3.4.78 \(\int \frac {x \sqrt {1-x^2}}{a+b x^2+c x^4} \, dx\) [378]

Optimal. Leaf size=182 \[ -\frac {\sqrt {b+2 c-\sqrt {b^2-4 a c}} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {1-x^2}}{\sqrt {b+2 c-\sqrt {b^2-4 a c}}}\right )}{\sqrt {2} \sqrt {c} \sqrt {b^2-4 a c}}+\frac {\sqrt {b+2 c+\sqrt {b^2-4 a c}} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {1-x^2}}{\sqrt {b+2 c+\sqrt {b^2-4 a c}}}\right )}{\sqrt {2} \sqrt {c} \sqrt {b^2-4 a c}} \]

[Out]

-1/2*arctanh(2^(1/2)*c^(1/2)*(-x^2+1)^(1/2)/(b+2*c-(-4*a*c+b^2)^(1/2))^(1/2))*(b+2*c-(-4*a*c+b^2)^(1/2))^(1/2)
*2^(1/2)/c^(1/2)/(-4*a*c+b^2)^(1/2)+1/2*arctanh(2^(1/2)*c^(1/2)*(-x^2+1)^(1/2)/(b+2*c+(-4*a*c+b^2)^(1/2))^(1/2
))*(b+2*c+(-4*a*c+b^2)^(1/2))^(1/2)*2^(1/2)/c^(1/2)/(-4*a*c+b^2)^(1/2)

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Rubi [A]
time = 0.18, antiderivative size = 182, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {1261, 713, 1144, 214} \begin {gather*} \frac {\sqrt {\sqrt {b^2-4 a c}+b+2 c} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {1-x^2}}{\sqrt {\sqrt {b^2-4 a c}+b+2 c}}\right )}{\sqrt {2} \sqrt {c} \sqrt {b^2-4 a c}}-\frac {\sqrt {-\sqrt {b^2-4 a c}+b+2 c} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {1-x^2}}{\sqrt {-\sqrt {b^2-4 a c}+b+2 c}}\right )}{\sqrt {2} \sqrt {c} \sqrt {b^2-4 a c}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x*Sqrt[1 - x^2])/(a + b*x^2 + c*x^4),x]

[Out]

-((Sqrt[b + 2*c - Sqrt[b^2 - 4*a*c]]*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[1 - x^2])/Sqrt[b + 2*c - Sqrt[b^2 - 4*a*c]]
])/(Sqrt[2]*Sqrt[c]*Sqrt[b^2 - 4*a*c])) + (Sqrt[b + 2*c + Sqrt[b^2 - 4*a*c]]*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[1 -
x^2])/Sqrt[b + 2*c + Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*Sqrt[c]*Sqrt[b^2 - 4*a*c])

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 713

Int[Sqrt[(d_.) + (e_.)*(x_)]/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[2*e, Subst[Int[x^2/(c*d^2
- b*d*e + a*e^2 - (2*c*d - b*e)*x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^
2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0]

Rule 1144

Int[((d_.)*(x_))^(m_)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[(
d^2/2)*(b/q + 1), Int[(d*x)^(m - 2)/(b/2 + q/2 + c*x^2), x], x] - Dist[(d^2/2)*(b/q - 1), Int[(d*x)^(m - 2)/(b
/2 - q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d}, x] && NeQ[b^2 - 4*a*c, 0] && GeQ[m, 2]

Rule 1261

Int[(x_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2, Subst[
Int[(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x]

Rubi steps

\begin {align*} \int \frac {x \sqrt {1-x^2}}{a+b x^2+c x^4} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {\sqrt {1-x}}{a+b x+c x^2} \, dx,x,x^2\right )\\ &=-\text {Subst}\left (\int \frac {x^2}{a+b+c+(-b-2 c) x^2+c x^4} \, dx,x,\sqrt {1-x^2}\right )\\ &=\frac {1}{2} \left (-1-\frac {b+2 c}{\sqrt {b^2-4 a c}}\right ) \text {Subst}\left (\int \frac {1}{\frac {1}{2} (-b-2 c)-\frac {1}{2} \sqrt {b^2-4 a c}+c x^2} \, dx,x,\sqrt {1-x^2}\right )-\frac {1}{2} \left (1-\frac {b+2 c}{\sqrt {b^2-4 a c}}\right ) \text {Subst}\left (\int \frac {1}{\frac {1}{2} (-b-2 c)+\frac {1}{2} \sqrt {b^2-4 a c}+c x^2} \, dx,x,\sqrt {1-x^2}\right )\\ &=-\frac {\sqrt {b+2 c-\sqrt {b^2-4 a c}} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {1-x^2}}{\sqrt {b+2 c-\sqrt {b^2-4 a c}}}\right )}{\sqrt {2} \sqrt {c} \sqrt {b^2-4 a c}}+\frac {\sqrt {b+2 c+\sqrt {b^2-4 a c}} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {1-x^2}}{\sqrt {b+2 c+\sqrt {b^2-4 a c}}}\right )}{\sqrt {2} \sqrt {c} \sqrt {b^2-4 a c}}\\ \end {align*}

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Mathematica [A]
time = 0.24, size = 169, normalized size = 0.93 \begin {gather*} \frac {\sqrt {-b-2 c-\sqrt {b^2-4 a c}} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {1-x^2}}{\sqrt {-b-2 c-\sqrt {b^2-4 a c}}}\right )-\sqrt {-b-2 c+\sqrt {b^2-4 a c}} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {1-x^2}}{\sqrt {-b-2 c+\sqrt {b^2-4 a c}}}\right )}{\sqrt {2} \sqrt {c} \sqrt {b^2-4 a c}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[-b - 2*c - Sqrt[b^2 - 4*a*c]]*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[1 - x^2])/Sqrt[-b - 2*c - Sqrt[b^2 - 4*a*c]]]
- Sqrt[-b - 2*c + Sqrt[b^2 - 4*a*c]]*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[1 - x^2])/Sqrt[-b - 2*c + Sqrt[b^2 - 4*a*c]
]])/(Sqrt[2]*Sqrt[c]*Sqrt[b^2 - 4*a*c])

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(359\) vs. \(2(143)=286\).
time = 0.13, size = 360, normalized size = 1.98

method	result	size
default	\(2 a \left (\frac {\left (-2 \sqrt {-4 a c +b^{2}}\, a -b \sqrt {-4 a c +b^{2}}+4 a c -b^{2}\right ) \arctan \left (\frac {\frac {2 a \left (\sqrt {-x^{2}+1}-1\right )^{2}}{x^{2}}+2 \sqrt {-4 a c +b^{2}}+2 a +2 b}{2 \sqrt {4 a c -2 b^{2}-2 \sqrt {-4 a c +b^{2}}\, a -2 b \sqrt {-4 a c +b^{2}}-2 a b}}\right )}{2 a \left (4 a c -b^{2}\right ) \sqrt {4 a c -2 b^{2}-2 \sqrt {-4 a c +b^{2}}\, a -2 b \sqrt {-4 a c +b^{2}}-2 a b}}-\frac {\left (2 \sqrt {-4 a c +b^{2}}\, a +b \sqrt {-4 a c +b^{2}}+4 a c -b^{2}\right ) \arctan \left (\frac {-\frac {2 a \left (\sqrt {-x^{2}+1}-1\right )^{2}}{x^{2}}+2 \sqrt {-4 a c +b^{2}}-2 a -2 b}{2 \sqrt {4 a c -2 b^{2}+2 \sqrt {-4 a c +b^{2}}\, a +2 b \sqrt {-4 a c +b^{2}}-2 a b}}\right )}{2 a \left (4 a c -b^{2}\right ) \sqrt {4 a c -2 b^{2}+2 \sqrt {-4 a c +b^{2}}\, a +2 b \sqrt {-4 a c +b^{2}}-2 a b}}\right )\)	\(360\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 871 vs. \(2 (143) = 286\).
time = 0.81, size = 871, normalized size = 4.79 \begin {gather*} -\frac {1}{2} \, \sqrt {\frac {1}{2}} \sqrt {\frac {b + 2 \, c - \frac {b^{2} c - 4 \, a c^{2}}{\sqrt {b^{2} c^{2} - 4 \, a c^{3}}}}{b^{2} c - 4 \, a c^{2}}} \log \left (\frac {b x^{2} + \frac {{\left (b^{2} c - 4 \, a c^{2}\right )} x^{2}}{\sqrt {b^{2} c^{2} - 4 \, a c^{3}}} + \sqrt {\frac {1}{2}} {\left ({\left (b^{2} - 4 \, a c\right )} x^{2} + \frac {{\left (b^{3} c - 4 \, a b c^{2}\right )} x^{2}}{\sqrt {b^{2} c^{2} - 4 \, a c^{3}}}\right )} \sqrt {\frac {b + 2 \, c - \frac {b^{2} c - 4 \, a c^{2}}{\sqrt {b^{2} c^{2} - 4 \, a c^{3}}}}{b^{2} c - 4 \, a c^{2}}} - 2 \, \sqrt {-x^{2} + 1} a + 2 \, a}{x^{2}}\right ) + \frac {1}{2} \, \sqrt {\frac {1}{2}} \sqrt {\frac {b + 2 \, c - \frac {b^{2} c - 4 \, a c^{2}}{\sqrt {b^{2} c^{2} - 4 \, a c^{3}}}}{b^{2} c - 4 \, a c^{2}}} \log \left (\frac {b x^{2} + \frac {{\left (b^{2} c - 4 \, a c^{2}\right )} x^{2}}{\sqrt {b^{2} c^{2} - 4 \, a c^{3}}} - \sqrt {\frac {1}{2}} {\left ({\left (b^{2} - 4 \, a c\right )} x^{2} + \frac {{\left (b^{3} c - 4 \, a b c^{2}\right )} x^{2}}{\sqrt {b^{2} c^{2} - 4 \, a c^{3}}}\right )} \sqrt {\frac {b + 2 \, c - \frac {b^{2} c - 4 \, a c^{2}}{\sqrt {b^{2} c^{2} - 4 \, a c^{3}}}}{b^{2} c - 4 \, a c^{2}}} - 2 \, \sqrt {-x^{2} + 1} a + 2 \, a}{x^{2}}\right ) - \frac {1}{2} \, \sqrt {\frac {1}{2}} \sqrt {\frac {b + 2 \, c + \frac {b^{2} c - 4 \, a c^{2}}{\sqrt {b^{2} c^{2} - 4 \, a c^{3}}}}{b^{2} c - 4 \, a c^{2}}} \log \left (\frac {b x^{2} - \frac {{\left (b^{2} c - 4 \, a c^{2}\right )} x^{2}}{\sqrt {b^{2} c^{2} - 4 \, a c^{3}}} + \sqrt {\frac {1}{2}} {\left ({\left (b^{2} - 4 \, a c\right )} x^{2} - \frac {{\left (b^{3} c - 4 \, a b c^{2}\right )} x^{2}}{\sqrt {b^{2} c^{2} - 4 \, a c^{3}}}\right )} \sqrt {\frac {b + 2 \, c + \frac {b^{2} c - 4 \, a c^{2}}{\sqrt {b^{2} c^{2} - 4 \, a c^{3}}}}{b^{2} c - 4 \, a c^{2}}} - 2 \, \sqrt {-x^{2} + 1} a + 2 \, a}{x^{2}}\right ) + \frac {1}{2} \, \sqrt {\frac {1}{2}} \sqrt {\frac {b + 2 \, c + \frac {b^{2} c - 4 \, a c^{2}}{\sqrt {b^{2} c^{2} - 4 \, a c^{3}}}}{b^{2} c - 4 \, a c^{2}}} \log \left (\frac {b x^{2} - \frac {{\left (b^{2} c - 4 \, a c^{2}\right )} x^{2}}{\sqrt {b^{2} c^{2} - 4 \, a c^{3}}} - \sqrt {\frac {1}{2}} {\left ({\left (b^{2} - 4 \, a c\right )} x^{2} - \frac {{\left (b^{3} c - 4 \, a b c^{2}\right )} x^{2}}{\sqrt {b^{2} c^{2} - 4 \, a c^{3}}}\right )} \sqrt {\frac {b + 2 \, c + \frac {b^{2} c - 4 \, a c^{2}}{\sqrt {b^{2} c^{2} - 4 \, a c^{3}}}}{b^{2} c - 4 \, a c^{2}}} - 2 \, \sqrt {-x^{2} + 1} a + 2 \, a}{x^{2}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*sqrt(1/2)*sqrt((b + 2*c - (b^2*c - 4*a*c^2)/sqrt(b^2*c^2 - 4*a*c^3))/(b^2*c - 4*a*c^2))*log((b*x^2 + (b^2
*c - 4*a*c^2)*x^2/sqrt(b^2*c^2 - 4*a*c^3) + sqrt(1/2)*((b^2 - 4*a*c)*x^2 + (b^3*c - 4*a*b*c^2)*x^2/sqrt(b^2*c^
2 - 4*a*c^3))*sqrt((b + 2*c - (b^2*c - 4*a*c^2)/sqrt(b^2*c^2 - 4*a*c^3))/(b^2*c - 4*a*c^2)) - 2*sqrt(-x^2 + 1)
*a + 2*a)/x^2) + 1/2*sqrt(1/2)*sqrt((b + 2*c - (b^2*c - 4*a*c^2)/sqrt(b^2*c^2 - 4*a*c^3))/(b^2*c - 4*a*c^2))*l
og((b*x^2 + (b^2*c - 4*a*c^2)*x^2/sqrt(b^2*c^2 - 4*a*c^3) - sqrt(1/2)*((b^2 - 4*a*c)*x^2 + (b^3*c - 4*a*b*c^2)
*x^2/sqrt(b^2*c^2 - 4*a*c^3))*sqrt((b + 2*c - (b^2*c - 4*a*c^2)/sqrt(b^2*c^2 - 4*a*c^3))/(b^2*c - 4*a*c^2)) -
2*sqrt(-x^2 + 1)*a + 2*a)/x^2) - 1/2*sqrt(1/2)*sqrt((b + 2*c + (b^2*c - 4*a*c^2)/sqrt(b^2*c^2 - 4*a*c^3))/(b^2
*c - 4*a*c^2))*log((b*x^2 - (b^2*c - 4*a*c^2)*x^2/sqrt(b^2*c^2 - 4*a*c^3) + sqrt(1/2)*((b^2 - 4*a*c)*x^2 - (b^
3*c - 4*a*b*c^2)*x^2/sqrt(b^2*c^2 - 4*a*c^3))*sqrt((b + 2*c + (b^2*c - 4*a*c^2)/sqrt(b^2*c^2 - 4*a*c^3))/(b^2*
c - 4*a*c^2)) - 2*sqrt(-x^2 + 1)*a + 2*a)/x^2) + 1/2*sqrt(1/2)*sqrt((b + 2*c + (b^2*c - 4*a*c^2)/sqrt(b^2*c^2
- 4*a*c^3))/(b^2*c - 4*a*c^2))*log((b*x^2 - (b^2*c - 4*a*c^2)*x^2/sqrt(b^2*c^2 - 4*a*c^3) - sqrt(1/2)*((b^2 -
4*a*c)*x^2 - (b^3*c - 4*a*b*c^2)*x^2/sqrt(b^2*c^2 - 4*a*c^3))*sqrt((b + 2*c + (b^2*c - 4*a*c^2)/sqrt(b^2*c^2 -
4*a*c^3))/(b^2*c - 4*a*c^2)) - 2*sqrt(-x^2 + 1)*a + 2*a)/x^2)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x \sqrt {- \left (x - 1\right ) \left (x + 1\right )}}{a + b x^{2} + c x^{4}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x*sqrt(-(x - 1)*(x + 1))/(a + b*x**2 + c*x**4), x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 591 vs. \(2 (143) = 286\).
time = 5.10, size = 591, normalized size = 3.25 \begin {gather*} -\frac {{\left (2 \, b^{2} c^{2} - 8 \, a c^{3} - \sqrt {2} \sqrt {b^{2} - 4 \, a c} \sqrt {-b c - 2 \, c^{2} - \sqrt {b^{2} - 4 \, a c} c} b^{2} + 4 \, \sqrt {2} \sqrt {b^{2} - 4 \, a c} \sqrt {-b c - 2 \, c^{2} - \sqrt {b^{2} - 4 \, a c} c} a c - 2 \, \sqrt {2} \sqrt {b^{2} - 4 \, a c} \sqrt {-b c - 2 \, c^{2} - \sqrt {b^{2} - 4 \, a c} c} b c - 5 \, \sqrt {2} \sqrt {b^{2} - 4 \, a c} \sqrt {-b c - 2 \, c^{2} - \sqrt {b^{2} - 4 \, a c} c} c^{2} - 2 \, {\left (b^{2} - 4 \, a c\right )} c^{2}\right )} \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} \sqrt {-x^{2} + 1}}{\sqrt {-\frac {b + 2 \, c + \sqrt {{\left (b + 2 \, c\right )}^{2} - 4 \, {\left (a + b + c\right )} c}}{c}}}\right )}{2 \, {\left (b^{4} - 8 \, a b^{2} c + 2 \, b^{3} c + 16 \, a^{2} c^{2} - 8 \, a b c^{2} + 5 \, b^{2} c^{2} - 20 \, a c^{3}\right )} {\left | c \right |}} + \frac {{\left (2 \, b^{2} c^{2} - 8 \, a c^{3} - \sqrt {2} \sqrt {b^{2} - 4 \, a c} \sqrt {-b c - 2 \, c^{2} + \sqrt {b^{2} - 4 \, a c} c} b^{2} + 4 \, \sqrt {2} \sqrt {b^{2} - 4 \, a c} \sqrt {-b c - 2 \, c^{2} + \sqrt {b^{2} - 4 \, a c} c} a c - 2 \, \sqrt {2} \sqrt {b^{2} - 4 \, a c} \sqrt {-b c - 2 \, c^{2} + \sqrt {b^{2} - 4 \, a c} c} b c - 5 \, \sqrt {2} \sqrt {b^{2} - 4 \, a c} \sqrt {-b c - 2 \, c^{2} + \sqrt {b^{2} - 4 \, a c} c} c^{2} - 2 \, {\left (b^{2} - 4 \, a c\right )} c^{2}\right )} \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} \sqrt {-x^{2} + 1}}{\sqrt {-\frac {b + 2 \, c - \sqrt {{\left (b + 2 \, c\right )}^{2} - 4 \, {\left (a + b + c\right )} c}}{c}}}\right )}{2 \, {\left (b^{4} - 8 \, a b^{2} c + 2 \, b^{3} c + 16 \, a^{2} c^{2} - 8 \, a b c^{2} + 5 \, b^{2} c^{2} - 20 \, a c^{3}\right )} {\left | c \right |}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(2*b^2*c^2 - 8*a*c^3 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(-b*c - 2*c^2 - sqrt(b^2 - 4*a*c)*c)*b^2 + 4*sqrt(2)
*sqrt(b^2 - 4*a*c)*sqrt(-b*c - 2*c^2 - sqrt(b^2 - 4*a*c)*c)*a*c - 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(-b*c - 2*c^
2 - sqrt(b^2 - 4*a*c)*c)*b*c - 5*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(-b*c - 2*c^2 - sqrt(b^2 - 4*a*c)*c)*c^2 - 2*(b
^2 - 4*a*c)*c^2)*arctan(2*sqrt(1/2)*sqrt(-x^2 + 1)/sqrt(-(b + 2*c + sqrt((b + 2*c)^2 - 4*(a + b + c)*c))/c))/(
(b^4 - 8*a*b^2*c + 2*b^3*c + 16*a^2*c^2 - 8*a*b*c^2 + 5*b^2*c^2 - 20*a*c^3)*abs(c)) + 1/2*(2*b^2*c^2 - 8*a*c^3
- sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(-b*c - 2*c^2 + sqrt(b^2 - 4*a*c)*c)*b^2 + 4*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(-
b*c - 2*c^2 + sqrt(b^2 - 4*a*c)*c)*a*c - 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(-b*c - 2*c^2 + sqrt(b^2 - 4*a*c)*c)*
b*c - 5*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(-b*c - 2*c^2 + sqrt(b^2 - 4*a*c)*c)*c^2 - 2*(b^2 - 4*a*c)*c^2)*arctan(2
*sqrt(1/2)*sqrt(-x^2 + 1)/sqrt(-(b + 2*c - sqrt((b + 2*c)^2 - 4*(a + b + c)*c))/c))/((b^4 - 8*a*b^2*c + 2*b^3*
c + 16*a^2*c^2 - 8*a*b*c^2 + 5*b^2*c^2 - 20*a*c^3)*abs(c))

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Mupad [B]
time = 1.29, size = 649, normalized size = 3.57 \begin {gather*} \frac {\ln \left (\frac {\frac {\left (x\,\sqrt {-\frac {b-\sqrt {b^2-4\,a\,c}}{2\,c}}-1\right )\,1{}\mathrm {i}}{\sqrt {\frac {b-\sqrt {b^2-4\,a\,c}}{2\,c}+1}}-\sqrt {1-x^2}\,1{}\mathrm {i}}{x-\sqrt {-\frac {b-\sqrt {b^2-4\,a\,c}}{2\,c}}}\right )\,\left (4\,a\,c+b\,\sqrt {b^2-4\,a\,c}+2\,c\,\sqrt {b^2-4\,a\,c}-b^2\right )}{4\,c\,\sqrt {\frac {b-\sqrt {b^2-4\,a\,c}}{2\,c}+1}\,\left (4\,a\,c-b^2\right )}-\frac {\ln \left (\frac {\frac {\left (x\,\sqrt {-\frac {b+\sqrt {b^2-4\,a\,c}}{2\,c}}-1\right )\,1{}\mathrm {i}}{\sqrt {\frac {b+\sqrt {b^2-4\,a\,c}}{2\,c}+1}}-\sqrt {1-x^2}\,1{}\mathrm {i}}{x-\sqrt {-\frac {b+\sqrt {b^2-4\,a\,c}}{2\,c}}}\right )\,\left (b\,\sqrt {b^2-4\,a\,c}-4\,a\,c+2\,c\,\sqrt {b^2-4\,a\,c}+b^2\right )}{4\,c\,\left (4\,a\,c-b^2\right )\,\sqrt {\frac {b+\sqrt {b^2-4\,a\,c}}{2\,c}+1}}+\frac {\ln \left (\frac {\frac {\left (x\,\sqrt {-\frac {b-\sqrt {b^2-4\,a\,c}}{2\,c}}+1\right )\,1{}\mathrm {i}}{\sqrt {\frac {b-\sqrt {b^2-4\,a\,c}}{2\,c}+1}}+\sqrt {1-x^2}\,1{}\mathrm {i}}{x+\sqrt {-\frac {b-\sqrt {b^2-4\,a\,c}}{2\,c}}}\right )\,\left (4\,a\,c+b\,\sqrt {b^2-4\,a\,c}+2\,c\,\sqrt {b^2-4\,a\,c}-b^2\right )}{4\,c\,\sqrt {\frac {b-\sqrt {b^2-4\,a\,c}}{2\,c}+1}\,\left (4\,a\,c-b^2\right )}-\frac {\ln \left (\frac {\frac {\left (x\,\sqrt {-\frac {b+\sqrt {b^2-4\,a\,c}}{2\,c}}+1\right )\,1{}\mathrm {i}}{\sqrt {\frac {b+\sqrt {b^2-4\,a\,c}}{2\,c}+1}}+\sqrt {1-x^2}\,1{}\mathrm {i}}{x+\sqrt {-\frac {b+\sqrt {b^2-4\,a\,c}}{2\,c}}}\right )\,\left (b\,\sqrt {b^2-4\,a\,c}-4\,a\,c+2\,c\,\sqrt {b^2-4\,a\,c}+b^2\right )}{4\,c\,\left (4\,a\,c-b^2\right )\,\sqrt {\frac {b+\sqrt {b^2-4\,a\,c}}{2\,c}+1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(log((((x*(-(b - (b^2 - 4*a*c)^(1/2))/(2*c))^(1/2) - 1)*1i)/((b - (b^2 - 4*a*c)^(1/2))/(2*c) + 1)^(1/2) - (1 -
x^2)^(1/2)*1i)/(x - (-(b - (b^2 - 4*a*c)^(1/2))/(2*c))^(1/2)))*(4*a*c + b*(b^2 - 4*a*c)^(1/2) + 2*c*(b^2 - 4*
a*c)^(1/2) - b^2))/(4*c*((b - (b^2 - 4*a*c)^(1/2))/(2*c) + 1)^(1/2)*(4*a*c - b^2)) - (log((((x*(-(b + (b^2 - 4
*a*c)^(1/2))/(2*c))^(1/2) - 1)*1i)/((b + (b^2 - 4*a*c)^(1/2))/(2*c) + 1)^(1/2) - (1 - x^2)^(1/2)*1i)/(x - (-(b
+ (b^2 - 4*a*c)^(1/2))/(2*c))^(1/2)))*(b*(b^2 - 4*a*c)^(1/2) - 4*a*c + 2*c*(b^2 - 4*a*c)^(1/2) + b^2))/(4*c*(
4*a*c - b^2)*((b + (b^2 - 4*a*c)^(1/2))/(2*c) + 1)^(1/2)) + (log((((x*(-(b - (b^2 - 4*a*c)^(1/2))/(2*c))^(1/2)
+ 1)*1i)/((b - (b^2 - 4*a*c)^(1/2))/(2*c) + 1)^(1/2) + (1 - x^2)^(1/2)*1i)/(x + (-(b - (b^2 - 4*a*c)^(1/2))/(
2*c))^(1/2)))*(4*a*c + b*(b^2 - 4*a*c)^(1/2) + 2*c*(b^2 - 4*a*c)^(1/2) - b^2))/(4*c*((b - (b^2 - 4*a*c)^(1/2))
/(2*c) + 1)^(1/2)*(4*a*c - b^2)) - (log((((x*(-(b + (b^2 - 4*a*c)^(1/2))/(2*c))^(1/2) + 1)*1i)/((b + (b^2 - 4*
a*c)^(1/2))/(2*c) + 1)^(1/2) + (1 - x^2)^(1/2)*1i)/(x + (-(b + (b^2 - 4*a*c)^(1/2))/(2*c))^(1/2)))*(b*(b^2 - 4
*a*c)^(1/2) - 4*a*c + 2*c*(b^2 - 4*a*c)^(1/2) + b^2))/(4*c*(4*a*c - b^2)*((b + (b^2 - 4*a*c)^(1/2))/(2*c) + 1)
^(1/2))

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