3.763 \(\int \frac{1}{\sqrt{x} (a+c x^4)^3} \, dx\)

Optimal. Leaf size=329 \[ \frac{15 \sqrt{x}}{64 a^2 \left (a+c x^4\right )}+\frac{\sqrt{x}}{8 a \left (a+c x^4\right )^2}+\frac{105 \log \left (-\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{512 \sqrt{2} (-a)^{23/8} \sqrt [8]{c}}-\frac{105 \log \left (\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{512 \sqrt{2} (-a)^{23/8} \sqrt [8]{c}}+\frac{105 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{256 \sqrt{2} (-a)^{23/8} \sqrt [8]{c}}-\frac{105 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}+1\right )}{256 \sqrt{2} (-a)^{23/8} \sqrt [8]{c}}-\frac{105 \tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{256 (-a)^{23/8} \sqrt [8]{c}}-\frac{105 \tanh ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{256 (-a)^{23/8} \sqrt [8]{c}} \]

[Out]

Sqrt[x]/(8*a*(a + c*x^4)^2) + (15*Sqrt[x])/(64*a^2*(a + c*x^4)) + (105*ArcTan[1 - (Sqrt[2]*c^(1/8)*Sqrt[x])/(-
a)^(1/8)])/(256*Sqrt[2]*(-a)^(23/8)*c^(1/8)) - (105*ArcTan[1 + (Sqrt[2]*c^(1/8)*Sqrt[x])/(-a)^(1/8)])/(256*Sqr
t[2]*(-a)^(23/8)*c^(1/8)) - (105*ArcTan[(c^(1/8)*Sqrt[x])/(-a)^(1/8)])/(256*(-a)^(23/8)*c^(1/8)) - (105*ArcTan
h[(c^(1/8)*Sqrt[x])/(-a)^(1/8)])/(256*(-a)^(23/8)*c^(1/8)) + (105*Log[(-a)^(1/4) - Sqrt[2]*(-a)^(1/8)*c^(1/8)*
Sqrt[x] + c^(1/4)*x])/(512*Sqrt[2]*(-a)^(23/8)*c^(1/8)) - (105*Log[(-a)^(1/4) + Sqrt[2]*(-a)^(1/8)*c^(1/8)*Sqr
t[x] + c^(1/4)*x])/(512*Sqrt[2]*(-a)^(23/8)*c^(1/8))

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Rubi [A]  time = 0.302557, antiderivative size = 329, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 12, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.8, Rules used = {290, 329, 214, 212, 208, 205, 211, 1165, 628, 1162, 617, 204} \[ \frac{15 \sqrt{x}}{64 a^2 \left (a+c x^4\right )}+\frac{\sqrt{x}}{8 a \left (a+c x^4\right )^2}+\frac{105 \log \left (-\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{512 \sqrt{2} (-a)^{23/8} \sqrt [8]{c}}-\frac{105 \log \left (\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{512 \sqrt{2} (-a)^{23/8} \sqrt [8]{c}}+\frac{105 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{256 \sqrt{2} (-a)^{23/8} \sqrt [8]{c}}-\frac{105 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}+1\right )}{256 \sqrt{2} (-a)^{23/8} \sqrt [8]{c}}-\frac{105 \tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{256 (-a)^{23/8} \sqrt [8]{c}}-\frac{105 \tanh ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{256 (-a)^{23/8} \sqrt [8]{c}} \]

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[x]*(a + c*x^4)^3),x]

[Out]

Sqrt[x]/(8*a*(a + c*x^4)^2) + (15*Sqrt[x])/(64*a^2*(a + c*x^4)) + (105*ArcTan[1 - (Sqrt[2]*c^(1/8)*Sqrt[x])/(-
a)^(1/8)])/(256*Sqrt[2]*(-a)^(23/8)*c^(1/8)) - (105*ArcTan[1 + (Sqrt[2]*c^(1/8)*Sqrt[x])/(-a)^(1/8)])/(256*Sqr
t[2]*(-a)^(23/8)*c^(1/8)) - (105*ArcTan[(c^(1/8)*Sqrt[x])/(-a)^(1/8)])/(256*(-a)^(23/8)*c^(1/8)) - (105*ArcTan
h[(c^(1/8)*Sqrt[x])/(-a)^(1/8)])/(256*(-a)^(23/8)*c^(1/8)) + (105*Log[(-a)^(1/4) - Sqrt[2]*(-a)^(1/8)*c^(1/8)*
Sqrt[x] + c^(1/4)*x])/(512*Sqrt[2]*(-a)^(23/8)*c^(1/8)) - (105*Log[(-a)^(1/4) + Sqrt[2]*(-a)^(1/8)*c^(1/8)*Sqr
t[x] + c^(1/4)*x])/(512*Sqrt[2]*(-a)^(23/8)*c^(1/8))

Rule 290

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(
a*c*n*(p + 1)), x] + Dist[(m + n*(p + 1) + 1)/(a*n*(p + 1)), Int[(c*x)^m*(a + b*x^n)^(p + 1), x], x] /; FreeQ[
{a, b, c, m}, x] && IGtQ[n, 0] && LtQ[p, -1] && IntBinomialQ[a, b, c, n, m, p, x]

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 )^3} \, dx &=\frac{\sqrt{x}}{8 a \left (a+c x^4\right )^2}+\frac{15 \int \frac{1}{\sqrt{x} \left (a+c x^4\right )^2} \, dx}{16 a}\\ &=\frac{\sqrt{x}}{8 a \left (a+c x^4\right )^2}+\frac{15 \sqrt{x}}{64 a^2 \left (a+c x^4\right )}+\frac{105 \int \frac{1}{\sqrt{x} \left (a+c x^4\right )} \, dx}{128 a^2}\\ &=\frac{\sqrt{x}}{8 a \left (a+c x^4\right )^2}+\frac{15 \sqrt{x}}{64 a^2 \left (a+c x^4\right )}+\frac{105 \operatorname{Subst}\left (\int \frac{1}{a+c x^8} \, dx,x,\sqrt{x}\right )}{64 a^2}\\ &=\frac{\sqrt{x}}{8 a \left (a+c x^4\right )^2}+\frac{15 \sqrt{x}}{64 a^2 \left (a+c x^4\right )}-\frac{105 \operatorname{Subst}\left (\int \frac{1}{\sqrt{-a}-\sqrt{c} x^4} \, dx,x,\sqrt{x}\right )}{128 (-a)^{5/2}}-\frac{105 \operatorname{Subst}\left (\int \frac{1}{\sqrt{-a}+\sqrt{c} x^4} \, dx,x,\sqrt{x}\right )}{128 (-a)^{5/2}}\\ &=\frac{\sqrt{x}}{8 a \left (a+c x^4\right )^2}+\frac{15 \sqrt{x}}{64 a^2 \left (a+c x^4\right )}-\frac{105 \operatorname{Subst}\left (\int \frac{1}{\sqrt [4]{-a}-\sqrt [4]{c} x^2} \, dx,x,\sqrt{x}\right )}{256 (-a)^{11/4}}-\frac{105 \operatorname{Subst}\left (\int \frac{1}{\sqrt [4]{-a}+\sqrt [4]{c} x^2} \, dx,x,\sqrt{x}\right )}{256 (-a)^{11/4}}-\frac{105 \operatorname{Subst}\left (\int \frac{\sqrt [4]{-a}-\sqrt [4]{c} x^2}{\sqrt{-a}+\sqrt{c} x^4} \, dx,x,\sqrt{x}\right )}{256 (-a)^{11/4}}-\frac{105 \operatorname{Subst}\left (\int \frac{\sqrt [4]{-a}+\sqrt [4]{c} x^2}{\sqrt{-a}+\sqrt{c} x^4} \, dx,x,\sqrt{x}\right )}{256 (-a)^{11/4}}\\ &=\frac{\sqrt{x}}{8 a \left (a+c x^4\right )^2}+\frac{15 \sqrt{x}}{64 a^2 \left (a+c x^4\right )}-\frac{105 \tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{256 (-a)^{23/8} \sqrt [8]{c}}-\frac{105 \tanh ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{256 (-a)^{23/8} \sqrt [8]{c}}-\frac{105 \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 )}{512 (-a)^{11/4} \sqrt [4]{c}}-\frac{105 \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 )}{512 (-a)^{11/4} \sqrt [4]{c}}+\frac{105 \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 )}{512 \sqrt{2} (-a)^{23/8} \sqrt [8]{c}}+\frac{105 \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 )}{512 \sqrt{2} (-a)^{23/8} \sqrt [8]{c}}\\ &=\frac{\sqrt{x}}{8 a \left (a+c x^4\right )^2}+\frac{15 \sqrt{x}}{64 a^2 \left (a+c x^4\right )}-\frac{105 \tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{256 (-a)^{23/8} \sqrt [8]{c}}-\frac{105 \tanh ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{256 (-a)^{23/8} \sqrt [8]{c}}+\frac{105 \log \left (\sqrt [4]{-a}-\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{c} x\right )}{512 \sqrt{2} (-a)^{23/8} \sqrt [8]{c}}-\frac{105 \log \left (\sqrt [4]{-a}+\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{c} x\right )}{512 \sqrt{2} (-a)^{23/8} \sqrt [8]{c}}-\frac{105 \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{256 \sqrt{2} (-a)^{23/8} \sqrt [8]{c}}+\frac{105 \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{256 \sqrt{2} (-a)^{23/8} \sqrt [8]{c}}\\ &=\frac{\sqrt{x}}{8 a \left (a+c x^4\right )^2}+\frac{15 \sqrt{x}}{64 a^2 \left (a+c x^4\right )}+\frac{105 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{256 \sqrt{2} (-a)^{23/8} \sqrt [8]{c}}-\frac{105 \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{256 \sqrt{2} (-a)^{23/8} \sqrt [8]{c}}-\frac{105 \tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{256 (-a)^{23/8} \sqrt [8]{c}}-\frac{105 \tanh ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{256 (-a)^{23/8} \sqrt [8]{c}}+\frac{105 \log \left (\sqrt [4]{-a}-\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{c} x\right )}{512 \sqrt{2} (-a)^{23/8} \sqrt [8]{c}}-\frac{105 \log \left (\sqrt [4]{-a}+\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{c} x\right )}{512 \sqrt{2} (-a)^{23/8} \sqrt [8]{c}}\\ \end{align*}

Mathematica [C]  time = 0.0161036, size = 62, normalized size = 0.19 \[ \frac{\sqrt{x} \left (105 \left (a+c x^4\right )^2 \, _2F_1\left (\frac{1}{8},1;\frac{9}{8};-\frac{c x^4}{a}\right )+a \left (23 a+15 c x^4\right )\right )}{64 a^3 \left (a+c x^4\right )^2} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[x]*(a + c*x^4)^3),x]

[Out]

(Sqrt[x]*(a*(23*a + 15*c*x^4) + 105*(a + c*x^4)^2*Hypergeometric2F1[1/8, 1, 9/8, -((c*x^4)/a)]))/(64*a^3*(a +
c*x^4)^2)

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Maple [C]  time = 0.019, size = 62, normalized size = 0.2 \begin{align*} 2\,{\frac{1}{ \left ( c{x}^{4}+a \right ) ^{2}} \left ({\frac{23\,\sqrt{x}}{128\,a}}+{\frac{15\,c{x}^{9/2}}{128\,{a}^{2}}} \right ) }+{\frac{105}{512\,{a}^{2}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)^3/x^(1/2),x)

[Out]

2*(23/128*x^(1/2)/a+15/128/a^2*c*x^(9/2))/(c*x^4+a)^2+105/512/a^2/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*} -105 \, c \int \frac{x^{\frac{7}{2}}}{128 \,{\left (a^{3} c x^{4} + a^{4}\right )}}\,{d x} + \frac{105 \, c^{2} x^{\frac{17}{2}} + 225 \, a c x^{\frac{9}{2}} + 128 \, a^{2} \sqrt{x}}{64 \,{\left (a^{3} c^{2} x^{8} + 2 \, a^{4} c x^{4} + a^{5}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*x^4+a)^3/x^(1/2),x, algorithm="maxima")

[Out]

-105*c*integrate(1/128*x^(7/2)/(a^3*c*x^4 + a^4), x) + 1/64*(105*c^2*x^(17/2) + 225*a*c*x^(9/2) + 128*a^2*sqrt
(x))/(a^3*c^2*x^8 + 2*a^4*c*x^4 + a^5)

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Fricas [B]  time = 1.72044, size = 1615, normalized size = 4.91 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*x^4+a)^3/x^(1/2),x, algorithm="fricas")

[Out]

1/1024*(420*sqrt(2)*(a^2*c^2*x^8 + 2*a^3*c*x^4 + a^4)*(-1/(a^23*c))^(1/8)*arctan(sqrt(2)*sqrt(a^6*(-1/(a^23*c)
)^(1/4) + sqrt(2)*a^3*sqrt(x)*(-1/(a^23*c))^(1/8) + x)*a^20*c*(-1/(a^23*c))^(7/8) - sqrt(2)*a^20*c*sqrt(x)*(-1
/(a^23*c))^(7/8) + 1) + 420*sqrt(2)*(a^2*c^2*x^8 + 2*a^3*c*x^4 + a^4)*(-1/(a^23*c))^(1/8)*arctan(sqrt(2)*sqrt(
a^6*(-1/(a^23*c))^(1/4) - sqrt(2)*a^3*sqrt(x)*(-1/(a^23*c))^(1/8) + x)*a^20*c*(-1/(a^23*c))^(7/8) - sqrt(2)*a^
20*c*sqrt(x)*(-1/(a^23*c))^(7/8) - 1) + 105*sqrt(2)*(a^2*c^2*x^8 + 2*a^3*c*x^4 + a^4)*(-1/(a^23*c))^(1/8)*log(
a^6*(-1/(a^23*c))^(1/4) + sqrt(2)*a^3*sqrt(x)*(-1/(a^23*c))^(1/8) + x) - 105*sqrt(2)*(a^2*c^2*x^8 + 2*a^3*c*x^
4 + a^4)*(-1/(a^23*c))^(1/8)*log(a^6*(-1/(a^23*c))^(1/4) - sqrt(2)*a^3*sqrt(x)*(-1/(a^23*c))^(1/8) + x) + 840*
(a^2*c^2*x^8 + 2*a^3*c*x^4 + a^4)*(-1/(a^23*c))^(1/8)*arctan(sqrt(a^6*(-1/(a^23*c))^(1/4) + x)*a^20*c*(-1/(a^2
3*c))^(7/8) - a^20*c*sqrt(x)*(-1/(a^23*c))^(7/8)) + 210*(a^2*c^2*x^8 + 2*a^3*c*x^4 + a^4)*(-1/(a^23*c))^(1/8)*
log(a^3*(-1/(a^23*c))^(1/8) + sqrt(x)) - 210*(a^2*c^2*x^8 + 2*a^3*c*x^4 + a^4)*(-1/(a^23*c))^(1/8)*log(-a^3*(-
1/(a^23*c))^(1/8) + sqrt(x)) + 16*(15*c*x^4 + 23*a)*sqrt(x))/(a^2*c^2*x^8 + 2*a^3*c*x^4 + a^4)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*x**4+a)**3/x**(1/2),x)

[Out]

Timed out

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Giac [B]  time = 1.47006, size = 626, normalized size = 1.9 \begin{align*} \frac{105 \, \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 )}{512 \, a^{3}} + \frac{105 \, \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 )}{512 \, a^{3}} + \frac{105 \, \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 )}{512 \, a^{3}} + \frac{105 \, \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 )}{512 \, a^{3}} + \frac{105 \, \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 )}{1024 \, a^{3}} - \frac{105 \, \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 )}{1024 \, a^{3}} + \frac{105 \, \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 )}{1024 \, a^{3}} - \frac{105 \, \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 )}{1024 \, a^{3}} + \frac{15 \, c x^{\frac{9}{2}} + 23 \, a \sqrt{x}}{64 \,{\left (c x^{4} + a\right )}^{2} a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*x^4+a)^3/x^(1/2),x, algorithm="giac")

[Out]

105/512*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^3 + 105/512*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^3 + 105/512*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^3 + 105/512*sqrt(-sqrt(2) + 2)*(a/c)^(1/8)*arctan(-(sqr
t(sqrt(2) + 2)*(a/c)^(1/8) - 2*sqrt(x))/(sqrt(-sqrt(2) + 2)*(a/c)^(1/8)))/a^3 + 105/1024*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^3 - 105/1024*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^3 + 105/1024*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^3 - 105/1024*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^3 + 1/64*(15*c*x^(9/2) + 23*a*sqrt(x))/((c*x
^4 + a)^2*a^2)