3.4.92 \(\int \frac {1}{x^4 \sqrt {d+e x^2} (a+b x^2+c x^4)} \, dx\) [392]

Optimal. Leaf size=341 \[ -\frac {\sqrt {d+e x^2}}{3 a d x^3}+\frac {b \sqrt {d+e x^2}}{a^2 d x}+\frac {2 e \sqrt {d+e x^2}}{3 a d^2 x}+\frac {c \left (b+\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\right ) \tan ^{-1}\left (\frac {\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e} x}{\sqrt {b-\sqrt {b^2-4 a c}} \sqrt {d+e x^2}}\right )}{a^2 \sqrt {b-\sqrt {b^2-4 a c}} \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}+\frac {c \left (b-\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\right ) \tan ^{-1}\left (\frac {\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e} x}{\sqrt {b+\sqrt {b^2-4 a c}} \sqrt {d+e x^2}}\right )}{a^2 \sqrt {b+\sqrt {b^2-4 a c}} \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.50, antiderivative size = 341, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 6, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {1317, 277, 270, 1706, 385, 211} \begin {gather*} \frac {c \left (\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}+b\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 (b-\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\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 {b \sqrt {d+e x^2}}{a^2 d x}+\frac {2 e \sqrt {d+e x^2}}{3 a d^2 x}-\frac {\sqrt {d+e x^2}}{3 a d x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^4*Sqrt[d + e*x^2]*(a + b*x^2 + c*x^4)),x]

[Out]

-1/3*Sqrt[d + e*x^2]/(a*d*x^3) + (b*Sqrt[d + e*x^2])/(a^2*d*x) + (2*e*Sqrt[d + e*x^2])/(3*a*d^2*x) + (c*(b + (
b^2 - 2*a*c)/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 - (b^2
- 2*a*c)/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]]*Sq
rt[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 1317

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

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]

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]
time = 10.53, size = 320, normalized size = 0.94 \begin {gather*} \frac {\frac {3 b \sqrt {d+e x^2}}{d x}-\frac {a \left (d-2 e x^2\right ) \sqrt {d+e x^2}}{d^2 x^3}+\frac {3 c \left (b+\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\right ) \tan ^{-1}\left (\frac {\sqrt {2 c d-b e+\sqrt {b^2-4 a c} e} x}{\sqrt {b-\sqrt {b^2-4 a c}} \sqrt {d+e x^2}}\right )}{\sqrt {b-\sqrt {b^2-4 a c}} \sqrt {2 c d+\left (-b+\sqrt {b^2-4 a c}\right ) e}}+\frac {3 c \left (b+\frac {-b^2+2 a c}{\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 )}{\sqrt {b+\sqrt {b^2-4 a c}} \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}}{3 a^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^4*Sqrt[d + e*x^2]*(a + b*x^2 + c*x^4)),x]

[Out]

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

________________________________________________________________________________________

Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.14, size = 248, normalized size = 0.73

method	result	size
risch	\(-\frac {\sqrt {e \,x^{2}+d}\, \left (-2 a e \,x^{2}-3 b d \,x^{2}+a d \right )}{3 d^{2} a^{2} x^{3}}-\frac {\sqrt {e}\, \left (\munderset {\textit {\_R} =\RootOf \left (c \,\textit {\_Z}^{4}+\left (4 e b -4 c d \right ) \textit {\_Z}^{3}+\left (16 a \,e^{2}-8 d e b +6 c \,d^{2}\right ) \textit {\_Z}^{2}+\left (4 d^{2} e b -4 c \,d^{3}\right ) \textit {\_Z} +d^{4} c \right )}{\sum }\frac {\left (b c \,\textit {\_R}^{2}+2 \left (-2 a c e +2 b^{2} e -b c d \right ) \textit {\_R} +b c \,d^{2}\right ) \ln \left (\left (\sqrt {e \,x^{2}+d}-\sqrt {e}\, x \right )^{2}-\textit {\_R} \right )}{c \,\textit {\_R}^{3}+3 \textit {\_R}^{2} b e -3 \textit {\_R}^{2} c d +8 \textit {\_R} a \,e^{2}-4 \textit {\_R} b d e +3 c \,d^{2} \textit {\_R} +d^{2} e b -c \,d^{3}}\right )}{2 a^{2}}\)	\(225\)
default	\(\frac {\sqrt {e}\, \left (\munderset {\textit {\_R} =\RootOf \left (c \,\textit {\_Z}^{4}+\left (4 e b -4 c d \right ) \textit {\_Z}^{3}+\left (16 a \,e^{2}-8 d e b +6 c \,d^{2}\right ) \textit {\_Z}^{2}+\left (4 d^{2} e b -4 c \,d^{3}\right ) \textit {\_Z} +d^{4} c \right )}{\sum }\frac {\left (-b c \,\textit {\_R}^{2}+2 \left (2 a c e -2 b^{2} e +b c d \right ) \textit {\_R} -b c \,d^{2}\right ) \ln \left (\left (\sqrt {e \,x^{2}+d}-\sqrt {e}\, x \right )^{2}-\textit {\_R} \right )}{c \,\textit {\_R}^{3}+3 \textit {\_R}^{2} b e -3 \textit {\_R}^{2} c d +8 \textit {\_R} a \,e^{2}-4 \textit {\_R} b d e +3 c \,d^{2} \textit {\_R} +d^{2} e b -c \,d^{3}}\right )}{2 a^{2}}+\frac {b \sqrt {e \,x^{2}+d}}{a^{2} d x}+\frac {-\frac {\sqrt {e \,x^{2}+d}}{3 d \,x^{3}}+\frac {2 e \sqrt {e \,x^{2}+d}}{3 d^{2} x}}{a}\)	\(248\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2/a^2*e^(1/2)*sum((-b*c*_R^2+2*(2*a*c*e-2*b^2*e+b*c*d)*_R-b*c*d^2)/(_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*(e*x^2+d)^(1/2)/a^2/d/x+1/a*(-1/3/d/x^3*(e*x^
2+d)^(1/2)+2/3*e/d^2/x*(e*x^2+d)^(1/2))

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 8220 vs. \(2 (301) = 602\).
time = 62.84, size = 8220, normalized size = 24.11 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/12*(3*sqrt(1/2)*a^2*d^2*x^3*sqrt(-((b^5*c - 5*a*b^3*c^2 + 5*a^2*b*c^3)*d - (b^6 - 6*a*b^4*c + 9*a^2*b^2*c^2
- 2*a^3*c^3)*e + ((a^5*b^2*c - 4*a^6*c^2)*d^2 - (a^5*b^3 - 4*a^6*b*c)*d*e + (a^6*b^2 - 4*a^7*c)*e^2)*sqrt(((b
^8*c^2 - 6*a*b^6*c^3 + 11*a^2*b^4*c^4 - 6*a^3*b^2*c^5 + a^4*c^6)*d^2 - 2*(b^9*c - 7*a*b^7*c^2 + 16*a^2*b^5*c^3
- 13*a^3*b^3*c^4 + 3*a^4*b*c^5)*d*e + (b^10 - 8*a*b^8*c + 22*a^2*b^6*c^2 - 24*a^3*b^4*c^3 + 9*a^4*b^2*c^4)*e^
2)/((a^10*b^2*c^2 - 4*a^11*c^3)*d^4 - 2*(a^10*b^3*c - 4*a^11*b*c^2)*d^3*e + (a^10*b^4 - 2*a^11*b^2*c - 8*a^12*
c^2)*d^2*e^2 - 2*(a^11*b^3 - 4*a^12*b*c)*d*e^3 + (a^12*b^2 - 4*a^13*c)*e^4)))/((a^5*b^2*c - 4*a^6*c^2)*d^2 - (
a^5*b^3 - 4*a^6*b*c)*d*e + (a^6*b^2 - 4*a^7*c)*e^2))*log(((b^5*c^4 - 3*a*b^3*c^5 + a^2*b*c^6)*d^2*x^2 + 4*(a*b
^5*c^3 - 4*a^2*b^3*c^4 + 3*a^3*b*c^5)*x^2*e^2 - 2*(a*b^4*c^4 - 3*a^2*b^2*c^5 + a^3*c^6)*d^2 + 2*sqrt(1/2)*((a*
b^7*c^2 - 7*a^2*b^5*c^3 + 13*a^3*b^3*c^4 - 4*a^4*b*c^5)*d^2*x - (2*a*b^8*c - 16*a^2*b^6*c^2 + 39*a^3*b^4*c^3 -
29*a^4*b^2*c^4 + 4*a^5*c^5)*d*x*e + (a*b^9 - 9*a^2*b^7*c + 27*a^3*b^5*c^2 - 31*a^4*b^3*c^3 + 12*a^5*b*c^4)*x*
e^2 - ((a^6*b^4*c^2 - 6*a^7*b^2*c^3 + 8*a^8*c^4)*d^3*x - (2*a^6*b^5*c - 13*a^7*b^3*c^2 + 20*a^8*b*c^3)*d^2*x*e
+ (a^6*b^6 - 6*a^7*b^4*c + 6*a^8*b^2*c^2 + 8*a^9*c^3)*d*x*e^2 - (a^7*b^5 - 7*a^8*b^3*c + 12*a^9*b*c^2)*x*e^3)
*sqrt(((b^8*c^2 - 6*a*b^6*c^3 + 11*a^2*b^4*c^4 - 6*a^3*b^2*c^5 + a^4*c^6)*d^2 - 2*(b^9*c - 7*a*b^7*c^2 + 16*a^
2*b^5*c^3 - 13*a^3*b^3*c^4 + 3*a^4*b*c^5)*d*e + (b^10 - 8*a*b^8*c + 22*a^2*b^6*c^2 - 24*a^3*b^4*c^3 + 9*a^4*b^
2*c^4)*e^2)/((a^10*b^2*c^2 - 4*a^11*c^3)*d^4 - 2*(a^10*b^3*c - 4*a^11*b*c^2)*d^3*e + (a^10*b^4 - 2*a^11*b^2*c
- 8*a^12*c^2)*d^2*e^2 - 2*(a^11*b^3 - 4*a^12*b*c)*d*e^3 + (a^12*b^2 - 4*a^13*c)*e^4)))*sqrt(x^2*e + d)*sqrt(-(
(b^5*c - 5*a*b^3*c^2 + 5*a^2*b*c^3)*d - (b^6 - 6*a*b^4*c + 9*a^2*b^2*c^2 - 2*a^3*c^3)*e + ((a^5*b^2*c - 4*a^6*
c^2)*d^2 - (a^5*b^3 - 4*a^6*b*c)*d*e + (a^6*b^2 - 4*a^7*c)*e^2)*sqrt(((b^8*c^2 - 6*a*b^6*c^3 + 11*a^2*b^4*c^4
- 6*a^3*b^2*c^5 + a^4*c^6)*d^2 - 2*(b^9*c - 7*a*b^7*c^2 + 16*a^2*b^5*c^3 - 13*a^3*b^3*c^4 + 3*a^4*b*c^5)*d*e +
(b^10 - 8*a*b^8*c + 22*a^2*b^6*c^2 - 24*a^3*b^4*c^3 + 9*a^4*b^2*c^4)*e^2)/((a^10*b^2*c^2 - 4*a^11*c^3)*d^4 -
2*(a^10*b^3*c - 4*a^11*b*c^2)*d^3*e + (a^10*b^4 - 2*a^11*b^2*c - 8*a^12*c^2)*d^2*e^2 - 2*(a^11*b^3 - 4*a^12*b*
c)*d*e^3 + (a^12*b^2 - 4*a^13*c)*e^4)))/((a^5*b^2*c - 4*a^6*c^2)*d^2 - (a^5*b^3 - 4*a^6*b*c)*d*e + (a^6*b^2 -
4*a^7*c)*e^2)) - ((b^6*c^3 - 9*a^2*b^2*c^5 + 4*a^3*c^6)*d*x^2 - 2*(a*b^5*c^3 - 4*a^2*b^3*c^4 + 3*a^3*b*c^5)*d)
*e + ((a^5*b^2*c^4 - 4*a^6*c^5)*d^3*x^2 - (a^5*b^3*c^3 - 4*a^6*b*c^4)*d^2*x^2*e + (a^6*b^2*c^3 - 4*a^7*c^4)*d*
x^2*e^2)*sqrt(((b^8*c^2 - 6*a*b^6*c^3 + 11*a^2*b^4*c^4 - 6*a^3*b^2*c^5 + a^4*c^6)*d^2 - 2*(b^9*c - 7*a*b^7*c^2
+ 16*a^2*b^5*c^3 - 13*a^3*b^3*c^4 + 3*a^4*b*c^5)*d*e + (b^10 - 8*a*b^8*c + 22*a^2*b^6*c^2 - 24*a^3*b^4*c^3 +
9*a^4*b^2*c^4)*e^2)/((a^10*b^2*c^2 - 4*a^11*c^3)*d^4 - 2*(a^10*b^3*c - 4*a^11*b*c^2)*d^3*e + (a^10*b^4 - 2*a^1
1*b^2*c - 8*a^12*c^2)*d^2*e^2 - 2*(a^11*b^3 - 4*a^12*b*c)*d*e^3 + (a^12*b^2 - 4*a^13*c)*e^4)))/x^2) - 3*sqrt(1
/2)*a^2*d^2*x^3*sqrt(-((b^5*c - 5*a*b^3*c^2 + 5*a^2*b*c^3)*d - (b^6 - 6*a*b^4*c + 9*a^2*b^2*c^2 - 2*a^3*c^3)*e
+ ((a^5*b^2*c - 4*a^6*c^2)*d^2 - (a^5*b^3 - 4*a^6*b*c)*d*e + (a^6*b^2 - 4*a^7*c)*e^2)*sqrt(((b^8*c^2 - 6*a*b^
6*c^3 + 11*a^2*b^4*c^4 - 6*a^3*b^2*c^5 + a^4*c^6)*d^2 - 2*(b^9*c - 7*a*b^7*c^2 + 16*a^2*b^5*c^3 - 13*a^3*b^3*c
^4 + 3*a^4*b*c^5)*d*e + (b^10 - 8*a*b^8*c + 22*a^2*b^6*c^2 - 24*a^3*b^4*c^3 + 9*a^4*b^2*c^4)*e^2)/((a^10*b^2*c
^2 - 4*a^11*c^3)*d^4 - 2*(a^10*b^3*c - 4*a^11*b*c^2)*d^3*e + (a^10*b^4 - 2*a^11*b^2*c - 8*a^12*c^2)*d^2*e^2 -
2*(a^11*b^3 - 4*a^12*b*c)*d*e^3 + (a^12*b^2 - 4*a^13*c)*e^4)))/((a^5*b^2*c - 4*a^6*c^2)*d^2 - (a^5*b^3 - 4*a^6
*b*c)*d*e + (a^6*b^2 - 4*a^7*c)*e^2))*log(((b^5*c^4 - 3*a*b^3*c^5 + a^2*b*c^6)*d^2*x^2 + 4*(a*b^5*c^3 - 4*a^2*
b^3*c^4 + 3*a^3*b*c^5)*x^2*e^2 - 2*(a*b^4*c^4 - 3*a^2*b^2*c^5 + a^3*c^6)*d^2 - 2*sqrt(1/2)*((a*b^7*c^2 - 7*a^2
*b^5*c^3 + 13*a^3*b^3*c^4 - 4*a^4*b*c^5)*d^2*x - (2*a*b^8*c - 16*a^2*b^6*c^2 + 39*a^3*b^4*c^3 - 29*a^4*b^2*c^4
+ 4*a^5*c^5)*d*x*e + (a*b^9 - 9*a^2*b^7*c + 27*a^3*b^5*c^2 - 31*a^4*b^3*c^3 + 12*a^5*b*c^4)*x*e^2 - ((a^6*b^4
*c^2 - 6*a^7*b^2*c^3 + 8*a^8*c^4)*d^3*x - (2*a^6*b^5*c - 13*a^7*b^3*c^2 + 20*a^8*b*c^3)*d^2*x*e + (a^6*b^6 - 6
*a^7*b^4*c + 6*a^8*b^2*c^2 + 8*a^9*c^3)*d*x*e^2 - (a^7*b^5 - 7*a^8*b^3*c + 12*a^9*b*c^2)*x*e^3)*sqrt(((b^8*c^2
- 6*a*b^6*c^3 + 11*a^2*b^4*c^4 - 6*a^3*b^2*c^5 + a^4*c^6)*d^2 - 2*(b^9*c - 7*a*b^7*c^2 + 16*a^2*b^5*c^3 - 13*
a^3*b^3*c^4 + 3*a^4*b*c^5)*d*e + (b^10 - 8*a*b^8*c + 22*a^2*b^6*c^2 - 24*a^3*b^4*c^3 + 9*a^4*b^2*c^4)*e^2)/((a
^10*b^2*c^2 - 4*a^11*c^3)*d^4 - 2*(a^10*b^3*c - 4*a^11*b*c^2)*d^3*e + (a^10*b^4 - 2*a^11*b^2*c - 8*a^12*c^2)*d
^2*e^2 - 2*(a^11*b^3 - 4*a^12*b*c)*d*e^3 + (a^12*b^2 - 4*a^13*c)*e^4)))*sqrt(x^2*e + d)*sqrt(-((b^5*c - 5*a*b^
3*c^2 + 5*a^2*b*c^3)*d - (b^6 - 6*a*b^4*c + 9*a^2*b^2*c^2 - 2*a^3*c^3)*e + ((a^5*b^2*c - 4*a^6*c^2)*d^2 - (a^5
*b^3 - 4*a^6*b*c)*d*e + (a^6*b^2 - 4*a^7*c)*e^2...

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^{4} \sqrt {d + e x^{2}} \left (a + b x^{2} + c x^{4}\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________