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]

1/2*arctan(c^(1/8)*x^(1/2)/(-a)^(1/8))/(-a)^(1/8)/c^(7/8)-1/2*arctanh(c^(1/8)*x^(1/2)/(-a)^(1/8))/(-a)^(1/8)/c
^(7/8)+1/4*arctan(-1+c^(1/8)*2^(1/2)*x^(1/2)/(-a)^(1/8))/(-a)^(1/8)/c^(7/8)*2^(1/2)+1/4*arctan(1+c^(1/8)*2^(1/
2)*x^(1/2)/(-a)^(1/8))/(-a)^(1/8)/c^(7/8)*2^(1/2)+1/8*ln((-a)^(1/4)+c^(1/4)*x-(-a)^(1/8)*c^(1/8)*2^(1/2)*x^(1/
2))/(-a)^(1/8)/c^(7/8)*2^(1/2)-1/8*ln((-a)^(1/4)+c^(1/4)*x+(-a)^(1/8)*c^(1/8)*2^(1/2)*x^(1/2))/(-a)^(1/8)/c^(7
/8)*2^(1/2)

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Rubi [A]  time = 0.23, antiderivative size = 287, normalized size of antiderivative = 1.00, 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 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 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]

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 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 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 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 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 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 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]

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*}

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Mathematica [C]  time = 0.01, size = 29, normalized size = 0.10 \[ \frac {2 x^{7/2} \, _2F_1\left (\frac {7}{8},1;\frac {15}{8};-\frac {c x^4}{a}\right )}{7 a} \]

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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fricas [B]  time = 0.62, size = 426, normalized size = 1.48 \[ -\frac {1}{2} \, \sqrt {2} \left (-\frac {1}{a c^{7}}\right )^{\frac {1}{8}} \arctan \left (\sqrt {2} \sqrt {\sqrt {2} a c^{6} \sqrt {x} \left (-\frac {1}{a c^{7}}\right )^{\frac {7}{8}} - a c^{5} \left (-\frac {1}{a c^{7}}\right )^{\frac {3}{4}} + x} c \left (-\frac {1}{a c^{7}}\right )^{\frac {1}{8}} - \sqrt {2} c \sqrt {x} \left (-\frac {1}{a c^{7}}\right )^{\frac {1}{8}} + 1\right ) - \frac {1}{2} \, \sqrt {2} \left (-\frac {1}{a c^{7}}\right )^{\frac {1}{8}} \arctan \left (\sqrt {2} \sqrt {-\sqrt {2} a c^{6} \sqrt {x} \left (-\frac {1}{a c^{7}}\right )^{\frac {7}{8}} - a c^{5} \left (-\frac {1}{a c^{7}}\right )^{\frac {3}{4}} + x} c \left (-\frac {1}{a c^{7}}\right )^{\frac {1}{8}} - \sqrt {2} c \sqrt {x} \left (-\frac {1}{a c^{7}}\right )^{\frac {1}{8}} - 1\right ) + \frac {1}{8} \, \sqrt {2} \left (-\frac {1}{a c^{7}}\right )^{\frac {1}{8}} \log \left (\sqrt {2} a c^{6} \sqrt {x} \left (-\frac {1}{a c^{7}}\right )^{\frac {7}{8}} - a c^{5} \left (-\frac {1}{a c^{7}}\right )^{\frac {3}{4}} + x\right ) - \frac {1}{8} \, \sqrt {2} \left (-\frac {1}{a c^{7}}\right )^{\frac {1}{8}} \log \left (-\sqrt {2} a c^{6} \sqrt {x} \left (-\frac {1}{a c^{7}}\right )^{\frac {7}{8}} - a c^{5} \left (-\frac {1}{a c^{7}}\right )^{\frac {3}{4}} + x\right ) - \left (-\frac {1}{a c^{7}}\right )^{\frac {1}{8}} \arctan \left (\sqrt {-a c^{5} \left (-\frac {1}{a c^{7}}\right )^{\frac {3}{4}} + x} c \left (-\frac {1}{a c^{7}}\right )^{\frac {1}{8}} - c \sqrt {x} \left (-\frac {1}{a c^{7}}\right )^{\frac {1}{8}}\right ) + \frac {1}{4} \, \left (-\frac {1}{a c^{7}}\right )^{\frac {1}{8}} \log \left (a c^{6} \left (-\frac {1}{a c^{7}}\right )^{\frac {7}{8}} + \sqrt {x}\right ) - \frac {1}{4} \, \left (-\frac {1}{a c^{7}}\right )^{\frac {1}{8}} \log \left (-a c^{6} \left (-\frac {1}{a c^{7}}\right )^{\frac {7}{8}} + \sqrt {x}\right ) \]

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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giac [B]  time = 0.53, size = 445, normalized size = 1.55 \[ \frac {\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 )}{2 \, a \sqrt {-2 \, \sqrt {2} + 4}} + \frac {\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 )}{2 \, a \sqrt {-2 \, \sqrt {2} + 4}} + \frac {\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 )}{2 \, a \sqrt {2 \, \sqrt {2} + 4}} + \frac {\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 )}{2 \, a \sqrt {2 \, \sqrt {2} + 4}} - \frac {\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 )}{4 \, a \sqrt {-2 \, \sqrt {2} + 4}} + \frac {\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 )}{4 \, a \sqrt {-2 \, \sqrt {2} + 4}} - \frac {\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 )}{4 \, a \sqrt {2 \, \sqrt {2} + 4}} + \frac {\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 )}{4 \, a \sqrt {2 \, \sqrt {2} + 4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 0.02, size = 29, normalized size = 0.10 \[ \frac {\ln \left (-\RootOf \left (c \,\textit {\_Z}^{8}+a \right )+\sqrt {x}\right )}{4 c \RootOf \left (c \,\textit {\_Z}^{8}+a \right )} \]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{\frac {5}{2}}}{c x^{4} + a}\,{d x} \]

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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mupad [B]  time = 1.11, size = 118, normalized size = 0.41 \[ \frac {\mathrm {atan}\left (\frac {c^{1/8}\,\sqrt {x}}{{\left (-a\right )}^{1/8}}\right )}{2\,{\left (-a\right )}^{1/8}\,c^{7/8}}+\frac {\mathrm {atan}\left (\frac {c^{1/8}\,\sqrt {x}\,1{}\mathrm {i}}{{\left (-a\right )}^{1/8}}\right )\,1{}\mathrm {i}}{2\,{\left (-a\right )}^{1/8}\,c^{7/8}}+\frac {\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,c^{1/8}\,\sqrt {x}\,\left (\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right )}{{\left (-a\right )}^{1/8}}\right )\,\left (\frac {1}{4}-\frac {1}{4}{}\mathrm {i}\right )}{{\left (-a\right )}^{1/8}\,c^{7/8}}+\frac {\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,c^{1/8}\,\sqrt {x}\,\left (\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right )}{{\left (-a\right )}^{1/8}}\right )\,\left (\frac {1}{4}+\frac {1}{4}{}\mathrm {i}\right )}{{\left (-a\right )}^{1/8}\,c^{7/8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(5/2)/(a + c*x^4),x)

[Out]

atan((c^(1/8)*x^(1/2))/(-a)^(1/8))/(2*(-a)^(1/8)*c^(7/8)) + (atan((c^(1/8)*x^(1/2)*1i)/(-a)^(1/8))*1i)/(2*(-a)
^(1/8)*c^(7/8)) + (2^(1/2)*atan((2^(1/2)*c^(1/8)*x^(1/2)*(1/2 - 1i/2))/(-a)^(1/8))*(1/4 - 1i/4))/((-a)^(1/8)*c
^(7/8)) + (2^(1/2)*atan((2^(1/2)*c^(1/8)*x^(1/2)*(1/2 + 1i/2))/(-a)^(1/8))*(1/4 + 1i/4))/((-a)^(1/8)*c^(7/8))

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sympy [A]  time = 94.13, size = 452, normalized size = 1.57 \[ \begin {cases} \frac {\tilde {\infty }}{\sqrt {x}} & \text {for}\: a = 0 \wedge c = 0 \\\frac {2 x^{\frac {7}{2}}}{7 a} & \text {for}\: c = 0 \\- \frac {2}{c \sqrt {x}} & \text {for}\: a = 0 \\- \frac {\left (-1\right )^{\frac {7}{8}} \log {\left (- \sqrt [8]{-1} \sqrt [8]{a} \sqrt [8]{\frac {1}{c}} + \sqrt {x} \right )}}{4 \sqrt [8]{a} c \sqrt [8]{\frac {1}{c}}} + \frac {\left (-1\right )^{\frac {7}{8}} \log {\left (\sqrt [8]{-1} \sqrt [8]{a} \sqrt [8]{\frac {1}{c}} + \sqrt {x} \right )}}{4 \sqrt [8]{a} c \sqrt [8]{\frac {1}{c}}} - \frac {\left (-1\right )^{\frac {7}{8}} \sqrt {2} \log {\left (- 4 \sqrt [8]{-1} \sqrt {2} \sqrt [8]{a} \sqrt {x} \sqrt [8]{\frac {1}{c}} + 4 \sqrt [4]{-1} \sqrt [4]{a} \sqrt [4]{\frac {1}{c}} + 4 x \right )}}{8 \sqrt [8]{a} c \sqrt [8]{\frac {1}{c}}} + \frac {\left (-1\right )^{\frac {7}{8}} \sqrt {2} \log {\left (4 \sqrt [8]{-1} \sqrt {2} \sqrt [8]{a} \sqrt {x} \sqrt [8]{\frac {1}{c}} + 4 \sqrt [4]{-1} \sqrt [4]{a} \sqrt [4]{\frac {1}{c}} + 4 x \right )}}{8 \sqrt [8]{a} c \sqrt [8]{\frac {1}{c}}} + \frac {\left (-1\right )^{\frac {7}{8}} \operatorname {atan}{\left (\frac {\left (-1\right )^{\frac {7}{8}} \sqrt {x}}{\sqrt [8]{a} \sqrt [8]{\frac {1}{c}}} \right )}}{2 \sqrt [8]{a} c \sqrt [8]{\frac {1}{c}}} - \frac {\left (-1\right )^{\frac {7}{8}} \sqrt {2} \operatorname {atan}{\left (1 - \frac {\left (-1\right )^{\frac {7}{8}} \sqrt {2} \sqrt {x}}{\sqrt [8]{a} \sqrt [8]{\frac {1}{c}}} \right )}}{4 \sqrt [8]{a} c \sqrt [8]{\frac {1}{c}}} + \frac {\left (-1\right )^{\frac {7}{8}} \sqrt {2} \operatorname {atan}{\left (1 + \frac {\left (-1\right )^{\frac {7}{8}} \sqrt {2} \sqrt {x}}{\sqrt [8]{a} \sqrt [8]{\frac {1}{c}}} \right )}}{4 \sqrt [8]{a} c \sqrt [8]{\frac {1}{c}}} & \text {otherwise} \end {cases} \]

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)*log(-(-1)**(1/8)*a**(1/8)*(1/c)**(1/8) + sqrt(x))/(4*a**(1/8)*c*(1/c)**(1/8)) + (-1)**(7/8)*log((-1)**
(1/8)*a**(1/8)*(1/c)**(1/8) + sqrt(x))/(4*a**(1/8)*c*(1/c)**(1/8)) - (-1)**(7/8)*sqrt(2)*log(-4*(-1)**(1/8)*sq
rt(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*(1/c)**(1/8)) +
(-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*(1/c)**(1/8)) + (-1)**(7/8)*atan((-1)**(7/8)*sqrt(x)/(a**(1/8)*(1/c)**(1/8)))/(2*a**
(1/8)*c*(1/c)**(1/8)) - (-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*(1/c)**(1/8)) + (-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*(1/c)**(1/8)), True))

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