Integrand size = 20, antiderivative size = 158 \[ \int \left (d+e x^2\right ) \sqrt {a-c x^4} \, dx=\frac {1}{15} x \left (5 d+3 e x^2\right ) \sqrt {a-c x^4}+\frac {2 a^{7/4} e \sqrt {1-\frac {c x^4}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{5 c^{3/4} \sqrt {a-c x^4}}+\frac {2 a^{5/4} \left (5 d-\frac {3 \sqrt {a} e}{\sqrt {c}}\right ) \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{15 \sqrt [4]{c} \sqrt {a-c x^4}} \] Output:
1/15*x*(3*e*x^2+5*d)*(-c*x^4+a)^(1/2)+2/5*a^(7/4)*e*(1-c*x^4/a)^(1/2)*Elli pticE(c^(1/4)*x/a^(1/4),I)/c^(3/4)/(-c*x^4+a)^(1/2)+2/15*a^(5/4)*(5*d-3*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 = 7.92 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.49 \[ \int \left (d+e x^2\right ) \sqrt {a-c x^4} \, dx=\frac {\sqrt {a-c x^4} \left (3 d x \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {1}{4},\frac {5}{4},\frac {c x^4}{a}\right )+e x^3 \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {3}{4},\frac {7}{4},\frac {c x^4}{a}\right )\right )}{3 \sqrt {1-\frac {c x^4}{a}}} \] Input:
Integrate[(d + e*x^2)*Sqrt[a - c*x^4],x]
Output:
(Sqrt[a - c*x^4]*(3*d*x*Hypergeometric2F1[-1/2, 1/4, 5/4, (c*x^4)/a] + e*x ^3*Hypergeometric2F1[-1/2, 3/4, 7/4, (c*x^4)/a]))/(3*Sqrt[1 - (c*x^4)/a])
Time = 0.58 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.01, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.450, Rules used = {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.
| \(\displaystyle \int \sqrt {a-c x^4} \left (d+e x^2\right ) \, dx\) |
| \(\Big \downarrow \) 1491 |
| \(\displaystyle \frac {1}{15} \int \frac {2 a \left (3 e x^2+5 d\right )}{\sqrt {a-c x^4}}dx+\frac {1}{15} x \sqrt {a-c x^4} \left (5 d+3 e x^2\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2}{15} a \int \frac {3 e x^2+5 d}{\sqrt {a-c x^4}}dx+\frac {1}{15} x \sqrt {a-c x^4} \left (5 d+3 e x^2\right )\) |
| \(\Big \downarrow \) 1513 |
| \(\displaystyle \frac {2}{15} a \left (\left (5 d-\frac {3 \sqrt {a} e}{\sqrt {c}}\right ) \int \frac {1}{\sqrt {a-c x^4}}dx+\frac {3 \sqrt {a} e \int \frac {\sqrt {c} x^2+\sqrt {a}}{\sqrt {a} \sqrt {a-c x^4}}dx}{\sqrt {c}}\right )+\frac {1}{15} x \sqrt {a-c x^4} \left (5 d+3 e x^2\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2}{15} a \left (\left (5 d-\frac {3 \sqrt {a} e}{\sqrt {c}}\right ) \int \frac {1}{\sqrt {a-c x^4}}dx+\frac {3 e \int \frac {\sqrt {c} x^2+\sqrt {a}}{\sqrt {a-c x^4}}dx}{\sqrt {c}}\right )+\frac {1}{15} x \sqrt {a-c x^4} \left (5 d+3 e x^2\right )\) |
| \(\Big \downarrow \) 765 |
| \(\displaystyle \frac {2}{15} a \left (\frac {\sqrt {1-\frac {c x^4}{a}} \left (5 d-\frac {3 \sqrt {a} e}{\sqrt {c}}\right ) \int \frac {1}{\sqrt {1-\frac {c x^4}{a}}}dx}{\sqrt {a-c x^4}}+\frac {3 e \int \frac {\sqrt {c} x^2+\sqrt {a}}{\sqrt {a-c x^4}}dx}{\sqrt {c}}\right )+\frac {1}{15} x \sqrt {a-c x^4} \left (5 d+3 e x^2\right )\) |
| \(\Big \downarrow \) 762 |
| \(\displaystyle \frac {2}{15} a \left (\frac {3 e \int \frac {\sqrt {c} x^2+\sqrt {a}}{\sqrt {a-c x^4}}dx}{\sqrt {c}}+\frac {\sqrt [4]{a} \sqrt {1-\frac {c x^4}{a}} \left (5 d-\frac {3 \sqrt {a} e}{\sqrt {c}}\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt [4]{c} \sqrt {a-c x^4}}\right )+\frac {1}{15} x \sqrt {a-c x^4} \left (5 d+3 e x^2\right )\) |
| \(\Big \downarrow \) 1390 |
| \(\displaystyle \frac {2}{15} a \left (\frac {3 e \sqrt {1-\frac {c x^4}{a}} \int \frac {\sqrt {c} x^2+\sqrt {a}}{\sqrt {1-\frac {c x^4}{a}}}dx}{\sqrt {c} \sqrt {a-c x^4}}+\frac {\sqrt [4]{a} \sqrt {1-\frac {c x^4}{a}} \left (5 d-\frac {3 \sqrt {a} e}{\sqrt {c}}\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt [4]{c} \sqrt {a-c x^4}}\right )+\frac {1}{15} x \sqrt {a-c x^4} \left (5 d+3 e x^2\right )\) |
| \(\Big \downarrow \) 1389 |
| \(\displaystyle \frac {2}{15} a \left (\frac {3 \sqrt {a} e \sqrt {1-\frac {c x^4}{a}} \int \frac {\sqrt {\frac {\sqrt {c} x^2}{\sqrt {a}}+1}}{\sqrt {1-\frac {\sqrt {c} x^2}{\sqrt {a}}}}dx}{\sqrt {c} \sqrt {a-c x^4}}+\frac {\sqrt [4]{a} \sqrt {1-\frac {c x^4}{a}} \left (5 d-\frac {3 \sqrt {a} e}{\sqrt {c}}\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt [4]{c} \sqrt {a-c x^4}}\right )+\frac {1}{15} x \sqrt {a-c x^4} \left (5 d+3 e x^2\right )\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {2}{15} a \left (\frac {3 a^{3/4} e \sqrt {1-\frac {c x^4}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{c^{3/4} \sqrt {a-c x^4}}+\frac {\sqrt [4]{a} \sqrt {1-\frac {c x^4}{a}} \left (5 d-\frac {3 \sqrt {a} e}{\sqrt {c}}\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt [4]{c} \sqrt {a-c x^4}}\right )+\frac {1}{15} x \sqrt {a-c x^4} \left (5 d+3 e x^2\right )\) |
Input:
Int[(d + e*x^2)*Sqrt[a - c*x^4],x]
Output:
(x*(5*d + 3*e*x^2)*Sqrt[a - c*x^4])/15 + (2*a*((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[A rcSin[(c^(1/4)*x)/a^(1/4)], -1])/(c^(1/4)*Sqrt[a - c*x^4])))/15
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 = 1.08 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.15
| method | result | size |
| risch | \(\frac {x \left (3 e \,x^{2}+5 d \right ) \sqrt {-c \,x^{4}+a}}{15}+\frac {2 a \left (\frac {5 d \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}-\frac {3 e \sqrt {a}\, \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )\right )}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}\, \sqrt {c}}\right )}{15}\) | \(182\) |
| elliptic | \(\frac {e \,x^{3} \sqrt {-c \,x^{4}+a}}{5}+\frac {d x \sqrt {-c \,x^{4}+a}}{3}+\frac {2 a d \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{3 \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}-\frac {2 a^{\frac {3}{2}} e \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )\right )}{5 \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}\, \sqrt {c}}\) | \(186\) |
| default | \(d \left (\frac {x \sqrt {-c \,x^{4}+a}}{3}+\frac {2 a \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{3 \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}\right )+e \left (\frac {x^{3} \sqrt {-c \,x^{4}+a}}{5}-\frac {2 a^{\frac {3}{2}} \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )\right )}{5 \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}\, \sqrt {c}}\right )\) | \(188\) |
Input:
int((e*x^2+d)*(-c*x^4+a)^(1/2),x,method=_RETURNVERBOSE)
Output:
1/15*x*(3*e*x^2+5*d)*(-c*x^4+a)^(1/2)+2/15*a*(5*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)-3*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)-Ell ipticE(x*(1/a^(1/2)*c^(1/2))^(1/2),I)))
Time = 0.07 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.69 \[ \int \left (d+e x^2\right ) \sqrt {a-c x^4} \, dx=-\frac {6 \, a \sqrt {-c} e x \left (\frac {a}{c}\right )^{\frac {3}{4}} E(\arcsin \left (\frac {\left (\frac {a}{c}\right )^{\frac {1}{4}}}{x}\right )\,|\,-1) - 2 \, {\left (5 \, c d + 3 \, a e\right )} \sqrt {-c} x \left (\frac {a}{c}\right )^{\frac {3}{4}} F(\arcsin \left (\frac {\left (\frac {a}{c}\right )^{\frac {1}{4}}}{x}\right )\,|\,-1) - {\left (3 \, c e x^{4} + 5 \, c d x^{2} - 6 \, a e\right )} \sqrt {-c x^{4} + a}}{15 \, c x} \] Input:
integrate((e*x^2+d)*(-c*x^4+a)^(1/2),x, algorithm="fricas")
Output:
-1/15*(6*a*sqrt(-c)*e*x*(a/c)^(3/4)*elliptic_e(arcsin((a/c)^(1/4)/x), -1) - 2*(5*c*d + 3*a*e)*sqrt(-c)*x*(a/c)^(3/4)*elliptic_f(arcsin((a/c)^(1/4)/x ), -1) - (3*c*e*x^4 + 5*c*d*x^2 - 6*a*e)*sqrt(-c*x^4 + a))/(c*x)
Time = 1.00 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.54 \[ \int \left (d+e x^2\right ) \sqrt {a-c x^4} \, dx=\frac {\sqrt {a} d x \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {1}{4} \\ \frac {5}{4} \end {matrix}\middle | {\frac {c x^{4} e^{2 i \pi }}{a}} \right )}}{4 \Gamma \left (\frac {5}{4}\right )} + \frac {\sqrt {a} e x^{3} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {c x^{4} e^{2 i \pi }}{a}} \right )}}{4 \Gamma \left (\frac {7}{4}\right )} \] Input:
integrate((e*x**2+d)*(-c*x**4+a)**(1/2),x)
Output:
sqrt(a)*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)) + sqrt(a)*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))
\[ \int \left (d+e x^2\right ) \sqrt {a-c x^4} \, dx=\int { \sqrt {-c x^{4} + a} {\left (e x^{2} + d\right )} \,d x } \] Input:
integrate((e*x^2+d)*(-c*x^4+a)^(1/2),x, algorithm="maxima")
Output:
integrate(sqrt(-c*x^4 + a)*(e*x^2 + d), x)
\[ \int \left (d+e x^2\right ) \sqrt {a-c x^4} \, dx=\int { \sqrt {-c x^{4} + a} {\left (e x^{2} + d\right )} \,d x } \] Input:
integrate((e*x^2+d)*(-c*x^4+a)^(1/2),x, algorithm="giac")
Output:
integrate(sqrt(-c*x^4 + a)*(e*x^2 + d), x)
Timed out. \[ \int \left (d+e x^2\right ) \sqrt {a-c x^4} \, dx=\int \sqrt {a-c\,x^4}\,\left (e\,x^2+d\right ) \,d x \] Input:
int((a - c*x^4)^(1/2)*(d + e*x^2),x)
Output:
int((a - c*x^4)^(1/2)*(d + e*x^2), x)
\[ \int \left (d+e x^2\right ) \sqrt {a-c x^4} \, dx=\frac {\sqrt {-c \,x^{4}+a}\, d x}{3}+\frac {\sqrt {-c \,x^{4}+a}\, e \,x^{3}}{5}+\frac {2 \left (\int \frac {\sqrt {-c \,x^{4}+a}}{-c \,x^{4}+a}d x \right ) a d}{3}+\frac {2 \left (\int \frac {\sqrt {-c \,x^{4}+a}\, x^{2}}{-c \,x^{4}+a}d x \right ) a e}{5} \] Input:
int((e*x^2+d)*(-c*x^4+a)^(1/2),x)
Output:
(5*sqrt(a - c*x**4)*d*x + 3*sqrt(a - c*x**4)*e*x**3 + 10*int(sqrt(a - c*x* *4)/(a - c*x**4),x)*a*d + 6*int((sqrt(a - c*x**4)*x**2)/(a - c*x**4),x)*a* e)/15