\(\int \frac {1}{\sqrt {x} (a+c x^4)} \, dx\) [743]

Optimal result

Integrand size = 15, antiderivative size = 287 \[ \int \frac {1}{\sqrt {x} \left (a+c x^4\right )} \, dx=\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 \sqrt {2} (-a)^{7/8} \sqrt [8]{c}}-\frac {\arctan \left (1+\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 \sqrt {2} (-a)^{7/8} \sqrt [8]{c}}-\frac {\arctan \left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{7/8} \sqrt [8]{c}}-\frac {\text {arctanh}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{7/8} \sqrt [8]{c}}+\frac {\log \left (\sqrt [4]{-a}-\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{c} x\right )}{4 \sqrt {2} (-a)^{7/8} \sqrt [8]{c}}-\frac {\log \left (\sqrt [4]{-a}+\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{c} x\right )}{4 \sqrt {2} (-a)^{7/8} \sqrt [8]{c}} \]

[Out]

-1/2*arctan(c^(1/8)*x^(1/2)/(-a)^(1/8))/(-a)^(7/8)/c^(1/8)-1/2*arctanh(c^(1/8)*x^(1/2)/(-a)^(1/8))/(-a)^(7/8)/
c^(1/8)-1/4*arctan(-1+c^(1/8)*2^(1/2)*x^(1/2)/(-a)^(1/8))/(-a)^(7/8)/c^(1/8)*2^(1/2)-1/4*arctan(1+c^(1/8)*2^(1
/2)*x^(1/2)/(-a)^(1/8))/(-a)^(7/8)/c^(1/8)*2^(1/2)+1/8*ln((-a)^(1/4)+c^(1/4)*x-(-a)^(1/8)*c^(1/8)*2^(1/2)*x^(1
/2))/(-a)^(7/8)/c^(1/8)*2^(1/2)-1/8*ln((-a)^(1/4)+c^(1/4)*x+(-a)^(1/8)*c^(1/8)*2^(1/2)*x^(1/2))/(-a)^(7/8)/c^(
1/8)*2^(1/2)

Rubi [A] (verified)

Time = 0.17 (sec) , antiderivative size = 287, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.733, Rules used = {335, 220, 218, 214, 211, 217, 1179, 642, 1176, 631, 210} \[ \int \frac {1}{\sqrt {x} \left (a+c x^4\right )} \, dx=\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 \sqrt {2} (-a)^{7/8} \sqrt [8]{c}}-\frac {\arctan \left (\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}+1\right )}{2 \sqrt {2} (-a)^{7/8} \sqrt [8]{c}}-\frac {\arctan \left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{7/8} \sqrt [8]{c}}-\frac {\text {arctanh}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{7/8} \sqrt [8]{c}}+\frac {\log \left (-\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{4 \sqrt {2} (-a)^{7/8} \sqrt [8]{c}}-\frac {\log \left (\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{4 \sqrt {2} (-a)^{7/8} \sqrt [8]{c}} \]

[In]

Int[1/(Sqrt[x]*(a + c*x^4)),x]

[Out]

ArcTan[1 - (Sqrt[2]*c^(1/8)*Sqrt[x])/(-a)^(1/8)]/(2*Sqrt[2]*(-a)^(7/8)*c^(1/8)) - ArcTan[1 + (Sqrt[2]*c^(1/8)*
Sqrt[x])/(-a)^(1/8)]/(2*Sqrt[2]*(-a)^(7/8)*c^(1/8)) - ArcTan[(c^(1/8)*Sqrt[x])/(-a)^(1/8)]/(2*(-a)^(7/8)*c^(1/
8)) - ArcTanh[(c^(1/8)*Sqrt[x])/(-a)^(1/8)]/(2*(-a)^(7/8)*c^(1/8)) + Log[(-a)^(1/4) - Sqrt[2]*(-a)^(1/8)*c^(1/
8)*Sqrt[x] + c^(1/4)*x]/(4*Sqrt[2]*(-a)^(7/8)*c^(1/8)) - Log[(-a)^(1/4) + Sqrt[2]*(-a)^(1/8)*c^(1/8)*Sqrt[x] +
c^(1/4)*x]/(4*Sqrt[2]*(-a)^(7/8)*c^(1/8))

Rule 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 211

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/Rt[a/b, 2]], x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 214

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], x] /; FreeQ[{a, b},
x] && NegQ[a/b]

Rule 217

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]}, Di
st[1/(2*r), Int[(r - s*x^2)/(a + b*x^4), x], x] + Dist[1/(2*r), Int[(r + s*x^2)/(a + b*x^4), x], x]] /; FreeQ[
{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] && AtomQ[SplitProduct[SumBaseQ, b
]]))

Rule 218

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]},
Dist[r/(2*a), Int[1/(r - s*x^2), x], x] + Dist[r/(2*a), Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] &&  !Gt
Q[a/b, 0]

Rule 220

Int[((a_) + (b_.)*(x_)^(n_))^(-1), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]
}, Dist[r/(2*a), Int[1/(r - s*x^(n/2)), x], x] + Dist[r/(2*a), Int[1/(r + s*x^(n/2)), x], x]] /; FreeQ[{a, b},
x] && IGtQ[n/4, 1] &&  !GtQ[a/b, 0]

Rule 335

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/c, Subst[I
nt[x^(k*(m + 1) - 1)*(a + b*(x^(k*n)/c^n))^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0]
&& FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 631

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[a*(c/b^2)]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[d*(Log[RemoveContent[a + b*x +
c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 1176

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

Rule 1179

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

Rubi steps \begin{align*} \text {integral}& = 2 \text {Subst}\left (\int \frac {1}{a+c x^8} \, dx,x,\sqrt {x}\right ) \\ & = -\frac {\text {Subst}\left (\int \frac {1}{\sqrt {-a}-\sqrt {c} x^4} \, dx,x,\sqrt {x}\right )}{\sqrt {-a}}-\frac {\text {Subst}\left (\int \frac {1}{\sqrt {-a}+\sqrt {c} x^4} \, dx,x,\sqrt {x}\right )}{\sqrt {-a}} \\ & = -\frac {\text {Subst}\left (\int \frac {1}{\sqrt [4]{-a}-\sqrt [4]{c} x^2} \, dx,x,\sqrt {x}\right )}{2 (-a)^{3/4}}-\frac {\text {Subst}\left (\int \frac {1}{\sqrt [4]{-a}+\sqrt [4]{c} x^2} \, dx,x,\sqrt {x}\right )}{2 (-a)^{3/4}}-\frac {\text {Subst}\left (\int \frac {\sqrt [4]{-a}-\sqrt [4]{c} x^2}{\sqrt {-a}+\sqrt {c} x^4} \, dx,x,\sqrt {x}\right )}{2 (-a)^{3/4}}-\frac {\text {Subst}\left (\int \frac {\sqrt [4]{-a}+\sqrt [4]{c} x^2}{\sqrt {-a}+\sqrt {c} x^4} \, dx,x,\sqrt {x}\right )}{2 (-a)^{3/4}} \\ & = -\frac {\tan ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{7/8} \sqrt [8]{c}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{7/8} \sqrt [8]{c}}-\frac {\text {Subst}\left (\int \frac {1}{\frac {\sqrt [4]{-a}}{\sqrt [4]{c}}-\frac {\sqrt {2} \sqrt [8]{-a} x}{\sqrt [8]{c}}+x^2} \, dx,x,\sqrt {x}\right )}{4 (-a)^{3/4} \sqrt [4]{c}}-\frac {\text {Subst}\left (\int \frac {1}{\frac {\sqrt [4]{-a}}{\sqrt [4]{c}}+\frac {\sqrt {2} \sqrt [8]{-a} x}{\sqrt [8]{c}}+x^2} \, dx,x,\sqrt {x}\right )}{4 (-a)^{3/4} \sqrt [4]{c}}+\frac {\text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [8]{-a}}{\sqrt [8]{c}}+2 x}{-\frac {\sqrt [4]{-a}}{\sqrt [4]{c}}-\frac {\sqrt {2} \sqrt [8]{-a} x}{\sqrt [8]{c}}-x^2} \, dx,x,\sqrt {x}\right )}{4 \sqrt {2} (-a)^{7/8} \sqrt [8]{c}}+\frac {\text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [8]{-a}}{\sqrt [8]{c}}-2 x}{-\frac {\sqrt [4]{-a}}{\sqrt [4]{c}}+\frac {\sqrt {2} \sqrt [8]{-a} x}{\sqrt [8]{c}}-x^2} \, dx,x,\sqrt {x}\right )}{4 \sqrt {2} (-a)^{7/8} \sqrt [8]{c}} \\ & = -\frac {\tan ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{7/8} \sqrt [8]{c}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{7/8} \sqrt [8]{c}}+\frac {\log \left (\sqrt [4]{-a}-\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{c} x\right )}{4 \sqrt {2} (-a)^{7/8} \sqrt [8]{c}}-\frac {\log \left (\sqrt [4]{-a}+\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{c} x\right )}{4 \sqrt {2} (-a)^{7/8} \sqrt [8]{c}}-\frac {\text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 \sqrt {2} (-a)^{7/8} \sqrt [8]{c}}+\frac {\text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 \sqrt {2} (-a)^{7/8} \sqrt [8]{c}} \\ & = \frac {\tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 \sqrt {2} (-a)^{7/8} \sqrt [8]{c}}-\frac {\tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 \sqrt {2} (-a)^{7/8} \sqrt [8]{c}}-\frac {\tan ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{7/8} \sqrt [8]{c}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{7/8} \sqrt [8]{c}}+\frac {\log \left (\sqrt [4]{-a}-\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{c} x\right )}{4 \sqrt {2} (-a)^{7/8} \sqrt [8]{c}}-\frac {\log \left (\sqrt [4]{-a}+\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{c} x\right )}{4 \sqrt {2} (-a)^{7/8} \sqrt [8]{c}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.58 (sec) , antiderivative size = 239, normalized size of antiderivative = 0.83 \[ \int \frac {1}{\sqrt {x} \left (a+c x^4\right )} \, dx=\frac {-\sqrt {2+\sqrt {2}} \arctan \left (\frac {\sqrt {1-\frac {1}{\sqrt {2}}} \left (\sqrt [4]{a}-\sqrt [4]{c} x\right )}{\sqrt [8]{a} \sqrt [8]{c} \sqrt {x}}\right )-\sqrt {2-\sqrt {2}} \arctan \left (\frac {\sqrt {1+\frac {1}{\sqrt {2}}} \left (\sqrt [4]{a}-\sqrt [4]{c} x\right )}{\sqrt [8]{a} \sqrt [8]{c} \sqrt {x}}\right )+\sqrt {2+\sqrt {2}} \text {arctanh}\left (\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{a} \sqrt [8]{c} \sqrt {x}}{\sqrt [4]{a}+\sqrt [4]{c} x}\right )+\sqrt {2-\sqrt {2}} \text {arctanh}\left (\frac {\sqrt [8]{a} \sqrt [8]{c} \sqrt {-\left (\left (-2+\sqrt {2}\right ) x\right )}}{\sqrt [4]{a}+\sqrt [4]{c} x}\right )}{4 a^{7/8} \sqrt [8]{c}} \]

[In]

Integrate[1/(Sqrt[x]*(a + c*x^4)),x]

[Out]

(-(Sqrt[2 + Sqrt[2]]*ArcTan[(Sqrt[1 - 1/Sqrt[2]]*(a^(1/4) - c^(1/4)*x))/(a^(1/8)*c^(1/8)*Sqrt[x])]) - Sqrt[2 -
Sqrt[2]]*ArcTan[(Sqrt[1 + 1/Sqrt[2]]*(a^(1/4) - c^(1/4)*x))/(a^(1/8)*c^(1/8)*Sqrt[x])] + Sqrt[2 + Sqrt[2]]*Ar
cTanh[(Sqrt[2 + Sqrt[2]]*a^(1/8)*c^(1/8)*Sqrt[x])/(a^(1/4) + c^(1/4)*x)] + Sqrt[2 - Sqrt[2]]*ArcTanh[(a^(1/8)*
c^(1/8)*Sqrt[-((-2 + Sqrt[2])*x)])/(a^(1/4) + c^(1/4)*x)])/(4*a^(7/8)*c^(1/8))

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 3.94 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.10

method	result	size
derivativedivides	\(\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (c \,\textit {\_Z}^{8}+a \right )}{\sum }\frac {\ln \left (\sqrt {x}-\textit {\_R} \right )}{\textit {\_R}^{7}}}{4 c}\)	\(29\)
default	\(\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (c \,\textit {\_Z}^{8}+a \right )}{\sum }\frac {\ln \left (\sqrt {x}-\textit {\_R} \right )}{\textit {\_R}^{7}}}{4 c}\)	\(29\)

[In]

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

[Out]

1/4/c*sum(1/_R^7*ln(x^(1/2)-_R),_R=RootOf(_Z^8*c+a))

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.28 (sec) , antiderivative size = 264, normalized size of antiderivative = 0.92 \[ \int \frac {1}{\sqrt {x} \left (a+c x^4\right )} \, dx=\left (\frac {1}{8} i + \frac {1}{8}\right ) \, \sqrt {2} \left (-\frac {1}{a^{7} c}\right )^{\frac {1}{8}} \log \left (\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {2} a \left (-\frac {1}{a^{7} c}\right )^{\frac {1}{8}} + \sqrt {x}\right ) - \left (\frac {1}{8} i - \frac {1}{8}\right ) \, \sqrt {2} \left (-\frac {1}{a^{7} c}\right )^{\frac {1}{8}} \log \left (-\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} a \left (-\frac {1}{a^{7} c}\right )^{\frac {1}{8}} + \sqrt {x}\right ) + \left (\frac {1}{8} i - \frac {1}{8}\right ) \, \sqrt {2} \left (-\frac {1}{a^{7} c}\right )^{\frac {1}{8}} \log \left (\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} a \left (-\frac {1}{a^{7} c}\right )^{\frac {1}{8}} + \sqrt {x}\right ) - \left (\frac {1}{8} i + \frac {1}{8}\right ) \, \sqrt {2} \left (-\frac {1}{a^{7} c}\right )^{\frac {1}{8}} \log \left (-\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {2} a \left (-\frac {1}{a^{7} c}\right )^{\frac {1}{8}} + \sqrt {x}\right ) + \frac {1}{4} \, \left (-\frac {1}{a^{7} c}\right )^{\frac {1}{8}} \log \left (a \left (-\frac {1}{a^{7} c}\right )^{\frac {1}{8}} + \sqrt {x}\right ) + \frac {1}{4} i \, \left (-\frac {1}{a^{7} c}\right )^{\frac {1}{8}} \log \left (i \, a \left (-\frac {1}{a^{7} c}\right )^{\frac {1}{8}} + \sqrt {x}\right ) - \frac {1}{4} i \, \left (-\frac {1}{a^{7} c}\right )^{\frac {1}{8}} \log \left (-i \, a \left (-\frac {1}{a^{7} c}\right )^{\frac {1}{8}} + \sqrt {x}\right ) - \frac {1}{4} \, \left (-\frac {1}{a^{7} c}\right )^{\frac {1}{8}} \log \left (-a \left (-\frac {1}{a^{7} c}\right )^{\frac {1}{8}} + \sqrt {x}\right ) \]

[In]

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

[Out]

(1/8*I + 1/8)*sqrt(2)*(-1/(a^7*c))^(1/8)*log((1/2*I + 1/2)*sqrt(2)*a*(-1/(a^7*c))^(1/8) + sqrt(x)) - (1/8*I -
1/8)*sqrt(2)*(-1/(a^7*c))^(1/8)*log(-(1/2*I - 1/2)*sqrt(2)*a*(-1/(a^7*c))^(1/8) + sqrt(x)) + (1/8*I - 1/8)*sqr
t(2)*(-1/(a^7*c))^(1/8)*log((1/2*I - 1/2)*sqrt(2)*a*(-1/(a^7*c))^(1/8) + sqrt(x)) - (1/8*I + 1/8)*sqrt(2)*(-1/
(a^7*c))^(1/8)*log(-(1/2*I + 1/2)*sqrt(2)*a*(-1/(a^7*c))^(1/8) + sqrt(x)) + 1/4*(-1/(a^7*c))^(1/8)*log(a*(-1/(
a^7*c))^(1/8) + sqrt(x)) + 1/4*I*(-1/(a^7*c))^(1/8)*log(I*a*(-1/(a^7*c))^(1/8) + sqrt(x)) - 1/4*I*(-1/(a^7*c))
^(1/8)*log(-I*a*(-1/(a^7*c))^(1/8) + sqrt(x)) - 1/4*(-1/(a^7*c))^(1/8)*log(-a*(-1/(a^7*c))^(1/8) + sqrt(x))

Sympy [A] (verification not implemented)

Time = 11.20 (sec) , antiderivative size = 289, normalized size of antiderivative = 1.01 \[ \int \frac {1}{\sqrt {x} \left (a+c x^4\right )} \, dx=\begin {cases} \frac {\tilde {\infty }}{x^{\frac {7}{2}}} & \text {for}\: a = 0 \wedge c = 0 \\- \frac {2}{7 c x^{\frac {7}{2}}} & \text {for}\: a = 0 \\\frac {2 \sqrt {x}}{a} & \text {for}\: c = 0 \\- \frac {\sqrt [8]{- \frac {a}{c}} \log {\left (\sqrt {x} - \sqrt [8]{- \frac {a}{c}} \right )}}{4 a} + \frac {\sqrt [8]{- \frac {a}{c}} \log {\left (\sqrt {x} + \sqrt [8]{- \frac {a}{c}} \right )}}{4 a} - \frac {\sqrt {2} \sqrt [8]{- \frac {a}{c}} \log {\left (- 4 \sqrt {2} \sqrt {x} \sqrt [8]{- \frac {a}{c}} + 4 x + 4 \sqrt [4]{- \frac {a}{c}} \right )}}{8 a} + \frac {\sqrt {2} \sqrt [8]{- \frac {a}{c}} \log {\left (4 \sqrt {2} \sqrt {x} \sqrt [8]{- \frac {a}{c}} + 4 x + 4 \sqrt [4]{- \frac {a}{c}} \right )}}{8 a} + \frac {\sqrt [8]{- \frac {a}{c}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [8]{- \frac {a}{c}}} \right )}}{2 a} + \frac {\sqrt {2} \sqrt [8]{- \frac {a}{c}} \operatorname {atan}{\left (\frac {\sqrt {2} \sqrt {x}}{\sqrt [8]{- \frac {a}{c}}} - 1 \right )}}{4 a} + \frac {\sqrt {2} \sqrt [8]{- \frac {a}{c}} \operatorname {atan}{\left (\frac {\sqrt {2} \sqrt {x}}{\sqrt [8]{- \frac {a}{c}}} + 1 \right )}}{4 a} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((zoo/x**(7/2), Eq(a, 0) & Eq(c, 0)), (-2/(7*c*x**(7/2)), Eq(a, 0)), (2*sqrt(x)/a, Eq(c, 0)), (-(-a/c
)**(1/8)*log(sqrt(x) - (-a/c)**(1/8))/(4*a) + (-a/c)**(1/8)*log(sqrt(x) + (-a/c)**(1/8))/(4*a) - sqrt(2)*(-a/c
)**(1/8)*log(-4*sqrt(2)*sqrt(x)*(-a/c)**(1/8) + 4*x + 4*(-a/c)**(1/4))/(8*a) + sqrt(2)*(-a/c)**(1/8)*log(4*sqr
t(2)*sqrt(x)*(-a/c)**(1/8) + 4*x + 4*(-a/c)**(1/4))/(8*a) + (-a/c)**(1/8)*atan(sqrt(x)/(-a/c)**(1/8))/(2*a) +
sqrt(2)*(-a/c)**(1/8)*atan(sqrt(2)*sqrt(x)/(-a/c)**(1/8) - 1)/(4*a) + sqrt(2)*(-a/c)**(1/8)*atan(sqrt(2)*sqrt(
x)/(-a/c)**(1/8) + 1)/(4*a), True))

Maxima [F]

\[ \int \frac {1}{\sqrt {x} \left (a+c x^4\right )} \, dx=\int { \frac {1}{{\left (c x^{4} + a\right )} \sqrt {x}} \,d x } \]

[In]

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

[Out]

-c*integrate(x^(7/2)/(a*c*x^4 + a^2), x) + 2*sqrt(x)/a

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 445 vs. \(2 (190) = 380\).

Time = 0.34 (sec) , antiderivative size = 445, normalized size of antiderivative = 1.55 \[ \int \frac {1}{\sqrt {x} \left (a+c x^4\right )} \, dx=\frac {\left (\frac {a}{c}\right )^{\frac {1}{8}} \arctan \left (\frac {\sqrt {-\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}} + 2 \, \sqrt {x}}{\sqrt {\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}}}\right )}{2 \, a \sqrt {-2 \, \sqrt {2} + 4}} + \frac {\left (\frac {a}{c}\right )^{\frac {1}{8}} \arctan \left (-\frac {\sqrt {-\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}} - 2 \, \sqrt {x}}{\sqrt {\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}}}\right )}{2 \, a \sqrt {-2 \, \sqrt {2} + 4}} + \frac {\left (\frac {a}{c}\right )^{\frac {1}{8}} \arctan \left (\frac {\sqrt {\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}} + 2 \, \sqrt {x}}{\sqrt {-\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}}}\right )}{2 \, a \sqrt {2 \, \sqrt {2} + 4}} + \frac {\left (\frac {a}{c}\right )^{\frac {1}{8}} \arctan \left (-\frac {\sqrt {\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}} - 2 \, \sqrt {x}}{\sqrt {-\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}}}\right )}{2 \, a \sqrt {2 \, \sqrt {2} + 4}} + \frac {\left (\frac {a}{c}\right )^{\frac {1}{8}} \log \left (\sqrt {x} \sqrt {\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}} + x + \left (\frac {a}{c}\right )^{\frac {1}{4}}\right )}{4 \, a \sqrt {-2 \, \sqrt {2} + 4}} - \frac {\left (\frac {a}{c}\right )^{\frac {1}{8}} \log \left (-\sqrt {x} \sqrt {\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}} + x + \left (\frac {a}{c}\right )^{\frac {1}{4}}\right )}{4 \, a \sqrt {-2 \, \sqrt {2} + 4}} + \frac {\left (\frac {a}{c}\right )^{\frac {1}{8}} \log \left (\sqrt {x} \sqrt {-\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}} + x + \left (\frac {a}{c}\right )^{\frac {1}{4}}\right )}{4 \, a \sqrt {2 \, \sqrt {2} + 4}} - \frac {\left (\frac {a}{c}\right )^{\frac {1}{8}} \log \left (-\sqrt {x} \sqrt {-\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}} + x + \left (\frac {a}{c}\right )^{\frac {1}{4}}\right )}{4 \, a \sqrt {2 \, \sqrt {2} + 4}} \]

[In]

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

[Out]

1/2*(a/c)^(1/8)*arctan((sqrt(-sqrt(2) + 2)*(a/c)^(1/8) + 2*sqrt(x))/(sqrt(sqrt(2) + 2)*(a/c)^(1/8)))/(a*sqrt(-
2*sqrt(2) + 4)) + 1/2*(a/c)^(1/8)*arctan(-(sqrt(-sqrt(2) + 2)*(a/c)^(1/8) - 2*sqrt(x))/(sqrt(sqrt(2) + 2)*(a/c
)^(1/8)))/(a*sqrt(-2*sqrt(2) + 4)) + 1/2*(a/c)^(1/8)*arctan((sqrt(sqrt(2) + 2)*(a/c)^(1/8) + 2*sqrt(x))/(sqrt(
-sqrt(2) + 2)*(a/c)^(1/8)))/(a*sqrt(2*sqrt(2) + 4)) + 1/2*(a/c)^(1/8)*arctan(-(sqrt(sqrt(2) + 2)*(a/c)^(1/8) -
2*sqrt(x))/(sqrt(-sqrt(2) + 2)*(a/c)^(1/8)))/(a*sqrt(2*sqrt(2) + 4)) + 1/4*(a/c)^(1/8)*log(sqrt(x)*sqrt(sqrt(
2) + 2)*(a/c)^(1/8) + x + (a/c)^(1/4))/(a*sqrt(-2*sqrt(2) + 4)) - 1/4*(a/c)^(1/8)*log(-sqrt(x)*sqrt(sqrt(2) +
2)*(a/c)^(1/8) + x + (a/c)^(1/4))/(a*sqrt(-2*sqrt(2) + 4)) + 1/4*(a/c)^(1/8)*log(sqrt(x)*sqrt(-sqrt(2) + 2)*(a
/c)^(1/8) + x + (a/c)^(1/4))/(a*sqrt(2*sqrt(2) + 4)) - 1/4*(a/c)^(1/8)*log(-sqrt(x)*sqrt(-sqrt(2) + 2)*(a/c)^(
1/8) + x + (a/c)^(1/4))/(a*sqrt(2*sqrt(2) + 4))

Mupad [B] (verification not implemented)

Time = 5.87 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.41 \[ \int \frac {1}{\sqrt {x} \left (a+c x^4\right )} \, dx=-\frac {\mathrm {atan}\left (\frac {c^{1/8}\,\sqrt {x}}{{\left (-a\right )}^{1/8}}\right )}{2\,{\left (-a\right )}^{7/8}\,c^{1/8}}+\frac {\mathrm {atan}\left (\frac {c^{1/8}\,\sqrt {x}\,1{}\mathrm {i}}{{\left (-a\right )}^{1/8}}\right )\,1{}\mathrm {i}}{2\,{\left (-a\right )}^{7/8}\,c^{1/8}}+\frac {\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,c^{1/8}\,\sqrt {x}\,\left (\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right )}{{\left (-a\right )}^{1/8}}\right )\,\left (-\frac {1}{4}-\frac {1}{4}{}\mathrm {i}\right )}{{\left (-a\right )}^{7/8}\,c^{1/8}}+\frac {\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,c^{1/8}\,\sqrt {x}\,\left (\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right )}{{\left (-a\right )}^{1/8}}\right )\,\left (-\frac {1}{4}+\frac {1}{4}{}\mathrm {i}\right )}{{\left (-a\right )}^{7/8}\,c^{1/8}} \]

[In]

int(1/(x^(1/2)*(a + c*x^4)),x)

[Out]

(atan((c^(1/8)*x^(1/2)*1i)/(-a)^(1/8))*1i)/(2*(-a)^(7/8)*c^(1/8)) - atan((c^(1/8)*x^(1/2))/(-a)^(1/8))/(2*(-a)
^(7/8)*c^(1/8)) - (2^(1/2)*atan((2^(1/2)*c^(1/8)*x^(1/2)*(1/2 - 1i/2))/(-a)^(1/8))*(1/4 + 1i/4))/((-a)^(7/8)*c
^(1/8)) - (2^(1/2)*atan((2^(1/2)*c^(1/8)*x^(1/2)*(1/2 + 1i/2))/(-a)^(1/8))*(1/4 - 1i/4))/((-a)^(7/8)*c^(1/8))