3.4.57 \(\int \frac {x \sqrt {d+e x^2}}{a+b x^2+c x^4} \, dx\) [357]

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

[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)

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Rubi [A]
time = 0.24, 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 = {1261, 713, 1144, 214} \begin {gather*} \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}} \end {gather*}

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

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Mathematica [C] Result contains complex when optimal does not.
time = 0.57, size = 256, normalized size = 1.27 \begin {gather*} \frac {\frac {\left (-2 i c d+\left (i b+\sqrt {-b^2+4 a c}\right ) e\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x^2}}{\sqrt {-2 c d+b e-i \sqrt {-b^2+4 a c} e}}\right )}{\sqrt {-2 c d+\left (b-i \sqrt {-b^2+4 a c}\right ) e}}+\frac {\left (2 i c d+\left (-i b+\sqrt {-b^2+4 a c}\right ) e\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x^2}}{\sqrt {-2 c d+b e+i \sqrt {-b^2+4 a c} e}}\right )}{\sqrt {-2 c d+\left (b+i \sqrt {-b^2+4 a c}\right ) e}}}{\sqrt {2} \sqrt {c} \sqrt {-b^2+4 a c}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((((-2*I)*c*d + (I*b + Sqrt[-b^2 + 4*a*c])*e)*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x^2])/Sqrt[-2*c*d + b*e - I*S
qrt[-b^2 + 4*a*c]*e]])/Sqrt[-2*c*d + (b - I*Sqrt[-b^2 + 4*a*c])*e] + (((2*I)*c*d + ((-I)*b + Sqrt[-b^2 + 4*a*c
])*e)*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x^2])/Sqrt[-2*c*d + b*e + I*Sqrt[-b^2 + 4*a*c]*e]])/Sqrt[-2*c*d + (b
+ I*Sqrt[-b^2 + 4*a*c])*e])/(Sqrt[2]*Sqrt[c]*Sqrt[-b^2 + 4*a*c])

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.12, size = 177, normalized size = 0.88

method	result	size
default	\(-\frac {e \left (\munderset {\textit {\_R} =\RootOf \left (c \,\textit {\_Z}^{8}+\left (4 e b -4 c d \right ) \textit {\_Z}^{6}+\left (16 a \,e^{2}-8 d e b +6 c \,d^{2}\right ) \textit {\_Z}^{4}+\left (4 d^{2} e b -4 c \,d^{3}\right ) \textit {\_Z}^{2}+d^{4} c \right )}{\sum }\frac {\left (-\textit {\_R}^{6}-\textit {\_R}^{4} d +\textit {\_R}^{2} d^{2}+d^{3}\right ) \ln \left (\sqrt {e \,x^{2}+d}-\sqrt {e}\, x -\textit {\_R} \right )}{\textit {\_R}^{7} c +3 \textit {\_R}^{5} b e -3 \textit {\_R}^{5} c d +8 \textit {\_R}^{3} a \,e^{2}-4 \textit {\_R}^{3} b d e +3 \textit {\_R}^{3} c \,d^{2}+\textit {\_R} b \,d^{2} e -\textit {\_R} c \,d^{3}}\right )}{4}\)	\(177\)

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,method=_RETURNVERBOSE)

[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*x^2+d)^(1/2)-e^(1/2)*x-_R),_R=RootOf(c*_Z^8+(4*b*e-4*c*d)*_Z^6+(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+d^4*c))

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

[Out]

integrate(sqrt(x^2*e + d)*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. 1037 vs. \(2 (169) = 338\).
time = 6.19, size = 1037, normalized size = 5.13 \begin {gather*} -\frac {1}{4} \, \sqrt {\frac {1}{2}} \sqrt {\frac {2 \, c d - b e + \frac {{\left (b^{2} c - 4 \, a c^{2}\right )} e}{\sqrt {b^{2} c^{2} - 4 \, a c^{3}}}}{b^{2} c - 4 \, a c^{2}}} \log \left (\frac {2 \, b d e + 2 \, \sqrt {\frac {1}{2}} \sqrt {x^{2} e + d} {\left ({\left (b^{2} - 4 \, a c\right )} e + \frac {{\left (b^{3} c - 4 \, a b c^{2}\right )} e}{\sqrt {b^{2} c^{2} - 4 \, a c^{3}}}\right )} \sqrt {\frac {2 \, c d - b e + \frac {{\left (b^{2} c - 4 \, a c^{2}\right )} e}{\sqrt {b^{2} c^{2} - 4 \, a c^{3}}}}{b^{2} c - 4 \, a c^{2}}} + {\left (b x^{2} - 2 \, a\right )} e^{2} + \frac {{\left ({\left (b^{2} c - 4 \, a c^{2}\right )} x^{2} e + 2 \, {\left (b^{2} c - 4 \, a c^{2}\right )} d\right )} e}{\sqrt {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 + \frac {{\left (b^{2} c - 4 \, a c^{2}\right )} e}{\sqrt {b^{2} c^{2} - 4 \, a c^{3}}}}{b^{2} c - 4 \, a c^{2}}} \log \left (\frac {2 \, b d e - 2 \, \sqrt {\frac {1}{2}} \sqrt {x^{2} e + d} {\left ({\left (b^{2} - 4 \, a c\right )} e + \frac {{\left (b^{3} c - 4 \, a b c^{2}\right )} e}{\sqrt {b^{2} c^{2} - 4 \, a c^{3}}}\right )} \sqrt {\frac {2 \, c d - b e + \frac {{\left (b^{2} c - 4 \, a c^{2}\right )} e}{\sqrt {b^{2} c^{2} - 4 \, a c^{3}}}}{b^{2} c - 4 \, a c^{2}}} + {\left (b x^{2} - 2 \, a\right )} e^{2} + \frac {{\left ({\left (b^{2} c - 4 \, a c^{2}\right )} x^{2} e + 2 \, {\left (b^{2} c - 4 \, a c^{2}\right )} d\right )} e}{\sqrt {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 - \frac {{\left (b^{2} c - 4 \, a c^{2}\right )} e}{\sqrt {b^{2} c^{2} - 4 \, a c^{3}}}}{b^{2} c - 4 \, a c^{2}}} \log \left (\frac {2 \, b d e + 2 \, \sqrt {\frac {1}{2}} \sqrt {x^{2} e + d} {\left ({\left (b^{2} - 4 \, a c\right )} e - \frac {{\left (b^{3} c - 4 \, a b c^{2}\right )} e}{\sqrt {b^{2} c^{2} - 4 \, a c^{3}}}\right )} \sqrt {\frac {2 \, c d - b e - \frac {{\left (b^{2} c - 4 \, a c^{2}\right )} e}{\sqrt {b^{2} c^{2} - 4 \, a c^{3}}}}{b^{2} c - 4 \, a c^{2}}} + {\left (b x^{2} - 2 \, a\right )} e^{2} - \frac {{\left ({\left (b^{2} c - 4 \, a c^{2}\right )} x^{2} e + 2 \, {\left (b^{2} c - 4 \, a c^{2}\right )} d\right )} e}{\sqrt {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 - \frac {{\left (b^{2} c - 4 \, a c^{2}\right )} e}{\sqrt {b^{2} c^{2} - 4 \, a c^{3}}}}{b^{2} c - 4 \, a c^{2}}} \log \left (\frac {2 \, b d e - 2 \, \sqrt {\frac {1}{2}} \sqrt {x^{2} e + d} {\left ({\left (b^{2} - 4 \, a c\right )} e - \frac {{\left (b^{3} c - 4 \, a b c^{2}\right )} e}{\sqrt {b^{2} c^{2} - 4 \, a c^{3}}}\right )} \sqrt {\frac {2 \, c d - b e - \frac {{\left (b^{2} c - 4 \, a c^{2}\right )} e}{\sqrt {b^{2} c^{2} - 4 \, a c^{3}}}}{b^{2} c - 4 \, a c^{2}}} + {\left (b x^{2} - 2 \, a\right )} e^{2} - \frac {{\left ({\left (b^{2} c - 4 \, a c^{2}\right )} x^{2} e + 2 \, {\left (b^{2} c - 4 \, a c^{2}\right )} d\right )} e}{\sqrt {b^{2} c^{2} - 4 \, a c^{3}}}}{x^{2}}\right ) \end {gather*}

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

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

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)

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Giac [A]
time = 3.96, size = 228, normalized size = 1.13 \begin {gather*} -\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 |}} \end {gather*}

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))

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Mupad [B]
time = 1.72, size = 717, normalized size = 3.55 \begin {gather*} -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 )}} \end {gather*}

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)

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