3.4.74 \(\int \frac {(d+e x^2)^{3/2}}{x^2 (a+b x^2+c x^4)} \, dx\) [374]

Optimal. Leaf size=260 \[ -\frac {d \sqrt {d+e x^2}}{a x}-\frac {\left (2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e\right )^{3/2} \tan ^{-1}\left (\frac {\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e} x}{\sqrt {b-\sqrt {b^2-4 a c}} \sqrt {d+e x^2}}\right )}{\sqrt {b^2-4 a c} \left (b-\sqrt {b^2-4 a c}\right )^{3/2}}+\frac {\left (2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e\right )^{3/2} \tan ^{-1}\left (\frac {\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e} x}{\sqrt {b+\sqrt {b^2-4 a c}} \sqrt {d+e x^2}}\right )}{\sqrt {b^2-4 a c} \left (b+\sqrt {b^2-4 a c}\right )^{3/2}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.58, antiderivative size = 432, normalized size of antiderivative = 1.66, number of steps used = 16, number of rules used = 8, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.276, Rules used = {1309, 283, 223, 212, 1706, 399, 385, 211} \begin {gather*} -\frac {\sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )} \left (\frac {b d-2 a e}{\sqrt {b^2-4 a c}}+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 )}{2 a \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {\sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )} \left (d-\frac {b d-2 a e}{\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 )}{2 a \sqrt {\sqrt {b^2-4 a c}+b}}-\frac {\sqrt {e} \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right ) \left (d-\frac {b d-2 a e}{\sqrt {b^2-4 a c}}\right )}{2 a}-\frac {\sqrt {e} \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right ) \left (\frac {b d-2 a e}{\sqrt {b^2-4 a c}}+d\right )}{2 a}-\frac {d \sqrt {d+e x^2}}{a x}+\frac {d \sqrt {e} \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{a} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 283

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

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 399

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

Rubi steps

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

________________________________________________________________________________________

Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(7789\) vs. \(2(260)=520\).
time = 16.26, size = 7789, normalized size = 29.96 \begin {gather*} \text {Result too large to show} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Result too large to show

________________________________________________________________________________________

Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.16, size = 351, normalized size = 1.35

method	result	size
risch	\(-\frac {d \sqrt {e \,x^{2}+d}}{a x}-\frac {\sqrt {e}\, \left (\munderset {\textit {\_R} =\RootOf \left (c \,\textit {\_Z}^{4}+\left (4 e b -4 c d \right ) \textit {\_Z}^{3}+\left (16 a \,e^{2}-8 d e b +6 c \,d^{2}\right ) \textit {\_Z}^{2}+\left (4 d^{2} e b -4 c \,d^{3}\right ) \textit {\_Z} +d^{4} c \right )}{\sum }\frac {\left (\left (a \,e^{2}-c \,d^{2}\right ) \textit {\_R}^{2}+2 d \left (3 a \,e^{2}-2 d e b +c \,d^{2}\right ) \textit {\_R} +d^{2} e^{2} a -d^{4} c \right ) \ln \left (\left (\sqrt {e \,x^{2}+d}-\sqrt {e}\, x \right )^{2}-\textit {\_R} \right )}{c \,\textit {\_R}^{3}+3 \textit {\_R}^{2} b e -3 \textit {\_R}^{2} c d +8 \textit {\_R} a \,e^{2}-4 \textit {\_R} b d e +3 c \,d^{2} \textit {\_R} +d^{2} e b -c \,d^{3}}\right )}{2 a}\)	\(224\)
default	\(\frac {\sqrt {e}\, \left (\frac {\left (\sqrt {e \,x^{2}+d}-\sqrt {e}\, x \right )^{2}}{2}-2 \left (\munderset {\textit {\_R} =\RootOf \left (c \,\textit {\_Z}^{4}+\left (4 e b -4 c d \right ) \textit {\_Z}^{3}+\left (16 a \,e^{2}-8 d e b +6 c \,d^{2}\right ) \textit {\_Z}^{2}+\left (4 d^{2} e b -4 c \,d^{3}\right ) \textit {\_Z} +d^{4} c \right )}{\sum }\frac {\left (\left (a \,e^{2}-c \,d^{2}\right ) \textit {\_R}^{2}+2 d \left (3 a \,e^{2}-2 d e b +c \,d^{2}\right ) \textit {\_R} +d^{2} e^{2} a -d^{4} c \right ) \ln \left (\left (\sqrt {e \,x^{2}+d}-\sqrt {e}\, x \right )^{2}-\textit {\_R} \right )}{c \,\textit {\_R}^{3}+3 \textit {\_R}^{2} b e -3 \textit {\_R}^{2} c d +8 \textit {\_R} a \,e^{2}-4 \textit {\_R} b d e +3 c \,d^{2} \textit {\_R} +d^{2} e b -c \,d^{3}}\right )-\frac {d^{2}}{2 \left (\sqrt {e \,x^{2}+d}-\sqrt {e}\, x \right )^{2}}+6 d \ln \left (\sqrt {e \,x^{2}+d}-\sqrt {e}\, x \right )\right )}{4 a}+\frac {-\frac {\left (e \,x^{2}+d \right )^{\frac {5}{2}}}{d x}+\frac {4 e \left (\frac {x \left (e \,x^{2}+d \right )^{\frac {3}{2}}}{4}+\frac {3 d \left (\frac {x \sqrt {e \,x^{2}+d}}{2}+\frac {d \ln \left (\sqrt {e}\, x +\sqrt {e \,x^{2}+d}\right )}{2 \sqrt {e}}\right )}{4}\right )}{d}}{a}\)	\(351\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4/a*e^(1/2)*(1/2*((e*x^2+d)^(1/2)-e^(1/2)*x)^2-2*sum(((a*e^2-c*d^2)*_R^2+2*d*(3*a*e^2-2*b*d*e+c*d^2)*_R+d^2*
e^2*a-d^4*c)/(_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*d^2/((e*x^2+d)^(1/2)-e^(1/2)*x)^2+6*d*ln((e*x^2+d)^(1/2)-e^(1/2)*x))+1/a*(-1/d/x*(e*x^2+d)^(5/2)+4*
e/d*(1/4*x*(e*x^2+d)^(3/2)+3/4*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)))))

________________________________________________________________________________________

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)/x^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^2), x)

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 4096 vs. \(2 (225) = 450\).
time = 8.47, size = 4096, normalized size = 15.75 \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)/x^2/(c*x^4+b*x^2+a),x, algorithm="fricas")

[Out]

1/4*(sqrt(1/2)*a*x*sqrt(-(3*a^2*b*d*e^2 + (b^3 - 3*a*b*c)*d^3 - 2*a^3*e^3 - 3*(a*b^2 - 2*a^2*c)*d^2*e + (a^3*b
^2 - 4*a^4*c)*sqrt(-(18*a^3*b*d^3*e^3 - (b^4 - 2*a*b^2*c + a^2*c^2)*d^6 - 9*a^4*d^2*e^4 + 6*(a*b^3 - a^2*b*c)*
d^5*e - 3*(5*a^2*b^2 - 2*a^3*c)*d^4*e^2)/(a^6*b^2 - 4*a^7*c)))/(a^3*b^2 - 4*a^4*c))*log(-((b^3*c - a*b*c^2)*d^
6*x^2 - 12*a^4*d*x^2*e^5 - 2*(a*b^2*c - a^2*c^2)*d^6 + 2*sqrt(1/2)*((a*b^4 - 5*a^2*b^2*c + 4*a^3*c^2)*d^4*x -
3*(a^2*b^3 - 4*a^3*b*c)*d^3*x*e + 3*(a^3*b^2 - 4*a^4*c)*d^2*x*e^2 - ((a^4*b^3 - 4*a^5*b*c)*d*x - 2*(a^5*b^2 -
4*a^6*c)*x*e)*sqrt(-(18*a^3*b*d^3*e^3 - (b^4 - 2*a*b^2*c + a^2*c^2)*d^6 - 9*a^4*d^2*e^4 + 6*(a*b^3 - a^2*b*c)*
d^5*e - 3*(5*a^2*b^2 - 2*a^3*c)*d^4*e^2)/(a^6*b^2 - 4*a^7*c)))*sqrt(x^2*e + d)*sqrt(-(3*a^2*b*d*e^2 + (b^3 - 3
*a*b*c)*d^3 - 2*a^3*e^3 - 3*(a*b^2 - 2*a^2*c)*d^2*e + (a^3*b^2 - 4*a^4*c)*sqrt(-(18*a^3*b*d^3*e^3 - (b^4 - 2*a
*b^2*c + a^2*c^2)*d^6 - 9*a^4*d^2*e^4 + 6*(a*b^3 - a^2*b*c)*d^5*e - 3*(5*a^2*b^2 - 2*a^3*c)*d^4*e^2)/(a^6*b^2
- 4*a^7*c)))/(a^3*b^2 - 4*a^4*c)) + 3*(9*a^3*b*d^2*x^2 - 2*a^4*d^2)*e^4 + 2*(6*a^3*b*d^3 - (11*a^2*b^2 + 4*a^3
*c)*d^3*x^2)*e^3 + 2*((4*a*b^3 + 5*a^2*b*c)*d^4*x^2 - 2*(2*a^2*b^2 + a^3*c)*d^4)*e^2 - ((b^4 + 6*a*b^2*c - 4*a
^2*c^2)*d^5*x^2 - 2*(a*b^3 + 2*a^2*b*c)*d^5)*e + ((a^3*b^2*c - 4*a^4*c^2)*d^3*x^2 - (a^3*b^3 - 4*a^4*b*c)*d^2*
x^2*e + (a^4*b^2 - 4*a^5*c)*d*x^2*e^2)*sqrt(-(18*a^3*b*d^3*e^3 - (b^4 - 2*a*b^2*c + a^2*c^2)*d^6 - 9*a^4*d^2*e
^4 + 6*(a*b^3 - a^2*b*c)*d^5*e - 3*(5*a^2*b^2 - 2*a^3*c)*d^4*e^2)/(a^6*b^2 - 4*a^7*c)))/x^2) - sqrt(1/2)*a*x*s
qrt(-(3*a^2*b*d*e^2 + (b^3 - 3*a*b*c)*d^3 - 2*a^3*e^3 - 3*(a*b^2 - 2*a^2*c)*d^2*e + (a^3*b^2 - 4*a^4*c)*sqrt(-
(18*a^3*b*d^3*e^3 - (b^4 - 2*a*b^2*c + a^2*c^2)*d^6 - 9*a^4*d^2*e^4 + 6*(a*b^3 - a^2*b*c)*d^5*e - 3*(5*a^2*b^2
- 2*a^3*c)*d^4*e^2)/(a^6*b^2 - 4*a^7*c)))/(a^3*b^2 - 4*a^4*c))*log(-((b^3*c - a*b*c^2)*d^6*x^2 - 12*a^4*d*x^2
*e^5 - 2*(a*b^2*c - a^2*c^2)*d^6 - 2*sqrt(1/2)*((a*b^4 - 5*a^2*b^2*c + 4*a^3*c^2)*d^4*x - 3*(a^2*b^3 - 4*a^3*b
*c)*d^3*x*e + 3*(a^3*b^2 - 4*a^4*c)*d^2*x*e^2 - ((a^4*b^3 - 4*a^5*b*c)*d*x - 2*(a^5*b^2 - 4*a^6*c)*x*e)*sqrt(-
(18*a^3*b*d^3*e^3 - (b^4 - 2*a*b^2*c + a^2*c^2)*d^6 - 9*a^4*d^2*e^4 + 6*(a*b^3 - a^2*b*c)*d^5*e - 3*(5*a^2*b^2
- 2*a^3*c)*d^4*e^2)/(a^6*b^2 - 4*a^7*c)))*sqrt(x^2*e + d)*sqrt(-(3*a^2*b*d*e^2 + (b^3 - 3*a*b*c)*d^3 - 2*a^3*
e^3 - 3*(a*b^2 - 2*a^2*c)*d^2*e + (a^3*b^2 - 4*a^4*c)*sqrt(-(18*a^3*b*d^3*e^3 - (b^4 - 2*a*b^2*c + a^2*c^2)*d^
6 - 9*a^4*d^2*e^4 + 6*(a*b^3 - a^2*b*c)*d^5*e - 3*(5*a^2*b^2 - 2*a^3*c)*d^4*e^2)/(a^6*b^2 - 4*a^7*c)))/(a^3*b^
2 - 4*a^4*c)) + 3*(9*a^3*b*d^2*x^2 - 2*a^4*d^2)*e^4 + 2*(6*a^3*b*d^3 - (11*a^2*b^2 + 4*a^3*c)*d^3*x^2)*e^3 + 2
*((4*a*b^3 + 5*a^2*b*c)*d^4*x^2 - 2*(2*a^2*b^2 + a^3*c)*d^4)*e^2 - ((b^4 + 6*a*b^2*c - 4*a^2*c^2)*d^5*x^2 - 2*
(a*b^3 + 2*a^2*b*c)*d^5)*e + ((a^3*b^2*c - 4*a^4*c^2)*d^3*x^2 - (a^3*b^3 - 4*a^4*b*c)*d^2*x^2*e + (a^4*b^2 - 4
*a^5*c)*d*x^2*e^2)*sqrt(-(18*a^3*b*d^3*e^3 - (b^4 - 2*a*b^2*c + a^2*c^2)*d^6 - 9*a^4*d^2*e^4 + 6*(a*b^3 - a^2*
b*c)*d^5*e - 3*(5*a^2*b^2 - 2*a^3*c)*d^4*e^2)/(a^6*b^2 - 4*a^7*c)))/x^2) + sqrt(1/2)*a*x*sqrt(-(3*a^2*b*d*e^2
+ (b^3 - 3*a*b*c)*d^3 - 2*a^3*e^3 - 3*(a*b^2 - 2*a^2*c)*d^2*e - (a^3*b^2 - 4*a^4*c)*sqrt(-(18*a^3*b*d^3*e^3 -
(b^4 - 2*a*b^2*c + a^2*c^2)*d^6 - 9*a^4*d^2*e^4 + 6*(a*b^3 - a^2*b*c)*d^5*e - 3*(5*a^2*b^2 - 2*a^3*c)*d^4*e^2)
/(a^6*b^2 - 4*a^7*c)))/(a^3*b^2 - 4*a^4*c))*log(-((b^3*c - a*b*c^2)*d^6*x^2 - 12*a^4*d*x^2*e^5 - 2*(a*b^2*c -
a^2*c^2)*d^6 + 2*sqrt(1/2)*((a*b^4 - 5*a^2*b^2*c + 4*a^3*c^2)*d^4*x - 3*(a^2*b^3 - 4*a^3*b*c)*d^3*x*e + 3*(a^3
*b^2 - 4*a^4*c)*d^2*x*e^2 + ((a^4*b^3 - 4*a^5*b*c)*d*x - 2*(a^5*b^2 - 4*a^6*c)*x*e)*sqrt(-(18*a^3*b*d^3*e^3 -
(b^4 - 2*a*b^2*c + a^2*c^2)*d^6 - 9*a^4*d^2*e^4 + 6*(a*b^3 - a^2*b*c)*d^5*e - 3*(5*a^2*b^2 - 2*a^3*c)*d^4*e^2)
/(a^6*b^2 - 4*a^7*c)))*sqrt(x^2*e + d)*sqrt(-(3*a^2*b*d*e^2 + (b^3 - 3*a*b*c)*d^3 - 2*a^3*e^3 - 3*(a*b^2 - 2*a
^2*c)*d^2*e - (a^3*b^2 - 4*a^4*c)*sqrt(-(18*a^3*b*d^3*e^3 - (b^4 - 2*a*b^2*c + a^2*c^2)*d^6 - 9*a^4*d^2*e^4 +
6*(a*b^3 - a^2*b*c)*d^5*e - 3*(5*a^2*b^2 - 2*a^3*c)*d^4*e^2)/(a^6*b^2 - 4*a^7*c)))/(a^3*b^2 - 4*a^4*c)) + 3*(9
*a^3*b*d^2*x^2 - 2*a^4*d^2)*e^4 + 2*(6*a^3*b*d^3 - (11*a^2*b^2 + 4*a^3*c)*d^3*x^2)*e^3 + 2*((4*a*b^3 + 5*a^2*b
*c)*d^4*x^2 - 2*(2*a^2*b^2 + a^3*c)*d^4)*e^2 - ((b^4 + 6*a*b^2*c - 4*a^2*c^2)*d^5*x^2 - 2*(a*b^3 + 2*a^2*b*c)*
d^5)*e - ((a^3*b^2*c - 4*a^4*c^2)*d^3*x^2 - (a^3*b^3 - 4*a^4*b*c)*d^2*x^2*e + (a^4*b^2 - 4*a^5*c)*d*x^2*e^2)*s
qrt(-(18*a^3*b*d^3*e^3 - (b^4 - 2*a*b^2*c + a^2*c^2)*d^6 - 9*a^4*d^2*e^4 + 6*(a*b^3 - a^2*b*c)*d^5*e - 3*(5*a^
2*b^2 - 2*a^3*c)*d^4*e^2)/(a^6*b^2 - 4*a^7*c)))/x^2) - sqrt(1/2)*a*x*sqrt(-(3*a^2*b*d*e^2 + (b^3 - 3*a*b*c)*d^
3 - 2*a^3*e^3 - 3*(a*b^2 - 2*a^2*c)*d^2*e - (a^3*b^2 - 4*a^4*c)*sqrt(-(18*a^3*b*d^3*e^3 - (b^4 - 2*a*b^2*c + a
^2*c^2)*d^6 - 9*a^4*d^2*e^4 + 6*(a*b^3 - a^2*b*c)*d^5*e - 3*(5*a^2*b^2 - 2*a^3*c)*d^4*e^2)/(a^6*b^2 - 4*a^7*c)
))/(a^3*b^2 - 4*a^4*c))*log(-((b^3*c - a*b*c^2)*d^6*x^2 - 12*a^4*d*x^2*e^5 - 2*(a*b^2*c - a^2*c^2)*d^6 - 2*sqr
t(1/2)*((a*b^4 - 5*a^2*b^2*c + 4*a^3*c^2)*d^4*x...

________________________________________________________________________________________

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

[Out]

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

[Out]

Timed out

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________