3.338 \(\int \frac{1}{x^{3/2} (b x^2+c x^4)^2} \, dx\)

Optimal. Leaf size=258 \[ -\frac{13 c^2}{2 b^4 \sqrt{x}}-\frac{13 c^{9/4} \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{8 \sqrt{2} b^{17/4}}+\frac{13 c^{9/4} \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{8 \sqrt{2} b^{17/4}}+\frac{13 c^{9/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{4 \sqrt{2} b^{17/4}}-\frac{13 c^{9/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{4 \sqrt{2} b^{17/4}}+\frac{13 c}{10 b^3 x^{5/2}}-\frac{13}{18 b^2 x^{9/2}}+\frac{1}{2 b x^{9/2} \left (b+c x^2\right )} \]

[Out]

-13/(18*b^2*x^(9/2)) + (13*c)/(10*b^3*x^(5/2)) - (13*c^2)/(2*b^4*Sqrt[x]) + 1/(2*b*x^(9/2)*(b + c*x^2)) + (13*
c^(9/4)*ArcTan[1 - (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)])/(4*Sqrt[2]*b^(17/4)) - (13*c^(9/4)*ArcTan[1 + (Sqrt[2]*
c^(1/4)*Sqrt[x])/b^(1/4)])/(4*Sqrt[2]*b^(17/4)) - (13*c^(9/4)*Log[Sqrt[b] - Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] +
Sqrt[c]*x])/(8*Sqrt[2]*b^(17/4)) + (13*c^(9/4)*Log[Sqrt[b] + Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(8*
Sqrt[2]*b^(17/4))

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Rubi [A]  time = 0.234889, antiderivative size = 258, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 10, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.526, Rules used = {1584, 290, 325, 329, 297, 1162, 617, 204, 1165, 628} \[ -\frac{13 c^2}{2 b^4 \sqrt{x}}-\frac{13 c^{9/4} \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{8 \sqrt{2} b^{17/4}}+\frac{13 c^{9/4} \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{8 \sqrt{2} b^{17/4}}+\frac{13 c^{9/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{4 \sqrt{2} b^{17/4}}-\frac{13 c^{9/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{4 \sqrt{2} b^{17/4}}+\frac{13 c}{10 b^3 x^{5/2}}-\frac{13}{18 b^2 x^{9/2}}+\frac{1}{2 b x^{9/2} \left (b+c x^2\right )} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^(3/2)*(b*x^2 + c*x^4)^2),x]

[Out]

-13/(18*b^2*x^(9/2)) + (13*c)/(10*b^3*x^(5/2)) - (13*c^2)/(2*b^4*Sqrt[x]) + 1/(2*b*x^(9/2)*(b + c*x^2)) + (13*
c^(9/4)*ArcTan[1 - (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)])/(4*Sqrt[2]*b^(17/4)) - (13*c^(9/4)*ArcTan[1 + (Sqrt[2]*
c^(1/4)*Sqrt[x])/b^(1/4)])/(4*Sqrt[2]*b^(17/4)) - (13*c^(9/4)*Log[Sqrt[b] - Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] +
Sqrt[c]*x])/(8*Sqrt[2]*b^(17/4)) + (13*c^(9/4)*Log[Sqrt[b] + Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(8*
Sqrt[2]*b^(17/4))

Rule 1584

Int[(u_.)*(x_)^(m_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(m + n*p)*(a + b*x^(q -
p))^n, x] /; FreeQ[{a, b, m, p, q}, x] && IntegerQ[n] && PosQ[q - p]

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 325

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 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 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]

Rubi steps

\begin{align*} \int \frac{1}{x^{3/2} \left (b x^2+c x^4\right )^2} \, dx &=\int \frac{1}{x^{11/2} \left (b+c x^2\right )^2} \, dx\\ &=\frac{1}{2 b x^{9/2} \left (b+c x^2\right )}+\frac{13 \int \frac{1}{x^{11/2} \left (b+c x^2\right )} \, dx}{4 b}\\ &=-\frac{13}{18 b^2 x^{9/2}}+\frac{1}{2 b x^{9/2} \left (b+c x^2\right )}-\frac{(13 c) \int \frac{1}{x^{7/2} \left (b+c x^2\right )} \, dx}{4 b^2}\\ &=-\frac{13}{18 b^2 x^{9/2}}+\frac{13 c}{10 b^3 x^{5/2}}+\frac{1}{2 b x^{9/2} \left (b+c x^2\right )}+\frac{\left (13 c^2\right ) \int \frac{1}{x^{3/2} \left (b+c x^2\right )} \, dx}{4 b^3}\\ &=-\frac{13}{18 b^2 x^{9/2}}+\frac{13 c}{10 b^3 x^{5/2}}-\frac{13 c^2}{2 b^4 \sqrt{x}}+\frac{1}{2 b x^{9/2} \left (b+c x^2\right )}-\frac{\left (13 c^3\right ) \int \frac{\sqrt{x}}{b+c x^2} \, dx}{4 b^4}\\ &=-\frac{13}{18 b^2 x^{9/2}}+\frac{13 c}{10 b^3 x^{5/2}}-\frac{13 c^2}{2 b^4 \sqrt{x}}+\frac{1}{2 b x^{9/2} \left (b+c x^2\right )}-\frac{\left (13 c^3\right ) \operatorname{Subst}\left (\int \frac{x^2}{b+c x^4} \, dx,x,\sqrt{x}\right )}{2 b^4}\\ &=-\frac{13}{18 b^2 x^{9/2}}+\frac{13 c}{10 b^3 x^{5/2}}-\frac{13 c^2}{2 b^4 \sqrt{x}}+\frac{1}{2 b x^{9/2} \left (b+c x^2\right )}+\frac{\left (13 c^{5/2}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{b}-\sqrt{c} x^2}{b+c x^4} \, dx,x,\sqrt{x}\right )}{4 b^4}-\frac{\left (13 c^{5/2}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{b}+\sqrt{c} x^2}{b+c x^4} \, dx,x,\sqrt{x}\right )}{4 b^4}\\ &=-\frac{13}{18 b^2 x^{9/2}}+\frac{13 c}{10 b^3 x^{5/2}}-\frac{13 c^2}{2 b^4 \sqrt{x}}+\frac{1}{2 b x^{9/2} \left (b+c x^2\right )}-\frac{\left (13 c^2\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{\sqrt{b}}{\sqrt{c}}-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{c}}+x^2} \, dx,x,\sqrt{x}\right )}{8 b^4}-\frac{\left (13 c^2\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{\sqrt{b}}{\sqrt{c}}+\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{c}}+x^2} \, dx,x,\sqrt{x}\right )}{8 b^4}-\frac{\left (13 c^{9/4}\right ) \operatorname{Subst}\left (\int \frac{\frac{\sqrt{2} \sqrt [4]{b}}{\sqrt [4]{c}}+2 x}{-\frac{\sqrt{b}}{\sqrt{c}}-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{c}}-x^2} \, dx,x,\sqrt{x}\right )}{8 \sqrt{2} b^{17/4}}-\frac{\left (13 c^{9/4}\right ) \operatorname{Subst}\left (\int \frac{\frac{\sqrt{2} \sqrt [4]{b}}{\sqrt [4]{c}}-2 x}{-\frac{\sqrt{b}}{\sqrt{c}}+\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{c}}-x^2} \, dx,x,\sqrt{x}\right )}{8 \sqrt{2} b^{17/4}}\\ &=-\frac{13}{18 b^2 x^{9/2}}+\frac{13 c}{10 b^3 x^{5/2}}-\frac{13 c^2}{2 b^4 \sqrt{x}}+\frac{1}{2 b x^{9/2} \left (b+c x^2\right )}-\frac{13 c^{9/4} \log \left (\sqrt{b}-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{c} x\right )}{8 \sqrt{2} b^{17/4}}+\frac{13 c^{9/4} \log \left (\sqrt{b}+\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{c} x\right )}{8 \sqrt{2} b^{17/4}}-\frac{\left (13 c^{9/4}\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{4 \sqrt{2} b^{17/4}}+\frac{\left (13 c^{9/4}\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{4 \sqrt{2} b^{17/4}}\\ &=-\frac{13}{18 b^2 x^{9/2}}+\frac{13 c}{10 b^3 x^{5/2}}-\frac{13 c^2}{2 b^4 \sqrt{x}}+\frac{1}{2 b x^{9/2} \left (b+c x^2\right )}+\frac{13 c^{9/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{4 \sqrt{2} b^{17/4}}-\frac{13 c^{9/4} \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{4 \sqrt{2} b^{17/4}}-\frac{13 c^{9/4} \log \left (\sqrt{b}-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{c} x\right )}{8 \sqrt{2} b^{17/4}}+\frac{13 c^{9/4} \log \left (\sqrt{b}+\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{c} x\right )}{8 \sqrt{2} b^{17/4}}\\ \end{align*}

Mathematica [C]  time = 0.0067058, size = 29, normalized size = 0.11 \[ -\frac{2 \, _2F_1\left (-\frac{9}{4},2;-\frac{5}{4};-\frac{c x^2}{b}\right )}{9 b^2 x^{9/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Hypergeometric2F1[-9/4, 2, -5/4, -((c*x^2)/b)])/(9*b^2*x^(9/2))

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Maple [A]  time = 0.06, size = 189, normalized size = 0.7 \begin{align*} -{\frac{2}{9\,{b}^{2}}{x}^{-{\frac{9}{2}}}}-6\,{\frac{{c}^{2}}{{b}^{4}\sqrt{x}}}+{\frac{4\,c}{5\,{b}^{3}}{x}^{-{\frac{5}{2}}}}-{\frac{{c}^{3}}{2\,{b}^{4} \left ( c{x}^{2}+b \right ) }{x}^{{\frac{3}{2}}}}-{\frac{13\,{c}^{2}\sqrt{2}}{16\,{b}^{4}}\ln \left ({ \left ( x-\sqrt [4]{{\frac{b}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{b}{c}}} \right ) \left ( x+\sqrt [4]{{\frac{b}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{b}{c}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}-{\frac{13\,{c}^{2}\sqrt{2}}{8\,{b}^{4}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}-{\frac{13\,{c}^{2}\sqrt{2}}{8\,{b}^{4}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^(3/2)/(c*x^4+b*x^2)^2,x)

[Out]

-2/9/b^2/x^(9/2)-6*c^2/b^4/x^(1/2)+4/5*c/b^3/x^(5/2)-1/2*c^3/b^4*x^(3/2)/(c*x^2+b)-13/16*c^2/b^4/(b/c)^(1/4)*2
^(1/2)*ln((x-(b/c)^(1/4)*x^(1/2)*2^(1/2)+(b/c)^(1/2))/(x+(b/c)^(1/4)*x^(1/2)*2^(1/2)+(b/c)^(1/2)))-13/8*c^2/b^
4/(b/c)^(1/4)*2^(1/2)*arctan(2^(1/2)/(b/c)^(1/4)*x^(1/2)+1)-13/8*c^2/b^4/(b/c)^(1/4)*2^(1/2)*arctan(2^(1/2)/(b
/c)^(1/4)*x^(1/2)-1)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.42345, size = 648, normalized size = 2.51 \begin{align*} \frac{2340 \,{\left (b^{4} c x^{7} + b^{5} x^{5}\right )} \left (-\frac{c^{9}}{b^{17}}\right )^{\frac{1}{4}} \arctan \left (-\frac{2197 \, b^{4} c^{7} \sqrt{x} \left (-\frac{c^{9}}{b^{17}}\right )^{\frac{1}{4}} - \sqrt{-4826809 \, b^{9} c^{9} \sqrt{-\frac{c^{9}}{b^{17}}} + 4826809 \, c^{14} x} b^{4} \left (-\frac{c^{9}}{b^{17}}\right )^{\frac{1}{4}}}{2197 \, c^{9}}\right ) - 585 \,{\left (b^{4} c x^{7} + b^{5} x^{5}\right )} \left (-\frac{c^{9}}{b^{17}}\right )^{\frac{1}{4}} \log \left (2197 \, b^{13} \left (-\frac{c^{9}}{b^{17}}\right )^{\frac{3}{4}} + 2197 \, c^{7} \sqrt{x}\right ) + 585 \,{\left (b^{4} c x^{7} + b^{5} x^{5}\right )} \left (-\frac{c^{9}}{b^{17}}\right )^{\frac{1}{4}} \log \left (-2197 \, b^{13} \left (-\frac{c^{9}}{b^{17}}\right )^{\frac{3}{4}} + 2197 \, c^{7} \sqrt{x}\right ) - 4 \,{\left (585 \, c^{3} x^{6} + 468 \, b c^{2} x^{4} - 52 \, b^{2} c x^{2} + 20 \, b^{3}\right )} \sqrt{x}}{360 \,{\left (b^{4} c x^{7} + b^{5} x^{5}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/360*(2340*(b^4*c*x^7 + b^5*x^5)*(-c^9/b^17)^(1/4)*arctan(-1/2197*(2197*b^4*c^7*sqrt(x)*(-c^9/b^17)^(1/4) - s
qrt(-4826809*b^9*c^9*sqrt(-c^9/b^17) + 4826809*c^14*x)*b^4*(-c^9/b^17)^(1/4))/c^9) - 585*(b^4*c*x^7 + b^5*x^5)
*(-c^9/b^17)^(1/4)*log(2197*b^13*(-c^9/b^17)^(3/4) + 2197*c^7*sqrt(x)) + 585*(b^4*c*x^7 + b^5*x^5)*(-c^9/b^17)
^(1/4)*log(-2197*b^13*(-c^9/b^17)^(3/4) + 2197*c^7*sqrt(x)) - 4*(585*c^3*x^6 + 468*b*c^2*x^4 - 52*b^2*c*x^2 +
20*b^3)*sqrt(x))/(b^4*c*x^7 + b^5*x^5)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [A]  time = 1.15583, size = 296, normalized size = 1.15 \begin{align*} -\frac{c^{3} x^{\frac{3}{2}}}{2 \,{\left (c x^{2} + b\right )} b^{4}} - \frac{13 \, \sqrt{2} \left (b c^{3}\right )^{\frac{3}{4}} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{b}{c}\right )^{\frac{1}{4}} + 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{b}{c}\right )^{\frac{1}{4}}}\right )}{8 \, b^{5}} - \frac{13 \, \sqrt{2} \left (b c^{3}\right )^{\frac{3}{4}} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{b}{c}\right )^{\frac{1}{4}} - 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{b}{c}\right )^{\frac{1}{4}}}\right )}{8 \, b^{5}} + \frac{13 \, \sqrt{2} \left (b c^{3}\right )^{\frac{3}{4}} \log \left (\sqrt{2} \sqrt{x} \left (\frac{b}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{b}{c}}\right )}{16 \, b^{5}} - \frac{13 \, \sqrt{2} \left (b c^{3}\right )^{\frac{3}{4}} \log \left (-\sqrt{2} \sqrt{x} \left (\frac{b}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{b}{c}}\right )}{16 \, b^{5}} - \frac{2 \,{\left (135 \, c^{2} x^{4} - 18 \, b c x^{2} + 5 \, b^{2}\right )}}{45 \, b^{4} x^{\frac{9}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*c^3*x^(3/2)/((c*x^2 + b)*b^4) - 13/8*sqrt(2)*(b*c^3)^(3/4)*arctan(1/2*sqrt(2)*(sqrt(2)*(b/c)^(1/4) + 2*sq
rt(x))/(b/c)^(1/4))/b^5 - 13/8*sqrt(2)*(b*c^3)^(3/4)*arctan(-1/2*sqrt(2)*(sqrt(2)*(b/c)^(1/4) - 2*sqrt(x))/(b/
c)^(1/4))/b^5 + 13/16*sqrt(2)*(b*c^3)^(3/4)*log(sqrt(2)*sqrt(x)*(b/c)^(1/4) + x + sqrt(b/c))/b^5 - 13/16*sqrt(
2)*(b*c^3)^(3/4)*log(-sqrt(2)*sqrt(x)*(b/c)^(1/4) + x + sqrt(b/c))/b^5 - 2/45*(135*c^2*x^4 - 18*b*c*x^2 + 5*b^
2)/(b^4*x^(9/2))