3.357 \(\int \frac {x \sqrt {d+e x^2}}{a+b x^2+c x^4} \, dx\)

Optimal. Leaf size=202 \[ \frac {\sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x^2}}{\sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}\right )}{\sqrt {2} \sqrt {c} \sqrt {b^2-4 a c}}-\frac {\sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x^2}}{\sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}\right )}{\sqrt {2} \sqrt {c} \sqrt {b^2-4 a c}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.36, antiderivative size = 202, 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 = {1247, 699, 1130, 208} \[ \frac {\sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x^2}}{\sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}\right )}{\sqrt {2} \sqrt {c} \sqrt {b^2-4 a c}}-\frac {\sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x^2}}{\sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}\right )}{\sqrt {2} \sqrt {c} \sqrt {b^2-4 a c}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e]*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x^2])/Sqrt[2*c*d - (b - Sqrt[b^
2 - 4*a*c])*e]])/(Sqrt[2]*Sqrt[c]*Sqrt[b^2 - 4*a*c])) + (Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]*ArcTanh[(Sqrt
[2]*Sqrt[c]*Sqrt[d + e*x^2])/Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]])/(Sqrt[2]*Sqrt[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 699

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 1130

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*(b/q + 1))/2, Int[(d*x)^(m - 2)/(b/2 + q/2 + c*x^2), x], x] - Dist[(d^2*(b/q - 1))/2, 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 1247

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

________________________________________________________________________________________

Mathematica [A]  time = 0.33, size = 179, normalized size = 0.89 \[ \frac {\sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x^2}}{\sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}\right )-\sqrt {e \sqrt {b^2-4 a c}-b e+2 c d} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x^2}}{\sqrt {e \sqrt {b^2-4 a c}-b e+2 c d}}\right )}{\sqrt {2} \sqrt {c} \sqrt {b^2-4 a c}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [B]  time = 19.82, size = 1085, normalized size = 5.37 \[ -\frac {1}{4} \, \sqrt {\frac {1}{2}} \sqrt {\frac {2 \, c d - b e + {\left (b^{2} c - 4 \, a c^{2}\right )} \sqrt {\frac {e^{2}}{b^{2} c^{2} - 4 \, a c^{3}}}}{b^{2} c - 4 \, a c^{2}}} \log \left (\frac {b e^{2} x^{2} + 2 \, b d e - 2 \, a e^{2} + 2 \, \sqrt {\frac {1}{2}} \sqrt {e x^{2} + d} {\left ({\left (b^{2} - 4 \, a c\right )} e + {\left (b^{3} c - 4 \, a b c^{2}\right )} \sqrt {\frac {e^{2}}{b^{2} c^{2} - 4 \, a c^{3}}}\right )} \sqrt {\frac {2 \, c d - b e + {\left (b^{2} c - 4 \, a c^{2}\right )} \sqrt {\frac {e^{2}}{b^{2} c^{2} - 4 \, a c^{3}}}}{b^{2} c - 4 \, a c^{2}}} + {\left ({\left (b^{2} c - 4 \, a c^{2}\right )} e x^{2} + 2 \, {\left (b^{2} c - 4 \, a c^{2}\right )} d\right )} \sqrt {\frac {e^{2}}{b^{2} c^{2} - 4 \, a c^{3}}}}{x^{2}}\right ) + \frac {1}{4} \, \sqrt {\frac {1}{2}} \sqrt {\frac {2 \, c d - b e + {\left (b^{2} c - 4 \, a c^{2}\right )} \sqrt {\frac {e^{2}}{b^{2} c^{2} - 4 \, a c^{3}}}}{b^{2} c - 4 \, a c^{2}}} \log \left (\frac {b e^{2} x^{2} + 2 \, b d e - 2 \, a e^{2} - 2 \, \sqrt {\frac {1}{2}} \sqrt {e x^{2} + d} {\left ({\left (b^{2} - 4 \, a c\right )} e + {\left (b^{3} c - 4 \, a b c^{2}\right )} \sqrt {\frac {e^{2}}{b^{2} c^{2} - 4 \, a c^{3}}}\right )} \sqrt {\frac {2 \, c d - b e + {\left (b^{2} c - 4 \, a c^{2}\right )} \sqrt {\frac {e^{2}}{b^{2} c^{2} - 4 \, a c^{3}}}}{b^{2} c - 4 \, a c^{2}}} + {\left ({\left (b^{2} c - 4 \, a c^{2}\right )} e x^{2} + 2 \, {\left (b^{2} c - 4 \, a c^{2}\right )} d\right )} \sqrt {\frac {e^{2}}{b^{2} c^{2} - 4 \, a c^{3}}}}{x^{2}}\right ) - \frac {1}{4} \, \sqrt {\frac {1}{2}} \sqrt {\frac {2 \, c d - b e - {\left (b^{2} c - 4 \, a c^{2}\right )} \sqrt {\frac {e^{2}}{b^{2} c^{2} - 4 \, a c^{3}}}}{b^{2} c - 4 \, a c^{2}}} \log \left (\frac {b e^{2} x^{2} + 2 \, b d e - 2 \, a e^{2} + 2 \, \sqrt {\frac {1}{2}} \sqrt {e x^{2} + d} {\left ({\left (b^{2} - 4 \, a c\right )} e - {\left (b^{3} c - 4 \, a b c^{2}\right )} \sqrt {\frac {e^{2}}{b^{2} c^{2} - 4 \, a c^{3}}}\right )} \sqrt {\frac {2 \, c d - b e - {\left (b^{2} c - 4 \, a c^{2}\right )} \sqrt {\frac {e^{2}}{b^{2} c^{2} - 4 \, a c^{3}}}}{b^{2} c - 4 \, a c^{2}}} - {\left ({\left (b^{2} c - 4 \, a c^{2}\right )} e x^{2} + 2 \, {\left (b^{2} c - 4 \, a c^{2}\right )} d\right )} \sqrt {\frac {e^{2}}{b^{2} c^{2} - 4 \, a c^{3}}}}{x^{2}}\right ) + \frac {1}{4} \, \sqrt {\frac {1}{2}} \sqrt {\frac {2 \, c d - b e - {\left (b^{2} c - 4 \, a c^{2}\right )} \sqrt {\frac {e^{2}}{b^{2} c^{2} - 4 \, a c^{3}}}}{b^{2} c - 4 \, a c^{2}}} \log \left (\frac {b e^{2} x^{2} + 2 \, b d e - 2 \, a e^{2} - 2 \, \sqrt {\frac {1}{2}} \sqrt {e x^{2} + d} {\left ({\left (b^{2} - 4 \, a c\right )} e - {\left (b^{3} c - 4 \, a b c^{2}\right )} \sqrt {\frac {e^{2}}{b^{2} c^{2} - 4 \, a c^{3}}}\right )} \sqrt {\frac {2 \, c d - b e - {\left (b^{2} c - 4 \, a c^{2}\right )} \sqrt {\frac {e^{2}}{b^{2} c^{2} - 4 \, a c^{3}}}}{b^{2} c - 4 \, a c^{2}}} - {\left ({\left (b^{2} c - 4 \, a c^{2}\right )} e x^{2} + 2 \, {\left (b^{2} c - 4 \, a c^{2}\right )} d\right )} \sqrt {\frac {e^{2}}{b^{2} c^{2} - 4 \, a c^{3}}}}{x^{2}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*sqrt(1/2)*sqrt((2*c*d - b*e + (b^2*c - 4*a*c^2)*sqrt(e^2/(b^2*c^2 - 4*a*c^3)))/(b^2*c - 4*a*c^2))*log((b*
e^2*x^2 + 2*b*d*e - 2*a*e^2 + 2*sqrt(1/2)*sqrt(e*x^2 + d)*((b^2 - 4*a*c)*e + (b^3*c - 4*a*b*c^2)*sqrt(e^2/(b^2
*c^2 - 4*a*c^3)))*sqrt((2*c*d - b*e + (b^2*c - 4*a*c^2)*sqrt(e^2/(b^2*c^2 - 4*a*c^3)))/(b^2*c - 4*a*c^2)) + ((
b^2*c - 4*a*c^2)*e*x^2 + 2*(b^2*c - 4*a*c^2)*d)*sqrt(e^2/(b^2*c^2 - 4*a*c^3)))/x^2) + 1/4*sqrt(1/2)*sqrt((2*c*
d - b*e + (b^2*c - 4*a*c^2)*sqrt(e^2/(b^2*c^2 - 4*a*c^3)))/(b^2*c - 4*a*c^2))*log((b*e^2*x^2 + 2*b*d*e - 2*a*e
^2 - 2*sqrt(1/2)*sqrt(e*x^2 + d)*((b^2 - 4*a*c)*e + (b^3*c - 4*a*b*c^2)*sqrt(e^2/(b^2*c^2 - 4*a*c^3)))*sqrt((2
*c*d - b*e + (b^2*c - 4*a*c^2)*sqrt(e^2/(b^2*c^2 - 4*a*c^3)))/(b^2*c - 4*a*c^2)) + ((b^2*c - 4*a*c^2)*e*x^2 +
2*(b^2*c - 4*a*c^2)*d)*sqrt(e^2/(b^2*c^2 - 4*a*c^3)))/x^2) - 1/4*sqrt(1/2)*sqrt((2*c*d - b*e - (b^2*c - 4*a*c^
2)*sqrt(e^2/(b^2*c^2 - 4*a*c^3)))/(b^2*c - 4*a*c^2))*log((b*e^2*x^2 + 2*b*d*e - 2*a*e^2 + 2*sqrt(1/2)*sqrt(e*x
^2 + d)*((b^2 - 4*a*c)*e - (b^3*c - 4*a*b*c^2)*sqrt(e^2/(b^2*c^2 - 4*a*c^3)))*sqrt((2*c*d - b*e - (b^2*c - 4*a
*c^2)*sqrt(e^2/(b^2*c^2 - 4*a*c^3)))/(b^2*c - 4*a*c^2)) - ((b^2*c - 4*a*c^2)*e*x^2 + 2*(b^2*c - 4*a*c^2)*d)*sq
rt(e^2/(b^2*c^2 - 4*a*c^3)))/x^2) + 1/4*sqrt(1/2)*sqrt((2*c*d - b*e - (b^2*c - 4*a*c^2)*sqrt(e^2/(b^2*c^2 - 4*
a*c^3)))/(b^2*c - 4*a*c^2))*log((b*e^2*x^2 + 2*b*d*e - 2*a*e^2 - 2*sqrt(1/2)*sqrt(e*x^2 + d)*((b^2 - 4*a*c)*e
- (b^3*c - 4*a*b*c^2)*sqrt(e^2/(b^2*c^2 - 4*a*c^3)))*sqrt((2*c*d - b*e - (b^2*c - 4*a*c^2)*sqrt(e^2/(b^2*c^2 -
4*a*c^3)))/(b^2*c - 4*a*c^2)) - ((b^2*c - 4*a*c^2)*e*x^2 + 2*(b^2*c - 4*a*c^2)*d)*sqrt(e^2/(b^2*c^2 - 4*a*c^3
)))/x^2)

________________________________________________________________________________________

giac [A]  time = 0.55, size = 228, normalized size = 1.13 \[ -\frac {\sqrt {-4 \, c^{2} d + 2 \, {\left (b c - \sqrt {b^{2} - 4 \, a c} c\right )} e} \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} \sqrt {x^{2} e + d}}{\sqrt {-\frac {2 \, c d - b e + \sqrt {{\left (2 \, c d - b e\right )}^{2} - 4 \, {\left (c d^{2} - b d e + a e^{2}\right )} c}}{c}}}\right )}{2 \, \sqrt {b^{2} - 4 \, a c} {\left | c \right |}} + \frac {\sqrt {-4 \, c^{2} d + 2 \, {\left (b c + \sqrt {b^{2} - 4 \, a c} c\right )} e} \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} \sqrt {x^{2} e + d}}{\sqrt {-\frac {2 \, c d - b e - \sqrt {{\left (2 \, c d - b e\right )}^{2} - 4 \, {\left (c d^{2} - b d e + a e^{2}\right )} c}}{c}}}\right )}{2 \, \sqrt {b^{2} - 4 \, a c} {\left | c \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*sqrt(-4*c^2*d + 2*(b*c - sqrt(b^2 - 4*a*c)*c)*e)*arctan(2*sqrt(1/2)*sqrt(x^2*e + d)/sqrt(-(2*c*d - b*e +
sqrt((2*c*d - b*e)^2 - 4*(c*d^2 - b*d*e + a*e^2)*c))/c))/(sqrt(b^2 - 4*a*c)*abs(c)) + 1/2*sqrt(-4*c^2*d + 2*(b
*c + sqrt(b^2 - 4*a*c)*c)*e)*arctan(2*sqrt(1/2)*sqrt(x^2*e + d)/sqrt(-(2*c*d - b*e - sqrt((2*c*d - b*e)^2 - 4*
(c*d^2 - b*d*e + a*e^2)*c))/c))/(sqrt(b^2 - 4*a*c)*abs(c))

________________________________________________________________________________________

maple [C]  time = 0.02, size = 177, normalized size = 0.88 \[ \frac {e \left (\RootOf \left (\textit {\_Z}^{8} c +\left (4 b e -4 c d \right ) \textit {\_Z}^{6}+c \,d^{4}+\left (16 a \,e^{2}-8 d e b +6 c \,d^{2}\right ) \textit {\_Z}^{4}+\left (4 b \,d^{2} e -4 c \,d^{3}\right ) \textit {\_Z}^{2}\right )^{6}+\RootOf \left (\textit {\_Z}^{8} c +\left (4 b e -4 c d \right ) \textit {\_Z}^{6}+c \,d^{4}+\left (16 a \,e^{2}-8 d e b +6 c \,d^{2}\right ) \textit {\_Z}^{4}+\left (4 b \,d^{2} e -4 c \,d^{3}\right ) \textit {\_Z}^{2}\right )^{4} d -\RootOf \left (\textit {\_Z}^{8} c +\left (4 b e -4 c d \right ) \textit {\_Z}^{6}+c \,d^{4}+\left (16 a \,e^{2}-8 d e b +6 c \,d^{2}\right ) \textit {\_Z}^{4}+\left (4 b \,d^{2} e -4 c \,d^{3}\right ) \textit {\_Z}^{2}\right )^{2} d^{2}-d^{3}\right ) \ln \left (-\sqrt {e}\, x -\RootOf \left (\textit {\_Z}^{8} c +\left (4 b e -4 c d \right ) \textit {\_Z}^{6}+c \,d^{4}+\left (16 a \,e^{2}-8 d e b +6 c \,d^{2}\right ) \textit {\_Z}^{4}+\left (4 b \,d^{2} e -4 c \,d^{3}\right ) \textit {\_Z}^{2}\right )+\sqrt {e \,x^{2}+d}\right )}{4 \RootOf \left (\textit {\_Z}^{8} c +\left (4 b e -4 c d \right ) \textit {\_Z}^{6}+c \,d^{4}+\left (16 a \,e^{2}-8 d e b +6 c \,d^{2}\right ) \textit {\_Z}^{4}+\left (4 b \,d^{2} e -4 c \,d^{3}\right ) \textit {\_Z}^{2}\right )^{7} c +12 \RootOf \left (\textit {\_Z}^{8} c +\left (4 b e -4 c d \right ) \textit {\_Z}^{6}+c \,d^{4}+\left (16 a \,e^{2}-8 d e b +6 c \,d^{2}\right ) \textit {\_Z}^{4}+\left (4 b \,d^{2} e -4 c \,d^{3}\right ) \textit {\_Z}^{2}\right )^{5} b e -12 \RootOf \left (\textit {\_Z}^{8} c +\left (4 b e -4 c d \right ) \textit {\_Z}^{6}+c \,d^{4}+\left (16 a \,e^{2}-8 d e b +6 c \,d^{2}\right ) \textit {\_Z}^{4}+\left (4 b \,d^{2} e -4 c \,d^{3}\right ) \textit {\_Z}^{2}\right )^{5} c d +32 \RootOf \left (\textit {\_Z}^{8} c +\left (4 b e -4 c d \right ) \textit {\_Z}^{6}+c \,d^{4}+\left (16 a \,e^{2}-8 d e b +6 c \,d^{2}\right ) \textit {\_Z}^{4}+\left (4 b \,d^{2} e -4 c \,d^{3}\right ) \textit {\_Z}^{2}\right )^{3} a \,e^{2}-16 \RootOf \left (\textit {\_Z}^{8} c +\left (4 b e -4 c d \right ) \textit {\_Z}^{6}+c \,d^{4}+\left (16 a \,e^{2}-8 d e b +6 c \,d^{2}\right ) \textit {\_Z}^{4}+\left (4 b \,d^{2} e -4 c \,d^{3}\right ) \textit {\_Z}^{2}\right )^{3} b d e +12 \RootOf \left (\textit {\_Z}^{8} c +\left (4 b e -4 c d \right ) \textit {\_Z}^{6}+c \,d^{4}+\left (16 a \,e^{2}-8 d e b +6 c \,d^{2}\right ) \textit {\_Z}^{4}+\left (4 b \,d^{2} e -4 c \,d^{3}\right ) \textit {\_Z}^{2}\right )^{3} c \,d^{2}+4 \RootOf \left (\textit {\_Z}^{8} c +\left (4 b e -4 c d \right ) \textit {\_Z}^{6}+c \,d^{4}+\left (16 a \,e^{2}-8 d e b +6 c \,d^{2}\right ) \textit {\_Z}^{4}+\left (4 b \,d^{2} e -4 c \,d^{3}\right ) \textit {\_Z}^{2}\right ) b \,d^{2} e -4 \RootOf \left (\textit {\_Z}^{8} c +\left (4 b e -4 c d \right ) \textit {\_Z}^{6}+c \,d^{4}+\left (16 a \,e^{2}-8 d e b +6 c \,d^{2}\right ) \textit {\_Z}^{4}+\left (4 b \,d^{2} e -4 c \,d^{3}\right ) \textit {\_Z}^{2}\right ) c \,d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*e*sum((_R^6+_R^4*d-_R^2*d^2-d^3)/(_R^7*c+3*_R^5*b*e-3*_R^5*c*d+8*_R^3*a*e^2-4*_R^3*b*d*e+3*_R^3*c*d^2+_R*b
*d^2*e-_R*c*d^3)*ln(-e^(1/2)*x-_R+(e*x^2+d)^(1/2)),_R=RootOf(_Z^8*c+(4*b*e-4*c*d)*_Z^6+c*d^4+(16*a*e^2-8*b*d*e
+6*c*d^2)*_Z^4+(4*b*d^2*e-4*c*d^3)*_Z^2))

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {e x^{2} + d} x}{c x^{4} + b x^{2} + a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(e*x^2 + d)*x/(c*x^4 + b*x^2 + a), x)

________________________________________________________________________________________

mupad [B]  time = 1.72, size = 717, normalized size = 3.55 \[ -2\,\mathrm {atanh}\left (\frac {2\,\left (\sqrt {e\,x^2+d}\,\left (-2\,b^2\,c\,e^4+4\,b\,c^2\,d\,e^3-4\,c^3\,d^2\,e^2+4\,a\,c^2\,e^4\right )+\frac {\sqrt {e\,x^2+d}\,\left (8\,b^3\,c^2\,e^3-16\,d\,b^2\,c^3\,e^2-32\,a\,b\,c^3\,e^3+64\,a\,d\,c^4\,e^2\right )\,\left (b^3\,e+e\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}+8\,a\,c^2\,d-2\,b^2\,c\,d-4\,a\,b\,c\,e\right )}{8\,\left (16\,a^2\,c^3-8\,a\,b^2\,c^2+b^4\,c\right )}\right )\,\sqrt {-\frac {b^3\,e+e\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}+8\,a\,c^2\,d-2\,b^2\,c\,d-4\,a\,b\,c\,e}{8\,\left (16\,a^2\,c^3-8\,a\,b^2\,c^2+b^4\,c\right )}}}{2\,c^2\,d^2\,e^3-2\,b\,c\,d\,e^4+2\,a\,c\,e^5}\right )\,\sqrt {-\frac {b^3\,e+e\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}+8\,a\,c^2\,d-2\,b^2\,c\,d-4\,a\,b\,c\,e}{8\,\left (16\,a^2\,c^3-8\,a\,b^2\,c^2+b^4\,c\right )}}-2\,\mathrm {atanh}\left (\frac {2\,\left (\sqrt {e\,x^2+d}\,\left (-2\,b^2\,c\,e^4+4\,b\,c^2\,d\,e^3-4\,c^3\,d^2\,e^2+4\,a\,c^2\,e^4\right )-\frac {\sqrt {e\,x^2+d}\,\left (8\,b^3\,c^2\,e^3-16\,d\,b^2\,c^3\,e^2-32\,a\,b\,c^3\,e^3+64\,a\,d\,c^4\,e^2\right )\,\left (e\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}-b^3\,e-8\,a\,c^2\,d+2\,b^2\,c\,d+4\,a\,b\,c\,e\right )}{8\,\left (16\,a^2\,c^3-8\,a\,b^2\,c^2+b^4\,c\right )}\right )\,\sqrt {\frac {e\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}-b^3\,e-8\,a\,c^2\,d+2\,b^2\,c\,d+4\,a\,b\,c\,e}{8\,\left (16\,a^2\,c^3-8\,a\,b^2\,c^2+b^4\,c\right )}}}{2\,c^2\,d^2\,e^3-2\,b\,c\,d\,e^4+2\,a\,c\,e^5}\right )\,\sqrt {\frac {e\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}-b^3\,e-8\,a\,c^2\,d+2\,b^2\,c\,d+4\,a\,b\,c\,e}{8\,\left (16\,a^2\,c^3-8\,a\,b^2\,c^2+b^4\,c\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

- 2*atanh((2*((d + e*x^2)^(1/2)*(4*a*c^2*e^4 - 2*b^2*c*e^4 - 4*c^3*d^2*e^2 + 4*b*c^2*d*e^3) + ((d + e*x^2)^(1/
2)*(8*b^3*c^2*e^3 - 16*b^2*c^3*d*e^2 - 32*a*b*c^3*e^3 + 64*a*c^4*d*e^2)*(b^3*e + e*(-(4*a*c - b^2)^3)^(1/2) +
8*a*c^2*d - 2*b^2*c*d - 4*a*b*c*e))/(8*(b^4*c + 16*a^2*c^3 - 8*a*b^2*c^2)))*(-(b^3*e + e*(-(4*a*c - b^2)^3)^(1
/2) + 8*a*c^2*d - 2*b^2*c*d - 4*a*b*c*e)/(8*(b^4*c + 16*a^2*c^3 - 8*a*b^2*c^2)))^(1/2))/(2*c^2*d^2*e^3 + 2*a*c
*e^5 - 2*b*c*d*e^4))*(-(b^3*e + e*(-(4*a*c - b^2)^3)^(1/2) + 8*a*c^2*d - 2*b^2*c*d - 4*a*b*c*e)/(8*(b^4*c + 16
*a^2*c^3 - 8*a*b^2*c^2)))^(1/2) - 2*atanh((2*((d + e*x^2)^(1/2)*(4*a*c^2*e^4 - 2*b^2*c*e^4 - 4*c^3*d^2*e^2 + 4
*b*c^2*d*e^3) - ((d + e*x^2)^(1/2)*(8*b^3*c^2*e^3 - 16*b^2*c^3*d*e^2 - 32*a*b*c^3*e^3 + 64*a*c^4*d*e^2)*(e*(-(
4*a*c - b^2)^3)^(1/2) - b^3*e - 8*a*c^2*d + 2*b^2*c*d + 4*a*b*c*e))/(8*(b^4*c + 16*a^2*c^3 - 8*a*b^2*c^2)))*((
e*(-(4*a*c - b^2)^3)^(1/2) - b^3*e - 8*a*c^2*d + 2*b^2*c*d + 4*a*b*c*e)/(8*(b^4*c + 16*a^2*c^3 - 8*a*b^2*c^2))
)^(1/2))/(2*c^2*d^2*e^3 + 2*a*c*e^5 - 2*b*c*d*e^4))*((e*(-(4*a*c - b^2)^3)^(1/2) - b^3*e - 8*a*c^2*d + 2*b^2*c
*d + 4*a*b*c*e)/(8*(b^4*c + 16*a^2*c^3 - 8*a*b^2*c^2)))^(1/2)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x \sqrt {d + e x^{2}}}{a + b x^{2} + c x^{4}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x*sqrt(d + e*x**2)/(a + b*x**2 + c*x**4), x)

________________________________________________________________________________________