3.739 \(\int \frac{x^{7/2}}{a+c x^4} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.288915, 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 = {321, 329, 214, 212, 208, 205, 211, 1165, 628, 1162, 617, 204} \[ \frac{\sqrt [8]{-a} \log \left (-\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{4 \sqrt{2} c^{9/8}}-\frac{\sqrt [8]{-a} \log \left (\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{4 \sqrt{2} c^{9/8}}+\frac{\sqrt [8]{-a} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{2 \sqrt{2} c^{9/8}}-\frac{\sqrt [8]{-a} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}+1\right )}{2 \sqrt{2} c^{9/8}}-\frac{\sqrt [8]{-a} \tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{2 c^{9/8}}-\frac{\sqrt [8]{-a} \tanh ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{2 c^{9/8}}+\frac{2 \sqrt{x}}{c} \]

Antiderivative was successfully verified.

[In]

Int[x^(7/2)/(a + c*x^4),x]

[Out]

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

Rule 321

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^n
)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && 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 214

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 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]

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])

Rubi steps

\begin{align*} \int \frac{x^{7/2}}{a+c x^4} \, dx &=\frac{2 \sqrt{x}}{c}-\frac{a \int \frac{1}{\sqrt{x} \left (a+c x^4\right )} \, dx}{c}\\ &=\frac{2 \sqrt{x}}{c}-\frac{(2 a) \operatorname{Subst}\left (\int \frac{1}{a+c x^8} \, dx,x,\sqrt{x}\right )}{c}\\ &=\frac{2 \sqrt{x}}{c}-\frac{\sqrt{-a} \operatorname{Subst}\left (\int \frac{1}{\sqrt{-a}-\sqrt{c} x^4} \, dx,x,\sqrt{x}\right )}{c}-\frac{\sqrt{-a} \operatorname{Subst}\left (\int \frac{1}{\sqrt{-a}+\sqrt{c} x^4} \, dx,x,\sqrt{x}\right )}{c}\\ &=\frac{2 \sqrt{x}}{c}-\frac{\sqrt [4]{-a} \operatorname{Subst}\left (\int \frac{1}{\sqrt [4]{-a}-\sqrt [4]{c} x^2} \, dx,x,\sqrt{x}\right )}{2 c}-\frac{\sqrt [4]{-a} \operatorname{Subst}\left (\int \frac{1}{\sqrt [4]{-a}+\sqrt [4]{c} x^2} \, dx,x,\sqrt{x}\right )}{2 c}-\frac{\sqrt [4]{-a} \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 c}-\frac{\sqrt [4]{-a} \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 c}\\ &=\frac{2 \sqrt{x}}{c}-\frac{\sqrt [8]{-a} \tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{2 c^{9/8}}-\frac{\sqrt [8]{-a} \tanh ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{2 c^{9/8}}-\frac{\sqrt [4]{-a} \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 c^{5/4}}-\frac{\sqrt [4]{-a} \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 c^{5/4}}+\frac{\sqrt [8]{-a} \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} c^{9/8}}+\frac{\sqrt [8]{-a} \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} c^{9/8}}\\ &=\frac{2 \sqrt{x}}{c}-\frac{\sqrt [8]{-a} \tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{2 c^{9/8}}-\frac{\sqrt [8]{-a} \tanh ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{2 c^{9/8}}+\frac{\sqrt [8]{-a} \log \left (\sqrt [4]{-a}-\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{c} x\right )}{4 \sqrt{2} c^{9/8}}-\frac{\sqrt [8]{-a} \log \left (\sqrt [4]{-a}+\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{c} x\right )}{4 \sqrt{2} c^{9/8}}-\frac{\sqrt [8]{-a} \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} c^{9/8}}+\frac{\sqrt [8]{-a} \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} c^{9/8}}\\ &=\frac{2 \sqrt{x}}{c}+\frac{\sqrt [8]{-a} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{2 \sqrt{2} c^{9/8}}-\frac{\sqrt [8]{-a} \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{2 \sqrt{2} c^{9/8}}-\frac{\sqrt [8]{-a} \tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{2 c^{9/8}}-\frac{\sqrt [8]{-a} \tanh ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{2 c^{9/8}}+\frac{\sqrt [8]{-a} \log \left (\sqrt [4]{-a}-\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{c} x\right )}{4 \sqrt{2} c^{9/8}}-\frac{\sqrt [8]{-a} \log \left (\sqrt [4]{-a}+\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{c} x\right )}{4 \sqrt{2} c^{9/8}}\\ \end{align*}

Mathematica [C]  time = 0.008233, size = 29, normalized size = 0.1 \[ -\frac{2 \sqrt{x} \left (\, _2F_1\left (\frac{1}{8},1;\frac{9}{8};-\frac{c x^4}{a}\right )-1\right )}{c} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(7/2)/(a + c*x^4),x]

[Out]

(-2*Sqrt[x]*(-1 + Hypergeometric2F1[1/8, 1, 9/8, -((c*x^4)/a)]))/c

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Maple [C]  time = 0.017, size = 39, normalized size = 0.1 \begin{align*} 2\,{\frac{\sqrt{x}}{c}}-{\frac{a}{4\,{c}^{2}}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{8}c+a \right ) }{\frac{1}{{{\it \_R}}^{7}}\ln \left ( \sqrt{x}-{\it \_R} \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(7/2)/(c*x^4+a),x)

[Out]

2*x^(1/2)/c-1/4*a/c^2*sum(1/_R^7*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*} \int \frac{x^{\frac{7}{2}}}{c x^{4} + a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(7/2)/(c*x^4+a),x, algorithm="maxima")

[Out]

integrate(x^(7/2)/(c*x^4 + a), x)

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Fricas [A]  time = 1.65465, size = 1031, normalized size = 3.47 \begin{align*} -\frac{4 \, \sqrt{2} c \left (-\frac{a}{c^{9}}\right )^{\frac{1}{8}} \arctan \left (\frac{\sqrt{2} \sqrt{c^{2} \left (-\frac{a}{c^{9}}\right )^{\frac{1}{4}} + \sqrt{2} c \sqrt{x} \left (-\frac{a}{c^{9}}\right )^{\frac{1}{8}} + x} c^{8} \left (-\frac{a}{c^{9}}\right )^{\frac{7}{8}} - \sqrt{2} c^{8} \sqrt{x} \left (-\frac{a}{c^{9}}\right )^{\frac{7}{8}} + a}{a}\right ) + 4 \, \sqrt{2} c \left (-\frac{a}{c^{9}}\right )^{\frac{1}{8}} \arctan \left (\frac{\sqrt{2} \sqrt{c^{2} \left (-\frac{a}{c^{9}}\right )^{\frac{1}{4}} - \sqrt{2} c \sqrt{x} \left (-\frac{a}{c^{9}}\right )^{\frac{1}{8}} + x} c^{8} \left (-\frac{a}{c^{9}}\right )^{\frac{7}{8}} - \sqrt{2} c^{8} \sqrt{x} \left (-\frac{a}{c^{9}}\right )^{\frac{7}{8}} - a}{a}\right ) + \sqrt{2} c \left (-\frac{a}{c^{9}}\right )^{\frac{1}{8}} \log \left (c^{2} \left (-\frac{a}{c^{9}}\right )^{\frac{1}{4}} + \sqrt{2} c \sqrt{x} \left (-\frac{a}{c^{9}}\right )^{\frac{1}{8}} + x\right ) - \sqrt{2} c \left (-\frac{a}{c^{9}}\right )^{\frac{1}{8}} \log \left (c^{2} \left (-\frac{a}{c^{9}}\right )^{\frac{1}{4}} - \sqrt{2} c \sqrt{x} \left (-\frac{a}{c^{9}}\right )^{\frac{1}{8}} + x\right ) + 8 \, c \left (-\frac{a}{c^{9}}\right )^{\frac{1}{8}} \arctan \left (\frac{\sqrt{c^{2} \left (-\frac{a}{c^{9}}\right )^{\frac{1}{4}} + x} c^{8} \left (-\frac{a}{c^{9}}\right )^{\frac{7}{8}} - c^{8} \sqrt{x} \left (-\frac{a}{c^{9}}\right )^{\frac{7}{8}}}{a}\right ) + 2 \, c \left (-\frac{a}{c^{9}}\right )^{\frac{1}{8}} \log \left (c \left (-\frac{a}{c^{9}}\right )^{\frac{1}{8}} + \sqrt{x}\right ) - 2 \, c \left (-\frac{a}{c^{9}}\right )^{\frac{1}{8}} \log \left (-c \left (-\frac{a}{c^{9}}\right )^{\frac{1}{8}} + \sqrt{x}\right ) - 16 \, \sqrt{x}}{8 \, c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(7/2)/(c*x^4+a),x, algorithm="fricas")

[Out]

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

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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**(7/2)/(c*x**4+a),x)

[Out]

Timed out

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Giac [B]  time = 1.28527, size = 601, normalized size = 2.02 \begin{align*} -\frac{\sqrt{\sqrt{2} + 2} \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 )}{4 \, c} - \frac{\sqrt{\sqrt{2} + 2} \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 )}{4 \, c} - \frac{\sqrt{-\sqrt{2} + 2} \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 )}{4 \, c} - \frac{\sqrt{-\sqrt{2} + 2} \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 )}{4 \, c} - \frac{\sqrt{\sqrt{2} + 2} \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 )}{8 \, c} + \frac{\sqrt{\sqrt{2} + 2} \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 )}{8 \, c} - \frac{\sqrt{-\sqrt{2} + 2} \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 )}{8 \, c} + \frac{\sqrt{-\sqrt{2} + 2} \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 )}{8 \, c} + \frac{2 \, \sqrt{x}}{c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(7/2)/(c*x^4+a),x, algorithm="giac")

[Out]

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