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

Optimal. Leaf size=299 \[ -\frac{c^{3/8} \log \left (-\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{4 \sqrt{2} (-a)^{11/8}}+\frac{c^{3/8} \log \left (\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{4 \sqrt{2} (-a)^{11/8}}-\frac{c^{3/8} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{2 \sqrt{2} (-a)^{11/8}}+\frac{c^{3/8} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}+1\right )}{2 \sqrt{2} (-a)^{11/8}}-\frac{c^{3/8} \tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{11/8}}-\frac{c^{3/8} \tanh ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{11/8}}-\frac{2}{3 a x^{3/2}} \]

[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))

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Rubi [A]  time = 0.285689, antiderivative size = 299, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 12, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.8, Rules used = {325, 329, 301, 211, 1165, 628, 1162, 617, 204, 212, 208, 205} \[ -\frac{c^{3/8} \log \left (-\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{4 \sqrt{2} (-a)^{11/8}}+\frac{c^{3/8} \log \left (\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{4 \sqrt{2} (-a)^{11/8}}-\frac{c^{3/8} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{2 \sqrt{2} (-a)^{11/8}}+\frac{c^{3/8} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}+1\right )}{2 \sqrt{2} (-a)^{11/8}}-\frac{c^{3/8} \tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{11/8}}-\frac{c^{3/8} \tanh ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{11/8}}-\frac{2}{3 a x^{3/2}} \]

Antiderivative was successfully verified.

[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 325

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 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 301

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 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])

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]

Rubi steps

\begin{align*} \int \frac{1}{x^{5/2} \left (a+c x^4\right )} \, dx &=-\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) \operatorname{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} \operatorname{Subst}\left (\int \frac{1}{\sqrt{-a}-\sqrt{c} x^4} \, dx,x,\sqrt{x}\right )}{a}-\frac{\sqrt{c} \operatorname{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} \operatorname{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} \operatorname{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} \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)^{5/4}}+\frac{\sqrt{c} \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)^{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} \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)^{5/4}}+\frac{\sqrt [4]{c} \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)^{5/4}}-\frac{c^{3/8} \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)^{11/8}}-\frac{c^{3/8} \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)^{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} \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)^{11/8}}-\frac{c^{3/8} \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)^{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 [C]  time = 0.0062103, size = 29, normalized size = 0.1 \[ -\frac{2 \, _2F_1\left (-\frac{3}{8},1;\frac{5}{8};-\frac{c x^4}{a}\right )}{3 a x^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^(5/2)*(a + c*x^4)),x]

[Out]

(-2*Hypergeometric2F1[-3/8, 1, 5/8, -((c*x^4)/a)])/(3*a*x^(3/2))

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Maple [C]  time = 0.008, size = 38, normalized size = 0.1 \begin{align*} -{\frac{2}{3\,a}{x}^{-{\frac{3}{2}}}}-{\frac{1}{4\,a}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{8}c+a \right ) }{\frac{1}{{{\it \_R}}^{3}}\ln \left ( \sqrt{x}-{\it \_R} \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3/a/x^(3/2)-1/4/a*sum(1/_R^3*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{3}{2}}}{a c x^{4} + a^{2}}\,{d x} - \frac{2}{3 \, a x^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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Fricas [B]  time = 1.68601, size = 1326, normalized size = 4.43 \begin{align*} -\frac{12 \, \sqrt{2} a x^{2} \left (-\frac{c^{3}}{a^{11}}\right )^{\frac{1}{8}} \arctan \left (-\frac{\sqrt{2} a^{4} c^{2} \sqrt{x} \left (-\frac{c^{3}}{a^{11}}\right )^{\frac{3}{8}} - \sqrt{2} \sqrt{\sqrt{2} a^{7} c^{2} \sqrt{x} \left (-\frac{c^{3}}{a^{11}}\right )^{\frac{5}{8}} - a^{3} c^{3} \left (-\frac{c^{3}}{a^{11}}\right )^{\frac{1}{4}} + c^{4} x} a^{4} \left (-\frac{c^{3}}{a^{11}}\right )^{\frac{3}{8}} - c^{3}}{c^{3}}\right ) + 12 \, \sqrt{2} a x^{2} \left (-\frac{c^{3}}{a^{11}}\right )^{\frac{1}{8}} \arctan \left (-\frac{\sqrt{2} a^{4} c^{2} \sqrt{x} \left (-\frac{c^{3}}{a^{11}}\right )^{\frac{3}{8}} - \sqrt{2} \sqrt{-\sqrt{2} a^{7} c^{2} \sqrt{x} \left (-\frac{c^{3}}{a^{11}}\right )^{\frac{5}{8}} - a^{3} c^{3} \left (-\frac{c^{3}}{a^{11}}\right )^{\frac{1}{4}} + c^{4} x} a^{4} \left (-\frac{c^{3}}{a^{11}}\right )^{\frac{3}{8}} + c^{3}}{c^{3}}\right ) + 3 \, \sqrt{2} a x^{2} \left (-\frac{c^{3}}{a^{11}}\right )^{\frac{1}{8}} \log \left (\sqrt{2} a^{7} c^{2} \sqrt{x} \left (-\frac{c^{3}}{a^{11}}\right )^{\frac{5}{8}} - a^{3} c^{3} \left (-\frac{c^{3}}{a^{11}}\right )^{\frac{1}{4}} + c^{4} x\right ) - 3 \, \sqrt{2} a x^{2} \left (-\frac{c^{3}}{a^{11}}\right )^{\frac{1}{8}} \log \left (-\sqrt{2} a^{7} c^{2} \sqrt{x} \left (-\frac{c^{3}}{a^{11}}\right )^{\frac{5}{8}} - a^{3} c^{3} \left (-\frac{c^{3}}{a^{11}}\right )^{\frac{1}{4}} + c^{4} x\right ) - 24 \, a x^{2} \left (-\frac{c^{3}}{a^{11}}\right )^{\frac{1}{8}} \arctan \left (-\frac{a^{4} c^{2} \sqrt{x} \left (-\frac{c^{3}}{a^{11}}\right )^{\frac{3}{8}} - \sqrt{-a^{3} c^{3} \left (-\frac{c^{3}}{a^{11}}\right )^{\frac{1}{4}} + c^{4} x} a^{4} \left (-\frac{c^{3}}{a^{11}}\right )^{\frac{3}{8}}}{c^{3}}\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 \, 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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/24*(12*sqrt(2)*a*x^2*(-c^3/a^11)^(1/8)*arctan(-(sqrt(2)*a^4*c^2*sqrt(x)*(-c^3/a^11)^(3/8) - sqrt(2)*sqrt(sq
rt(2)*a^7*c^2*sqrt(x)*(-c^3/a^11)^(5/8) - a^3*c^3*(-c^3/a^11)^(1/4) + c^4*x)*a^4*(-c^3/a^11)^(3/8) - c^3)/c^3)
+ 12*sqrt(2)*a*x^2*(-c^3/a^11)^(1/8)*arctan(-(sqrt(2)*a^4*c^2*sqrt(x)*(-c^3/a^11)^(3/8) - sqrt(2)*sqrt(-sqrt(
2)*a^7*c^2*sqrt(x)*(-c^3/a^11)^(5/8) - a^3*c^3*(-c^3/a^11)^(1/4) + c^4*x)*a^4*(-c^3/a^11)^(3/8) + c^3)/c^3) +
3*sqrt(2)*a*x^2*(-c^3/a^11)^(1/8)*log(sqrt(2)*a^7*c^2*sqrt(x)*(-c^3/a^11)^(5/8) - a^3*c^3*(-c^3/a^11)^(1/4) +
c^4*x) - 3*sqrt(2)*a*x^2*(-c^3/a^11)^(1/8)*log(-sqrt(2)*a^7*c^2*sqrt(x)*(-c^3/a^11)^(5/8) - a^3*c^3*(-c^3/a^11
)^(1/4) + c^4*x) - 24*a*x^2*(-c^3/a^11)^(1/8)*arctan(-(a^4*c^2*sqrt(x)*(-c^3/a^11)^(3/8) - sqrt(-a^3*c^3*(-c^3
/a^11)^(1/4) + c^4*x)*a^4*(-c^3/a^11)^(3/8))/c^3) - 6*a*x^2*(-c^3/a^11)^(1/8)*log(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)

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

[Out]

Timed out

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Giac [B]  time = 1.31541, size = 612, normalized size = 2.05 \begin{align*} \frac{c \sqrt{-\sqrt{2} + 2} \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 )}{4 \, a^{2}} + \frac{c \sqrt{-\sqrt{2} + 2} \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 )}{4 \, a^{2}} - \frac{c \sqrt{\sqrt{2} + 2} \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 )}{4 \, a^{2}} - \frac{c \sqrt{\sqrt{2} + 2} \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 )}{4 \, a^{2}} + \frac{c \sqrt{-\sqrt{2} + 2} \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 )}{8 \, a^{2}} - \frac{c \sqrt{-\sqrt{2} + 2} \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 )}{8 \, a^{2}} - \frac{c \sqrt{\sqrt{2} + 2} \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 )}{8 \, a^{2}} + \frac{c \sqrt{\sqrt{2} + 2} \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 )}{8 \, a^{2}} - \frac{2}{3 \, a x^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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