-\frac {b^{2} c \sqrt {b \,x^{4}+a}}{32 a \,x^{4}}-\frac {4 b^{2} d \sqrt {b \,x^{4}+a}}{77 a \,x^{3}}-\frac {e \,b^{2} \sqrt {b \,x^{4}+a}}{10 a \,x^{2}}-\frac {4 b^{2} f \sqrt {b \,x^{4}+a}}{15 a x}-\frac {4 b^{3} d \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{77 a \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}+\frac {4 i b^{\frac {5}{2}} f \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{15 \sqrt {a}\, \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}+\frac {b^{3} c \,\operatorname {arctanh}\left (\frac {\sqrt {a}}{\sqrt {b \,x^{4}+a}}\right )}{32 a^{\frac {3}{2}}}\) | \(411\) |
Input:
int((f*x^3+e*x^2+d*x+c)*(b*x^4+a)^(3/2)/x^13,x,method=_RETURNVERBOSE)
Output:
-1/110880*(b*x^4+a)^(1/2)*(29568*b^2*f*x^11+11088*b^2*e*x^10+5760*b^2*d*x^ 9+3465*b^2*c*x^8+27104*a*b*f*x^7+22176*a*b*e*x^6+18720*a*b*d*x^5+16170*a*b *c*x^4+12320*a^2*f*x^3+11088*a^2*e*x^2+10080*a^2*d*x+9240*a^2*c)/x^12/a-1/ 18480/a*b^3*(960*d/(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)-1155/2*c/a^(1/2)*ln((2*a+2*a^(1/2)*(b*x^4+a)^(1/2))/x^2) -4928*I*f*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)/b^(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)))
Time = 0.11 (sec) , antiderivative size = 250, normalized size of antiderivative = 0.52 \[ \int \frac {\left (c+d x+e x^2+f x^3\right ) \left (a+b x^4\right )^{3/2}}{x^{13}} \, dx=-\frac {59136 \, a^{\frac {3}{2}} b^{2} f x^{12} \left (-\frac {b}{a}\right )^{\frac {3}{4}} E(\arcsin \left (x \left (-\frac {b}{a}\right )^{\frac {1}{4}}\right )\,|\,-1) - 3465 \, \sqrt {a} b^{3} c x^{12} \log \left (-\frac {b x^{4} + 2 \, \sqrt {b x^{4} + a} \sqrt {a} + 2 \, a}{x^{4}}\right ) - 768 \, {\left (15 \, a b^{2} d + 77 \, a b^{2} f\right )} \sqrt {a} x^{12} \left (-\frac {b}{a}\right )^{\frac {3}{4}} F(\arcsin \left (x \left (-\frac {b}{a}\right )^{\frac {1}{4}}\right )\,|\,-1) + 2 \, {\left (29568 \, a b^{2} f x^{11} + 11088 \, a b^{2} e x^{10} + 5760 \, a b^{2} d x^{9} + 3465 \, a b^{2} c x^{8} + 27104 \, a^{2} b f x^{7} + 22176 \, a^{2} b e x^{6} + 18720 \, a^{2} b d x^{5} + 16170 \, a^{2} b c x^{4} + 12320 \, a^{3} f x^{3} + 11088 \, a^{3} e x^{2} + 10080 \, a^{3} d x + 9240 \, a^{3} c\right )} \sqrt {b x^{4} + a}}{221760 \, a^{2} x^{12}} \] Input:
integrate((f*x^3+e*x^2+d*x+c)*(b*x^4+a)^(3/2)/x^13,x, algorithm="fricas")
Output:
-1/221760*(59136*a^(3/2)*b^2*f*x^12*(-b/a)^(3/4)*elliptic_e(arcsin(x*(-b/a )^(1/4)), -1) - 3465*sqrt(a)*b^3*c*x^12*log(-(b*x^4 + 2*sqrt(b*x^4 + a)*sq rt(a) + 2*a)/x^4) - 768*(15*a*b^2*d + 77*a*b^2*f)*sqrt(a)*x^12*(-b/a)^(3/4 )*elliptic_f(arcsin(x*(-b/a)^(1/4)), -1) + 2*(29568*a*b^2*f*x^11 + 11088*a *b^2*e*x^10 + 5760*a*b^2*d*x^9 + 3465*a*b^2*c*x^8 + 27104*a^2*b*f*x^7 + 22 176*a^2*b*e*x^6 + 18720*a^2*b*d*x^5 + 16170*a^2*b*c*x^4 + 12320*a^3*f*x^3 + 11088*a^3*e*x^2 + 10080*a^3*d*x + 9240*a^3*c)*sqrt(b*x^4 + a))/(a^2*x^12 )
Result contains complex when optimal does not.
Time = 10.61 (sec) , antiderivative size = 403, normalized size of antiderivative = 0.84 \[ \int \frac {\left (c+d x+e x^2+f x^3\right ) \left (a+b x^4\right )^{3/2}}{x^{13}} \, dx=\frac {a^{\frac {3}{2}} d \Gamma \left (- \frac {11}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {11}{4}, - \frac {1}{2} \\ - \frac {7}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 x^{11} \Gamma \left (- \frac {7}{4}\right )} + \frac {a^{\frac {3}{2}} f \Gamma \left (- \frac {9}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {9}{4}, - \frac {1}{2} \\ - \frac {5}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 x^{9} \Gamma \left (- \frac {5}{4}\right )} + \frac {\sqrt {a} b d \Gamma \left (- \frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {7}{4}, - \frac {1}{2} \\ - \frac {3}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 x^{7} \Gamma \left (- \frac {3}{4}\right )} + \frac {\sqrt {a} b f \Gamma \left (- \frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {5}{4}, - \frac {1}{2} \\ - \frac {1}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 x^{5} \Gamma \left (- \frac {1}{4}\right )} - \frac {a^{2} c}{12 \sqrt {b} x^{14} \sqrt {\frac {a}{b x^{4}} + 1}} - \frac {11 a \sqrt {b} c}{48 x^{10} \sqrt {\frac {a}{b x^{4}} + 1}} - \frac {a \sqrt {b} e \sqrt {\frac {a}{b x^{4}} + 1}}{10 x^{8}} - \frac {17 b^{\frac {3}{2}} c}{96 x^{6} \sqrt {\frac {a}{b x^{4}} + 1}} - \frac {b^{\frac {3}{2}} e \sqrt {\frac {a}{b x^{4}} + 1}}{5 x^{4}} - \frac {b^{\frac {5}{2}} c}{32 a x^{2} \sqrt {\frac {a}{b x^{4}} + 1}} - \frac {b^{\frac {5}{2}} e \sqrt {\frac {a}{b x^{4}} + 1}}{10 a} + \frac {b^{3} c \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x^{2}} \right )}}{32 a^{\frac {3}{2}}} \] Input:
integrate((f*x**3+e*x**2+d*x+c)*(b*x**4+a)**(3/2)/x**13,x)
Output:
a**(3/2)*d*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)*f*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*d*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*f*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*c/(12*sqrt(b)*x**14*sqrt(a/(b* x**4) + 1)) - 11*a*sqrt(b)*c/(48*x**10*sqrt(a/(b*x**4) + 1)) - a*sqrt(b)*e *sqrt(a/(b*x**4) + 1)/(10*x**8) - 17*b**(3/2)*c/(96*x**6*sqrt(a/(b*x**4) + 1)) - b**(3/2)*e*sqrt(a/(b*x**4) + 1)/(5*x**4) - b**(5/2)*c/(32*a*x**2*sq rt(a/(b*x**4) + 1)) - b**(5/2)*e*sqrt(a/(b*x**4) + 1)/(10*a) + b**3*c*asin h(sqrt(a)/(sqrt(b)*x**2))/(32*a**(3/2))
\[ \int \frac {\left (c+d x+e x^2+f x^3\right ) \left (a+b x^4\right )^{3/2}}{x^{13}} \, dx=\int { \frac {{\left (b x^{4} + a\right )}^{\frac {3}{2}} {\left (f x^{3} + e x^{2} + d x + c\right )}}{x^{13}} \,d x } \] Input:
integrate((f*x^3+e*x^2+d*x+c)*(b*x^4+a)^(3/2)/x^13,x, algorithm="maxima")
Output:
-1/192*(3*b^3*log((sqrt(b*x^4 + a) - sqrt(a))/(sqrt(b*x^4 + a) + sqrt(a))) /a^(3/2) + 2*(3*(b*x^4 + a)^(5/2)*b^3 + 8*(b*x^4 + a)^(3/2)*a*b^3 - 3*sqrt (b*x^4 + a)*a^2*b^3)/((b*x^4 + a)^3*a - 3*(b*x^4 + a)^2*a^2 + 3*(b*x^4 + a )*a^3 - a^4))*c + 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^12, x)
\[ \int \frac {\left (c+d x+e x^2+f x^3\right ) \left (a+b x^4\right )^{3/2}}{x^{13}} \, dx=\int { \frac {{\left (b x^{4} + a\right )}^{\frac {3}{2}} {\left (f x^{3} + e x^{2} + d x + c\right )}}{x^{13}} \,d x } \] Input:
integrate((f*x^3+e*x^2+d*x+c)*(b*x^4+a)^(3/2)/x^13,x, algorithm="giac")
Output:
integrate((b*x^4 + a)^(3/2)*(f*x^3 + e*x^2 + d*x + c)/x^13, x)
Timed out. \[ \int \frac {\left (c+d x+e x^2+f x^3\right ) \left (a+b x^4\right )^{3/2}}{x^{13}} \, dx=\int \frac {{\left (b\,x^4+a\right )}^{3/2}\,\left (f\,x^3+e\,x^2+d\,x+c\right )}{x^{13}} \,d x \] Input:
int(((a + b*x^4)^(3/2)*(c + d*x + e*x^2 + f*x^3))/x^13,x)
Output:
int(((a + b*x^4)^(3/2)*(c + d*x + e*x^2 + f*x^3))/x^13, x)
\[ \int \frac {\left (c+d x+e x^2+f x^3\right ) \left (a+b x^4\right )^{3/2}}{x^{13}} \, dx=\int \frac {\left (f \,x^{3}+e \,x^{2}+d x +c \right ) \left (b \,x^{4}+a \right )^{\frac {3}{2}}}{x^{13}}d x \] Input:
int((f*x^3+e*x^2+d*x+c)*(b*x^4+a)^(3/2)/x^13,x)
Output:
int((f*x^3+e*x^2+d*x+c)*(b*x^4+a)^(3/2)/x^13,x)