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

Optimal. Leaf size=287 \[ \frac{\log \left (-\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{4 \sqrt{2} \sqrt [8]{-a} c^{7/8}}-\frac{\log \left (\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{4 \sqrt{2} \sqrt [8]{-a} c^{7/8}}-\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{2 \sqrt{2} \sqrt [8]{-a} c^{7/8}}+\frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}+1\right )}{2 \sqrt{2} \sqrt [8]{-a} c^{7/8}}+\frac{\tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{2 \sqrt [8]{-a} c^{7/8}}-\frac{\tanh ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{2 \sqrt [8]{-a} c^{7/8}} \]

[Out]

-ArcTan[1 - (Sqrt[2]*c^(1/8)*Sqrt[x])/(-a)^(1/8)]/(2*Sqrt[2]*(-a)^(1/8)*c^(7/8)) + ArcTan[1 + (Sqrt[2]*c^(1/8)
*Sqrt[x])/(-a)^(1/8)]/(2*Sqrt[2]*(-a)^(1/8)*c^(7/8)) + ArcTan[(c^(1/8)*Sqrt[x])/(-a)^(1/8)]/(2*(-a)^(1/8)*c^(7
/8)) - ArcTanh[(c^(1/8)*Sqrt[x])/(-a)^(1/8)]/(2*(-a)^(1/8)*c^(7/8)) + Log[(-a)^(1/4) - Sqrt[2]*(-a)^(1/8)*c^(1
/8)*Sqrt[x] + c^(1/4)*x]/(4*Sqrt[2]*(-a)^(1/8)*c^(7/8)) - Log[(-a)^(1/4) + Sqrt[2]*(-a)^(1/8)*c^(1/8)*Sqrt[x]
+ c^(1/4)*x]/(4*Sqrt[2]*(-a)^(1/8)*c^(7/8))

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Rubi [A]  time = 0.234791, antiderivative size = 287, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 11, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.733, Rules used = {329, 301, 297, 1162, 617, 204, 1165, 628, 298, 205, 208} \[ \frac{\log \left (-\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{4 \sqrt{2} \sqrt [8]{-a} c^{7/8}}-\frac{\log \left (\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{4 \sqrt{2} \sqrt [8]{-a} c^{7/8}}-\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{2 \sqrt{2} \sqrt [8]{-a} c^{7/8}}+\frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}+1\right )}{2 \sqrt{2} \sqrt [8]{-a} c^{7/8}}+\frac{\tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{2 \sqrt [8]{-a} c^{7/8}}-\frac{\tanh ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{2 \sqrt [8]{-a} c^{7/8}} \]

Antiderivative was successfully verified.

[In]

Int[x^(5/2)/(a + c*x^4),x]

[Out]

-ArcTan[1 - (Sqrt[2]*c^(1/8)*Sqrt[x])/(-a)^(1/8)]/(2*Sqrt[2]*(-a)^(1/8)*c^(7/8)) + ArcTan[1 + (Sqrt[2]*c^(1/8)
*Sqrt[x])/(-a)^(1/8)]/(2*Sqrt[2]*(-a)^(1/8)*c^(7/8)) + ArcTan[(c^(1/8)*Sqrt[x])/(-a)^(1/8)]/(2*(-a)^(1/8)*c^(7
/8)) - ArcTanh[(c^(1/8)*Sqrt[x])/(-a)^(1/8)]/(2*(-a)^(1/8)*c^(7/8)) + Log[(-a)^(1/4) - Sqrt[2]*(-a)^(1/8)*c^(1
/8)*Sqrt[x] + c^(1/4)*x]/(4*Sqrt[2]*(-a)^(1/8)*c^(7/8)) - Log[(-a)^(1/4) + Sqrt[2]*(-a)^(1/8)*c^(1/8)*Sqrt[x]
+ c^(1/4)*x]/(4*Sqrt[2]*(-a)^(1/8)*c^(7/8))

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{x^{5/2}}{a+c x^4} \, dx &=2 \operatorname{Subst}\left (\int \frac{x^6}{a+c x^8} \, dx,x,\sqrt{x}\right )\\ &=-\frac{\operatorname{Subst}\left (\int \frac{x^2}{\sqrt{-a}-\sqrt{c} x^4} \, dx,x,\sqrt{x}\right )}{\sqrt{c}}+\frac{\operatorname{Subst}\left (\int \frac{x^2}{\sqrt{-a}+\sqrt{c} x^4} \, dx,x,\sqrt{x}\right )}{\sqrt{c}}\\ &=-\frac{\operatorname{Subst}\left (\int \frac{1}{\sqrt [4]{-a}-\sqrt [4]{c} x^2} \, dx,x,\sqrt{x}\right )}{2 c^{3/4}}+\frac{\operatorname{Subst}\left (\int \frac{1}{\sqrt [4]{-a}+\sqrt [4]{c} x^2} \, dx,x,\sqrt{x}\right )}{2 c^{3/4}}-\frac{\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^{3/4}}+\frac{\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^{3/4}}\\ &=\frac{\tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{2 \sqrt [8]{-a} c^{7/8}}-\frac{\tanh ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{2 \sqrt [8]{-a} c^{7/8}}+\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 c}+\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 c}+\frac{\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} \sqrt [8]{-a} c^{7/8}}+\frac{\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} \sqrt [8]{-a} c^{7/8}}\\ &=\frac{\tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{2 \sqrt [8]{-a} c^{7/8}}-\frac{\tanh ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{2 \sqrt [8]{-a} c^{7/8}}+\frac{\log \left (\sqrt [4]{-a}-\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{c} x\right )}{4 \sqrt{2} \sqrt [8]{-a} c^{7/8}}-\frac{\log \left (\sqrt [4]{-a}+\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{c} x\right )}{4 \sqrt{2} \sqrt [8]{-a} c^{7/8}}+\frac{\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} \sqrt [8]{-a} c^{7/8}}-\frac{\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} \sqrt [8]{-a} c^{7/8}}\\ &=-\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{2 \sqrt{2} \sqrt [8]{-a} c^{7/8}}+\frac{\tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{2 \sqrt{2} \sqrt [8]{-a} c^{7/8}}+\frac{\tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{2 \sqrt [8]{-a} c^{7/8}}-\frac{\tanh ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{2 \sqrt [8]{-a} c^{7/8}}+\frac{\log \left (\sqrt [4]{-a}-\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{c} x\right )}{4 \sqrt{2} \sqrt [8]{-a} c^{7/8}}-\frac{\log \left (\sqrt [4]{-a}+\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{c} x\right )}{4 \sqrt{2} \sqrt [8]{-a} c^{7/8}}\\ \end{align*}

Mathematica [C]  time = 0.0055328, size = 29, normalized size = 0.1 \[ \frac{2 x^{7/2} \, _2F_1\left (\frac{7}{8},1;\frac{15}{8};-\frac{c x^4}{a}\right )}{7 a} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(5/2)/(a + c*x^4),x]

[Out]

(2*x^(7/2)*Hypergeometric2F1[7/8, 1, 15/8, -((c*x^4)/a)])/(7*a)

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Maple [C]  time = 0.015, size = 29, normalized size = 0.1 \begin{align*}{\frac{1}{4\,c}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{8}c+a \right ) }{\frac{1}{{\it \_R}}\ln \left ( \sqrt{x}-{\it \_R} \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4/c*sum(1/_R*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{5}{2}}}{c x^{4} + a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^(5/2)/(c*x^4 + a), x)

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Fricas [B]  time = 1.70178, size = 1150, normalized size = 4.01 \begin{align*} -\frac{1}{2} \, \sqrt{2} \left (-\frac{1}{a c^{7}}\right )^{\frac{1}{8}} \arctan \left (\sqrt{2} \sqrt{\sqrt{2} a c^{6} \sqrt{x} \left (-\frac{1}{a c^{7}}\right )^{\frac{7}{8}} - a c^{5} \left (-\frac{1}{a c^{7}}\right )^{\frac{3}{4}} + x} c \left (-\frac{1}{a c^{7}}\right )^{\frac{1}{8}} - \sqrt{2} c \sqrt{x} \left (-\frac{1}{a c^{7}}\right )^{\frac{1}{8}} + 1\right ) - \frac{1}{2} \, \sqrt{2} \left (-\frac{1}{a c^{7}}\right )^{\frac{1}{8}} \arctan \left (\sqrt{2} \sqrt{-\sqrt{2} a c^{6} \sqrt{x} \left (-\frac{1}{a c^{7}}\right )^{\frac{7}{8}} - a c^{5} \left (-\frac{1}{a c^{7}}\right )^{\frac{3}{4}} + x} c \left (-\frac{1}{a c^{7}}\right )^{\frac{1}{8}} - \sqrt{2} c \sqrt{x} \left (-\frac{1}{a c^{7}}\right )^{\frac{1}{8}} - 1\right ) + \frac{1}{8} \, \sqrt{2} \left (-\frac{1}{a c^{7}}\right )^{\frac{1}{8}} \log \left (\sqrt{2} a c^{6} \sqrt{x} \left (-\frac{1}{a c^{7}}\right )^{\frac{7}{8}} - a c^{5} \left (-\frac{1}{a c^{7}}\right )^{\frac{3}{4}} + x\right ) - \frac{1}{8} \, \sqrt{2} \left (-\frac{1}{a c^{7}}\right )^{\frac{1}{8}} \log \left (-\sqrt{2} a c^{6} \sqrt{x} \left (-\frac{1}{a c^{7}}\right )^{\frac{7}{8}} - a c^{5} \left (-\frac{1}{a c^{7}}\right )^{\frac{3}{4}} + x\right ) - \left (-\frac{1}{a c^{7}}\right )^{\frac{1}{8}} \arctan \left (\sqrt{-a c^{5} \left (-\frac{1}{a c^{7}}\right )^{\frac{3}{4}} + x} c \left (-\frac{1}{a c^{7}}\right )^{\frac{1}{8}} - c \sqrt{x} \left (-\frac{1}{a c^{7}}\right )^{\frac{1}{8}}\right ) + \frac{1}{4} \, \left (-\frac{1}{a c^{7}}\right )^{\frac{1}{8}} \log \left (a c^{6} \left (-\frac{1}{a c^{7}}\right )^{\frac{7}{8}} + \sqrt{x}\right ) - \frac{1}{4} \, \left (-\frac{1}{a c^{7}}\right )^{\frac{1}{8}} \log \left (-a c^{6} \left (-\frac{1}{a c^{7}}\right )^{\frac{7}{8}} + \sqrt{x}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 141.239, size = 464, normalized size = 1.62 \begin{align*} \begin{cases} \frac{\tilde{\infty }}{\sqrt{x}} & \text{for}\: a = 0 \wedge c = 0 \\\frac{2 x^{\frac{7}{2}}}{7 a} & \text{for}\: c = 0 \\- \frac{2}{c \sqrt{x}} & \text{for}\: a = 0 \\\frac{\left (-1\right )^{\frac{7}{8}} c^{174} \left (\frac{1}{c}\right )^{\frac{1399}{8}} \log{\left (\sqrt [8]{-1} \sqrt [8]{a} \sqrt [8]{\frac{1}{c}} + \sqrt{x} \right )}}{4 \sqrt [8]{a}} - \frac{\left (-1\right )^{\frac{7}{8}} \sqrt{2} c^{174} \left (\frac{1}{c}\right )^{\frac{1399}{8}} \operatorname{atan}{\left (1 - \frac{\left (-1\right )^{\frac{7}{8}} \sqrt{2} \sqrt{x}}{\sqrt [8]{a} \sqrt [8]{\frac{1}{c}}} \right )}}{4 \sqrt [8]{a}} - \frac{\left (-1\right )^{\frac{7}{8}} \log{\left (- \sqrt [8]{-1} \sqrt [8]{a} \sqrt [8]{\frac{1}{c}} + \sqrt{x} \right )}}{4 \sqrt [8]{a} c^{17} \left (\frac{1}{c}\right )^{\frac{129}{8}}} - \frac{\left (-1\right )^{\frac{7}{8}} \sqrt{2} \log{\left (- 4 \sqrt [8]{-1} \sqrt{2} \sqrt [8]{a} \sqrt{x} \sqrt [8]{\frac{1}{c}} + 4 \sqrt [4]{-1} \sqrt [4]{a} \sqrt [4]{\frac{1}{c}} + 4 x \right )}}{8 \sqrt [8]{a} c^{17} \left (\frac{1}{c}\right )^{\frac{129}{8}}} + \frac{\left (-1\right )^{\frac{7}{8}} \sqrt{2} \log{\left (4 \sqrt [8]{-1} \sqrt{2} \sqrt [8]{a} \sqrt{x} \sqrt [8]{\frac{1}{c}} + 4 \sqrt [4]{-1} \sqrt [4]{a} \sqrt [4]{\frac{1}{c}} + 4 x \right )}}{8 \sqrt [8]{a} c^{17} \left (\frac{1}{c}\right )^{\frac{129}{8}}} + \frac{\left (-1\right )^{\frac{7}{8}} \operatorname{atan}{\left (\frac{\left (-1\right )^{\frac{7}{8}} \sqrt{x}}{\sqrt [8]{a} \sqrt [8]{\frac{1}{c}}} \right )}}{2 \sqrt [8]{a} c^{17} \left (\frac{1}{c}\right )^{\frac{129}{8}}} + \frac{\left (-1\right )^{\frac{7}{8}} \sqrt{2} \operatorname{atan}{\left (1 + \frac{\left (-1\right )^{\frac{7}{8}} \sqrt{2} \sqrt{x}}{\sqrt [8]{a} \sqrt [8]{\frac{1}{c}}} \right )}}{4 \sqrt [8]{a} c^{151} \left (\frac{1}{c}\right )^{\frac{1201}{8}}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((zoo/sqrt(x), Eq(a, 0) & Eq(c, 0)), (2*x**(7/2)/(7*a), Eq(c, 0)), (-2/(c*sqrt(x)), Eq(a, 0)), ((-1)*
*(7/8)*c**174*(1/c)**(1399/8)*log((-1)**(1/8)*a**(1/8)*(1/c)**(1/8) + sqrt(x))/(4*a**(1/8)) - (-1)**(7/8)*sqrt
(2)*c**174*(1/c)**(1399/8)*atan(1 - (-1)**(7/8)*sqrt(2)*sqrt(x)/(a**(1/8)*(1/c)**(1/8)))/(4*a**(1/8)) - (-1)**
(7/8)*log(-(-1)**(1/8)*a**(1/8)*(1/c)**(1/8) + sqrt(x))/(4*a**(1/8)*c**17*(1/c)**(129/8)) - (-1)**(7/8)*sqrt(2
)*log(-4*(-1)**(1/8)*sqrt(2)*a**(1/8)*sqrt(x)*(1/c)**(1/8) + 4*(-1)**(1/4)*a**(1/4)*(1/c)**(1/4) + 4*x)/(8*a**
(1/8)*c**17*(1/c)**(129/8)) + (-1)**(7/8)*sqrt(2)*log(4*(-1)**(1/8)*sqrt(2)*a**(1/8)*sqrt(x)*(1/c)**(1/8) + 4*
(-1)**(1/4)*a**(1/4)*(1/c)**(1/4) + 4*x)/(8*a**(1/8)*c**17*(1/c)**(129/8)) + (-1)**(7/8)*atan((-1)**(7/8)*sqrt
(x)/(a**(1/8)*(1/c)**(1/8)))/(2*a**(1/8)*c**17*(1/c)**(129/8)) + (-1)**(7/8)*sqrt(2)*atan(1 + (-1)**(7/8)*sqrt
(2)*sqrt(x)/(a**(1/8)*(1/c)**(1/8)))/(4*a**(1/8)*c**151*(1/c)**(1201/8)), True))

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Giac [B]  time = 1.33483, size = 590, normalized size = 2.06 \begin{align*} \frac{\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} + \frac{\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} + \frac{\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} + \frac{\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} - \frac{\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} + \frac{\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} - \frac{\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} + \frac{\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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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