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

Optimal. Leaf size=329 \[ \frac{13 x^{3/2}}{64 a^2 \left (a+c x^4\right )}-\frac{65 \log \left (-\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{512 \sqrt{2} (-a)^{21/8} c^{3/8}}+\frac{65 \log \left (\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{512 \sqrt{2} (-a)^{21/8} c^{3/8}}+\frac{65 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{256 \sqrt{2} (-a)^{21/8} c^{3/8}}-\frac{65 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}+1\right )}{256 \sqrt{2} (-a)^{21/8} c^{3/8}}+\frac{65 \tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{256 (-a)^{21/8} c^{3/8}}-\frac{65 \tanh ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{256 (-a)^{21/8} c^{3/8}}+\frac{x^{3/2}}{8 a \left (a+c x^4\right )^2} \]

[Out]

x^(3/2)/(8*a*(a + c*x^4)^2) + (13*x^(3/2))/(64*a^2*(a + c*x^4)) + (65*ArcTan[1 - (Sqrt[2]*c^(1/8)*Sqrt[x])/(-a
)^(1/8)])/(256*Sqrt[2]*(-a)^(21/8)*c^(3/8)) - (65*ArcTan[1 + (Sqrt[2]*c^(1/8)*Sqrt[x])/(-a)^(1/8)])/(256*Sqrt[
2]*(-a)^(21/8)*c^(3/8)) + (65*ArcTan[(c^(1/8)*Sqrt[x])/(-a)^(1/8)])/(256*(-a)^(21/8)*c^(3/8)) - (65*ArcTanh[(c
^(1/8)*Sqrt[x])/(-a)^(1/8)])/(256*(-a)^(21/8)*c^(3/8)) - (65*Log[(-a)^(1/4) - Sqrt[2]*(-a)^(1/8)*c^(1/8)*Sqrt[
x] + c^(1/4)*x])/(512*Sqrt[2]*(-a)^(21/8)*c^(3/8)) + (65*Log[(-a)^(1/4) + Sqrt[2]*(-a)^(1/8)*c^(1/8)*Sqrt[x] +
c^(1/4)*x])/(512*Sqrt[2]*(-a)^(21/8)*c^(3/8))

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Rubi [A]  time = 0.312949, 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, 300, 297, 1162, 617, 204, 1165, 628, 298, 205, 208} \[ \frac{13 x^{3/2}}{64 a^2 \left (a+c x^4\right )}-\frac{65 \log \left (-\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{512 \sqrt{2} (-a)^{21/8} c^{3/8}}+\frac{65 \log \left (\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{512 \sqrt{2} (-a)^{21/8} c^{3/8}}+\frac{65 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{256 \sqrt{2} (-a)^{21/8} c^{3/8}}-\frac{65 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}+1\right )}{256 \sqrt{2} (-a)^{21/8} c^{3/8}}+\frac{65 \tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{256 (-a)^{21/8} c^{3/8}}-\frac{65 \tanh ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{256 (-a)^{21/8} c^{3/8}}+\frac{x^{3/2}}{8 a \left (a+c x^4\right )^2} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[x]/(a + c*x^4)^3,x]

[Out]

x^(3/2)/(8*a*(a + c*x^4)^2) + (13*x^(3/2))/(64*a^2*(a + c*x^4)) + (65*ArcTan[1 - (Sqrt[2]*c^(1/8)*Sqrt[x])/(-a
)^(1/8)])/(256*Sqrt[2]*(-a)^(21/8)*c^(3/8)) - (65*ArcTan[1 + (Sqrt[2]*c^(1/8)*Sqrt[x])/(-a)^(1/8)])/(256*Sqrt[
2]*(-a)^(21/8)*c^(3/8)) + (65*ArcTan[(c^(1/8)*Sqrt[x])/(-a)^(1/8)])/(256*(-a)^(21/8)*c^(3/8)) - (65*ArcTanh[(c
^(1/8)*Sqrt[x])/(-a)^(1/8)])/(256*(-a)^(21/8)*c^(3/8)) - (65*Log[(-a)^(1/4) - Sqrt[2]*(-a)^(1/8)*c^(1/8)*Sqrt[
x] + c^(1/4)*x])/(512*Sqrt[2]*(-a)^(21/8)*c^(3/8)) + (65*Log[(-a)^(1/4) + Sqrt[2]*(-a)^(1/8)*c^(1/8)*Sqrt[x] +
c^(1/4)*x])/(512*Sqrt[2]*(-a)^(21/8)*c^(3/8))

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 300

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

Rule 297

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

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])

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 298

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

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 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]

Rubi steps

\begin{align*} \int \frac{\sqrt{x}}{\left (a+c x^4\right )^3} \, dx &=\frac{x^{3/2}}{8 a \left (a+c x^4\right )^2}+\frac{13 \int \frac{\sqrt{x}}{\left (a+c x^4\right )^2} \, dx}{16 a}\\ &=\frac{x^{3/2}}{8 a \left (a+c x^4\right )^2}+\frac{13 x^{3/2}}{64 a^2 \left (a+c x^4\right )}+\frac{65 \int \frac{\sqrt{x}}{a+c x^4} \, dx}{128 a^2}\\ &=\frac{x^{3/2}}{8 a \left (a+c x^4\right )^2}+\frac{13 x^{3/2}}{64 a^2 \left (a+c x^4\right )}+\frac{65 \operatorname{Subst}\left (\int \frac{x^2}{a+c x^8} \, dx,x,\sqrt{x}\right )}{64 a^2}\\ &=\frac{x^{3/2}}{8 a \left (a+c x^4\right )^2}+\frac{13 x^{3/2}}{64 a^2 \left (a+c x^4\right )}-\frac{65 \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{-a}-\sqrt{c} x^4} \, dx,x,\sqrt{x}\right )}{128 (-a)^{5/2}}-\frac{65 \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{-a}+\sqrt{c} x^4} \, dx,x,\sqrt{x}\right )}{128 (-a)^{5/2}}\\ &=\frac{x^{3/2}}{8 a \left (a+c x^4\right )^2}+\frac{13 x^{3/2}}{64 a^2 \left (a+c x^4\right )}-\frac{65 \operatorname{Subst}\left (\int \frac{1}{\sqrt [4]{-a}-\sqrt [4]{c} x^2} \, dx,x,\sqrt{x}\right )}{256 (-a)^{5/2} \sqrt [4]{c}}+\frac{65 \operatorname{Subst}\left (\int \frac{1}{\sqrt [4]{-a}+\sqrt [4]{c} x^2} \, dx,x,\sqrt{x}\right )}{256 (-a)^{5/2} \sqrt [4]{c}}+\frac{65 \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)^{5/2} \sqrt [4]{c}}-\frac{65 \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)^{5/2} \sqrt [4]{c}}\\ &=\frac{x^{3/2}}{8 a \left (a+c x^4\right )^2}+\frac{13 x^{3/2}}{64 a^2 \left (a+c x^4\right )}+\frac{65 \tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{256 (-a)^{21/8} c^{3/8}}-\frac{65 \tanh ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{256 (-a)^{21/8} c^{3/8}}-\frac{65 \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)^{5/2} \sqrt{c}}-\frac{65 \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)^{5/2} \sqrt{c}}-\frac{65 \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)^{21/8} c^{3/8}}-\frac{65 \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)^{21/8} c^{3/8}}\\ &=\frac{x^{3/2}}{8 a \left (a+c x^4\right )^2}+\frac{13 x^{3/2}}{64 a^2 \left (a+c x^4\right )}+\frac{65 \tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{256 (-a)^{21/8} c^{3/8}}-\frac{65 \tanh ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{256 (-a)^{21/8} c^{3/8}}-\frac{65 \log \left (\sqrt [4]{-a}-\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{c} x\right )}{512 \sqrt{2} (-a)^{21/8} c^{3/8}}+\frac{65 \log \left (\sqrt [4]{-a}+\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{c} x\right )}{512 \sqrt{2} (-a)^{21/8} c^{3/8}}-\frac{65 \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)^{21/8} c^{3/8}}+\frac{65 \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)^{21/8} c^{3/8}}\\ &=\frac{x^{3/2}}{8 a \left (a+c x^4\right )^2}+\frac{13 x^{3/2}}{64 a^2 \left (a+c x^4\right )}+\frac{65 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{256 \sqrt{2} (-a)^{21/8} c^{3/8}}-\frac{65 \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{256 \sqrt{2} (-a)^{21/8} c^{3/8}}+\frac{65 \tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{256 (-a)^{21/8} c^{3/8}}-\frac{65 \tanh ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{256 (-a)^{21/8} c^{3/8}}-\frac{65 \log \left (\sqrt [4]{-a}-\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{c} x\right )}{512 \sqrt{2} (-a)^{21/8} c^{3/8}}+\frac{65 \log \left (\sqrt [4]{-a}+\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{c} x\right )}{512 \sqrt{2} (-a)^{21/8} c^{3/8}}\\ \end{align*}

Mathematica [C]  time = 0.005762, size = 29, normalized size = 0.09 \[ \frac{2 x^{3/2} \, _2F_1\left (\frac{3}{8},3;\frac{11}{8};-\frac{c x^4}{a}\right )}{3 a^3} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[x]/(a + c*x^4)^3,x]

[Out]

(2*x^(3/2)*Hypergeometric2F1[3/8, 3, 11/8, -((c*x^4)/a)])/(3*a^3)

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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{21\,{x}^{3/2}}{128\,a}}+{\frac{13\,c{x}^{11/2}}{128\,{a}^{2}}} \right ) }+{\frac{65}{512\,{a}^{2}c}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{8}c+a \right ) }{\frac{1}{{{\it \_R}}^{5}}\ln \left ( \sqrt{x}-{\it \_R} \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*(21/128*x^(3/2)/a+13/128/a^2*c*x^(11/2))/(c*x^4+a)^2+65/512/a^2/c*sum(1/_R^5*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*} \frac{13 \, c x^{\frac{11}{2}} + 21 \, a x^{\frac{3}{2}}}{64 \,{\left (a^{2} c^{2} x^{8} + 2 \, a^{3} c x^{4} + a^{4}\right )}} + 65 \, \int \frac{\sqrt{x}}{128 \,{\left (a^{2} c x^{4} + a^{3}\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/64*(13*c*x^(11/2) + 21*a*x^(3/2))/(a^2*c^2*x^8 + 2*a^3*c*x^4 + a^4) + 65*integrate(1/128*sqrt(x)/(a^2*c*x^4
+ a^3), x)

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Fricas [B]  time = 1.74663, size = 1747, normalized size = 5.31 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/1024*(260*sqrt(2)*(a^2*c^2*x^8 + 2*a^3*c*x^4 + a^4)*(-1/(a^21*c^3))^(1/8)*arctan(sqrt(2)*sqrt(a^16*c^2*(-1/
(a^21*c^3))^(3/4) + sqrt(2)*a^8*c*sqrt(x)*(-1/(a^21*c^3))^(3/8) + x)*a^13*c^2*(-1/(a^21*c^3))^(5/8) - sqrt(2)*
a^13*c^2*sqrt(x)*(-1/(a^21*c^3))^(5/8) + 1) + 260*sqrt(2)*(a^2*c^2*x^8 + 2*a^3*c*x^4 + a^4)*(-1/(a^21*c^3))^(1
/8)*arctan(sqrt(2)*sqrt(a^16*c^2*(-1/(a^21*c^3))^(3/4) - sqrt(2)*a^8*c*sqrt(x)*(-1/(a^21*c^3))^(3/8) + x)*a^13
*c^2*(-1/(a^21*c^3))^(5/8) - sqrt(2)*a^13*c^2*sqrt(x)*(-1/(a^21*c^3))^(5/8) - 1) - 65*sqrt(2)*(a^2*c^2*x^8 + 2
*a^3*c*x^4 + a^4)*(-1/(a^21*c^3))^(1/8)*log(a^16*c^2*(-1/(a^21*c^3))^(3/4) + sqrt(2)*a^8*c*sqrt(x)*(-1/(a^21*c
^3))^(3/8) + x) + 65*sqrt(2)*(a^2*c^2*x^8 + 2*a^3*c*x^4 + a^4)*(-1/(a^21*c^3))^(1/8)*log(a^16*c^2*(-1/(a^21*c^
3))^(3/4) - sqrt(2)*a^8*c*sqrt(x)*(-1/(a^21*c^3))^(3/8) + x) - 520*(a^2*c^2*x^8 + 2*a^3*c*x^4 + a^4)*(-1/(a^21
*c^3))^(1/8)*arctan(sqrt(a^16*c^2*(-1/(a^21*c^3))^(3/4) + x)*a^13*c^2*(-1/(a^21*c^3))^(5/8) - a^13*c^2*sqrt(x)
*(-1/(a^21*c^3))^(5/8)) + 130*(a^2*c^2*x^8 + 2*a^3*c*x^4 + a^4)*(-1/(a^21*c^3))^(1/8)*log(a^8*c*(-1/(a^21*c^3)
)^(3/8) + sqrt(x)) - 130*(a^2*c^2*x^8 + 2*a^3*c*x^4 + a^4)*(-1/(a^21*c^3))^(1/8)*log(-a^8*c*(-1/(a^21*c^3))^(3
/8) + sqrt(x)) - 16*(13*c*x^5 + 21*a*x)*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(x**(1/2)/(c*x**4+a)**3,x)

[Out]

Timed out

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Giac [B]  time = 1.51253, size = 626, normalized size = 1.9 \begin{align*} -\frac{65 \, \sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{3}{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{65 \, \sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{3}{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{65 \, \sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{3}{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{65 \, \sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{3}{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{65 \, \sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{3}{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{65 \, \sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{3}{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{65 \, \sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{3}{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{65 \, \sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{3}{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{13 \, c x^{\frac{11}{2}} + 21 \, a x^{\frac{3}{2}}}{64 \,{\left (c x^{4} + a\right )}^{2} a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-65/512*sqrt(-sqrt(2) + 2)*(a/c)^(3/8)*arctan((sqrt(-sqrt(2) + 2)*(a/c)^(1/8) + 2*sqrt(x))/(sqrt(sqrt(2) + 2)*
(a/c)^(1/8)))/a^3 - 65/512*sqrt(-sqrt(2) + 2)*(a/c)^(3/8)*arctan(-(sqrt(-sqrt(2) + 2)*(a/c)^(1/8) - 2*sqrt(x))
/(sqrt(sqrt(2) + 2)*(a/c)^(1/8)))/a^3 + 65/512*sqrt(sqrt(2) + 2)*(a/c)^(3/8)*arctan((sqrt(sqrt(2) + 2)*(a/c)^(
1/8) + 2*sqrt(x))/(sqrt(-sqrt(2) + 2)*(a/c)^(1/8)))/a^3 + 65/512*sqrt(sqrt(2) + 2)*(a/c)^(3/8)*arctan(-(sqrt(s
qrt(2) + 2)*(a/c)^(1/8) - 2*sqrt(x))/(sqrt(-sqrt(2) + 2)*(a/c)^(1/8)))/a^3 + 65/1024*sqrt(-sqrt(2) + 2)*(a/c)^
(3/8)*log(sqrt(x)*sqrt(sqrt(2) + 2)*(a/c)^(1/8) + x + (a/c)^(1/4))/a^3 - 65/1024*sqrt(-sqrt(2) + 2)*(a/c)^(3/8
)*log(-sqrt(x)*sqrt(sqrt(2) + 2)*(a/c)^(1/8) + x + (a/c)^(1/4))/a^3 - 65/1024*sqrt(sqrt(2) + 2)*(a/c)^(3/8)*lo
g(sqrt(x)*sqrt(-sqrt(2) + 2)*(a/c)^(1/8) + x + (a/c)^(1/4))/a^3 + 65/1024*sqrt(sqrt(2) + 2)*(a/c)^(3/8)*log(-s
qrt(x)*sqrt(-sqrt(2) + 2)*(a/c)^(1/8) + x + (a/c)^(1/4))/a^3 + 1/64*(13*c*x^(11/2) + 21*a*x^(3/2))/((c*x^4 + a
)^2*a^2)