3.8.54 \(\int \frac {1}{x^{3/2} (a+c x^4)^2} \, dx\) [754]

Optimal. Leaf size=320 \[ -\frac {9}{4 a^2 \sqrt {x}}+\frac {1}{4 a \sqrt {x} \left (a+c x^4\right )}+\frac {9 \sqrt [8]{c} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{16 \sqrt {2} (-a)^{17/8}}-\frac {9 \sqrt [8]{c} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{16 \sqrt {2} (-a)^{17/8}}-\frac {9 \sqrt [8]{c} \tan ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{16 (-a)^{17/8}}+\frac {9 \sqrt [8]{c} \tanh ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{16 (-a)^{17/8}}-\frac {9 \sqrt [8]{c} \log \left (\sqrt [4]{-a}-\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{c} x\right )}{32 \sqrt {2} (-a)^{17/8}}+\frac {9 \sqrt [8]{c} \log \left (\sqrt [4]{-a}+\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{c} x\right )}{32 \sqrt {2} (-a)^{17/8}} \]

[Out]

-9/16*c^(1/8)*arctan(c^(1/8)*x^(1/2)/(-a)^(1/8))/(-a)^(17/8)+9/16*c^(1/8)*arctanh(c^(1/8)*x^(1/2)/(-a)^(1/8))/
(-a)^(17/8)-9/32*c^(1/8)*arctan(-1+c^(1/8)*2^(1/2)*x^(1/2)/(-a)^(1/8))/(-a)^(17/8)*2^(1/2)-9/32*c^(1/8)*arctan
(1+c^(1/8)*2^(1/2)*x^(1/2)/(-a)^(1/8))/(-a)^(17/8)*2^(1/2)-9/64*c^(1/8)*ln((-a)^(1/4)+c^(1/4)*x-(-a)^(1/8)*c^(
1/8)*2^(1/2)*x^(1/2))/(-a)^(17/8)*2^(1/2)+9/64*c^(1/8)*ln((-a)^(1/4)+c^(1/4)*x+(-a)^(1/8)*c^(1/8)*2^(1/2)*x^(1
/2))/(-a)^(17/8)*2^(1/2)-9/4/a^2/x^(1/2)+1/4/a/(c*x^4+a)/x^(1/2)

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Rubi [A]
time = 0.21, antiderivative size = 320, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 13, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.867, Rules used = {296, 331, 335, 307, 303, 1176, 631, 210, 1179, 642, 304, 211, 214} \begin {gather*} -\frac {9}{4 a^2 \sqrt {x}}+\frac {9 \sqrt [8]{c} \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{16 \sqrt {2} (-a)^{17/8}}-\frac {9 \sqrt [8]{c} \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}+1\right )}{16 \sqrt {2} (-a)^{17/8}}-\frac {9 \sqrt [8]{c} \text {ArcTan}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{16 (-a)^{17/8}}+\frac {1}{4 a \sqrt {x} \left (a+c x^4\right )}-\frac {9 \sqrt [8]{c} \log \left (-\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{32 \sqrt {2} (-a)^{17/8}}+\frac {9 \sqrt [8]{c} \log \left (\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{32 \sqrt {2} (-a)^{17/8}}+\frac {9 \sqrt [8]{c} \tanh ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{16 (-a)^{17/8}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^(3/2)*(a + c*x^4)^2),x]

[Out]

-9/(4*a^2*Sqrt[x]) + 1/(4*a*Sqrt[x]*(a + c*x^4)) + (9*c^(1/8)*ArcTan[1 - (Sqrt[2]*c^(1/8)*Sqrt[x])/(-a)^(1/8)]
)/(16*Sqrt[2]*(-a)^(17/8)) - (9*c^(1/8)*ArcTan[1 + (Sqrt[2]*c^(1/8)*Sqrt[x])/(-a)^(1/8)])/(16*Sqrt[2]*(-a)^(17
/8)) - (9*c^(1/8)*ArcTan[(c^(1/8)*Sqrt[x])/(-a)^(1/8)])/(16*(-a)^(17/8)) + (9*c^(1/8)*ArcTanh[(c^(1/8)*Sqrt[x]
)/(-a)^(1/8)])/(16*(-a)^(17/8)) - (9*c^(1/8)*Log[(-a)^(1/4) - Sqrt[2]*(-a)^(1/8)*c^(1/8)*Sqrt[x] + c^(1/4)*x])
/(32*Sqrt[2]*(-a)^(17/8)) + (9*c^(1/8)*Log[(-a)^(1/4) + Sqrt[2]*(-a)^(1/8)*c^(1/8)*Sqrt[x] + c^(1/4)*x])/(32*S
qrt[2]*(-a)^(17/8))

Rule 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 211

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/Rt[a/b, 2]], x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 214

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], x] /; FreeQ[{a, b},
x] && NegQ[a/b]

Rule 296

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] /; Free
Q[{a, b, c, m}, x] && IGtQ[n, 0] && LtQ[p, -1] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 303

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 304

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 307

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 331

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*c
*(m + 1))), x] - Dist[b*((m + n*(p + 1) + 1)/(a*c^n*(m + 1))), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 335

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 631

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 642

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 1176

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 1179

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 {1}{x^{3/2} \left (a+c x^4\right )^2} \, dx &=\frac {1}{4 a \sqrt {x} \left (a+c x^4\right )}+\frac {9 \int \frac {1}{x^{3/2} \left (a+c x^4\right )} \, dx}{8 a}\\ &=-\frac {9}{4 a^2 \sqrt {x}}+\frac {1}{4 a \sqrt {x} \left (a+c x^4\right )}-\frac {(9 c) \int \frac {x^{5/2}}{a+c x^4} \, dx}{8 a^2}\\ &=-\frac {9}{4 a^2 \sqrt {x}}+\frac {1}{4 a \sqrt {x} \left (a+c x^4\right )}-\frac {(9 c) \text {Subst}\left (\int \frac {x^6}{a+c x^8} \, dx,x,\sqrt {x}\right )}{4 a^2}\\ &=-\frac {9}{4 a^2 \sqrt {x}}+\frac {1}{4 a \sqrt {x} \left (a+c x^4\right )}+\frac {\left (9 \sqrt {c}\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {-a}-\sqrt {c} x^4} \, dx,x,\sqrt {x}\right )}{8 a^2}-\frac {\left (9 \sqrt {c}\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {-a}+\sqrt {c} x^4} \, dx,x,\sqrt {x}\right )}{8 a^2}\\ &=-\frac {9}{4 a^2 \sqrt {x}}+\frac {1}{4 a \sqrt {x} \left (a+c x^4\right )}+\frac {\left (9 \sqrt [4]{c}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{-a}-\sqrt [4]{c} x^2} \, dx,x,\sqrt {x}\right )}{16 a^2}-\frac {\left (9 \sqrt [4]{c}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{-a}+\sqrt [4]{c} x^2} \, dx,x,\sqrt {x}\right )}{16 a^2}+\frac {\left (9 \sqrt [4]{c}\right ) \text {Subst}\left (\int \frac {\sqrt [4]{-a}-\sqrt [4]{c} x^2}{\sqrt {-a}+\sqrt {c} x^4} \, dx,x,\sqrt {x}\right )}{16 a^2}-\frac {\left (9 \sqrt [4]{c}\right ) \text {Subst}\left (\int \frac {\sqrt [4]{-a}+\sqrt [4]{c} x^2}{\sqrt {-a}+\sqrt {c} x^4} \, dx,x,\sqrt {x}\right )}{16 a^2}\\ &=-\frac {9}{4 a^2 \sqrt {x}}+\frac {1}{4 a \sqrt {x} \left (a+c x^4\right )}-\frac {9 \sqrt [8]{c} \tan ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{16 (-a)^{17/8}}+\frac {9 \sqrt [8]{c} \tanh ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{16 (-a)^{17/8}}-\frac {9 \text {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 a^2}-\frac {9 \text {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 a^2}-\frac {\left (9 \sqrt [8]{c}\right ) \text {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)^{17/8}}-\frac {\left (9 \sqrt [8]{c}\right ) \text {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)^{17/8}}\\ &=-\frac {9}{4 a^2 \sqrt {x}}+\frac {1}{4 a \sqrt {x} \left (a+c x^4\right )}-\frac {9 \sqrt [8]{c} \tan ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{16 (-a)^{17/8}}+\frac {9 \sqrt [8]{c} \tanh ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{16 (-a)^{17/8}}-\frac {9 \sqrt [8]{c} \log \left (\sqrt [4]{-a}-\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{c} x\right )}{32 \sqrt {2} (-a)^{17/8}}+\frac {9 \sqrt [8]{c} \log \left (\sqrt [4]{-a}+\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{c} x\right )}{32 \sqrt {2} (-a)^{17/8}}-\frac {\left (9 \sqrt [8]{c}\right ) \text {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)^{17/8}}+\frac {\left (9 \sqrt [8]{c}\right ) \text {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)^{17/8}}\\ &=-\frac {9}{4 a^2 \sqrt {x}}+\frac {1}{4 a \sqrt {x} \left (a+c x^4\right )}+\frac {9 \sqrt [8]{c} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{16 \sqrt {2} (-a)^{17/8}}-\frac {9 \sqrt [8]{c} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{16 \sqrt {2} (-a)^{17/8}}-\frac {9 \sqrt [8]{c} \tan ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{16 (-a)^{17/8}}+\frac {9 \sqrt [8]{c} \tanh ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{16 (-a)^{17/8}}-\frac {9 \sqrt [8]{c} \log \left (\sqrt [4]{-a}-\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{c} x\right )}{32 \sqrt {2} (-a)^{17/8}}+\frac {9 \sqrt [8]{c} \log \left (\sqrt [4]{-a}+\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{c} x\right )}{32 \sqrt {2} (-a)^{17/8}}\\ \end {align*}

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Mathematica [A]
time = 1.21, size = 287, normalized size = 0.90 \begin {gather*} \frac {-\frac {8 \sqrt [8]{a} \left (8 a+9 c x^4\right )}{\sqrt {x} \left (a+c x^4\right )}+9 \sqrt {2+\sqrt {2}} \sqrt [8]{c} \tan ^{-1}\left (\frac {\sqrt {1-\frac {1}{\sqrt {2}}} \left (\sqrt [4]{a}-\sqrt [4]{c} x\right )}{\sqrt [8]{a} \sqrt [8]{c} \sqrt {x}}\right )+9 \sqrt {2-\sqrt {2}} \sqrt [8]{c} \tan ^{-1}\left (\frac {\sqrt {1+\frac {1}{\sqrt {2}}} \left (\sqrt [4]{a}-\sqrt [4]{c} x\right )}{\sqrt [8]{a} \sqrt [8]{c} \sqrt {x}}\right )+9 \sqrt {2+\sqrt {2}} \sqrt [8]{c} \tanh ^{-1}\left (\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{a} \sqrt [8]{c} \sqrt {x}}{\sqrt [4]{a}+\sqrt [4]{c} x}\right )+9 \sqrt {2-\sqrt {2}} \sqrt [8]{c} \tanh ^{-1}\left (\frac {\sqrt [8]{a} \sqrt [8]{c} \sqrt {-\left (\left (-2+\sqrt {2}\right ) x\right )}}{\sqrt [4]{a}+\sqrt [4]{c} x}\right )}{32 a^{17/8}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^(3/2)*(a + c*x^4)^2),x]

[Out]

((-8*a^(1/8)*(8*a + 9*c*x^4))/(Sqrt[x]*(a + c*x^4)) + 9*Sqrt[2 + Sqrt[2]]*c^(1/8)*ArcTan[(Sqrt[1 - 1/Sqrt[2]]*
(a^(1/4) - c^(1/4)*x))/(a^(1/8)*c^(1/8)*Sqrt[x])] + 9*Sqrt[2 - Sqrt[2]]*c^(1/8)*ArcTan[(Sqrt[1 + 1/Sqrt[2]]*(a
^(1/4) - c^(1/4)*x))/(a^(1/8)*c^(1/8)*Sqrt[x])] + 9*Sqrt[2 + Sqrt[2]]*c^(1/8)*ArcTanh[(Sqrt[2 + Sqrt[2]]*a^(1/
8)*c^(1/8)*Sqrt[x])/(a^(1/4) + c^(1/4)*x)] + 9*Sqrt[2 - Sqrt[2]]*c^(1/8)*ArcTanh[(a^(1/8)*c^(1/8)*Sqrt[-((-2 +
Sqrt[2])*x)])/(a^(1/4) + c^(1/4)*x)])/(32*a^(17/8))

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.18, size = 59, normalized size = 0.18

method	result	size
risch	\(-\frac {2}{a^{2} \sqrt {x}}-\frac {c \,x^{\frac {7}{2}}}{4 a^{2} \left (x^{4} c +a \right )}-\frac {9 \left (\munderset {\textit {\_R} =\RootOf \left (c \,\textit {\_Z}^{8}+a \right )}{\sum }\frac {\ln \left (\sqrt {x}-\textit {\_R} \right )}{\textit {\_R}}\right )}{32 a^{2}}\)	\(56\)
derivativedivides	\(-\frac {2 c \left (\frac {x^{\frac {7}{2}}}{8 x^{4} c +8 a}+\frac {9 \left (\munderset {\textit {\_R} =\RootOf \left (c \,\textit {\_Z}^{8}+a \right )}{\sum }\frac {\ln \left (\sqrt {x}-\textit {\_R} \right )}{\textit {\_R}}\right )}{64 c}\right )}{a^{2}}-\frac {2}{a^{2} \sqrt {x}}\)	\(59\)
default	\(-\frac {2 c \left (\frac {x^{\frac {7}{2}}}{8 x^{4} c +8 a}+\frac {9 \left (\munderset {\textit {\_R} =\RootOf \left (c \,\textit {\_Z}^{8}+a \right )}{\sum }\frac {\ln \left (\sqrt {x}-\textit {\_R} \right )}{\textit {\_R}}\right )}{64 c}\right )}{a^{2}}-\frac {2}{a^{2} \sqrt {x}}\)	\(59\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^(3/2)/(c*x^4+a)^2,x,method=_RETURNVERBOSE)

[Out]

-2/a^2*c*(1/8*x^(7/2)/(c*x^4+a)+9/64/c*sum(1/_R*ln(x^(1/2)-_R),_R=RootOf(_Z^8*c+a)))-2/a^2/x^(1/2)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^(3/2)/(c*x^4+a)^2,x, algorithm="maxima")

[Out]

-9*c*integrate(1/8*x^(5/2)/(a^2*c*x^4 + a^3), x) - 1/4*(9*c*x^(7/2) + 8*a/sqrt(x))/(a^2*c*x^4 + a^3)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 575 vs. \(2 (215) = 430\).
time = 0.40, size = 575, normalized size = 1.80 \begin {gather*} \frac {36 \, \sqrt {2} {\left (a^{2} c x^{5} + a^{3} x\right )} \left (-\frac {c}{a^{17}}\right )^{\frac {1}{8}} \arctan \left (-\frac {4782969 \, \sqrt {2} a^{2} c \sqrt {x} \left (-\frac {c}{a^{17}}\right )^{\frac {1}{8}} - \sqrt {2} \sqrt {-22876792454961 \, \sqrt {2} a^{15} c \sqrt {x} \left (-\frac {c}{a^{17}}\right )^{\frac {7}{8}} - 22876792454961 \, a^{13} c \left (-\frac {c}{a^{17}}\right )^{\frac {3}{4}} + 22876792454961 \, c^{2} x} a^{2} \left (-\frac {c}{a^{17}}\right )^{\frac {1}{8}} + 4782969 \, c}{4782969 \, c}\right ) + 36 \, \sqrt {2} {\left (a^{2} c x^{5} + a^{3} x\right )} \left (-\frac {c}{a^{17}}\right )^{\frac {1}{8}} \arctan \left (-\frac {\sqrt {2} a^{2} c \sqrt {x} \left (-\frac {c}{a^{17}}\right )^{\frac {1}{8}} - \sqrt {2} \sqrt {\sqrt {2} a^{15} c \sqrt {x} \left (-\frac {c}{a^{17}}\right )^{\frac {7}{8}} - a^{13} c \left (-\frac {c}{a^{17}}\right )^{\frac {3}{4}} + c^{2} x} a^{2} \left (-\frac {c}{a^{17}}\right )^{\frac {1}{8}} - c}{c}\right ) - 9 \, \sqrt {2} {\left (a^{2} c x^{5} + a^{3} x\right )} \left (-\frac {c}{a^{17}}\right )^{\frac {1}{8}} \log \left (22876792454961 \, \sqrt {2} a^{15} c \sqrt {x} \left (-\frac {c}{a^{17}}\right )^{\frac {7}{8}} - 22876792454961 \, a^{13} c \left (-\frac {c}{a^{17}}\right )^{\frac {3}{4}} + 22876792454961 \, c^{2} x\right ) + 9 \, \sqrt {2} {\left (a^{2} c x^{5} + a^{3} x\right )} \left (-\frac {c}{a^{17}}\right )^{\frac {1}{8}} \log \left (-22876792454961 \, \sqrt {2} a^{15} c \sqrt {x} \left (-\frac {c}{a^{17}}\right )^{\frac {7}{8}} - 22876792454961 \, a^{13} c \left (-\frac {c}{a^{17}}\right )^{\frac {3}{4}} + 22876792454961 \, c^{2} x\right ) + 72 \, {\left (a^{2} c x^{5} + a^{3} x\right )} \left (-\frac {c}{a^{17}}\right )^{\frac {1}{8}} \arctan \left (-\frac {4782969 \, a^{2} c \sqrt {x} \left (-\frac {c}{a^{17}}\right )^{\frac {1}{8}} - \sqrt {-22876792454961 \, a^{13} c \left (-\frac {c}{a^{17}}\right )^{\frac {3}{4}} + 22876792454961 \, c^{2} x} a^{2} \left (-\frac {c}{a^{17}}\right )^{\frac {1}{8}}}{4782969 \, c}\right ) - 18 \, {\left (a^{2} c x^{5} + a^{3} x\right )} \left (-\frac {c}{a^{17}}\right )^{\frac {1}{8}} \log \left (4782969 \, a^{15} \left (-\frac {c}{a^{17}}\right )^{\frac {7}{8}} + 4782969 \, c \sqrt {x}\right ) + 18 \, {\left (a^{2} c x^{5} + a^{3} x\right )} \left (-\frac {c}{a^{17}}\right )^{\frac {1}{8}} \log \left (-4782969 \, a^{15} \left (-\frac {c}{a^{17}}\right )^{\frac {7}{8}} + 4782969 \, c \sqrt {x}\right ) - 16 \, {\left (9 \, c x^{4} + 8 \, a\right )} \sqrt {x}}{64 \, {\left (a^{2} c x^{5} + a^{3} x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^(3/2)/(c*x^4+a)^2,x, algorithm="fricas")

[Out]

1/64*(36*sqrt(2)*(a^2*c*x^5 + a^3*x)*(-c/a^17)^(1/8)*arctan(-1/4782969*(4782969*sqrt(2)*a^2*c*sqrt(x)*(-c/a^17
)^(1/8) - sqrt(2)*sqrt(-22876792454961*sqrt(2)*a^15*c*sqrt(x)*(-c/a^17)^(7/8) - 22876792454961*a^13*c*(-c/a^17
)^(3/4) + 22876792454961*c^2*x)*a^2*(-c/a^17)^(1/8) + 4782969*c)/c) + 36*sqrt(2)*(a^2*c*x^5 + a^3*x)*(-c/a^17)
^(1/8)*arctan(-(sqrt(2)*a^2*c*sqrt(x)*(-c/a^17)^(1/8) - sqrt(2)*sqrt(sqrt(2)*a^15*c*sqrt(x)*(-c/a^17)^(7/8) -
a^13*c*(-c/a^17)^(3/4) + c^2*x)*a^2*(-c/a^17)^(1/8) - c)/c) - 9*sqrt(2)*(a^2*c*x^5 + a^3*x)*(-c/a^17)^(1/8)*lo
g(22876792454961*sqrt(2)*a^15*c*sqrt(x)*(-c/a^17)^(7/8) - 22876792454961*a^13*c*(-c/a^17)^(3/4) + 228767924549
61*c^2*x) + 9*sqrt(2)*(a^2*c*x^5 + a^3*x)*(-c/a^17)^(1/8)*log(-22876792454961*sqrt(2)*a^15*c*sqrt(x)*(-c/a^17)
^(7/8) - 22876792454961*a^13*c*(-c/a^17)^(3/4) + 22876792454961*c^2*x) + 72*(a^2*c*x^5 + a^3*x)*(-c/a^17)^(1/8
)*arctan(-1/4782969*(4782969*a^2*c*sqrt(x)*(-c/a^17)^(1/8) - sqrt(-22876792454961*a^13*c*(-c/a^17)^(3/4) + 228
76792454961*c^2*x)*a^2*(-c/a^17)^(1/8))/c) - 18*(a^2*c*x^5 + a^3*x)*(-c/a^17)^(1/8)*log(4782969*a^15*(-c/a^17)
^(7/8) + 4782969*c*sqrt(x)) + 18*(a^2*c*x^5 + a^3*x)*(-c/a^17)^(1/8)*log(-4782969*a^15*(-c/a^17)^(7/8) + 47829
69*c*sqrt(x)) - 16*(9*c*x^4 + 8*a)*sqrt(x))/(a^2*c*x^5 + a^3*x)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**(3/2)/(c*x**4+a)**2,x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 481 vs. \(2 (215) = 430\).
time = 0.83, size = 481, normalized size = 1.50 \begin {gather*} -\frac {9 \, c \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 )}{16 \, a^{3} \sqrt {-2 \, \sqrt {2} + 4}} - \frac {9 \, c \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 )}{16 \, a^{3} \sqrt {-2 \, \sqrt {2} + 4}} - \frac {9 \, c \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 )}{16 \, a^{3} \sqrt {2 \, \sqrt {2} + 4}} - \frac {9 \, c \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 )}{16 \, a^{3} \sqrt {2 \, \sqrt {2} + 4}} + \frac {9 \, c \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 )}{32 \, a^{3} \sqrt {-2 \, \sqrt {2} + 4}} - \frac {9 \, c \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 )}{32 \, a^{3} \sqrt {-2 \, \sqrt {2} + 4}} + \frac {9 \, c \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 )}{32 \, a^{3} \sqrt {2 \, \sqrt {2} + 4}} - \frac {9 \, c \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 )}{32 \, a^{3} \sqrt {2 \, \sqrt {2} + 4}} - \frac {9 \, c x^{4} + 8 \, a}{4 \, {\left (c x^{\frac {9}{2}} + a \sqrt {x}\right )} a^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^(3/2)/(c*x^4+a)^2,x, algorithm="giac")

[Out]

-9/16*c*(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^3*
sqrt(-2*sqrt(2) + 4)) - 9/16*c*(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^3*sqrt(-2*sqrt(2) + 4)) - 9/16*c*(a/c)^(7/8)*arctan((sqrt(sqrt(2) + 2)*(a/c)^(1/8) + 2*s
qrt(x))/(sqrt(-sqrt(2) + 2)*(a/c)^(1/8)))/(a^3*sqrt(2*sqrt(2) + 4)) - 9/16*c*(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^3*sqrt(2*sqrt(2) + 4)) + 9/32*c*(a/c)^(7/8
)*log(sqrt(x)*sqrt(sqrt(2) + 2)*(a/c)^(1/8) + x + (a/c)^(1/4))/(a^3*sqrt(-2*sqrt(2) + 4)) - 9/32*c*(a/c)^(7/8)
*log(-sqrt(x)*sqrt(sqrt(2) + 2)*(a/c)^(1/8) + x + (a/c)^(1/4))/(a^3*sqrt(-2*sqrt(2) + 4)) + 9/32*c*(a/c)^(7/8)
*log(sqrt(x)*sqrt(-sqrt(2) + 2)*(a/c)^(1/8) + x + (a/c)^(1/4))/(a^3*sqrt(2*sqrt(2) + 4)) - 9/32*c*(a/c)^(7/8)*
log(-sqrt(x)*sqrt(-sqrt(2) + 2)*(a/c)^(1/8) + x + (a/c)^(1/4))/(a^3*sqrt(2*sqrt(2) + 4)) - 1/4*(9*c*x^4 + 8*a)
/((c*x^(9/2) + a*sqrt(x))*a^2)

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Mupad [B]
time = 1.14, size = 148, normalized size = 0.46 \begin {gather*} -\frac {\frac {2}{a}+\frac {9\,c\,x^4}{4\,a^2}}{a\,\sqrt {x}+c\,x^{9/2}}-\frac {9\,{\left (-c\right )}^{1/8}\,\mathrm {atan}\left (\frac {{\left (-c\right )}^{1/8}\,\sqrt {x}}{a^{1/8}}\right )}{16\,a^{17/8}}-\frac {{\left (-c\right )}^{1/8}\,\mathrm {atan}\left (\frac {{\left (-c\right )}^{1/8}\,\sqrt {x}\,1{}\mathrm {i}}{a^{1/8}}\right )\,9{}\mathrm {i}}{16\,a^{17/8}}+\frac {\sqrt {2}\,{\left (-c\right )}^{1/8}\,\mathrm {atan}\left (\frac {\sqrt {2}\,{\left (-c\right )}^{1/8}\,\sqrt {x}\,\left (\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right )}{a^{1/8}}\right )\,\left (-\frac {9}{32}+\frac {9}{32}{}\mathrm {i}\right )}{a^{17/8}}+\frac {\sqrt {2}\,{\left (-c\right )}^{1/8}\,\mathrm {atan}\left (\frac {\sqrt {2}\,{\left (-c\right )}^{1/8}\,\sqrt {x}\,\left (\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right )}{a^{1/8}}\right )\,\left (-\frac {9}{32}-\frac {9}{32}{}\mathrm {i}\right )}{a^{17/8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^(3/2)*(a + c*x^4)^2),x)

[Out]

- (2/a + (9*c*x^4)/(4*a^2))/(a*x^(1/2) + c*x^(9/2)) - (9*(-c)^(1/8)*atan(((-c)^(1/8)*x^(1/2))/a^(1/8)))/(16*a^
(17/8)) - ((-c)^(1/8)*atan(((-c)^(1/8)*x^(1/2)*1i)/a^(1/8))*9i)/(16*a^(17/8)) - (2^(1/2)*(-c)^(1/8)*atan((2^(1
/2)*(-c)^(1/8)*x^(1/2)*(1/2 - 1i/2))/a^(1/8))*(9/32 - 9i/32))/a^(17/8) - (2^(1/2)*(-c)^(1/8)*atan((2^(1/2)*(-c
)^(1/8)*x^(1/2)*(1/2 + 1i/2))/a^(1/8))*(9/32 + 9i/32))/a^(17/8)

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