\(\int \frac {1}{x^{5/2} (a+c x^4)} \, dx\) [745]

Optimal result

Integrand size = 15, antiderivative size = 299 \[ \int \frac {1}{x^{5/2} \left (a+c x^4\right )} \, dx=-\frac {2}{3 a x^{3/2}}-\frac {c^{3/8} \arctan \left (1-\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 \sqrt {2} (-a)^{11/8}}+\frac {c^{3/8} \arctan \left (1+\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 \sqrt {2} (-a)^{11/8}}-\frac {c^{3/8} \arctan \left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{11/8}}-\frac {c^{3/8} \text {arctanh}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{11/8}}-\frac {c^{3/8} \log \left (\sqrt [4]{-a}-\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{c} x\right )}{4 \sqrt {2} (-a)^{11/8}}+\frac {c^{3/8} \log \left (\sqrt [4]{-a}+\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{c} x\right )}{4 \sqrt {2} (-a)^{11/8}} \]

[Out]

-2/3/a/x^(3/2)-1/2*c^(3/8)*arctan(c^(1/8)*x^(1/2)/(-a)^(1/8))/(-a)^(11/8)-1/2*c^(3/8)*arctanh(c^(1/8)*x^(1/2)/
(-a)^(1/8))/(-a)^(11/8)+1/4*c^(3/8)*arctan(-1+c^(1/8)*2^(1/2)*x^(1/2)/(-a)^(1/8))/(-a)^(11/8)*2^(1/2)+1/4*c^(3
/8)*arctan(1+c^(1/8)*2^(1/2)*x^(1/2)/(-a)^(1/8))/(-a)^(11/8)*2^(1/2)-1/8*c^(3/8)*ln((-a)^(1/4)+c^(1/4)*x-(-a)^
(1/8)*c^(1/8)*2^(1/2)*x^(1/2))/(-a)^(11/8)*2^(1/2)+1/8*c^(3/8)*ln((-a)^(1/4)+c^(1/4)*x+(-a)^(1/8)*c^(1/8)*2^(1
/2)*x^(1/2))/(-a)^(11/8)*2^(1/2)

Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 299, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {331, 335, 307, 217, 1179, 642, 1176, 631, 210, 218, 214, 211} \[ \int \frac {1}{x^{5/2} \left (a+c x^4\right )} \, dx=-\frac {c^{3/8} \arctan \left (1-\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 \sqrt {2} (-a)^{11/8}}+\frac {c^{3/8} \arctan \left (\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}+1\right )}{2 \sqrt {2} (-a)^{11/8}}-\frac {c^{3/8} \arctan \left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{11/8}}-\frac {c^{3/8} \text {arctanh}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{11/8}}-\frac {c^{3/8} \log \left (-\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{4 \sqrt {2} (-a)^{11/8}}+\frac {c^{3/8} \log \left (\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{4 \sqrt {2} (-a)^{11/8}}-\frac {2}{3 a x^{3/2}} \]

[In]

Int[1/(x^(5/2)*(a + c*x^4)),x]

[Out]

-2/(3*a*x^(3/2)) - (c^(3/8)*ArcTan[1 - (Sqrt[2]*c^(1/8)*Sqrt[x])/(-a)^(1/8)])/(2*Sqrt[2]*(-a)^(11/8)) + (c^(3/
8)*ArcTan[1 + (Sqrt[2]*c^(1/8)*Sqrt[x])/(-a)^(1/8)])/(2*Sqrt[2]*(-a)^(11/8)) - (c^(3/8)*ArcTan[(c^(1/8)*Sqrt[x
])/(-a)^(1/8)])/(2*(-a)^(11/8)) - (c^(3/8)*ArcTanh[(c^(1/8)*Sqrt[x])/(-a)^(1/8)])/(2*(-a)^(11/8)) - (c^(3/8)*L
og[(-a)^(1/4) - Sqrt[2]*(-a)^(1/8)*c^(1/8)*Sqrt[x] + c^(1/4)*x])/(4*Sqrt[2]*(-a)^(11/8)) + (c^(3/8)*Log[(-a)^(
1/4) + Sqrt[2]*(-a)^(1/8)*c^(1/8)*Sqrt[x] + c^(1/4)*x])/(4*Sqrt[2]*(-a)^(11/8))

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 307

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

Rule 331

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*c
*(m + 1))), x] - Dist[b*((m + n*(p + 1) + 1)/(a*c^n*(m + 1))), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -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*} \text {integral}& = -\frac {2}{3 a x^{3/2}}-\frac {c \int \frac {x^{3/2}}{a+c x^4} \, dx}{a} \\ & = -\frac {2}{3 a x^{3/2}}-\frac {(2 c) \text {Subst}\left (\int \frac {x^4}{a+c x^8} \, dx,x,\sqrt {x}\right )}{a} \\ & = -\frac {2}{3 a x^{3/2}}+\frac {\sqrt {c} \text {Subst}\left (\int \frac {1}{\sqrt {-a}-\sqrt {c} x^4} \, dx,x,\sqrt {x}\right )}{a}-\frac {\sqrt {c} \text {Subst}\left (\int \frac {1}{\sqrt {-a}+\sqrt {c} x^4} \, dx,x,\sqrt {x}\right )}{a} \\ & = -\frac {2}{3 a x^{3/2}}-\frac {\sqrt {c} \text {Subst}\left (\int \frac {1}{\sqrt [4]{-a}-\sqrt [4]{c} x^2} \, dx,x,\sqrt {x}\right )}{2 (-a)^{5/4}}-\frac {\sqrt {c} \text {Subst}\left (\int \frac {1}{\sqrt [4]{-a}+\sqrt [4]{c} x^2} \, dx,x,\sqrt {x}\right )}{2 (-a)^{5/4}}+\frac {\sqrt {c} \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)^{5/4}}+\frac {\sqrt {c} \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)^{5/4}} \\ & = -\frac {2}{3 a x^{3/2}}-\frac {c^{3/8} \tan ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{11/8}}-\frac {c^{3/8} \tanh ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{11/8}}+\frac {\sqrt [4]{c} \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)^{5/4}}+\frac {\sqrt [4]{c} \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)^{5/4}}-\frac {c^{3/8} \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)^{11/8}}-\frac {c^{3/8} \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)^{11/8}} \\ & = -\frac {2}{3 a x^{3/2}}-\frac {c^{3/8} \tan ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{11/8}}-\frac {c^{3/8} \tanh ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{11/8}}-\frac {c^{3/8} \log \left (\sqrt [4]{-a}-\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{c} x\right )}{4 \sqrt {2} (-a)^{11/8}}+\frac {c^{3/8} \log \left (\sqrt [4]{-a}+\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{c} x\right )}{4 \sqrt {2} (-a)^{11/8}}+\frac {c^{3/8} \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)^{11/8}}-\frac {c^{3/8} \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)^{11/8}} \\ & = -\frac {2}{3 a x^{3/2}}-\frac {c^{3/8} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 \sqrt {2} (-a)^{11/8}}+\frac {c^{3/8} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 \sqrt {2} (-a)^{11/8}}-\frac {c^{3/8} \tan ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{11/8}}-\frac {c^{3/8} \tanh ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{11/8}}-\frac {c^{3/8} \log \left (\sqrt [4]{-a}-\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{c} x\right )}{4 \sqrt {2} (-a)^{11/8}}+\frac {c^{3/8} \log \left (\sqrt [4]{-a}+\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{c} x\right )}{4 \sqrt {2} (-a)^{11/8}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.71 (sec) , antiderivative size = 288, normalized size of antiderivative = 0.96 \[ \int \frac {1}{x^{5/2} \left (a+c x^4\right )} \, dx=-\frac {8 a^{3/8}+3 \sqrt {2-\sqrt {2}} c^{3/8} x^{3/2} \arctan \left (\frac {\sqrt {1-\frac {1}{\sqrt {2}}} \left (\sqrt [4]{a}-\sqrt [4]{c} x\right )}{\sqrt [8]{a} \sqrt [8]{c} \sqrt {x}}\right )-3 \sqrt {2+\sqrt {2}} c^{3/8} x^{3/2} \arctan \left (\frac {\sqrt {1+\frac {1}{\sqrt {2}}} \left (\sqrt [4]{a}-\sqrt [4]{c} x\right )}{\sqrt [8]{a} \sqrt [8]{c} \sqrt {x}}\right )-3 \sqrt {2-\sqrt {2}} c^{3/8} x^{3/2} \text {arctanh}\left (\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{a} \sqrt [8]{c} \sqrt {x}}{\sqrt [4]{a}+\sqrt [4]{c} x}\right )+3 \sqrt {2+\sqrt {2}} c^{3/8} x^{3/2} \text {arctanh}\left (\frac {\sqrt [8]{a} \sqrt [8]{c} \sqrt {-\left (\left (-2+\sqrt {2}\right ) x\right )}}{\sqrt [4]{a}+\sqrt [4]{c} x}\right )}{12 a^{11/8} x^{3/2}} \]

[In]

Integrate[1/(x^(5/2)*(a + c*x^4)),x]

[Out]

-1/12*(8*a^(3/8) + 3*Sqrt[2 - Sqrt[2]]*c^(3/8)*x^(3/2)*ArcTan[(Sqrt[1 - 1/Sqrt[2]]*(a^(1/4) - c^(1/4)*x))/(a^(
1/8)*c^(1/8)*Sqrt[x])] - 3*Sqrt[2 + Sqrt[2]]*c^(3/8)*x^(3/2)*ArcTan[(Sqrt[1 + 1/Sqrt[2]]*(a^(1/4) - c^(1/4)*x)
)/(a^(1/8)*c^(1/8)*Sqrt[x])] - 3*Sqrt[2 - Sqrt[2]]*c^(3/8)*x^(3/2)*ArcTanh[(Sqrt[2 + Sqrt[2]]*a^(1/8)*c^(1/8)*
Sqrt[x])/(a^(1/4) + c^(1/4)*x)] + 3*Sqrt[2 + Sqrt[2]]*c^(3/8)*x^(3/2)*ArcTanh[(a^(1/8)*c^(1/8)*Sqrt[-((-2 + Sq
rt[2])*x)])/(a^(1/4) + c^(1/4)*x)])/(a^(11/8)*x^(3/2))

Maple [C] (verified)

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

Time = 3.96 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.13

method	result	size
derivativedivides	\(-\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (c \,\textit {\_Z}^{8}+a \right )}{\sum }\frac {\ln \left (\sqrt {x}-\textit {\_R} \right )}{\textit {\_R}^{3}}}{4 a}-\frac {2}{3 a \,x^{\frac {3}{2}}}\)	\(38\)
default	\(-\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (c \,\textit {\_Z}^{8}+a \right )}{\sum }\frac {\ln \left (\sqrt {x}-\textit {\_R} \right )}{\textit {\_R}^{3}}}{4 a}-\frac {2}{3 a \,x^{\frac {3}{2}}}\)	\(38\)
risch	\(-\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (c \,\textit {\_Z}^{8}+a \right )}{\sum }\frac {\ln \left (\sqrt {x}-\textit {\_R} \right )}{\textit {\_R}^{3}}}{4 a}-\frac {2}{3 a \,x^{\frac {3}{2}}}\)	\(38\)

[In]

int(1/x^(5/2)/(c*x^4+a),x,method=_RETURNVERBOSE)

[Out]

-1/4/a*sum(1/_R^3*ln(x^(1/2)-_R),_R=RootOf(_Z^8*c+a))-2/3/a/x^(3/2)

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.28 (sec) , antiderivative size = 357, normalized size of antiderivative = 1.19 \[ \int \frac {1}{x^{5/2} \left (a+c x^4\right )} \, dx=-\frac {\left (3 i + 3\right ) \, \sqrt {2} a x^{2} \left (-\frac {c^{3}}{a^{11}}\right )^{\frac {1}{8}} \log \left (\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {2} a^{7} \left (-\frac {c^{3}}{a^{11}}\right )^{\frac {5}{8}} + c^{2} \sqrt {x}\right ) - \left (3 i - 3\right ) \, \sqrt {2} a x^{2} \left (-\frac {c^{3}}{a^{11}}\right )^{\frac {1}{8}} \log \left (-\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} a^{7} \left (-\frac {c^{3}}{a^{11}}\right )^{\frac {5}{8}} + c^{2} \sqrt {x}\right ) + \left (3 i - 3\right ) \, \sqrt {2} a x^{2} \left (-\frac {c^{3}}{a^{11}}\right )^{\frac {1}{8}} \log \left (\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} a^{7} \left (-\frac {c^{3}}{a^{11}}\right )^{\frac {5}{8}} + c^{2} \sqrt {x}\right ) - \left (3 i + 3\right ) \, \sqrt {2} a x^{2} \left (-\frac {c^{3}}{a^{11}}\right )^{\frac {1}{8}} \log \left (-\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {2} a^{7} \left (-\frac {c^{3}}{a^{11}}\right )^{\frac {5}{8}} + c^{2} \sqrt {x}\right ) - 6 \, a x^{2} \left (-\frac {c^{3}}{a^{11}}\right )^{\frac {1}{8}} \log \left (a^{7} \left (-\frac {c^{3}}{a^{11}}\right )^{\frac {5}{8}} + c^{2} \sqrt {x}\right ) - 6 i \, a x^{2} \left (-\frac {c^{3}}{a^{11}}\right )^{\frac {1}{8}} \log \left (i \, a^{7} \left (-\frac {c^{3}}{a^{11}}\right )^{\frac {5}{8}} + c^{2} \sqrt {x}\right ) + 6 i \, a x^{2} \left (-\frac {c^{3}}{a^{11}}\right )^{\frac {1}{8}} \log \left (-i \, a^{7} \left (-\frac {c^{3}}{a^{11}}\right )^{\frac {5}{8}} + c^{2} \sqrt {x}\right ) + 6 \, a x^{2} \left (-\frac {c^{3}}{a^{11}}\right )^{\frac {1}{8}} \log \left (-a^{7} \left (-\frac {c^{3}}{a^{11}}\right )^{\frac {5}{8}} + c^{2} \sqrt {x}\right ) + 16 \, \sqrt {x}}{24 \, a x^{2}} \]

[In]

integrate(1/x^(5/2)/(c*x^4+a),x, algorithm="fricas")

[Out]

-1/24*((3*I + 3)*sqrt(2)*a*x^2*(-c^3/a^11)^(1/8)*log((1/2*I + 1/2)*sqrt(2)*a^7*(-c^3/a^11)^(5/8) + c^2*sqrt(x)
) - (3*I - 3)*sqrt(2)*a*x^2*(-c^3/a^11)^(1/8)*log(-(1/2*I - 1/2)*sqrt(2)*a^7*(-c^3/a^11)^(5/8) + c^2*sqrt(x))
+ (3*I - 3)*sqrt(2)*a*x^2*(-c^3/a^11)^(1/8)*log((1/2*I - 1/2)*sqrt(2)*a^7*(-c^3/a^11)^(5/8) + c^2*sqrt(x)) - (
3*I + 3)*sqrt(2)*a*x^2*(-c^3/a^11)^(1/8)*log(-(1/2*I + 1/2)*sqrt(2)*a^7*(-c^3/a^11)^(5/8) + c^2*sqrt(x)) - 6*a
*x^2*(-c^3/a^11)^(1/8)*log(a^7*(-c^3/a^11)^(5/8) + c^2*sqrt(x)) - 6*I*a*x^2*(-c^3/a^11)^(1/8)*log(I*a^7*(-c^3/
a^11)^(5/8) + c^2*sqrt(x)) + 6*I*a*x^2*(-c^3/a^11)^(1/8)*log(-I*a^7*(-c^3/a^11)^(5/8) + c^2*sqrt(x)) + 6*a*x^2
*(-c^3/a^11)^(1/8)*log(-a^7*(-c^3/a^11)^(5/8) + c^2*sqrt(x)) + 16*sqrt(x))/(a*x^2)

Sympy [A] (verification not implemented)

Time = 36.02 (sec) , antiderivative size = 303, normalized size of antiderivative = 1.01 \[ \int \frac {1}{x^{5/2} \left (a+c x^4\right )} \, dx=\begin {cases} \frac {\tilde {\infty }}{x^{\frac {11}{2}}} & \text {for}\: a = 0 \wedge c = 0 \\- \frac {2}{11 c x^{\frac {11}{2}}} & \text {for}\: a = 0 \\- \frac {2}{3 a x^{\frac {3}{2}}} & \text {for}\: c = 0 \\- \frac {\log {\left (\sqrt {x} - \sqrt [8]{- \frac {a}{c}} \right )}}{4 a \left (- \frac {a}{c}\right )^{\frac {3}{8}}} + \frac {\log {\left (\sqrt {x} + \sqrt [8]{- \frac {a}{c}} \right )}}{4 a \left (- \frac {a}{c}\right )^{\frac {3}{8}}} + \frac {\sqrt {2} \log {\left (- 4 \sqrt {2} \sqrt {x} \sqrt [8]{- \frac {a}{c}} + 4 x + 4 \sqrt [4]{- \frac {a}{c}} \right )}}{8 a \left (- \frac {a}{c}\right )^{\frac {3}{8}}} - \frac {\sqrt {2} \log {\left (4 \sqrt {2} \sqrt {x} \sqrt [8]{- \frac {a}{c}} + 4 x + 4 \sqrt [4]{- \frac {a}{c}} \right )}}{8 a \left (- \frac {a}{c}\right )^{\frac {3}{8}}} + \frac {\operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [8]{- \frac {a}{c}}} \right )}}{2 a \left (- \frac {a}{c}\right )^{\frac {3}{8}}} - \frac {\sqrt {2} \operatorname {atan}{\left (\frac {\sqrt {2} \sqrt {x}}{\sqrt [8]{- \frac {a}{c}}} - 1 \right )}}{4 a \left (- \frac {a}{c}\right )^{\frac {3}{8}}} - \frac {\sqrt {2} \operatorname {atan}{\left (\frac {\sqrt {2} \sqrt {x}}{\sqrt [8]{- \frac {a}{c}}} + 1 \right )}}{4 a \left (- \frac {a}{c}\right )^{\frac {3}{8}}} - \frac {2}{3 a x^{\frac {3}{2}}} & \text {otherwise} \end {cases} \]

[In]

integrate(1/x**(5/2)/(c*x**4+a),x)

[Out]

Piecewise((zoo/x**(11/2), Eq(a, 0) & Eq(c, 0)), (-2/(11*c*x**(11/2)), Eq(a, 0)), (-2/(3*a*x**(3/2)), Eq(c, 0))
, (-log(sqrt(x) - (-a/c)**(1/8))/(4*a*(-a/c)**(3/8)) + log(sqrt(x) + (-a/c)**(1/8))/(4*a*(-a/c)**(3/8)) + sqrt
(2)*log(-4*sqrt(2)*sqrt(x)*(-a/c)**(1/8) + 4*x + 4*(-a/c)**(1/4))/(8*a*(-a/c)**(3/8)) - sqrt(2)*log(4*sqrt(2)*
sqrt(x)*(-a/c)**(1/8) + 4*x + 4*(-a/c)**(1/4))/(8*a*(-a/c)**(3/8)) + atan(sqrt(x)/(-a/c)**(1/8))/(2*a*(-a/c)**
(3/8)) - sqrt(2)*atan(sqrt(2)*sqrt(x)/(-a/c)**(1/8) - 1)/(4*a*(-a/c)**(3/8)) - sqrt(2)*atan(sqrt(2)*sqrt(x)/(-
a/c)**(1/8) + 1)/(4*a*(-a/c)**(3/8)) - 2/(3*a*x**(3/2)), True))

Maxima [F]

\[ \int \frac {1}{x^{5/2} \left (a+c x^4\right )} \, dx=\int { \frac {1}{{\left (c x^{4} + a\right )} x^{\frac {5}{2}}} \,d x } \]

[In]

integrate(1/x^(5/2)/(c*x^4+a),x, algorithm="maxima")

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 461 vs. \(2 (198) = 396\).

Time = 0.42 (sec) , antiderivative size = 461, normalized size of antiderivative = 1.54 \[ \int \frac {1}{x^{5/2} \left (a+c x^4\right )} \, dx=\frac {c \left (\frac {a}{c}\right )^{\frac {5}{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^{2} \sqrt {2 \, \sqrt {2} + 4}} + \frac {c \left (\frac {a}{c}\right )^{\frac {5}{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^{2} \sqrt {2 \, \sqrt {2} + 4}} - \frac {c \left (\frac {a}{c}\right )^{\frac {5}{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^{2} \sqrt {-2 \, \sqrt {2} + 4}} - \frac {c \left (\frac {a}{c}\right )^{\frac {5}{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^{2} \sqrt {-2 \, \sqrt {2} + 4}} + \frac {c \left (\frac {a}{c}\right )^{\frac {5}{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^{2} \sqrt {2 \, \sqrt {2} + 4}} - \frac {c \left (\frac {a}{c}\right )^{\frac {5}{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^{2} \sqrt {2 \, \sqrt {2} + 4}} - \frac {c \left (\frac {a}{c}\right )^{\frac {5}{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^{2} \sqrt {-2 \, \sqrt {2} + 4}} + \frac {c \left (\frac {a}{c}\right )^{\frac {5}{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^{2} \sqrt {-2 \, \sqrt {2} + 4}} - \frac {2}{3 \, a x^{\frac {3}{2}}} \]

[In]

integrate(1/x^(5/2)/(c*x^4+a),x, algorithm="giac")

[Out]

1/2*c*(a/c)^(5/8)*arctan((sqrt(-sqrt(2) + 2)*(a/c)^(1/8) + 2*sqrt(x))/(sqrt(sqrt(2) + 2)*(a/c)^(1/8)))/(a^2*sq
rt(2*sqrt(2) + 4)) + 1/2*c*(a/c)^(5/8)*arctan(-(sqrt(-sqrt(2) + 2)*(a/c)^(1/8) - 2*sqrt(x))/(sqrt(sqrt(2) + 2)
*(a/c)^(1/8)))/(a^2*sqrt(2*sqrt(2) + 4)) - 1/2*c*(a/c)^(5/8)*arctan((sqrt(sqrt(2) + 2)*(a/c)^(1/8) + 2*sqrt(x)
)/(sqrt(-sqrt(2) + 2)*(a/c)^(1/8)))/(a^2*sqrt(-2*sqrt(2) + 4)) - 1/2*c*(a/c)^(5/8)*arctan(-(sqrt(sqrt(2) + 2)*
(a/c)^(1/8) - 2*sqrt(x))/(sqrt(-sqrt(2) + 2)*(a/c)^(1/8)))/(a^2*sqrt(-2*sqrt(2) + 4)) + 1/4*c*(a/c)^(5/8)*log(
sqrt(x)*sqrt(sqrt(2) + 2)*(a/c)^(1/8) + x + (a/c)^(1/4))/(a^2*sqrt(2*sqrt(2) + 4)) - 1/4*c*(a/c)^(5/8)*log(-sq
rt(x)*sqrt(sqrt(2) + 2)*(a/c)^(1/8) + x + (a/c)^(1/4))/(a^2*sqrt(2*sqrt(2) + 4)) - 1/4*c*(a/c)^(5/8)*log(sqrt(
x)*sqrt(-sqrt(2) + 2)*(a/c)^(1/8) + x + (a/c)^(1/4))/(a^2*sqrt(-2*sqrt(2) + 4)) + 1/4*c*(a/c)^(5/8)*log(-sqrt(
x)*sqrt(-sqrt(2) + 2)*(a/c)^(1/8) + x + (a/c)^(1/4))/(a^2*sqrt(-2*sqrt(2) + 4)) - 2/3/(a*x^(3/2))

Mupad [B] (verification not implemented)

Time = 5.69 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.42 \[ \int \frac {1}{x^{5/2} \left (a+c x^4\right )} \, dx=\frac {{\left (-c\right )}^{3/8}\,\mathrm {atan}\left (\frac {{\left (-c\right )}^{1/8}\,\sqrt {x}}{a^{1/8}}\right )}{2\,a^{11/8}}-\frac {2}{3\,a\,x^{3/2}}-\frac {{\left (-c\right )}^{3/8}\,\mathrm {atan}\left (\frac {{\left (-c\right )}^{1/8}\,\sqrt {x}\,1{}\mathrm {i}}{a^{1/8}}\right )\,1{}\mathrm {i}}{2\,a^{11/8}}+\frac {\sqrt {2}\,{\left (-c\right )}^{3/8}\,\mathrm {atan}\left (\frac {\sqrt {2}\,{\left (-c\right )}^{1/8}\,\sqrt {x}\,\left (\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right )}{a^{1/8}}\right )\,\left (-\frac {1}{4}-\frac {1}{4}{}\mathrm {i}\right )}{a^{11/8}}+\frac {\sqrt {2}\,{\left (-c\right )}^{3/8}\,\mathrm {atan}\left (\frac {\sqrt {2}\,{\left (-c\right )}^{1/8}\,\sqrt {x}\,\left (\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right )}{a^{1/8}}\right )\,\left (-\frac {1}{4}+\frac {1}{4}{}\mathrm {i}\right )}{a^{11/8}} \]

[In]

int(1/(x^(5/2)*(a + c*x^4)),x)

[Out]

((-c)^(3/8)*atan(((-c)^(1/8)*x^(1/2))/a^(1/8)))/(2*a^(11/8)) - 2/(3*a*x^(3/2)) - ((-c)^(3/8)*atan(((-c)^(1/8)*
x^(1/2)*1i)/a^(1/8))*1i)/(2*a^(11/8)) - (2^(1/2)*(-c)^(3/8)*atan((2^(1/2)*(-c)^(1/8)*x^(1/2)*(1/2 - 1i/2))/a^(
1/8))*(1/4 + 1i/4))/a^(11/8) - (2^(1/2)*(-c)^(3/8)*atan((2^(1/2)*(-c)^(1/8)*x^(1/2)*(1/2 + 1i/2))/a^(1/8))*(1/
4 - 1i/4))/a^(11/8)