3.4.65 \(\int \frac {\sqrt {d+e x^2}}{x^4 (a+b x^2+c x^4)} \, dx\) [365]

Optimal. Leaf size=373 \[ -\frac {\sqrt {d+e x^2}}{3 a x^3}+\frac {2 e \sqrt {d+e x^2}}{3 a d x}+\frac {(b d-a e) \sqrt {d+e x^2}}{a^2 d x}+\frac {c \left (b d-a e+\frac {b^2 d-2 a c d-a b e}{\sqrt {b^2-4 a c}}\right ) \tan ^{-1}\left (\frac {\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e} x}{\sqrt {b-\sqrt {b^2-4 a c}} \sqrt {d+e x^2}}\right )}{a^2 \sqrt {b-\sqrt {b^2-4 a c}} \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}+\frac {c \left (b d-a e-\frac {b^2 d-2 a c d-a b e}{\sqrt {b^2-4 a c}}\right ) \tan ^{-1}\left (\frac {\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e} x}{\sqrt {b+\sqrt {b^2-4 a c}} \sqrt {d+e x^2}}\right )}{a^2 \sqrt {b+\sqrt {b^2-4 a c}} \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \]

[Out]

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

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Rubi [A]
time = 1.71, antiderivative size = 373, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 7, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {1309, 277, 270, 6860, 1706, 385, 211} \begin {gather*} \frac {c \left (\frac {-a b e-2 a c d+b^2 d}{\sqrt {b^2-4 a c}}-a e+b d\right ) \text {ArcTan}\left (\frac {x \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}{\sqrt {b-\sqrt {b^2-4 a c}} \sqrt {d+e x^2}}\right )}{a^2 \sqrt {b-\sqrt {b^2-4 a c}} \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}+\frac {c \left (-\frac {-a b e-2 a c d+b^2 d}{\sqrt {b^2-4 a c}}-a e+b d\right ) \text {ArcTan}\left (\frac {x \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}{\sqrt {\sqrt {b^2-4 a c}+b} \sqrt {d+e x^2}}\right )}{a^2 \sqrt {\sqrt {b^2-4 a c}+b} \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}+\frac {\sqrt {d+e x^2} (b d-a e)}{a^2 d x}+\frac {2 e \sqrt {d+e x^2}}{3 a d x}-\frac {\sqrt {d+e x^2}}{3 a x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/3*Sqrt[d + e*x^2]/(a*x^3) + (2*e*Sqrt[d + e*x^2])/(3*a*d*x) + ((b*d - a*e)*Sqrt[d + e*x^2])/(a^2*d*x) + (c*
(b*d - a*e + (b^2*d - 2*a*c*d - a*b*e)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e]*x)/(
Sqrt[b - Sqrt[b^2 - 4*a*c]]*Sqrt[d + e*x^2])])/(a^2*Sqrt[b - Sqrt[b^2 - 4*a*c]]*Sqrt[2*c*d - (b - Sqrt[b^2 - 4
*a*c])*e]) + (c*(b*d - a*e - (b^2*d - 2*a*c*d - a*b*e)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2*c*d - (b + Sqrt[b^2 -
4*a*c])*e]*x)/(Sqrt[b + Sqrt[b^2 - 4*a*c]]*Sqrt[d + e*x^2])])/(a^2*Sqrt[b + Sqrt[b^2 - 4*a*c]]*Sqrt[2*c*d - (
b + Sqrt[b^2 - 4*a*c])*e])

Rule 211

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/Rt[a/b, 2]], x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*
c*(m + 1))), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

Rule 277

Int[(x_)^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x^(m + 1)*((a + b*x^n)^(p + 1)/(a*(m + 1))), x]
- Dist[b*((m + n*(p + 1) + 1)/(a*(m + 1))), Int[x^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, m, n, p}, x]
&& ILtQ[Simplify[(m + 1)/n + p + 1], 0] && NeQ[m, -1]

Rule 385

Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Subst[Int[1/(c - (b*c - a*d)*x^n), x]
, x, x/(a + b*x^n)^(1/n)] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[n*p + 1, 0] && IntegerQ[n]

Rule 1309

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

Rule 1706

Int[(Px_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandInteg
rand[Px*(d + e*x^2)^q*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, e, q}, x] && PolyQ[Px, x^2] && NeQ[b
^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[p]

Rule 6860

Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a +
b*x^n + c*x^(2*n)), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]

Rubi steps

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

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(7777\) vs. \(2(373)=746\).
time = 16.31, size = 7777, normalized size = 20.85 \begin {gather*} \text {Result too large to show} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Result too large to show

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.15, size = 318, normalized size = 0.85

method	result	size
risch	\(-\frac {\sqrt {e \,x^{2}+d}\, \left (a e \,x^{2}-3 b d \,x^{2}+a d \right )}{3 a^{2} x^{3} d}+\frac {\sqrt {e}\, \left (\munderset {\textit {\_R} =\RootOf \left (c \,\textit {\_Z}^{4}+\left (4 e b -4 c d \right ) \textit {\_Z}^{3}+\left (16 a \,e^{2}-8 d e b +6 c \,d^{2}\right ) \textit {\_Z}^{2}+\left (4 d^{2} e b -4 c \,d^{3}\right ) \textit {\_Z} +d^{4} c \right )}{\sum }\frac {\left (c \left (a e -b d \right ) \textit {\_R}^{2}+2 \left (2 a b \,e^{2}+a c d e -2 b^{2} d e +b c \,d^{2}\right ) \textit {\_R} +a \,d^{2} e c -b c \,d^{3}\right ) \ln \left (\left (\sqrt {e \,x^{2}+d}-\sqrt {e}\, x \right )^{2}-\textit {\_R} \right )}{c \,\textit {\_R}^{3}+3 \textit {\_R}^{2} b e -3 \textit {\_R}^{2} c d +8 \textit {\_R} a \,e^{2}-4 \textit {\_R} b d e +3 c \,d^{2} \textit {\_R} +d^{2} e b -c \,d^{3}}\right )}{2 a^{2}}\)	\(248\)
default	\(\frac {\sqrt {e}\, \left (-b \ln \left (\sqrt {e \,x^{2}+d}-\sqrt {e}\, x \right )+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (c \,\textit {\_Z}^{4}+\left (4 e b -4 c d \right ) \textit {\_Z}^{3}+\left (16 a \,e^{2}-8 d e b +6 c \,d^{2}\right ) \textit {\_Z}^{2}+\left (4 d^{2} e b -4 c \,d^{3}\right ) \textit {\_Z} +d^{4} c \right )}{\sum }\frac {\left (c \left (a e -b d \right ) \textit {\_R}^{2}+2 \left (2 a b \,e^{2}+a c d e -2 b^{2} d e +b c \,d^{2}\right ) \textit {\_R} +a \,d^{2} e c -b c \,d^{3}\right ) \ln \left (\left (\sqrt {e \,x^{2}+d}-\sqrt {e}\, x \right )^{2}-\textit {\_R} \right )}{c \,\textit {\_R}^{3}+3 \textit {\_R}^{2} b e -3 \textit {\_R}^{2} c d +8 \textit {\_R} a \,e^{2}-4 \textit {\_R} b d e +3 c \,d^{2} \textit {\_R} +d^{2} e b -c \,d^{3}}\right )}{2}\right )}{a^{2}}-\frac {b \left (-\frac {\left (e \,x^{2}+d \right )^{\frac {3}{2}}}{d x}+\frac {2 e \left (\frac {x \sqrt {e \,x^{2}+d}}{2}+\frac {d \ln \left (\sqrt {e}\, x +\sqrt {e \,x^{2}+d}\right )}{2 \sqrt {e}}\right )}{d}\right )}{a^{2}}-\frac {\left (e \,x^{2}+d \right )^{\frac {3}{2}}}{3 a d \,x^{3}}\)	\(318\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 4132 vs. \(2 (338) = 676\).
time = 6.47, size = 4132, normalized size = 11.08 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/12*(3*sqrt(1/2)*a^2*d*x^3*sqrt(-((b^5 - 5*a*b^3*c + 5*a^2*b*c^2)*d - (a*b^4 - 4*a^2*b^2*c + 2*a^3*c^2)*e +
(a^5*b^2 - 4*a^6*c)*sqrt(((b^8 - 6*a*b^6*c + 11*a^2*b^4*c^2 - 6*a^3*b^2*c^3 + a^4*c^4)*d^2 - 2*(a*b^7 - 5*a^2*
b^5*c + 7*a^3*b^3*c^2 - 2*a^4*b*c^3)*d*e + (a^2*b^6 - 4*a^3*b^4*c + 4*a^4*b^2*c^2)*e^2)/(a^10*b^2 - 4*a^11*c))
)/(a^5*b^2 - 4*a^6*c))*log(-((b^5*c^2 - 3*a*b^3*c^3 + a^2*b*c^4)*d^2*x^2 + (a^5*b^2*c^2 - 4*a^6*c^3)*d*x^2*sqr
t(((b^8 - 6*a*b^6*c + 11*a^2*b^4*c^2 - 6*a^3*b^2*c^3 + a^4*c^4)*d^2 - 2*(a*b^7 - 5*a^2*b^5*c + 7*a^3*b^3*c^2 -
2*a^4*b*c^3)*d*e + (a^2*b^6 - 4*a^3*b^4*c + 4*a^4*b^2*c^2)*e^2)/(a^10*b^2 - 4*a^11*c)) + 4*(a^2*b^3*c^2 - 2*a
^3*b*c^3)*x^2*e^2 - 2*(a*b^4*c^2 - 3*a^2*b^2*c^3 + a^3*c^4)*d^2 + 2*sqrt(1/2)*((a*b^7 - 7*a^2*b^5*c + 13*a^3*b
^3*c^2 - 4*a^4*b*c^3)*d*x - (a^2*b^6 - 6*a^3*b^4*c + 8*a^4*b^2*c^2)*x*e - (a^6*b^4 - 6*a^7*b^2*c + 8*a^8*c^2)*
x*sqrt(((b^8 - 6*a*b^6*c + 11*a^2*b^4*c^2 - 6*a^3*b^2*c^3 + a^4*c^4)*d^2 - 2*(a*b^7 - 5*a^2*b^5*c + 7*a^3*b^3*
c^2 - 2*a^4*b*c^3)*d*e + (a^2*b^6 - 4*a^3*b^4*c + 4*a^4*b^2*c^2)*e^2)/(a^10*b^2 - 4*a^11*c)))*sqrt(x^2*e + d)*
sqrt(-((b^5 - 5*a*b^3*c + 5*a^2*b*c^2)*d - (a*b^4 - 4*a^2*b^2*c + 2*a^3*c^2)*e + (a^5*b^2 - 4*a^6*c)*sqrt(((b^
8 - 6*a*b^6*c + 11*a^2*b^4*c^2 - 6*a^3*b^2*c^3 + a^4*c^4)*d^2 - 2*(a*b^7 - 5*a^2*b^5*c + 7*a^3*b^3*c^2 - 2*a^4
*b*c^3)*d*e + (a^2*b^6 - 4*a^3*b^4*c + 4*a^4*b^2*c^2)*e^2)/(a^10*b^2 - 4*a^11*c)))/(a^5*b^2 - 4*a^6*c)) - ((5*
a*b^4*c^2 - 14*a^2*b^2*c^3 + 4*a^3*c^4)*d*x^2 - 2*(a^2*b^3*c^2 - 2*a^3*b*c^3)*d)*e)/x^2) - 3*sqrt(1/2)*a^2*d*x
^3*sqrt(-((b^5 - 5*a*b^3*c + 5*a^2*b*c^2)*d - (a*b^4 - 4*a^2*b^2*c + 2*a^3*c^2)*e + (a^5*b^2 - 4*a^6*c)*sqrt((
(b^8 - 6*a*b^6*c + 11*a^2*b^4*c^2 - 6*a^3*b^2*c^3 + a^4*c^4)*d^2 - 2*(a*b^7 - 5*a^2*b^5*c + 7*a^3*b^3*c^2 - 2*
a^4*b*c^3)*d*e + (a^2*b^6 - 4*a^3*b^4*c + 4*a^4*b^2*c^2)*e^2)/(a^10*b^2 - 4*a^11*c)))/(a^5*b^2 - 4*a^6*c))*log
(-((b^5*c^2 - 3*a*b^3*c^3 + a^2*b*c^4)*d^2*x^2 + (a^5*b^2*c^2 - 4*a^6*c^3)*d*x^2*sqrt(((b^8 - 6*a*b^6*c + 11*a
^2*b^4*c^2 - 6*a^3*b^2*c^3 + a^4*c^4)*d^2 - 2*(a*b^7 - 5*a^2*b^5*c + 7*a^3*b^3*c^2 - 2*a^4*b*c^3)*d*e + (a^2*b
^6 - 4*a^3*b^4*c + 4*a^4*b^2*c^2)*e^2)/(a^10*b^2 - 4*a^11*c)) + 4*(a^2*b^3*c^2 - 2*a^3*b*c^3)*x^2*e^2 - 2*(a*b
^4*c^2 - 3*a^2*b^2*c^3 + a^3*c^4)*d^2 - 2*sqrt(1/2)*((a*b^7 - 7*a^2*b^5*c + 13*a^3*b^3*c^2 - 4*a^4*b*c^3)*d*x
- (a^2*b^6 - 6*a^3*b^4*c + 8*a^4*b^2*c^2)*x*e - (a^6*b^4 - 6*a^7*b^2*c + 8*a^8*c^2)*x*sqrt(((b^8 - 6*a*b^6*c +
11*a^2*b^4*c^2 - 6*a^3*b^2*c^3 + a^4*c^4)*d^2 - 2*(a*b^7 - 5*a^2*b^5*c + 7*a^3*b^3*c^2 - 2*a^4*b*c^3)*d*e + (
a^2*b^6 - 4*a^3*b^4*c + 4*a^4*b^2*c^2)*e^2)/(a^10*b^2 - 4*a^11*c)))*sqrt(x^2*e + d)*sqrt(-((b^5 - 5*a*b^3*c +
5*a^2*b*c^2)*d - (a*b^4 - 4*a^2*b^2*c + 2*a^3*c^2)*e + (a^5*b^2 - 4*a^6*c)*sqrt(((b^8 - 6*a*b^6*c + 11*a^2*b^4
*c^2 - 6*a^3*b^2*c^3 + a^4*c^4)*d^2 - 2*(a*b^7 - 5*a^2*b^5*c + 7*a^3*b^3*c^2 - 2*a^4*b*c^3)*d*e + (a^2*b^6 - 4
*a^3*b^4*c + 4*a^4*b^2*c^2)*e^2)/(a^10*b^2 - 4*a^11*c)))/(a^5*b^2 - 4*a^6*c)) - ((5*a*b^4*c^2 - 14*a^2*b^2*c^3
+ 4*a^3*c^4)*d*x^2 - 2*(a^2*b^3*c^2 - 2*a^3*b*c^3)*d)*e)/x^2) + 3*sqrt(1/2)*a^2*d*x^3*sqrt(-((b^5 - 5*a*b^3*c
+ 5*a^2*b*c^2)*d - (a*b^4 - 4*a^2*b^2*c + 2*a^3*c^2)*e - (a^5*b^2 - 4*a^6*c)*sqrt(((b^8 - 6*a*b^6*c + 11*a^2*
b^4*c^2 - 6*a^3*b^2*c^3 + a^4*c^4)*d^2 - 2*(a*b^7 - 5*a^2*b^5*c + 7*a^3*b^3*c^2 - 2*a^4*b*c^3)*d*e + (a^2*b^6
- 4*a^3*b^4*c + 4*a^4*b^2*c^2)*e^2)/(a^10*b^2 - 4*a^11*c)))/(a^5*b^2 - 4*a^6*c))*log(-((b^5*c^2 - 3*a*b^3*c^3
+ a^2*b*c^4)*d^2*x^2 - (a^5*b^2*c^2 - 4*a^6*c^3)*d*x^2*sqrt(((b^8 - 6*a*b^6*c + 11*a^2*b^4*c^2 - 6*a^3*b^2*c^3
+ a^4*c^4)*d^2 - 2*(a*b^7 - 5*a^2*b^5*c + 7*a^3*b^3*c^2 - 2*a^4*b*c^3)*d*e + (a^2*b^6 - 4*a^3*b^4*c + 4*a^4*b
^2*c^2)*e^2)/(a^10*b^2 - 4*a^11*c)) + 4*(a^2*b^3*c^2 - 2*a^3*b*c^3)*x^2*e^2 - 2*(a*b^4*c^2 - 3*a^2*b^2*c^3 + a
^3*c^4)*d^2 + 2*sqrt(1/2)*((a*b^7 - 7*a^2*b^5*c + 13*a^3*b^3*c^2 - 4*a^4*b*c^3)*d*x - (a^2*b^6 - 6*a^3*b^4*c +
8*a^4*b^2*c^2)*x*e + (a^6*b^4 - 6*a^7*b^2*c + 8*a^8*c^2)*x*sqrt(((b^8 - 6*a*b^6*c + 11*a^2*b^4*c^2 - 6*a^3*b^
2*c^3 + a^4*c^4)*d^2 - 2*(a*b^7 - 5*a^2*b^5*c + 7*a^3*b^3*c^2 - 2*a^4*b*c^3)*d*e + (a^2*b^6 - 4*a^3*b^4*c + 4*
a^4*b^2*c^2)*e^2)/(a^10*b^2 - 4*a^11*c)))*sqrt(x^2*e + d)*sqrt(-((b^5 - 5*a*b^3*c + 5*a^2*b*c^2)*d - (a*b^4 -
4*a^2*b^2*c + 2*a^3*c^2)*e - (a^5*b^2 - 4*a^6*c)*sqrt(((b^8 - 6*a*b^6*c + 11*a^2*b^4*c^2 - 6*a^3*b^2*c^3 + a^4
*c^4)*d^2 - 2*(a*b^7 - 5*a^2*b^5*c + 7*a^3*b^3*c^2 - 2*a^4*b*c^3)*d*e + (a^2*b^6 - 4*a^3*b^4*c + 4*a^4*b^2*c^2
)*e^2)/(a^10*b^2 - 4*a^11*c)))/(a^5*b^2 - 4*a^6*c)) - ((5*a*b^4*c^2 - 14*a^2*b^2*c^3 + 4*a^3*c^4)*d*x^2 - 2*(a
^2*b^3*c^2 - 2*a^3*b*c^3)*d)*e)/x^2) - 3*sqrt(1/2)*a^2*d*x^3*sqrt(-((b^5 - 5*a*b^3*c + 5*a^2*b*c^2)*d - (a*b^4
- 4*a^2*b^2*c + 2*a^3*c^2)*e - (a^5*b^2 - 4*a^6*c)*sqrt(((b^8 - 6*a*b^6*c + 11*a^2*b^4*c^2 - 6*a^3*b^2*c^3 +
a^4*c^4)*d^2 - 2*(a*b^7 - 5*a^2*b^5*c + 7*a^3*b^3*c^2 - 2*a^4*b*c^3)*d*e + (a^2*b^6 - 4*a^3*b^4*c + 4*a^4*b^2*
c^2)*e^2)/(a^10*b^2 - 4*a^11*c)))/(a^5*b^2 - 4*a^6*c))*log(-((b^5*c^2 - 3*a*b^3*c^3 + a^2*b*c^4)*d^2*x^2 - (a^
5*b^2*c^2 - 4*a^6*c^3)*d*x^2*sqrt(((b^8 - 6*a*b...

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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