3.4.46 \(\int \frac {1}{x (d+e x^2) (a+b x^2+c x^4)^{3/2}} \, dx\) [346]

Optimal. Leaf size=266 \[ \frac {b^2-2 a c+b c x^2}{a \left (b^2-4 a c\right ) d \sqrt {a+b x^2+c x^4}}+\frac {e \left (b c d-b^2 e+2 a c e+c (2 c d-b e) x^2\right )}{\left (b^2-4 a c\right ) d \left (c d^2-b d e+a e^2\right ) \sqrt {a+b x^2+c x^4}}-\frac {\tanh ^{-1}\left (\frac {2 a+b x^2}{2 \sqrt {a} \sqrt {a+b x^2+c x^4}}\right )}{2 a^{3/2} d}-\frac {e^3 \tanh ^{-1}\left (\frac {b d-2 a e+(2 c d-b e) x^2}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x^2+c x^4}}\right )}{2 d \left (c d^2-b d e+a e^2\right )^{3/2}} \]

[Out]

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

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Rubi [A]
time = 0.24, antiderivative size = 266, 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 = {1265, 974, 754, 12, 738, 212} \begin {gather*} -\frac {\tanh ^{-1}\left (\frac {2 a+b x^2}{2 \sqrt {a} \sqrt {a+b x^2+c x^4}}\right )}{2 a^{3/2} d}+\frac {e \left (2 a c e+b^2 (-e)+c x^2 (2 c d-b e)+b c d\right )}{d \left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4} \left (a e^2-b d e+c d^2\right )}+\frac {-2 a c+b^2+b c x^2}{a d \left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}}-\frac {e^3 \tanh ^{-1}\left (\frac {-2 a e+x^2 (2 c d-b e)+b d}{2 \sqrt {a+b x^2+c x^4} \sqrt {a e^2-b d e+c d^2}}\right )}{2 d \left (a e^2-b d e+c d^2\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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 738

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[-2, Subst[Int[1/(4*c*d
^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, (2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a,
b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[2*c*d - b*e, 0]

Rule 754

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(b
*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e)*x)*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e +
a*e^2))), x] + Dist[1/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^m*Simp[b*c*d*e*(2*p - m
+ 2) + b^2*e^2*(m + p + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3) - c*e*(2*c*d - b*e)*(m + 2*p + 4)*x
, x]*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b
*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && LtQ[p, -1] && IntQuadraticQ[a, b, c, d, e, m, p, x]

Rule 974

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :
> Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] &
& NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && (IntegerQ[p] || (ILtQ[m, 0] &&
ILtQ[n, 0])) &&  !(IGtQ[m, 0] || IGtQ[n, 0])

Rule 1265

Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2,
Subst[Int[x^((m - 1)/2)*(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x] &&
IntegerQ[(m - 1)/2]

Rubi steps

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

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Mathematica [A]
time = 1.12, size = 247, normalized size = 0.93 \begin {gather*} \frac {b^3 e-b c \left (3 a e+c d x^2\right )+2 a c^2 \left (d-e x^2\right )+b^2 c \left (-d+e x^2\right )}{a \left (-b^2+4 a c\right ) \left (c d^2+e (-b d+a e)\right ) \sqrt {a+b x^2+c x^4}}-\frac {e^3 \sqrt {-c d^2+b d e-a e^2} \tan ^{-1}\left (\frac {\sqrt {c} \left (d+e x^2\right )-e \sqrt {a+b x^2+c x^4}}{\sqrt {-c d^2+e (b d-a e)}}\right )}{d \left (c d^2+e (-b d+a e)\right )^2}+\frac {\tanh ^{-1}\left (\frac {\sqrt {c} x^2-\sqrt {a+b x^2+c x^4}}{\sqrt {a}}\right )}{a^{3/2} d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(562\) vs. \(2(242)=484\).
time = 0.34, size = 563, normalized size = 2.12

method	result	size
default	\(-\frac {e \left (\frac {2 c e \ln \left (\frac {\frac {2 a \,e^{2}-2 d e b +2 c \,d^{2}}{e^{2}}+\frac {\left (e b -2 c d \right ) \left (x^{2}+\frac {d}{e}\right )}{e}+2 \sqrt {\frac {a \,e^{2}-d e b +c \,d^{2}}{e^{2}}}\, \sqrt {c \left (x^{2}+\frac {d}{e}\right )^{2}+\frac {\left (e b -2 c d \right ) \left (x^{2}+\frac {d}{e}\right )}{e}+\frac {a \,e^{2}-d e b +c \,d^{2}}{e^{2}}}}{x^{2}+\frac {d}{e}}\right )}{\left (e \sqrt {-4 a c +b^{2}}-e b +2 c d \right ) \left (e \sqrt {-4 a c +b^{2}}+e b -2 c d \right ) \sqrt {\frac {a \,e^{2}-d e b +c \,d^{2}}{e^{2}}}}-\frac {2 c \sqrt {\left (x^{2}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right )^{2} c +\sqrt {-4 a c +b^{2}}\, \left (x^{2}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}}{\left (-4 a c +b^{2}\right ) \left (e \sqrt {-4 a c +b^{2}}-e b +2 c d \right ) \left (x^{2}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}+\frac {2 c \sqrt {\left (x^{2}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right )^{2} c -\sqrt {-4 a c +b^{2}}\, \left (x^{2}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}}{\left (-4 a c +b^{2}\right ) \left (e \sqrt {-4 a c +b^{2}}+e b -2 c d \right ) \left (x^{2}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}\right )}{d}+\frac {\frac {1}{2 a \sqrt {c \,x^{4}+b \,x^{2}+a}}-\frac {b \left (2 c \,x^{2}+b \right )}{2 a \left (4 a c -b^{2}\right ) \sqrt {c \,x^{4}+b \,x^{2}+a}}-\frac {\ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right )}{2 a^{\frac {3}{2}}}}{d}\)	\(563\)
elliptic	\(-\frac {2 c \,e^{2} \ln \left (\frac {\frac {2 a \,e^{2}-2 d e b +2 c \,d^{2}}{e^{2}}+\frac {\left (e b -2 c d \right ) \left (x^{2}+\frac {d}{e}\right )}{e}+2 \sqrt {\frac {a \,e^{2}-d e b +c \,d^{2}}{e^{2}}}\, \sqrt {c \left (x^{2}+\frac {d}{e}\right )^{2}+\frac {\left (e b -2 c d \right ) \left (x^{2}+\frac {d}{e}\right )}{e}+\frac {a \,e^{2}-d e b +c \,d^{2}}{e^{2}}}}{x^{2}+\frac {d}{e}}\right )}{\left (e \sqrt {-4 a c +b^{2}}-e b +2 c d \right ) \left (e \sqrt {-4 a c +b^{2}}+e b -2 c d \right ) d \sqrt {\frac {a \,e^{2}-d e b +c \,d^{2}}{e^{2}}}}+\frac {2 c \ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right )}{\left (-b +\sqrt {-4 a c +b^{2}}\right ) \left (b +\sqrt {-4 a c +b^{2}}\right ) d \sqrt {a}}-\frac {4 c^{2} \sqrt {\left (x^{2}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right )^{2} c +\sqrt {-4 a c +b^{2}}\, \left (x^{2}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}}{\left (-4 a c +b^{2}\right ) \left (e \sqrt {-4 a c +b^{2}}-e b +2 c d \right ) \left (-b +\sqrt {-4 a c +b^{2}}\right ) \left (x^{2}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}-\frac {4 c^{2} \sqrt {\left (x^{2}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right )^{2} c -\sqrt {-4 a c +b^{2}}\, \left (x^{2}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}}{\left (-4 a c +b^{2}\right ) \left (e \sqrt {-4 a c +b^{2}}+e b -2 c d \right ) \left (b +\sqrt {-4 a c +b^{2}}\right ) \left (x^{2}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}\)	\(565\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1203 vs. \(2 (247) = 494\).
time = 2.48, size = 4917, normalized size = 18.48 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/4*((a^3*b^2 - 4*a^4*c + (a^2*b^2*c - 4*a^3*c^2)*x^4 + (a^2*b^3 - 4*a^3*b*c)*x^2)*sqrt(c*d^2 - b*d*e + a*e^2
)*e^3*log(-(8*c^2*d^2*x^4 + 8*b*c*d^2*x^2 + (b^2 + 4*a*c)*d^2 - 4*sqrt(c*x^4 + b*x^2 + a)*(2*c*d*x^2 + b*d - (
b*x^2 + 2*a)*e)*sqrt(c*d^2 - b*d*e + a*e^2) + ((b^2 + 4*a*c)*x^4 + 8*a*b*x^2 + 8*a^2)*e^2 - 2*(4*b*c*d*x^4 + (
3*b^2 + 4*a*c)*d*x^2 + 4*a*b*d)*e)/(x^4*e^2 + 2*d*x^2*e + d^2)) + ((b^2*c^3 - 4*a*c^4)*d^4*x^4 + (b^3*c^2 - 4*
a*b*c^3)*d^4*x^2 + (a*b^2*c^2 - 4*a^2*c^3)*d^4 + (a^3*b^2 - 4*a^4*c + (a^2*b^2*c - 4*a^3*c^2)*x^4 + (a^2*b^3 -
4*a^3*b*c)*x^2)*e^4 - 2*((a*b^3*c - 4*a^2*b*c^2)*d*x^4 + (a*b^4 - 4*a^2*b^2*c)*d*x^2 + (a^2*b^3 - 4*a^3*b*c)*
d)*e^3 + ((b^4*c - 2*a*b^2*c^2 - 8*a^2*c^3)*d^2*x^4 + (b^5 - 2*a*b^3*c - 8*a^2*b*c^2)*d^2*x^2 + (a*b^4 - 2*a^2
*b^2*c - 8*a^3*c^2)*d^2)*e^2 - 2*((b^3*c^2 - 4*a*b*c^3)*d^3*x^4 + (b^4*c - 4*a*b^2*c^2)*d^3*x^2 + (a*b^3*c - 4
*a^2*b*c^2)*d^3)*e)*sqrt(a)*log(-((b^2 + 4*a*c)*x^4 + 8*a*b*x^2 - 4*sqrt(c*x^4 + b*x^2 + a)*(b*x^2 + 2*a)*sqrt
(a) + 8*a^2)/x^4) + 4*(a*b*c^3*d^4*x^2 + (a*b^2*c^2 - 2*a^2*c^3)*d^4 - ((a^2*b^2*c - 2*a^3*c^2)*d*x^2 + (a^2*b
^3 - 3*a^3*b*c)*d)*e^3 + ((a*b^3*c - a^2*b*c^2)*d^2*x^2 + (a*b^4 - 2*a^2*b^2*c - 2*a^3*c^2)*d^2)*e^2 - (2*(a*b
^2*c^2 - a^2*c^3)*d^3*x^2 + (2*a*b^3*c - 5*a^2*b*c^2)*d^3)*e)*sqrt(c*x^4 + b*x^2 + a))/((a^2*b^2*c^3 - 4*a^3*c
^4)*d^5*x^4 + (a^2*b^3*c^2 - 4*a^3*b*c^3)*d^5*x^2 + (a^3*b^2*c^2 - 4*a^4*c^3)*d^5 + ((a^4*b^2*c - 4*a^5*c^2)*d
*x^4 + (a^4*b^3 - 4*a^5*b*c)*d*x^2 + (a^5*b^2 - 4*a^6*c)*d)*e^4 - 2*((a^3*b^3*c - 4*a^4*b*c^2)*d^2*x^4 + (a^3*
b^4 - 4*a^4*b^2*c)*d^2*x^2 + (a^4*b^3 - 4*a^5*b*c)*d^2)*e^3 + ((a^2*b^4*c - 2*a^3*b^2*c^2 - 8*a^4*c^3)*d^3*x^4
+ (a^2*b^5 - 2*a^3*b^3*c - 8*a^4*b*c^2)*d^3*x^2 + (a^3*b^4 - 2*a^4*b^2*c - 8*a^5*c^2)*d^3)*e^2 - 2*((a^2*b^3*
c^2 - 4*a^3*b*c^3)*d^4*x^4 + (a^2*b^4*c - 4*a^3*b^2*c^2)*d^4*x^2 + (a^3*b^3*c - 4*a^4*b*c^2)*d^4)*e), -1/4*(2*
(a^3*b^2 - 4*a^4*c + (a^2*b^2*c - 4*a^3*c^2)*x^4 + (a^2*b^3 - 4*a^3*b*c)*x^2)*sqrt(-c*d^2 + b*d*e - a*e^2)*arc
tan(-1/2*sqrt(c*x^4 + b*x^2 + a)*(2*c*d*x^2 + b*d - (b*x^2 + 2*a)*e)*sqrt(-c*d^2 + b*d*e - a*e^2)/(c^2*d^2*x^4
+ b*c*d^2*x^2 + a*c*d^2 + (a*c*x^4 + a*b*x^2 + a^2)*e^2 - (b*c*d*x^4 + b^2*d*x^2 + a*b*d)*e))*e^3 - ((b^2*c^3
- 4*a*c^4)*d^4*x^4 + (b^3*c^2 - 4*a*b*c^3)*d^4*x^2 + (a*b^2*c^2 - 4*a^2*c^3)*d^4 + (a^3*b^2 - 4*a^4*c + (a^2*
b^2*c - 4*a^3*c^2)*x^4 + (a^2*b^3 - 4*a^3*b*c)*x^2)*e^4 - 2*((a*b^3*c - 4*a^2*b*c^2)*d*x^4 + (a*b^4 - 4*a^2*b^
2*c)*d*x^2 + (a^2*b^3 - 4*a^3*b*c)*d)*e^3 + ((b^4*c - 2*a*b^2*c^2 - 8*a^2*c^3)*d^2*x^4 + (b^5 - 2*a*b^3*c - 8*
a^2*b*c^2)*d^2*x^2 + (a*b^4 - 2*a^2*b^2*c - 8*a^3*c^2)*d^2)*e^2 - 2*((b^3*c^2 - 4*a*b*c^3)*d^3*x^4 + (b^4*c -
4*a*b^2*c^2)*d^3*x^2 + (a*b^3*c - 4*a^2*b*c^2)*d^3)*e)*sqrt(a)*log(-((b^2 + 4*a*c)*x^4 + 8*a*b*x^2 - 4*sqrt(c*
x^4 + b*x^2 + a)*(b*x^2 + 2*a)*sqrt(a) + 8*a^2)/x^4) - 4*(a*b*c^3*d^4*x^2 + (a*b^2*c^2 - 2*a^2*c^3)*d^4 - ((a^
2*b^2*c - 2*a^3*c^2)*d*x^2 + (a^2*b^3 - 3*a^3*b*c)*d)*e^3 + ((a*b^3*c - a^2*b*c^2)*d^2*x^2 + (a*b^4 - 2*a^2*b^
2*c - 2*a^3*c^2)*d^2)*e^2 - (2*(a*b^2*c^2 - a^2*c^3)*d^3*x^2 + (2*a*b^3*c - 5*a^2*b*c^2)*d^3)*e)*sqrt(c*x^4 +
b*x^2 + a))/((a^2*b^2*c^3 - 4*a^3*c^4)*d^5*x^4 + (a^2*b^3*c^2 - 4*a^3*b*c^3)*d^5*x^2 + (a^3*b^2*c^2 - 4*a^4*c^
3)*d^5 + ((a^4*b^2*c - 4*a^5*c^2)*d*x^4 + (a^4*b^3 - 4*a^5*b*c)*d*x^2 + (a^5*b^2 - 4*a^6*c)*d)*e^4 - 2*((a^3*b
^3*c - 4*a^4*b*c^2)*d^2*x^4 + (a^3*b^4 - 4*a^4*b^2*c)*d^2*x^2 + (a^4*b^3 - 4*a^5*b*c)*d^2)*e^3 + ((a^2*b^4*c -
2*a^3*b^2*c^2 - 8*a^4*c^3)*d^3*x^4 + (a^2*b^5 - 2*a^3*b^3*c - 8*a^4*b*c^2)*d^3*x^2 + (a^3*b^4 - 2*a^4*b^2*c -
8*a^5*c^2)*d^3)*e^2 - 2*((a^2*b^3*c^2 - 4*a^3*b*c^3)*d^4*x^4 + (a^2*b^4*c - 4*a^3*b^2*c^2)*d^4*x^2 + (a^3*b^3
*c - 4*a^4*b*c^2)*d^4)*e), 1/4*((a^3*b^2 - 4*a^4*c + (a^2*b^2*c - 4*a^3*c^2)*x^4 + (a^2*b^3 - 4*a^3*b*c)*x^2)*
sqrt(c*d^2 - b*d*e + a*e^2)*e^3*log(-(8*c^2*d^2*x^4 + 8*b*c*d^2*x^2 + (b^2 + 4*a*c)*d^2 - 4*sqrt(c*x^4 + b*x^2
+ a)*(2*c*d*x^2 + b*d - (b*x^2 + 2*a)*e)*sqrt(c*d^2 - b*d*e + a*e^2) + ((b^2 + 4*a*c)*x^4 + 8*a*b*x^2 + 8*a^2
)*e^2 - 2*(4*b*c*d*x^4 + (3*b^2 + 4*a*c)*d*x^2 + 4*a*b*d)*e)/(x^4*e^2 + 2*d*x^2*e + d^2)) + 2*((b^2*c^3 - 4*a*
c^4)*d^4*x^4 + (b^3*c^2 - 4*a*b*c^3)*d^4*x^2 + (a*b^2*c^2 - 4*a^2*c^3)*d^4 + (a^3*b^2 - 4*a^4*c + (a^2*b^2*c -
4*a^3*c^2)*x^4 + (a^2*b^3 - 4*a^3*b*c)*x^2)*e^4 - 2*((a*b^3*c - 4*a^2*b*c^2)*d*x^4 + (a*b^4 - 4*a^2*b^2*c)*d*
x^2 + (a^2*b^3 - 4*a^3*b*c)*d)*e^3 + ((b^4*c - 2*a*b^2*c^2 - 8*a^2*c^3)*d^2*x^4 + (b^5 - 2*a*b^3*c - 8*a^2*b*c
^2)*d^2*x^2 + (a*b^4 - 2*a^2*b^2*c - 8*a^3*c^2)*d^2)*e^2 - 2*((b^3*c^2 - 4*a*b*c^3)*d^3*x^4 + (b^4*c - 4*a*b^2
*c^2)*d^3*x^2 + (a*b^3*c - 4*a^2*b*c^2)*d^3)*e)*sqrt(-a)*arctan(1/2*sqrt(c*x^4 + b*x^2 + a)*(b*x^2 + 2*a)*sqrt
(-a)/(a*c*x^4 + a*b*x^2 + a^2)) + 4*(a*b*c^3*d^4*x^2 + (a*b^2*c^2 - 2*a^2*c^3)*d^4 - ((a^2*b^2*c - 2*a^3*c^2)*
d*x^2 + (a^2*b^3 - 3*a^3*b*c)*d)*e^3 + ((a*b^3*c - a^2*b*c^2)*d^2*x^2 + (a*b^4 - 2*a^2*b^2*c - 2*a^3*c^2)*d^2)
*e^2 - (2*(a*b^2*c^2 - a^2*c^3)*d^3*x^2 + (2*a*b^3*c - 5*a^2*b*c^2)*d^3)*e)*sqrt(c*x^4 + b*x^2 + a))/((a^2*b^2
*c^3 - 4*a^3*c^4)*d^5*x^4 + (a^2*b^3*c^2 - 4*a^...

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Error: Bad Argument Type

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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