3.376 \(\int \frac {x^5 \sqrt {1-x^2}}{a+b x^2+c x^4} \, dx\)
Optimal. Leaf size=281 \[ \frac {\left (-\frac {-3 a b c-2 a c^2+b^3+b^2 c}{\sqrt {b^2-4 a c}}-a c+b^2+b c\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {1-x^2}}{\sqrt {-\sqrt {b^2-4 a c}+b+2 c}}\right )}{\sqrt {2} c^{5/2} \sqrt {-\sqrt {b^2-4 a c}+b+2 c}}+\frac {\left (\frac {-3 a b c-2 a c^2+b^3+b^2 c}{\sqrt {b^2-4 a c}}-a c+b^2+b c\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {1-x^2}}{\sqrt {\sqrt {b^2-4 a c}+b+2 c}}\right )}{\sqrt {2} c^{5/2} \sqrt {\sqrt {b^2-4 a c}+b+2 c}}-\frac {b \sqrt {1-x^2}}{c^2}-\frac {\left (1-x^2\right )^{3/2}}{3 c} \]
[Out]
-1/3*(-x^2+1)^(3/2)/c-b*(-x^2+1)^(1/2)/c^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-a*c+b*c+(3*a*b*c+2*a*c^2-b^3-b^2*c)/(-4*a*c+b^2)^(1/2))/c^(5/2)*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^2-a*c+b*c+(-3*a*b*c-
2*a*c^2+b^3+b^2*c)/(-4*a*c+b^2)^(1/2))/c^(5/2)*2^(1/2)/(b+2*c+(-4*a*c+b^2)^(1/2))^(1/2)
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Rubi [A] time = 7.34, antiderivative size = 281, normalized size of antiderivative = 1.00,
number of steps used = 7, number of rules used = 5, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used =
{1251, 897, 1287, 1166, 208} \[ \frac {\left (-\frac {-3 a b c-2 a c^2+b^2 c+b^3}{\sqrt {b^2-4 a c}}-a c+b^2+b c\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {1-x^2}}{\sqrt {-\sqrt {b^2-4 a c}+b+2 c}}\right )}{\sqrt {2} c^{5/2} \sqrt {-\sqrt {b^2-4 a c}+b+2 c}}+\frac {\left (\frac {-3 a b c-2 a c^2+b^2 c+b^3}{\sqrt {b^2-4 a c}}-a c+b^2+b c\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {1-x^2}}{\sqrt {\sqrt {b^2-4 a c}+b+2 c}}\right )}{\sqrt {2} c^{5/2} \sqrt {\sqrt {b^2-4 a c}+b+2 c}}-\frac {b \sqrt {1-x^2}}{c^2}-\frac {\left (1-x^2\right )^{3/2}}{3 c} \]
Antiderivative was successfully verified.
[In]
Int[(x^5*Sqrt[1 - x^2])/(a + b*x^2 + c*x^4),x]
[Out]
-((b*Sqrt[1 - x^2])/c^2) - (1 - x^2)^(3/2)/(3*c) + ((b^2 - a*c + b*c - (b^3 - 3*a*b*c + b^2*c - 2*a*c^2)/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^(5/2)*Sqr
t[b + 2*c - Sqrt[b^2 - 4*a*c]]) + ((b^2 - a*c + b*c + (b^3 - 3*a*b*c + b^2*c - 2*a*c^2)/Sqrt[b^2 - 4*a*c])*Arc
Tanh[(Sqrt[2]*Sqrt[c]*Sqrt[1 - x^2])/Sqrt[b + 2*c + Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*c^(5/2)*Sqrt[b + 2*c + Sqrt[
b^2 - 4*a*c]])
Rule 208
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]
Rule 897
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :
> With[{q = Denominator[m]}, Dist[q/e, Subst[Int[x^(q*(m + 1) - 1)*((e*f - d*g)/e + (g*x^q)/e)^n*((c*d^2 - b*d
*e + a*e^2)/e^2 - ((2*c*d - b*e)*x^q)/e^2 + (c*x^(2*q))/e^2)^p, x], x, (d + e*x)^(1/q)], x]] /; FreeQ[{a, b, c
, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegersQ[n,
p] && FractionQ[m]
Rule 1166
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 1251
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]
Rule 1287
Int[(((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.))/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> Int[Ex
pandIntegrand[((f*x)^m*(d + e*x^2)^q)/(a + b*x^2 + c*x^4), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b^
2 - 4*a*c, 0] && IntegerQ[q] && IntegerQ[m]
Rubi steps
\begin {align*} \int \frac {x^5 \sqrt {1-x^2}}{a+b x^2+c x^4} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {\sqrt {1-x} x^2}{a+b x+c x^2} \, dx,x,x^2\right )\\ &=-\operatorname {Subst}\left (\int \frac {x^2 \left (1-x^2\right )^2}{a+b+c+(-b-2 c) x^2+c x^4} \, dx,x,\sqrt {1-x^2}\right )\\ &=-\operatorname {Subst}\left (\int \left (\frac {b}{c^2}+\frac {x^2}{c}-\frac {b (a+b+c)-\left (b^2-a c+b c\right ) x^2}{c^2 \left (a+b+c+(-b-2 c) x^2+c x^4\right )}\right ) \, dx,x,\sqrt {1-x^2}\right )\\ &=-\frac {b \sqrt {1-x^2}}{c^2}-\frac {\left (1-x^2\right )^{3/2}}{3 c}+\frac {\operatorname {Subst}\left (\int \frac {b (a+b+c)+\left (-b^2+a c-b c\right ) x^2}{a+b+c+(-b-2 c) x^2+c x^4} \, dx,x,\sqrt {1-x^2}\right )}{c^2}\\ &=-\frac {b \sqrt {1-x^2}}{c^2}-\frac {\left (1-x^2\right )^{3/2}}{3 c}-\frac {\left (b^2-a c+b c-\frac {b^3-3 a b c+b^2 c-2 a c^2}{\sqrt {b^2-4 a c}}\right ) \operatorname {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^2}-\frac {\left (b^2-a c+b c+\frac {b^3-3 a b c+b^2 c-2 a c^2}{\sqrt {b^2-4 a c}}\right ) \operatorname {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^2}\\ &=-\frac {b \sqrt {1-x^2}}{c^2}-\frac {\left (1-x^2\right )^{3/2}}{3 c}+\frac {\left (b^2-a c+b c-\frac {b^3-3 a b c+b^2 c-2 a c^2}{\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^{5/2} \sqrt {b+2 c-\sqrt {b^2-4 a c}}}+\frac {\left (b^2-a c+b c+\frac {b^3-3 a b c+b^2 c-2 a c^2}{\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^{5/2} \sqrt {b+2 c+\sqrt {b^2-4 a c}}}\\ \end {align*}
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Mathematica [A] time = 0.54, size = 354, normalized size = 1.26 \[ \frac {-\frac {3 \sqrt {2} \left (b^2 \left (\sqrt {b^2-4 a c}+c\right )+b c \left (\sqrt {b^2-4 a c}-3 a\right )-a c \left (\sqrt {b^2-4 a c}+2 c\right )+b^3\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {1-x^2}}{\sqrt {-\sqrt {b^2-4 a c}-b-2 c}}\right )}{\sqrt {b^2-4 a c} \sqrt {-\sqrt {b^2-4 a c}-b-2 c}}-\frac {3 \sqrt {2} \left (b^2 \left (\sqrt {b^2-4 a c}-c\right )+b c \left (\sqrt {b^2-4 a c}+3 a\right )+a c \left (2 c-\sqrt {b^2-4 a c}\right )-b^3\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {1-x^2}}{\sqrt {\sqrt {b^2-4 a c}-b-2 c}}\right )}{\sqrt {b^2-4 a c} \sqrt {\sqrt {b^2-4 a c}-b-2 c}}-6 b \sqrt {c} \sqrt {1-x^2}-2 c^{3/2} \left (1-x^2\right )^{3/2}}{6 c^{5/2}} \]
Antiderivative was successfully verified.
[In]
Integrate[(x^5*Sqrt[1 - x^2])/(a + b*x^2 + c*x^4),x]
[Out]
(-6*b*Sqrt[c]*Sqrt[1 - x^2] - 2*c^(3/2)*(1 - x^2)^(3/2) - (3*Sqrt[2]*(b^3 + b*c*(-3*a + Sqrt[b^2 - 4*a*c]) + b
^2*(c + Sqrt[b^2 - 4*a*c]) - a*c*(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 - 4*a*c]*Sqrt[-b - 2*c - Sqrt[b^2 - 4*a*c]]) - (3*Sqrt[2]*(-b^3 + a*c*(2*c
- Sqrt[b^2 - 4*a*c]) + b*c*(3*a + Sqrt[b^2 - 4*a*c]) + 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 - 4*a*c]*Sqrt[-b - 2*c + Sqrt[b^2 - 4*a*c]]))/(6
*c^(5/2))
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fricas [B] time = 16.52, size = 3615, normalized size = 12.86 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^5*(-x^2+1)^(1/2)/(c*x^4+b*x^2+a),x, algorithm="fricas")
[Out]
-1/6*(3*sqrt(1/2)*c^2*sqrt((b^5 + 2*a^2*c^3 + (5*a^2*b - 4*a*b^2)*c^2 - (5*a*b^3 - b^4)*c - (b^2*c^5 - 4*a*c^6
)*sqrt((b^8 + (a^4 - 4*a^3*b + 4*a^2*b^2)*c^4 - 2*(3*a^3*b^2 - 7*a^2*b^3 + 2*a*b^4)*c^3 + (11*a^2*b^4 - 10*a*b
^5 + b^6)*c^2 - 2*(3*a*b^6 - b^7)*c)/(b^2*c^10 - 4*a*c^11)))/(b^2*c^5 - 4*a*c^6))*log(-(2*a^3*b^4 + (a^2*b^2*c
^5 - 4*a^3*c^6)*x^2*sqrt((b^8 + (a^4 - 4*a^3*b + 4*a^2*b^2)*c^4 - 2*(3*a^3*b^2 - 7*a^2*b^3 + 2*a*b^4)*c^3 + (1
1*a^2*b^4 - 10*a*b^5 + b^6)*c^2 - 2*(3*a*b^6 - b^7)*c)/(b^2*c^10 - 4*a*c^11)) + 2*(a^5 - 2*a^4*b)*c^2 + (a^2*b
^5 + (a^4*b - 2*a^3*b^2)*c^2 - (3*a^3*b^3 - a^2*b^4)*c)*x^2 - 2*(3*a^4*b^2 - a^3*b^3)*c + sqrt(1/2)*((b^5*c^5
- 7*a*b^3*c^6 + 12*a^2*b*c^7)*x^2*sqrt((b^8 + (a^4 - 4*a^3*b + 4*a^2*b^2)*c^4 - 2*(3*a^3*b^2 - 7*a^2*b^3 + 2*a
*b^4)*c^3 + (11*a^2*b^4 - 10*a*b^5 + b^6)*c^2 - 2*(3*a*b^6 - b^7)*c)/(b^2*c^10 - 4*a*c^11)) + (b^8 + 4*(a^4 -
2*a^3*b)*c^4 - (17*a^3*b^2 - 14*a^2*b^3)*c^3 + (20*a^2*b^4 - 7*a*b^5)*c^2 - (8*a*b^6 - b^7)*c)*x^2)*sqrt((b^5
+ 2*a^2*c^3 + (5*a^2*b - 4*a*b^2)*c^2 - (5*a*b^3 - b^4)*c - (b^2*c^5 - 4*a*c^6)*sqrt((b^8 + (a^4 - 4*a^3*b + 4
*a^2*b^2)*c^4 - 2*(3*a^3*b^2 - 7*a^2*b^3 + 2*a*b^4)*c^3 + (11*a^2*b^4 - 10*a*b^5 + b^6)*c^2 - 2*(3*a*b^6 - b^7
)*c)/(b^2*c^10 - 4*a*c^11)))/(b^2*c^5 - 4*a*c^6)) - 2*(a^3*b^4 + (a^5 - 2*a^4*b)*c^2 - (3*a^4*b^2 - a^3*b^3)*c
)*sqrt(-x^2 + 1))/x^2) - 3*sqrt(1/2)*c^2*sqrt((b^5 + 2*a^2*c^3 + (5*a^2*b - 4*a*b^2)*c^2 - (5*a*b^3 - b^4)*c -
(b^2*c^5 - 4*a*c^6)*sqrt((b^8 + (a^4 - 4*a^3*b + 4*a^2*b^2)*c^4 - 2*(3*a^3*b^2 - 7*a^2*b^3 + 2*a*b^4)*c^3 + (
11*a^2*b^4 - 10*a*b^5 + b^6)*c^2 - 2*(3*a*b^6 - b^7)*c)/(b^2*c^10 - 4*a*c^11)))/(b^2*c^5 - 4*a*c^6))*log(-(2*a
^3*b^4 + (a^2*b^2*c^5 - 4*a^3*c^6)*x^2*sqrt((b^8 + (a^4 - 4*a^3*b + 4*a^2*b^2)*c^4 - 2*(3*a^3*b^2 - 7*a^2*b^3
+ 2*a*b^4)*c^3 + (11*a^2*b^4 - 10*a*b^5 + b^6)*c^2 - 2*(3*a*b^6 - b^7)*c)/(b^2*c^10 - 4*a*c^11)) + 2*(a^5 - 2*
a^4*b)*c^2 + (a^2*b^5 + (a^4*b - 2*a^3*b^2)*c^2 - (3*a^3*b^3 - a^2*b^4)*c)*x^2 - 2*(3*a^4*b^2 - a^3*b^3)*c - s
qrt(1/2)*((b^5*c^5 - 7*a*b^3*c^6 + 12*a^2*b*c^7)*x^2*sqrt((b^8 + (a^4 - 4*a^3*b + 4*a^2*b^2)*c^4 - 2*(3*a^3*b^
2 - 7*a^2*b^3 + 2*a*b^4)*c^3 + (11*a^2*b^4 - 10*a*b^5 + b^6)*c^2 - 2*(3*a*b^6 - b^7)*c)/(b^2*c^10 - 4*a*c^11))
+ (b^8 + 4*(a^4 - 2*a^3*b)*c^4 - (17*a^3*b^2 - 14*a^2*b^3)*c^3 + (20*a^2*b^4 - 7*a*b^5)*c^2 - (8*a*b^6 - b^7)
*c)*x^2)*sqrt((b^5 + 2*a^2*c^3 + (5*a^2*b - 4*a*b^2)*c^2 - (5*a*b^3 - b^4)*c - (b^2*c^5 - 4*a*c^6)*sqrt((b^8 +
(a^4 - 4*a^3*b + 4*a^2*b^2)*c^4 - 2*(3*a^3*b^2 - 7*a^2*b^3 + 2*a*b^4)*c^3 + (11*a^2*b^4 - 10*a*b^5 + b^6)*c^2
- 2*(3*a*b^6 - b^7)*c)/(b^2*c^10 - 4*a*c^11)))/(b^2*c^5 - 4*a*c^6)) - 2*(a^3*b^4 + (a^5 - 2*a^4*b)*c^2 - (3*a
^4*b^2 - a^3*b^3)*c)*sqrt(-x^2 + 1))/x^2) - 3*sqrt(1/2)*c^2*sqrt((b^5 + 2*a^2*c^3 + (5*a^2*b - 4*a*b^2)*c^2 -
(5*a*b^3 - b^4)*c + (b^2*c^5 - 4*a*c^6)*sqrt((b^8 + (a^4 - 4*a^3*b + 4*a^2*b^2)*c^4 - 2*(3*a^3*b^2 - 7*a^2*b^3
+ 2*a*b^4)*c^3 + (11*a^2*b^4 - 10*a*b^5 + b^6)*c^2 - 2*(3*a*b^6 - b^7)*c)/(b^2*c^10 - 4*a*c^11)))/(b^2*c^5 -
4*a*c^6))*log(-(2*a^3*b^4 - (a^2*b^2*c^5 - 4*a^3*c^6)*x^2*sqrt((b^8 + (a^4 - 4*a^3*b + 4*a^2*b^2)*c^4 - 2*(3*a
^3*b^2 - 7*a^2*b^3 + 2*a*b^4)*c^3 + (11*a^2*b^4 - 10*a*b^5 + b^6)*c^2 - 2*(3*a*b^6 - b^7)*c)/(b^2*c^10 - 4*a*c
^11)) + 2*(a^5 - 2*a^4*b)*c^2 + (a^2*b^5 + (a^4*b - 2*a^3*b^2)*c^2 - (3*a^3*b^3 - a^2*b^4)*c)*x^2 - 2*(3*a^4*b
^2 - a^3*b^3)*c + sqrt(1/2)*((b^5*c^5 - 7*a*b^3*c^6 + 12*a^2*b*c^7)*x^2*sqrt((b^8 + (a^4 - 4*a^3*b + 4*a^2*b^2
)*c^4 - 2*(3*a^3*b^2 - 7*a^2*b^3 + 2*a*b^4)*c^3 + (11*a^2*b^4 - 10*a*b^5 + b^6)*c^2 - 2*(3*a*b^6 - b^7)*c)/(b^
2*c^10 - 4*a*c^11)) - (b^8 + 4*(a^4 - 2*a^3*b)*c^4 - (17*a^3*b^2 - 14*a^2*b^3)*c^3 + (20*a^2*b^4 - 7*a*b^5)*c^
2 - (8*a*b^6 - b^7)*c)*x^2)*sqrt((b^5 + 2*a^2*c^3 + (5*a^2*b - 4*a*b^2)*c^2 - (5*a*b^3 - b^4)*c + (b^2*c^5 - 4
*a*c^6)*sqrt((b^8 + (a^4 - 4*a^3*b + 4*a^2*b^2)*c^4 - 2*(3*a^3*b^2 - 7*a^2*b^3 + 2*a*b^4)*c^3 + (11*a^2*b^4 -
10*a*b^5 + b^6)*c^2 - 2*(3*a*b^6 - b^7)*c)/(b^2*c^10 - 4*a*c^11)))/(b^2*c^5 - 4*a*c^6)) - 2*(a^3*b^4 + (a^5 -
2*a^4*b)*c^2 - (3*a^4*b^2 - a^3*b^3)*c)*sqrt(-x^2 + 1))/x^2) + 3*sqrt(1/2)*c^2*sqrt((b^5 + 2*a^2*c^3 + (5*a^2*
b - 4*a*b^2)*c^2 - (5*a*b^3 - b^4)*c + (b^2*c^5 - 4*a*c^6)*sqrt((b^8 + (a^4 - 4*a^3*b + 4*a^2*b^2)*c^4 - 2*(3*
a^3*b^2 - 7*a^2*b^3 + 2*a*b^4)*c^3 + (11*a^2*b^4 - 10*a*b^5 + b^6)*c^2 - 2*(3*a*b^6 - b^7)*c)/(b^2*c^10 - 4*a*
c^11)))/(b^2*c^5 - 4*a*c^6))*log(-(2*a^3*b^4 - (a^2*b^2*c^5 - 4*a^3*c^6)*x^2*sqrt((b^8 + (a^4 - 4*a^3*b + 4*a^
2*b^2)*c^4 - 2*(3*a^3*b^2 - 7*a^2*b^3 + 2*a*b^4)*c^3 + (11*a^2*b^4 - 10*a*b^5 + b^6)*c^2 - 2*(3*a*b^6 - b^7)*c
)/(b^2*c^10 - 4*a*c^11)) + 2*(a^5 - 2*a^4*b)*c^2 + (a^2*b^5 + (a^4*b - 2*a^3*b^2)*c^2 - (3*a^3*b^3 - a^2*b^4)*
c)*x^2 - 2*(3*a^4*b^2 - a^3*b^3)*c - sqrt(1/2)*((b^5*c^5 - 7*a*b^3*c^6 + 12*a^2*b*c^7)*x^2*sqrt((b^8 + (a^4 -
4*a^3*b + 4*a^2*b^2)*c^4 - 2*(3*a^3*b^2 - 7*a^2*b^3 + 2*a*b^4)*c^3 + (11*a^2*b^4 - 10*a*b^5 + b^6)*c^2 - 2*(3*
a*b^6 - b^7)*c)/(b^2*c^10 - 4*a*c^11)) - (b^8 + 4*(a^4 - 2*a^3*b)*c^4 - (17*a^3*b^2 - 14*a^2*b^3)*c^3 + (20*a^
2*b^4 - 7*a*b^5)*c^2 - (8*a*b^6 - b^7)*c)*x^2)*sqrt((b^5 + 2*a^2*c^3 + (5*a^2*b - 4*a*b^2)*c^2 - (5*a*b^3 - b^
4)*c + (b^2*c^5 - 4*a*c^6)*sqrt((b^8 + (a^4 - 4*a^3*b + 4*a^2*b^2)*c^4 - 2*(3*a^3*b^2 - 7*a^2*b^3 + 2*a*b^4)*c
^3 + (11*a^2*b^4 - 10*a*b^5 + b^6)*c^2 - 2*(3*a*b^6 - b^7)*c)/(b^2*c^10 - 4*a*c^11)))/(b^2*c^5 - 4*a*c^6)) - 2
*(a^3*b^4 + (a^5 - 2*a^4*b)*c^2 - (3*a^4*b^2 - a^3*b^3)*c)*sqrt(-x^2 + 1))/x^2) - 2*(c*x^2 - 3*b - c)*sqrt(-x^
2 + 1))/c^2
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giac [B] time = 4.48, size = 4637, normalized size = 16.50 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^5*(-x^2+1)^(1/2)/(c*x^4+b*x^2+a),x, algorithm="giac")
[Out]
-1/8*(2*b^6*c^4 - 14*a*b^4*c^5 + 6*b^5*c^5 + 24*a^2*b^2*c^6 - 40*a*b^3*c^6 + 4*b^4*c^6 + 64*a^2*b*c^7 - 24*a*b
^2*c^7 + 32*a^2*c^8 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(-b*c - 2*c^2 - sqrt(b^2 - 4*a*c)*c)*b^6*c^2 + 7*sqrt(2)*s
qrt(b^2 - 4*a*c)*sqrt(-b*c - 2*c^2 - sqrt(b^2 - 4*a*c)*c)*a*b^4*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^5*c^3 - 12*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(-b*c - 2*c^2 - sqrt(b^2 - 4*a*c)*c)*a
^2*b^2*c^4 + 26*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^4 - 13*sqrt(2)*sqrt
(b^2 - 4*a*c)*sqrt(-b*c - 2*c^2 - sqrt(b^2 - 4*a*c)*c)*b^4*c^4 - 32*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^5 + 43*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^5 - 19*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(-b*c - 2*c^2 - sqrt(b^2 - 4*a*c)*c)*b^3*c^5 - 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^6 + 48*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(-b*c - 2*c^2 - sq
rt(b^2 - 4*a*c)*c)*a*b*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^2*c^6 + 2
0*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(-b*c - 2*c^2 - sqrt(b^2 - 4*a*c)*c)*a*c^7 - 2*(b^2 - 4*a*c)*b^4*c^4 + 6*(b^2
- 4*a*c)*a*b^2*c^5 - 6*(b^2 - 4*a*c)*b^3*c^5 + 16*(b^2 - 4*a*c)*a*b*c^6 - 4*(b^2 - 4*a*c)*b^2*c^6 + 8*(b^2 - 4
*a*c)*a*c^7 - (2*b^6*c^2 - 18*a*b^4*c^3 + 2*b^5*c^3 + 48*a^2*b^2*c^4 - 16*a*b^3*c^4 - 32*a^3*c^5 + 32*a^2*b*c^
5 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(-b*c - 2*c^2 - sqrt(b^2 - 4*a*c)*c)*b^6 + 9*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(
-b*c - 2*c^2 - sqrt(b^2 - 4*a*c)*c)*a*b^4*c - 3*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(-b*c - 2*c^2 - sqrt(b^2 - 4*a*c
)*c)*b^5*c - 24*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(-b*c - 2*c^2 - sqrt(b^2 - 4*a*c)*c)*a^2*b^2*c^2 + 18*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^2 - 7*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(-b*c - 2
*c^2 - sqrt(b^2 - 4*a*c)*c)*b^4*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^
3*c^3 - 24*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^3 + 33*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^3 - 5*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(-b*c - 2*c^2 -
sqrt(b^2 - 4*a*c)*c)*b^3*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^2*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*c^4 - 2*(b^2 - 4*a*c)*b^4*c^2 + 10*
(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)*c^2 - 2
*(sqrt(2)*sqrt(-b*c - 2*c^2 - sqrt(b^2 - 4*a*c)*c)*a*b^5*c^2 + sqrt(2)*sqrt(-b*c - 2*c^2 - sqrt(b^2 - 4*a*c)*c
)*b^6*c^2 - 8*sqrt(2)*sqrt(-b*c - 2*c^2 - sqrt(b^2 - 4*a*c)*c)*a^2*b^3*c^3 - 6*sqrt(2)*sqrt(-b*c - 2*c^2 - sqr
t(b^2 - 4*a*c)*c)*a*b^4*c^3 + 3*sqrt(2)*sqrt(-b*c - 2*c^2 - sqrt(b^2 - 4*a*c)*c)*b^5*c^3 + 2*a*b^5*c^3 + 2*b^6
*c^3 + 16*sqrt(2)*sqrt(-b*c - 2*c^2 - sqrt(b^2 - 4*a*c)*c)*a^3*b*c^4 + 8*sqrt(2)*sqrt(-b*c - 2*c^2 - sqrt(b^2
- 4*a*c)*c)*a^2*b^2*c^4 - 11*sqrt(2)*sqrt(-b*c - 2*c^2 - sqrt(b^2 - 4*a*c)*c)*a*b^3*c^4 - 16*a^2*b^3*c^4 + 7*s
qrt(2)*sqrt(-b*c - 2*c^2 - sqrt(b^2 - 4*a*c)*c)*b^4*c^4 - 16*a*b^4*c^4 + 2*b^5*c^4 - 4*sqrt(2)*sqrt(-b*c - 2*c
^2 - sqrt(b^2 - 4*a*c)*c)*a^2*b*c^5 + 32*a^3*b*c^5 - 28*sqrt(2)*sqrt(-b*c - 2*c^2 - sqrt(b^2 - 4*a*c)*c)*a*b^2
*c^5 + 32*a^2*b^2*c^5 + 5*sqrt(2)*sqrt(-b*c - 2*c^2 - sqrt(b^2 - 4*a*c)*c)*b^3*c^5 - 16*a*b^3*c^5 - 20*sqrt(2)
*sqrt(-b*c - 2*c^2 - sqrt(b^2 - 4*a*c)*c)*a*b*c^6 + 32*a^2*b*c^6 - 2*(b^2 - 4*a*c)*a*b^3*c^3 - 2*(b^2 - 4*a*c)
*b^4*c^3 + 8*(b^2 - 4*a*c)*a^2*b*c^4 + 8*(b^2 - 4*a*c)*a*b^2*c^4 - 2*(b^2 - 4*a*c)*b^3*c^4 + 8*(b^2 - 4*a*c)*a
*b*c^5)*abs(c))*arctan(2*sqrt(1/2)*sqrt(-x^2 + 1)/sqrt(-(b*c^3 + 2*c^4 + sqrt(-4*(a*c^3 + b*c^3 + c^4)*c^4 + (
b*c^3 + 2*c^4)^2))/c^4))/((a*b^4*c^4 + b^5*c^4 - 8*a^2*b^2*c^5 - 6*a*b^3*c^5 + 3*b^4*c^5 + 16*a^3*c^6 + 8*a^2*
b*c^6 - 11*a*b^2*c^6 + 7*b^3*c^6 - 4*a^2*c^7 - 28*a*b*c^7 + 5*b^2*c^7 - 20*a*c^8)*c^2) + 1/8*(2*b^6*c^4 - 14*a
*b^4*c^5 + 6*b^5*c^5 + 24*a^2*b^2*c^6 - 40*a*b^3*c^6 + 4*b^4*c^6 + 64*a^2*b*c^7 - 24*a*b^2*c^7 + 32*a^2*c^8 -
sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(-b*c - 2*c^2 + sqrt(b^2 - 4*a*c)*c)*b^6*c^2 + 7*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(
-b*c - 2*c^2 + sqrt(b^2 - 4*a*c)*c)*a*b^4*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^5*c^3 - 12*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(-b*c - 2*c^2 + sqrt(b^2 - 4*a*c)*c)*a^2*b^2*c^4 + 26*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^4 - 13*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(-b*
c - 2*c^2 + sqrt(b^2 - 4*a*c)*c)*b^4*c^4 - 32*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^5 + 43*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^5 - 19*sqrt(2)*sq
rt(b^2 - 4*a*c)*sqrt(-b*c - 2*c^2 + sqrt(b^2 - 4*a*c)*c)*b^3*c^5 - 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^6 + 48*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(-b*c - 2*c^2 + sqrt(b^2 - 4*a*c)*c)*a*b
*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^2*c^6 + 20*sqrt(2)*sqrt(b^2 - 4
*a*c)*sqrt(-b*c - 2*c^2 + sqrt(b^2 - 4*a*c)*c)*a*c^7 - 2*(b^2 - 4*a*c)*b^4*c^4 + 6*(b^2 - 4*a*c)*a*b^2*c^5 - 6
*(b^2 - 4*a*c)*b^3*c^5 + 16*(b^2 - 4*a*c)*a*b*c^6 - 4*(b^2 - 4*a*c)*b^2*c^6 + 8*(b^2 - 4*a*c)*a*c^7 - (2*b^6*c
^2 - 18*a*b^4*c^3 + 2*b^5*c^3 + 48*a^2*b^2*c^4 - 16*a*b^3*c^4 - 32*a^3*c^5 + 32*a^2*b*c^5 - sqrt(2)*sqrt(b^2 -
4*a*c)*sqrt(-b*c - 2*c^2 + sqrt(b^2 - 4*a*c)*c)*b^6 + 9*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(-b*c - 2*c^2 + sqrt(b^
2 - 4*a*c)*c)*a*b^4*c - 3*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(-b*c - 2*c^2 + sqrt(b^2 - 4*a*c)*c)*b^5*c - 24*sqrt(2
)*sqrt(b^2 - 4*a*c)*sqrt(-b*c - 2*c^2 + sqrt(b^2 - 4*a*c)*c)*a^2*b^2*c^2 + 18*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^2 - 7*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(-b*c - 2*c^2 + sqrt(b^2 - 4*a*
c)*c)*b^4*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^3*c^3 - 24*sqrt(2)*sqr
t(b^2 - 4*a*c)*sqrt(-b*c - 2*c^2 + sqrt(b^2 - 4*a*c)*c)*a^2*b*c^3 + 33*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^3 - 5*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(-b*c - 2*c^2 + sqrt(b^2 - 4*a*c)*c)*b
^3*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^2*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*c^4 - 2*(b^2 - 4*a*c)*b^4*c^2 + 10*(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)*c^2 + 2*(sqrt(2)*sqrt(-b*c -
2*c^2 + sqrt(b^2 - 4*a*c)*c)*a*b^5*c^2 + sqrt(2)*sqrt(-b*c - 2*c^2 + sqrt(b^2 - 4*a*c)*c)*b^6*c^2 - 8*sqrt(2)*
sqrt(-b*c - 2*c^2 + sqrt(b^2 - 4*a*c)*c)*a^2*b^3*c^3 - 6*sqrt(2)*sqrt(-b*c - 2*c^2 + sqrt(b^2 - 4*a*c)*c)*a*b^
4*c^3 + 3*sqrt(2)*sqrt(-b*c - 2*c^2 + sqrt(b^2 - 4*a*c)*c)*b^5*c^3 - 2*a*b^5*c^3 - 2*b^6*c^3 + 16*sqrt(2)*sqrt
(-b*c - 2*c^2 + sqrt(b^2 - 4*a*c)*c)*a^3*b*c^4 + 8*sqrt(2)*sqrt(-b*c - 2*c^2 + sqrt(b^2 - 4*a*c)*c)*a^2*b^2*c^
4 - 11*sqrt(2)*sqrt(-b*c - 2*c^2 + sqrt(b^2 - 4*a*c)*c)*a*b^3*c^4 + 16*a^2*b^3*c^4 + 7*sqrt(2)*sqrt(-b*c - 2*c
^2 + sqrt(b^2 - 4*a*c)*c)*b^4*c^4 + 16*a*b^4*c^4 - 2*b^5*c^4 - 4*sqrt(2)*sqrt(-b*c - 2*c^2 + sqrt(b^2 - 4*a*c)
*c)*a^2*b*c^5 - 32*a^3*b*c^5 - 28*sqrt(2)*sqrt(-b*c - 2*c^2 + sqrt(b^2 - 4*a*c)*c)*a*b^2*c^5 - 32*a^2*b^2*c^5
+ 5*sqrt(2)*sqrt(-b*c - 2*c^2 + sqrt(b^2 - 4*a*c)*c)*b^3*c^5 + 16*a*b^3*c^5 - 20*sqrt(2)*sqrt(-b*c - 2*c^2 + s
qrt(b^2 - 4*a*c)*c)*a*b*c^6 - 32*a^2*b*c^6 + 2*(b^2 - 4*a*c)*a*b^3*c^3 + 2*(b^2 - 4*a*c)*b^4*c^3 - 8*(b^2 - 4*
a*c)*a^2*b*c^4 - 8*(b^2 - 4*a*c)*a*b^2*c^4 + 2*(b^2 - 4*a*c)*b^3*c^4 - 8*(b^2 - 4*a*c)*a*b*c^5)*abs(c))*arctan
(2*sqrt(1/2)*sqrt(-x^2 + 1)/sqrt(-(b*c^3 + 2*c^4 - sqrt(-4*(a*c^3 + b*c^3 + c^4)*c^4 + (b*c^3 + 2*c^4)^2))/c^4
))/((a*b^4*c^4 + b^5*c^4 - 8*a^2*b^2*c^5 - 6*a*b^3*c^5 + 3*b^4*c^5 + 16*a^3*c^6 + 8*a^2*b*c^6 - 11*a*b^2*c^6 +
7*b^3*c^6 - 4*a^2*c^7 - 28*a*b*c^7 + 5*b^2*c^7 - 20*a*c^8)*c^2) - 1/3*((-x^2 + 1)^(3/2)*c^2 + 3*sqrt(-x^2 + 1
)*b*c)/c^3
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maple [B] time = 0.10, size = 2134, normalized size = 7.59 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x^5*(-x^2+1)^(1/2)/(c*x^4+b*x^2+a),x)
[Out]
-1/3*(-x^2+1)^(3/2)/c+4/c*a^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*((-x^2+1)^(1/2)-1)^2/x^2*a+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))*(-4*a*c+b^2)^(1/2)-2/c^2*a/(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*((-x^2+1)^(1/2)-1)^2/x^2*a+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^2*(-4*a*c+b^2)^(1/2)-2/c*a/(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*((-x^2+1)^(
1/2)-1)^2/x^2*a+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))*(-4*a*c+b^2)^(1/2)*b+8/c*a^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*((-x^2+1)^(1/2)-1)^2/x^2*a+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+8*a^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*((-x^2+1)^(1/2)-1)^2/x^2*a+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^2*a/(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*((-x^2+1)^(1/2)-1)^2/x^2*a+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^3-2/c*a/(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*((-x^2+1)^(1/2)-1)^2/x
^2*a+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^
2+4/c*a^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*
((-x^2+1)^(1/2)-1)^2/x^2*a+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))*(-4*a*c+b^2)^(1/2)-2/c^2*a/(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*((-x^2+1)^(1/2)-1)^2/x^2*a+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^2*(-4*a*c+b^2)^(1/2)-2/c*a/(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*((-x^2+1)^(1/2)-1)^2/x^2*a+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))*(-4*a*c+b^2)^(
1/2)*b-8/c*a^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*((-x^2+1)^(1/2)-1)^2/x^2*a+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-8*a^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*((-x^2+1)^(1/2)-1)^2/x^2*a+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^2*a/(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*((-x^2+1)^(1/2)-1)^2/x^2*a+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^3+2/c*a/(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*((-x^2+1)^(1/2)-1)^2/x^2*a+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^2-2/c^2*b/(2/x^2-2/x^2*(-x^
2+1)^(1/2))
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {-x^{2} + 1} x^{5}}{c x^{4} + b x^{2} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^5*(-x^2+1)^(1/2)/(c*x^4+b*x^2+a),x, algorithm="maxima")
[Out]
integrate(sqrt(-x^2 + 1)*x^5/(c*x^4 + b*x^2 + a), x)
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mupad [B] time = 1.45, size = 917, normalized size = 3.26 \[ \sqrt {1-x^2}\,\left (\frac {2}{3\,c}-\frac {\frac {b}{c}+1}{c}+\frac {x^2}{3\,c}\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^3\,c+b^4-b^3\,\sqrt {b^2-4\,a\,c}+4\,a^2\,c^2+2\,a\,c^2\,\sqrt {b^2-4\,a\,c}-b^2\,c\,\sqrt {b^2-4\,a\,c}-4\,a\,b\,c^2-5\,a\,b^2\,c+3\,a\,b\,c\,\sqrt {b^2-4\,a\,c}\right )}{4\,c^3\,\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^3\,c+b^4+b^3\,\sqrt {b^2-4\,a\,c}+4\,a^2\,c^2-2\,a\,c^2\,\sqrt {b^2-4\,a\,c}+b^2\,c\,\sqrt {b^2-4\,a\,c}-4\,a\,b\,c^2-5\,a\,b^2\,c-3\,a\,b\,c\,\sqrt {b^2-4\,a\,c}\right )}{4\,c^3\,\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 (b^3\,c+b^4+b^3\,\sqrt {b^2-4\,a\,c}+4\,a^2\,c^2-2\,a\,c^2\,\sqrt {b^2-4\,a\,c}+b^2\,c\,\sqrt {b^2-4\,a\,c}-4\,a\,b\,c^2-5\,a\,b^2\,c-3\,a\,b\,c\,\sqrt {b^2-4\,a\,c}\right )}{4\,c^3\,\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 (b^3\,c+b^4-b^3\,\sqrt {b^2-4\,a\,c}+4\,a^2\,c^2+2\,a\,c^2\,\sqrt {b^2-4\,a\,c}-b^2\,c\,\sqrt {b^2-4\,a\,c}-4\,a\,b\,c^2-5\,a\,b^2\,c+3\,a\,b\,c\,\sqrt {b^2-4\,a\,c}\right )}{4\,c^3\,\sqrt {\frac {b-\sqrt {b^2-4\,a\,c}}{2\,c}+1}\,\left (4\,a\,c-b^2\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((x^5*(1 - x^2)^(1/2))/(a + b*x^2 + c*x^4),x)
[Out]
(1 - x^2)^(1/2)*(2/(3*c) - (b/c + 1)/c + x^2/(3*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)))*(b^3*c + b^4 - b^3*(b^2 - 4*a*c)^(1/2) + 4*a^2*c^2 + 2*a*c^2*(b^2 - 4*a*c)^(1/2) - b^2*c*(b^2 - 4*a*c)
^(1/2) - 4*a*b*c^2 - 5*a*b^2*c + 3*a*b*c*(b^2 - 4*a*c)^(1/2)))/(4*c^3*((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^3*c + b^4 + b^3*(b^2 -
4*a*c)^(1/2) + 4*a^2*c^2 - 2*a*c^2*(b^2 - 4*a*c)^(1/2) + b^2*c*(b^2 - 4*a*c)^(1/2) - 4*a*b*c^2 - 5*a*b^2*c -
3*a*b*c*(b^2 - 4*a*c)^(1/2)))/(4*c^3*(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)))*(b^3*c + b^4 + b^3*(b^2 - 4*a*c)^(1/2) + 4*a^2*c^2 - 2*a*c
^2*(b^2 - 4*a*c)^(1/2) + b^2*c*(b^2 - 4*a*c)^(1/2) - 4*a*b*c^2 - 5*a*b^2*c - 3*a*b*c*(b^2 - 4*a*c)^(1/2)))/(4*
c^3*(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)))*(b^3*c + b^4 - b^3*(b^2 - 4*a*c)^(1/2) + 4*a^2*c^2 + 2*a*c^2*(b^2 - 4*a*c)^(1/2) - b^2*c*(b
^2 - 4*a*c)^(1/2) - 4*a*b*c^2 - 5*a*b^2*c + 3*a*b*c*(b^2 - 4*a*c)^(1/2)))/(4*c^3*((b - (b^2 - 4*a*c)^(1/2))/(2
*c) + 1)^(1/2)*(4*a*c - b^2))
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{5} \sqrt {- \left (x - 1\right ) \left (x + 1\right )}}{a + b x^{2} + c x^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x**5*(-x**2+1)**(1/2)/(c*x**4+b*x**2+a),x)
[Out]
Integral(x**5*sqrt(-(x - 1)*(x + 1))/(a + b*x**2 + c*x**4), x)
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