3.4.63 \(\int \frac {\sqrt {d+e x^2}}{a+b x^2+c x^4} \, dx\) [363]

Optimal. Leaf size=240 \[ \frac {\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e} \tan ^{-1}\left (\frac {\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e} x}{\sqrt {b-\sqrt {b^2-4 a c}} \sqrt {d+e x^2}}\right )}{\sqrt {b^2-4 a c} \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e} \tan ^{-1}\left (\frac {\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e} x}{\sqrt {b+\sqrt {b^2-4 a c}} \sqrt {d+e x^2}}\right )}{\sqrt {b^2-4 a c} \sqrt {b+\sqrt {b^2-4 a c}}} \]

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

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Rubi [A]
time = 0.23, antiderivative size = 240, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 6, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {1188, 399, 223, 212, 385, 211} \begin {gather*} \frac {\sqrt {2 c d-e \left (b-\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^2-4 a c} \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {\sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\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 {b^2-4 a c} \sqrt {\sqrt {b^2-4 a c}+b}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
time = 8.66, size = 250, normalized size = 1.04 \begin {gather*} \frac {1}{2} e^{3/2} \text {RootSum}\left [c d^4-4 c d^3 \text {$\#$1}^2+4 b d^2 e \text {$\#$1}^2+6 c d^2 \text {$\#$1}^4-8 b d e \text {$\#$1}^4+16 a e^2 \text {$\#$1}^4-4 c d \text {$\#$1}^6+4 b e \text {$\#$1}^6+c \text {$\#$1}^8\&,\frac {d^2 \log \left (-\sqrt {e} x+\sqrt {d+e x^2}-\text {$\#$1}\right )+2 d \log \left (-\sqrt {e} x+\sqrt {d+e x^2}-\text {$\#$1}\right ) \text {$\#$1}^2+\log \left (-\sqrt {e} x+\sqrt {d+e x^2}-\text {$\#$1}\right ) \text {$\#$1}^4}{c d^3-b d^2 e-3 c d^2 \text {$\#$1}^2+4 b d e \text {$\#$1}^2-8 a e^2 \text {$\#$1}^2+3 c d \text {$\#$1}^4-3 b e \text {$\#$1}^4-c \text {$\#$1}^6}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(e^(3/2)*RootSum[c*d^4 - 4*c*d^3*#1^2 + 4*b*d^2*e*#1^2 + 6*c*d^2*#1^4 - 8*b*d*e*#1^4 + 16*a*e^2*#1^4 - 4*c*d*#
1^6 + 4*b*e*#1^6 + c*#1^8 & , (d^2*Log[-(Sqrt[e]*x) + Sqrt[d + e*x^2] - #1] + 2*d*Log[-(Sqrt[e]*x) + Sqrt[d +
e*x^2] - #1]*#1^2 + Log[-(Sqrt[e]*x) + Sqrt[d + e*x^2] - #1]*#1^4)/(c*d^3 - b*d^2*e - 3*c*d^2*#1^2 + 4*b*d*e*#
1^2 - 8*a*e^2*#1^2 + 3*c*d*#1^4 - 3*b*e*#1^4 - c*#1^6) & ])/2

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.12, size = 161, normalized size = 0.67

method	result	size
default	\(-\frac {e^{\frac {3}{2}} \left (\munderset {\textit {\_R} =\RootOf \left (c \,\textit {\_Z}^{4}+\left (4 e b -4 c d \right ) \textit {\_Z}^{3}+\left (16 a \,e^{2}-8 d e b +6 c \,d^{2}\right ) \textit {\_Z}^{2}+\left (4 d^{2} e b -4 c \,d^{3}\right ) \textit {\_Z} +d^{4} c \right )}{\sum }\frac {\left (\textit {\_R}^{2}+2 \textit {\_R} d +d^{2}\right ) \ln \left (\left (\sqrt {e \,x^{2}+d}-\sqrt {e}\, x \right )^{2}-\textit {\_R} \right )}{c \,\textit {\_R}^{3}+3 \textit {\_R}^{2} b e -3 \textit {\_R}^{2} c d +8 \textit {\_R} a \,e^{2}-4 \textit {\_R} b d e +3 c \,d^{2} \textit {\_R} +d^{2} e b -c \,d^{3}}\right )}{2}\)	\(161\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*e^(3/2)*sum((_R^2+2*_R*d+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))

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

[Out]

integrate(sqrt(x^2*e + d)/(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. 1007 vs. \(2 (206) = 412\).
time = 0.87, size = 1007, normalized size = 4.20 \begin {gather*} \frac {1}{4} \, \sqrt {\frac {1}{2}} \sqrt {-\frac {b d - 2 \, a e + {\left (a b^{2} - 4 \, a^{2} c\right )} \sqrt {\frac {d^{2}}{a^{2} b^{2} - 4 \, a^{3} c}}}{a b^{2} - 4 \, a^{2} c}} \log \left (-\frac {b d^{2} x^{2} - 4 \, a d x^{2} e + {\left (a b^{2} - 4 \, a^{2} c\right )} d \sqrt {\frac {d^{2}}{a^{2} b^{2} - 4 \, a^{3} c}} x^{2} + 4 \, \sqrt {\frac {1}{2}} {\left (a^{2} b^{2} - 4 \, a^{3} c\right )} \sqrt {x^{2} e + d} \sqrt {\frac {d^{2}}{a^{2} b^{2} - 4 \, a^{3} c}} x \sqrt {-\frac {b d - 2 \, a e + {\left (a b^{2} - 4 \, a^{2} c\right )} \sqrt {\frac {d^{2}}{a^{2} b^{2} - 4 \, a^{3} c}}}{a b^{2} - 4 \, a^{2} c}} - 2 \, a d^{2}}{x^{2}}\right ) - \frac {1}{4} \, \sqrt {\frac {1}{2}} \sqrt {-\frac {b d - 2 \, a e + {\left (a b^{2} - 4 \, a^{2} c\right )} \sqrt {\frac {d^{2}}{a^{2} b^{2} - 4 \, a^{3} c}}}{a b^{2} - 4 \, a^{2} c}} \log \left (-\frac {b d^{2} x^{2} - 4 \, a d x^{2} e + {\left (a b^{2} - 4 \, a^{2} c\right )} d \sqrt {\frac {d^{2}}{a^{2} b^{2} - 4 \, a^{3} c}} x^{2} - 4 \, \sqrt {\frac {1}{2}} {\left (a^{2} b^{2} - 4 \, a^{3} c\right )} \sqrt {x^{2} e + d} \sqrt {\frac {d^{2}}{a^{2} b^{2} - 4 \, a^{3} c}} x \sqrt {-\frac {b d - 2 \, a e + {\left (a b^{2} - 4 \, a^{2} c\right )} \sqrt {\frac {d^{2}}{a^{2} b^{2} - 4 \, a^{3} c}}}{a b^{2} - 4 \, a^{2} c}} - 2 \, a d^{2}}{x^{2}}\right ) - \frac {1}{4} \, \sqrt {\frac {1}{2}} \sqrt {-\frac {b d - 2 \, a e - {\left (a b^{2} - 4 \, a^{2} c\right )} \sqrt {\frac {d^{2}}{a^{2} b^{2} - 4 \, a^{3} c}}}{a b^{2} - 4 \, a^{2} c}} \log \left (-\frac {b d^{2} x^{2} - 4 \, a d x^{2} e - {\left (a b^{2} - 4 \, a^{2} c\right )} d \sqrt {\frac {d^{2}}{a^{2} b^{2} - 4 \, a^{3} c}} x^{2} + 4 \, \sqrt {\frac {1}{2}} {\left (a^{2} b^{2} - 4 \, a^{3} c\right )} \sqrt {x^{2} e + d} \sqrt {\frac {d^{2}}{a^{2} b^{2} - 4 \, a^{3} c}} x \sqrt {-\frac {b d - 2 \, a e - {\left (a b^{2} - 4 \, a^{2} c\right )} \sqrt {\frac {d^{2}}{a^{2} b^{2} - 4 \, a^{3} c}}}{a b^{2} - 4 \, a^{2} c}} - 2 \, a d^{2}}{x^{2}}\right ) + \frac {1}{4} \, \sqrt {\frac {1}{2}} \sqrt {-\frac {b d - 2 \, a e - {\left (a b^{2} - 4 \, a^{2} c\right )} \sqrt {\frac {d^{2}}{a^{2} b^{2} - 4 \, a^{3} c}}}{a b^{2} - 4 \, a^{2} c}} \log \left (-\frac {b d^{2} x^{2} - 4 \, a d x^{2} e - {\left (a b^{2} - 4 \, a^{2} c\right )} d \sqrt {\frac {d^{2}}{a^{2} b^{2} - 4 \, a^{3} c}} x^{2} - 4 \, \sqrt {\frac {1}{2}} {\left (a^{2} b^{2} - 4 \, a^{3} c\right )} \sqrt {x^{2} e + d} \sqrt {\frac {d^{2}}{a^{2} b^{2} - 4 \, a^{3} c}} x \sqrt {-\frac {b d - 2 \, a e - {\left (a b^{2} - 4 \, a^{2} c\right )} \sqrt {\frac {d^{2}}{a^{2} b^{2} - 4 \, a^{3} c}}}{a b^{2} - 4 \, a^{2} c}} - 2 \, a d^{2}}{x^{2}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

[Out]

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

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