3.738 \(\int \frac{x^{9/2}}{a+c x^4} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.376223, 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 = {321, 329, 300, 297, 1162, 617, 204, 1165, 628, 298, 205, 208} \[ -\frac{(-a)^{3/8} \log \left (-\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{4 \sqrt{2} c^{11/8}}+\frac{(-a)^{3/8} \log \left (\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{4 \sqrt{2} c^{11/8}}+\frac{(-a)^{3/8} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{2 \sqrt{2} c^{11/8}}-\frac{(-a)^{3/8} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}+1\right )}{2 \sqrt{2} c^{11/8}}+\frac{(-a)^{3/8} \tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{2 c^{11/8}}-\frac{(-a)^{3/8} \tanh ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{2 c^{11/8}}+\frac{2 x^{3/2}}{3 c} \]

Antiderivative was successfully verified.

[In]

Int[x^(9/2)/(a + c*x^4),x]

[Out]

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

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 300

Int[(x_)^(m_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{r = Numerator[Rt[-(a/b), 2]], s = Denominator[Rt[-(
a/b), 2]]}, Dist[r/(2*a), Int[x^m/(r + s*x^(n/2)), x], x] + Dist[r/(2*a), Int[x^m/(r - s*x^(n/2)), x], x]] /;
FreeQ[{a, b}, x] && IGtQ[n/4, 0] && IGtQ[m, 0] && LtQ[m, n/2] &&  !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{x^{9/2}}{a+c x^4} \, dx &=\frac{2 x^{3/2}}{3 c}-\frac{a \int \frac{\sqrt{x}}{a+c x^4} \, dx}{c}\\ &=\frac{2 x^{3/2}}{3 c}-\frac{(2 a) \operatorname{Subst}\left (\int \frac{x^2}{a+c x^8} \, dx,x,\sqrt{x}\right )}{c}\\ &=\frac{2 x^{3/2}}{3 c}-\frac{\sqrt{-a} \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{-a}-\sqrt{c} x^4} \, dx,x,\sqrt{x}\right )}{c}-\frac{\sqrt{-a} \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{-a}+\sqrt{c} x^4} \, dx,x,\sqrt{x}\right )}{c}\\ &=\frac{2 x^{3/2}}{3 c}-\frac{\sqrt{-a} \operatorname{Subst}\left (\int \frac{1}{\sqrt [4]{-a}-\sqrt [4]{c} x^2} \, dx,x,\sqrt{x}\right )}{2 c^{5/4}}+\frac{\sqrt{-a} \operatorname{Subst}\left (\int \frac{1}{\sqrt [4]{-a}+\sqrt [4]{c} x^2} \, dx,x,\sqrt{x}\right )}{2 c^{5/4}}+\frac{\sqrt{-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^{5/4}}-\frac{\sqrt{-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^{5/4}}\\ &=\frac{2 x^{3/2}}{3 c}+\frac{(-a)^{3/8} \tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{2 c^{11/8}}-\frac{(-a)^{3/8} \tanh ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{2 c^{11/8}}-\frac{\sqrt{-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^{3/2}}-\frac{\sqrt{-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^{3/2}}-\frac{(-a)^{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} c^{11/8}}-\frac{(-a)^{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} c^{11/8}}\\ &=\frac{2 x^{3/2}}{3 c}+\frac{(-a)^{3/8} \tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{2 c^{11/8}}-\frac{(-a)^{3/8} \tanh ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{2 c^{11/8}}-\frac{(-a)^{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} c^{11/8}}+\frac{(-a)^{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} c^{11/8}}-\frac{(-a)^{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} c^{11/8}}+\frac{(-a)^{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} c^{11/8}}\\ &=\frac{2 x^{3/2}}{3 c}+\frac{(-a)^{3/8} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{2 \sqrt{2} c^{11/8}}-\frac{(-a)^{3/8} \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{2 \sqrt{2} c^{11/8}}+\frac{(-a)^{3/8} \tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{2 c^{11/8}}-\frac{(-a)^{3/8} \tanh ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{2 c^{11/8}}-\frac{(-a)^{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} c^{11/8}}+\frac{(-a)^{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} c^{11/8}}\\ \end{align*}

Mathematica [C]  time = 0.007594, size = 31, normalized size = 0.1 \[ -\frac{2 x^{3/2} \left (\, _2F_1\left (\frac{3}{8},1;\frac{11}{8};-\frac{c x^4}{a}\right )-1\right )}{3 c} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(9/2)/(a + c*x^4),x]

[Out]

(-2*x^(3/2)*(-1 + Hypergeometric2F1[3/8, 1, 11/8, -((c*x^4)/a)]))/(3*c)

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Maple [C]  time = 0.023, size = 39, normalized size = 0.1 \begin{align*}{\frac{2}{3\,c}{x}^{{\frac{3}{2}}}}-{\frac{a}{4\,{c}^{2}}\sum _{{\it \_R}={\it RootOf} \left ( c{{\it \_Z}}^{8}+a \right ) }{\frac{1}{{{\it \_R}}^{5}}\ln \left ( \sqrt{x}-{\it \_R} \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(9/2)/(c*x^4+a),x)

[Out]

2/3*x^(3/2)/c-1/4*a/c^2*sum(1/_R^5*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*} -a \int \frac{\sqrt{x}}{c^{2} x^{4} + a c}\,{d x} + \frac{2 \, x^{\frac{3}{2}}}{3 \, c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(9/2)/(c*x^4+a),x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 1.69676, size = 1223, normalized size = 4.09 \begin{align*} \frac{12 \, \sqrt{2} c \left (-\frac{a^{3}}{c^{11}}\right )^{\frac{1}{8}} \arctan \left (-\frac{\sqrt{2} a c^{7} \sqrt{x} \left (-\frac{a^{3}}{c^{11}}\right )^{\frac{5}{8}} - \sqrt{2} \sqrt{c^{8} \left (-\frac{a^{3}}{c^{11}}\right )^{\frac{3}{4}} + \sqrt{2} a c^{4} \sqrt{x} \left (-\frac{a^{3}}{c^{11}}\right )^{\frac{3}{8}} + a^{2} x} c^{7} \left (-\frac{a^{3}}{c^{11}}\right )^{\frac{5}{8}} - a^{3}}{a^{3}}\right ) + 12 \, \sqrt{2} c \left (-\frac{a^{3}}{c^{11}}\right )^{\frac{1}{8}} \arctan \left (-\frac{\sqrt{2} a c^{7} \sqrt{x} \left (-\frac{a^{3}}{c^{11}}\right )^{\frac{5}{8}} - \sqrt{2} \sqrt{c^{8} \left (-\frac{a^{3}}{c^{11}}\right )^{\frac{3}{4}} - \sqrt{2} a c^{4} \sqrt{x} \left (-\frac{a^{3}}{c^{11}}\right )^{\frac{3}{8}} + a^{2} x} c^{7} \left (-\frac{a^{3}}{c^{11}}\right )^{\frac{5}{8}} + a^{3}}{a^{3}}\right ) - 3 \, \sqrt{2} c \left (-\frac{a^{3}}{c^{11}}\right )^{\frac{1}{8}} \log \left (c^{8} \left (-\frac{a^{3}}{c^{11}}\right )^{\frac{3}{4}} + \sqrt{2} a c^{4} \sqrt{x} \left (-\frac{a^{3}}{c^{11}}\right )^{\frac{3}{8}} + a^{2} x\right ) + 3 \, \sqrt{2} c \left (-\frac{a^{3}}{c^{11}}\right )^{\frac{1}{8}} \log \left (c^{8} \left (-\frac{a^{3}}{c^{11}}\right )^{\frac{3}{4}} - \sqrt{2} a c^{4} \sqrt{x} \left (-\frac{a^{3}}{c^{11}}\right )^{\frac{3}{8}} + a^{2} x\right ) - 24 \, c \left (-\frac{a^{3}}{c^{11}}\right )^{\frac{1}{8}} \arctan \left (-\frac{a c^{7} \sqrt{x} \left (-\frac{a^{3}}{c^{11}}\right )^{\frac{5}{8}} - \sqrt{c^{8} \left (-\frac{a^{3}}{c^{11}}\right )^{\frac{3}{4}} + a^{2} x} c^{7} \left (-\frac{a^{3}}{c^{11}}\right )^{\frac{5}{8}}}{a^{3}}\right ) + 6 \, c \left (-\frac{a^{3}}{c^{11}}\right )^{\frac{1}{8}} \log \left (c^{4} \left (-\frac{a^{3}}{c^{11}}\right )^{\frac{3}{8}} + a \sqrt{x}\right ) - 6 \, c \left (-\frac{a^{3}}{c^{11}}\right )^{\frac{1}{8}} \log \left (-c^{4} \left (-\frac{a^{3}}{c^{11}}\right )^{\frac{3}{8}} + a \sqrt{x}\right ) + 16 \, x^{\frac{3}{2}}}{24 \, c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(9/2)/(c*x^4+a),x, algorithm="fricas")

[Out]

1/24*(12*sqrt(2)*c*(-a^3/c^11)^(1/8)*arctan(-(sqrt(2)*a*c^7*sqrt(x)*(-a^3/c^11)^(5/8) - sqrt(2)*sqrt(c^8*(-a^3
/c^11)^(3/4) + sqrt(2)*a*c^4*sqrt(x)*(-a^3/c^11)^(3/8) + a^2*x)*c^7*(-a^3/c^11)^(5/8) - a^3)/a^3) + 12*sqrt(2)
*c*(-a^3/c^11)^(1/8)*arctan(-(sqrt(2)*a*c^7*sqrt(x)*(-a^3/c^11)^(5/8) - sqrt(2)*sqrt(c^8*(-a^3/c^11)^(3/4) - s
qrt(2)*a*c^4*sqrt(x)*(-a^3/c^11)^(3/8) + a^2*x)*c^7*(-a^3/c^11)^(5/8) + a^3)/a^3) - 3*sqrt(2)*c*(-a^3/c^11)^(1
/8)*log(c^8*(-a^3/c^11)^(3/4) + sqrt(2)*a*c^4*sqrt(x)*(-a^3/c^11)^(3/8) + a^2*x) + 3*sqrt(2)*c*(-a^3/c^11)^(1/
8)*log(c^8*(-a^3/c^11)^(3/4) - sqrt(2)*a*c^4*sqrt(x)*(-a^3/c^11)^(3/8) + a^2*x) - 24*c*(-a^3/c^11)^(1/8)*arcta
n(-(a*c^7*sqrt(x)*(-a^3/c^11)^(5/8) - sqrt(c^8*(-a^3/c^11)^(3/4) + a^2*x)*c^7*(-a^3/c^11)^(5/8))/a^3) + 6*c*(-
a^3/c^11)^(1/8)*log(c^4*(-a^3/c^11)^(3/8) + a*sqrt(x)) - 6*c*(-a^3/c^11)^(1/8)*log(-c^4*(-a^3/c^11)^(3/8) + a*
sqrt(x)) + 16*x^(3/2))/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**(9/2)/(c*x**4+a),x)

[Out]

Timed out

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Giac [B]  time = 1.35655, size = 601, normalized size = 2.01 \begin{align*} \frac{\sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{3}{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{3}{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{3}{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{3}{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{3}{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{3}{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{3}{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{3}{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 \, x^{\frac{3}{2}}}{3 \, c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(9/2)/(c*x^4+a),x, algorithm="giac")

[Out]

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