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

Optimal. Leaf size=308 \[ -\frac {3 \log \left (-\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{32 \sqrt {2} (-a)^{5/8} c^{11/8}}+\frac {3 \log \left (\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{32 \sqrt {2} (-a)^{5/8} c^{11/8}}+\frac {3 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{16 \sqrt {2} (-a)^{5/8} c^{11/8}}-\frac {3 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}+1\right )}{16 \sqrt {2} (-a)^{5/8} c^{11/8}}+\frac {3 \tan ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{16 (-a)^{5/8} c^{11/8}}-\frac {3 \tanh ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{16 (-a)^{5/8} c^{11/8}}-\frac {x^{3/2}}{4 c \left (a+c x^4\right )} \]

[Out]

-1/4*x^(3/2)/c/(c*x^4+a)+3/16*arctan(c^(1/8)*x^(1/2)/(-a)^(1/8))/(-a)^(5/8)/c^(11/8)-3/16*arctanh(c^(1/8)*x^(1
/2)/(-a)^(1/8))/(-a)^(5/8)/c^(11/8)-3/32*arctan(-1+c^(1/8)*2^(1/2)*x^(1/2)/(-a)^(1/8))/(-a)^(5/8)/c^(11/8)*2^(
1/2)-3/32*arctan(1+c^(1/8)*2^(1/2)*x^(1/2)/(-a)^(1/8))/(-a)^(5/8)/c^(11/8)*2^(1/2)-3/64*ln((-a)^(1/4)+c^(1/4)*
x-(-a)^(1/8)*c^(1/8)*2^(1/2)*x^(1/2))/(-a)^(5/8)/c^(11/8)*2^(1/2)+3/64*ln((-a)^(1/4)+c^(1/4)*x+(-a)^(1/8)*c^(1
/8)*2^(1/2)*x^(1/2))/(-a)^(5/8)/c^(11/8)*2^(1/2)

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Rubi [A]  time = 0.25, antiderivative size = 308, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 12, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {288, 329, 300, 297, 1162, 617, 204, 1165, 628, 298, 205, 208} \[ -\frac {3 \log \left (-\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{32 \sqrt {2} (-a)^{5/8} c^{11/8}}+\frac {3 \log \left (\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{32 \sqrt {2} (-a)^{5/8} c^{11/8}}+\frac {3 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{16 \sqrt {2} (-a)^{5/8} c^{11/8}}-\frac {3 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}+1\right )}{16 \sqrt {2} (-a)^{5/8} c^{11/8}}+\frac {3 \tan ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{16 (-a)^{5/8} c^{11/8}}-\frac {3 \tanh ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{16 (-a)^{5/8} c^{11/8}}-\frac {x^{3/2}}{4 c \left (a+c x^4\right )} \]

Antiderivative was successfully verified.

[In]

Int[x^(9/2)/(a + c*x^4)^2,x]

[Out]

-x^(3/2)/(4*c*(a + c*x^4)) + (3*ArcTan[1 - (Sqrt[2]*c^(1/8)*Sqrt[x])/(-a)^(1/8)])/(16*Sqrt[2]*(-a)^(5/8)*c^(11
/8)) - (3*ArcTan[1 + (Sqrt[2]*c^(1/8)*Sqrt[x])/(-a)^(1/8)])/(16*Sqrt[2]*(-a)^(5/8)*c^(11/8)) + (3*ArcTan[(c^(1
/8)*Sqrt[x])/(-a)^(1/8)])/(16*(-a)^(5/8)*c^(11/8)) - (3*ArcTanh[(c^(1/8)*Sqrt[x])/(-a)^(1/8)])/(16*(-a)^(5/8)*
c^(11/8)) - (3*Log[(-a)^(1/4) - Sqrt[2]*(-a)^(1/8)*c^(1/8)*Sqrt[x] + c^(1/4)*x])/(32*Sqrt[2]*(-a)^(5/8)*c^(11/
8)) + (3*Log[(-a)^(1/4) + Sqrt[2]*(-a)^(1/8)*c^(1/8)*Sqrt[x] + c^(1/4)*x])/(32*Sqrt[2]*(-a)^(5/8)*c^(11/8))

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 288

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*n*(p + 1)), x] - Dist[(c^n*(m - n + 1))/(b*n*(p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^(p + 1), x
], x] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m + 1, n] &&  !ILtQ[(m + n*(p + 1) + 1)/n, 0]
&& IntBinomialQ[a, b, c, n, m, p, x]

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 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 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^{9/2}}{\left (a+c x^4\right )^2} \, dx &=-\frac {x^{3/2}}{4 c \left (a+c x^4\right )}+\frac {3 \int \frac {\sqrt {x}}{a+c x^4} \, dx}{8 c}\\ &=-\frac {x^{3/2}}{4 c \left (a+c x^4\right )}+\frac {3 \operatorname {Subst}\left (\int \frac {x^2}{a+c x^8} \, dx,x,\sqrt {x}\right )}{4 c}\\ &=-\frac {x^{3/2}}{4 c \left (a+c x^4\right )}-\frac {3 \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {-a}-\sqrt {c} x^4} \, dx,x,\sqrt {x}\right )}{8 \sqrt {-a} c}-\frac {3 \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {-a}+\sqrt {c} x^4} \, dx,x,\sqrt {x}\right )}{8 \sqrt {-a} c}\\ &=-\frac {x^{3/2}}{4 c \left (a+c x^4\right )}-\frac {3 \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{-a}-\sqrt [4]{c} x^2} \, dx,x,\sqrt {x}\right )}{16 \sqrt {-a} c^{5/4}}+\frac {3 \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{-a}+\sqrt [4]{c} x^2} \, dx,x,\sqrt {x}\right )}{16 \sqrt {-a} c^{5/4}}+\frac {3 \operatorname {Subst}\left (\int \frac {\sqrt [4]{-a}-\sqrt [4]{c} x^2}{\sqrt {-a}+\sqrt {c} x^4} \, dx,x,\sqrt {x}\right )}{16 \sqrt {-a} c^{5/4}}-\frac {3 \operatorname {Subst}\left (\int \frac {\sqrt [4]{-a}+\sqrt [4]{c} x^2}{\sqrt {-a}+\sqrt {c} x^4} \, dx,x,\sqrt {x}\right )}{16 \sqrt {-a} c^{5/4}}\\ &=-\frac {x^{3/2}}{4 c \left (a+c x^4\right )}+\frac {3 \tan ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{16 (-a)^{5/8} c^{11/8}}-\frac {3 \tanh ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{16 (-a)^{5/8} c^{11/8}}-\frac {3 \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 )}{32 \sqrt {-a} c^{3/2}}-\frac {3 \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 )}{32 \sqrt {-a} c^{3/2}}-\frac {3 \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 )}{32 \sqrt {2} (-a)^{5/8} c^{11/8}}-\frac {3 \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 )}{32 \sqrt {2} (-a)^{5/8} c^{11/8}}\\ &=-\frac {x^{3/2}}{4 c \left (a+c x^4\right )}+\frac {3 \tan ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{16 (-a)^{5/8} c^{11/8}}-\frac {3 \tanh ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{16 (-a)^{5/8} c^{11/8}}-\frac {3 \log \left (\sqrt [4]{-a}-\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{c} x\right )}{32 \sqrt {2} (-a)^{5/8} c^{11/8}}+\frac {3 \log \left (\sqrt [4]{-a}+\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{c} x\right )}{32 \sqrt {2} (-a)^{5/8} c^{11/8}}-\frac {3 \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{16 \sqrt {2} (-a)^{5/8} c^{11/8}}+\frac {3 \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{16 \sqrt {2} (-a)^{5/8} c^{11/8}}\\ &=-\frac {x^{3/2}}{4 c \left (a+c x^4\right )}+\frac {3 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{16 \sqrt {2} (-a)^{5/8} c^{11/8}}-\frac {3 \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{16 \sqrt {2} (-a)^{5/8} c^{11/8}}+\frac {3 \tan ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{16 (-a)^{5/8} c^{11/8}}-\frac {3 \tanh ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{16 (-a)^{5/8} c^{11/8}}-\frac {3 \log \left (\sqrt [4]{-a}-\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{c} x\right )}{32 \sqrt {2} (-a)^{5/8} c^{11/8}}+\frac {3 \log \left (\sqrt [4]{-a}+\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{c} x\right )}{32 \sqrt {2} (-a)^{5/8} c^{11/8}}\\ \end {align*}

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Mathematica [C]  time = 0.01, size = 54, normalized size = 0.18 \[ \frac {2 x^{3/2} \, _2F_1\left (\frac {3}{8},2;\frac {11}{8};-\frac {c x^4}{a}\right )}{5 a c}-\frac {2 x^{3/2}}{5 c \left (a+c x^4\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(9/2)/(a + c*x^4)^2,x]

[Out]

(-2*x^(3/2))/(5*c*(a + c*x^4)) + (2*x^(3/2)*Hypergeometric2F1[3/8, 2, 11/8, -((c*x^4)/a)])/(5*a*c)

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fricas [B]  time = 0.78, size = 570, normalized size = 1.85 \[ -\frac {12 \, \sqrt {2} {\left (c^{2} x^{4} + a c\right )} \left (-\frac {1}{a^{5} c^{11}}\right )^{\frac {1}{8}} \arctan \left (\sqrt {2} \sqrt {a^{4} c^{8} \left (-\frac {1}{a^{5} c^{11}}\right )^{\frac {3}{4}} + \sqrt {2} a^{2} c^{4} \sqrt {x} \left (-\frac {1}{a^{5} c^{11}}\right )^{\frac {3}{8}} + x} a^{3} c^{7} \left (-\frac {1}{a^{5} c^{11}}\right )^{\frac {5}{8}} - \sqrt {2} a^{3} c^{7} \sqrt {x} \left (-\frac {1}{a^{5} c^{11}}\right )^{\frac {5}{8}} + 1\right ) + 12 \, \sqrt {2} {\left (c^{2} x^{4} + a c\right )} \left (-\frac {1}{a^{5} c^{11}}\right )^{\frac {1}{8}} \arctan \left (\sqrt {2} \sqrt {a^{4} c^{8} \left (-\frac {1}{a^{5} c^{11}}\right )^{\frac {3}{4}} - \sqrt {2} a^{2} c^{4} \sqrt {x} \left (-\frac {1}{a^{5} c^{11}}\right )^{\frac {3}{8}} + x} a^{3} c^{7} \left (-\frac {1}{a^{5} c^{11}}\right )^{\frac {5}{8}} - \sqrt {2} a^{3} c^{7} \sqrt {x} \left (-\frac {1}{a^{5} c^{11}}\right )^{\frac {5}{8}} - 1\right ) - 3 \, \sqrt {2} {\left (c^{2} x^{4} + a c\right )} \left (-\frac {1}{a^{5} c^{11}}\right )^{\frac {1}{8}} \log \left (a^{4} c^{8} \left (-\frac {1}{a^{5} c^{11}}\right )^{\frac {3}{4}} + \sqrt {2} a^{2} c^{4} \sqrt {x} \left (-\frac {1}{a^{5} c^{11}}\right )^{\frac {3}{8}} + x\right ) + 3 \, \sqrt {2} {\left (c^{2} x^{4} + a c\right )} \left (-\frac {1}{a^{5} c^{11}}\right )^{\frac {1}{8}} \log \left (a^{4} c^{8} \left (-\frac {1}{a^{5} c^{11}}\right )^{\frac {3}{4}} - \sqrt {2} a^{2} c^{4} \sqrt {x} \left (-\frac {1}{a^{5} c^{11}}\right )^{\frac {3}{8}} + x\right ) - 24 \, {\left (c^{2} x^{4} + a c\right )} \left (-\frac {1}{a^{5} c^{11}}\right )^{\frac {1}{8}} \arctan \left (\sqrt {a^{4} c^{8} \left (-\frac {1}{a^{5} c^{11}}\right )^{\frac {3}{4}} + x} a^{3} c^{7} \left (-\frac {1}{a^{5} c^{11}}\right )^{\frac {5}{8}} - a^{3} c^{7} \sqrt {x} \left (-\frac {1}{a^{5} c^{11}}\right )^{\frac {5}{8}}\right ) + 6 \, {\left (c^{2} x^{4} + a c\right )} \left (-\frac {1}{a^{5} c^{11}}\right )^{\frac {1}{8}} \log \left (a^{2} c^{4} \left (-\frac {1}{a^{5} c^{11}}\right )^{\frac {3}{8}} + \sqrt {x}\right ) - 6 \, {\left (c^{2} x^{4} + a c\right )} \left (-\frac {1}{a^{5} c^{11}}\right )^{\frac {1}{8}} \log \left (-a^{2} c^{4} \left (-\frac {1}{a^{5} c^{11}}\right )^{\frac {3}{8}} + \sqrt {x}\right ) + 16 \, x^{\frac {3}{2}}}{64 \, {\left (c^{2} x^{4} + a c\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/64*(12*sqrt(2)*(c^2*x^4 + a*c)*(-1/(a^5*c^11))^(1/8)*arctan(sqrt(2)*sqrt(a^4*c^8*(-1/(a^5*c^11))^(3/4) + sq
rt(2)*a^2*c^4*sqrt(x)*(-1/(a^5*c^11))^(3/8) + x)*a^3*c^7*(-1/(a^5*c^11))^(5/8) - sqrt(2)*a^3*c^7*sqrt(x)*(-1/(
a^5*c^11))^(5/8) + 1) + 12*sqrt(2)*(c^2*x^4 + a*c)*(-1/(a^5*c^11))^(1/8)*arctan(sqrt(2)*sqrt(a^4*c^8*(-1/(a^5*
c^11))^(3/4) - sqrt(2)*a^2*c^4*sqrt(x)*(-1/(a^5*c^11))^(3/8) + x)*a^3*c^7*(-1/(a^5*c^11))^(5/8) - sqrt(2)*a^3*
c^7*sqrt(x)*(-1/(a^5*c^11))^(5/8) - 1) - 3*sqrt(2)*(c^2*x^4 + a*c)*(-1/(a^5*c^11))^(1/8)*log(a^4*c^8*(-1/(a^5*
c^11))^(3/4) + sqrt(2)*a^2*c^4*sqrt(x)*(-1/(a^5*c^11))^(3/8) + x) + 3*sqrt(2)*(c^2*x^4 + a*c)*(-1/(a^5*c^11))^
(1/8)*log(a^4*c^8*(-1/(a^5*c^11))^(3/4) - sqrt(2)*a^2*c^4*sqrt(x)*(-1/(a^5*c^11))^(3/8) + x) - 24*(c^2*x^4 + a
*c)*(-1/(a^5*c^11))^(1/8)*arctan(sqrt(a^4*c^8*(-1/(a^5*c^11))^(3/4) + x)*a^3*c^7*(-1/(a^5*c^11))^(5/8) - a^3*c
^7*sqrt(x)*(-1/(a^5*c^11))^(5/8)) + 6*(c^2*x^4 + a*c)*(-1/(a^5*c^11))^(1/8)*log(a^2*c^4*(-1/(a^5*c^11))^(3/8)
+ sqrt(x)) - 6*(c^2*x^4 + a*c)*(-1/(a^5*c^11))^(1/8)*log(-a^2*c^4*(-1/(a^5*c^11))^(3/8) + sqrt(x)) + 16*x^(3/2
))/(c^2*x^4 + a*c)

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giac [B]  time = 0.73, size = 486, normalized size = 1.58 \[ -\frac {x^{\frac {3}{2}}}{4 \, {\left (c x^{4} + a\right )} c} - \frac {3 \, \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 )}{16 \, a c \sqrt {2 \, \sqrt {2} + 4}} - \frac {3 \, \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 )}{16 \, a c \sqrt {2 \, \sqrt {2} + 4}} + \frac {3 \, \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 )}{16 \, a c \sqrt {-2 \, \sqrt {2} + 4}} + \frac {3 \, \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 )}{16 \, a c \sqrt {-2 \, \sqrt {2} + 4}} + \frac {3 \, \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 )}{32 \, a c \sqrt {2 \, \sqrt {2} + 4}} - \frac {3 \, \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 )}{32 \, a c \sqrt {2 \, \sqrt {2} + 4}} - \frac {3 \, \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 )}{32 \, a c \sqrt {-2 \, \sqrt {2} + 4}} + \frac {3 \, \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 )}{32 \, a c \sqrt {-2 \, \sqrt {2} + 4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*x^(3/2)/((c*x^4 + a)*c) - 3/16*(a/c)^(3/8)*arctan((sqrt(-sqrt(2) + 2)*(a/c)^(1/8) + 2*sqrt(x))/(sqrt(sqrt
(2) + 2)*(a/c)^(1/8)))/(a*c*sqrt(2*sqrt(2) + 4)) - 3/16*(a/c)^(3/8)*arctan(-(sqrt(-sqrt(2) + 2)*(a/c)^(1/8) -
2*sqrt(x))/(sqrt(sqrt(2) + 2)*(a/c)^(1/8)))/(a*c*sqrt(2*sqrt(2) + 4)) + 3/16*(a/c)^(3/8)*arctan((sqrt(sqrt(2)
+ 2)*(a/c)^(1/8) + 2*sqrt(x))/(sqrt(-sqrt(2) + 2)*(a/c)^(1/8)))/(a*c*sqrt(-2*sqrt(2) + 4)) + 3/16*(a/c)^(3/8)*
arctan(-(sqrt(sqrt(2) + 2)*(a/c)^(1/8) - 2*sqrt(x))/(sqrt(-sqrt(2) + 2)*(a/c)^(1/8)))/(a*c*sqrt(-2*sqrt(2) + 4
)) + 3/32*(a/c)^(3/8)*log(sqrt(x)*sqrt(sqrt(2) + 2)*(a/c)^(1/8) + x + (a/c)^(1/4))/(a*c*sqrt(2*sqrt(2) + 4)) -
3/32*(a/c)^(3/8)*log(-sqrt(x)*sqrt(sqrt(2) + 2)*(a/c)^(1/8) + x + (a/c)^(1/4))/(a*c*sqrt(2*sqrt(2) + 4)) - 3/
32*(a/c)^(3/8)*log(sqrt(x)*sqrt(-sqrt(2) + 2)*(a/c)^(1/8) + x + (a/c)^(1/4))/(a*c*sqrt(-2*sqrt(2) + 4)) + 3/32
*(a/c)^(3/8)*log(-sqrt(x)*sqrt(-sqrt(2) + 2)*(a/c)^(1/8) + x + (a/c)^(1/4))/(a*c*sqrt(-2*sqrt(2) + 4))

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maple [C]  time = 0.02, size = 47, normalized size = 0.15 \[ -\frac {x^{\frac {3}{2}}}{4 \left (c \,x^{4}+a \right ) c}+\frac {3 \ln \left (-\RootOf \left (c \,\textit {\_Z}^{8}+a \right )+\sqrt {x}\right )}{32 c^{2} \RootOf \left (c \,\textit {\_Z}^{8}+a \right )^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*x^(3/2)/c/(c*x^4+a)+3/32/c^2*sum(1/_R^5*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 \[ -\frac {x^{\frac {3}{2}}}{4 \, {\left (c^{2} x^{4} + a c\right )}} + 3 \, \int \frac {\sqrt {x}}{8 \, {\left (c^{2} x^{4} + a c\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*x^(3/2)/(c^2*x^4 + a*c) + 3*integrate(1/8*sqrt(x)/(c^2*x^4 + a*c), x)

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mupad [B]  time = 1.06, size = 135, normalized size = 0.44 \[ \frac {3\,\mathrm {atan}\left (\frac {c^{1/8}\,\sqrt {x}}{{\left (-a\right )}^{1/8}}\right )}{16\,{\left (-a\right )}^{5/8}\,c^{11/8}}-\frac {x^{3/2}}{4\,c\,\left (c\,x^4+a\right )}+\frac {\mathrm {atan}\left (\frac {c^{1/8}\,\sqrt {x}\,1{}\mathrm {i}}{{\left (-a\right )}^{1/8}}\right )\,3{}\mathrm {i}}{16\,{\left (-a\right )}^{5/8}\,c^{11/8}}+\frac {\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,c^{1/8}\,\sqrt {x}\,\left (\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right )}{{\left (-a\right )}^{1/8}}\right )\,\left (-\frac {3}{32}+\frac {3}{32}{}\mathrm {i}\right )}{{\left (-a\right )}^{5/8}\,c^{11/8}}+\frac {\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,c^{1/8}\,\sqrt {x}\,\left (\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right )}{{\left (-a\right )}^{1/8}}\right )\,\left (-\frac {3}{32}-\frac {3}{32}{}\mathrm {i}\right )}{{\left (-a\right )}^{5/8}\,c^{11/8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(9/2)/(a + c*x^4)^2,x)

[Out]

(3*atan((c^(1/8)*x^(1/2))/(-a)^(1/8)))/(16*(-a)^(5/8)*c^(11/8)) - x^(3/2)/(4*c*(a + c*x^4)) + (atan((c^(1/8)*x
^(1/2)*1i)/(-a)^(1/8))*3i)/(16*(-a)^(5/8)*c^(11/8)) - (2^(1/2)*atan((2^(1/2)*c^(1/8)*x^(1/2)*(1/2 - 1i/2))/(-a
)^(1/8))*(3/32 - 3i/32))/((-a)^(5/8)*c^(11/8)) - (2^(1/2)*atan((2^(1/2)*c^(1/8)*x^(1/2)*(1/2 + 1i/2))/(-a)^(1/
8))*(3/32 + 3i/32))/((-a)^(5/8)*c^(11/8))

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

[Out]

Timed out

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