3.4.97 \(\int \frac {1}{(d+e x^2)^{3/2} (a+b x^2+c x^4)} \, dx\) [397]

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

[Out]

e^2*x/d/(a*e^2-b*d*e+c*d^2)/(e*x^2+d)^(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))*(e+(b*e-2*c*d)/(-4*a*c+b^2)^(1/2))/(a*e^2-b*d*e+c*d^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))*(e+(-b*e+2*c*d)/(-4*a*c+b^2)^(1/2))/(a*e^2-b*d*e+c*d^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 = 0.56, antiderivative size = 341, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {1186, 197, 1706, 385, 211} \begin {gather*} -\frac {c \left (e-\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\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 )}{\sqrt {b-\sqrt {b^2-4 a c}} \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )} \left (a e^2-b d e+c d^2\right )}-\frac {c \left (\frac {2 c d-b e}{\sqrt {b^2-4 a c}}+e\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 )}{\sqrt {\sqrt {b^2-4 a c}+b} \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )} \left (a e^2-b d e+c d^2\right )}+\frac {e^2 x}{d \sqrt {d+e x^2} \left (a e^2-b d e+c d^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((d + e*x^2)^(3/2)*(a + b*x^2 + c*x^4)),x]

[Out]

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

Rule 197

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

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 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 1186

Int[((d_) + (e_.)*(x_)^2)^(q_)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> Dist[e^2/(c*d^2 - b*d*e + a*
e^2), Int[(d + e*x^2)^q, x], x] + Dist[1/(c*d^2 - b*d*e + a*e^2), Int[(d + e*x^2)^(q + 1)*((c*d - b*e - c*e*x^
2)/(a + b*x^2 + c*x^4)), 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] &&  !IntegerQ[q] && LtQ[q, -1]

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

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
time = 15.38, size = 2061, normalized size = 6.04 \begin {gather*} \text {Result too large to show} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[1/((d + e*x^2)^(3/2)*(a + b*x^2 + c*x^4)),x]

[Out]

(2*c*x*(45*Sqrt[-(((-b + Sqrt[b^2 - 4*a*c])*(2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)*x^2*(d + e*x^2))/(d^2*(b - Sq
rt[b^2 - 4*a*c] + 2*c*x^2)^2))] + (30*e*x^2*Sqrt[-(((-b + Sqrt[b^2 - 4*a*c])*(2*c*d + (-b + Sqrt[b^2 - 4*a*c])
*e)*x^2*(d + e*x^2))/(d^2*(b - Sqrt[b^2 - 4*a*c] + 2*c*x^2)^2))])/d - 45*ArcSin[Sqrt[-(((2*c*d + (-b + Sqrt[b^
2 - 4*a*c])*e)*x^2)/(d*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2)))]] - (30*e*x^2*ArcSin[Sqrt[-(((2*c*d + (-b + Sqrt[b
^2 - 4*a*c])*e)*x^2)/(d*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2)))]])/d - (45*(2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)*x
^2*ArcSin[Sqrt[-(((2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)*x^2)/(d*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2)))]])/(d*(-b
+ Sqrt[b^2 - 4*a*c] - 2*c*x^2)) - (30*e*(2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)*x^4*ArcSin[Sqrt[-(((2*c*d + (-b +
Sqrt[b^2 - 4*a*c])*e)*x^2)/(d*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2)))]])/(d^2*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2)
) + 4*(-(((2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)*x^2)/(d*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2))))^(5/2)*Sqrt[((-b +
Sqrt[b^2 - 4*a*c])*(d + e*x^2))/(d*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2))]*Hypergeometric2F1[2, 2, 7/2, -(((2*c*
d + (-b + Sqrt[b^2 - 4*a*c])*e)*x^2)/(d*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2)))] + (4*e*x^2*(-(((2*c*d + (-b + Sq
rt[b^2 - 4*a*c])*e)*x^2)/(d*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2))))^(5/2)*Sqrt[((-b + Sqrt[b^2 - 4*a*c])*(d + e*
x^2))/(d*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2))]*Hypergeometric2F1[2, 2, 7/2, -(((2*c*d + (-b + Sqrt[b^2 - 4*a*c]
)*e)*x^2)/(d*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2)))])/d))/(15*Sqrt[b^2 - 4*a*c]*d*(-(((2*c*d + (-b + Sqrt[b^2 -
4*a*c])*e)*x^2)/(d*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2))))^(3/2)*(b - Sqrt[b^2 - 4*a*c] + 2*c*x^2)*Sqrt[d + e*x^
2]*Sqrt[((-b + Sqrt[b^2 - 4*a*c])*(d + e*x^2))/(d*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2))]) - (2*c*x*Sqrt[((b + Sq
rt[b^2 - 4*a*c])*(d + e*x^2))/(d*(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2))]*(45*Sqrt[-(((b + Sqrt[b^2 - 4*a*c])*(-2*c
*d + (b + Sqrt[b^2 - 4*a*c])*e)*x^2*(d + e*x^2))/(d^2*(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)^2))] + (30*e*x^2*Sqrt[
-(((b + Sqrt[b^2 - 4*a*c])*(-2*c*d + (b + Sqrt[b^2 - 4*a*c])*e)*x^2*(d + e*x^2))/(d^2*(b + Sqrt[b^2 - 4*a*c] +
2*c*x^2)^2))])/d - 45*ArcSin[Sqrt[((2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)*x^2)/(d*(b + Sqrt[b^2 - 4*a*c] + 2*c*x
^2))]] - (30*e*x^2*ArcSin[Sqrt[((2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)*x^2)/(d*(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2))
]])/d + (45*(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)*x^2*ArcSin[Sqrt[((2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)*x^2)/(d*(
b + Sqrt[b^2 - 4*a*c] + 2*c*x^2))]])/(d*(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)) - (30*e*(-2*c*d + (b + Sqrt[b^2 - 4
*a*c])*e)*x^4*ArcSin[Sqrt[((2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)*x^2)/(d*(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2))]])/(
d^2*(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)) + 4*(((2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)*x^2)/(d*(b + Sqrt[b^2 - 4*a*c
] + 2*c*x^2)))^(5/2)*Sqrt[((b + Sqrt[b^2 - 4*a*c])*(d + e*x^2))/(d*(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2))]*Hyperge
ometric2F1[2, 2, 7/2, ((2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)*x^2)/(d*(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2))] + (4*e*
x^2*(((2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)*x^2)/(d*(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)))^(5/2)*Sqrt[((b + Sqrt[b^
2 - 4*a*c])*(d + e*x^2))/(d*(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2))]*Hypergeometric2F1[2, 2, 7/2, ((2*c*d - (b + Sq
rt[b^2 - 4*a*c])*e)*x^2)/(d*(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2))])/d))/(15*Sqrt[b^2 - 4*a*c]*(b + Sqrt[b^2 - 4*a
*c])*(((2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)*x^2)/(d*(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)))^(3/2)*(d + e*x^2)^(3/2)
)

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.12, size = 240, normalized size = 0.70

method	result	size
default	\(-64 e^{\frac {3}{2}} \left (-\frac {\munderset {\textit {\_R} =\RootOf \left (c \,\textit {\_Z}^{4}+\left (4 e b -4 c d \right ) \textit {\_Z}^{3}+\left (16 a \,e^{2}-8 d e b +6 c \,d^{2}\right ) \textit {\_Z}^{2}+\left (4 d^{2} e b -4 c \,d^{3}\right ) \textit {\_Z} +d^{4} c \right )}{\sum }\frac {\left (c \,\textit {\_R}^{2}+2 \left (2 e b -3 c d \right ) \textit {\_R} +c \,d^{2}\right ) \ln \left (\left (\sqrt {e \,x^{2}+d}-\sqrt {e}\, x \right )^{2}-\textit {\_R} \right )}{c \,\textit {\_R}^{3}+3 \textit {\_R}^{2} b e -3 \textit {\_R}^{2} c d +8 \textit {\_R} a \,e^{2}-4 \textit {\_R} b d e +3 c \,d^{2} \textit {\_R} +d^{2} e b -c \,d^{3}}}{8 \left (16 a \,e^{2}-16 d e b +16 c \,d^{2}\right )}-\frac {1}{2 \left (16 a \,e^{2}-16 d e b +16 c \,d^{2}\right ) \left (\left (\sqrt {e \,x^{2}+d}-\sqrt {e}\, x \right )^{2}+d \right )}\right )\)	\(240\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(e*x^2+d)^(3/2)/(c*x^4+b*x^2+a),x,method=_RETURNVERBOSE)

[Out]

-64*e^(3/2)*(-1/8/(16*a*e^2-16*b*d*e+16*c*d^2)*sum((c*_R^2+2*(2*b*e-3*c*d)*_R+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))-1/2/(16*a*e^2-16*b*d*e+16*c*d^2
)/(((e*x^2+d)^(1/2)-e^(1/2)*x)^2+d))

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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(1/(e*x^2+d)^(3/2)/(c*x^4+b*x^2+a),x, algorithm="maxima")

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 17065 vs. \(2 (309) = 618\).
time = 225.02, size = 17065, normalized size = 50.04 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x^2+d)^(3/2)/(c*x^4+b*x^2+a),x, algorithm="fricas")

[Out]

1/4*(sqrt(1/2)*(c*d^4 + a*d*x^2*e^3 - (b*d^2*x^2 - a*d^2)*e^2 + (c*d^3*x^2 - b*d^3)*e)*sqrt(-(b*c^3*d^3 - 3*(b
^2*c^2 - 2*a*c^3)*d^2*e + 3*(b^3*c - 3*a*b*c^2)*d*e^2 - (b^4 - 4*a*b^2*c + 2*a^2*c^2)*e^3 + ((a*b^2*c^3 - 4*a^
2*c^4)*d^6 - 3*(a*b^3*c^2 - 4*a^2*b*c^3)*d^5*e + 3*(a*b^4*c - 3*a^2*b^2*c^2 - 4*a^3*c^3)*d^4*e^2 - (a*b^5 + 2*
a^2*b^3*c - 24*a^3*b*c^2)*d^3*e^3 + 3*(a^2*b^4 - 3*a^3*b^2*c - 4*a^4*c^2)*d^2*e^4 - 3*(a^3*b^3 - 4*a^4*b*c)*d*
e^5 + (a^4*b^2 - 4*a^5*c)*e^6)*sqrt((c^6*d^6 - 6*b*c^5*d^5*e + 3*(5*b^2*c^4 - 2*a*c^5)*d^4*e^2 - 2*(10*b^3*c^3
- 11*a*b*c^4)*d^3*e^3 + 3*(5*b^4*c^2 - 10*a*b^2*c^3 + 3*a^2*c^4)*d^2*e^4 - 6*(b^5*c - 3*a*b^3*c^2 + 2*a^2*b*c
^3)*d*e^5 + (b^6 - 4*a*b^4*c + 4*a^2*b^2*c^2)*e^6)/((a^2*b^2*c^6 - 4*a^3*c^7)*d^12 - 6*(a^2*b^3*c^5 - 4*a^3*b*
c^6)*d^11*e + 3*(5*a^2*b^4*c^4 - 18*a^3*b^2*c^5 - 8*a^4*c^6)*d^10*e^2 - 10*(2*a^2*b^5*c^3 - 5*a^3*b^3*c^4 - 12
*a^4*b*c^5)*d^9*e^3 + 15*(a^2*b^6*c^2 - 15*a^4*b^2*c^4 - 4*a^5*c^5)*d^8*e^4 - 6*(a^2*b^7*c + 6*a^3*b^5*c^2 - 3
0*a^4*b^3*c^3 - 40*a^5*b*c^4)*d^7*e^5 + (a^2*b^8 + 26*a^3*b^6*c - 30*a^4*b^4*c^2 - 340*a^5*b^2*c^3 - 80*a^6*c^
4)*d^6*e^6 - 6*(a^3*b^7 + 6*a^4*b^5*c - 30*a^5*b^3*c^2 - 40*a^6*b*c^3)*d^5*e^7 + 15*(a^4*b^6 - 15*a^6*b^2*c^2
- 4*a^7*c^3)*d^4*e^8 - 10*(2*a^5*b^5 - 5*a^6*b^3*c - 12*a^7*b*c^2)*d^3*e^9 + 3*(5*a^6*b^4 - 18*a^7*b^2*c - 8*a
^8*c^2)*d^2*e^10 - 6*(a^7*b^3 - 4*a^8*b*c)*d*e^11 + (a^8*b^2 - 4*a^9*c)*e^12)))/((a*b^2*c^3 - 4*a^2*c^4)*d^6 -
3*(a*b^3*c^2 - 4*a^2*b*c^3)*d^5*e + 3*(a*b^4*c - 3*a^2*b^2*c^2 - 4*a^3*c^3)*d^4*e^2 - (a*b^5 + 2*a^2*b^3*c -
24*a^3*b*c^2)*d^3*e^3 + 3*(a^2*b^4 - 3*a^3*b^2*c - 4*a^4*c^2)*d^2*e^4 - 3*(a^3*b^3 - 4*a^4*b*c)*d*e^5 + (a^4*b
^2 - 4*a^5*c)*e^6))*log(-(b*c^5*d^4*x^2 - 2*a*c^5*d^4 + 4*(a*b^3*c^2 - 2*a^2*b*c^3)*x^2*e^4 + 2*sqrt(1/2)*(2*(
a*b^2*c^4 - 4*a^2*c^5)*d^4*x*e - 7*(a*b^3*c^3 - 4*a^2*b*c^4)*d^3*x*e^2 + 3*(3*a*b^4*c^2 - 14*a^2*b^2*c^3 + 8*a
^3*c^4)*d^2*x*e^3 - (5*a*b^5*c - 27*a^2*b^3*c^2 + 28*a^3*b*c^3)*d*x*e^4 + (a*b^6 - 6*a^2*b^4*c + 8*a^3*b^2*c^2
)*x*e^5 + (2*(a^2*b^2*c^5 - 4*a^3*c^6)*d^8*x - 8*(a^2*b^3*c^4 - 4*a^3*b*c^5)*d^7*x*e + (13*a^2*b^4*c^3 - 48*a^
3*b^2*c^4 - 16*a^4*c^5)*d^6*x*e^2 - (11*a^2*b^5*c^2 - 32*a^3*b^3*c^3 - 48*a^4*b*c^4)*d^5*x*e^3 + 5*(a^2*b^6*c
- a^3*b^4*c^2 - 12*a^4*b^2*c^3)*d^4*x*e^4 - (a^2*b^7 + 6*a^3*b^5*c - 40*a^4*b^3*c^2)*d^3*x*e^5 + (3*a^3*b^6 -
9*a^4*b^4*c - 16*a^5*b^2*c^2 + 16*a^6*c^3)*d^2*x*e^6 - (3*a^4*b^5 - 16*a^5*b^3*c + 16*a^6*b*c^2)*d*x*e^7 + (a^
5*b^4 - 6*a^6*b^2*c + 8*a^7*c^2)*x*e^8)*sqrt((c^6*d^6 - 6*b*c^5*d^5*e + 3*(5*b^2*c^4 - 2*a*c^5)*d^4*e^2 - 2*(1
0*b^3*c^3 - 11*a*b*c^4)*d^3*e^3 + 3*(5*b^4*c^2 - 10*a*b^2*c^3 + 3*a^2*c^4)*d^2*e^4 - 6*(b^5*c - 3*a*b^3*c^2 +
2*a^2*b*c^3)*d*e^5 + (b^6 - 4*a*b^4*c + 4*a^2*b^2*c^2)*e^6)/((a^2*b^2*c^6 - 4*a^3*c^7)*d^12 - 6*(a^2*b^3*c^5 -
4*a^3*b*c^6)*d^11*e + 3*(5*a^2*b^4*c^4 - 18*a^3*b^2*c^5 - 8*a^4*c^6)*d^10*e^2 - 10*(2*a^2*b^5*c^3 - 5*a^3*b^3
*c^4 - 12*a^4*b*c^5)*d^9*e^3 + 15*(a^2*b^6*c^2 - 15*a^4*b^2*c^4 - 4*a^5*c^5)*d^8*e^4 - 6*(a^2*b^7*c + 6*a^3*b^
5*c^2 - 30*a^4*b^3*c^3 - 40*a^5*b*c^4)*d^7*e^5 + (a^2*b^8 + 26*a^3*b^6*c - 30*a^4*b^4*c^2 - 340*a^5*b^2*c^3 -
80*a^6*c^4)*d^6*e^6 - 6*(a^3*b^7 + 6*a^4*b^5*c - 30*a^5*b^3*c^2 - 40*a^6*b*c^3)*d^5*e^7 + 15*(a^4*b^6 - 15*a^6
*b^2*c^2 - 4*a^7*c^3)*d^4*e^8 - 10*(2*a^5*b^5 - 5*a^6*b^3*c - 12*a^7*b*c^2)*d^3*e^9 + 3*(5*a^6*b^4 - 18*a^7*b^
2*c - 8*a^8*c^2)*d^2*e^10 - 6*(a^7*b^3 - 4*a^8*b*c)*d*e^11 + (a^8*b^2 - 4*a^9*c)*e^12)))*sqrt(x^2*e + d)*sqrt(
-(b*c^3*d^3 - 3*(b^2*c^2 - 2*a*c^3)*d^2*e + 3*(b^3*c - 3*a*b*c^2)*d*e^2 - (b^4 - 4*a*b^2*c + 2*a^2*c^2)*e^3 +
((a*b^2*c^3 - 4*a^2*c^4)*d^6 - 3*(a*b^3*c^2 - 4*a^2*b*c^3)*d^5*e + 3*(a*b^4*c - 3*a^2*b^2*c^2 - 4*a^3*c^3)*d^4
*e^2 - (a*b^5 + 2*a^2*b^3*c - 24*a^3*b*c^2)*d^3*e^3 + 3*(a^2*b^4 - 3*a^3*b^2*c - 4*a^4*c^2)*d^2*e^4 - 3*(a^3*b
^3 - 4*a^4*b*c)*d*e^5 + (a^4*b^2 - 4*a^5*c)*e^6)*sqrt((c^6*d^6 - 6*b*c^5*d^5*e + 3*(5*b^2*c^4 - 2*a*c^5)*d^4*e
^2 - 2*(10*b^3*c^3 - 11*a*b*c^4)*d^3*e^3 + 3*(5*b^4*c^2 - 10*a*b^2*c^3 + 3*a^2*c^4)*d^2*e^4 - 6*(b^5*c - 3*a*b
^3*c^2 + 2*a^2*b*c^3)*d*e^5 + (b^6 - 4*a*b^4*c + 4*a^2*b^2*c^2)*e^6)/((a^2*b^2*c^6 - 4*a^3*c^7)*d^12 - 6*(a^2*
b^3*c^5 - 4*a^3*b*c^6)*d^11*e + 3*(5*a^2*b^4*c^4 - 18*a^3*b^2*c^5 - 8*a^4*c^6)*d^10*e^2 - 10*(2*a^2*b^5*c^3 -
5*a^3*b^3*c^4 - 12*a^4*b*c^5)*d^9*e^3 + 15*(a^2*b^6*c^2 - 15*a^4*b^2*c^4 - 4*a^5*c^5)*d^8*e^4 - 6*(a^2*b^7*c +
6*a^3*b^5*c^2 - 30*a^4*b^3*c^3 - 40*a^5*b*c^4)*d^7*e^5 + (a^2*b^8 + 26*a^3*b^6*c - 30*a^4*b^4*c^2 - 340*a^5*b
^2*c^3 - 80*a^6*c^4)*d^6*e^6 - 6*(a^3*b^7 + 6*a^4*b^5*c - 30*a^5*b^3*c^2 - 40*a^6*b*c^3)*d^5*e^7 + 15*(a^4*b^6
- 15*a^6*b^2*c^2 - 4*a^7*c^3)*d^4*e^8 - 10*(2*a^5*b^5 - 5*a^6*b^3*c - 12*a^7*b*c^2)*d^3*e^9 + 3*(5*a^6*b^4 -
18*a^7*b^2*c - 8*a^8*c^2)*d^2*e^10 - 6*(a^7*b^3 - 4*a^8*b*c)*d*e^11 + (a^8*b^2 - 4*a^9*c)*e^12)))/((a*b^2*c^3
- 4*a^2*c^4)*d^6 - 3*(a*b^3*c^2 - 4*a^2*b*c^3)*d^5*e + 3*(a*b^4*c - 3*a^2*b^2*c^2 - 4*a^3*c^3)*d^4*e^2 - (a*b^
5 + 2*a^2*b^3*c - 24*a^3*b*c^2)*d^3*e^3 + 3*(a^2*b^4 - 3*a^3*b^2*c - 4*a^4*c^2)*d^2*e^4 - 3*(a^3*b^3 - 4*a^4*b
*c)*d*e^5 + (a^4*b^2 - 4*a^5*c)*e^6)) - ((b^4*c...

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (d + e x^{2}\right )^{\frac {3}{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/(e*x**2+d)**(3/2)/(c*x**4+b*x**2+a),x)

[Out]

Integral(1/((d + e*x**2)**(3/2)*(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(1/(e*x^2+d)^(3/2)/(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 {1}{{\left (e\,x^2+d\right )}^{3/2}\,\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/((d + e*x^2)^(3/2)*(a + b*x^2 + c*x^4)),x)

[Out]

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

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