3.4.77 \(\int \frac {x^3 \sqrt {1-x^2}}{a+b x^2+c x^4} \, dx\) [377]

Optimal. Leaf size=229 \[ \frac {\sqrt {1-x^2}}{c}-\frac {\left (b+c-\frac {b^2-2 a c+b c}{\sqrt {b^2-4 a c}}\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {1-x^2}}{\sqrt {b+2 c-\sqrt {b^2-4 a c}}}\right )}{\sqrt {2} c^{3/2} \sqrt {b+2 c-\sqrt {b^2-4 a c}}}-\frac {\left (b+c+\frac {b^2-2 a c+b c}{\sqrt {b^2-4 a c}}\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {1-x^2}}{\sqrt {b+2 c+\sqrt {b^2-4 a c}}}\right )}{\sqrt {2} c^{3/2} \sqrt {b+2 c+\sqrt {b^2-4 a c}}} \]

[Out]

(-x^2+1)^(1/2)/c-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+c+(2*a*c-b^2-
b*c)/(-4*a*c+b^2)^(1/2))/c^(3/2)*2^(1/2)/(b+2*c-(-4*a*c+b^2)^(1/2))^(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+c+(-2*a*c+b^2+b*c)/(-4*a*c+b^2)^(1/2))/c^(3/2)*2^(1/2)/(b+2*c+(-4*
a*c+b^2)^(1/2))^(1/2)

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Rubi [A]
time = 1.20, antiderivative size = 229, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {1265, 838, 840, 1180, 214} \begin {gather*} -\frac {\left (-\frac {-2 a c+b^2+b c}{\sqrt {b^2-4 a c}}+b+c\right ) \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} c^{3/2} \sqrt {-\sqrt {b^2-4 a c}+b+2 c}}-\frac {\left (\frac {-2 a c+b^2+b c}{\sqrt {b^2-4 a c}}+b+c\right ) \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} c^{3/2} \sqrt {\sqrt {b^2-4 a c}+b+2 c}}+\frac {\sqrt {1-x^2}}{c} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x^3*Sqrt[1 - x^2])/(a + b*x^2 + c*x^4),x]

[Out]

Sqrt[1 - x^2]/c - ((b + c - (b^2 - 2*a*c + b*c)/Sqrt[b^2 - 4*a*c])*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[1 - x^2])/Sqr
t[b + 2*c - Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*c^(3/2)*Sqrt[b + 2*c - Sqrt[b^2 - 4*a*c]]) - ((b + c + (b^2 - 2*a*c
+ b*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]
*c^(3/2)*Sqrt[b + 2*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 838

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[g*
((d + e*x)^m/(c*m)), x] + Dist[1/c, Int[(d + e*x)^(m - 1)*(Simp[c*d*f - a*e*g + (g*c*d - b*e*g + c*e*f)*x, x]/
(a + b*x + c*x^2)), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*
e^2, 0] && FractionQ[m] && GtQ[m, 0]

Rule 840

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

Rule 1180

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

Rule 1265

Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2,
Subst[Int[x^((m - 1)/2)*(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] &&
IntegerQ[(m - 1)/2]

Rubi steps

\begin {align*} \int \frac {x^3 \sqrt {1-x^2}}{a+b x^2+c x^4} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {\sqrt {1-x} x}{a+b x+c x^2} \, dx,x,x^2\right )\\ &=\frac {\sqrt {1-x^2}}{c}+\frac {\text {Subst}\left (\int \frac {a+(b+c) x}{\sqrt {1-x} \left (a+b x+c x^2\right )} \, dx,x,x^2\right )}{2 c}\\ &=\frac {\sqrt {1-x^2}}{c}+\frac {\text {Subst}\left (\int \frac {-a-b-c+(b+c) x^2}{a+b+c+(-b-2 c) x^2+c x^4} \, dx,x,\sqrt {1-x^2}\right )}{c}\\ &=\frac {\sqrt {1-x^2}}{c}+\frac {\left (b+c-\frac {b^2-2 a c+b 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 )}{2 c}+\frac {\left (b+c+\frac {b^2-2 a c+b 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 )}{2 c}\\ &=\frac {\sqrt {1-x^2}}{c}-\frac {\left (b+c-\frac {b^2-2 a c+b c}{\sqrt {b^2-4 a c}}\right ) \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} c^{3/2} \sqrt {b+2 c-\sqrt {b^2-4 a c}}}-\frac {\left (b+c+\frac {b^2-2 a c+b c}{\sqrt {b^2-4 a c}}\right ) \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} c^{3/2} \sqrt {b+2 c+\sqrt {b^2-4 a c}}}\\ \end {align*}

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Mathematica [A]
time = 0.58, size = 285, normalized size = 1.24 \begin {gather*} \frac {\sqrt {1-x^2}}{c}+\frac {\left (b^2-2 a c+b c+b \sqrt {b^2-4 a c}+c \sqrt {b^2-4 a c}\right ) \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} c^{3/2} \sqrt {b^2-4 a c} \sqrt {-b-2 c-\sqrt {b^2-4 a c}}}+\frac {\left (-b^2+2 a c-b c+b \sqrt {b^2-4 a c}+c \sqrt {b^2-4 a c}\right ) \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} c^{3/2} \sqrt {b^2-4 a c} \sqrt {-b-2 c+\sqrt {b^2-4 a c}}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Sqrt[1 - x^2]/c + ((b^2 - 2*a*c + b*c + b*Sqrt[b^2 - 4*a*c] + c*Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*Sqr
t[1 - x^2])/Sqrt[-b - 2*c - Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*c^(3/2)*Sqrt[b^2 - 4*a*c]*Sqrt[-b - 2*c - Sqrt[b^2 -
4*a*c]]) + ((-b^2 + 2*a*c - b*c + b*Sqrt[b^2 - 4*a*c] + 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]*c^(3/2)*Sqrt[b^2 - 4*a*c]*Sqrt[-b - 2*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. \(382\) vs. \(2(189)=378\).
time = 0.15, size = 383, normalized size = 1.67

method	result	size
default	\(\frac {2 a \left (-\frac {\left (-b \sqrt {-4 a c +b^{2}}-2 \sqrt {-4 a c +b^{2}}\, c +4 a c -b^{2}\right ) \arctan \left (\frac {-\frac {2 a \left (\sqrt {-x^{2}+1}-1\right )^{2}}{x^{2}}+2 \sqrt {-4 a c +b^{2}}-2 a -2 b}{2 \sqrt {4 a c -2 b^{2}+2 \sqrt {-4 a c +b^{2}}\, a +2 b \sqrt {-4 a c +b^{2}}-2 a b}}\right )}{\left (8 a c -2 b^{2}\right ) \sqrt {4 a c -2 b^{2}+2 \sqrt {-4 a c +b^{2}}\, a +2 b \sqrt {-4 a c +b^{2}}-2 a b}}+\frac {\left (b \sqrt {-4 a c +b^{2}}+2 \sqrt {-4 a c +b^{2}}\, c +4 a c -b^{2}\right ) \arctan \left (\frac {\frac {2 a \left (\sqrt {-x^{2}+1}-1\right )^{2}}{x^{2}}+2 \sqrt {-4 a c +b^{2}}+2 a +2 b}{2 \sqrt {4 a c -2 b^{2}-2 \sqrt {-4 a c +b^{2}}\, a -2 b \sqrt {-4 a c +b^{2}}-2 a b}}\right )}{\left (8 a c -2 b^{2}\right ) \sqrt {4 a c -2 b^{2}-2 \sqrt {-4 a c +b^{2}}\, a -2 b \sqrt {-4 a c +b^{2}}-2 a b}}\right )}{c}+\frac {2}{c \left (\frac {\left (\sqrt {-x^{2}+1}-1\right )^{2}}{x^{2}}+1\right )}\)	\(383\)
risch	\(-\frac {x^{2}-1}{c \sqrt {-x^{2}+1}}+\frac {2 a \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 ) \sqrt {-4 a c +b^{2}}\, b}{c \left (8 a c -2 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 {4 a \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 ) \sqrt {-4 a c +b^{2}}}{\left (8 a c -2 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 {8 a^{2} \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 )}{\left (8 a c -2 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 {2 a \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 ) b^{2}}{c \left (8 a c -2 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 {2 a \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 ) \sqrt {-4 a c +b^{2}}\, b}{c \left (8 a c -2 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 {4 a \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 ) \sqrt {-4 a c +b^{2}}}{\left (8 a c -2 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 {8 a^{2} \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 )}{\left (8 a c -2 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 {2 a \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 ) b^{2}}{c \left (8 a c -2 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}}\)	\(1205\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/c*a*(-(-b*(-4*a*c+b^2)^(1/2)-2*(-4*a*c+b^2)^(1/2)*c+4*a*c-b^2)/(8*a*c-2*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))+(b*(-4*a*c+b^2)^(1/2)+2*(-4*a*c+b^
2)^(1/2)*c+4*a*c-b^2)/(8*a*c-2*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)*ar
ctan(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)))+2/c/(((-x^2+1)^(1/2)-1)^2/x^2+1)

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

[Out]

integrate(sqrt(-x^2 + 1)*x^3/(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. 2053 vs. \(2 (187) = 374\).
time = 1.49, size = 2053, normalized size = 8.97 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(sqrt(1/2)*c*sqrt((b^3 - 2*a*c^2 - (3*a*b - b^2)*c - (b^2*c^3 - 4*a*c^4)*sqrt((b^4 + (a^2 - 2*a*b + b^2)*c
^2 - 2*(a*b^2 - b^3)*c)/(b^2*c^6 - 4*a*c^7)))/(b^2*c^3 - 4*a*c^4))*log((2*a^2*b^2 + (a*b^2*c^3 - 4*a^2*c^4)*x^
2*sqrt((b^4 + (a^2 - 2*a*b + b^2)*c^2 - 2*(a*b^2 - b^3)*c)/(b^2*c^6 - 4*a*c^7)) + (a*b^3 - (a^2*b - a*b^2)*c)*
x^2 - 2*(a^3 - a^2*b)*c + sqrt(1/2)*((b^4*c^3 - 6*a*b^2*c^4 + 8*a^2*c^5)*x^2*sqrt((b^4 + (a^2 - 2*a*b + b^2)*c
^2 - 2*(a*b^2 - b^3)*c)/(b^2*c^6 - 4*a*c^7)) + (b^5 + 4*(a^2*b - a*b^2)*c^2 - (5*a*b^3 - b^4)*c)*x^2)*sqrt((b^
3 - 2*a*c^2 - (3*a*b - b^2)*c - (b^2*c^3 - 4*a*c^4)*sqrt((b^4 + (a^2 - 2*a*b + b^2)*c^2 - 2*(a*b^2 - b^3)*c)/(
b^2*c^6 - 4*a*c^7)))/(b^2*c^3 - 4*a*c^4)) - 2*(a^2*b^2 - (a^3 - a^2*b)*c)*sqrt(-x^2 + 1))/x^2) - sqrt(1/2)*c*s
qrt((b^3 - 2*a*c^2 - (3*a*b - b^2)*c - (b^2*c^3 - 4*a*c^4)*sqrt((b^4 + (a^2 - 2*a*b + b^2)*c^2 - 2*(a*b^2 - b^
3)*c)/(b^2*c^6 - 4*a*c^7)))/(b^2*c^3 - 4*a*c^4))*log((2*a^2*b^2 + (a*b^2*c^3 - 4*a^2*c^4)*x^2*sqrt((b^4 + (a^2
- 2*a*b + b^2)*c^2 - 2*(a*b^2 - b^3)*c)/(b^2*c^6 - 4*a*c^7)) + (a*b^3 - (a^2*b - a*b^2)*c)*x^2 - 2*(a^3 - a^2
*b)*c - sqrt(1/2)*((b^4*c^3 - 6*a*b^2*c^4 + 8*a^2*c^5)*x^2*sqrt((b^4 + (a^2 - 2*a*b + b^2)*c^2 - 2*(a*b^2 - b^
3)*c)/(b^2*c^6 - 4*a*c^7)) + (b^5 + 4*(a^2*b - a*b^2)*c^2 - (5*a*b^3 - b^4)*c)*x^2)*sqrt((b^3 - 2*a*c^2 - (3*a
*b - b^2)*c - (b^2*c^3 - 4*a*c^4)*sqrt((b^4 + (a^2 - 2*a*b + b^2)*c^2 - 2*(a*b^2 - b^3)*c)/(b^2*c^6 - 4*a*c^7)
))/(b^2*c^3 - 4*a*c^4)) - 2*(a^2*b^2 - (a^3 - a^2*b)*c)*sqrt(-x^2 + 1))/x^2) - sqrt(1/2)*c*sqrt((b^3 - 2*a*c^2
- (3*a*b - b^2)*c + (b^2*c^3 - 4*a*c^4)*sqrt((b^4 + (a^2 - 2*a*b + b^2)*c^2 - 2*(a*b^2 - b^3)*c)/(b^2*c^6 - 4
*a*c^7)))/(b^2*c^3 - 4*a*c^4))*log((2*a^2*b^2 - (a*b^2*c^3 - 4*a^2*c^4)*x^2*sqrt((b^4 + (a^2 - 2*a*b + b^2)*c^
2 - 2*(a*b^2 - b^3)*c)/(b^2*c^6 - 4*a*c^7)) + (a*b^3 - (a^2*b - a*b^2)*c)*x^2 - 2*(a^3 - a^2*b)*c + sqrt(1/2)*
((b^4*c^3 - 6*a*b^2*c^4 + 8*a^2*c^5)*x^2*sqrt((b^4 + (a^2 - 2*a*b + b^2)*c^2 - 2*(a*b^2 - b^3)*c)/(b^2*c^6 - 4
*a*c^7)) - (b^5 + 4*(a^2*b - a*b^2)*c^2 - (5*a*b^3 - b^4)*c)*x^2)*sqrt((b^3 - 2*a*c^2 - (3*a*b - b^2)*c + (b^2
*c^3 - 4*a*c^4)*sqrt((b^4 + (a^2 - 2*a*b + b^2)*c^2 - 2*(a*b^2 - b^3)*c)/(b^2*c^6 - 4*a*c^7)))/(b^2*c^3 - 4*a*
c^4)) - 2*(a^2*b^2 - (a^3 - a^2*b)*c)*sqrt(-x^2 + 1))/x^2) + sqrt(1/2)*c*sqrt((b^3 - 2*a*c^2 - (3*a*b - b^2)*c
+ (b^2*c^3 - 4*a*c^4)*sqrt((b^4 + (a^2 - 2*a*b + b^2)*c^2 - 2*(a*b^2 - b^3)*c)/(b^2*c^6 - 4*a*c^7)))/(b^2*c^3
- 4*a*c^4))*log((2*a^2*b^2 - (a*b^2*c^3 - 4*a^2*c^4)*x^2*sqrt((b^4 + (a^2 - 2*a*b + b^2)*c^2 - 2*(a*b^2 - b^3
)*c)/(b^2*c^6 - 4*a*c^7)) + (a*b^3 - (a^2*b - a*b^2)*c)*x^2 - 2*(a^3 - a^2*b)*c - sqrt(1/2)*((b^4*c^3 - 6*a*b^
2*c^4 + 8*a^2*c^5)*x^2*sqrt((b^4 + (a^2 - 2*a*b + b^2)*c^2 - 2*(a*b^2 - b^3)*c)/(b^2*c^6 - 4*a*c^7)) - (b^5 +
4*(a^2*b - a*b^2)*c^2 - (5*a*b^3 - b^4)*c)*x^2)*sqrt((b^3 - 2*a*c^2 - (3*a*b - b^2)*c + (b^2*c^3 - 4*a*c^4)*sq
rt((b^4 + (a^2 - 2*a*b + b^2)*c^2 - 2*(a*b^2 - b^3)*c)/(b^2*c^6 - 4*a*c^7)))/(b^2*c^3 - 4*a*c^4)) - 2*(a^2*b^2
- (a^3 - a^2*b)*c)*sqrt(-x^2 + 1))/x^2) + 2*sqrt(-x^2 + 1))/c

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{3} \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**3*(-x**2+1)**(1/2)/(c*x**4+b*x**2+a),x)

[Out]

Integral(x**3*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. 4060 vs. \(2 (187) = 374\).
time = 6.05, size = 4060, normalized size = 17.73 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(-x^2 + 1)/c + 1/8*(2*b^5*c^4 - 12*a*b^3*c^5 + 6*b^4*c^5 + 16*a^2*b*c^6 - 32*a*b^2*c^6 + 4*b^3*c^6 + 32*a^
2*c^7 - 16*a*b*c^7 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(-b*c - 2*c^2 - sqrt(b^2 - 4*a*c)*c)*b^5*c^2 + 6*sqrt(2)*sq
rt(b^2 - 4*a*c)*sqrt(-b*c - 2*c^2 - sqrt(b^2 - 4*a*c)*c)*a*b^3*c^3 - 5*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(-b*c - 2
*c^2 - sqrt(b^2 - 4*a*c)*c)*b^4*c^3 - 8*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(-b*c - 2*c^2 - sqrt(b^2 - 4*a*c)*c)*a^2
*b*c^4 + 20*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(-b*c - 2*c^2 - sqrt(b^2 - 4*a*c)*c)*a*b^2*c^4 - 13*sqrt(2)*sqrt(b^2
- 4*a*c)*sqrt(-b*c - 2*c^2 - sqrt(b^2 - 4*a*c)*c)*b^3*c^4 - 16*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(-b*c - 2*c^2 -
sqrt(b^2 - 4*a*c)*c)*a^2*c^5 + 26*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(-b*c - 2*c^2 - sqrt(b^2 - 4*a*c)*c)*a*b*c^5 -
19*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(-b*c - 2*c^2 - sqrt(b^2 - 4*a*c)*c)*b^2*c^5 + 20*sqrt(2)*sqrt(b^2 - 4*a*c)*
sqrt(-b*c - 2*c^2 - sqrt(b^2 - 4*a*c)*c)*a*c^6 - 10*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(-b*c - 2*c^2 - sqrt(b^2 - 4
*a*c)*c)*b*c^6 - 2*(b^2 - 4*a*c)*b^3*c^4 + 4*(b^2 - 4*a*c)*a*b*c^5 - 6*(b^2 - 4*a*c)*b^2*c^5 + 8*(b^2 - 4*a*c)
*a*c^6 - 4*(b^2 - 4*a*c)*b*c^6 - (2*b^5*c^2 - 16*a*b^3*c^3 + 2*b^4*c^3 + 32*a^2*b*c^4 - 16*a*b^2*c^4 + 32*a^2*
c^5 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(-b*c - 2*c^2 - sqrt(b^2 - 4*a*c)*c)*b^5 + 8*sqrt(2)*sqrt(b^2 - 4*a*c)*sqr
t(-b*c - 2*c^2 - sqrt(b^2 - 4*a*c)*c)*a*b^3*c - 3*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(-b*c - 2*c^2 - sqrt(b^2 - 4*a
*c)*c)*b^4*c - 16*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(-b*c - 2*c^2 - sqrt(b^2 - 4*a*c)*c)*a^2*b*c^2 + 16*sqrt(2)*sq
rt(b^2 - 4*a*c)*sqrt(-b*c - 2*c^2 - sqrt(b^2 - 4*a*c)*c)*a*b^2*c^2 - 7*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(-b*c - 2
*c^2 - sqrt(b^2 - 4*a*c)*c)*b^3*c^2 - 16*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(-b*c - 2*c^2 - sqrt(b^2 - 4*a*c)*c)*a^
2*c^3 + 28*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(-b*c - 2*c^2 - sqrt(b^2 - 4*a*c)*c)*a*b*c^3 - 5*sqrt(2)*sqrt(b^2 - 4
*a*c)*sqrt(-b*c - 2*c^2 - sqrt(b^2 - 4*a*c)*c)*b^2*c^3 + 20*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(-b*c - 2*c^2 - sqrt
(b^2 - 4*a*c)*c)*a*c^4 - 2*(b^2 - 4*a*c)*b^3*c^2 + 8*(b^2 - 4*a*c)*a*b*c^3 - 2*(b^2 - 4*a*c)*b^2*c^3 + 8*(b^2
- 4*a*c)*a*c^4)*c^2 - 2*(sqrt(2)*sqrt(-b*c - 2*c^2 - sqrt(b^2 - 4*a*c)*c)*a*b^4*c^2 + sqrt(2)*sqrt(-b*c - 2*c^
2 - sqrt(b^2 - 4*a*c)*c)*b^5*c^2 - 8*sqrt(2)*sqrt(-b*c - 2*c^2 - sqrt(b^2 - 4*a*c)*c)*a^2*b^2*c^3 - 6*sqrt(2)*
sqrt(-b*c - 2*c^2 - sqrt(b^2 - 4*a*c)*c)*a*b^3*c^3 + 3*sqrt(2)*sqrt(-b*c - 2*c^2 - sqrt(b^2 - 4*a*c)*c)*b^4*c^
3 + 2*a*b^4*c^3 + 2*b^5*c^3 + 16*sqrt(2)*sqrt(-b*c - 2*c^2 - sqrt(b^2 - 4*a*c)*c)*a^3*c^4 + 8*sqrt(2)*sqrt(-b*
c - 2*c^2 - sqrt(b^2 - 4*a*c)*c)*a^2*b*c^4 - 11*sqrt(2)*sqrt(-b*c - 2*c^2 - sqrt(b^2 - 4*a*c)*c)*a*b^2*c^4 - 1
6*a^2*b^2*c^4 + 7*sqrt(2)*sqrt(-b*c - 2*c^2 - sqrt(b^2 - 4*a*c)*c)*b^3*c^4 - 16*a*b^3*c^4 + 2*b^4*c^4 - 4*sqrt
(2)*sqrt(-b*c - 2*c^2 - sqrt(b^2 - 4*a*c)*c)*a^2*c^5 + 32*a^3*c^5 - 28*sqrt(2)*sqrt(-b*c - 2*c^2 - sqrt(b^2 -
4*a*c)*c)*a*b*c^5 + 32*a^2*b*c^5 + 5*sqrt(2)*sqrt(-b*c - 2*c^2 - sqrt(b^2 - 4*a*c)*c)*b^2*c^5 - 16*a*b^2*c^5 -
20*sqrt(2)*sqrt(-b*c - 2*c^2 - sqrt(b^2 - 4*a*c)*c)*a*c^6 + 32*a^2*c^6 - 2*(b^2 - 4*a*c)*a*b^2*c^3 - 2*(b^2 -
4*a*c)*b^3*c^3 + 8*(b^2 - 4*a*c)*a^2*c^4 + 8*(b^2 - 4*a*c)*a*b*c^4 - 2*(b^2 - 4*a*c)*b^2*c^4 + 8*(b^2 - 4*a*c
)*a*c^5)*abs(c))*arctan(2*sqrt(1/2)*sqrt(-x^2 + 1)/sqrt(-(b*c + 2*c^2 + sqrt(-4*(a*c + b*c + c^2)*c^2 + (b*c +
2*c^2)^2))/c^2))/((a*b^4*c^3 + b^5*c^3 - 8*a^2*b^2*c^4 - 6*a*b^3*c^4 + 3*b^4*c^4 + 16*a^3*c^5 + 8*a^2*b*c^5 -
11*a*b^2*c^5 + 7*b^3*c^5 - 4*a^2*c^6 - 28*a*b*c^6 + 5*b^2*c^6 - 20*a*c^7)*c^2) + 1/8*(2*b^5*c^4 - 12*a*b^3*c^
5 + 6*b^4*c^5 + 16*a^2*b*c^6 - 32*a*b^2*c^6 + 4*b^3*c^6 + 32*a^2*c^7 - 16*a*b*c^7 - sqrt(2)*sqrt(b^2 - 4*a*c)*
sqrt(-b*c - 2*c^2 + sqrt(b^2 - 4*a*c)*c)*b^5*c^2 + 6*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(-b*c - 2*c^2 + sqrt(b^2 -
4*a*c)*c)*a*b^3*c^3 - 5*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(-b*c - 2*c^2 + sqrt(b^2 - 4*a*c)*c)*b^4*c^3 - 8*sqrt(2)
*sqrt(b^2 - 4*a*c)*sqrt(-b*c - 2*c^2 + sqrt(b^2 - 4*a*c)*c)*a^2*b*c^4 + 20*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(-b*c
- 2*c^2 + sqrt(b^2 - 4*a*c)*c)*a*b^2*c^4 - 13*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(-b*c - 2*c^2 + sqrt(b^2 - 4*a*c)
*c)*b^3*c^4 - 16*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(-b*c - 2*c^2 + sqrt(b^2 - 4*a*c)*c)*a^2*c^5 + 26*sqrt(2)*sqrt(
b^2 - 4*a*c)*sqrt(-b*c - 2*c^2 + sqrt(b^2 - 4*a*c)*c)*a*b*c^5 - 19*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(-b*c - 2*c^2
+ sqrt(b^2 - 4*a*c)*c)*b^2*c^5 + 20*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(-b*c - 2*c^2 + sqrt(b^2 - 4*a*c)*c)*a*c^6
- 10*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(-b*c - 2*c^2 + sqrt(b^2 - 4*a*c)*c)*b*c^6 - 2*(b^2 - 4*a*c)*b^3*c^4 + 4*(b
^2 - 4*a*c)*a*b*c^5 - 6*(b^2 - 4*a*c)*b^2*c^5 + 8*(b^2 - 4*a*c)*a*c^6 - 4*(b^2 - 4*a*c)*b*c^6 - (2*b^5*c^2 - 1
6*a*b^3*c^3 + 2*b^4*c^3 + 32*a^2*b*c^4 - 16*a*b^2*c^4 + 32*a^2*c^5 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(-b*c - 2*c
^2 + sqrt(b^2 - 4*a*c)*c)*b^5 + 8*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(-b*c - 2*c^2 + sqrt(b^2 - 4*a*c)*c)*a*b^3*c -
3*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(-b*c - 2*c^2 + sqrt(b^2 - 4*a*c)*c)*b^4*c - 16*sqrt(2)*sqrt(b^2 - 4*a*c)*sqr
t(-b*c - 2*c^2 + sqrt(b^2 - 4*a*c)*c)*a^2*b*c^2...

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Mupad [B]
time = 1.31, size = 776, normalized size = 3.39 \begin {gather*} \frac {\sqrt {1-x^2}}{c}-\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^2-b^2\,c-b^3+b^2\,\sqrt {b^2-4\,a\,c}+4\,a\,b\,c-2\,a\,c\,\sqrt {b^2-4\,a\,c}+b\,c\,\sqrt {b^2-4\,a\,c}\right )}{4\,c^2\,\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^2\,c-4\,a\,c^2+b^3+b^2\,\sqrt {b^2-4\,a\,c}-4\,a\,b\,c-2\,a\,c\,\sqrt {b^2-4\,a\,c}+b\,c\,\sqrt {b^2-4\,a\,c}\right )}{4\,c^2\,\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^2-b^2\,c-b^3+b^2\,\sqrt {b^2-4\,a\,c}+4\,a\,b\,c-2\,a\,c\,\sqrt {b^2-4\,a\,c}+b\,c\,\sqrt {b^2-4\,a\,c}\right )}{4\,c^2\,\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^2\,c-4\,a\,c^2+b^3+b^2\,\sqrt {b^2-4\,a\,c}-4\,a\,b\,c-2\,a\,c\,\sqrt {b^2-4\,a\,c}+b\,c\,\sqrt {b^2-4\,a\,c}\right )}{4\,c^2\,\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^3*(1 - x^2)^(1/2))/(a + b*x^2 + c*x^4),x)

[Out]

(1 - x^2)^(1/2)/c - (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^2 - b^2*c - b^3 + b
^2*(b^2 - 4*a*c)^(1/2) + 4*a*b*c - 2*a*c*(b^2 - 4*a*c)^(1/2) + b*c*(b^2 - 4*a*c)^(1/2)))/(4*c^2*((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^2*c - 4*a*c^2 + b^3 + b^2*(b^2 - 4*a*c)^(1/2) - 4*a*b*c - 2*a*c*(b^2 - 4*a*c)^(1/2) + b*c*(b^2 - 4*a*c)^(1
/2)))/(4*c^2*(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^2 - b^2*c - b^3 + b^2*(b^2 - 4*a*c)^(1/2) + 4*a*b*c - 2*a*c*(b^2 - 4*a*c)^(
1/2) + b*c*(b^2 - 4*a*c)^(1/2)))/(4*c^2*((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^2*c - 4*a*c^2 + b^3 + b^2*(b^2 - 4*a*c)^(1/2) - 4*a*
b*c - 2*a*c*(b^2 - 4*a*c)^(1/2) + b*c*(b^2 - 4*a*c)^(1/2)))/(4*c^2*(4*a*c - b^2)*((b + (b^2 - 4*a*c)^(1/2))/(2
*c) + 1)^(1/2))

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