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

Optimal. Leaf size=287 \[ \frac{\log \left (-\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{4 \sqrt{2} (-a)^{7/8} \sqrt [8]{c}}-\frac{\log \left (\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{4 \sqrt{2} (-a)^{7/8} \sqrt [8]{c}}+\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{2 \sqrt{2} (-a)^{7/8} \sqrt [8]{c}}-\frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}+1\right )}{2 \sqrt{2} (-a)^{7/8} \sqrt [8]{c}}-\frac{\tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{7/8} \sqrt [8]{c}}-\frac{\tanh ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{7/8} \sqrt [8]{c}} \]

[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))

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Rubi [A]  time = 0.223721, antiderivative size = 287, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 11, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.733, Rules used = {329, 214, 212, 208, 205, 211, 1165, 628, 1162, 617, 204} \[ \frac{\log \left (-\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{4 \sqrt{2} (-a)^{7/8} \sqrt [8]{c}}-\frac{\log \left (\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{4 \sqrt{2} (-a)^{7/8} \sqrt [8]{c}}+\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{2 \sqrt{2} (-a)^{7/8} \sqrt [8]{c}}-\frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}+1\right )}{2 \sqrt{2} (-a)^{7/8} \sqrt [8]{c}}-\frac{\tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{7/8} \sqrt [8]{c}}-\frac{\tanh ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{7/8} \sqrt [8]{c}} \]

Antiderivative was successfully verified.

[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 329

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 214

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 212

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] &&
!GtQ[a/b, 0]

Rule 208

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

Rule 205

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

Rule 211

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 1165

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]

Rule 628

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 1162

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 617

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 204

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

Rubi steps

\begin{align*} \int \frac{1}{\sqrt{x} \left (a+c x^4\right )} \, dx &=2 \operatorname{Subst}\left (\int \frac{1}{a+c x^8} \, dx,x,\sqrt{x}\right )\\ &=-\frac{\operatorname{Subst}\left (\int \frac{1}{\sqrt{-a}-\sqrt{c} x^4} \, dx,x,\sqrt{x}\right )}{\sqrt{-a}}-\frac{\operatorname{Subst}\left (\int \frac{1}{\sqrt{-a}+\sqrt{c} x^4} \, dx,x,\sqrt{x}\right )}{\sqrt{-a}}\\ &=-\frac{\operatorname{Subst}\left (\int \frac{1}{\sqrt [4]{-a}-\sqrt [4]{c} x^2} \, dx,x,\sqrt{x}\right )}{2 (-a)^{3/4}}-\frac{\operatorname{Subst}\left (\int \frac{1}{\sqrt [4]{-a}+\sqrt [4]{c} x^2} \, dx,x,\sqrt{x}\right )}{2 (-a)^{3/4}}-\frac{\operatorname{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{\operatorname{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{\operatorname{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{\operatorname{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{\operatorname{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{\operatorname{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{\operatorname{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{\operatorname{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 [C]  time = 0.0055695, size = 27, normalized size = 0.09 \[ \frac{2 \sqrt{x} \, _2F_1\left (\frac{1}{8},1;\frac{9}{8};-\frac{c x^4}{a}\right )}{a} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[x]*(a + c*x^4)),x]

[Out]

(2*Sqrt[x]*Hypergeometric2F1[1/8, 1, 9/8, -((c*x^4)/a)])/a

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Maple [C]  time = 0.003, size = 29, normalized size = 0.1 \begin{align*}{\frac{1}{4\,c}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{8}c+a \right ) }{\frac{1}{{{\it \_R}}^{7}}\ln \left ( \sqrt{x}-{\it \_R} \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -c \int \frac{x^{\frac{7}{2}}}{a c x^{4} + a^{2}}\,{d x} + \frac{2 \, \sqrt{x}}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Fricas [B]  time = 1.95437, size = 1131, normalized size = 3.94 \begin{align*} \frac{1}{2} \, \sqrt{2} \left (-\frac{1}{a^{7} c}\right )^{\frac{1}{8}} \arctan \left (\sqrt{2} \sqrt{a^{2} \left (-\frac{1}{a^{7} c}\right )^{\frac{1}{4}} + \sqrt{2} a \sqrt{x} \left (-\frac{1}{a^{7} c}\right )^{\frac{1}{8}} + x} a^{6} c \left (-\frac{1}{a^{7} c}\right )^{\frac{7}{8}} - \sqrt{2} a^{6} c \sqrt{x} \left (-\frac{1}{a^{7} c}\right )^{\frac{7}{8}} + 1\right ) + \frac{1}{2} \, \sqrt{2} \left (-\frac{1}{a^{7} c}\right )^{\frac{1}{8}} \arctan \left (\sqrt{2} \sqrt{a^{2} \left (-\frac{1}{a^{7} c}\right )^{\frac{1}{4}} - \sqrt{2} a \sqrt{x} \left (-\frac{1}{a^{7} c}\right )^{\frac{1}{8}} + x} a^{6} c \left (-\frac{1}{a^{7} c}\right )^{\frac{7}{8}} - \sqrt{2} a^{6} c \sqrt{x} \left (-\frac{1}{a^{7} c}\right )^{\frac{7}{8}} - 1\right ) + \frac{1}{8} \, \sqrt{2} \left (-\frac{1}{a^{7} c}\right )^{\frac{1}{8}} \log \left (a^{2} \left (-\frac{1}{a^{7} c}\right )^{\frac{1}{4}} + \sqrt{2} a \sqrt{x} \left (-\frac{1}{a^{7} c}\right )^{\frac{1}{8}} + x\right ) - \frac{1}{8} \, \sqrt{2} \left (-\frac{1}{a^{7} c}\right )^{\frac{1}{8}} \log \left (a^{2} \left (-\frac{1}{a^{7} c}\right )^{\frac{1}{4}} - \sqrt{2} a \sqrt{x} \left (-\frac{1}{a^{7} c}\right )^{\frac{1}{8}} + x\right ) + \left (-\frac{1}{a^{7} c}\right )^{\frac{1}{8}} \arctan \left (\sqrt{a^{2} \left (-\frac{1}{a^{7} c}\right )^{\frac{1}{4}} + x} a^{6} c \left (-\frac{1}{a^{7} c}\right )^{\frac{7}{8}} - a^{6} c \sqrt{x} \left (-\frac{1}{a^{7} c}\right )^{\frac{7}{8}}\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} \, \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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*sqrt(2)*(-1/(a^7*c))^(1/8)*arctan(sqrt(2)*sqrt(a^2*(-1/(a^7*c))^(1/4) + sqrt(2)*a*sqrt(x)*(-1/(a^7*c))^(1/
8) + x)*a^6*c*(-1/(a^7*c))^(7/8) - sqrt(2)*a^6*c*sqrt(x)*(-1/(a^7*c))^(7/8) + 1) + 1/2*sqrt(2)*(-1/(a^7*c))^(1
/8)*arctan(sqrt(2)*sqrt(a^2*(-1/(a^7*c))^(1/4) - sqrt(2)*a*sqrt(x)*(-1/(a^7*c))^(1/8) + x)*a^6*c*(-1/(a^7*c))^
(7/8) - sqrt(2)*a^6*c*sqrt(x)*(-1/(a^7*c))^(7/8) - 1) + 1/8*sqrt(2)*(-1/(a^7*c))^(1/8)*log(a^2*(-1/(a^7*c))^(1
/4) + sqrt(2)*a*sqrt(x)*(-1/(a^7*c))^(1/8) + x) - 1/8*sqrt(2)*(-1/(a^7*c))^(1/8)*log(a^2*(-1/(a^7*c))^(1/4) -
sqrt(2)*a*sqrt(x)*(-1/(a^7*c))^(1/8) + x) + (-1/(a^7*c))^(1/8)*arctan(sqrt(a^2*(-1/(a^7*c))^(1/4) + x)*a^6*c*(
-1/(a^7*c))^(7/8) - a^6*c*sqrt(x)*(-1/(a^7*c))^(7/8)) + 1/4*(-1/(a^7*c))^(1/8)*log(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))

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Sympy [A]  time = 73.6814, size = 464, normalized size = 1.62 \begin{align*} \begin{cases} \frac{\tilde{\infty }}{x^{\frac{7}{2}}} & \text{for}\: a = 0 \wedge c = 0 \\\frac{2 \sqrt{x}}{a} & \text{for}\: c = 0 \\- \frac{2}{7 c x^{\frac{7}{2}}} & \text{for}\: a = 0 \\- \frac{\sqrt [8]{-1} \log{\left (- \sqrt [8]{-1} \sqrt [8]{a} \sqrt [8]{\frac{1}{c}} + \sqrt{x} \right )}}{4 a^{\frac{7}{8}} c^{18} \left (\frac{1}{c}\right )^{\frac{143}{8}}} + \frac{\sqrt [8]{-1} \log{\left (\sqrt [8]{-1} \sqrt [8]{a} \sqrt [8]{\frac{1}{c}} + \sqrt{x} \right )}}{4 a^{\frac{7}{8}} c^{18} \left (\frac{1}{c}\right )^{\frac{143}{8}}} - \frac{\sqrt [8]{-1} \sqrt{2} \log{\left (- 4 \sqrt [8]{-1} \sqrt{2} \sqrt [8]{a} \sqrt{x} \sqrt [8]{\frac{1}{c}} + 4 \sqrt [4]{-1} \sqrt [4]{a} \sqrt [4]{\frac{1}{c}} + 4 x \right )}}{8 a^{\frac{7}{8}} c^{18} \left (\frac{1}{c}\right )^{\frac{143}{8}}} + \frac{\sqrt [8]{-1} \sqrt{2} \log{\left (4 \sqrt [8]{-1} \sqrt{2} \sqrt [8]{a} \sqrt{x} \sqrt [8]{\frac{1}{c}} + 4 \sqrt [4]{-1} \sqrt [4]{a} \sqrt [4]{\frac{1}{c}} + 4 x \right )}}{8 a^{\frac{7}{8}} c^{18} \left (\frac{1}{c}\right )^{\frac{143}{8}}} - \frac{\sqrt [8]{-1} \operatorname{atan}{\left (\frac{\left (-1\right )^{\frac{7}{8}} \sqrt{x}}{\sqrt [8]{a} \sqrt [8]{\frac{1}{c}}} \right )}}{2 a^{\frac{7}{8}} c^{18} \left (\frac{1}{c}\right )^{\frac{143}{8}}} + \frac{\sqrt [8]{-1} \sqrt{2} \operatorname{atan}{\left (1 - \frac{\left (-1\right )^{\frac{7}{8}} \sqrt{2} \sqrt{x}}{\sqrt [8]{a} \sqrt [8]{\frac{1}{c}}} \right )}}{4 a^{\frac{7}{8}} c^{18} \left (\frac{1}{c}\right )^{\frac{143}{8}}} - \frac{\sqrt [8]{-1} \sqrt{2} \operatorname{atan}{\left (1 + \frac{\left (-1\right )^{\frac{7}{8}} \sqrt{2} \sqrt{x}}{\sqrt [8]{a} \sqrt [8]{\frac{1}{c}}} \right )}}{4 a^{\frac{7}{8}} c^{18} \left (\frac{1}{c}\right )^{\frac{143}{8}}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((zoo/x**(7/2), Eq(a, 0) & Eq(c, 0)), (2*sqrt(x)/a, Eq(c, 0)), (-2/(7*c*x**(7/2)), Eq(a, 0)), (-(-1)*
*(1/8)*log(-(-1)**(1/8)*a**(1/8)*(1/c)**(1/8) + sqrt(x))/(4*a**(7/8)*c**18*(1/c)**(143/8)) + (-1)**(1/8)*log((
-1)**(1/8)*a**(1/8)*(1/c)**(1/8) + sqrt(x))/(4*a**(7/8)*c**18*(1/c)**(143/8)) - (-1)**(1/8)*sqrt(2)*log(-4*(-1
)**(1/8)*sqrt(2)*a**(1/8)*sqrt(x)*(1/c)**(1/8) + 4*(-1)**(1/4)*a**(1/4)*(1/c)**(1/4) + 4*x)/(8*a**(7/8)*c**18*
(1/c)**(143/8)) + (-1)**(1/8)*sqrt(2)*log(4*(-1)**(1/8)*sqrt(2)*a**(1/8)*sqrt(x)*(1/c)**(1/8) + 4*(-1)**(1/4)*
a**(1/4)*(1/c)**(1/4) + 4*x)/(8*a**(7/8)*c**18*(1/c)**(143/8)) - (-1)**(1/8)*atan((-1)**(7/8)*sqrt(x)/(a**(1/8
)*(1/c)**(1/8)))/(2*a**(7/8)*c**18*(1/c)**(143/8)) + (-1)**(1/8)*sqrt(2)*atan(1 - (-1)**(7/8)*sqrt(2)*sqrt(x)/
(a**(1/8)*(1/c)**(1/8)))/(4*a**(7/8)*c**18*(1/c)**(143/8)) - (-1)**(1/8)*sqrt(2)*atan(1 + (-1)**(7/8)*sqrt(2)*
sqrt(x)/(a**(1/8)*(1/c)**(1/8)))/(4*a**(7/8)*c**18*(1/c)**(143/8)), True))

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Giac [B]  time = 1.25838, size = 590, normalized size = 2.06 \begin{align*} \frac{\sqrt{\sqrt{2} + 2} \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 )}{4 \, a} + \frac{\sqrt{\sqrt{2} + 2} \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 )}{4 \, a} + \frac{\sqrt{-\sqrt{2} + 2} \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 )}{4 \, a} + \frac{\sqrt{-\sqrt{2} + 2} \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 )}{4 \, a} + \frac{\sqrt{\sqrt{2} + 2} \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 )}{8 \, a} - \frac{\sqrt{\sqrt{2} + 2} \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 )}{8 \, a} + \frac{\sqrt{-\sqrt{2} + 2} \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 )}{8 \, a} - \frac{\sqrt{-\sqrt{2} + 2} \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 )}{8 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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