Integrand size = 20, antiderivative size = 188 \[ \int \frac {d+e x^2}{\left (a-c x^4\right )^{5/2}} \, dx=\frac {x \left (d+e x^2\right )}{6 a \left (a-c x^4\right )^{3/2}}+\frac {x \left (5 d+3 e x^2\right )}{12 a^2 \sqrt {a-c x^4}}-\frac {e \sqrt {1-\frac {c x^4}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{4 a^{5/4} c^{3/4} \sqrt {a-c x^4}}+\frac {\left (5 \sqrt {c} d+3 \sqrt {a} e\right ) \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{12 a^{7/4} c^{3/4} \sqrt {a-c x^4}} \] Output:
1/6*x*(e*x^2+d)/a/(-c*x^4+a)^(3/2)+1/12*x*(3*e*x^2+5*d)/a^2/(-c*x^4+a)^(1/ 2)-1/4*e*(1-c*x^4/a)^(1/2)*EllipticE(c^(1/4)*x/a^(1/4),I)/a^(5/4)/c^(3/4)/ (-c*x^4+a)^(1/2)+1/12*(5*c^(1/2)*d+3*a^(1/2)*e)*(1-c*x^4/a)^(1/2)*Elliptic F(c^(1/4)*x/a^(1/4),I)/a^(7/4)/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 = 10.13 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.66 \[ \int \frac {d+e x^2}{\left (a-c x^4\right )^{5/2}} \, dx=\frac {d x \left (7 a-5 c x^4\right )+5 d x \left (a-c x^4\right ) \sqrt {1-\frac {c x^4}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},\frac {c x^4}{a}\right )+4 e x^3 \left (a-c x^4\right ) \sqrt {1-\frac {c x^4}{a}} \operatorname {Hypergeometric2F1}\left (\frac {3}{4},\frac {5}{2},\frac {7}{4},\frac {c x^4}{a}\right )}{12 a^2 \left (a-c x^4\right )^{3/2}} \] Input:
Integrate[(d + e*x^2)/(a - c*x^4)^(5/2),x]
Output:
(d*x*(7*a - 5*c*x^4) + 5*d*x*(a - c*x^4)*Sqrt[1 - (c*x^4)/a]*Hypergeometri c2F1[1/4, 1/2, 5/4, (c*x^4)/a] + 4*e*x^3*(a - c*x^4)*Sqrt[1 - (c*x^4)/a]*H ypergeometric2F1[3/4, 5/2, 7/4, (c*x^4)/a])/(12*a^2*(a - c*x^4)^(3/2))
Time = 0.69 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.06, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.550, Rules used = {1493, 25, 1493, 25, 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.
| \(\displaystyle \int \frac {d+e x^2}{\left (a-c x^4\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 1493 |
| \(\displaystyle \frac {x \left (d+e x^2\right )}{6 a \left (a-c x^4\right )^{3/2}}-\frac {\int -\frac {3 e x^2+5 d}{\left (a-c x^4\right )^{3/2}}dx}{6 a}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {3 e x^2+5 d}{\left (a-c x^4\right )^{3/2}}dx}{6 a}+\frac {x \left (d+e x^2\right )}{6 a \left (a-c x^4\right )^{3/2}}\) |
| \(\Big \downarrow \) 1493 |
| \(\displaystyle \frac {\frac {x \left (5 d+3 e x^2\right )}{2 a \sqrt {a-c x^4}}-\frac {\int -\frac {5 d-3 e x^2}{\sqrt {a-c x^4}}dx}{2 a}}{6 a}+\frac {x \left (d+e x^2\right )}{6 a \left (a-c x^4\right )^{3/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int \frac {5 d-3 e x^2}{\sqrt {a-c x^4}}dx}{2 a}+\frac {x \left (5 d+3 e x^2\right )}{2 a \sqrt {a-c x^4}}}{6 a}+\frac {x \left (d+e x^2\right )}{6 a \left (a-c x^4\right )^{3/2}}\) |
| \(\Big \downarrow \) 1513 |
| \(\displaystyle \frac {\frac {\left (\frac {3 \sqrt {a} e}{\sqrt {c}}+5 d\right ) \int \frac {1}{\sqrt {a-c x^4}}dx-\frac {3 \sqrt {a} e \int \frac {\sqrt {c} x^2+\sqrt {a}}{\sqrt {a} \sqrt {a-c x^4}}dx}{\sqrt {c}}}{2 a}+\frac {x \left (5 d+3 e x^2\right )}{2 a \sqrt {a-c x^4}}}{6 a}+\frac {x \left (d+e x^2\right )}{6 a \left (a-c x^4\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\left (\frac {3 \sqrt {a} e}{\sqrt {c}}+5 d\right ) \int \frac {1}{\sqrt {a-c x^4}}dx-\frac {3 e \int \frac {\sqrt {c} x^2+\sqrt {a}}{\sqrt {a-c x^4}}dx}{\sqrt {c}}}{2 a}+\frac {x \left (5 d+3 e x^2\right )}{2 a \sqrt {a-c x^4}}}{6 a}+\frac {x \left (d+e x^2\right )}{6 a \left (a-c x^4\right )^{3/2}}\) |
| \(\Big \downarrow \) 765 |
| \(\displaystyle \frac {\frac {\frac {\sqrt {1-\frac {c x^4}{a}} \left (\frac {3 \sqrt {a} e}{\sqrt {c}}+5 d\right ) \int \frac {1}{\sqrt {1-\frac {c x^4}{a}}}dx}{\sqrt {a-c x^4}}-\frac {3 e \int \frac {\sqrt {c} x^2+\sqrt {a}}{\sqrt {a-c x^4}}dx}{\sqrt {c}}}{2 a}+\frac {x \left (5 d+3 e x^2\right )}{2 a \sqrt {a-c x^4}}}{6 a}+\frac {x \left (d+e x^2\right )}{6 a \left (a-c x^4\right )^{3/2}}\) |
| \(\Big \downarrow \) 762 |
| \(\displaystyle \frac {\frac {\frac {\sqrt [4]{a} \sqrt {1-\frac {c x^4}{a}} \left (\frac {3 \sqrt {a} e}{\sqrt {c}}+5 d\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt [4]{c} \sqrt {a-c x^4}}-\frac {3 e \int \frac {\sqrt {c} x^2+\sqrt {a}}{\sqrt {a-c x^4}}dx}{\sqrt {c}}}{2 a}+\frac {x \left (5 d+3 e x^2\right )}{2 a \sqrt {a-c x^4}}}{6 a}+\frac {x \left (d+e x^2\right )}{6 a \left (a-c x^4\right )^{3/2}}\) |
| \(\Big \downarrow \) 1390 |
| \(\displaystyle \frac {\frac {\frac {\sqrt [4]{a} \sqrt {1-\frac {c x^4}{a}} \left (\frac {3 \sqrt {a} e}{\sqrt {c}}+5 d\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt [4]{c} \sqrt {a-c x^4}}-\frac {3 e \sqrt {1-\frac {c x^4}{a}} \int \frac {\sqrt {c} x^2+\sqrt {a}}{\sqrt {1-\frac {c x^4}{a}}}dx}{\sqrt {c} \sqrt {a-c x^4}}}{2 a}+\frac {x \left (5 d+3 e x^2\right )}{2 a \sqrt {a-c x^4}}}{6 a}+\frac {x \left (d+e x^2\right )}{6 a \left (a-c x^4\right )^{3/2}}\) |
| \(\Big \downarrow \) 1389 |
| \(\displaystyle \frac {\frac {\frac {\sqrt [4]{a} \sqrt {1-\frac {c x^4}{a}} \left (\frac {3 \sqrt {a} e}{\sqrt {c}}+5 d\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt [4]{c} \sqrt {a-c x^4}}-\frac {3 \sqrt {a} e \sqrt {1-\frac {c x^4}{a}} \int \frac {\sqrt {\frac {\sqrt {c} x^2}{\sqrt {a}}+1}}{\sqrt {1-\frac {\sqrt {c} x^2}{\sqrt {a}}}}dx}{\sqrt {c} \sqrt {a-c x^4}}}{2 a}+\frac {x \left (5 d+3 e x^2\right )}{2 a \sqrt {a-c x^4}}}{6 a}+\frac {x \left (d+e x^2\right )}{6 a \left (a-c x^4\right )^{3/2}}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {\frac {\frac {\sqrt [4]{a} \sqrt {1-\frac {c x^4}{a}} \left (\frac {3 \sqrt {a} e}{\sqrt {c}}+5 d\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt [4]{c} \sqrt {a-c x^4}}-\frac {3 a^{3/4} e \sqrt {1-\frac {c x^4}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{c^{3/4} \sqrt {a-c x^4}}}{2 a}+\frac {x \left (5 d+3 e x^2\right )}{2 a \sqrt {a-c x^4}}}{6 a}+\frac {x \left (d+e x^2\right )}{6 a \left (a-c x^4\right )^{3/2}}\) |
Input:
Int[(d + e*x^2)/(a - c*x^4)^(5/2),x]
Output:
(x*(d + e*x^2))/(6*a*(a - c*x^4)^(3/2)) + ((x*(5*d + 3*e*x^2))/(2*a*Sqrt[a - c*x^4]) + ((-3*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)*(5*d + (3*Sqrt[a]*e )/Sqrt[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]))/(2*a))/(6*a)
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 + e*x^2)*((a + c*x^4)^(p + 1)/(4*a*(p + 1))), x] + Simp[1/(4*a*(p + 1) ) Int[Simp[d*(4*p + 5) + e*(4*p + 7)*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] && LtQ[p, -1] && Integer Q[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 = 0.72 (sec) , antiderivative size = 245, normalized size of antiderivative = 1.30
| method | result | size |
| elliptic | \(\frac {\left (\frac {e \,x^{3}}{6 a \,c^{2}}+\frac {d x}{6 a \,c^{2}}\right ) \sqrt {-c \,x^{4}+a}}{\left (x^{4}-\frac {a}{c}\right )^{2}}+\frac {2 c \left (\frac {e \,x^{3}}{8 a^{2} c}+\frac {5 d x}{24 a^{2} c}\right )}{\sqrt {-\left (x^{4}-\frac {a}{c}\right ) c}}+\frac {5 d \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{12 a^{2} \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}+\frac {e \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )\right )}{4 a^{\frac {3}{2}} \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}\, \sqrt {c}}\) | \(245\) |
| default | \(d \left (\frac {x \sqrt {-c \,x^{4}+a}}{6 a \,c^{2} \left (x^{4}-\frac {a}{c}\right )^{2}}+\frac {5 x}{12 a^{2} \sqrt {-\left (x^{4}-\frac {a}{c}\right ) c}}+\frac {5 \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{12 a^{2} \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}\right )+e \left (\frac {x^{3} \sqrt {-c \,x^{4}+a}}{6 a \,c^{2} \left (x^{4}-\frac {a}{c}\right )^{2}}+\frac {x^{3}}{4 a^{2} \sqrt {-\left (x^{4}-\frac {a}{c}\right ) c}}+\frac {\sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )\right )}{4 a^{\frac {3}{2}} \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}\, \sqrt {c}}\right )\) | \(270\) |
Input:
int((e*x^2+d)/(-c*x^4+a)^(5/2),x,method=_RETURNVERBOSE)
Output:
(1/6/a*e/c^2*x^3+1/6*d/a/c^2*x)*(-c*x^4+a)^(1/2)/(x^4-1/c*a)^2+2*c*(1/8/a^ 2/c*e*x^3+5/24/a^2/c*d*x)/(-(x^4-1/c*a)*c)^(1/2)+5/12*d/a^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)*EllipticF(x*(1/a^(1/2)*c^(1/2))^(1/2),I)+1/4/a^(3/2)*e/ (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 = 192, normalized size of antiderivative = 1.02 \[ \int \frac {d+e x^2}{\left (a-c x^4\right )^{5/2}} \, dx=-\frac {3 \, {\left (c^{2} e x^{8} - 2 \, a c e x^{4} + a^{2} e\right )} \sqrt {a} \left (\frac {c}{a}\right )^{\frac {3}{4}} E(\arcsin \left (x \left (\frac {c}{a}\right )^{\frac {1}{4}}\right )\,|\,-1) - {\left ({\left (5 \, c^{2} d + 3 \, c^{2} e\right )} x^{8} - 2 \, {\left (5 \, a c d + 3 \, a c e\right )} x^{4} + 5 \, a^{2} d + 3 \, a^{2} e\right )} \sqrt {a} \left (\frac {c}{a}\right )^{\frac {3}{4}} F(\arcsin \left (x \left (\frac {c}{a}\right )^{\frac {1}{4}}\right )\,|\,-1) + {\left (3 \, c^{2} e x^{7} + 5 \, c^{2} d x^{5} - 5 \, a c e x^{3} - 7 \, a c d x\right )} \sqrt {-c x^{4} + a}}{12 \, {\left (a^{2} c^{3} x^{8} - 2 \, a^{3} c^{2} x^{4} + a^{4} c\right )}} \] Input:
integrate((e*x^2+d)/(-c*x^4+a)^(5/2),x, algorithm="fricas")
Output:
-1/12*(3*(c^2*e*x^8 - 2*a*c*e*x^4 + a^2*e)*sqrt(a)*(c/a)^(3/4)*elliptic_e( arcsin(x*(c/a)^(1/4)), -1) - ((5*c^2*d + 3*c^2*e)*x^8 - 2*(5*a*c*d + 3*a*c *e)*x^4 + 5*a^2*d + 3*a^2*e)*sqrt(a)*(c/a)^(3/4)*elliptic_f(arcsin(x*(c/a) ^(1/4)), -1) + (3*c^2*e*x^7 + 5*c^2*d*x^5 - 5*a*c*e*x^3 - 7*a*c*d*x)*sqrt( -c*x^4 + a))/(a^2*c^3*x^8 - 2*a^3*c^2*x^4 + a^4*c)
Time = 9.89 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.44 \[ \int \frac {d+e x^2}{\left (a-c x^4\right )^{5/2}} \, dx=\frac {d x \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {5}{2} \\ \frac {5}{4} \end {matrix}\middle | {\frac {c x^{4} e^{2 i \pi }}{a}} \right )}}{4 a^{\frac {5}{2}} \Gamma \left (\frac {5}{4}\right )} + \frac {e x^{3} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{4}, \frac {5}{2} \\ \frac {7}{4} \end {matrix}\middle | {\frac {c x^{4} e^{2 i \pi }}{a}} \right )}}{4 a^{\frac {5}{2}} \Gamma \left (\frac {7}{4}\right )} \] Input:
integrate((e*x**2+d)/(-c*x**4+a)**(5/2),x)
Output:
d*x*gamma(1/4)*hyper((1/4, 5/2), (5/4,), c*x**4*exp_polar(2*I*pi)/a)/(4*a* *(5/2)*gamma(5/4)) + e*x**3*gamma(3/4)*hyper((3/4, 5/2), (7/4,), c*x**4*ex p_polar(2*I*pi)/a)/(4*a**(5/2)*gamma(7/4))
\[ \int \frac {d+e x^2}{\left (a-c x^4\right )^{5/2}} \, dx=\int { \frac {e x^{2} + d}{{\left (-c x^{4} + a\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate((e*x^2+d)/(-c*x^4+a)^(5/2),x, algorithm="maxima")
Output:
integrate((e*x^2 + d)/(-c*x^4 + a)^(5/2), x)
\[ \int \frac {d+e x^2}{\left (a-c x^4\right )^{5/2}} \, dx=\int { \frac {e x^{2} + d}{{\left (-c x^{4} + a\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate((e*x^2+d)/(-c*x^4+a)^(5/2),x, algorithm="giac")
Output:
integrate((e*x^2 + d)/(-c*x^4 + a)^(5/2), x)
Timed out. \[ \int \frac {d+e x^2}{\left (a-c x^4\right )^{5/2}} \, dx=\int \frac {e\,x^2+d}{{\left (a-c\,x^4\right )}^{5/2}} \,d x \] Input:
int((d + e*x^2)/(a - c*x^4)^(5/2),x)
Output:
int((d + e*x^2)/(a - c*x^4)^(5/2), x)
\[ \int \frac {d+e x^2}{\left (a-c x^4\right )^{5/2}} \, dx=\left (\int \frac {\sqrt {-c \,x^{4}+a}}{-c^{3} x^{12}+3 a \,c^{2} x^{8}-3 a^{2} c \,x^{4}+a^{3}}d x \right ) d +\left (\int \frac {\sqrt {-c \,x^{4}+a}\, x^{2}}{-c^{3} x^{12}+3 a \,c^{2} x^{8}-3 a^{2} c \,x^{4}+a^{3}}d x \right ) e \] Input:
int((e*x^2+d)/(-c*x^4+a)^(5/2),x)
Output:
int(sqrt(a - c*x**4)/(a**3 - 3*a**2*c*x**4 + 3*a*c**2*x**8 - c**3*x**12),x )*d + int((sqrt(a - c*x**4)*x**2)/(a**3 - 3*a**2*c*x**4 + 3*a*c**2*x**8 - c**3*x**12),x)*e