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

Optimal. Leaf size=329 \[ \frac {11 x^{5/2}}{64 a^2 \left (a+c x^4\right )}-\frac {33 \log \left (-\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{512 \sqrt {2} (-a)^{19/8} c^{5/8}}+\frac {33 \log \left (\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{512 \sqrt {2} (-a)^{19/8} c^{5/8}}-\frac {33 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{256 \sqrt {2} (-a)^{19/8} c^{5/8}}+\frac {33 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}+1\right )}{256 \sqrt {2} (-a)^{19/8} c^{5/8}}-\frac {33 \tan ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{256 (-a)^{19/8} c^{5/8}}-\frac {33 \tanh ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{256 (-a)^{19/8} c^{5/8}}+\frac {x^{5/2}}{8 a \left (a+c x^4\right )^2} \]

[Out]

1/8*x^(5/2)/a/(c*x^4+a)^2+11/64*x^(5/2)/a^2/(c*x^4+a)-33/256*arctan(c^(1/8)*x^(1/2)/(-a)^(1/8))/(-a)^(19/8)/c^
(5/8)-33/256*arctanh(c^(1/8)*x^(1/2)/(-a)^(1/8))/(-a)^(19/8)/c^(5/8)+33/512*arctan(-1+c^(1/8)*2^(1/2)*x^(1/2)/
(-a)^(1/8))/(-a)^(19/8)/c^(5/8)*2^(1/2)+33/512*arctan(1+c^(1/8)*2^(1/2)*x^(1/2)/(-a)^(1/8))/(-a)^(19/8)/c^(5/8
)*2^(1/2)-33/1024*ln((-a)^(1/4)+c^(1/4)*x-(-a)^(1/8)*c^(1/8)*2^(1/2)*x^(1/2))/(-a)^(19/8)/c^(5/8)*2^(1/2)+33/1
024*ln((-a)^(1/4)+c^(1/4)*x+(-a)^(1/8)*c^(1/8)*2^(1/2)*x^(1/2))/(-a)^(19/8)/c^(5/8)*2^(1/2)

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Rubi [A]  time = 0.30, antiderivative size = 329, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 12, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {290, 329, 301, 211, 1165, 628, 1162, 617, 204, 212, 208, 205} \[ \frac {11 x^{5/2}}{64 a^2 \left (a+c x^4\right )}-\frac {33 \log \left (-\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{512 \sqrt {2} (-a)^{19/8} c^{5/8}}+\frac {33 \log \left (\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{512 \sqrt {2} (-a)^{19/8} c^{5/8}}-\frac {33 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{256 \sqrt {2} (-a)^{19/8} c^{5/8}}+\frac {33 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}+1\right )}{256 \sqrt {2} (-a)^{19/8} c^{5/8}}-\frac {33 \tan ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{256 (-a)^{19/8} c^{5/8}}-\frac {33 \tanh ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{256 (-a)^{19/8} c^{5/8}}+\frac {x^{5/2}}{8 a \left (a+c x^4\right )^2} \]

Antiderivative was successfully verified.

[In]

Int[x^(3/2)/(a + c*x^4)^3,x]

[Out]

x^(5/2)/(8*a*(a + c*x^4)^2) + (11*x^(5/2))/(64*a^2*(a + c*x^4)) - (33*ArcTan[1 - (Sqrt[2]*c^(1/8)*Sqrt[x])/(-a
)^(1/8)])/(256*Sqrt[2]*(-a)^(19/8)*c^(5/8)) + (33*ArcTan[1 + (Sqrt[2]*c^(1/8)*Sqrt[x])/(-a)^(1/8)])/(256*Sqrt[
2]*(-a)^(19/8)*c^(5/8)) - (33*ArcTan[(c^(1/8)*Sqrt[x])/(-a)^(1/8)])/(256*(-a)^(19/8)*c^(5/8)) - (33*ArcTanh[(c
^(1/8)*Sqrt[x])/(-a)^(1/8)])/(256*(-a)^(19/8)*c^(5/8)) - (33*Log[(-a)^(1/4) - Sqrt[2]*(-a)^(1/8)*c^(1/8)*Sqrt[
x] + c^(1/4)*x])/(512*Sqrt[2]*(-a)^(19/8)*c^(5/8)) + (33*Log[(-a)^(1/4) + Sqrt[2]*(-a)^(1/8)*c^(1/8)*Sqrt[x] +
c^(1/4)*x])/(512*Sqrt[2]*(-a)^(19/8)*c^(5/8))

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 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 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 290

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(
a*c*n*(p + 1)), x] + Dist[(m + n*(p + 1) + 1)/(a*n*(p + 1)), Int[(c*x)^m*(a + b*x^n)^(p + 1), x], x] /; FreeQ[
{a, b, c, m}, x] && IGtQ[n, 0] && LtQ[p, -1] && 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 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^{3/2}}{\left (a+c x^4\right )^3} \, dx &=\frac {x^{5/2}}{8 a \left (a+c x^4\right )^2}+\frac {11 \int \frac {x^{3/2}}{\left (a+c x^4\right )^2} \, dx}{16 a}\\ &=\frac {x^{5/2}}{8 a \left (a+c x^4\right )^2}+\frac {11 x^{5/2}}{64 a^2 \left (a+c x^4\right )}+\frac {33 \int \frac {x^{3/2}}{a+c x^4} \, dx}{128 a^2}\\ &=\frac {x^{5/2}}{8 a \left (a+c x^4\right )^2}+\frac {11 x^{5/2}}{64 a^2 \left (a+c x^4\right )}+\frac {33 \operatorname {Subst}\left (\int \frac {x^4}{a+c x^8} \, dx,x,\sqrt {x}\right )}{64 a^2}\\ &=\frac {x^{5/2}}{8 a \left (a+c x^4\right )^2}+\frac {11 x^{5/2}}{64 a^2 \left (a+c x^4\right )}-\frac {33 \operatorname {Subst}\left (\int \frac {1}{\sqrt {-a}-\sqrt {c} x^4} \, dx,x,\sqrt {x}\right )}{128 a^2 \sqrt {c}}+\frac {33 \operatorname {Subst}\left (\int \frac {1}{\sqrt {-a}+\sqrt {c} x^4} \, dx,x,\sqrt {x}\right )}{128 a^2 \sqrt {c}}\\ &=\frac {x^{5/2}}{8 a \left (a+c x^4\right )^2}+\frac {11 x^{5/2}}{64 a^2 \left (a+c x^4\right )}-\frac {33 \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{-a}-\sqrt [4]{c} x^2} \, dx,x,\sqrt {x}\right )}{256 (-a)^{9/4} \sqrt {c}}-\frac {33 \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{-a}+\sqrt [4]{c} x^2} \, dx,x,\sqrt {x}\right )}{256 (-a)^{9/4} \sqrt {c}}+\frac {33 \operatorname {Subst}\left (\int \frac {\sqrt [4]{-a}-\sqrt [4]{c} x^2}{\sqrt {-a}+\sqrt {c} x^4} \, dx,x,\sqrt {x}\right )}{256 (-a)^{9/4} \sqrt {c}}+\frac {33 \operatorname {Subst}\left (\int \frac {\sqrt [4]{-a}+\sqrt [4]{c} x^2}{\sqrt {-a}+\sqrt {c} x^4} \, dx,x,\sqrt {x}\right )}{256 (-a)^{9/4} \sqrt {c}}\\ &=\frac {x^{5/2}}{8 a \left (a+c x^4\right )^2}+\frac {11 x^{5/2}}{64 a^2 \left (a+c x^4\right )}-\frac {33 \tan ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{256 (-a)^{19/8} c^{5/8}}-\frac {33 \tanh ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{256 (-a)^{19/8} c^{5/8}}+\frac {33 \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 )}{512 (-a)^{9/4} c^{3/4}}+\frac {33 \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 )}{512 (-a)^{9/4} c^{3/4}}-\frac {33 \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 )}{512 \sqrt {2} (-a)^{19/8} c^{5/8}}-\frac {33 \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 )}{512 \sqrt {2} (-a)^{19/8} c^{5/8}}\\ &=\frac {x^{5/2}}{8 a \left (a+c x^4\right )^2}+\frac {11 x^{5/2}}{64 a^2 \left (a+c x^4\right )}-\frac {33 \tan ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{256 (-a)^{19/8} c^{5/8}}-\frac {33 \tanh ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{256 (-a)^{19/8} c^{5/8}}-\frac {33 \log \left (\sqrt [4]{-a}-\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{c} x\right )}{512 \sqrt {2} (-a)^{19/8} c^{5/8}}+\frac {33 \log \left (\sqrt [4]{-a}+\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{c} x\right )}{512 \sqrt {2} (-a)^{19/8} c^{5/8}}+\frac {33 \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{256 \sqrt {2} (-a)^{19/8} c^{5/8}}-\frac {33 \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{256 \sqrt {2} (-a)^{19/8} c^{5/8}}\\ &=\frac {x^{5/2}}{8 a \left (a+c x^4\right )^2}+\frac {11 x^{5/2}}{64 a^2 \left (a+c x^4\right )}-\frac {33 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{256 \sqrt {2} (-a)^{19/8} c^{5/8}}+\frac {33 \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{256 \sqrt {2} (-a)^{19/8} c^{5/8}}-\frac {33 \tan ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{256 (-a)^{19/8} c^{5/8}}-\frac {33 \tanh ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{256 (-a)^{19/8} c^{5/8}}-\frac {33 \log \left (\sqrt [4]{-a}-\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{c} x\right )}{512 \sqrt {2} (-a)^{19/8} c^{5/8}}+\frac {33 \log \left (\sqrt [4]{-a}+\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{c} x\right )}{512 \sqrt {2} (-a)^{19/8} c^{5/8}}\\ \end {align*}

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Mathematica [C]  time = 0.01, size = 29, normalized size = 0.09 \[ \frac {2 x^{5/2} \, _2F_1\left (\frac {5}{8},3;\frac {13}{8};-\frac {c x^4}{a}\right )}{5 a^3} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(3/2)/(a + c*x^4)^3,x]

[Out]

(2*x^(5/2)*Hypergeometric2F1[5/8, 3, 13/8, -((c*x^4)/a)])/(5*a^3)

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fricas [B]  time = 0.72, size = 674, normalized size = 2.05 \[ \frac {132 \, \sqrt {2} {\left (a^{2} c^{2} x^{8} + 2 \, a^{3} c x^{4} + a^{4}\right )} \left (-\frac {1}{a^{19} c^{5}}\right )^{\frac {1}{8}} \arctan \left (\sqrt {2} \sqrt {\sqrt {2} a^{12} c^{3} \sqrt {x} \left (-\frac {1}{a^{19} c^{5}}\right )^{\frac {5}{8}} - a^{5} c \left (-\frac {1}{a^{19} c^{5}}\right )^{\frac {1}{4}} + x} a^{7} c^{2} \left (-\frac {1}{a^{19} c^{5}}\right )^{\frac {3}{8}} - \sqrt {2} a^{7} c^{2} \sqrt {x} \left (-\frac {1}{a^{19} c^{5}}\right )^{\frac {3}{8}} + 1\right ) + 132 \, \sqrt {2} {\left (a^{2} c^{2} x^{8} + 2 \, a^{3} c x^{4} + a^{4}\right )} \left (-\frac {1}{a^{19} c^{5}}\right )^{\frac {1}{8}} \arctan \left (\sqrt {2} \sqrt {-\sqrt {2} a^{12} c^{3} \sqrt {x} \left (-\frac {1}{a^{19} c^{5}}\right )^{\frac {5}{8}} - a^{5} c \left (-\frac {1}{a^{19} c^{5}}\right )^{\frac {1}{4}} + x} a^{7} c^{2} \left (-\frac {1}{a^{19} c^{5}}\right )^{\frac {3}{8}} - \sqrt {2} a^{7} c^{2} \sqrt {x} \left (-\frac {1}{a^{19} c^{5}}\right )^{\frac {3}{8}} - 1\right ) + 33 \, \sqrt {2} {\left (a^{2} c^{2} x^{8} + 2 \, a^{3} c x^{4} + a^{4}\right )} \left (-\frac {1}{a^{19} c^{5}}\right )^{\frac {1}{8}} \log \left (\sqrt {2} a^{12} c^{3} \sqrt {x} \left (-\frac {1}{a^{19} c^{5}}\right )^{\frac {5}{8}} - a^{5} c \left (-\frac {1}{a^{19} c^{5}}\right )^{\frac {1}{4}} + x\right ) - 33 \, \sqrt {2} {\left (a^{2} c^{2} x^{8} + 2 \, a^{3} c x^{4} + a^{4}\right )} \left (-\frac {1}{a^{19} c^{5}}\right )^{\frac {1}{8}} \log \left (-\sqrt {2} a^{12} c^{3} \sqrt {x} \left (-\frac {1}{a^{19} c^{5}}\right )^{\frac {5}{8}} - a^{5} c \left (-\frac {1}{a^{19} c^{5}}\right )^{\frac {1}{4}} + x\right ) - 264 \, {\left (a^{2} c^{2} x^{8} + 2 \, a^{3} c x^{4} + a^{4}\right )} \left (-\frac {1}{a^{19} c^{5}}\right )^{\frac {1}{8}} \arctan \left (\sqrt {-a^{5} c \left (-\frac {1}{a^{19} c^{5}}\right )^{\frac {1}{4}} + x} a^{7} c^{2} \left (-\frac {1}{a^{19} c^{5}}\right )^{\frac {3}{8}} - a^{7} c^{2} \sqrt {x} \left (-\frac {1}{a^{19} c^{5}}\right )^{\frac {3}{8}}\right ) - 66 \, {\left (a^{2} c^{2} x^{8} + 2 \, a^{3} c x^{4} + a^{4}\right )} \left (-\frac {1}{a^{19} c^{5}}\right )^{\frac {1}{8}} \log \left (a^{12} c^{3} \left (-\frac {1}{a^{19} c^{5}}\right )^{\frac {5}{8}} + \sqrt {x}\right ) + 66 \, {\left (a^{2} c^{2} x^{8} + 2 \, a^{3} c x^{4} + a^{4}\right )} \left (-\frac {1}{a^{19} c^{5}}\right )^{\frac {1}{8}} \log \left (-a^{12} c^{3} \left (-\frac {1}{a^{19} c^{5}}\right )^{\frac {5}{8}} + \sqrt {x}\right ) + 16 \, {\left (11 \, c x^{6} + 19 \, a x^{2}\right )} \sqrt {x}}{1024 \, {\left (a^{2} c^{2} x^{8} + 2 \, a^{3} c x^{4} + a^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/1024*(132*sqrt(2)*(a^2*c^2*x^8 + 2*a^3*c*x^4 + a^4)*(-1/(a^19*c^5))^(1/8)*arctan(sqrt(2)*sqrt(sqrt(2)*a^12*c
^3*sqrt(x)*(-1/(a^19*c^5))^(5/8) - a^5*c*(-1/(a^19*c^5))^(1/4) + x)*a^7*c^2*(-1/(a^19*c^5))^(3/8) - sqrt(2)*a^
7*c^2*sqrt(x)*(-1/(a^19*c^5))^(3/8) + 1) + 132*sqrt(2)*(a^2*c^2*x^8 + 2*a^3*c*x^4 + a^4)*(-1/(a^19*c^5))^(1/8)
*arctan(sqrt(2)*sqrt(-sqrt(2)*a^12*c^3*sqrt(x)*(-1/(a^19*c^5))^(5/8) - a^5*c*(-1/(a^19*c^5))^(1/4) + x)*a^7*c^
2*(-1/(a^19*c^5))^(3/8) - sqrt(2)*a^7*c^2*sqrt(x)*(-1/(a^19*c^5))^(3/8) - 1) + 33*sqrt(2)*(a^2*c^2*x^8 + 2*a^3
*c*x^4 + a^4)*(-1/(a^19*c^5))^(1/8)*log(sqrt(2)*a^12*c^3*sqrt(x)*(-1/(a^19*c^5))^(5/8) - a^5*c*(-1/(a^19*c^5))
^(1/4) + x) - 33*sqrt(2)*(a^2*c^2*x^8 + 2*a^3*c*x^4 + a^4)*(-1/(a^19*c^5))^(1/8)*log(-sqrt(2)*a^12*c^3*sqrt(x)
*(-1/(a^19*c^5))^(5/8) - a^5*c*(-1/(a^19*c^5))^(1/4) + x) - 264*(a^2*c^2*x^8 + 2*a^3*c*x^4 + a^4)*(-1/(a^19*c^
5))^(1/8)*arctan(sqrt(-a^5*c*(-1/(a^19*c^5))^(1/4) + x)*a^7*c^2*(-1/(a^19*c^5))^(3/8) - a^7*c^2*sqrt(x)*(-1/(a
^19*c^5))^(3/8)) - 66*(a^2*c^2*x^8 + 2*a^3*c*x^4 + a^4)*(-1/(a^19*c^5))^(1/8)*log(a^12*c^3*(-1/(a^19*c^5))^(5/
8) + sqrt(x)) + 66*(a^2*c^2*x^8 + 2*a^3*c*x^4 + a^4)*(-1/(a^19*c^5))^(1/8)*log(-a^12*c^3*(-1/(a^19*c^5))^(5/8)
+ sqrt(x)) + 16*(11*c*x^6 + 19*a*x^2)*sqrt(x))/(a^2*c^2*x^8 + 2*a^3*c*x^4 + a^4)

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giac [B]  time = 0.86, size = 472, normalized size = 1.43 \[ -\frac {33 \, \left (\frac {a}{c}\right )^{\frac {5}{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 )}{256 \, a^{3} \sqrt {2 \, \sqrt {2} + 4}} - \frac {33 \, \left (\frac {a}{c}\right )^{\frac {5}{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 )}{256 \, a^{3} \sqrt {2 \, \sqrt {2} + 4}} + \frac {33 \, \left (\frac {a}{c}\right )^{\frac {5}{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 )}{256 \, a^{3} \sqrt {-2 \, \sqrt {2} + 4}} + \frac {33 \, \left (\frac {a}{c}\right )^{\frac {5}{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 )}{256 \, a^{3} \sqrt {-2 \, \sqrt {2} + 4}} - \frac {33 \, \left (\frac {a}{c}\right )^{\frac {5}{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 )}{512 \, a^{3} \sqrt {2 \, \sqrt {2} + 4}} + \frac {33 \, \left (\frac {a}{c}\right )^{\frac {5}{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 )}{512 \, a^{3} \sqrt {2 \, \sqrt {2} + 4}} + \frac {33 \, \left (\frac {a}{c}\right )^{\frac {5}{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 )}{512 \, a^{3} \sqrt {-2 \, \sqrt {2} + 4}} - \frac {33 \, \left (\frac {a}{c}\right )^{\frac {5}{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 )}{512 \, a^{3} \sqrt {-2 \, \sqrt {2} + 4}} + \frac {11 \, c x^{\frac {13}{2}} + 19 \, a x^{\frac {5}{2}}}{64 \, {\left (c x^{4} + a\right )}^{2} a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-33/256*(a/c)^(5/8)*arctan((sqrt(-sqrt(2) + 2)*(a/c)^(1/8) + 2*sqrt(x))/(sqrt(sqrt(2) + 2)*(a/c)^(1/8)))/(a^3*
sqrt(2*sqrt(2) + 4)) - 33/256*(a/c)^(5/8)*arctan(-(sqrt(-sqrt(2) + 2)*(a/c)^(1/8) - 2*sqrt(x))/(sqrt(sqrt(2) +
2)*(a/c)^(1/8)))/(a^3*sqrt(2*sqrt(2) + 4)) + 33/256*(a/c)^(5/8)*arctan((sqrt(sqrt(2) + 2)*(a/c)^(1/8) + 2*sqr
t(x))/(sqrt(-sqrt(2) + 2)*(a/c)^(1/8)))/(a^3*sqrt(-2*sqrt(2) + 4)) + 33/256*(a/c)^(5/8)*arctan(-(sqrt(sqrt(2)
+ 2)*(a/c)^(1/8) - 2*sqrt(x))/(sqrt(-sqrt(2) + 2)*(a/c)^(1/8)))/(a^3*sqrt(-2*sqrt(2) + 4)) - 33/512*(a/c)^(5/8
)*log(sqrt(x)*sqrt(sqrt(2) + 2)*(a/c)^(1/8) + x + (a/c)^(1/4))/(a^3*sqrt(2*sqrt(2) + 4)) + 33/512*(a/c)^(5/8)*
log(-sqrt(x)*sqrt(sqrt(2) + 2)*(a/c)^(1/8) + x + (a/c)^(1/4))/(a^3*sqrt(2*sqrt(2) + 4)) + 33/512*(a/c)^(5/8)*l
og(sqrt(x)*sqrt(-sqrt(2) + 2)*(a/c)^(1/8) + x + (a/c)^(1/4))/(a^3*sqrt(-2*sqrt(2) + 4)) - 33/512*(a/c)^(5/8)*l
og(-sqrt(x)*sqrt(-sqrt(2) + 2)*(a/c)^(1/8) + x + (a/c)^(1/4))/(a^3*sqrt(-2*sqrt(2) + 4)) + 1/64*(11*c*x^(13/2)
+ 19*a*x^(5/2))/((c*x^4 + a)^2*a^2)

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maple [C]  time = 0.02, size = 62, normalized size = 0.19 \[ \frac {33 \ln \left (-\RootOf \left (c \,\textit {\_Z}^{8}+a \right )+\sqrt {x}\right )}{512 a^{2} c \RootOf \left (c \,\textit {\_Z}^{8}+a \right )^{3}}+\frac {\frac {11 c \,x^{\frac {13}{2}}}{64 a^{2}}+\frac {19 x^{\frac {5}{2}}}{64 a}}{\left (c \,x^{4}+a \right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*(19/128*x^(5/2)/a+11/128/a^2*c*x^(13/2))/(c*x^4+a)^2+33/512/a^2/c*sum(1/_R^3*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 \[ \frac {11 \, c x^{\frac {13}{2}} + 19 \, a x^{\frac {5}{2}}}{64 \, {\left (a^{2} c^{2} x^{8} + 2 \, a^{3} c x^{4} + a^{4}\right )}} + 33 \, \int \frac {x^{\frac {3}{2}}}{128 \, {\left (a^{2} c x^{4} + a^{3}\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/64*(11*c*x^(13/2) + 19*a*x^(5/2))/(a^2*c^2*x^8 + 2*a^3*c*x^4 + a^4) + 33*integrate(1/128*x^(3/2)/(a^2*c*x^4
+ a^3), x)

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mupad [B]  time = 0.14, size = 157, normalized size = 0.48 \[ \frac {\frac {19\,x^{5/2}}{64\,a}+\frac {11\,c\,x^{13/2}}{64\,a^2}}{a^2+2\,a\,c\,x^4+c^2\,x^8}-\frac {33\,\mathrm {atan}\left (\frac {c^{1/8}\,\sqrt {x}}{{\left (-a\right )}^{1/8}}\right )}{256\,{\left (-a\right )}^{19/8}\,c^{5/8}}+\frac {\mathrm {atan}\left (\frac {c^{1/8}\,\sqrt {x}\,1{}\mathrm {i}}{{\left (-a\right )}^{1/8}}\right )\,33{}\mathrm {i}}{256\,{\left (-a\right )}^{19/8}\,c^{5/8}}+\frac {\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,c^{1/8}\,\sqrt {x}\,\left (\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right )}{{\left (-a\right )}^{1/8}}\right )\,\left (\frac {33}{512}+\frac {33}{512}{}\mathrm {i}\right )}{{\left (-a\right )}^{19/8}\,c^{5/8}}+\frac {\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,c^{1/8}\,\sqrt {x}\,\left (\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right )}{{\left (-a\right )}^{1/8}}\right )\,\left (\frac {33}{512}-\frac {33}{512}{}\mathrm {i}\right )}{{\left (-a\right )}^{19/8}\,c^{5/8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(3/2)/(a + c*x^4)^3,x)

[Out]

((19*x^(5/2))/(64*a) + (11*c*x^(13/2))/(64*a^2))/(a^2 + c^2*x^8 + 2*a*c*x^4) - (33*atan((c^(1/8)*x^(1/2))/(-a)
^(1/8)))/(256*(-a)^(19/8)*c^(5/8)) + (atan((c^(1/8)*x^(1/2)*1i)/(-a)^(1/8))*33i)/(256*(-a)^(19/8)*c^(5/8)) + (
2^(1/2)*atan((2^(1/2)*c^(1/8)*x^(1/2)*(1/2 - 1i/2))/(-a)^(1/8))*(33/512 + 33i/512))/((-a)^(19/8)*c^(5/8)) + (2
^(1/2)*atan((2^(1/2)*c^(1/8)*x^(1/2)*(1/2 + 1i/2))/(-a)^(1/8))*(33/512 - 33i/512))/((-a)^(19/8)*c^(5/8))

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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