Integrand size = 21, antiderivative size = 264 \[ \int \frac {\left (d+e x^2\right )^2}{\sqrt {a+c x^4}} \, dx=\frac {e^2 x \sqrt {a+c x^4}}{3 c}+\frac {2 d e x \sqrt {a+c x^4}}{\sqrt {c} \left (\sqrt {a}+\sqrt {c} x^2\right )}-\frac {2 \sqrt [4]{a} d e \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{c^{3/4} \sqrt {a+c x^4}}+\frac {\left (3 c d^2+6 \sqrt {a} \sqrt {c} d e-a e^2\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{6 \sqrt [4]{a} c^{5/4} \sqrt {a+c x^4}} \] Output:
1/3*e^2*x*(c*x^4+a)^(1/2)/c+2*d*e*x*(c*x^4+a)^(1/2)/c^(1/2)/(a^(1/2)+c^(1/ 2)*x^2)-2*a^(1/4)*d*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)+1/6*(3*c*d^2+6*a^(1/2)*c^(1/2)*d*e-a*e^2)*(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))/a^(1/4)/c^(5/4)/(c*x^4+a)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.07 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.45 \[ \int \frac {\left (d+e x^2\right )^2}{\sqrt {a+c x^4}} \, dx=\frac {\left (3 c d^2-a e^2\right ) x \sqrt {1+\frac {c x^4}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},-\frac {c x^4}{a}\right )+e x \left (e \left (a+c x^4\right )+2 c d x^2 \sqrt {1+\frac {c x^4}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},-\frac {c x^4}{a}\right )\right )}{3 c \sqrt {a+c x^4}} \] Input:
Integrate[(d + e*x^2)^2/Sqrt[a + c*x^4],x]
Output:
((3*c*d^2 - a*e^2)*x*Sqrt[1 + (c*x^4)/a]*Hypergeometric2F1[1/4, 1/2, 5/4, -((c*x^4)/a)] + e*x*(e*(a + c*x^4) + 2*c*d*x^2*Sqrt[1 + (c*x^4)/a]*Hyperge ometric2F1[1/2, 3/4, 7/4, -((c*x^4)/a)]))/(3*c*Sqrt[a + c*x^4])
Time = 0.61 (sec) , antiderivative size = 272, normalized size of antiderivative = 1.03, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {1519, 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.
| \(\displaystyle \int \frac {\left (d+e x^2\right )^2}{\sqrt {a+c x^4}} \, dx\) |
| \(\Big \downarrow \) 1519 |
| \(\displaystyle \frac {\int \frac {3 c d^2+6 c e x^2 d-a e^2}{\sqrt {c x^4+a}}dx}{3 c}+\frac {e^2 x \sqrt {a+c x^4}}{3 c}\) |
| \(\Big \downarrow \) 1512 |
| \(\displaystyle \frac {\left (6 \sqrt {a} \sqrt {c} d e-a e^2+3 c d^2\right ) \int \frac {1}{\sqrt {c x^4+a}}dx-6 \sqrt {a} \sqrt {c} d e \int \frac {\sqrt {a}-\sqrt {c} x^2}{\sqrt {a} \sqrt {c x^4+a}}dx}{3 c}+\frac {e^2 x \sqrt {a+c x^4}}{3 c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\left (6 \sqrt {a} \sqrt {c} d e-a e^2+3 c d^2\right ) \int \frac {1}{\sqrt {c x^4+a}}dx-6 \sqrt {c} d e \int \frac {\sqrt {a}-\sqrt {c} x^2}{\sqrt {c x^4+a}}dx}{3 c}+\frac {e^2 x \sqrt {a+c x^4}}{3 c}\) |
| \(\Big \downarrow \) 761 |
| \(\displaystyle \frac {\frac {\left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \left (6 \sqrt {a} \sqrt {c} d e-a e^2+3 c d^2\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{a} \sqrt [4]{c} \sqrt {a+c x^4}}-6 \sqrt {c} d e \int \frac {\sqrt {a}-\sqrt {c} x^2}{\sqrt {c x^4+a}}dx}{3 c}+\frac {e^2 x \sqrt {a+c x^4}}{3 c}\) |
| \(\Big \downarrow \) 1510 |
| \(\displaystyle \frac {\frac {\left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \left (6 \sqrt {a} \sqrt {c} d e-a e^2+3 c d^2\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{a} \sqrt [4]{c} \sqrt {a+c x^4}}-6 \sqrt {c} d e \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{\sqrt [4]{c} \sqrt {a+c x^4}}-\frac {x \sqrt {a+c x^4}}{\sqrt {a}+\sqrt {c} x^2}\right )}{3 c}+\frac {e^2 x \sqrt {a+c x^4}}{3 c}\) |
Input:
Int[(d + e*x^2)^2/Sqrt[a + c*x^4],x]
Output:
(e^2*x*Sqrt[a + c*x^4])/(3*c) + (-6*Sqrt[c]*d*e*(-((x*Sqrt[a + c*x^4])/(Sq rt[a] + Sqrt[c]*x^2)) + (a^(1/4)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/ (Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticE[2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/ (c^(1/4)*Sqrt[a + c*x^4])) + ((3*c*d^2 + 6*Sqrt[a]*Sqrt[c]*d*e - a*e^2)*(S qrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*Elliptic F[2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/(2*a^(1/4)*c^(1/4)*Sqrt[a + c*x^4]) )/(3*c)
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)/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]
Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (c_.)*(x_)^4)^(p_), x_Symbol] :> Sim p[e^q*x^(2*q - 3)*((a + c*x^4)^(p + 1)/(c*(4*p + 2*q + 1))), x] + Simp[1/(c *(4*p + 2*q + 1)) Int[(a + c*x^4)^p*ExpandToSum[c*(4*p + 2*q + 1)*(d + e* x^2)^q - a*(2*q - 3)*e^q*x^(2*q - 4) - c*(4*p + 2*q + 1)*e^q*x^(2*q), x], x ], x] /; FreeQ[{a, c, d, e, p}, x] && NeQ[c*d^2 + a*e^2, 0] && IGtQ[q, 1]
Result contains complex when optimal does not.
Time = 1.80 (sec) , antiderivative size = 200, normalized size of antiderivative = 0.76
| method | result | size |
| elliptic | \(\frac {e^{2} x \sqrt {c \,x^{4}+a}}{3 c}+\frac {\left (d^{2}-\frac {a \,e^{2}}{3 c}\right ) \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}+\frac {2 i d e \sqrt {a}\, \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )\right )}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}\, \sqrt {c}}\) | \(200\) |
| default | \(\frac {d^{2} \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}+e^{2} \left (\frac {x \sqrt {c \,x^{4}+a}}{3 c}-\frac {a \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{3 c \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}\right )+\frac {2 i d e \sqrt {a}\, \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )\right )}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}\, \sqrt {c}}\) | \(266\) |
| risch | \(\frac {e^{2} x \sqrt {c \,x^{4}+a}}{3 c}-\frac {\frac {a \,e^{2} \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}-\frac {3 c \,d^{2} \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}-\frac {6 i e \sqrt {c}\, d \sqrt {a}\, \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )\right )}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}}{3 c}\) | \(271\) |
Input:
int((e*x^2+d)^2/(c*x^4+a)^(1/2),x,method=_RETURNVERBOSE)
Output:
1/3*e^2*x*(c*x^4+a)^(1/2)/c+(d^2-1/3*a*e^2/c)/(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)+2*I*d*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)-Ellip ticE(x*(I/a^(1/2)*c^(1/2))^(1/2),I))
Time = 0.08 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.43 \[ \int \frac {\left (d+e x^2\right )^2}{\sqrt {a+c x^4}} \, dx=\frac {6 \, a \sqrt {c} d e x \left (-\frac {a}{c}\right )^{\frac {3}{4}} E(\arcsin \left (\frac {\left (-\frac {a}{c}\right )^{\frac {1}{4}}}{x}\right )\,|\,-1) + {\left (3 \, c d^{2} - 6 \, a d e - a e^{2}\right )} \sqrt {c} x \left (-\frac {a}{c}\right )^{\frac {3}{4}} F(\arcsin \left (\frac {\left (-\frac {a}{c}\right )^{\frac {1}{4}}}{x}\right )\,|\,-1) + {\left (a e^{2} x^{2} + 6 \, a d e\right )} \sqrt {c x^{4} + a}}{3 \, a c x} \] Input:
integrate((e*x^2+d)^2/(c*x^4+a)^(1/2),x, algorithm="fricas")
Output:
1/3*(6*a*sqrt(c)*d*e*x*(-a/c)^(3/4)*elliptic_e(arcsin((-a/c)^(1/4)/x), -1) + (3*c*d^2 - 6*a*d*e - a*e^2)*sqrt(c)*x*(-a/c)^(3/4)*elliptic_f(arcsin((- a/c)^(1/4)/x), -1) + (a*e^2*x^2 + 6*a*d*e)*sqrt(c*x^4 + a))/(a*c*x)
Result contains complex when optimal does not.
Time = 1.30 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.47 \[ \int \frac {\left (d+e x^2\right )^2}{\sqrt {a+c x^4}} \, dx=\frac {d^{2} x \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {1}{2} \\ \frac {5}{4} \end {matrix}\middle | {\frac {c x^{4} e^{i \pi }}{a}} \right )}}{4 \sqrt {a} \Gamma \left (\frac {5}{4}\right )} + \frac {d e x^{3} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {c x^{4} e^{i \pi }}{a}} \right )}}{2 \sqrt {a} \Gamma \left (\frac {7}{4}\right )} + \frac {e^{2} x^{5} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {5}{4} \\ \frac {9}{4} \end {matrix}\middle | {\frac {c x^{4} e^{i \pi }}{a}} \right )}}{4 \sqrt {a} \Gamma \left (\frac {9}{4}\right )} \] Input:
integrate((e*x**2+d)**2/(c*x**4+a)**(1/2),x)
Output:
d**2*x*gamma(1/4)*hyper((1/4, 1/2), (5/4,), c*x**4*exp_polar(I*pi)/a)/(4*s qrt(a)*gamma(5/4)) + d*e*x**3*gamma(3/4)*hyper((1/2, 3/4), (7/4,), c*x**4* exp_polar(I*pi)/a)/(2*sqrt(a)*gamma(7/4)) + e**2*x**5*gamma(5/4)*hyper((1/ 2, 5/4), (9/4,), c*x**4*exp_polar(I*pi)/a)/(4*sqrt(a)*gamma(9/4))
\[ \int \frac {\left (d+e x^2\right )^2}{\sqrt {a+c x^4}} \, dx=\int { \frac {{\left (e x^{2} + d\right )}^{2}}{\sqrt {c x^{4} + a}} \,d x } \] Input:
integrate((e*x^2+d)^2/(c*x^4+a)^(1/2),x, algorithm="maxima")
Output:
integrate((e*x^2 + d)^2/sqrt(c*x^4 + a), x)
\[ \int \frac {\left (d+e x^2\right )^2}{\sqrt {a+c x^4}} \, dx=\int { \frac {{\left (e x^{2} + d\right )}^{2}}{\sqrt {c x^{4} + a}} \,d x } \] Input:
integrate((e*x^2+d)^2/(c*x^4+a)^(1/2),x, algorithm="giac")
Output:
integrate((e*x^2 + d)^2/sqrt(c*x^4 + a), x)
Timed out. \[ \int \frac {\left (d+e x^2\right )^2}{\sqrt {a+c x^4}} \, dx=\int \frac {{\left (e\,x^2+d\right )}^2}{\sqrt {c\,x^4+a}} \,d x \] Input:
int((d + e*x^2)^2/(a + c*x^4)^(1/2),x)
Output:
int((d + e*x^2)^2/(a + c*x^4)^(1/2), x)
\[ \int \frac {\left (d+e x^2\right )^2}{\sqrt {a+c x^4}} \, dx=\frac {\sqrt {c \,x^{4}+a}\, e^{2} x -\left (\int \frac {\sqrt {c \,x^{4}+a}}{c \,x^{4}+a}d x \right ) a \,e^{2}+3 \left (\int \frac {\sqrt {c \,x^{4}+a}}{c \,x^{4}+a}d x \right ) c \,d^{2}+6 \left (\int \frac {\sqrt {c \,x^{4}+a}\, x^{2}}{c \,x^{4}+a}d x \right ) c d e}{3 c} \] Input:
int((e*x^2+d)^2/(c*x^4+a)^(1/2),x)
Output:
(sqrt(a + c*x**4)*e**2*x - int(sqrt(a + c*x**4)/(a + c*x**4),x)*a*e**2 + 3 *int(sqrt(a + c*x**4)/(a + c*x**4),x)*c*d**2 + 6*int((sqrt(a + c*x**4)*x** 2)/(a + c*x**4),x)*c*d*e)/(3*c)