3.4.66 \(\int \frac {\sqrt {d+e x^2}}{x^6 (a+b x^2+c x^4)} \, dx\) [366]

Optimal. Leaf size=512 \[ -\frac {\sqrt {d+e x^2}}{5 a x^5}+\frac {4 e \sqrt {d+e x^2}}{15 a d x^3}+\frac {(b d-a e) \sqrt {d+e x^2}}{3 a^2 d x^3}-\frac {8 e^2 \sqrt {d+e x^2}}{15 a d^2 x}-\frac {2 e (b d-a e) \sqrt {d+e x^2}}{3 a^2 d^2 x}-\frac {\left (b^2 d-a c d-a b e\right ) \sqrt {d+e x^2}}{a^3 d x}-\frac {c \left (b^2 d-a c d-a b e+\frac {b^3 d-3 a b c d-a b^2 e+2 a^2 c e}{\sqrt {b^2-4 a c}}\right ) \tan ^{-1}\left (\frac {\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e} x}{\sqrt {b-\sqrt {b^2-4 a c}} \sqrt {d+e x^2}}\right )}{a^3 \sqrt {b-\sqrt {b^2-4 a c}} \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}-\frac {c \left (b^2 d-a c d-a b e-\frac {b^3 d-3 a b c d-a b^2 e+2 a^2 c e}{\sqrt {b^2-4 a c}}\right ) \tan ^{-1}\left (\frac {\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e} x}{\sqrt {b+\sqrt {b^2-4 a c}} \sqrt {d+e x^2}}\right )}{a^3 \sqrt {b+\sqrt {b^2-4 a c}} \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \]

[Out]

-1/5*(e*x^2+d)^(1/2)/a/x^5+4/15*e*(e*x^2+d)^(1/2)/a/d/x^3+1/3*(-a*e+b*d)*(e*x^2+d)^(1/2)/a^2/d/x^3-8/15*e^2*(e
*x^2+d)^(1/2)/a/d^2/x-2/3*e*(-a*e+b*d)*(e*x^2+d)^(1/2)/a^2/d^2/x-(-a*b*e-a*c*d+b^2*d)*(e*x^2+d)^(1/2)/a^3/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^2*d-a*c*d-a
*b*e+(2*a^2*c*e-a*b^2*e-3*a*b*c*d+b^3*d)/(-4*a*c+b^2)^(1/2))/a^3/(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^2*d-a*c*d-a*b*e+(-2*a^2*c*e+a*b^2*e+3*a*b*c*d-b^3*d)/(-4*a*c+b^2)^(1/2))/a^3/(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 = 3.50, antiderivative size = 512, normalized size of antiderivative = 1.00, number of steps used = 15, 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 {\sqrt {d+e x^2} \left (-a b e-a c d+b^2 d\right )}{a^3 d x}-\frac {2 e \sqrt {d+e x^2} (b d-a e)}{3 a^2 d^2 x}+\frac {\sqrt {d+e x^2} (b d-a e)}{3 a^2 d x^3}-\frac {c \left (\frac {2 a^2 c e-a b^2 e-3 a b c d+b^3 d}{\sqrt {b^2-4 a c}}-a b e-a c d+b^2 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^3 \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 {2 a^2 c e-a b^2 e-3 a b c d+b^3 d}{\sqrt {b^2-4 a c}}-a b e-a c d+b^2 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^3 \sqrt {\sqrt {b^2-4 a c}+b} \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}-\frac {8 e^2 \sqrt {d+e x^2}}{15 a d^2 x}-\frac {\sqrt {d+e x^2}}{5 a x^5}+\frac {4 e \sqrt {d+e x^2}}{15 a d x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[d + e*x^2]/(x^6*(a + b*x^2 + c*x^4)),x]

[Out]

-1/5*Sqrt[d + e*x^2]/(a*x^5) + (4*e*Sqrt[d + e*x^2])/(15*a*d*x^3) + ((b*d - a*e)*Sqrt[d + e*x^2])/(3*a^2*d*x^3
) - (8*e^2*Sqrt[d + e*x^2])/(15*a*d^2*x) - (2*e*(b*d - a*e)*Sqrt[d + e*x^2])/(3*a^2*d^2*x) - ((b^2*d - a*c*d -
a*b*e)*Sqrt[d + e*x^2])/(a^3*d*x) - (c*(b^2*d - a*c*d - a*b*e + (b^3*d - 3*a*b*c*d - a*b^2*e + 2*a^2*c*e)/Sqr
t[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^3*Sqrt[b - Sqrt[b^2 - 4*a*c]]*Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e]) - (c*(b^2*d - a*c*d - a*b*e - (
b^3*d - 3*a*b*c*d - a*b^2*e + 2*a^2*c*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^3*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^6 \left (a+b x^2+c x^4\right )} \, dx &=-\frac {\int \frac {b d-a e+c d x^2}{x^4 \sqrt {d+e x^2} \left (a+b x^2+c x^4\right )} \, dx}{a}+\frac {d \int \frac {1}{x^6 \sqrt {d+e x^2}} \, dx}{a}\\ &=-\frac {\sqrt {d+e x^2}}{5 a x^5}-\frac {\int \left (\frac {b d-a e}{a x^4 \sqrt {d+e x^2}}+\frac {-b^2 d+a c d+a b e}{a^2 x^2 \sqrt {d+e x^2}}+\frac {b^3 d-2 a b c d-a b^2 e+a^2 c e+c \left (b^2 d-a c d-a b e\right ) x^2}{a^2 \sqrt {d+e x^2} \left (a+b x^2+c x^4\right )}\right ) \, dx}{a}-\frac {(4 e) \int \frac {1}{x^4 \sqrt {d+e x^2}} \, dx}{5 a}\\ &=-\frac {\sqrt {d+e x^2}}{5 a x^5}+\frac {4 e \sqrt {d+e x^2}}{15 a d x^3}-\frac {\int \frac {b^3 d-2 a b c d-a b^2 e+a^2 c e+c \left (b^2 d-a c d-a b e\right ) x^2}{\sqrt {d+e x^2} \left (a+b x^2+c x^4\right )} \, dx}{a^3}+\frac {\left (8 e^2\right ) \int \frac {1}{x^2 \sqrt {d+e x^2}} \, dx}{15 a d}-\frac {(b d-a e) \int \frac {1}{x^4 \sqrt {d+e x^2}} \, dx}{a^2}+\frac {\left (b^2 d-a c d-a b e\right ) \int \frac {1}{x^2 \sqrt {d+e x^2}} \, dx}{a^3}\\ &=-\frac {\sqrt {d+e x^2}}{5 a x^5}+\frac {4 e \sqrt {d+e x^2}}{15 a d x^3}+\frac {(b d-a e) \sqrt {d+e x^2}}{3 a^2 d x^3}-\frac {8 e^2 \sqrt {d+e x^2}}{15 a d^2 x}-\frac {\left (b^2 d-a c d-a b e\right ) \sqrt {d+e x^2}}{a^3 d x}-\frac {\int \left (\frac {c \left (b^2 d-a c d-a b e\right )+\frac {c \left (b^3 d-3 a b c d-a b^2 e+2 a^2 c 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 \left (b^2 d-a c d-a b e\right )-\frac {c \left (b^3 d-3 a b c d-a b^2 e+2 a^2 c 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^3}+\frac {(2 e (b d-a e)) \int \frac {1}{x^2 \sqrt {d+e x^2}} \, dx}{3 a^2 d}\\ &=-\frac {\sqrt {d+e x^2}}{5 a x^5}+\frac {4 e \sqrt {d+e x^2}}{15 a d x^3}+\frac {(b d-a e) \sqrt {d+e x^2}}{3 a^2 d x^3}-\frac {8 e^2 \sqrt {d+e x^2}}{15 a d^2 x}-\frac {2 e (b d-a e) \sqrt {d+e x^2}}{3 a^2 d^2 x}-\frac {\left (b^2 d-a c d-a b e\right ) \sqrt {d+e x^2}}{a^3 d x}-\frac {\left (c \left (b^2 d-a c d-a b e-\frac {b^3 d-3 a b c d-a b^2 e+2 a^2 c 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^3}-\frac {\left (c \left (b^2 d-a c d-a b e+\frac {b^3 d-3 a b c d-a b^2 e+2 a^2 c 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^3}\\ &=-\frac {\sqrt {d+e x^2}}{5 a x^5}+\frac {4 e \sqrt {d+e x^2}}{15 a d x^3}+\frac {(b d-a e) \sqrt {d+e x^2}}{3 a^2 d x^3}-\frac {8 e^2 \sqrt {d+e x^2}}{15 a d^2 x}-\frac {2 e (b d-a e) \sqrt {d+e x^2}}{3 a^2 d^2 x}-\frac {\left (b^2 d-a c d-a b e\right ) \sqrt {d+e x^2}}{a^3 d x}-\frac {\left (c \left (b^2 d-a c d-a b e-\frac {b^3 d-3 a b c d-a b^2 e+2 a^2 c 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^3}-\frac {\left (c \left (b^2 d-a c d-a b e+\frac {b^3 d-3 a b c d-a b^2 e+2 a^2 c 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^3}\\ &=-\frac {\sqrt {d+e x^2}}{5 a x^5}+\frac {4 e \sqrt {d+e x^2}}{15 a d x^3}+\frac {(b d-a e) \sqrt {d+e x^2}}{3 a^2 d x^3}-\frac {8 e^2 \sqrt {d+e x^2}}{15 a d^2 x}-\frac {2 e (b d-a e) \sqrt {d+e x^2}}{3 a^2 d^2 x}-\frac {\left (b^2 d-a c d-a b e\right ) \sqrt {d+e x^2}}{a^3 d x}-\frac {c \left (b^2 d-a c d-a b e+\frac {b^3 d-3 a b c d-a b^2 e+2 a^2 c 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^3 \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^2 d-a c d-a b e-\frac {b^3 d-3 a b c d-a b^2 e+2 a^2 c 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^3 \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. \(10933\) vs. \(2(512)=1024\).
time = 16.41, size = 10933, normalized size = 21.35 \begin {gather*} \text {Result too large to show} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[d + e*x^2]/(x^6*(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 = 0.16, size = 417, normalized size = 0.81

method	result	size
risch	\(-\frac {\sqrt {e \,x^{2}+d}\, \left (-2 a^{2} e^{2} x^{4}-5 a b d e \,x^{4}-15 a c \,d^{2} x^{4}+15 b^{2} d^{2} x^{4}+a^{2} d e \,x^{2}-5 a b \,d^{2} x^{2}+3 a^{2} d^{2}\right )}{15 d^{2} a^{3} x^{5}}-\frac {\sqrt {e}\, \left (\munderset {\textit {\_R} =\RootOf \left (c \,\textit {\_Z}^{4}+\left (4 e b -4 c d \right ) \textit {\_Z}^{3}+\left (16 a \,e^{2}-8 d e b +6 c \,d^{2}\right ) \textit {\_Z}^{2}+\left (4 d^{2} e b -4 c \,d^{3}\right ) \textit {\_Z} +d^{4} c \right )}{\sum }\frac {\left (c \left (a b e +a c d -b^{2} d \right ) \textit {\_R}^{2}+2 \left (-2 a^{2} c \,e^{2}+2 a \,b^{2} e^{2}+3 a b c d e -a \,c^{2} d^{2}-2 b^{3} d e +b^{2} c \,d^{2}\right ) \textit {\_R} +a b c \,d^{2} e +a \,c^{2} d^{3}-b^{2} c \,d^{3}\right ) \ln \left (\left (\sqrt {e \,x^{2}+d}-\sqrt {e}\, x \right )^{2}-\textit {\_R} \right )}{c \,\textit {\_R}^{3}+3 \textit {\_R}^{2} b e -3 \textit {\_R}^{2} c d +8 \textit {\_R} a \,e^{2}-4 \textit {\_R} b d e +3 c \,d^{2} \textit {\_R} +d^{2} e b -c \,d^{3}}\right )}{2 a^{3}}\)	\(342\)
default	\(-\frac {\sqrt {e}\, \left (\left (a c -b^{2}\right ) \ln \left (\sqrt {e \,x^{2}+d}-\sqrt {e}\, x \right )-\frac {\left (\munderset {\textit {\_R} =\RootOf \left (c \,\textit {\_Z}^{4}+\left (4 e b -4 c d \right ) \textit {\_Z}^{3}+\left (16 a \,e^{2}-8 d e b +6 c \,d^{2}\right ) \textit {\_Z}^{2}+\left (4 d^{2} e b -4 c \,d^{3}\right ) \textit {\_Z} +d^{4} c \right )}{\sum }\frac {\left (c \left (-a b e -a c d +b^{2} d \right ) \textit {\_R}^{2}+2 \left (2 a^{2} c \,e^{2}-2 a \,b^{2} e^{2}-3 a b c d e +a \,c^{2} d^{2}+2 b^{3} d e -b^{2} c \,d^{2}\right ) \textit {\_R} -a b c \,d^{2} e -a \,c^{2} d^{3}+b^{2} c \,d^{3}\right ) \ln \left (\left (\sqrt {e \,x^{2}+d}-\sqrt {e}\, x \right )^{2}-\textit {\_R} \right )}{c \,\textit {\_R}^{3}+3 \textit {\_R}^{2} b e -3 \textit {\_R}^{2} c d +8 \textit {\_R} a \,e^{2}-4 \textit {\_R} b d e +3 c \,d^{2} \textit {\_R} +d^{2} e b -c \,d^{3}}\right )}{2}\right )}{a^{3}}+\frac {-\frac {\left (e \,x^{2}+d \right )^{\frac {3}{2}}}{5 d \,x^{5}}+\frac {2 e \left (e \,x^{2}+d \right )^{\frac {3}{2}}}{15 d^{2} x^{3}}}{a}+\frac {\left (-a c +b^{2}\right ) \left (-\frac {\left (e \,x^{2}+d \right )^{\frac {3}{2}}}{d x}+\frac {2 e \left (\frac {x \sqrt {e \,x^{2}+d}}{2}+\frac {d \ln \left (\sqrt {e}\, x +\sqrt {e \,x^{2}+d}\right )}{2 \sqrt {e}}\right )}{d}\right )}{a^{3}}+\frac {b \left (e \,x^{2}+d \right )^{\frac {3}{2}}}{3 a^{2} d \,x^{3}}\)	\(417\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x^2+d)^(1/2)/x^6/(c*x^4+b*x^2+a),x,method=_RETURNVERBOSE)

[Out]

-1/a^3*e^(1/2)*((a*c-b^2)*ln((e*x^2+d)^(1/2)-e^(1/2)*x)-1/2*sum((c*(-a*b*e-a*c*d+b^2*d)*_R^2+2*(2*a^2*c*e^2-2*
a*b^2*e^2-3*a*b*c*d*e+a*c^2*d^2+2*b^3*d*e-b^2*c*d^2)*_R-a*b*c*d^2*e-a*c^2*d^3+b^2*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)))+1/a*(-1/5/d/x^5*(e*x^2+d)^
(3/2)+2/15*e/d^2/x^3*(e*x^2+d)^(3/2))+(-a*c+b^2)/a^3*(-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*b/a^2/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^6/(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^6), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 5811 vs. \(2 (472) = 944\).
time = 15.80, size = 5811, normalized size = 11.35 \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^6/(c*x^4+b*x^2+a),x, algorithm="fricas")

[Out]

1/60*(15*sqrt(1/2)*a^3*d^2*x^5*sqrt(-((b^7 - 7*a*b^5*c + 14*a^2*b^3*c^2 - 7*a^3*b*c^3)*d - (a*b^6 - 6*a^2*b^4*
c + 9*a^3*b^2*c^2 - 2*a^4*c^3)*e + (a^7*b^2 - 4*a^8*c)*sqrt(((b^12 - 10*a*b^10*c + 37*a^2*b^8*c^2 - 62*a^3*b^6
*c^3 + 46*a^4*b^4*c^4 - 12*a^5*b^2*c^5 + a^6*c^6)*d^2 - 2*(a*b^11 - 9*a^2*b^9*c + 29*a^3*b^7*c^2 - 40*a^4*b^5*
c^3 + 22*a^5*b^3*c^4 - 3*a^6*b*c^5)*d*e + (a^2*b^10 - 8*a^3*b^8*c + 22*a^4*b^6*c^2 - 24*a^5*b^4*c^3 + 9*a^6*b^
2*c^4)*e^2)/(a^14*b^2 - 4*a^15*c)))/(a^7*b^2 - 4*a^8*c))*log(((b^7*c^3 - 5*a*b^5*c^4 + 6*a^2*b^3*c^5 - a^3*b*c
^6)*d^2*x^2 + (a^7*b^2*c^3 - 4*a^8*c^4)*d*x^2*sqrt(((b^12 - 10*a*b^10*c + 37*a^2*b^8*c^2 - 62*a^3*b^6*c^3 + 46
*a^4*b^4*c^4 - 12*a^5*b^2*c^5 + a^6*c^6)*d^2 - 2*(a*b^11 - 9*a^2*b^9*c + 29*a^3*b^7*c^2 - 40*a^4*b^5*c^3 + 22*
a^5*b^3*c^4 - 3*a^6*b*c^5)*d*e + (a^2*b^10 - 8*a^3*b^8*c + 22*a^4*b^6*c^2 - 24*a^5*b^4*c^3 + 9*a^6*b^2*c^4)*e^
2)/(a^14*b^2 - 4*a^15*c)) + 4*(a^2*b^5*c^3 - 4*a^3*b^3*c^4 + 3*a^4*b*c^5)*x^2*e^2 - 2*(a*b^6*c^3 - 5*a^2*b^4*c
^4 + 6*a^3*b^2*c^5 - a^4*c^6)*d^2 + 2*sqrt(1/2)*((a*b^10 - 10*a^2*b^8*c + 35*a^3*b^6*c^2 - 51*a^4*b^4*c^3 + 29
*a^5*b^2*c^4 - 4*a^6*c^5)*d*x - (a^2*b^9 - 9*a^3*b^7*c + 27*a^4*b^5*c^2 - 31*a^5*b^3*c^3 + 12*a^6*b*c^4)*x*e -
(a^8*b^5 - 7*a^9*b^3*c + 12*a^10*b*c^2)*x*sqrt(((b^12 - 10*a*b^10*c + 37*a^2*b^8*c^2 - 62*a^3*b^6*c^3 + 46*a^
4*b^4*c^4 - 12*a^5*b^2*c^5 + a^6*c^6)*d^2 - 2*(a*b^11 - 9*a^2*b^9*c + 29*a^3*b^7*c^2 - 40*a^4*b^5*c^3 + 22*a^5
*b^3*c^4 - 3*a^6*b*c^5)*d*e + (a^2*b^10 - 8*a^3*b^8*c + 22*a^4*b^6*c^2 - 24*a^5*b^4*c^3 + 9*a^6*b^2*c^4)*e^2)/
(a^14*b^2 - 4*a^15*c)))*sqrt(x^2*e + d)*sqrt(-((b^7 - 7*a*b^5*c + 14*a^2*b^3*c^2 - 7*a^3*b*c^3)*d - (a*b^6 - 6
*a^2*b^4*c + 9*a^3*b^2*c^2 - 2*a^4*c^3)*e + (a^7*b^2 - 4*a^8*c)*sqrt(((b^12 - 10*a*b^10*c + 37*a^2*b^8*c^2 - 6
2*a^3*b^6*c^3 + 46*a^4*b^4*c^4 - 12*a^5*b^2*c^5 + a^6*c^6)*d^2 - 2*(a*b^11 - 9*a^2*b^9*c + 29*a^3*b^7*c^2 - 40
*a^4*b^5*c^3 + 22*a^5*b^3*c^4 - 3*a^6*b*c^5)*d*e + (a^2*b^10 - 8*a^3*b^8*c + 22*a^4*b^6*c^2 - 24*a^5*b^4*c^3 +
9*a^6*b^2*c^4)*e^2)/(a^14*b^2 - 4*a^15*c)))/(a^7*b^2 - 4*a^8*c)) - ((5*a*b^6*c^3 - 24*a^2*b^4*c^4 + 27*a^3*b^
2*c^5 - 4*a^4*c^6)*d*x^2 - 2*(a^2*b^5*c^3 - 4*a^3*b^3*c^4 + 3*a^4*b*c^5)*d)*e)/x^2) - 15*sqrt(1/2)*a^3*d^2*x^5
*sqrt(-((b^7 - 7*a*b^5*c + 14*a^2*b^3*c^2 - 7*a^3*b*c^3)*d - (a*b^6 - 6*a^2*b^4*c + 9*a^3*b^2*c^2 - 2*a^4*c^3)
*e + (a^7*b^2 - 4*a^8*c)*sqrt(((b^12 - 10*a*b^10*c + 37*a^2*b^8*c^2 - 62*a^3*b^6*c^3 + 46*a^4*b^4*c^4 - 12*a^5
*b^2*c^5 + a^6*c^6)*d^2 - 2*(a*b^11 - 9*a^2*b^9*c + 29*a^3*b^7*c^2 - 40*a^4*b^5*c^3 + 22*a^5*b^3*c^4 - 3*a^6*b
*c^5)*d*e + (a^2*b^10 - 8*a^3*b^8*c + 22*a^4*b^6*c^2 - 24*a^5*b^4*c^3 + 9*a^6*b^2*c^4)*e^2)/(a^14*b^2 - 4*a^15
*c)))/(a^7*b^2 - 4*a^8*c))*log(((b^7*c^3 - 5*a*b^5*c^4 + 6*a^2*b^3*c^5 - a^3*b*c^6)*d^2*x^2 + (a^7*b^2*c^3 - 4
*a^8*c^4)*d*x^2*sqrt(((b^12 - 10*a*b^10*c + 37*a^2*b^8*c^2 - 62*a^3*b^6*c^3 + 46*a^4*b^4*c^4 - 12*a^5*b^2*c^5
+ a^6*c^6)*d^2 - 2*(a*b^11 - 9*a^2*b^9*c + 29*a^3*b^7*c^2 - 40*a^4*b^5*c^3 + 22*a^5*b^3*c^4 - 3*a^6*b*c^5)*d*e
+ (a^2*b^10 - 8*a^3*b^8*c + 22*a^4*b^6*c^2 - 24*a^5*b^4*c^3 + 9*a^6*b^2*c^4)*e^2)/(a^14*b^2 - 4*a^15*c)) + 4*
(a^2*b^5*c^3 - 4*a^3*b^3*c^4 + 3*a^4*b*c^5)*x^2*e^2 - 2*(a*b^6*c^3 - 5*a^2*b^4*c^4 + 6*a^3*b^2*c^5 - a^4*c^6)*
d^2 - 2*sqrt(1/2)*((a*b^10 - 10*a^2*b^8*c + 35*a^3*b^6*c^2 - 51*a^4*b^4*c^3 + 29*a^5*b^2*c^4 - 4*a^6*c^5)*d*x
- (a^2*b^9 - 9*a^3*b^7*c + 27*a^4*b^5*c^2 - 31*a^5*b^3*c^3 + 12*a^6*b*c^4)*x*e - (a^8*b^5 - 7*a^9*b^3*c + 12*a
^10*b*c^2)*x*sqrt(((b^12 - 10*a*b^10*c + 37*a^2*b^8*c^2 - 62*a^3*b^6*c^3 + 46*a^4*b^4*c^4 - 12*a^5*b^2*c^5 + a
^6*c^6)*d^2 - 2*(a*b^11 - 9*a^2*b^9*c + 29*a^3*b^7*c^2 - 40*a^4*b^5*c^3 + 22*a^5*b^3*c^4 - 3*a^6*b*c^5)*d*e +
(a^2*b^10 - 8*a^3*b^8*c + 22*a^4*b^6*c^2 - 24*a^5*b^4*c^3 + 9*a^6*b^2*c^4)*e^2)/(a^14*b^2 - 4*a^15*c)))*sqrt(x
^2*e + d)*sqrt(-((b^7 - 7*a*b^5*c + 14*a^2*b^3*c^2 - 7*a^3*b*c^3)*d - (a*b^6 - 6*a^2*b^4*c + 9*a^3*b^2*c^2 - 2
*a^4*c^3)*e + (a^7*b^2 - 4*a^8*c)*sqrt(((b^12 - 10*a*b^10*c + 37*a^2*b^8*c^2 - 62*a^3*b^6*c^3 + 46*a^4*b^4*c^4
- 12*a^5*b^2*c^5 + a^6*c^6)*d^2 - 2*(a*b^11 - 9*a^2*b^9*c + 29*a^3*b^7*c^2 - 40*a^4*b^5*c^3 + 22*a^5*b^3*c^4
- 3*a^6*b*c^5)*d*e + (a^2*b^10 - 8*a^3*b^8*c + 22*a^4*b^6*c^2 - 24*a^5*b^4*c^3 + 9*a^6*b^2*c^4)*e^2)/(a^14*b^2
- 4*a^15*c)))/(a^7*b^2 - 4*a^8*c)) - ((5*a*b^6*c^3 - 24*a^2*b^4*c^4 + 27*a^3*b^2*c^5 - 4*a^4*c^6)*d*x^2 - 2*(
a^2*b^5*c^3 - 4*a^3*b^3*c^4 + 3*a^4*b*c^5)*d)*e)/x^2) + 15*sqrt(1/2)*a^3*d^2*x^5*sqrt(-((b^7 - 7*a*b^5*c + 14*
a^2*b^3*c^2 - 7*a^3*b*c^3)*d - (a*b^6 - 6*a^2*b^4*c + 9*a^3*b^2*c^2 - 2*a^4*c^3)*e - (a^7*b^2 - 4*a^8*c)*sqrt(
((b^12 - 10*a*b^10*c + 37*a^2*b^8*c^2 - 62*a^3*b^6*c^3 + 46*a^4*b^4*c^4 - 12*a^5*b^2*c^5 + a^6*c^6)*d^2 - 2*(a
*b^11 - 9*a^2*b^9*c + 29*a^3*b^7*c^2 - 40*a^4*b^5*c^3 + 22*a^5*b^3*c^4 - 3*a^6*b*c^5)*d*e + (a^2*b^10 - 8*a^3*
b^8*c + 22*a^4*b^6*c^2 - 24*a^5*b^4*c^3 + 9*a^6*b^2*c^4)*e^2)/(a^14*b^2 - 4*a^15*c)))/(a^7*b^2 - 4*a^8*c))*log
(((b^7*c^3 - 5*a*b^5*c^4 + 6*a^2*b^3*c^5 - a^3*b*c^6)*d^2*x^2 - (a^7*b^2*c^3 - 4*a^8*c^4)*d*x^2*sqrt(((b^12 -
10*a*b^10*c + 37*a^2*b^8*c^2 - 62*a^3*b^6*c^3 +...

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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^{6} \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**6/(c*x**4+b*x**2+a),x)

[Out]

Integral(sqrt(d + e*x**2)/(x**6*(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^6/(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^6\,\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^6*(a + b*x^2 + c*x^4)),x)

[Out]

int((d + e*x^2)^(1/2)/(x^6*(a + b*x^2 + c*x^4)), x)

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