3.8.63 \(\int \frac {1}{\sqrt {x} (a+c x^4)^3} \, dx\) [763]

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

[Out]

-105/256*arctan(c^(1/8)*x^(1/2)/(-a)^(1/8))/(-a)^(23/8)/c^(1/8)-105/256*arctanh(c^(1/8)*x^(1/2)/(-a)^(1/8))/(-
a)^(23/8)/c^(1/8)-105/512*arctan(-1+c^(1/8)*2^(1/2)*x^(1/2)/(-a)^(1/8))/(-a)^(23/8)/c^(1/8)*2^(1/2)-105/512*ar
ctan(1+c^(1/8)*2^(1/2)*x^(1/2)/(-a)^(1/8))/(-a)^(23/8)/c^(1/8)*2^(1/2)+105/1024*ln((-a)^(1/4)+c^(1/4)*x-(-a)^(
1/8)*c^(1/8)*2^(1/2)*x^(1/2))/(-a)^(23/8)/c^(1/8)*2^(1/2)-105/1024*ln((-a)^(1/4)+c^(1/4)*x+(-a)^(1/8)*c^(1/8)*
2^(1/2)*x^(1/2))/(-a)^(23/8)/c^(1/8)*2^(1/2)+1/8*x^(1/2)/a/(c*x^4+a)^2+15/64*x^(1/2)/a^2/(c*x^4+a)

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Rubi [A]
time = 0.22, antiderivative size = 329, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 12, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {296, 335, 220, 218, 214, 211, 217, 1179, 642, 1176, 631, 210} \begin {gather*} \frac {15 \sqrt {x}}{64 a^2 \left (a+c x^4\right )}+\frac {105 \text {ArcTan}\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 \text {ArcTan}\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 \text {ArcTan}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{256 (-a)^{23/8} \sqrt [8]{c}}+\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 \tanh ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{256 (-a)^{23/8} \sqrt [8]{c}} \end {gather*}

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 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 296

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] /; Free
Q[{a, b, c, m}, x] && IGtQ[n, 0] && LtQ[p, -1] && IntBinomialQ[a, b, c, n, m, p, x]

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*} \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 \text {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 \text {Subst}\left (\int \frac {1}{\sqrt {-a}-\sqrt {c} x^4} \, dx,x,\sqrt {x}\right )}{128 (-a)^{5/2}}-\frac {105 \text {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 \text {Subst}\left (\int \frac {1}{\sqrt [4]{-a}-\sqrt [4]{c} x^2} \, dx,x,\sqrt {x}\right )}{256 (-a)^{11/4}}-\frac {105 \text {Subst}\left (\int \frac {1}{\sqrt [4]{-a}+\sqrt [4]{c} x^2} \, dx,x,\sqrt {x}\right )}{256 (-a)^{11/4}}-\frac {105 \text {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 \text {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 \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 )}{512 (-a)^{11/4} \sqrt [4]{c}}-\frac {105 \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 )}{512 (-a)^{11/4} \sqrt [4]{c}}+\frac {105 \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 )}{512 \sqrt {2} (-a)^{23/8} \sqrt [8]{c}}+\frac {105 \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 )}{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 \text {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 \text {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*}

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Mathematica [A]
time = 1.01, size = 287, normalized size = 0.87 \begin {gather*} \frac {\frac {8 a^{7/8} \sqrt {x} \left (23 a+15 c x^4\right )}{\left (a+c x^4\right )^2}-\frac {105 \sqrt {2+\sqrt {2}} \tan ^{-1}\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 [8]{c}}-\frac {105 \sqrt {2-\sqrt {2}} \tan ^{-1}\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 [8]{c}}+\frac {105 \sqrt {2+\sqrt {2}} \tanh ^{-1}\left (\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{a} \sqrt [8]{c} \sqrt {x}}{\sqrt [4]{a}+\sqrt [4]{c} x}\right )}{\sqrt [8]{c}}+\frac {105 \sqrt {2-\sqrt {2}} \tanh ^{-1}\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 )}{\sqrt [8]{c}}}{512 a^{23/8}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((8*a^(7/8)*Sqrt[x]*(23*a + 15*c*x^4))/(a + c*x^4)^2 - (105*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])])/c^(1/8) - (105*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])])/c^(1/8) + (105*Sqrt[2 + Sqrt[2]]*ArcTanh[(Sqrt[2 + Sqrt[2]]*a
^(1/8)*c^(1/8)*Sqrt[x])/(a^(1/4) + c^(1/4)*x)])/c^(1/8) + (105*Sqrt[2 - Sqrt[2]]*ArcTanh[(a^(1/8)*c^(1/8)*Sqrt
[-((-2 + Sqrt[2])*x)])/(a^(1/4) + c^(1/4)*x)])/c^(1/8))/(512*a^(23/8))

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.14, size = 62, normalized size = 0.19

method	result	size
derivativedivides	\(\frac {\frac {23 \sqrt {x}}{64 a}+\frac {15 c \,x^{\frac {9}{2}}}{64 a^{2}}}{\left (x^{4} c +a \right )^{2}}+\frac {105 \left (\munderset {\textit {\_R} =\RootOf \left (c \,\textit {\_Z}^{8}+a \right )}{\sum }\frac {\ln \left (\sqrt {x}-\textit {\_R} \right )}{\textit {\_R}^{7}}\right )}{512 a^{2} c}\)	\(62\)
default	\(\frac {\frac {23 \sqrt {x}}{64 a}+\frac {15 c \,x^{\frac {9}{2}}}{64 a^{2}}}{\left (x^{4} c +a \right )^{2}}+\frac {105 \left (\munderset {\textit {\_R} =\RootOf \left (c \,\textit {\_Z}^{8}+a \right )}{\sum }\frac {\ln \left (\sqrt {x}-\textit {\_R} \right )}{\textit {\_R}^{7}}\right )}{512 a^{2} c}\)	\(62\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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 = 0.00, 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/(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] Leaf count of result is larger than twice the leaf count of optimal. 631 vs. \(2 (224) = 448\).
time = 0.39, size = 631, normalized size = 1.92 \begin {gather*} \frac {420 \, \sqrt {2} {\left (a^{2} c^{2} x^{8} + 2 \, a^{3} c x^{4} + a^{4}\right )} \left (-\frac {1}{a^{23} c}\right )^{\frac {1}{8}} \arctan \left (\sqrt {2} \sqrt {a^{6} \left (-\frac {1}{a^{23} c}\right )^{\frac {1}{4}} + \sqrt {2} a^{3} \sqrt {x} \left (-\frac {1}{a^{23} c}\right )^{\frac {1}{8}} + x} a^{20} c \left (-\frac {1}{a^{23} c}\right )^{\frac {7}{8}} - \sqrt {2} a^{20} c \sqrt {x} \left (-\frac {1}{a^{23} c}\right )^{\frac {7}{8}} + 1\right ) + 420 \, \sqrt {2} {\left (a^{2} c^{2} x^{8} + 2 \, a^{3} c x^{4} + a^{4}\right )} \left (-\frac {1}{a^{23} c}\right )^{\frac {1}{8}} \arctan \left (\sqrt {2} \sqrt {a^{6} \left (-\frac {1}{a^{23} c}\right )^{\frac {1}{4}} - \sqrt {2} a^{3} \sqrt {x} \left (-\frac {1}{a^{23} c}\right )^{\frac {1}{8}} + x} a^{20} c \left (-\frac {1}{a^{23} c}\right )^{\frac {7}{8}} - \sqrt {2} a^{20} c \sqrt {x} \left (-\frac {1}{a^{23} c}\right )^{\frac {7}{8}} - 1\right ) + 105 \, \sqrt {2} {\left (a^{2} c^{2} x^{8} + 2 \, a^{3} c x^{4} + a^{4}\right )} \left (-\frac {1}{a^{23} c}\right )^{\frac {1}{8}} \log \left (a^{6} \left (-\frac {1}{a^{23} c}\right )^{\frac {1}{4}} + \sqrt {2} a^{3} \sqrt {x} \left (-\frac {1}{a^{23} c}\right )^{\frac {1}{8}} + x\right ) - 105 \, \sqrt {2} {\left (a^{2} c^{2} x^{8} + 2 \, a^{3} c x^{4} + a^{4}\right )} \left (-\frac {1}{a^{23} c}\right )^{\frac {1}{8}} \log \left (a^{6} \left (-\frac {1}{a^{23} c}\right )^{\frac {1}{4}} - \sqrt {2} a^{3} \sqrt {x} \left (-\frac {1}{a^{23} c}\right )^{\frac {1}{8}} + x\right ) + 840 \, {\left (a^{2} c^{2} x^{8} + 2 \, a^{3} c x^{4} + a^{4}\right )} \left (-\frac {1}{a^{23} c}\right )^{\frac {1}{8}} \arctan \left (\sqrt {a^{6} \left (-\frac {1}{a^{23} c}\right )^{\frac {1}{4}} + x} a^{20} c \left (-\frac {1}{a^{23} c}\right )^{\frac {7}{8}} - a^{20} c \sqrt {x} \left (-\frac {1}{a^{23} c}\right )^{\frac {7}{8}}\right ) + 210 \, {\left (a^{2} c^{2} x^{8} + 2 \, a^{3} c x^{4} + a^{4}\right )} \left (-\frac {1}{a^{23} c}\right )^{\frac {1}{8}} \log \left (a^{3} \left (-\frac {1}{a^{23} c}\right )^{\frac {1}{8}} + \sqrt {x}\right ) - 210 \, {\left (a^{2} c^{2} x^{8} + 2 \, a^{3} c x^{4} + a^{4}\right )} \left (-\frac {1}{a^{23} c}\right )^{\frac {1}{8}} \log \left (-a^{3} \left (-\frac {1}{a^{23} c}\right )^{\frac {1}{8}} + \sqrt {x}\right ) + 16 \, {\left (15 \, c x^{4} + 23 \, a\right )} \sqrt {x}}{1024 \, {\left (a^{2} c^{2} x^{8} + 2 \, a^{3} c x^{4} + a^{4}\right )}} \end {gather*}

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

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 472 vs. \(2 (224) = 448\).
time = 0.79, size = 472, normalized size = 1.43 \begin {gather*} \frac {105 \, \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 )}{256 \, a^{3} \sqrt {-2 \, \sqrt {2} + 4}} + \frac {105 \, \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 )}{256 \, a^{3} \sqrt {-2 \, \sqrt {2} + 4}} + \frac {105 \, \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 )}{256 \, a^{3} \sqrt {2 \, \sqrt {2} + 4}} + \frac {105 \, \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 )}{256 \, a^{3} \sqrt {2 \, \sqrt {2} + 4}} + \frac {105 \, \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 )}{512 \, a^{3} \sqrt {-2 \, \sqrt {2} + 4}} - \frac {105 \, \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 )}{512 \, a^{3} \sqrt {-2 \, \sqrt {2} + 4}} + \frac {105 \, \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 )}{512 \, a^{3} \sqrt {2 \, \sqrt {2} + 4}} - \frac {105 \, \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 )}{512 \, a^{3} \sqrt {2 \, \sqrt {2} + 4}} + \frac {15 \, c x^{\frac {9}{2}} + 23 \, a \sqrt {x}}{64 \, {\left (c x^{4} + a\right )}^{2} a^{2}} \end {gather*}

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]

105/256*(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*
sqrt(-2*sqrt(2) + 4)) + 105/256*(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*sqrt(-2*sqrt(2) + 4)) + 105/256*(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*sqrt(2*sqrt(2) + 4)) + 105/256*(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*sqrt(2*sqrt(2) + 4)) + 105/512*(a/c)^
(1/8)*log(sqrt(x)*sqrt(sqrt(2) + 2)*(a/c)^(1/8) + x + (a/c)^(1/4))/(a^3*sqrt(-2*sqrt(2) + 4)) - 105/512*(a/c)^
(1/8)*log(-sqrt(x)*sqrt(sqrt(2) + 2)*(a/c)^(1/8) + x + (a/c)^(1/4))/(a^3*sqrt(-2*sqrt(2) + 4)) + 105/512*(a/c)
^(1/8)*log(sqrt(x)*sqrt(-sqrt(2) + 2)*(a/c)^(1/8) + x + (a/c)^(1/4))/(a^3*sqrt(2*sqrt(2) + 4)) - 105/512*(a/c)
^(1/8)*log(-sqrt(x)*sqrt(-sqrt(2) + 2)*(a/c)^(1/8) + x + (a/c)^(1/4))/(a^3*sqrt(2*sqrt(2) + 4)) + 1/64*(15*c*x
^(9/2) + 23*a*sqrt(x))/((c*x^4 + a)^2*a^2)

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Mupad [B]
time = 0.14, size = 157, normalized size = 0.48 \begin {gather*} \frac {\frac {23\,\sqrt {x}}{64\,a}+\frac {15\,c\,x^{9/2}}{64\,a^2}}{a^2+2\,a\,c\,x^4+c^2\,x^8}-\frac {105\,\mathrm {atan}\left (\frac {c^{1/8}\,\sqrt {x}}{{\left (-a\right )}^{1/8}}\right )}{256\,{\left (-a\right )}^{23/8}\,c^{1/8}}+\frac {\mathrm {atan}\left (\frac {c^{1/8}\,\sqrt {x}\,1{}\mathrm {i}}{{\left (-a\right )}^{1/8}}\right )\,105{}\mathrm {i}}{256\,{\left (-a\right )}^{23/8}\,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 {105}{512}-\frac {105}{512}{}\mathrm {i}\right )}{{\left (-a\right )}^{23/8}\,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 {105}{512}+\frac {105}{512}{}\mathrm {i}\right )}{{\left (-a\right )}^{23/8}\,c^{1/8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^(1/2)*(a + c*x^4)^3),x)

[Out]

((23*x^(1/2))/(64*a) + (15*c*x^(9/2))/(64*a^2))/(a^2 + c^2*x^8 + 2*a*c*x^4) - (105*atan((c^(1/8)*x^(1/2))/(-a)
^(1/8)))/(256*(-a)^(23/8)*c^(1/8)) + (atan((c^(1/8)*x^(1/2)*1i)/(-a)^(1/8))*105i)/(256*(-a)^(23/8)*c^(1/8)) -
(2^(1/2)*atan((2^(1/2)*c^(1/8)*x^(1/2)*(1/2 - 1i/2))/(-a)^(1/8))*(105/512 + 105i/512))/((-a)^(23/8)*c^(1/8)) -
(2^(1/2)*atan((2^(1/2)*c^(1/8)*x^(1/2)*(1/2 + 1i/2))/(-a)^(1/8))*(105/512 - 105i/512))/((-a)^(23/8)*c^(1/8))

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