Integrand size = 19, antiderivative size = 261 \[ \int \left (d+e x^2\right ) \sqrt {a+c x^4} \, dx=\frac {2 a e x \sqrt {a+c x^4}}{5 \sqrt {c} \left (\sqrt {a}+\sqrt {c} x^2\right )}+\frac {1}{15} x \left (5 d+3 e x^2\right ) \sqrt {a+c x^4}-\frac {2 a^{5/4} e \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{5 c^{3/4} \sqrt {a+c x^4}}+\frac {a^{3/4} \left (5 \sqrt {c} d+3 \sqrt {a} e\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{15 c^{3/4} \sqrt {a+c x^4}} \] Output:
2/5*a*e*x*(c*x^4+a)^(1/2)/c^(1/2)/(a^(1/2)+c^(1/2)*x^2)+1/15*x*(3*e*x^2+5* d)*(c*x^4+a)^(1/2)-2/5*a^(5/4)*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)+1/15*a^(3/4)*(5*c^(1/2)*d+3*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 = 7.95 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.30 \[ \int \left (d+e x^2\right ) \sqrt {a+c x^4} \, dx=\frac {\sqrt {a+c x^4} \left (3 d x \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {1}{4},\frac {5}{4},-\frac {c x^4}{a}\right )+e x^3 \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {3}{4},\frac {7}{4},-\frac {c x^4}{a}\right )\right )}{3 \sqrt {1+\frac {c x^4}{a}}} \] 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 = 0.56 (sec) , antiderivative size = 263, normalized size of antiderivative = 1.01, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {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.
| \(\displaystyle \int \sqrt {a+c x^4} \left (d+e x^2\right ) \, dx\) |
| \(\Big \downarrow \) 1491 |
| \(\displaystyle \frac {1}{15} \int \frac {2 a \left (3 e x^2+5 d\right )}{\sqrt {c x^4+a}}dx+\frac {1}{15} x \sqrt {a+c x^4} \left (5 d+3 e x^2\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2}{15} a \int \frac {3 e x^2+5 d}{\sqrt {c x^4+a}}dx+\frac {1}{15} x \sqrt {a+c x^4} \left (5 d+3 e x^2\right )\) |
| \(\Big \downarrow \) 1512 |
| \(\displaystyle \frac {2}{15} a \left (\left (\frac {3 \sqrt {a} e}{\sqrt {c}}+5 d\right ) \int \frac {1}{\sqrt {c x^4+a}}dx-\frac {3 \sqrt {a} e \int \frac {\sqrt {a}-\sqrt {c} x^2}{\sqrt {a} \sqrt {c x^4+a}}dx}{\sqrt {c}}\right )+\frac {1}{15} x \sqrt {a+c x^4} \left (5 d+3 e x^2\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2}{15} a \left (\left (\frac {3 \sqrt {a} e}{\sqrt {c}}+5 d\right ) \int \frac {1}{\sqrt {c x^4+a}}dx-\frac {3 e \int \frac {\sqrt {a}-\sqrt {c} x^2}{\sqrt {c x^4+a}}dx}{\sqrt {c}}\right )+\frac {1}{15} x \sqrt {a+c x^4} \left (5 d+3 e x^2\right )\) |
| \(\Big \downarrow \) 761 |
| \(\displaystyle \frac {2}{15} a \left (\frac {\left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \left (\frac {3 \sqrt {a} e}{\sqrt {c}}+5 d\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{a} \sqrt [4]{c} \sqrt {a+c x^4}}-\frac {3 e \int \frac {\sqrt {a}-\sqrt {c} x^2}{\sqrt {c x^4+a}}dx}{\sqrt {c}}\right )+\frac {1}{15} x \sqrt {a+c x^4} \left (5 d+3 e x^2\right )\) |
| \(\Big \downarrow \) 1510 |
| \(\displaystyle \frac {2}{15} a \left (\frac {\left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \left (\frac {3 \sqrt {a} e}{\sqrt {c}}+5 d\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{a} \sqrt [4]{c} \sqrt {a+c x^4}}-\frac {3 e \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{\sqrt [4]{c} \sqrt {a+c x^4}}-\frac {x \sqrt {a+c x^4}}{\sqrt {a}+\sqrt {c} x^2}\right )}{\sqrt {c}}\right )+\frac {1}{15} x \sqrt {a+c x^4} \left (5 d+3 e x^2\right )\) |
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*e*(-((x*Sqrt[a + c*x^4] )/(Sqrt[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])))/Sqrt[c] + ((5*d + (3*Sqrt[a]*e)/Sqrt[c])* (Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*Ellipt icF[2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/(2*a^(1/4)*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[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 = 1.05 (sec) , antiderivative size = 196, normalized size of antiderivative = 0.75
| method | result | size |
| risch | \(\frac {x \left (3 e \,x^{2}+5 d \right ) \sqrt {c \,x^{4}+a}}{15}+\frac {2 a \left (\frac {5 d \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}+\frac {3 i e \sqrt {a}\, \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )\right )}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}\, \sqrt {c}}\right )}{15}\) | \(196\) |
| elliptic | \(\frac {e \,x^{3} \sqrt {c \,x^{4}+a}}{5}+\frac {d x \sqrt {c \,x^{4}+a}}{3}+\frac {2 a d \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{3 \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}+\frac {2 i a^{\frac {3}{2}} e \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )\right )}{5 \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}\, \sqrt {c}}\) | \(199\) |
| default | \(d \left (\frac {x \sqrt {c \,x^{4}+a}}{3}+\frac {2 a \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{3 \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}\right )+e \left (\frac {x^{3} \sqrt {c \,x^{4}+a}}{5}+\frac {2 i a^{\frac {3}{2}} \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )\right )}{5 \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}\, \sqrt {c}}\right )\) | \(201\) |
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/(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)+3*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)-Elli pticE(x*(I/a^(1/2)*c^(1/2))^(1/2),I)))
Time = 0.07 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.41 \[ \int \left (d+e x^2\right ) \sqrt {a+c x^4} \, dx=\frac {6 \, a \sqrt {c} e x \left (-\frac {a}{c}\right )^{\frac {3}{4}} E(\arcsin \left (\frac {\left (-\frac {a}{c}\right )^{\frac {1}{4}}}{x}\right )\,|\,-1) + 2 \, {\left (5 \, c d - 3 \, a e\right )} \sqrt {c} x \left (-\frac {a}{c}\right )^{\frac {3}{4}} F(\arcsin \left (\frac {\left (-\frac {a}{c}\right )^{\frac {1}{4}}}{x}\right )\,|\,-1) + {\left (3 \, c e x^{4} + 5 \, c d x^{2} + 6 \, a e\right )} \sqrt {c x^{4} + a}}{15 \, c x} \] 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)
Result contains complex when optimal does not.
Time = 0.92 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.31 \[ \int \left (d+e x^2\right ) \sqrt {a+c x^4} \, dx=\frac {\sqrt {a} d x \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {1}{4} \\ \frac {5}{4} \end {matrix}\middle | {\frac {c x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac {5}{4}\right )} + \frac {\sqrt {a} e x^{3} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {c x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac {7}{4}\right )} \] 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(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(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. \[ \int \left (d+e x^2\right ) \sqrt {a+c x^4} \, dx=\int \sqrt {c\,x^4+a}\,\left (e\,x^2+d\right ) \,d x \] 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