3.9.17 \(\int \frac {1}{(a+b x^4) \sqrt {c+d x^4}} \, dx\) [817]

Optimal. Leaf size=638 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {\frac {\sqrt {-a} \left (\frac {b c}{a}-d\right )}{\sqrt {b}}} x}{\sqrt {c+d x^4}}\right )}{4 a \sqrt {-\frac {b c-a d}{\sqrt {-a} \sqrt {b}}}}+\frac {\tan ^{-1}\left (\frac {\sqrt {\frac {b c-a d}{\sqrt {-a} \sqrt {b}}} x}{\sqrt {c+d x^4}}\right )}{4 a \sqrt {\frac {b c-a d}{\sqrt {-a} \sqrt {b}}}}+\frac {d^{3/4} \left (\sqrt {c}+\sqrt {d} x^2\right ) \sqrt {\frac {c+d x^4}{\left (\sqrt {c}+\sqrt {d} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{d} x}{\sqrt [4]{c}}\right )|\frac {1}{2}\right )}{2 \sqrt [4]{c} (b c+a d) \sqrt {c+d x^4}}+\frac {\left (\sqrt {b} \sqrt {c}+\sqrt {-a} \sqrt {d}\right ) \left (\sqrt {c}+\sqrt {d} x^2\right ) \sqrt {\frac {c+d x^4}{\left (\sqrt {c}+\sqrt {d} x^2\right )^2}} \Pi \left (-\frac {\left (\sqrt {b} \sqrt {c}-\sqrt {-a} \sqrt {d}\right )^2}{4 \sqrt {-a} \sqrt {b} \sqrt {c} \sqrt {d}};2 \tan ^{-1}\left (\frac {\sqrt [4]{d} x}{\sqrt [4]{c}}\right )|\frac {1}{2}\right )}{8 a \sqrt [4]{c} \left (\sqrt {b} \sqrt {c}-\sqrt {-a} \sqrt {d}\right ) \sqrt [4]{d} \sqrt {c+d x^4}}+\frac {\left (\sqrt {b} \sqrt {c}-\sqrt {-a} \sqrt {d}\right ) \left (\sqrt {c}+\sqrt {d} x^2\right ) \sqrt {\frac {c+d x^4}{\left (\sqrt {c}+\sqrt {d} x^2\right )^2}} \Pi \left (\frac {\left (\sqrt {b} \sqrt {c}+\sqrt {-a} \sqrt {d}\right )^2}{4 \sqrt {-a} \sqrt {b} \sqrt {c} \sqrt {d}};2 \tan ^{-1}\left (\frac {\sqrt [4]{d} x}{\sqrt [4]{c}}\right )|\frac {1}{2}\right )}{8 a \sqrt [4]{c} \left (\sqrt {b} \sqrt {c}+\sqrt {-a} \sqrt {d}\right ) \sqrt [4]{d} \sqrt {c+d x^4}} \]

[Out]

1/4*arctan(x*((b*c/a-d)*(-a)^(1/2)/b^(1/2))^(1/2)/(d*x^4+c)^(1/2))/a/((a*d-b*c)/(-a)^(1/2)/b^(1/2))^(1/2)+1/4*
arctan(x*((-a*d+b*c)/(-a)^(1/2)/b^(1/2))^(1/2)/(d*x^4+c)^(1/2))/a/((-a*d+b*c)/(-a)^(1/2)/b^(1/2))^(1/2)+1/2*d^
(3/4)*(cos(2*arctan(d^(1/4)*x/c^(1/4)))^2)^(1/2)/cos(2*arctan(d^(1/4)*x/c^(1/4)))*EllipticF(sin(2*arctan(d^(1/
4)*x/c^(1/4))),1/2*2^(1/2))*(c^(1/2)+x^2*d^(1/2))*((d*x^4+c)/(c^(1/2)+x^2*d^(1/2))^2)^(1/2)/c^(1/4)/(a*d+b*c)/
(d*x^4+c)^(1/2)+1/8*(cos(2*arctan(d^(1/4)*x/c^(1/4)))^2)^(1/2)/cos(2*arctan(d^(1/4)*x/c^(1/4)))*EllipticPi(sin
(2*arctan(d^(1/4)*x/c^(1/4))),1/4*(b^(1/2)*c^(1/2)+(-a)^(1/2)*d^(1/2))^2/(-a)^(1/2)/b^(1/2)/c^(1/2)/d^(1/2),1/
2*2^(1/2))*(c^(1/2)+x^2*d^(1/2))*(b^(1/2)*c^(1/2)-(-a)^(1/2)*d^(1/2))*((d*x^4+c)/(c^(1/2)+x^2*d^(1/2))^2)^(1/2
)/a/c^(1/4)/d^(1/4)/(b^(1/2)*c^(1/2)+(-a)^(1/2)*d^(1/2))/(d*x^4+c)^(1/2)+1/8*(cos(2*arctan(d^(1/4)*x/c^(1/4)))
^2)^(1/2)/cos(2*arctan(d^(1/4)*x/c^(1/4)))*EllipticPi(sin(2*arctan(d^(1/4)*x/c^(1/4))),-1/4*(b^(1/2)*c^(1/2)-(
-a)^(1/2)*d^(1/2))^2/(-a)^(1/2)/b^(1/2)/c^(1/2)/d^(1/2),1/2*2^(1/2))*(c^(1/2)+x^2*d^(1/2))*(b^(1/2)*c^(1/2)+(-
a)^(1/2)*d^(1/2))*((d*x^4+c)/(c^(1/2)+x^2*d^(1/2))^2)^(1/2)/a/c^(1/4)/d^(1/4)/(b^(1/2)*c^(1/2)-(-a)^(1/2)*d^(1
/2))/(d*x^4+c)^(1/2)

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Rubi [A]
time = 0.50, antiderivative size = 742, normalized size of antiderivative = 1.16, number of steps used = 7, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {418, 1231, 226, 1721} \begin {gather*} -\frac {\sqrt [4]{b} \text {ArcTan}\left (\frac {x \sqrt {b c-a d}}{\sqrt [4]{-a} \sqrt [4]{b} \sqrt {c+d x^4}}\right )}{4 (-a)^{3/4} \sqrt {b c-a d}}-\frac {\sqrt [4]{b} \text {ArcTan}\left (\frac {x \sqrt {a d-b c}}{\sqrt [4]{-a} \sqrt [4]{b} \sqrt {c+d x^4}}\right )}{4 (-a)^{3/4} \sqrt {a d-b c}}+\frac {\sqrt [4]{d} \left (\sqrt {c}+\sqrt {d} x^2\right ) \sqrt {\frac {c+d x^4}{\left (\sqrt {c}+\sqrt {d} x^2\right )^2}} \left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {-a}}+\sqrt {d}\right ) F\left (2 \text {ArcTan}\left (\frac {\sqrt [4]{d} x}{\sqrt [4]{c}}\right )|\frac {1}{2}\right )}{4 \sqrt [4]{c} \sqrt {c+d x^4} (a d+b c)}+\frac {\sqrt [4]{d} \left (\sqrt {c}+\sqrt {d} x^2\right ) \sqrt {\frac {c+d x^4}{\left (\sqrt {c}+\sqrt {d} x^2\right )^2}} \left (\sqrt {-a} \sqrt {b} \sqrt {c}+a \sqrt {d}\right ) F\left (2 \text {ArcTan}\left (\frac {\sqrt [4]{d} x}{\sqrt [4]{c}}\right )|\frac {1}{2}\right )}{4 a \sqrt [4]{c} \sqrt {c+d x^4} (a d+b c)}+\frac {\left (\sqrt {c}+\sqrt {d} x^2\right ) \sqrt {\frac {c+d x^4}{\left (\sqrt {c}+\sqrt {d} x^2\right )^2}} \left (\sqrt {b} \sqrt {c}-\sqrt {-a} \sqrt {d}\right )^2 \Pi \left (\frac {\left (\sqrt {b} \sqrt {c}+\sqrt {-a} \sqrt {d}\right )^2}{4 \sqrt {-a} \sqrt {b} \sqrt {c} \sqrt {d}};2 \text {ArcTan}\left (\frac {\sqrt [4]{d} x}{\sqrt [4]{c}}\right )|\frac {1}{2}\right )}{8 a \sqrt [4]{c} \sqrt [4]{d} \sqrt {c+d x^4} (a d+b c)}+\frac {\left (\sqrt {c}+\sqrt {d} x^2\right ) \sqrt {\frac {c+d x^4}{\left (\sqrt {c}+\sqrt {d} x^2\right )^2}} \left (\sqrt {-a} \sqrt {d}+\sqrt {b} \sqrt {c}\right )^2 \Pi \left (-\frac {\left (\sqrt {b} \sqrt {c}-\sqrt {-a} \sqrt {d}\right )^2}{4 \sqrt {-a} \sqrt {b} \sqrt {c} \sqrt {d}};2 \text {ArcTan}\left (\frac {\sqrt [4]{d} x}{\sqrt [4]{c}}\right )|\frac {1}{2}\right )}{8 a \sqrt [4]{c} \sqrt [4]{d} \sqrt {c+d x^4} (a d+b c)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((a + b*x^4)*Sqrt[c + d*x^4]),x]

[Out]

-1/4*(b^(1/4)*ArcTan[(Sqrt[b*c - a*d]*x)/((-a)^(1/4)*b^(1/4)*Sqrt[c + d*x^4])])/((-a)^(3/4)*Sqrt[b*c - a*d]) -
(b^(1/4)*ArcTan[(Sqrt[-(b*c) + a*d]*x)/((-a)^(1/4)*b^(1/4)*Sqrt[c + d*x^4])])/(4*(-a)^(3/4)*Sqrt[-(b*c) + a*d
]) + (((Sqrt[b]*Sqrt[c])/Sqrt[-a] + Sqrt[d])*d^(1/4)*(Sqrt[c] + Sqrt[d]*x^2)*Sqrt[(c + d*x^4)/(Sqrt[c] + Sqrt[
d]*x^2)^2]*EllipticF[2*ArcTan[(d^(1/4)*x)/c^(1/4)], 1/2])/(4*c^(1/4)*(b*c + a*d)*Sqrt[c + d*x^4]) + ((Sqrt[-a]
*Sqrt[b]*Sqrt[c] + a*Sqrt[d])*d^(1/4)*(Sqrt[c] + Sqrt[d]*x^2)*Sqrt[(c + d*x^4)/(Sqrt[c] + Sqrt[d]*x^2)^2]*Elli
pticF[2*ArcTan[(d^(1/4)*x)/c^(1/4)], 1/2])/(4*a*c^(1/4)*(b*c + a*d)*Sqrt[c + d*x^4]) + ((Sqrt[b]*Sqrt[c] + Sqr
t[-a]*Sqrt[d])^2*(Sqrt[c] + Sqrt[d]*x^2)*Sqrt[(c + d*x^4)/(Sqrt[c] + Sqrt[d]*x^2)^2]*EllipticPi[-1/4*(Sqrt[b]*
Sqrt[c] - Sqrt[-a]*Sqrt[d])^2/(Sqrt[-a]*Sqrt[b]*Sqrt[c]*Sqrt[d]), 2*ArcTan[(d^(1/4)*x)/c^(1/4)], 1/2])/(8*a*c^
(1/4)*d^(1/4)*(b*c + a*d)*Sqrt[c + d*x^4]) + ((Sqrt[b]*Sqrt[c] - Sqrt[-a]*Sqrt[d])^2*(Sqrt[c] + Sqrt[d]*x^2)*S
qrt[(c + d*x^4)/(Sqrt[c] + Sqrt[d]*x^2)^2]*EllipticPi[(Sqrt[b]*Sqrt[c] + Sqrt[-a]*Sqrt[d])^2/(4*Sqrt[-a]*Sqrt[
b]*Sqrt[c]*Sqrt[d]), 2*ArcTan[(d^(1/4)*x)/c^(1/4)], 1/2])/(8*a*c^(1/4)*d^(1/4)*(b*c + a*d)*Sqrt[c + d*x^4])

Rule 226

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]

Rule 418

Int[1/(Sqrt[(a_) + (b_.)*(x_)^4]*((c_) + (d_.)*(x_)^4)), x_Symbol] :> Dist[1/(2*c), Int[1/(Sqrt[a + b*x^4]*(1
- Rt[-d/c, 2]*x^2)), x], x] + Dist[1/(2*c), Int[1/(Sqrt[a + b*x^4]*(1 + Rt[-d/c, 2]*x^2)), x], x] /; FreeQ[{a,
b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 1231

Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> With[{q = Rt[c/a, 2]}, Dist[(c*d + a*e*q
)/(c*d^2 - a*e^2), Int[1/Sqrt[a + c*x^4], x], x] - Dist[(a*e*(e + d*q))/(c*d^2 - a*e^2), Int[(1 + q*x^2)/((d +
e*x^2)*Sqrt[a + c*x^4]), x], x]] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && NeQ[c*d^2 - a*e^2, 0]
&& PosQ[c/a]

Rule 1721

Int[((A_) + (B_.)*(x_)^2)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> With[{q = Rt[B/A, 2]
}, Simp[(-(B*d - A*e))*(ArcTan[Rt[c*(d/e) + a*(e/d), 2]*(x/Sqrt[a + c*x^4])]/(2*d*e*Rt[c*(d/e) + a*(e/d), 2]))
, x] + Simp[(B*d + A*e)*(A + B*x^2)*(Sqrt[A^2*((a + c*x^4)/(a*(A + B*x^2)^2))]/(4*d*e*A*q*Sqrt[a + c*x^4]))*El
lipticPi[Cancel[-(B*d - A*e)^2/(4*d*e*A*B)], 2*ArcTan[q*x], 1/2], x]] /; FreeQ[{a, c, d, e, A, B}, x] && NeQ[c
*d^2 + a*e^2, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[c/a] && EqQ[c*A^2 - a*B^2, 0]

Rubi steps

\begin {align*} \int \frac {1}{\left (a+b x^4\right ) \sqrt {c+d x^4}} \, dx &=\frac {\int \frac {1}{\left (1-\frac {\sqrt {b} x^2}{\sqrt {-a}}\right ) \sqrt {c+d x^4}} \, dx}{2 a}+\frac {\int \frac {1}{\left (1+\frac {\sqrt {b} x^2}{\sqrt {-a}}\right ) \sqrt {c+d x^4}} \, dx}{2 a}\\ &=\frac {\left (\sqrt {b} \sqrt {c} \left (\sqrt {b} \sqrt {c}-\sqrt {-a} \sqrt {d}\right )\right ) \int \frac {1+\frac {\sqrt {d} x^2}{\sqrt {c}}}{\left (1-\frac {\sqrt {b} x^2}{\sqrt {-a}}\right ) \sqrt {c+d x^4}} \, dx}{2 a (b c+a d)}+\frac {\left (\sqrt {b} \sqrt {c} \left (\sqrt {b} \sqrt {c}+\sqrt {-a} \sqrt {d}\right )\right ) \int \frac {1+\frac {\sqrt {d} x^2}{\sqrt {c}}}{\left (1+\frac {\sqrt {b} x^2}{\sqrt {-a}}\right ) \sqrt {c+d x^4}} \, dx}{2 a (b c+a d)}+\frac {\left (\left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {-a}}+\sqrt {d}\right ) \sqrt {d}\right ) \int \frac {1}{\sqrt {c+d x^4}} \, dx}{2 (b c+a d)}+\frac {\left (\left (\sqrt {-a} \sqrt {b} \sqrt {c}+a \sqrt {d}\right ) \sqrt {d}\right ) \int \frac {1}{\sqrt {c+d x^4}} \, dx}{2 a (b c+a d)}\\ &=-\frac {\sqrt [4]{b} \tan ^{-1}\left (\frac {\sqrt {b c-a d} x}{\sqrt [4]{-a} \sqrt [4]{b} \sqrt {c+d x^4}}\right )}{4 (-a)^{3/4} \sqrt {b c-a d}}-\frac {\sqrt [4]{b} \tan ^{-1}\left (\frac {\sqrt {-b c+a d} x}{\sqrt [4]{-a} \sqrt [4]{b} \sqrt {c+d x^4}}\right )}{4 (-a)^{3/4} \sqrt {-b c+a d}}+\frac {\left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {-a}}+\sqrt {d}\right ) \sqrt [4]{d} \left (\sqrt {c}+\sqrt {d} x^2\right ) \sqrt {\frac {c+d x^4}{\left (\sqrt {c}+\sqrt {d} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{d} x}{\sqrt [4]{c}}\right )|\frac {1}{2}\right )}{4 \sqrt [4]{c} (b c+a d) \sqrt {c+d x^4}}+\frac {\left (\sqrt {-a} \sqrt {b} \sqrt {c}+a \sqrt {d}\right ) \sqrt [4]{d} \left (\sqrt {c}+\sqrt {d} x^2\right ) \sqrt {\frac {c+d x^4}{\left (\sqrt {c}+\sqrt {d} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{d} x}{\sqrt [4]{c}}\right )|\frac {1}{2}\right )}{4 a \sqrt [4]{c} (b c+a d) \sqrt {c+d x^4}}+\frac {\left (\sqrt {b} \sqrt {c}+\sqrt {-a} \sqrt {d}\right )^2 \left (\sqrt {c}+\sqrt {d} x^2\right ) \sqrt {\frac {c+d x^4}{\left (\sqrt {c}+\sqrt {d} x^2\right )^2}} \Pi \left (-\frac {\left (\sqrt {b} \sqrt {c}-\sqrt {-a} \sqrt {d}\right )^2}{4 \sqrt {-a} \sqrt {b} \sqrt {c} \sqrt {d}};2 \tan ^{-1}\left (\frac {\sqrt [4]{d} x}{\sqrt [4]{c}}\right )|\frac {1}{2}\right )}{8 a \sqrt [4]{c} \sqrt [4]{d} (b c+a d) \sqrt {c+d x^4}}+\frac {\left (\sqrt {b} \sqrt {c}-\sqrt {-a} \sqrt {d}\right )^2 \left (\sqrt {c}+\sqrt {d} x^2\right ) \sqrt {\frac {c+d x^4}{\left (\sqrt {c}+\sqrt {d} x^2\right )^2}} \Pi \left (\frac {\left (\sqrt {b} \sqrt {c}+\sqrt {-a} \sqrt {d}\right )^2}{4 \sqrt {-a} \sqrt {b} \sqrt {c} \sqrt {d}};2 \tan ^{-1}\left (\frac {\sqrt [4]{d} x}{\sqrt [4]{c}}\right )|\frac {1}{2}\right )}{8 a \sqrt [4]{c} \sqrt [4]{d} (b c+a d) \sqrt {c+d x^4}}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
time = 10.04, size = 161, normalized size = 0.25 \begin {gather*} -\frac {5 a c x F_1\left (\frac {1}{4};\frac {1}{2},1;\frac {5}{4};-\frac {d x^4}{c},-\frac {b x^4}{a}\right )}{\left (a+b x^4\right ) \sqrt {c+d x^4} \left (-5 a c F_1\left (\frac {1}{4};\frac {1}{2},1;\frac {5}{4};-\frac {d x^4}{c},-\frac {b x^4}{a}\right )+2 x^4 \left (2 b c F_1\left (\frac {5}{4};\frac {1}{2},2;\frac {9}{4};-\frac {d x^4}{c},-\frac {b x^4}{a}\right )+a d F_1\left (\frac {5}{4};\frac {3}{2},1;\frac {9}{4};-\frac {d x^4}{c},-\frac {b x^4}{a}\right )\right )\right )} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[1/((a + b*x^4)*Sqrt[c + d*x^4]),x]

[Out]

(-5*a*c*x*AppellF1[1/4, 1/2, 1, 5/4, -((d*x^4)/c), -((b*x^4)/a)])/((a + b*x^4)*Sqrt[c + d*x^4]*(-5*a*c*AppellF
1[1/4, 1/2, 1, 5/4, -((d*x^4)/c), -((b*x^4)/a)] + 2*x^4*(2*b*c*AppellF1[5/4, 1/2, 2, 9/4, -((d*x^4)/c), -((b*x
^4)/a)] + a*d*AppellF1[5/4, 3/2, 1, 9/4, -((d*x^4)/c), -((b*x^4)/a)])))

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.34, size = 191, normalized size = 0.30

method	result	size
default	\(\frac {\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{4} b +a \right )}{\sum }\frac {-\frac {\arctanh \left (\frac {2 d \,x^{2} \underline {\hspace {1.25 ex}}\alpha ^{2}+2 c}{2 \sqrt {\frac {-a d +b c}{b}}\, \sqrt {d \,x^{4}+c}}\right )}{\sqrt {\frac {-a d +b c}{b}}}+\frac {2 \underline {\hspace {1.25 ex}}\alpha ^{3} b \sqrt {1-\frac {i \sqrt {d}\, x^{2}}{\sqrt {c}}}\, \sqrt {1+\frac {i \sqrt {d}\, x^{2}}{\sqrt {c}}}\, \EllipticPi \left (x \sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}, \frac {i \sqrt {c}\, \underline {\hspace {1.25 ex}}\alpha ^{2} b}{\sqrt {d}\, a}, \frac {\sqrt {-\frac {i \sqrt {d}}{\sqrt {c}}}}{\sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}}\right )}{\sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}\, a \sqrt {d \,x^{4}+c}}}{\underline {\hspace {1.25 ex}}\alpha ^{3}}}{8 b}\)	\(191\)
elliptic	\(\frac {\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{4} b +a \right )}{\sum }\frac {-\frac {\arctanh \left (\frac {2 d \,x^{2} \underline {\hspace {1.25 ex}}\alpha ^{2}+2 c}{2 \sqrt {\frac {-a d +b c}{b}}\, \sqrt {d \,x^{4}+c}}\right )}{\sqrt {\frac {-a d +b c}{b}}}+\frac {2 \underline {\hspace {1.25 ex}}\alpha ^{3} b \sqrt {1-\frac {i \sqrt {d}\, x^{2}}{\sqrt {c}}}\, \sqrt {1+\frac {i \sqrt {d}\, x^{2}}{\sqrt {c}}}\, \EllipticPi \left (x \sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}, \frac {i \sqrt {c}\, \underline {\hspace {1.25 ex}}\alpha ^{2} b}{\sqrt {d}\, a}, \frac {\sqrt {-\frac {i \sqrt {d}}{\sqrt {c}}}}{\sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}}\right )}{\sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}\, a \sqrt {d \,x^{4}+c}}}{\underline {\hspace {1.25 ex}}\alpha ^{3}}}{8 b}\)	\(191\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b*x^4+a)/(d*x^4+c)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/8/b*sum(1/_alpha^3*(-1/((-a*d+b*c)/b)^(1/2)*arctanh(1/2*(2*_alpha^2*d*x^2+2*c)/((-a*d+b*c)/b)^(1/2)/(d*x^4+c
)^(1/2))+2/(I/c^(1/2)*d^(1/2))^(1/2)*_alpha^3*b/a*(1-I/c^(1/2)*d^(1/2)*x^2)^(1/2)*(1+I/c^(1/2)*d^(1/2)*x^2)^(1
/2)/(d*x^4+c)^(1/2)*EllipticPi(x*(I/c^(1/2)*d^(1/2))^(1/2),I*c^(1/2)/d^(1/2)*_alpha^2/a*b,(-I/c^(1/2)*d^(1/2))
^(1/2)/(I/c^(1/2)*d^(1/2))^(1/2))),_alpha=RootOf(_Z^4*b+a))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x^4+a)/(d*x^4+c)^(1/2),x, algorithm="maxima")

[Out]

integrate(1/((b*x^4 + a)*sqrt(d*x^4 + c)), x)

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Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x^4+a)/(d*x^4+c)^(1/2),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (a + b x^{4}\right ) \sqrt {c + d x^{4}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x**4+a)/(d*x**4+c)**(1/2),x)

[Out]

Integral(1/((a + b*x**4)*sqrt(c + d*x**4)), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x^4+a)/(d*x^4+c)^(1/2),x, algorithm="giac")

[Out]

integrate(1/((b*x^4 + a)*sqrt(d*x^4 + c)), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {1}{\left (b\,x^4+a\right )\,\sqrt {d\,x^4+c}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((a + b*x^4)*(c + d*x^4)^(1/2)),x)

[Out]

int(1/((a + b*x^4)*(c + d*x^4)^(1/2)), x)

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