3.3.18 \(\int \frac {\sqrt {x}}{(b x^2+c x^4)^2} \, dx\)

Optimal. Leaf size=243 \[ \frac {9 c^{5/4} \log \left (-\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {b}+\sqrt {c} x\right )}{8 \sqrt {2} b^{13/4}}-\frac {9 c^{5/4} \log \left (\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {b}+\sqrt {c} x\right )}{8 \sqrt {2} b^{13/4}}-\frac {9 c^{5/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{4 \sqrt {2} b^{13/4}}+\frac {9 c^{5/4} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}+1\right )}{4 \sqrt {2} b^{13/4}}+\frac {9 c}{2 b^3 \sqrt {x}}-\frac {9}{10 b^2 x^{5/2}}+\frac {1}{2 b x^{5/2} \left (b+c x^2\right )} \]

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Rubi [A]  time = 0.22, antiderivative size = 243, normalized size of antiderivative = 1.00, number of steps used = 14, 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} \begin {gather*} \frac {9 c^{5/4} \log \left (-\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {b}+\sqrt {c} x\right )}{8 \sqrt {2} b^{13/4}}-\frac {9 c^{5/4} \log \left (\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {b}+\sqrt {c} x\right )}{8 \sqrt {2} b^{13/4}}-\frac {9 c^{5/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{4 \sqrt {2} b^{13/4}}+\frac {9 c^{5/4} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}+1\right )}{4 \sqrt {2} b^{13/4}}+\frac {9 c}{2 b^3 \sqrt {x}}-\frac {9}{10 b^2 x^{5/2}}+\frac {1}{2 b x^{5/2} \left (b+c x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[x]/(b*x^2 + c*x^4)^2,x]

[Out]

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

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 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 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 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 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]

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]

Rubi steps

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

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Mathematica [C]  time = 0.01, size = 29, normalized size = 0.12 \begin {gather*} -\frac {2 \, _2F_1\left (-\frac {5}{4},2;-\frac {1}{4};-\frac {c x^2}{b}\right )}{5 b^2 x^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Hypergeometric2F1[-5/4, 2, -1/4, -((c*x^2)/b)])/(5*b^2*x^(5/2))

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IntegrateAlgebraic [A]  time = 0.35, size = 160, normalized size = 0.66 \begin {gather*} -\frac {9 c^{5/4} \tan ^{-1}\left (\frac {\frac {\sqrt [4]{b}}{\sqrt {2} \sqrt [4]{c}}-\frac {\sqrt [4]{c} x}{\sqrt {2} \sqrt [4]{b}}}{\sqrt {x}}\right )}{4 \sqrt {2} b^{13/4}}-\frac {9 c^{5/4} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}}{\sqrt {b}+\sqrt {c} x}\right )}{4 \sqrt {2} b^{13/4}}+\frac {-4 b^2+36 b c x^2+45 c^2 x^4}{10 b^3 x^{5/2} \left (b+c x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[Sqrt[x]/(b*x^2 + c*x^4)^2,x]

[Out]

(-4*b^2 + 36*b*c*x^2 + 45*c^2*x^4)/(10*b^3*x^(5/2)*(b + c*x^2)) - (9*c^(5/4)*ArcTan[(b^(1/4)/(Sqrt[2]*c^(1/4))
- (c^(1/4)*x)/(Sqrt[2]*b^(1/4)))/Sqrt[x]])/(4*Sqrt[2]*b^(13/4)) - (9*c^(5/4)*ArcTanh[(Sqrt[2]*b^(1/4)*c^(1/4)
*Sqrt[x])/(Sqrt[b] + Sqrt[c]*x)])/(4*Sqrt[2]*b^(13/4))

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fricas [A]  time = 1.04, size = 251, normalized size = 1.03 \begin {gather*} -\frac {180 \, {\left (b^{3} c x^{5} + b^{4} x^{3}\right )} \left (-\frac {c^{5}}{b^{13}}\right )^{\frac {1}{4}} \arctan \left (-\frac {729 \, b^{3} c^{4} \sqrt {x} \left (-\frac {c^{5}}{b^{13}}\right )^{\frac {1}{4}} - \sqrt {-531441 \, b^{7} c^{5} \sqrt {-\frac {c^{5}}{b^{13}}} + 531441 \, c^{8} x} b^{3} \left (-\frac {c^{5}}{b^{13}}\right )^{\frac {1}{4}}}{729 \, c^{5}}\right ) - 45 \, {\left (b^{3} c x^{5} + b^{4} x^{3}\right )} \left (-\frac {c^{5}}{b^{13}}\right )^{\frac {1}{4}} \log \left (729 \, b^{10} \left (-\frac {c^{5}}{b^{13}}\right )^{\frac {3}{4}} + 729 \, c^{4} \sqrt {x}\right ) + 45 \, {\left (b^{3} c x^{5} + b^{4} x^{3}\right )} \left (-\frac {c^{5}}{b^{13}}\right )^{\frac {1}{4}} \log \left (-729 \, b^{10} \left (-\frac {c^{5}}{b^{13}}\right )^{\frac {3}{4}} + 729 \, c^{4} \sqrt {x}\right ) - 4 \, {\left (45 \, c^{2} x^{4} + 36 \, b c x^{2} - 4 \, b^{2}\right )} \sqrt {x}}{40 \, {\left (b^{3} c x^{5} + b^{4} x^{3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/40*(180*(b^3*c*x^5 + b^4*x^3)*(-c^5/b^13)^(1/4)*arctan(-1/729*(729*b^3*c^4*sqrt(x)*(-c^5/b^13)^(1/4) - sqrt
(-531441*b^7*c^5*sqrt(-c^5/b^13) + 531441*c^8*x)*b^3*(-c^5/b^13)^(1/4))/c^5) - 45*(b^3*c*x^5 + b^4*x^3)*(-c^5/
b^13)^(1/4)*log(729*b^10*(-c^5/b^13)^(3/4) + 729*c^4*sqrt(x)) + 45*(b^3*c*x^5 + b^4*x^3)*(-c^5/b^13)^(1/4)*log
(-729*b^10*(-c^5/b^13)^(3/4) + 729*c^4*sqrt(x)) - 4*(45*c^2*x^4 + 36*b*c*x^2 - 4*b^2)*sqrt(x))/(b^3*c*x^5 + b^
4*x^3)

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giac [A]  time = 0.19, size = 220, normalized size = 0.91 \begin {gather*} \frac {c^{2} x^{\frac {3}{2}}}{2 \, {\left (c x^{2} + b\right )} b^{3}} + \frac {9 \, \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^{4} c} + \frac {9 \, \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^{4} c} - \frac {9 \, \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^{4} c} + \frac {9 \, \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^{4} c} + \frac {2 \, {\left (10 \, c x^{2} - b\right )}}{5 \, b^{3} x^{\frac {5}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*c^2*x^(3/2)/((c*x^2 + b)*b^3) + 9/8*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^4*c) + 9/8*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^4*c) - 9/16*sqrt(2)*(b*c^3)^(3/4)*log(sqrt(2)*sqrt(x)*(b/c)^(1/4) + x + sqrt(b/c))/(b^4*c) + 9/1
6*sqrt(2)*(b*c^3)^(3/4)*log(-sqrt(2)*sqrt(x)*(b/c)^(1/4) + x + sqrt(b/c))/(b^4*c) + 2/5*(10*c*x^2 - b)/(b^3*x^
(5/2))

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maple [A]  time = 0.02, size = 172, normalized size = 0.71 \begin {gather*} \frac {c^{2} x^{\frac {3}{2}}}{2 \left (c \,x^{2}+b \right ) b^{3}}+\frac {9 \sqrt {2}\, c \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}-1\right )}{8 \left (\frac {b}{c}\right )^{\frac {1}{4}} b^{3}}+\frac {9 \sqrt {2}\, c \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}+1\right )}{8 \left (\frac {b}{c}\right )^{\frac {1}{4}} b^{3}}+\frac {9 \sqrt {2}\, c \ln \left (\frac {x -\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {b}{c}}}{x +\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {b}{c}}}\right )}{16 \left (\frac {b}{c}\right )^{\frac {1}{4}} b^{3}}+\frac {4 c}{b^{3} \sqrt {x}}-\frac {2}{5 b^{2} x^{\frac {5}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 2.97, size = 221, normalized size = 0.91 \begin {gather*} \frac {45 \, c^{2} x^{4} + 36 \, b c x^{2} - 4 \, b^{2}}{10 \, {\left (b^{3} c x^{\frac {9}{2}} + b^{4} x^{\frac {5}{2}}\right )}} + \frac {9 \, c^{2} {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} b^{\frac {1}{4}} c^{\frac {1}{4}} + 2 \, \sqrt {c} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {b} \sqrt {c}}}\right )}{\sqrt {\sqrt {b} \sqrt {c}} \sqrt {c}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} b^{\frac {1}{4}} c^{\frac {1}{4}} - 2 \, \sqrt {c} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {b} \sqrt {c}}}\right )}{\sqrt {\sqrt {b} \sqrt {c}} \sqrt {c}} - \frac {\sqrt {2} \log \left (\sqrt {2} b^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {x} + \sqrt {c} x + \sqrt {b}\right )}{b^{\frac {1}{4}} c^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (-\sqrt {2} b^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {x} + \sqrt {c} x + \sqrt {b}\right )}{b^{\frac {1}{4}} c^{\frac {3}{4}}}\right )}}{16 \, b^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*(45*c^2*x^4 + 36*b*c*x^2 - 4*b^2)/(b^3*c*x^(9/2) + b^4*x^(5/2)) + 9/16*c^2*(2*sqrt(2)*arctan(1/2*sqrt(2)*
(sqrt(2)*b^(1/4)*c^(1/4) + 2*sqrt(c)*sqrt(x))/sqrt(sqrt(b)*sqrt(c)))/(sqrt(sqrt(b)*sqrt(c))*sqrt(c)) + 2*sqrt(
2)*arctan(-1/2*sqrt(2)*(sqrt(2)*b^(1/4)*c^(1/4) - 2*sqrt(c)*sqrt(x))/sqrt(sqrt(b)*sqrt(c)))/(sqrt(sqrt(b)*sqrt
(c))*sqrt(c)) - sqrt(2)*log(sqrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x + sqrt(b))/(b^(1/4)*c^(3/4)) + sqrt(2)
*log(-sqrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x + sqrt(b))/(b^(1/4)*c^(3/4)))/b^3

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mupad [B]  time = 4.37, size = 87, normalized size = 0.36 \begin {gather*} \frac {\frac {18\,c\,x^2}{5\,b^2}-\frac {2}{5\,b}+\frac {9\,c^2\,x^4}{2\,b^3}}{b\,x^{5/2}+c\,x^{9/2}}-\frac {9\,{\left (-c\right )}^{5/4}\,\mathrm {atan}\left (\frac {{\left (-c\right )}^{1/4}\,\sqrt {x}}{b^{1/4}}\right )}{4\,b^{13/4}}+\frac {9\,{\left (-c\right )}^{5/4}\,\mathrm {atanh}\left (\frac {{\left (-c\right )}^{1/4}\,\sqrt {x}}{b^{1/4}}\right )}{4\,b^{13/4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(1/2)/(b*x^2 + c*x^4)^2,x)

[Out]

((18*c*x^2)/(5*b^2) - 2/(5*b) + (9*c^2*x^4)/(2*b^3))/(b*x^(5/2) + c*x^(9/2)) - (9*(-c)^(5/4)*atan(((-c)^(1/4)*
x^(1/2))/b^(1/4)))/(4*b^(13/4)) + (9*(-c)^(5/4)*atanh(((-c)^(1/4)*x^(1/2))/b^(1/4)))/(4*b^(13/4))

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sympy [F(-1)]  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(x**(1/2)/(c*x**4+b*x**2)**2,x)

[Out]

Timed out

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