3.744 \(\int \frac{1}{x^{3/2} (a+c x^4)} \, dx\)

Optimal. Leaf size=297 \[ \frac{\sqrt [8]{c} \log \left (-\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{4 \sqrt{2} (-a)^{9/8}}-\frac{\sqrt [8]{c} \log \left (\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{4 \sqrt{2} (-a)^{9/8}}-\frac{\sqrt [8]{c} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{2 \sqrt{2} (-a)^{9/8}}+\frac{\sqrt [8]{c} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}+1\right )}{2 \sqrt{2} (-a)^{9/8}}+\frac{\sqrt [8]{c} \tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{9/8}}-\frac{\sqrt [8]{c} \tanh ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{9/8}}-\frac{2}{a \sqrt{x}} \]

[Out]

-2/(a*Sqrt[x]) - (c^(1/8)*ArcTan[1 - (Sqrt[2]*c^(1/8)*Sqrt[x])/(-a)^(1/8)])/(2*Sqrt[2]*(-a)^(9/8)) + (c^(1/8)*
ArcTan[1 + (Sqrt[2]*c^(1/8)*Sqrt[x])/(-a)^(1/8)])/(2*Sqrt[2]*(-a)^(9/8)) + (c^(1/8)*ArcTan[(c^(1/8)*Sqrt[x])/(
-a)^(1/8)])/(2*(-a)^(9/8)) - (c^(1/8)*ArcTanh[(c^(1/8)*Sqrt[x])/(-a)^(1/8)])/(2*(-a)^(9/8)) + (c^(1/8)*Log[(-a
)^(1/4) - Sqrt[2]*(-a)^(1/8)*c^(1/8)*Sqrt[x] + c^(1/4)*x])/(4*Sqrt[2]*(-a)^(9/8)) - (c^(1/8)*Log[(-a)^(1/4) +
Sqrt[2]*(-a)^(1/8)*c^(1/8)*Sqrt[x] + c^(1/4)*x])/(4*Sqrt[2]*(-a)^(9/8))

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Rubi [A]  time = 0.273812, antiderivative size = 297, 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, 297, 1162, 617, 204, 1165, 628, 298, 205, 208} \[ \frac{\sqrt [8]{c} \log \left (-\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{4 \sqrt{2} (-a)^{9/8}}-\frac{\sqrt [8]{c} \log \left (\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{4 \sqrt{2} (-a)^{9/8}}-\frac{\sqrt [8]{c} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{2 \sqrt{2} (-a)^{9/8}}+\frac{\sqrt [8]{c} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}+1\right )}{2 \sqrt{2} (-a)^{9/8}}+\frac{\sqrt [8]{c} \tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{9/8}}-\frac{\sqrt [8]{c} \tanh ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{9/8}}-\frac{2}{a \sqrt{x}} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^(3/2)*(a + c*x^4)),x]

[Out]

-2/(a*Sqrt[x]) - (c^(1/8)*ArcTan[1 - (Sqrt[2]*c^(1/8)*Sqrt[x])/(-a)^(1/8)])/(2*Sqrt[2]*(-a)^(9/8)) + (c^(1/8)*
ArcTan[1 + (Sqrt[2]*c^(1/8)*Sqrt[x])/(-a)^(1/8)])/(2*Sqrt[2]*(-a)^(9/8)) + (c^(1/8)*ArcTan[(c^(1/8)*Sqrt[x])/(
-a)^(1/8)])/(2*(-a)^(9/8)) - (c^(1/8)*ArcTanh[(c^(1/8)*Sqrt[x])/(-a)^(1/8)])/(2*(-a)^(9/8)) + (c^(1/8)*Log[(-a
)^(1/4) - Sqrt[2]*(-a)^(1/8)*c^(1/8)*Sqrt[x] + c^(1/4)*x])/(4*Sqrt[2]*(-a)^(9/8)) - (c^(1/8)*Log[(-a)^(1/4) +
Sqrt[2]*(-a)^(1/8)*c^(1/8)*Sqrt[x] + c^(1/4)*x])/(4*Sqrt[2]*(-a)^(9/8))

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 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{1}{x^{3/2} \left (a+c x^4\right )} \, dx &=-\frac{2}{a \sqrt{x}}-\frac{c \int \frac{x^{5/2}}{a+c x^4} \, dx}{a}\\ &=-\frac{2}{a \sqrt{x}}-\frac{(2 c) \operatorname{Subst}\left (\int \frac{x^6}{a+c x^8} \, dx,x,\sqrt{x}\right )}{a}\\ &=-\frac{2}{a \sqrt{x}}+\frac{\sqrt{c} \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{-a}-\sqrt{c} x^4} \, dx,x,\sqrt{x}\right )}{a}-\frac{\sqrt{c} \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{-a}+\sqrt{c} x^4} \, dx,x,\sqrt{x}\right )}{a}\\ &=-\frac{2}{a \sqrt{x}}+\frac{\sqrt [4]{c} \operatorname{Subst}\left (\int \frac{1}{\sqrt [4]{-a}-\sqrt [4]{c} x^2} \, dx,x,\sqrt{x}\right )}{2 a}-\frac{\sqrt [4]{c} \operatorname{Subst}\left (\int \frac{1}{\sqrt [4]{-a}+\sqrt [4]{c} x^2} \, dx,x,\sqrt{x}\right )}{2 a}+\frac{\sqrt [4]{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}-\frac{\sqrt [4]{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}\\ &=-\frac{2}{a \sqrt{x}}+\frac{\sqrt [8]{c} \tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{9/8}}-\frac{\sqrt [8]{c} \tanh ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{9/8}}-\frac{\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}-\frac{\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}+\frac{\sqrt [8]{c} \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)^{9/8}}+\frac{\sqrt [8]{c} \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)^{9/8}}\\ &=-\frac{2}{a \sqrt{x}}+\frac{\sqrt [8]{c} \tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{9/8}}-\frac{\sqrt [8]{c} \tanh ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{9/8}}+\frac{\sqrt [8]{c} \log \left (\sqrt [4]{-a}-\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{c} x\right )}{4 \sqrt{2} (-a)^{9/8}}-\frac{\sqrt [8]{c} \log \left (\sqrt [4]{-a}+\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{c} x\right )}{4 \sqrt{2} (-a)^{9/8}}+\frac{\sqrt [8]{c} \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)^{9/8}}-\frac{\sqrt [8]{c} \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)^{9/8}}\\ &=-\frac{2}{a \sqrt{x}}-\frac{\sqrt [8]{c} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{2 \sqrt{2} (-a)^{9/8}}+\frac{\sqrt [8]{c} \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{2 \sqrt{2} (-a)^{9/8}}+\frac{\sqrt [8]{c} \tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{9/8}}-\frac{\sqrt [8]{c} \tanh ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{9/8}}+\frac{\sqrt [8]{c} \log \left (\sqrt [4]{-a}-\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{c} x\right )}{4 \sqrt{2} (-a)^{9/8}}-\frac{\sqrt [8]{c} \log \left (\sqrt [4]{-a}+\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{c} x\right )}{4 \sqrt{2} (-a)^{9/8}}\\ \end{align*}

Mathematica [C]  time = 0.0053997, size = 27, normalized size = 0.09 \[ -\frac{2 \, _2F_1\left (-\frac{1}{8},1;\frac{7}{8};-\frac{c x^4}{a}\right )}{a \sqrt{x}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^(3/2)*(a + c*x^4)),x]

[Out]

(-2*Hypergeometric2F1[-1/8, 1, 7/8, -((c*x^4)/a)])/(a*Sqrt[x])

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Maple [C]  time = 0.007, size = 38, normalized size = 0.1 \begin{align*} -{\frac{1}{4\,a}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{8}c+a \right ) }{\frac{1}{{\it \_R}}\ln \left ( \sqrt{x}-{\it \_R} \right ) }}-2\,{\frac{1}{a\sqrt{x}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -c \int \frac{x^{\frac{5}{2}}}{a c x^{4} + a^{2}}\,{d x} - \frac{2}{a \sqrt{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-c*integrate(x^(5/2)/(a*c*x^4 + a^2), x) - 2/(a*sqrt(x))

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Fricas [B]  time = 1.68229, size = 1127, normalized size = 3.79 \begin{align*} \frac{4 \, \sqrt{2} a x \left (-\frac{c}{a^{9}}\right )^{\frac{1}{8}} \arctan \left (-\frac{\sqrt{2} a c \sqrt{x} \left (-\frac{c}{a^{9}}\right )^{\frac{1}{8}} - \sqrt{2} \sqrt{\sqrt{2} a^{8} c \sqrt{x} \left (-\frac{c}{a^{9}}\right )^{\frac{7}{8}} - a^{7} c \left (-\frac{c}{a^{9}}\right )^{\frac{3}{4}} + c^{2} x} a \left (-\frac{c}{a^{9}}\right )^{\frac{1}{8}} - c}{c}\right ) + 4 \, \sqrt{2} a x \left (-\frac{c}{a^{9}}\right )^{\frac{1}{8}} \arctan \left (-\frac{\sqrt{2} a c \sqrt{x} \left (-\frac{c}{a^{9}}\right )^{\frac{1}{8}} - \sqrt{2} \sqrt{-\sqrt{2} a^{8} c \sqrt{x} \left (-\frac{c}{a^{9}}\right )^{\frac{7}{8}} - a^{7} c \left (-\frac{c}{a^{9}}\right )^{\frac{3}{4}} + c^{2} x} a \left (-\frac{c}{a^{9}}\right )^{\frac{1}{8}} + c}{c}\right ) - \sqrt{2} a x \left (-\frac{c}{a^{9}}\right )^{\frac{1}{8}} \log \left (\sqrt{2} a^{8} c \sqrt{x} \left (-\frac{c}{a^{9}}\right )^{\frac{7}{8}} - a^{7} c \left (-\frac{c}{a^{9}}\right )^{\frac{3}{4}} + c^{2} x\right ) + \sqrt{2} a x \left (-\frac{c}{a^{9}}\right )^{\frac{1}{8}} \log \left (-\sqrt{2} a^{8} c \sqrt{x} \left (-\frac{c}{a^{9}}\right )^{\frac{7}{8}} - a^{7} c \left (-\frac{c}{a^{9}}\right )^{\frac{3}{4}} + c^{2} x\right ) + 8 \, a x \left (-\frac{c}{a^{9}}\right )^{\frac{1}{8}} \arctan \left (-\frac{a c \sqrt{x} \left (-\frac{c}{a^{9}}\right )^{\frac{1}{8}} - \sqrt{-a^{7} c \left (-\frac{c}{a^{9}}\right )^{\frac{3}{4}} + c^{2} x} a \left (-\frac{c}{a^{9}}\right )^{\frac{1}{8}}}{c}\right ) - 2 \, a x \left (-\frac{c}{a^{9}}\right )^{\frac{1}{8}} \log \left (a^{8} \left (-\frac{c}{a^{9}}\right )^{\frac{7}{8}} + c \sqrt{x}\right ) + 2 \, a x \left (-\frac{c}{a^{9}}\right )^{\frac{1}{8}} \log \left (-a^{8} \left (-\frac{c}{a^{9}}\right )^{\frac{7}{8}} + c \sqrt{x}\right ) - 16 \, \sqrt{x}}{8 \, a x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*(4*sqrt(2)*a*x*(-c/a^9)^(1/8)*arctan(-(sqrt(2)*a*c*sqrt(x)*(-c/a^9)^(1/8) - sqrt(2)*sqrt(sqrt(2)*a^8*c*sqr
t(x)*(-c/a^9)^(7/8) - a^7*c*(-c/a^9)^(3/4) + c^2*x)*a*(-c/a^9)^(1/8) - c)/c) + 4*sqrt(2)*a*x*(-c/a^9)^(1/8)*ar
ctan(-(sqrt(2)*a*c*sqrt(x)*(-c/a^9)^(1/8) - sqrt(2)*sqrt(-sqrt(2)*a^8*c*sqrt(x)*(-c/a^9)^(7/8) - a^7*c*(-c/a^9
)^(3/4) + c^2*x)*a*(-c/a^9)^(1/8) + c)/c) - sqrt(2)*a*x*(-c/a^9)^(1/8)*log(sqrt(2)*a^8*c*sqrt(x)*(-c/a^9)^(7/8
) - a^7*c*(-c/a^9)^(3/4) + c^2*x) + sqrt(2)*a*x*(-c/a^9)^(1/8)*log(-sqrt(2)*a^8*c*sqrt(x)*(-c/a^9)^(7/8) - a^7
*c*(-c/a^9)^(3/4) + c^2*x) + 8*a*x*(-c/a^9)^(1/8)*arctan(-(a*c*sqrt(x)*(-c/a^9)^(1/8) - sqrt(-a^7*c*(-c/a^9)^(
3/4) + c^2*x)*a*(-c/a^9)^(1/8))/c) - 2*a*x*(-c/a^9)^(1/8)*log(a^8*(-c/a^9)^(7/8) + c*sqrt(x)) + 2*a*x*(-c/a^9)
^(1/8)*log(-a^8*(-c/a^9)^(7/8) + c*sqrt(x)) - 16*sqrt(x))/(a*x)

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

[Out]

Timed out

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Giac [B]  time = 1.33349, size = 612, normalized size = 2.06 \begin{align*} -\frac{c \sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{7}{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{7}{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{7}{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{7}{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{7}{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{7}{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{7}{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{7}{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}{a \sqrt{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*c*sqrt(sqrt(2) + 2)*(a/c)^(7/8)*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)^(7/8)*arctan(-(sqrt(-sqrt(2) + 2)*(a/c)^(1/8) - 2*sqrt(x))/(sq
rt(sqrt(2) + 2)*(a/c)^(1/8)))/a^2 - 1/4*c*sqrt(-sqrt(2) + 2)*(a/c)^(7/8)*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)^(7/8)*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)^(7/8)*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)^(7/8)*log(-sqr
t(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)^(7/8)*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)^(7/8)*log(-sqrt(x)*sqrt(
-sqrt(2) + 2)*(a/c)^(1/8) + x + (a/c)^(1/4))/a^2 - 2/(a*sqrt(x))