Integrand size = 24, antiderivative size = 1605 \[ \int \frac {1}{\left (c+d x+e x^2\right ) \sqrt {a+b x^4}} \, dx =\text {Too large to display} \]
1/2*b^(1/4)*e*(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)^(1/2)/cos(2*arctan(b^(1 /4)*x/a^(1/4)))*EllipticF(sin(2*arctan(b^(1/4)*x/a^(1/4))),1/2*2^(1/2))*(a ^(1/2)+x^2*b^(1/2))*(d-(-4*c*e+d^2)^(1/2))*((b*x^4+a)/(a^(1/2)+x^2*b^(1/2) )^2)^(1/2)/a^(1/4)/(-4*c*e+d^2)^(1/2)/(2*e^2*a^(1/2)+b^(1/2)*(d^2-2*c*e-d* (-4*c*e+d^2)^(1/2)))/(b*x^4+a)^(1/2)+1/2*e*(cos(2*arctan(b^(1/4)*x/a^(1/4) ))^2)^(1/2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticPi(sin(2*arctan(b^(1/ 4)*x/a^(1/4))),1/4*(2*e^2*a^(1/2)+b^(1/2)*(d^2-2*c*e-d*(-4*c*e+d^2)^(1/2)) )^2/e^2/a^(1/2)/b^(1/2)/(d-(-4*c*e+d^2)^(1/2))^2,1/2*2^(1/2))*(a^(1/2)+x^2 *b^(1/2))*(2*e^2*a^(1/2)-b^(1/2)*(d^2-2*c*e-d*(-4*c*e+d^2)^(1/2)))*((b*x^4 +a)/(a^(1/2)+x^2*b^(1/2))^2)^(1/2)/a^(1/4)/b^(1/4)/(d-(-4*c*e+d^2)^(1/2))/ (-4*c*e+d^2)^(1/2)/(2*e^2*a^(1/2)+b^(1/2)*(d^2-2*c*e-d*(-4*c*e+d^2)^(1/2)) )/(b*x^4+a)^(1/2)-1/2*b^(1/4)*e*(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)^(1/2) /cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(b^(1/4)*x/a^(1/4) )),1/2*2^(1/2))*(a^(1/2)+x^2*b^(1/2))*(d+(-4*c*e+d^2)^(1/2))*((b*x^4+a)/(a ^(1/2)+x^2*b^(1/2))^2)^(1/2)/a^(1/4)/(-4*c*e+d^2)^(1/2)/(2*e^2*a^(1/2)+b^( 1/2)*(d^2-2*c*e+d*(-4*c*e+d^2)^(1/2)))/(b*x^4+a)^(1/2)-1/2*e*(cos(2*arctan (b^(1/4)*x/a^(1/4)))^2)^(1/2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticPi( sin(2*arctan(b^(1/4)*x/a^(1/4))),1/4*(2*e^2*a^(1/2)+b^(1/2)*(d^2-2*c*e+d*( -4*c*e+d^2)^(1/2)))^2/e^2/a^(1/2)/b^(1/2)/(d+(-4*c*e+d^2)^(1/2))^2,1/2*2^( 1/2))*(a^(1/2)+x^2*b^(1/2))*(2*e^2*a^(1/2)-b^(1/2)*(d^2-2*c*e+d*(-4*c*e...
Result contains higher order function than in optimal. Order 9 vs. order 4 in optimal.
Time = 11.25 (sec) , antiderivative size = 455, normalized size of antiderivative = 0.28 \[ \int \frac {1}{\left (c+d x+e x^2\right ) \sqrt {a+b x^4}} \, dx=-\frac {i \sqrt {1+\frac {b x^4}{a}} \left (\left (-d^2+\sqrt {d^4-4 c d^2 e}\right ) \operatorname {EllipticPi}\left (\frac {2 i \sqrt {a} e^2}{\sqrt {b} \left (d^2-2 c e+\sqrt {d^4-4 c d^2 e}\right )},i \text {arcsinh}\left (\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}} x\right ),-1\right )+\left (d^2+\sqrt {d^4-4 c d^2 e}\right ) \operatorname {EllipticPi}\left (-\frac {2 i \sqrt {a} e^2}{\sqrt {b} \left (-d^2+2 c e+\sqrt {d^4-4 c d^2 e}\right )},i \text {arcsinh}\left (\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}} x\right ),-1\right )\right )}{2 \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}} c \sqrt {d^4-4 c d^2 e} \sqrt {a+b x^4}}+\sqrt {b} d \text {RootSum}\left [a^2 e^2-2 a \sqrt {b} d^2 \text {$\#$1}+4 a \sqrt {b} c e \text {$\#$1}+4 b c^2 \text {$\#$1}^2-2 a e^2 \text {$\#$1}^2+2 \sqrt {b} d^2 \text {$\#$1}^3-4 \sqrt {b} c e \text {$\#$1}^3+e^2 \text {$\#$1}^4\&,\frac {\log \left (-\sqrt {b} x^2+\sqrt {a+b x^4}-\text {$\#$1}\right ) \text {$\#$1}}{-a \sqrt {b} d^2+2 a \sqrt {b} c e+4 b c^2 \text {$\#$1}-2 a e^2 \text {$\#$1}+3 \sqrt {b} d^2 \text {$\#$1}^2-6 \sqrt {b} c e \text {$\#$1}^2+2 e^2 \text {$\#$1}^3}\&\right ] \]
((-1/2*I)*Sqrt[1 + (b*x^4)/a]*((-d^2 + Sqrt[d^4 - 4*c*d^2*e])*EllipticPi[( (2*I)*Sqrt[a]*e^2)/(Sqrt[b]*(d^2 - 2*c*e + Sqrt[d^4 - 4*c*d^2*e])), I*ArcS inh[Sqrt[(I*Sqrt[b])/Sqrt[a]]*x], -1] + (d^2 + Sqrt[d^4 - 4*c*d^2*e])*Elli pticPi[((-2*I)*Sqrt[a]*e^2)/(Sqrt[b]*(-d^2 + 2*c*e + Sqrt[d^4 - 4*c*d^2*e] )), I*ArcSinh[Sqrt[(I*Sqrt[b])/Sqrt[a]]*x], -1]))/(Sqrt[(I*Sqrt[b])/Sqrt[a ]]*c*Sqrt[d^4 - 4*c*d^2*e]*Sqrt[a + b*x^4]) + Sqrt[b]*d*RootSum[a^2*e^2 - 2*a*Sqrt[b]*d^2*#1 + 4*a*Sqrt[b]*c*e*#1 + 4*b*c^2*#1^2 - 2*a*e^2*#1^2 + 2* Sqrt[b]*d^2*#1^3 - 4*Sqrt[b]*c*e*#1^3 + e^2*#1^4 & , (Log[-(Sqrt[b]*x^2) + Sqrt[a + b*x^4] - #1]*#1)/(-(a*Sqrt[b]*d^2) + 2*a*Sqrt[b]*c*e + 4*b*c^2*# 1 - 2*a*e^2*#1 + 3*Sqrt[b]*d^2*#1^2 - 6*Sqrt[b]*c*e*#1^2 + 2*e^2*#1^3) & ]
Time = 8.07 (sec) , antiderivative size = 1590, normalized size of antiderivative = 0.99, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {7279, 2009}
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.
| \(\displaystyle \int \frac {1}{\sqrt {a+b x^4} \left (c+d x+e x^2\right )} \, dx\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle \int \left (\frac {2 e}{\sqrt {a+b x^4} \sqrt {d^2-4 c e} \left (-\sqrt {d^2-4 c e}+d+2 e x\right )}-\frac {2 e}{\sqrt {a+b x^4} \sqrt {d^2-4 c e} \left (\sqrt {d^2-4 c e}+d+2 e x\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\text {arctanh}\left (\frac {\sqrt {2} \sqrt {b d^4-4 b c e d^2-b \sqrt {d^2-4 c e} \left (d^2-2 c e\right ) d+2 a e^4+2 b c^2 e^2} x}{e \left (d-\sqrt {d^2-4 c e}\right ) \sqrt {b x^4+a}}\right ) e^2}{\sqrt {2} \sqrt {d^2-4 c e} \sqrt {2 a e^4+b \left (d^4-\sqrt {d^2-4 c e} d^3-4 c e d^2+2 c e \sqrt {d^2-4 c e} d+2 c^2 e^2\right )}}-\frac {\text {arctanh}\left (\frac {\sqrt {2} \sqrt {b d^4-4 b c e d^2+b \sqrt {d^2-4 c e} \left (d^2-2 c e\right ) d+2 a e^4+2 b c^2 e^2} x}{e \left (d+\sqrt {d^2-4 c e}\right ) \sqrt {b x^4+a}}\right ) e^2}{\sqrt {2} \sqrt {d^2-4 c e} \sqrt {2 a e^4+b \left (d^4+\sqrt {d^2-4 c e} d^3-4 c e d^2-2 c e \sqrt {d^2-4 c e} d+2 c^2 e^2\right )}}-\frac {\text {arctanh}\left (\frac {4 a e^2+b \left (d-\sqrt {d^2-4 c e}\right )^2 x^2}{2 \sqrt {2} \sqrt {b d^4-4 b c e d^2-b \sqrt {d^2-4 c e} \left (d^2-2 c e\right ) d+2 a e^4+2 b c^2 e^2} \sqrt {b x^4+a}}\right ) e^2}{\sqrt {2} \sqrt {d^2-4 c e} \sqrt {b d^4-4 b c e d^2-b \sqrt {d^2-4 c e} \left (d^2-2 c e\right ) d+2 a e^4+2 b c^2 e^2}}+\frac {\text {arctanh}\left (\frac {4 a e^2+b \left (d+\sqrt {d^2-4 c e}\right )^2 x^2}{2 \sqrt {2} \sqrt {b d^4-4 b c e d^2+b \sqrt {d^2-4 c e} \left (d^2-2 c e\right ) d+2 a e^4+2 b c^2 e^2} \sqrt {b x^4+a}}\right ) e^2}{\sqrt {2} \sqrt {d^2-4 c e} \sqrt {b d^4-4 b c e d^2+b \sqrt {d^2-4 c e} \left (d^2-2 c e\right ) d+2 a e^4+2 b c^2 e^2}}+\frac {\sqrt [4]{b} \left (d-\sqrt {d^2-4 c e}\right ) \left (\sqrt {b} x^2+\sqrt {a}\right ) \sqrt {\frac {b x^4+a}{\left (\sqrt {b} x^2+\sqrt {a}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right ) e}{2 \sqrt [4]{a} \sqrt {d^2-4 c e} \left (2 \sqrt {a} e^2+\sqrt {b} \left (d^2-\sqrt {d^2-4 c e} d-2 c e\right )\right ) \sqrt {b x^4+a}}-\frac {\sqrt [4]{b} \left (d+\sqrt {d^2-4 c e}\right ) \left (\sqrt {b} x^2+\sqrt {a}\right ) \sqrt {\frac {b x^4+a}{\left (\sqrt {b} x^2+\sqrt {a}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right ) e}{2 \sqrt [4]{a} \sqrt {d^2-4 c e} \left (2 \sqrt {a} e^2+\sqrt {b} \left (d^2+\sqrt {d^2-4 c e} d-2 c e\right )\right ) \sqrt {b x^4+a}}+\frac {\left (2 \sqrt {a} e^2-\sqrt {b} \left (d^2-\sqrt {d^2-4 c e} d-2 c e\right )\right ) \left (\sqrt {b} x^2+\sqrt {a}\right ) \sqrt {\frac {b x^4+a}{\left (\sqrt {b} x^2+\sqrt {a}\right )^2}} \operatorname {EllipticPi}\left (\frac {\sqrt {a} \left (4 e^2+\frac {\sqrt {b} \left (d-\sqrt {d^2-4 c e}\right )^2}{\sqrt {a}}\right )^2}{16 \sqrt {b} e^2 \left (d-\sqrt {d^2-4 c e}\right )^2},2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right ) e}{2 \sqrt [4]{a} \sqrt [4]{b} \sqrt {d^2-4 c e} \left (d-\sqrt {d^2-4 c e}\right ) \left (2 \sqrt {a} e^2+\sqrt {b} \left (d^2-\sqrt {d^2-4 c e} d-2 c e\right )\right ) \sqrt {b x^4+a}}-\frac {\left (2 \sqrt {a} e^2-\sqrt {b} \left (d^2+\sqrt {d^2-4 c e} d-2 c e\right )\right ) \left (\sqrt {b} x^2+\sqrt {a}\right ) \sqrt {\frac {b x^4+a}{\left (\sqrt {b} x^2+\sqrt {a}\right )^2}} \operatorname {EllipticPi}\left (\frac {\sqrt {a} \left (4 e^2+\frac {\sqrt {b} \left (d+\sqrt {d^2-4 c e}\right )^2}{\sqrt {a}}\right )^2}{16 \sqrt {b} e^2 \left (d+\sqrt {d^2-4 c e}\right )^2},2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right ) e}{2 \sqrt [4]{a} \sqrt [4]{b} \sqrt {d^2-4 c e} \left (d+\sqrt {d^2-4 c e}\right ) \left (2 \sqrt {a} e^2+\sqrt {b} \left (d^2+\sqrt {d^2-4 c e} d-2 c e\right )\right ) \sqrt {b x^4+a}}\) |
(e^2*ArcTanh[(Sqrt[2]*Sqrt[b*d^4 - 4*b*c*d^2*e + 2*b*c^2*e^2 + 2*a*e^4 - b *d*Sqrt[d^2 - 4*c*e]*(d^2 - 2*c*e)]*x)/(e*(d - Sqrt[d^2 - 4*c*e])*Sqrt[a + b*x^4])])/(Sqrt[2]*Sqrt[d^2 - 4*c*e]*Sqrt[2*a*e^4 + b*(d^4 - 4*c*d^2*e + 2*c^2*e^2 - d^3*Sqrt[d^2 - 4*c*e] + 2*c*d*e*Sqrt[d^2 - 4*c*e])]) - (e^2*Ar cTanh[(Sqrt[2]*Sqrt[b*d^4 - 4*b*c*d^2*e + 2*b*c^2*e^2 + 2*a*e^4 + b*d*Sqrt [d^2 - 4*c*e]*(d^2 - 2*c*e)]*x)/(e*(d + Sqrt[d^2 - 4*c*e])*Sqrt[a + b*x^4] )])/(Sqrt[2]*Sqrt[d^2 - 4*c*e]*Sqrt[2*a*e^4 + b*(d^4 - 4*c*d^2*e + 2*c^2*e ^2 + d^3*Sqrt[d^2 - 4*c*e] - 2*c*d*e*Sqrt[d^2 - 4*c*e])]) - (e^2*ArcTanh[( 4*a*e^2 + b*(d - Sqrt[d^2 - 4*c*e])^2*x^2)/(2*Sqrt[2]*Sqrt[b*d^4 - 4*b*c*d ^2*e + 2*b*c^2*e^2 + 2*a*e^4 - b*d*Sqrt[d^2 - 4*c*e]*(d^2 - 2*c*e)]*Sqrt[a + b*x^4])])/(Sqrt[2]*Sqrt[d^2 - 4*c*e]*Sqrt[b*d^4 - 4*b*c*d^2*e + 2*b*c^2 *e^2 + 2*a*e^4 - b*d*Sqrt[d^2 - 4*c*e]*(d^2 - 2*c*e)]) + (e^2*ArcTanh[(4*a *e^2 + b*(d + Sqrt[d^2 - 4*c*e])^2*x^2)/(2*Sqrt[2]*Sqrt[b*d^4 - 4*b*c*d^2* e + 2*b*c^2*e^2 + 2*a*e^4 + b*d*Sqrt[d^2 - 4*c*e]*(d^2 - 2*c*e)]*Sqrt[a + b*x^4])])/(Sqrt[2]*Sqrt[d^2 - 4*c*e]*Sqrt[b*d^4 - 4*b*c*d^2*e + 2*b*c^2*e^ 2 + 2*a*e^4 + b*d*Sqrt[d^2 - 4*c*e]*(d^2 - 2*c*e)]) + (b^(1/4)*e*(d - Sqrt [d^2 - 4*c*e])*(Sqrt[a] + Sqrt[b]*x^2)*Sqrt[(a + b*x^4)/(Sqrt[a] + Sqrt[b] *x^2)^2]*EllipticF[2*ArcTan[(b^(1/4)*x)/a^(1/4)], 1/2])/(2*a^(1/4)*Sqrt[d^ 2 - 4*c*e]*(2*Sqrt[a]*e^2 + Sqrt[b]*(d^2 - 2*c*e - d*Sqrt[d^2 - 4*c*e]))*S qrt[a + b*x^4]) - (b^(1/4)*e*(d + Sqrt[d^2 - 4*c*e])*(Sqrt[a] + Sqrt[b]...
3.3.26.3.1 Defintions of rubi rules used
Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[ {v = RationalFunctionExpand[u/(a + b*x^n + c*x^(2*n)), x]}, Int[v, x] /; Su mQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]
Result contains complex when optimal does not.
Time = 1.36 (sec) , antiderivative size = 1153, normalized size of antiderivative = 0.72
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(1153\) |
| elliptic | \(\text {Expression too large to display}\) | \(1153\) |
-1/2/(-4*c*e+d^2)^(1/2)/(1/2*b/e^4*d^4-1/2*b/e^4*d^3*(-4*c*e+d^2)^(1/2)-2* b/e^3*c*d^2+b/e^3*c*d*(-4*c*e+d^2)^(1/2)+b/e^2*c^2+a)^(1/2)*arctanh(1/2/(1 /2*b/e^4*d^4-1/2*b/e^4*d^3*(-4*c*e+d^2)^(1/2)-2*b/e^3*c*d^2+b/e^3*c*d*(-4* c*e+d^2)^(1/2)+b/e^2*c^2+a)^(1/2)/(b*x^4+a)^(1/2)*b*x^2/e^2*d^2-1/2/(1/2*b /e^4*d^4-1/2*b/e^4*d^3*(-4*c*e+d^2)^(1/2)-2*b/e^3*c*d^2+b/e^3*c*d*(-4*c*e+ d^2)^(1/2)+b/e^2*c^2+a)^(1/2)/(b*x^4+a)^(1/2)*b*x^2/e^2*d*(-4*c*e+d^2)^(1/ 2)-1/(1/2*b/e^4*d^4-1/2*b/e^4*d^3*(-4*c*e+d^2)^(1/2)-2*b/e^3*c*d^2+b/e^3*c *d*(-4*c*e+d^2)^(1/2)+b/e^2*c^2+a)^(1/2)/(b*x^4+a)^(1/2)*b*x^2/e*c+1/(1/2* b/e^4*d^4-1/2*b/e^4*d^3*(-4*c*e+d^2)^(1/2)-2*b/e^3*c*d^2+b/e^3*c*d*(-4*c*e +d^2)^(1/2)+b/e^2*c^2+a)^(1/2)/(b*x^4+a)^(1/2)*a)-2/(-4*c*e+d^2)^(1/2)/(I/ a^(1/2)*b^(1/2))^(1/2)*e/(-d+(-4*c*e+d^2)^(1/2))*(1-I/a^(1/2)*b^(1/2)*x^2) ^(1/2)*(1+I/a^(1/2)*b^(1/2)*x^2)^(1/2)/(b*x^4+a)^(1/2)*EllipticPi(x*(I/a^( 1/2)*b^(1/2))^(1/2),-4*I*a^(1/2)/b^(1/2)*e^2/(-d+(-4*c*e+d^2)^(1/2))^2,(-I /a^(1/2)*b^(1/2))^(1/2)/(I/a^(1/2)*b^(1/2))^(1/2))+1/2/(-4*c*e+d^2)^(1/2)/ (1/2*b/e^4*d^4+1/2*b/e^4*d^3*(-4*c*e+d^2)^(1/2)-2*b/e^3*c*d^2-b/e^3*c*d*(- 4*c*e+d^2)^(1/2)+b/e^2*c^2+a)^(1/2)*arctanh(1/2/(1/2*b/e^4*d^4+1/2*b/e^4*d ^3*(-4*c*e+d^2)^(1/2)-2*b/e^3*c*d^2-b/e^3*c*d*(-4*c*e+d^2)^(1/2)+b/e^2*c^2 +a)^(1/2)/(b*x^4+a)^(1/2)*b*x^2/e^2*d^2+1/2/(1/2*b/e^4*d^4+1/2*b/e^4*d^3*( -4*c*e+d^2)^(1/2)-2*b/e^3*c*d^2-b/e^3*c*d*(-4*c*e+d^2)^(1/2)+b/e^2*c^2+a)^ (1/2)/(b*x^4+a)^(1/2)*b*x^2/e^2*d*(-4*c*e+d^2)^(1/2)-1/(1/2*b/e^4*d^4+1...
Timed out. \[ \int \frac {1}{\left (c+d x+e x^2\right ) \sqrt {a+b x^4}} \, dx=\text {Timed out} \]
\[ \int \frac {1}{\left (c+d x+e x^2\right ) \sqrt {a+b x^4}} \, dx=\int \frac {1}{\sqrt {a + b x^{4}} \left (c + d x + e x^{2}\right )}\, dx \]
\[ \int \frac {1}{\left (c+d x+e x^2\right ) \sqrt {a+b x^4}} \, dx=\int { \frac {1}{\sqrt {b x^{4} + a} {\left (e x^{2} + d x + c\right )}} \,d x } \]
\[ \int \frac {1}{\left (c+d x+e x^2\right ) \sqrt {a+b x^4}} \, dx=\int { \frac {1}{\sqrt {b x^{4} + a} {\left (e x^{2} + d x + c\right )}} \,d x } \]
Timed out. \[ \int \frac {1}{\left (c+d x+e x^2\right ) \sqrt {a+b x^4}} \, dx=\int \frac {1}{\sqrt {b\,x^4+a}\,\left (e\,x^2+d\,x+c\right )} \,d x \]