3.4.73 \(\int \frac {(d+e x^2)^{3/2}}{a+b x^2+c x^4} \, dx\) [373]

Optimal. Leaf size=487 \[ \frac {\left (2 c^2 d^2+b \left (b-\sqrt {b^2-4 a c}\right ) e^2-2 c e \left (b d-\sqrt {b^2-4 a c} d+a e\right )\right ) \tan ^{-1}\left (\frac {\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e} x}{\sqrt {b-\sqrt {b^2-4 a c}} \sqrt {d+e x^2}}\right )}{c \sqrt {b^2-4 a c} \sqrt {b-\sqrt {b^2-4 a c}} \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}-\frac {\left (2 c^2 d^2+b \left (b+\sqrt {b^2-4 a c}\right ) e^2-2 c e \left (b d+\sqrt {b^2-4 a c} d+a e\right )\right ) \tan ^{-1}\left (\frac {\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e} x}{\sqrt {b+\sqrt {b^2-4 a c}} \sqrt {d+e x^2}}\right )}{c \sqrt {b^2-4 a c} \sqrt {b+\sqrt {b^2-4 a c}} \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}+\frac {\sqrt {e} \left (3 c d-\left (b-\sqrt {b^2-4 a c}\right ) e\right ) \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{2 c \sqrt {b^2-4 a c}}-\frac {\sqrt {e} \left (3 c d-\left (b+\sqrt {b^2-4 a c}\right ) e\right ) \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{2 c \sqrt {b^2-4 a c}} \]

[Out]

1/2*arctanh(x*e^(1/2)/(e*x^2+d)^(1/2))*(3*c*d-e*(b-(-4*a*c+b^2)^(1/2)))*e^(1/2)/c/(-4*a*c+b^2)^(1/2)-1/2*arcta
nh(x*e^(1/2)/(e*x^2+d)^(1/2))*(3*c*d-e*(b+(-4*a*c+b^2)^(1/2)))*e^(1/2)/c/(-4*a*c+b^2)^(1/2)+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))*(2*c^2*d^2+b*e^2*(b-(-4*a*c+b^2)^(
1/2))-2*c*e*(b*d+a*e-d*(-4*a*c+b^2)^(1/2)))/c/(-4*a*c+b^2)^(1/2)/(2*c*d-e*(b-(-4*a*c+b^2)^(1/2)))^(1/2)/(b-(-4
*a*c+b^2)^(1/2))^(1/2)-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))*(2*c^2*d^2+b*e^2*(b+(-4*a*c+b^2)^(1/2))-2*c*e*(b*d+a*e+d*(-4*a*c+b^2)^(1/2)))/c/(-4*a*c+b^2)^(1/2)/(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 = 1.07, antiderivative size = 487, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 7, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {1188, 427, 537, 223, 212, 385, 211} \begin {gather*} \frac {\left (-2 c e \left (-d \sqrt {b^2-4 a c}+a e+b d\right )+b e^2 \left (b-\sqrt {b^2-4 a c}\right )+2 c^2 d^2\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 )}{c \sqrt {b^2-4 a c} \sqrt {b-\sqrt {b^2-4 a c}} \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}-\frac {\left (-2 c e \left (d \sqrt {b^2-4 a c}+a e+b d\right )+b e^2 \left (\sqrt {b^2-4 a c}+b\right )+2 c^2 d^2\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 )}{c \sqrt {b^2-4 a c} \sqrt {\sqrt {b^2-4 a c}+b} \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}+\frac {\sqrt {e} \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right ) \left (3 c d-e \left (b-\sqrt {b^2-4 a c}\right )\right )}{2 c \sqrt {b^2-4 a c}}-\frac {\sqrt {e} \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right ) \left (3 c d-e \left (\sqrt {b^2-4 a c}+b\right )\right )}{2 c \sqrt {b^2-4 a c}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 223

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

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 427

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[d*x*(a + b*x^n)^(p + 1)*((c
+ d*x^n)^(q - 1)/(b*(n*(p + q) + 1))), x] + Dist[1/(b*(n*(p + q) + 1)), Int[(a + b*x^n)^p*(c + d*x^n)^(q - 2)
*Simp[c*(b*c*(n*(p + q) + 1) - a*d) + d*(b*c*(n*(p + 2*q - 1) + 1) - a*d*(n*(q - 1) + 1))*x^n, x], x], x] /; F
reeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d, 0] && GtQ[q, 1] && NeQ[n*(p + q) + 1, 0] &&  !IGtQ[p, 1] && IntB
inomialQ[a, b, c, d, n, p, q, x]

Rule 537

Int[((e_) + (f_.)*(x_)^(n_))/(((a_) + (b_.)*(x_)^(n_))*Sqrt[(c_) + (d_.)*(x_)^(n_)]), x_Symbol] :> Dist[f/b, I
nt[1/Sqrt[c + d*x^n], x], x] + Dist[(b*e - a*f)/b, Int[1/((a + b*x^n)*Sqrt[c + d*x^n]), x], x] /; FreeQ[{a, b,
c, d, e, f, n}, x]

Rule 1188

Int[((d_) + (e_.)*(x_)^2)^(q_)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{r = Rt[b^2 - 4*a*c, 2]
}, Dist[2*(c/r), Int[(d + e*x^2)^q/(b - r + 2*c*x^2), x], x] - Dist[2*(c/r), Int[(d + e*x^2)^q/(b + r + 2*c*x^
2), x], x]] /; FreeQ[{a, b, c, d, e, q}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] &&  !Integ
erQ[q]

Rubi steps

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

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(9290\) vs. \(2(487)=974\).
time = 16.13, size = 9290, normalized size = 19.08 \begin {gather*} \text {Result too large to show} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x^2)^(3/2)/(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 = 0.13, size = 215, normalized size = 0.44

method	result	size
default	\(-e^{\frac {3}{2}} \left (\frac {\ln \left (\sqrt {e \,x^{2}+d}-\sqrt {e}\, x \right )}{c}-\frac {\munderset {\textit {\_R} =\RootOf \left (c \,\textit {\_Z}^{4}+\left (4 e b -4 c d \right ) \textit {\_Z}^{3}+\left (16 a \,e^{2}-8 d e b +6 c \,d^{2}\right ) \textit {\_Z}^{2}+\left (4 d^{2} e b -4 c \,d^{3}\right ) \textit {\_Z} +d^{4} c \right )}{\sum }\frac {\left (\left (e b -2 c d \right ) \textit {\_R}^{2}+2 e \left (2 a e -b d \right ) \textit {\_R} +d^{2} e b -2 c \,d^{3}\right ) \ln \left (\left (\sqrt {e \,x^{2}+d}-\sqrt {e}\, x \right )^{2}-\textit {\_R} \right )}{c \,\textit {\_R}^{3}+3 \textit {\_R}^{2} b e -3 \textit {\_R}^{2} c d +8 \textit {\_R} a \,e^{2}-4 \textit {\_R} b d e +3 c \,d^{2} \textit {\_R} +d^{2} e b -c \,d^{3}}}{2 c}\right )\)	\(215\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

integrate((x^2*e + d)^(3/2)/(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. 3855 vs. \(2 (423) = 846\).
time = 15.37, size = 3855, normalized size = 7.92 \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)^(3/2)/(c*x^4+b*x^2+a),x, algorithm="fricas")

[Out]

-1/4*(sqrt(1/2)*c*sqrt(-(b*c^2*d^3 - 6*a*c^2*d^2*e + 3*a*b*c*d*e^2 - (a*b^2 - 2*a^2*c)*e^3 + (a*b^2*c^2 - 4*a^
2*c^3)*sqrt((c^4*d^6 - 6*a*c^3*d^4*e^2 + 2*a*b*c^2*d^3*e^3 + 9*a^2*c^2*d^2*e^4 - 6*a^2*b*c*d*e^5 + a^2*b^2*e^6
)/(a^2*b^2*c^4 - 4*a^3*c^5)))/(a*b^2*c^2 - 4*a^2*c^3))*log(-(b*c^3*d^6*x^2 - 2*a*c^3*d^6 - 4*a^3*b*x^2*e^6 + 2
*sqrt(1/2)*((a*b^2*c^2 - 4*a^2*c^3)*d^3*x*e - 3*(a^2*b^2*c - 4*a^3*c^2)*d*x*e^3 + (a^2*b^3 - 4*a^3*b*c)*x*e^4
- (2*(a^2*b^2*c^3 - 4*a^3*c^4)*d*x - (a^2*b^3*c^2 - 4*a^3*b*c^3)*x*e)*sqrt((c^4*d^6 - 6*a*c^3*d^4*e^2 + 2*a*b*
c^2*d^3*e^3 + 9*a^2*c^2*d^2*e^4 - 6*a^2*b*c*d*e^5 + a^2*b^2*e^6)/(a^2*b^2*c^4 - 4*a^3*c^5)))*sqrt(x^2*e + d)*s
qrt(-(b*c^2*d^3 - 6*a*c^2*d^2*e + 3*a*b*c*d*e^2 - (a*b^2 - 2*a^2*c)*e^3 + (a*b^2*c^2 - 4*a^2*c^3)*sqrt((c^4*d^
6 - 6*a*c^3*d^4*e^2 + 2*a*b*c^2*d^3*e^3 + 9*a^2*c^2*d^2*e^4 - 6*a^2*b*c*d*e^5 + a^2*b^2*e^6)/(a^2*b^2*c^4 - 4*
a^3*c^5)))/(a*b^2*c^2 - 4*a^2*c^3)) - (2*a^3*b*d - (5*a^2*b^2 + 12*a^3*c)*d*x^2)*e^5 - ((a*b^3 + 19*a^2*b*c)*d
^2*x^2 - 2*(a^2*b^2 + 3*a^3*c)*d^2)*e^4 - 4*(2*a^2*b*c*d^3 - (a*b^2*c + 2*a^2*c^2)*d^3*x^2)*e^3 + 2*(a*b*c^2*d
^4*x^2 + 2*a^2*c^2*d^4)*e^2 + (2*a*b*c^2*d^5 - (b^2*c^2 + 4*a*c^3)*d^5*x^2)*e + ((a*b^2*c^3 - 4*a^2*c^4)*d^3*x
^2 - (a*b^3*c^2 - 4*a^2*b*c^3)*d^2*x^2*e + (a^2*b^2*c^2 - 4*a^3*c^3)*d*x^2*e^2)*sqrt((c^4*d^6 - 6*a*c^3*d^4*e^
2 + 2*a*b*c^2*d^3*e^3 + 9*a^2*c^2*d^2*e^4 - 6*a^2*b*c*d*e^5 + a^2*b^2*e^6)/(a^2*b^2*c^4 - 4*a^3*c^5)))/x^2) -
sqrt(1/2)*c*sqrt(-(b*c^2*d^3 - 6*a*c^2*d^2*e + 3*a*b*c*d*e^2 - (a*b^2 - 2*a^2*c)*e^3 + (a*b^2*c^2 - 4*a^2*c^3)
*sqrt((c^4*d^6 - 6*a*c^3*d^4*e^2 + 2*a*b*c^2*d^3*e^3 + 9*a^2*c^2*d^2*e^4 - 6*a^2*b*c*d*e^5 + a^2*b^2*e^6)/(a^2
*b^2*c^4 - 4*a^3*c^5)))/(a*b^2*c^2 - 4*a^2*c^3))*log(-(b*c^3*d^6*x^2 - 2*a*c^3*d^6 - 4*a^3*b*x^2*e^6 - 2*sqrt(
1/2)*((a*b^2*c^2 - 4*a^2*c^3)*d^3*x*e - 3*(a^2*b^2*c - 4*a^3*c^2)*d*x*e^3 + (a^2*b^3 - 4*a^3*b*c)*x*e^4 - (2*(
a^2*b^2*c^3 - 4*a^3*c^4)*d*x - (a^2*b^3*c^2 - 4*a^3*b*c^3)*x*e)*sqrt((c^4*d^6 - 6*a*c^3*d^4*e^2 + 2*a*b*c^2*d^
3*e^3 + 9*a^2*c^2*d^2*e^4 - 6*a^2*b*c*d*e^5 + a^2*b^2*e^6)/(a^2*b^2*c^4 - 4*a^3*c^5)))*sqrt(x^2*e + d)*sqrt(-(
b*c^2*d^3 - 6*a*c^2*d^2*e + 3*a*b*c*d*e^2 - (a*b^2 - 2*a^2*c)*e^3 + (a*b^2*c^2 - 4*a^2*c^3)*sqrt((c^4*d^6 - 6*
a*c^3*d^4*e^2 + 2*a*b*c^2*d^3*e^3 + 9*a^2*c^2*d^2*e^4 - 6*a^2*b*c*d*e^5 + a^2*b^2*e^6)/(a^2*b^2*c^4 - 4*a^3*c^
5)))/(a*b^2*c^2 - 4*a^2*c^3)) - (2*a^3*b*d - (5*a^2*b^2 + 12*a^3*c)*d*x^2)*e^5 - ((a*b^3 + 19*a^2*b*c)*d^2*x^2
- 2*(a^2*b^2 + 3*a^3*c)*d^2)*e^4 - 4*(2*a^2*b*c*d^3 - (a*b^2*c + 2*a^2*c^2)*d^3*x^2)*e^3 + 2*(a*b*c^2*d^4*x^2
+ 2*a^2*c^2*d^4)*e^2 + (2*a*b*c^2*d^5 - (b^2*c^2 + 4*a*c^3)*d^5*x^2)*e + ((a*b^2*c^3 - 4*a^2*c^4)*d^3*x^2 - (
a*b^3*c^2 - 4*a^2*b*c^3)*d^2*x^2*e + (a^2*b^2*c^2 - 4*a^3*c^3)*d*x^2*e^2)*sqrt((c^4*d^6 - 6*a*c^3*d^4*e^2 + 2*
a*b*c^2*d^3*e^3 + 9*a^2*c^2*d^2*e^4 - 6*a^2*b*c*d*e^5 + a^2*b^2*e^6)/(a^2*b^2*c^4 - 4*a^3*c^5)))/x^2) + sqrt(1
/2)*c*sqrt(-(b*c^2*d^3 - 6*a*c^2*d^2*e + 3*a*b*c*d*e^2 - (a*b^2 - 2*a^2*c)*e^3 - (a*b^2*c^2 - 4*a^2*c^3)*sqrt(
(c^4*d^6 - 6*a*c^3*d^4*e^2 + 2*a*b*c^2*d^3*e^3 + 9*a^2*c^2*d^2*e^4 - 6*a^2*b*c*d*e^5 + a^2*b^2*e^6)/(a^2*b^2*c
^4 - 4*a^3*c^5)))/(a*b^2*c^2 - 4*a^2*c^3))*log(-(b*c^3*d^6*x^2 - 2*a*c^3*d^6 - 4*a^3*b*x^2*e^6 + 2*sqrt(1/2)*(
(a*b^2*c^2 - 4*a^2*c^3)*d^3*x*e - 3*(a^2*b^2*c - 4*a^3*c^2)*d*x*e^3 + (a^2*b^3 - 4*a^3*b*c)*x*e^4 + (2*(a^2*b^
2*c^3 - 4*a^3*c^4)*d*x - (a^2*b^3*c^2 - 4*a^3*b*c^3)*x*e)*sqrt((c^4*d^6 - 6*a*c^3*d^4*e^2 + 2*a*b*c^2*d^3*e^3
+ 9*a^2*c^2*d^2*e^4 - 6*a^2*b*c*d*e^5 + a^2*b^2*e^6)/(a^2*b^2*c^4 - 4*a^3*c^5)))*sqrt(x^2*e + d)*sqrt(-(b*c^2*
d^3 - 6*a*c^2*d^2*e + 3*a*b*c*d*e^2 - (a*b^2 - 2*a^2*c)*e^3 - (a*b^2*c^2 - 4*a^2*c^3)*sqrt((c^4*d^6 - 6*a*c^3*
d^4*e^2 + 2*a*b*c^2*d^3*e^3 + 9*a^2*c^2*d^2*e^4 - 6*a^2*b*c*d*e^5 + a^2*b^2*e^6)/(a^2*b^2*c^4 - 4*a^3*c^5)))/(
a*b^2*c^2 - 4*a^2*c^3)) - (2*a^3*b*d - (5*a^2*b^2 + 12*a^3*c)*d*x^2)*e^5 - ((a*b^3 + 19*a^2*b*c)*d^2*x^2 - 2*(
a^2*b^2 + 3*a^3*c)*d^2)*e^4 - 4*(2*a^2*b*c*d^3 - (a*b^2*c + 2*a^2*c^2)*d^3*x^2)*e^3 + 2*(a*b*c^2*d^4*x^2 + 2*a
^2*c^2*d^4)*e^2 + (2*a*b*c^2*d^5 - (b^2*c^2 + 4*a*c^3)*d^5*x^2)*e - ((a*b^2*c^3 - 4*a^2*c^4)*d^3*x^2 - (a*b^3*
c^2 - 4*a^2*b*c^3)*d^2*x^2*e + (a^2*b^2*c^2 - 4*a^3*c^3)*d*x^2*e^2)*sqrt((c^4*d^6 - 6*a*c^3*d^4*e^2 + 2*a*b*c^
2*d^3*e^3 + 9*a^2*c^2*d^2*e^4 - 6*a^2*b*c*d*e^5 + a^2*b^2*e^6)/(a^2*b^2*c^4 - 4*a^3*c^5)))/x^2) - sqrt(1/2)*c*
sqrt(-(b*c^2*d^3 - 6*a*c^2*d^2*e + 3*a*b*c*d*e^2 - (a*b^2 - 2*a^2*c)*e^3 - (a*b^2*c^2 - 4*a^2*c^3)*sqrt((c^4*d
^6 - 6*a*c^3*d^4*e^2 + 2*a*b*c^2*d^3*e^3 + 9*a^2*c^2*d^2*e^4 - 6*a^2*b*c*d*e^5 + a^2*b^2*e^6)/(a^2*b^2*c^4 - 4
*a^3*c^5)))/(a*b^2*c^2 - 4*a^2*c^3))*log(-(b*c^3*d^6*x^2 - 2*a*c^3*d^6 - 4*a^3*b*x^2*e^6 - 2*sqrt(1/2)*((a*b^2
*c^2 - 4*a^2*c^3)*d^3*x*e - 3*(a^2*b^2*c - 4*a^3*c^2)*d*x*e^3 + (a^2*b^3 - 4*a^3*b*c)*x*e^4 + (2*(a^2*b^2*c^3
- 4*a^3*c^4)*d*x - (a^2*b^3*c^2 - 4*a^3*b*c^3)*x*e)*sqrt((c^4*d^6 - 6*a*c^3*d^4*e^2 + 2*a*b*c^2*d^3*e^3 + 9*a^
2*c^2*d^2*e^4 - 6*a^2*b*c*d*e^5 + a^2*b^2*e^6)/(a^2*b^2*c^4 - 4*a^3*c^5)))*sqrt(x^2*e + d)*sqrt(-(b*c^2*d^3 -
6*a*c^2*d^2*e + 3*a*b*c*d*e^2 - (a*b^2 - 2*a^2*...

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((d + e*x**2)**(3/2)/(a + b*x**2 + c*x**4), x)

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Giac [A]
time = 7.77, size = 27, normalized size = 0.06 \begin {gather*} -\frac {e^{\frac {3}{2}} \log \left ({\left (x e^{\frac {1}{2}} - \sqrt {x^{2} e + d}\right )}^{2}\right )}{2 \, c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*e^(3/2)*log((x*e^(1/2) - sqrt(x^2*e + d))^2)/c

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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