Integrand size = 19, antiderivative size = 319 \[ \int \left (d+e x^2\right ) \left (a+c x^4\right )^{5/2} \, dx=\frac {8 a^3 e x \sqrt {a+c x^4}}{39 \sqrt {c} \left (\sqrt {a}+\sqrt {c} x^2\right )}+\frac {4 a^2 x \left (195 d+77 e x^2\right ) \sqrt {a+c x^4}}{3003}+\frac {10 a x \left (117 d+77 e x^2\right ) \left (a+c x^4\right )^{3/2}}{9009}+\frac {1}{143} x \left (13 d+11 e x^2\right ) \left (a+c x^4\right )^{5/2}-\frac {8 a^{13/4} e \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{39 c^{3/4} \sqrt {a+c x^4}}+\frac {4 a^{11/4} \left (195 \sqrt {c} d+77 \sqrt {a} e\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{3003 c^{3/4} \sqrt {a+c x^4}} \] Output:
8/39*a^3*e*x*(c*x^4+a)^(1/2)/c^(1/2)/(a^(1/2)+c^(1/2)*x^2)+4/3003*a^2*x*(7 7*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^(13/4)*e*(a^(1/2)+c^(1/2) *x^2)*((c*x^4+a)/(a^(1/2)+c^(1/2)*x^2)^2)^(1/2)*EllipticE(sin(2*arctan(c^( 1/4)*x/a^(1/4))),1/2*2^(1/2))/c^(3/4)/(c*x^4+a)^(1/2)+4/3003*a^(11/4)*(195 *c^(1/2)*d+77*a^(1/2)*e)*(a^(1/2)+c^(1/2)*x^2)*((c*x^4+a)/(a^(1/2)+c^(1/2) *x^2)^2)^(1/2)*InverseJacobiAM(2*arctan(c^(1/4)*x/a^(1/4)),1/2*2^(1/2))/c^ (3/4)/(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 = 0.25 \[ \int \left (d+e x^2\right ) \left (a+c x^4\right )^{5/2} \, dx=\frac {a^2 \sqrt {a+c x^4} \left (3 d x \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},\frac {1}{4},\frac {5}{4},-\frac {c x^4}{a}\right )+e x^3 \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},\frac {3}{4},\frac {7}{4},-\frac {c x^4}{a}\right )\right )}{3 \sqrt {1+\frac {c x^4}{a}}} \] 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 = 0.76 (sec) , antiderivative size = 327, normalized size of antiderivative = 1.03, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.526, Rules used = {1491, 27, 1491, 27, 1491, 27, 1512, 27, 761, 1510}
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 |
| \(\displaystyle \frac {5}{143} \int 2 a \left (11 e x^2+13 d\right ) \left (c x^4+a\right )^{3/2}dx+\frac {1}{143} x \left (a+c x^4\right )^{5/2} \left (13 d+11 e x^2\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {10}{143} a \int \left (11 e x^2+13 d\right ) \left (c x^4+a\right )^{3/2}dx+\frac {1}{143} x \left (a+c x^4\right )^{5/2} \left (13 d+11 e x^2\right )\) |
| \(\Big \downarrow \) 1491 |
| \(\displaystyle \frac {10}{143} a \left (\frac {1}{21} \int 2 a \left (77 e x^2+117 d\right ) \sqrt {c x^4+a}dx+\frac {1}{63} x \left (a+c x^4\right )^{3/2} \left (117 d+77 e x^2\right )\right )+\frac {1}{143} x \left (a+c x^4\right )^{5/2} \left (13 d+11 e x^2\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {10}{143} a \left (\frac {2}{21} a \int \left (77 e x^2+117 d\right ) \sqrt {c x^4+a}dx+\frac {1}{63} x \left (a+c x^4\right )^{3/2} \left (117 d+77 e x^2\right )\right )+\frac {1}{143} x \left (a+c x^4\right )^{5/2} \left (13 d+11 e x^2\right )\) |
| \(\Big \downarrow \) 1491 |
| \(\displaystyle \frac {10}{143} a \left (\frac {2}{21} a \left (\frac {1}{15} \int \frac {6 a \left (77 e x^2+195 d\right )}{\sqrt {c x^4+a}}dx+\frac {1}{5} x \sqrt {a+c x^4} \left (195 d+77 e x^2\right )\right )+\frac {1}{63} x \left (a+c x^4\right )^{3/2} \left (117 d+77 e x^2\right )\right )+\frac {1}{143} x \left (a+c x^4\right )^{5/2} \left (13 d+11 e x^2\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {10}{143} a \left (\frac {2}{21} a \left (\frac {2}{5} a \int \frac {77 e x^2+195 d}{\sqrt {c x^4+a}}dx+\frac {1}{5} x \sqrt {a+c x^4} \left (195 d+77 e x^2\right )\right )+\frac {1}{63} x \left (a+c x^4\right )^{3/2} \left (117 d+77 e x^2\right )\right )+\frac {1}{143} x \left (a+c x^4\right )^{5/2} \left (13 d+11 e x^2\right )\) |
| \(\Big \downarrow \) 1512 |
| \(\displaystyle \frac {10}{143} a \left (\frac {2}{21} a \left (\frac {2}{5} a \left (\left (\frac {77 \sqrt {a} e}{\sqrt {c}}+195 d\right ) \int \frac {1}{\sqrt {c x^4+a}}dx-\frac {77 \sqrt {a} e \int \frac {\sqrt {a}-\sqrt {c} x^2}{\sqrt {a} \sqrt {c x^4+a}}dx}{\sqrt {c}}\right )+\frac {1}{5} x \sqrt {a+c x^4} \left (195 d+77 e x^2\right )\right )+\frac {1}{63} x \left (a+c x^4\right )^{3/2} \left (117 d+77 e x^2\right )\right )+\frac {1}{143} x \left (a+c x^4\right )^{5/2} \left (13 d+11 e x^2\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {10}{143} a \left (\frac {2}{21} a \left (\frac {2}{5} a \left (\left (\frac {77 \sqrt {a} e}{\sqrt {c}}+195 d\right ) \int \frac {1}{\sqrt {c x^4+a}}dx-\frac {77 e \int \frac {\sqrt {a}-\sqrt {c} x^2}{\sqrt {c x^4+a}}dx}{\sqrt {c}}\right )+\frac {1}{5} x \sqrt {a+c x^4} \left (195 d+77 e x^2\right )\right )+\frac {1}{63} x \left (a+c x^4\right )^{3/2} \left (117 d+77 e x^2\right )\right )+\frac {1}{143} x \left (a+c x^4\right )^{5/2} \left (13 d+11 e x^2\right )\) |
| \(\Big \downarrow \) 761 |
| \(\displaystyle \frac {10}{143} a \left (\frac {2}{21} a \left (\frac {2}{5} a \left (\frac {\left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \left (\frac {77 \sqrt {a} e}{\sqrt {c}}+195 d\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{a} \sqrt [4]{c} \sqrt {a+c x^4}}-\frac {77 e \int \frac {\sqrt {a}-\sqrt {c} x^2}{\sqrt {c x^4+a}}dx}{\sqrt {c}}\right )+\frac {1}{5} x \sqrt {a+c x^4} \left (195 d+77 e x^2\right )\right )+\frac {1}{63} x \left (a+c x^4\right )^{3/2} \left (117 d+77 e x^2\right )\right )+\frac {1}{143} x \left (a+c x^4\right )^{5/2} \left (13 d+11 e x^2\right )\) |
| \(\Big \downarrow \) 1510 |
| \(\displaystyle \frac {10}{143} a \left (\frac {2}{21} a \left (\frac {2}{5} a \left (\frac {\left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \left (\frac {77 \sqrt {a} e}{\sqrt {c}}+195 d\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{a} \sqrt [4]{c} \sqrt {a+c x^4}}-\frac {77 e \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{\sqrt [4]{c} \sqrt {a+c x^4}}-\frac {x \sqrt {a+c x^4}}{\sqrt {a}+\sqrt {c} x^2}\right )}{\sqrt {c}}\right )+\frac {1}{5} x \sqrt {a+c x^4} \left (195 d+77 e x^2\right )\right )+\frac {1}{63} x \left (a+c x^4\right )^{3/2} \left (117 d+77 e x^2\right )\right )+\frac {1}{143} x \left (a+c x^4\right )^{5/2} \left (13 d+11 e x^2\right )\) |
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*e*(-((x*Sqrt[a + c*x^4])/(Sqrt[a] + Sqrt[c]*x^2)) + (a^(1/4)*(S qrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*Elliptic E[2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/(c^(1/4)*Sqrt[a + c*x^4])))/Sqrt[c] + ((195*d + (77*Sqrt[a]*e)/Sqrt[c])*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x ^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticF[2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/ 2])/(2*a^(1/4)*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[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 4]}, Simp[( 1 + q^2*x^2)*(Sqrt[(a + b*x^4)/(a*(1 + q^2*x^2)^2)]/(2*q*Sqrt[a + b*x^4]))* EllipticF[2*ArcTan[q*x], 1/2], x]] /; FreeQ[{a, b}, x] && PosQ[b/a]
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, 4]}, Simp[(-d)*x*(Sqrt[a + c*x^4]/(a*(1 + q^2*x^2))), x] + Simp[d* (1 + q^2*x^2)*(Sqrt[(a + c*x^4)/(a*(1 + q^2*x^2)^2)]/(q*Sqrt[a + c*x^4]))*E llipticE[2*ArcTan[q*x], 1/2], x] /; EqQ[e + d*q^2, 0]] /; FreeQ[{a, c, d, e }, x] && PosQ[c/a]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c/a, 2]}, Simp[(e + d*q)/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] /; NeQ[e + d*q, 0]] /; FreeQ[{a, c , d, e}, x] && PosQ[c/a]
Result contains complex when optimal does not.
Time = 1.08 (sec) , antiderivative size = 238, normalized size of antiderivative = 0.75
| method | result | size |
| risch | \(\frac {x \left (693 c^{2} e \,x^{10}+819 c^{2} d \,x^{8}+2156 a c e \,x^{6}+2808 a c d \,x^{4}+2387 a^{2} e \,x^{2}+4329 a^{2} d \right ) \sqrt {c \,x^{4}+a}}{9009}+\frac {8 a^{3} \left (\frac {195 d \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}+\frac {77 i e \sqrt {a}\, \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )\right )}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}\, \sqrt {c}}\right )}{3003}\) | \(238\) |
| default | \(d \left (\frac {c^{2} x^{9} \sqrt {c \,x^{4}+a}}{11}+\frac {24 a c \,x^{5} \sqrt {c \,x^{4}+a}}{77}+\frac {37 a^{2} x \sqrt {c \,x^{4}+a}}{77}+\frac {40 a^{3} \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{77 \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}\right )+e \left (\frac {c^{2} x^{11} \sqrt {c \,x^{4}+a}}{13}+\frac {28 a c \,x^{7} \sqrt {c \,x^{4}+a}}{117}+\frac {31 a^{2} x^{3} \sqrt {c \,x^{4}+a}}{117}+\frac {8 i a^{\frac {7}{2}} \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )\right )}{39 \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}\, \sqrt {c}}\right )\) | \(275\) |
| elliptic | \(\frac {e \,c^{2} x^{11} \sqrt {c \,x^{4}+a}}{13}+\frac {c^{2} d \,x^{9} \sqrt {c \,x^{4}+a}}{11}+\frac {28 a c e \,x^{7} \sqrt {c \,x^{4}+a}}{117}+\frac {24 a c d \,x^{5} \sqrt {c \,x^{4}+a}}{77}+\frac {31 a^{2} e \,x^{3} \sqrt {c \,x^{4}+a}}{117}+\frac {37 a^{2} d x \sqrt {c \,x^{4}+a}}{77}+\frac {40 d \,a^{3} \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{77 \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}+\frac {8 i e \,a^{\frac {7}{2}} \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )\right )}{39 \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}\, \sqrt {c}}\) | \(277\) |
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/(I/a^(1/2)*c^(1/2) )^(1/2)*(1-I/a^(1/2)*c^(1/2)*x^2)^(1/2)*(1+I/a^(1/2)*c^(1/2)*x^2)^(1/2)/(c *x^4+a)^(1/2)*EllipticF(x*(I/a^(1/2)*c^(1/2))^(1/2),I)+77*I*e*a^(1/2)/(I/a ^(1/2)*c^(1/2))^(1/2)*(1-I/a^(1/2)*c^(1/2)*x^2)^(1/2)*(1+I/a^(1/2)*c^(1/2) *x^2)^(1/2)/(c*x^4+a)^(1/2)/c^(1/2)*(EllipticF(x*(I/a^(1/2)*c^(1/2))^(1/2) ,I)-EllipticE(x*(I/a^(1/2)*c^(1/2))^(1/2),I)))
Time = 0.07 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.50 \[ \int \left (d+e x^2\right ) \left (a+c x^4\right )^{5/2} \, dx=\frac {1848 \, a^{3} \sqrt {c} e x \left (-\frac {a}{c}\right )^{\frac {3}{4}} E(\arcsin \left (\frac {\left (-\frac {a}{c}\right )^{\frac {1}{4}}}{x}\right )\,|\,-1) + 24 \, {\left (195 \, a^{2} c d - 77 \, a^{3} e\right )} \sqrt {c} x \left (-\frac {a}{c}\right )^{\frac {3}{4}} F(\arcsin \left (\frac {\left (-\frac {a}{c}\right )^{\frac {1}{4}}}{x}\right )\,|\,-1) + {\left (693 \, c^{3} e x^{12} + 819 \, c^{3} d x^{10} + 2156 \, a c^{2} e x^{8} + 2808 \, a c^{2} d x^{6} + 2387 \, a^{2} c e x^{4} + 4329 \, a^{2} c d x^{2} + 1848 \, a^{3} e\right )} \sqrt {c x^{4} + a}}{9009 \, c x} \] 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)
Result contains complex when optimal does not.
Time = 2.46 (sec) , antiderivative size = 262, normalized size of antiderivative = 0.82 \[ \int \left (d+e x^2\right ) \left (a+c x^4\right )^{5/2} \, dx=\frac {a^{\frac {5}{2}} d x \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {1}{4} \\ \frac {5}{4} \end {matrix}\middle | {\frac {c x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac {5}{4}\right )} + \frac {a^{\frac {5}{2}} e x^{3} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {c x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac {7}{4}\right )} + \frac {a^{\frac {3}{2}} c d x^{5} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {5}{4} \\ \frac {9}{4} \end {matrix}\middle | {\frac {c x^{4} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (\frac {9}{4}\right )} + \frac {a^{\frac {3}{2}} c e x^{7} \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {7}{4} \\ \frac {11}{4} \end {matrix}\middle | {\frac {c x^{4} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (\frac {11}{4}\right )} + \frac {\sqrt {a} c^{2} d x^{9} \Gamma \left (\frac {9}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {9}{4} \\ \frac {13}{4} \end {matrix}\middle | {\frac {c x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac {13}{4}\right )} + \frac {\sqrt {a} c^{2} e x^{11} \Gamma \left (\frac {11}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {11}{4} \\ \frac {15}{4} \end {matrix}\middle | {\frac {c x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac {15}{4}\right )} \] 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(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(I*pi)/a)/(4*gamma(7/4)) + a**(3/2)*c*d*x**5*gamma(5/4)*hy per((-1/2, 5/4), (9/4,), c*x**4*exp_polar(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(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(I*pi)/a)/(4*gamma(13/4)) + sqrt(a)*c**2*e*x**11*gam ma(11/4)*hyper((-1/2, 11/4), (15/4,), c*x**4*exp_polar(I*pi)/a)/(4*gamma(1 5/4))
\[ \int \left (d+e x^2\right ) \left (a+c x^4\right )^{5/2} \, dx=\int { {\left (c x^{4} + a\right )}^{\frac {5}{2}} {\left (e x^{2} + d\right )} \,d x } \] 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. \[ \int \left (d+e x^2\right ) \left (a+c x^4\right )^{5/2} \, dx=\int {\left (c\,x^4+a\right )}^{5/2}\,\left (e\,x^2+d\right ) \,d x \] 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