Integrand size = 20, antiderivative size = 216 \[ \int \left (d+e x^2\right ) \left (a-c x^4\right )^{5/2} \, dx=\frac {4 a^2 x \left (195 d+77 e x^2\right ) \sqrt {a-c x^4}}{3003}+\frac {10 a x \left (117 d+77 e x^2\right ) \left (a-c x^4\right )^{3/2}}{9009}+\frac {1}{143} x \left (13 d+11 e x^2\right ) \left (a-c x^4\right )^{5/2}+\frac {8 a^{15/4} e \sqrt {1-\frac {c x^4}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{39 c^{3/4} \sqrt {a-c x^4}}+\frac {8 a^{13/4} \left (195 d-\frac {77 \sqrt {a} e}{\sqrt {c}}\right ) \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{3003 \sqrt [4]{c} \sqrt {a-c x^4}} \] Output:
4/3003*a^2*x*(77*e*x^2+195*d)*(-c*x^4+a)^(1/2)+10/9009*a*x*(77*e*x^2+117*d )*(-c*x^4+a)^(3/2)+1/143*x*(11*e*x^2+13*d)*(-c*x^4+a)^(5/2)+8/39*a^(15/4)* e*(1-c*x^4/a)^(1/2)*EllipticE(c^(1/4)*x/a^(1/4),I)/c^(3/4)/(-c*x^4+a)^(1/2 )+8/3003*a^(13/4)*(195*d-77*a^(1/2)*e/c^(1/2))*(1-c*x^4/a)^(1/2)*EllipticF (c^(1/4)*x/a^(1/4),I)/c^(1/4)/(-c*x^4+a)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.05 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.37 \[ \int \left (d+e x^2\right ) \left (a-c x^4\right )^{5/2} \, dx=\frac {a^2 \sqrt {a-c x^4} \left (3 d x \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},\frac {1}{4},\frac {5}{4},\frac {c x^4}{a}\right )+e x^3 \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},\frac {3}{4},\frac {7}{4},\frac {c x^4}{a}\right )\right )}{3 \sqrt {1-\frac {c x^4}{a}}} \] Input:
Integrate[(d + e*x^2)*(a - c*x^4)^(5/2),x]
Output:
(a^2*Sqrt[a - c*x^4]*(3*d*x*Hypergeometric2F1[-5/2, 1/4, 5/4, (c*x^4)/a] + e*x^3*Hypergeometric2F1[-5/2, 3/4, 7/4, (c*x^4)/a]))/(3*Sqrt[1 - (c*x^4)/ a])
Time = 0.80 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.04, number of steps used = 13, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.650, Rules used = {1491, 27, 1491, 27, 1491, 27, 1513, 27, 765, 762, 1390, 1389, 327}
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.
| \(\displaystyle \int \left (a-c x^4\right )^{5/2} \left (d+e x^2\right ) \, dx\) |
| \(\Big \downarrow \) 1491 |
| \(\displaystyle \frac {5}{143} \int 2 a \left (11 e x^2+13 d\right ) \left (a-c x^4\right )^{3/2}dx+\frac {1}{143} x \left (a-c x^4\right )^{5/2} \left (13 d+11 e x^2\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {10}{143} a \int \left (11 e x^2+13 d\right ) \left (a-c x^4\right )^{3/2}dx+\frac {1}{143} x \left (a-c x^4\right )^{5/2} \left (13 d+11 e x^2\right )\) |
| \(\Big \downarrow \) 1491 |
| \(\displaystyle \frac {10}{143} a \left (\frac {1}{21} \int 2 a \left (77 e x^2+117 d\right ) \sqrt {a-c x^4}dx+\frac {1}{63} x \left (a-c x^4\right )^{3/2} \left (117 d+77 e x^2\right )\right )+\frac {1}{143} x \left (a-c x^4\right )^{5/2} \left (13 d+11 e x^2\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {10}{143} a \left (\frac {2}{21} a \int \left (77 e x^2+117 d\right ) \sqrt {a-c x^4}dx+\frac {1}{63} x \left (a-c x^4\right )^{3/2} \left (117 d+77 e x^2\right )\right )+\frac {1}{143} x \left (a-c x^4\right )^{5/2} \left (13 d+11 e x^2\right )\) |
| \(\Big \downarrow \) 1491 |
| \(\displaystyle \frac {10}{143} a \left (\frac {2}{21} a \left (\frac {1}{15} \int \frac {6 a \left (77 e x^2+195 d\right )}{\sqrt {a-c x^4}}dx+\frac {1}{5} x \sqrt {a-c x^4} \left (195 d+77 e x^2\right )\right )+\frac {1}{63} x \left (a-c x^4\right )^{3/2} \left (117 d+77 e x^2\right )\right )+\frac {1}{143} x \left (a-c x^4\right )^{5/2} \left (13 d+11 e x^2\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {10}{143} a \left (\frac {2}{21} a \left (\frac {2}{5} a \int \frac {77 e x^2+195 d}{\sqrt {a-c x^4}}dx+\frac {1}{5} x \sqrt {a-c x^4} \left (195 d+77 e x^2\right )\right )+\frac {1}{63} x \left (a-c x^4\right )^{3/2} \left (117 d+77 e x^2\right )\right )+\frac {1}{143} x \left (a-c x^4\right )^{5/2} \left (13 d+11 e x^2\right )\) |
| \(\Big \downarrow \) 1513 |
| \(\displaystyle \frac {10}{143} a \left (\frac {2}{21} a \left (\frac {2}{5} a \left (\left (195 d-\frac {77 \sqrt {a} e}{\sqrt {c}}\right ) \int \frac {1}{\sqrt {a-c x^4}}dx+\frac {77 \sqrt {a} e \int \frac {\sqrt {c} x^2+\sqrt {a}}{\sqrt {a} \sqrt {a-c x^4}}dx}{\sqrt {c}}\right )+\frac {1}{5} x \sqrt {a-c x^4} \left (195 d+77 e x^2\right )\right )+\frac {1}{63} x \left (a-c x^4\right )^{3/2} \left (117 d+77 e x^2\right )\right )+\frac {1}{143} x \left (a-c x^4\right )^{5/2} \left (13 d+11 e x^2\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {10}{143} a \left (\frac {2}{21} a \left (\frac {2}{5} a \left (\left (195 d-\frac {77 \sqrt {a} e}{\sqrt {c}}\right ) \int \frac {1}{\sqrt {a-c x^4}}dx+\frac {77 e \int \frac {\sqrt {c} x^2+\sqrt {a}}{\sqrt {a-c x^4}}dx}{\sqrt {c}}\right )+\frac {1}{5} x \sqrt {a-c x^4} \left (195 d+77 e x^2\right )\right )+\frac {1}{63} x \left (a-c x^4\right )^{3/2} \left (117 d+77 e x^2\right )\right )+\frac {1}{143} x \left (a-c x^4\right )^{5/2} \left (13 d+11 e x^2\right )\) |
| \(\Big \downarrow \) 765 |
| \(\displaystyle \frac {10}{143} a \left (\frac {2}{21} a \left (\frac {2}{5} a \left (\frac {\sqrt {1-\frac {c x^4}{a}} \left (195 d-\frac {77 \sqrt {a} e}{\sqrt {c}}\right ) \int \frac {1}{\sqrt {1-\frac {c x^4}{a}}}dx}{\sqrt {a-c x^4}}+\frac {77 e \int \frac {\sqrt {c} x^2+\sqrt {a}}{\sqrt {a-c x^4}}dx}{\sqrt {c}}\right )+\frac {1}{5} x \sqrt {a-c x^4} \left (195 d+77 e x^2\right )\right )+\frac {1}{63} x \left (a-c x^4\right )^{3/2} \left (117 d+77 e x^2\right )\right )+\frac {1}{143} x \left (a-c x^4\right )^{5/2} \left (13 d+11 e x^2\right )\) |
| \(\Big \downarrow \) 762 |
| \(\displaystyle \frac {10}{143} a \left (\frac {2}{21} a \left (\frac {2}{5} a \left (\frac {77 e \int \frac {\sqrt {c} x^2+\sqrt {a}}{\sqrt {a-c x^4}}dx}{\sqrt {c}}+\frac {\sqrt [4]{a} \sqrt {1-\frac {c x^4}{a}} \left (195 d-\frac {77 \sqrt {a} e}{\sqrt {c}}\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt [4]{c} \sqrt {a-c x^4}}\right )+\frac {1}{5} x \sqrt {a-c x^4} \left (195 d+77 e x^2\right )\right )+\frac {1}{63} x \left (a-c x^4\right )^{3/2} \left (117 d+77 e x^2\right )\right )+\frac {1}{143} x \left (a-c x^4\right )^{5/2} \left (13 d+11 e x^2\right )\) |
| \(\Big \downarrow \) 1390 |
| \(\displaystyle \frac {10}{143} a \left (\frac {2}{21} a \left (\frac {2}{5} a \left (\frac {77 e \sqrt {1-\frac {c x^4}{a}} \int \frac {\sqrt {c} x^2+\sqrt {a}}{\sqrt {1-\frac {c x^4}{a}}}dx}{\sqrt {c} \sqrt {a-c x^4}}+\frac {\sqrt [4]{a} \sqrt {1-\frac {c x^4}{a}} \left (195 d-\frac {77 \sqrt {a} e}{\sqrt {c}}\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt [4]{c} \sqrt {a-c x^4}}\right )+\frac {1}{5} x \sqrt {a-c x^4} \left (195 d+77 e x^2\right )\right )+\frac {1}{63} x \left (a-c x^4\right )^{3/2} \left (117 d+77 e x^2\right )\right )+\frac {1}{143} x \left (a-c x^4\right )^{5/2} \left (13 d+11 e x^2\right )\) |
| \(\Big \downarrow \) 1389 |
| \(\displaystyle \frac {10}{143} a \left (\frac {2}{21} a \left (\frac {2}{5} a \left (\frac {77 \sqrt {a} e \sqrt {1-\frac {c x^4}{a}} \int \frac {\sqrt {\frac {\sqrt {c} x^2}{\sqrt {a}}+1}}{\sqrt {1-\frac {\sqrt {c} x^2}{\sqrt {a}}}}dx}{\sqrt {c} \sqrt {a-c x^4}}+\frac {\sqrt [4]{a} \sqrt {1-\frac {c x^4}{a}} \left (195 d-\frac {77 \sqrt {a} e}{\sqrt {c}}\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt [4]{c} \sqrt {a-c x^4}}\right )+\frac {1}{5} x \sqrt {a-c x^4} \left (195 d+77 e x^2\right )\right )+\frac {1}{63} x \left (a-c x^4\right )^{3/2} \left (117 d+77 e x^2\right )\right )+\frac {1}{143} x \left (a-c x^4\right )^{5/2} \left (13 d+11 e x^2\right )\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {10}{143} a \left (\frac {2}{21} a \left (\frac {2}{5} a \left (\frac {77 a^{3/4} e \sqrt {1-\frac {c x^4}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{c^{3/4} \sqrt {a-c x^4}}+\frac {\sqrt [4]{a} \sqrt {1-\frac {c x^4}{a}} \left (195 d-\frac {77 \sqrt {a} e}{\sqrt {c}}\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt [4]{c} \sqrt {a-c x^4}}\right )+\frac {1}{5} x \sqrt {a-c x^4} \left (195 d+77 e x^2\right )\right )+\frac {1}{63} x \left (a-c x^4\right )^{3/2} \left (117 d+77 e x^2\right )\right )+\frac {1}{143} x \left (a-c x^4\right )^{5/2} \left (13 d+11 e x^2\right )\) |
Input:
Int[(d + e*x^2)*(a - c*x^4)^(5/2),x]
Output:
(x*(13*d + 11*e*x^2)*(a - c*x^4)^(5/2))/143 + (10*a*((x*(117*d + 77*e*x^2) *(a - c*x^4)^(3/2))/63 + (2*a*((x*(195*d + 77*e*x^2)*Sqrt[a - c*x^4])/5 + (2*a*((77*a^(3/4)*e*Sqrt[1 - (c*x^4)/a]*EllipticE[ArcSin[(c^(1/4)*x)/a^(1/ 4)], -1])/(c^(3/4)*Sqrt[a - c*x^4]) + (a^(1/4)*(195*d - (77*Sqrt[a]*e)/Sqr t[c])*Sqrt[1 - (c*x^4)/a]*EllipticF[ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/(c^( 1/4)*Sqrt[a - c*x^4])))/5))/21))/143
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ (Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) )], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[(1/(Sqrt[a]*Rt[-b/a, 4]) )*EllipticF[ArcSin[Rt[-b/a, 4]*x], -1], x] /; FreeQ[{a, b}, x] && NegQ[b/a] && GtQ[a, 0]
Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[Sqrt[1 + b*(x^4/a)]/Sqrt [a + b*x^4] Int[1/Sqrt[1 + b*(x^4/a)], x], x] /; FreeQ[{a, b}, x] && NegQ [b/a] && !GtQ[a, 0]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> Simp[d/Sq rt[a] Int[Sqrt[1 + e*(x^2/d)]/Sqrt[1 - e*(x^2/d)], x], x] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 + a*e^2, 0] && NegQ[c/a] && GtQ[a, 0]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> Simp[Sqrt [1 + c*(x^4/a)]/Sqrt[a + c*x^4] Int[(d + e*x^2)/Sqrt[1 + c*(x^4/a)], x], x] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 + a*e^2, 0] && NegQ[c/a] && !GtQ [a, 0] && !(LtQ[a, 0] && GtQ[c, 0])
Int[((d_) + (e_.)*(x_)^2)*((a_) + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[x*( d*(4*p + 3) + e*(4*p + 1)*x^2)*((a + c*x^4)^p/((4*p + 1)*(4*p + 3))), x] + Simp[2*(p/((4*p + 1)*(4*p + 3))) Int[Simp[2*a*d*(4*p + 3) + (2*a*e*(4*p + 1))*x^2, x]*(a + c*x^4)^(p - 1), x], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c *d^2 + a*e^2, 0] && GtQ[p, 0] && FractionQ[p] && IntegerQ[2*p]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[-c/a, 2]}, Simp[(d*q - e)/q Int[1/Sqrt[a + c*x^4], x], x] + Simp[e/q Int[(1 + q*x^2)/Sqrt[a + c*x^4], x], x]] /; FreeQ[{a, c, d, e}, x] && Neg Q[c/a] && NeQ[c*d^2 + a*e^2, 0]
Time = 1.11 (sec) , antiderivative size = 224, normalized size of antiderivative = 1.04
| method | result | size |
| risch | \(\frac {x \left (693 c^{2} e \,x^{10}+819 c^{2} d \,x^{8}-2156 a c e \,x^{6}-2808 a c d \,x^{4}+2387 a^{2} e \,x^{2}+4329 a^{2} d \right ) \sqrt {-c \,x^{4}+a}}{9009}+\frac {8 a^{3} \left (\frac {195 d \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}-\frac {77 e \sqrt {a}\, \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )\right )}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}\, \sqrt {c}}\right )}{3003}\) | \(224\) |
| default | \(d \left (\frac {c^{2} x^{9} \sqrt {-c \,x^{4}+a}}{11}-\frac {24 a c \,x^{5} \sqrt {-c \,x^{4}+a}}{77}+\frac {37 a^{2} x \sqrt {-c \,x^{4}+a}}{77}+\frac {40 a^{3} \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{77 \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}\right )+e \left (\frac {c^{2} x^{11} \sqrt {-c \,x^{4}+a}}{13}-\frac {28 a c \,x^{7} \sqrt {-c \,x^{4}+a}}{117}+\frac {31 a^{2} x^{3} \sqrt {-c \,x^{4}+a}}{117}-\frac {8 a^{\frac {7}{2}} \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )\right )}{39 \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}\, \sqrt {c}}\right )\) | \(266\) |
| elliptic | \(\frac {e \,c^{2} x^{11} \sqrt {-c \,x^{4}+a}}{13}+\frac {c^{2} d \,x^{9} \sqrt {-c \,x^{4}+a}}{11}-\frac {28 a c e \,x^{7} \sqrt {-c \,x^{4}+a}}{117}-\frac {24 a c d \,x^{5} \sqrt {-c \,x^{4}+a}}{77}+\frac {31 a^{2} e \,x^{3} \sqrt {-c \,x^{4}+a}}{117}+\frac {37 a^{2} d x \sqrt {-c \,x^{4}+a}}{77}+\frac {40 d \,a^{3} \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{77 \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}-\frac {8 e \,a^{\frac {7}{2}} \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )\right )}{39 \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}\, \sqrt {c}}\) | \(268\) |
Input:
int((e*x^2+d)*(-c*x^4+a)^(5/2),x,method=_RETURNVERBOSE)
Output:
1/9009*x*(693*c^2*e*x^10+819*c^2*d*x^8-2156*a*c*e*x^6-2808*a*c*d*x^4+2387* a^2*e*x^2+4329*a^2*d)*(-c*x^4+a)^(1/2)+8/3003*a^3*(195*d/(1/a^(1/2)*c^(1/2 ))^(1/2)*(1-1/a^(1/2)*c^(1/2)*x^2)^(1/2)*(1+1/a^(1/2)*c^(1/2)*x^2)^(1/2)/( -c*x^4+a)^(1/2)*EllipticF(x*(1/a^(1/2)*c^(1/2))^(1/2),I)-77*e*a^(1/2)/(1/a ^(1/2)*c^(1/2))^(1/2)*(1-1/a^(1/2)*c^(1/2)*x^2)^(1/2)*(1+1/a^(1/2)*c^(1/2) *x^2)^(1/2)/(-c*x^4+a)^(1/2)/c^(1/2)*(EllipticF(x*(1/a^(1/2)*c^(1/2))^(1/2 ),I)-EllipticE(x*(1/a^(1/2)*c^(1/2))^(1/2),I)))
Time = 0.08 (sec) , antiderivative size = 162, normalized size of antiderivative = 0.75 \[ \int \left (d+e x^2\right ) \left (a-c x^4\right )^{5/2} \, dx=-\frac {1848 \, a^{3} \sqrt {-c} e x \left (\frac {a}{c}\right )^{\frac {3}{4}} E(\arcsin \left (\frac {\left (\frac {a}{c}\right )^{\frac {1}{4}}}{x}\right )\,|\,-1) - 24 \, {\left (195 \, a^{2} c d + 77 \, a^{3} e\right )} \sqrt {-c} x \left (\frac {a}{c}\right )^{\frac {3}{4}} F(\arcsin \left (\frac {\left (\frac {a}{c}\right )^{\frac {1}{4}}}{x}\right )\,|\,-1) - {\left (693 \, c^{3} e x^{12} + 819 \, c^{3} d x^{10} - 2156 \, a c^{2} e x^{8} - 2808 \, a c^{2} d x^{6} + 2387 \, a^{2} c e x^{4} + 4329 \, a^{2} c d x^{2} - 1848 \, a^{3} e\right )} \sqrt {-c x^{4} + a}}{9009 \, c x} \] Input:
integrate((e*x^2+d)*(-c*x^4+a)^(5/2),x, algorithm="fricas")
Output:
-1/9009*(1848*a^3*sqrt(-c)*e*x*(a/c)^(3/4)*elliptic_e(arcsin((a/c)^(1/4)/x ), -1) - 24*(195*a^2*c*d + 77*a^3*e)*sqrt(-c)*x*(a/c)^(3/4)*elliptic_f(arc sin((a/c)^(1/4)/x), -1) - (693*c^3*e*x^12 + 819*c^3*d*x^10 - 2156*a*c^2*e* x^8 - 2808*a*c^2*d*x^6 + 2387*a^2*c*e*x^4 + 4329*a^2*c*d*x^2 - 1848*a^3*e) *sqrt(-c*x^4 + a))/(c*x)
Time = 2.55 (sec) , antiderivative size = 272, normalized size of antiderivative = 1.26 \[ \int \left (d+e x^2\right ) \left (a-c x^4\right )^{5/2} \, dx=\frac {a^{\frac {5}{2}} d x \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {1}{4} \\ \frac {5}{4} \end {matrix}\middle | {\frac {c x^{4} e^{2 i \pi }}{a}} \right )}}{4 \Gamma \left (\frac {5}{4}\right )} + \frac {a^{\frac {5}{2}} e x^{3} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {c x^{4} e^{2 i \pi }}{a}} \right )}}{4 \Gamma \left (\frac {7}{4}\right )} - \frac {a^{\frac {3}{2}} c d x^{5} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {5}{4} \\ \frac {9}{4} \end {matrix}\middle | {\frac {c x^{4} e^{2 i \pi }}{a}} \right )}}{2 \Gamma \left (\frac {9}{4}\right )} - \frac {a^{\frac {3}{2}} c e x^{7} \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {7}{4} \\ \frac {11}{4} \end {matrix}\middle | {\frac {c x^{4} e^{2 i \pi }}{a}} \right )}}{2 \Gamma \left (\frac {11}{4}\right )} + \frac {\sqrt {a} c^{2} d x^{9} \Gamma \left (\frac {9}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {9}{4} \\ \frac {13}{4} \end {matrix}\middle | {\frac {c x^{4} e^{2 i \pi }}{a}} \right )}}{4 \Gamma \left (\frac {13}{4}\right )} + \frac {\sqrt {a} c^{2} e x^{11} \Gamma \left (\frac {11}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {11}{4} \\ \frac {15}{4} \end {matrix}\middle | {\frac {c x^{4} e^{2 i \pi }}{a}} \right )}}{4 \Gamma \left (\frac {15}{4}\right )} \] Input:
integrate((e*x**2+d)*(-c*x**4+a)**(5/2),x)
Output:
a**(5/2)*d*x*gamma(1/4)*hyper((-1/2, 1/4), (5/4,), c*x**4*exp_polar(2*I*pi )/a)/(4*gamma(5/4)) + a**(5/2)*e*x**3*gamma(3/4)*hyper((-1/2, 3/4), (7/4,) , c*x**4*exp_polar(2*I*pi)/a)/(4*gamma(7/4)) - a**(3/2)*c*d*x**5*gamma(5/4 )*hyper((-1/2, 5/4), (9/4,), c*x**4*exp_polar(2*I*pi)/a)/(2*gamma(9/4)) - a**(3/2)*c*e*x**7*gamma(7/4)*hyper((-1/2, 7/4), (11/4,), c*x**4*exp_polar( 2*I*pi)/a)/(2*gamma(11/4)) + sqrt(a)*c**2*d*x**9*gamma(9/4)*hyper((-1/2, 9 /4), (13/4,), c*x**4*exp_polar(2*I*pi)/a)/(4*gamma(13/4)) + sqrt(a)*c**2*e *x**11*gamma(11/4)*hyper((-1/2, 11/4), (15/4,), c*x**4*exp_polar(2*I*pi)/a )/(4*gamma(15/4))
\[ \int \left (d+e x^2\right ) \left (a-c x^4\right )^{5/2} \, dx=\int { {\left (-c x^{4} + a\right )}^{\frac {5}{2}} {\left (e x^{2} + d\right )} \,d x } \] Input:
integrate((e*x^2+d)*(-c*x^4+a)^(5/2),x, algorithm="maxima")
Output:
integrate((-c*x^4 + a)^(5/2)*(e*x^2 + d), x)
\[ \int \left (d+e x^2\right ) \left (a-c x^4\right )^{5/2} \, dx=\int { {\left (-c x^{4} + a\right )}^{\frac {5}{2}} {\left (e x^{2} + d\right )} \,d x } \] Input:
integrate((e*x^2+d)*(-c*x^4+a)^(5/2),x, algorithm="giac")
Output:
integrate((-c*x^4 + a)^(5/2)*(e*x^2 + d), x)
Timed out. \[ \int \left (d+e x^2\right ) \left (a-c x^4\right )^{5/2} \, dx=\int {\left (a-c\,x^4\right )}^{5/2}\,\left (e\,x^2+d\right ) \,d x \] Input:
int((a - c*x^4)^(5/2)*(d + e*x^2),x)
Output:
int((a - c*x^4)^(5/2)*(d + e*x^2), x)
\[ \int \left (d+e x^2\right ) \left (a-c x^4\right )^{5/2} \, dx=\frac {37 \sqrt {-c \,x^{4}+a}\, a^{2} d x}{77}+\frac {31 \sqrt {-c \,x^{4}+a}\, a^{2} e \,x^{3}}{117}-\frac {24 \sqrt {-c \,x^{4}+a}\, a c d \,x^{5}}{77}-\frac {28 \sqrt {-c \,x^{4}+a}\, a c e \,x^{7}}{117}+\frac {\sqrt {-c \,x^{4}+a}\, c^{2} d \,x^{9}}{11}+\frac {\sqrt {-c \,x^{4}+a}\, c^{2} e \,x^{11}}{13}+\frac {40 \left (\int \frac {\sqrt {-c \,x^{4}+a}}{-c \,x^{4}+a}d x \right ) a^{3} d}{77}+\frac {8 \left (\int \frac {\sqrt {-c \,x^{4}+a}\, x^{2}}{-c \,x^{4}+a}d x \right ) a^{3} e}{39} \] Input:
int((e*x^2+d)*(-c*x^4+a)^(5/2),x)
Output:
(4329*sqrt(a - c*x**4)*a**2*d*x + 2387*sqrt(a - c*x**4)*a**2*e*x**3 - 2808 *sqrt(a - c*x**4)*a*c*d*x**5 - 2156*sqrt(a - c*x**4)*a*c*e*x**7 + 819*sqrt (a - c*x**4)*c**2*d*x**9 + 693*sqrt(a - c*x**4)*c**2*e*x**11 + 4680*int(sq rt(a - c*x**4)/(a - c*x**4),x)*a**3*d + 1848*int((sqrt(a - c*x**4)*x**2)/( a - c*x**4),x)*a**3*e)/9009