3.4.56 \(\int \frac {x^3 \sqrt {d+e x^2}}{a+b x^2+c x^4} \, dx\) [356]

Optimal. Leaf size=292 \[ \frac {\sqrt {d+e x^2}}{c}+\frac {\left (b c d-b^2 e+2 a c e-\sqrt {b^2-4 a c} (c d-b e)\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x^2}}{\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}\right )}{\sqrt {2} c^{3/2} \sqrt {b^2-4 a c} \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}-\frac {\left (b c d-b^2 e+2 a c e+\sqrt {b^2-4 a c} (c d-b e)\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x^2}}{\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}\right )}{\sqrt {2} c^{3/2} \sqrt {b^2-4 a c} \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \]

[Out]

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

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Rubi [A]
time = 2.43, antiderivative size = 292, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {1265, 838, 840, 1180, 214} \begin {gather*} \frac {\left (-\sqrt {b^2-4 a c} (c d-b e)+2 a c e+b^2 (-e)+b c d\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x^2}}{\sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}\right )}{\sqrt {2} c^{3/2} \sqrt {b^2-4 a c} \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}-\frac {\left (\sqrt {b^2-4 a c} (c d-b e)+2 a c e+b^2 (-e)+b c d\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x^2}}{\sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}\right )}{\sqrt {2} c^{3/2} \sqrt {b^2-4 a c} \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}+\frac {\sqrt {d+e x^2}}{c} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 214

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

Rule 838

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[g*
((d + e*x)^m/(c*m)), x] + Dist[1/c, Int[(d + e*x)^(m - 1)*(Simp[c*d*f - a*e*g + (g*c*d - b*e*g + c*e*f)*x, x]/
(a + b*x + c*x^2)), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*
e^2, 0] && FractionQ[m] && GtQ[m, 0]

Rule 840

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

Rule 1180

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

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

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Mathematica [C] Result contains complex when optimal does not.
time = 0.94, size = 350, normalized size = 1.20 \begin {gather*} \frac {2 \sqrt {c} \sqrt {d+e x^2}-\frac {\left (-i b c d-c \sqrt {-b^2+4 a c} d+i b^2 e-2 i a c e+b \sqrt {-b^2+4 a c} e\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x^2}}{\sqrt {-2 c d+b e-i \sqrt {-b^2+4 a c} e}}\right )}{\sqrt {-\frac {b^2}{2}+2 a c} \sqrt {-2 c d+\left (b-i \sqrt {-b^2+4 a c}\right ) e}}-\frac {\left (i b c d-c \sqrt {-b^2+4 a c} d-i b^2 e+2 i a c e+b \sqrt {-b^2+4 a c} e\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x^2}}{\sqrt {-2 c d+b e+i \sqrt {-b^2+4 a c} e}}\right )}{\sqrt {-\frac {b^2}{2}+2 a c} \sqrt {-2 c d+\left (b+i \sqrt {-b^2+4 a c}\right ) e}}}{2 c^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[c]*Sqrt[d + e*x^2] - (((-I)*b*c*d - c*Sqrt[-b^2 + 4*a*c]*d + I*b^2*e - (2*I)*a*c*e + b*Sqrt[-b^2 + 4*a
*c]*e)*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x^2])/Sqrt[-2*c*d + b*e - I*Sqrt[-b^2 + 4*a*c]*e]])/(Sqrt[-1/2*b^2 +
2*a*c]*Sqrt[-2*c*d + (b - I*Sqrt[-b^2 + 4*a*c])*e]) - ((I*b*c*d - c*Sqrt[-b^2 + 4*a*c]*d - I*b^2*e + (2*I)*a*
c*e + b*Sqrt[-b^2 + 4*a*c]*e)*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x^2])/Sqrt[-2*c*d + b*e + I*Sqrt[-b^2 + 4*a*c
]*e]])/(Sqrt[-1/2*b^2 + 2*a*c]*Sqrt[-2*c*d + (b + I*Sqrt[-b^2 + 4*a*c])*e]))/(2*c^(3/2))

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.14, size = 273, normalized size = 0.93

method	result	size
risch	\(\frac {\sqrt {e \,x^{2}+d}}{c}-\frac {\munderset {\textit {\_R} =\RootOf \left (c \,\textit {\_Z}^{8}+\left (4 e b -4 c d \right ) \textit {\_Z}^{6}+\left (16 a \,e^{2}-8 d e b +6 c \,d^{2}\right ) \textit {\_Z}^{4}+\left (4 d^{2} e b -4 c \,d^{3}\right ) \textit {\_Z}^{2}+d^{4} c \right )}{\sum }\frac {\left (\left (e b -c d \right ) \textit {\_R}^{6}+\left (4 a \,e^{2}-3 d e b +3 c \,d^{2}\right ) \textit {\_R}^{4}+d \left (-4 a \,e^{2}+3 d e b -3 c \,d^{2}\right ) \textit {\_R}^{2}-d^{3} e b +d^{4} c \right ) \ln \left (\sqrt {e \,x^{2}+d}-\sqrt {e}\, x -\textit {\_R} \right )}{\textit {\_R}^{7} c +3 \textit {\_R}^{5} b e -3 \textit {\_R}^{5} c d +8 \textit {\_R}^{3} a \,e^{2}-4 \textit {\_R}^{3} b d e +3 \textit {\_R}^{3} c \,d^{2}+\textit {\_R} b \,d^{2} e -\textit {\_R} c \,d^{3}}}{4 c}\)	\(241\)
default	\(\frac {\sqrt {e \,x^{2}+d}-\sqrt {e}\, x}{2 c}-\frac {\munderset {\textit {\_R} =\RootOf \left (c \,\textit {\_Z}^{8}+\left (4 e b -4 c d \right ) \textit {\_Z}^{6}+\left (16 a \,e^{2}-8 d e b +6 c \,d^{2}\right ) \textit {\_Z}^{4}+\left (4 d^{2} e b -4 c \,d^{3}\right ) \textit {\_Z}^{2}+d^{4} c \right )}{\sum }\frac {\left (\left (e b -c d \right ) \textit {\_R}^{6}+\left (4 a \,e^{2}-3 d e b +3 c \,d^{2}\right ) \textit {\_R}^{4}+d \left (-4 a \,e^{2}+3 d e b -3 c \,d^{2}\right ) \textit {\_R}^{2}-d^{3} e b +d^{4} c \right ) \ln \left (\sqrt {e \,x^{2}+d}-\sqrt {e}\, x -\textit {\_R} \right )}{\textit {\_R}^{7} c +3 \textit {\_R}^{5} b e -3 \textit {\_R}^{5} c d +8 \textit {\_R}^{3} a \,e^{2}-4 \textit {\_R}^{3} b d e +3 \textit {\_R}^{3} c \,d^{2}+\textit {\_R} b \,d^{2} e -\textit {\_R} c \,d^{3}}}{4 c}+\frac {d}{2 c \left (\sqrt {e \,x^{2}+d}-\sqrt {e}\, x \right )}\)	\(273\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2/c*((e*x^2+d)^(1/2)-e^(1/2)*x)-1/4/c*sum(((b*e-c*d)*_R^6+(4*a*e^2-3*b*d*e+3*c*d^2)*_R^4+d*(-4*a*e^2+3*b*d*e
-3*c*d^2)*_R^2-d^3*e*b+d^4*c)/(_R^7*c+3*_R^5*b*e-3*_R^5*c*d+8*_R^3*a*e^2-4*_R^3*b*d*e+3*_R^3*c*d^2+_R*b*d^2*e-
_R*c*d^3)*ln((e*x^2+d)^(1/2)-e^(1/2)*x-_R),_R=RootOf(c*_Z^8+(4*b*e-4*c*d)*_Z^6+(16*a*e^2-8*b*d*e+6*c*d^2)*_Z^4
+(4*b*d^2*e-4*c*d^3)*_Z^2+d^4*c))+1/2/c*d/((e*x^2+d)^(1/2)-e^(1/2)*x)

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

[Out]

integrate(sqrt(x^2*e + d)*x^3/(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. 2452 vs. \(2 (260) = 520\).
time = 29.46, size = 2452, normalized size = 8.40 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(sqrt(1/2)*c*sqrt(((b^2*c - 2*a*c^2)*d - (b^3 - 3*a*b*c)*e + (b^2*c^3 - 4*a*c^4)*sqrt((b^2*c^2*d^2 - 2*(b^
3*c - a*b*c^2)*d*e + (b^4 - 2*a*b^2*c + a^2*c^2)*e^2)/(b^2*c^6 - 4*a*c^7)))/(b^2*c^3 - 4*a*c^4))*log((2*a*b^2*
c*d^2 + 2*sqrt(1/2)*sqrt(x^2*e + d)*((b^4*c - 4*a*b^2*c^2)*d - (b^5 - 5*a*b^3*c + 4*a^2*b*c^2)*e - (b^4*c^3 -
6*a*b^2*c^4 + 8*a^2*c^5)*sqrt((b^2*c^2*d^2 - 2*(b^3*c - a*b*c^2)*d*e + (b^4 - 2*a*b^2*c + a^2*c^2)*e^2)/(b^2*c
^6 - 4*a*c^7)))*sqrt(((b^2*c - 2*a*c^2)*d - (b^3 - 3*a*b*c)*e + (b^2*c^3 - 4*a*c^4)*sqrt((b^2*c^2*d^2 - 2*(b^3
*c - a*b*c^2)*d*e + (b^4 - 2*a*b^2*c + a^2*c^2)*e^2)/(b^2*c^6 - 4*a*c^7)))/(b^2*c^3 - 4*a*c^4)) + (2*a^2*b^2 -
2*a^3*c - (a*b^3 - a^2*b*c)*x^2)*e^2 + (a*b^2*c*d*x^2 - 2*a*b^3*d)*e - ((a*b^2*c^3 - 4*a^2*c^4)*x^2*e + 2*(a*
b^2*c^3 - 4*a^2*c^4)*d)*sqrt((b^2*c^2*d^2 - 2*(b^3*c - a*b*c^2)*d*e + (b^4 - 2*a*b^2*c + a^2*c^2)*e^2)/(b^2*c^
6 - 4*a*c^7)))/x^2) - sqrt(1/2)*c*sqrt(((b^2*c - 2*a*c^2)*d - (b^3 - 3*a*b*c)*e + (b^2*c^3 - 4*a*c^4)*sqrt((b^
2*c^2*d^2 - 2*(b^3*c - a*b*c^2)*d*e + (b^4 - 2*a*b^2*c + a^2*c^2)*e^2)/(b^2*c^6 - 4*a*c^7)))/(b^2*c^3 - 4*a*c^
4))*log((2*a*b^2*c*d^2 - 2*sqrt(1/2)*sqrt(x^2*e + d)*((b^4*c - 4*a*b^2*c^2)*d - (b^5 - 5*a*b^3*c + 4*a^2*b*c^2
)*e - (b^4*c^3 - 6*a*b^2*c^4 + 8*a^2*c^5)*sqrt((b^2*c^2*d^2 - 2*(b^3*c - a*b*c^2)*d*e + (b^4 - 2*a*b^2*c + a^2
*c^2)*e^2)/(b^2*c^6 - 4*a*c^7)))*sqrt(((b^2*c - 2*a*c^2)*d - (b^3 - 3*a*b*c)*e + (b^2*c^3 - 4*a*c^4)*sqrt((b^2
*c^2*d^2 - 2*(b^3*c - a*b*c^2)*d*e + (b^4 - 2*a*b^2*c + a^2*c^2)*e^2)/(b^2*c^6 - 4*a*c^7)))/(b^2*c^3 - 4*a*c^4
)) + (2*a^2*b^2 - 2*a^3*c - (a*b^3 - a^2*b*c)*x^2)*e^2 + (a*b^2*c*d*x^2 - 2*a*b^3*d)*e - ((a*b^2*c^3 - 4*a^2*c
^4)*x^2*e + 2*(a*b^2*c^3 - 4*a^2*c^4)*d)*sqrt((b^2*c^2*d^2 - 2*(b^3*c - a*b*c^2)*d*e + (b^4 - 2*a*b^2*c + a^2*
c^2)*e^2)/(b^2*c^6 - 4*a*c^7)))/x^2) + sqrt(1/2)*c*sqrt(((b^2*c - 2*a*c^2)*d - (b^3 - 3*a*b*c)*e - (b^2*c^3 -
4*a*c^4)*sqrt((b^2*c^2*d^2 - 2*(b^3*c - a*b*c^2)*d*e + (b^4 - 2*a*b^2*c + a^2*c^2)*e^2)/(b^2*c^6 - 4*a*c^7)))/
(b^2*c^3 - 4*a*c^4))*log((2*a*b^2*c*d^2 + 2*sqrt(1/2)*sqrt(x^2*e + d)*((b^4*c - 4*a*b^2*c^2)*d - (b^5 - 5*a*b^
3*c + 4*a^2*b*c^2)*e + (b^4*c^3 - 6*a*b^2*c^4 + 8*a^2*c^5)*sqrt((b^2*c^2*d^2 - 2*(b^3*c - a*b*c^2)*d*e + (b^4
- 2*a*b^2*c + a^2*c^2)*e^2)/(b^2*c^6 - 4*a*c^7)))*sqrt(((b^2*c - 2*a*c^2)*d - (b^3 - 3*a*b*c)*e - (b^2*c^3 - 4
*a*c^4)*sqrt((b^2*c^2*d^2 - 2*(b^3*c - a*b*c^2)*d*e + (b^4 - 2*a*b^2*c + a^2*c^2)*e^2)/(b^2*c^6 - 4*a*c^7)))/(
b^2*c^3 - 4*a*c^4)) + (2*a^2*b^2 - 2*a^3*c - (a*b^3 - a^2*b*c)*x^2)*e^2 + (a*b^2*c*d*x^2 - 2*a*b^3*d)*e + ((a*
b^2*c^3 - 4*a^2*c^4)*x^2*e + 2*(a*b^2*c^3 - 4*a^2*c^4)*d)*sqrt((b^2*c^2*d^2 - 2*(b^3*c - a*b*c^2)*d*e + (b^4 -
2*a*b^2*c + a^2*c^2)*e^2)/(b^2*c^6 - 4*a*c^7)))/x^2) - sqrt(1/2)*c*sqrt(((b^2*c - 2*a*c^2)*d - (b^3 - 3*a*b*c
)*e - (b^2*c^3 - 4*a*c^4)*sqrt((b^2*c^2*d^2 - 2*(b^3*c - a*b*c^2)*d*e + (b^4 - 2*a*b^2*c + a^2*c^2)*e^2)/(b^2*
c^6 - 4*a*c^7)))/(b^2*c^3 - 4*a*c^4))*log((2*a*b^2*c*d^2 - 2*sqrt(1/2)*sqrt(x^2*e + d)*((b^4*c - 4*a*b^2*c^2)*
d - (b^5 - 5*a*b^3*c + 4*a^2*b*c^2)*e + (b^4*c^3 - 6*a*b^2*c^4 + 8*a^2*c^5)*sqrt((b^2*c^2*d^2 - 2*(b^3*c - a*b
*c^2)*d*e + (b^4 - 2*a*b^2*c + a^2*c^2)*e^2)/(b^2*c^6 - 4*a*c^7)))*sqrt(((b^2*c - 2*a*c^2)*d - (b^3 - 3*a*b*c)
*e - (b^2*c^3 - 4*a*c^4)*sqrt((b^2*c^2*d^2 - 2*(b^3*c - a*b*c^2)*d*e + (b^4 - 2*a*b^2*c + a^2*c^2)*e^2)/(b^2*c
^6 - 4*a*c^7)))/(b^2*c^3 - 4*a*c^4)) + (2*a^2*b^2 - 2*a^3*c - (a*b^3 - a^2*b*c)*x^2)*e^2 + (a*b^2*c*d*x^2 - 2*
a*b^3*d)*e + ((a*b^2*c^3 - 4*a^2*c^4)*x^2*e + 2*(a*b^2*c^3 - 4*a^2*c^4)*d)*sqrt((b^2*c^2*d^2 - 2*(b^3*c - a*b*
c^2)*d*e + (b^4 - 2*a*b^2*c + a^2*c^2)*e^2)/(b^2*c^6 - 4*a*c^7)))/x^2) + 4*sqrt(x^2*e + d))/c

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

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 619 vs. \(2 (260) = 520\).
time = 4.24, size = 619, normalized size = 2.12 \begin {gather*} \frac {\sqrt {x^{2} e + d}}{c} - \frac {{\left (2 \, b c^{4} d^{2} + {\left ({\left (b^{2} c - 4 \, a c^{2}\right )} d e - {\left (b^{3} - 4 \, a b c\right )} e^{2}\right )} c^{2} - {\left (3 \, b^{2} c^{3} - 4 \, a c^{4}\right )} d e - 2 \, {\left (\sqrt {b^{2} - 4 \, a c} c^{3} d^{2} - \sqrt {b^{2} - 4 \, a c} b c^{2} d e + \sqrt {b^{2} - 4 \, a c} a c^{2} e^{2}\right )} {\left | c \right |} + {\left (b^{3} c^{2} - 2 \, a b c^{3}\right )} e^{2}\right )} \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} \sqrt {x^{2} e + d}}{\sqrt {-\frac {2 \, c^{2} d - b c e + \sqrt {-4 \, {\left (c^{2} d^{2} - b c d e + a c e^{2}\right )} c^{2} + {\left (2 \, c^{2} d - b c e\right )}^{2}}}{c^{2}}}}\right )}{{\left (2 \, \sqrt {b^{2} - 4 \, a c} c^{2} d - {\left (b^{2} c - 4 \, a c^{2} + \sqrt {b^{2} - 4 \, a c} b c\right )} e\right )} \sqrt {-4 \, c^{2} d + 2 \, {\left (b c - \sqrt {b^{2} - 4 \, a c} c\right )} e} c^{2}} + \frac {{\left (2 \, b c^{4} d^{2} + {\left ({\left (b^{2} c - 4 \, a c^{2}\right )} d e - {\left (b^{3} - 4 \, a b c\right )} e^{2}\right )} c^{2} - {\left (3 \, b^{2} c^{3} - 4 \, a c^{4}\right )} d e + 2 \, {\left (\sqrt {b^{2} - 4 \, a c} c^{3} d^{2} - \sqrt {b^{2} - 4 \, a c} b c^{2} d e + \sqrt {b^{2} - 4 \, a c} a c^{2} e^{2}\right )} {\left | c \right |} + {\left (b^{3} c^{2} - 2 \, a b c^{3}\right )} e^{2}\right )} \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} \sqrt {x^{2} e + d}}{\sqrt {-\frac {2 \, c^{2} d - b c e - \sqrt {-4 \, {\left (c^{2} d^{2} - b c d e + a c e^{2}\right )} c^{2} + {\left (2 \, c^{2} d - b c e\right )}^{2}}}{c^{2}}}}\right )}{{\left (2 \, \sqrt {b^{2} - 4 \, a c} c^{2} d + {\left (b^{2} c - 4 \, a c^{2} - \sqrt {b^{2} - 4 \, a c} b c\right )} e\right )} \sqrt {-4 \, c^{2} d + 2 \, {\left (b c + \sqrt {b^{2} - 4 \, a c} c\right )} e} c^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 2.34, size = 2500, normalized size = 8.56 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(d + e*x^2)^(1/2)/c - atan(((((16*a^2*c^3*e^4 - 4*a*b^2*c^2*e^4 + 16*a*c^4*d^2*e^2 + 4*b^3*c^2*d*e^3 - 4*b^2*c
^3*d^2*e^2 - 16*a*b*c^3*d*e^3)/c - (2*(d + e*x^2)^(1/2)*((8*a^2*c^3*d - b^5*e - b^2*e*(-(4*a*c - b^2)^3)^(1/2)
+ b^4*c*d + 7*a*b^3*c*e + a*c*e*(-(4*a*c - b^2)^3)^(1/2) + b*c*d*(-(4*a*c - b^2)^3)^(1/2) - 6*a*b^2*c^2*d - 1
2*a^2*b*c^2*e)/(8*(16*a^2*c^5 + b^4*c^3 - 8*a*b^2*c^4)))^(1/2)*(4*b^3*c^3*e^3 - 8*b^2*c^4*d*e^2 - 16*a*b*c^4*e
^3 + 32*a*c^5*d*e^2))/c)*((8*a^2*c^3*d - b^5*e - b^2*e*(-(4*a*c - b^2)^3)^(1/2) + b^4*c*d + 7*a*b^3*c*e + a*c*
e*(-(4*a*c - b^2)^3)^(1/2) + b*c*d*(-(4*a*c - b^2)^3)^(1/2) - 6*a*b^2*c^2*d - 12*a^2*b*c^2*e)/(8*(16*a^2*c^5 +
b^4*c^3 - 8*a*b^2*c^4)))^(1/2) - (2*(d + e*x^2)^(1/2)*(b^4*e^4 + 2*a^2*c^2*e^4 - 2*a*c^3*d^2*e^2 + b^2*c^2*d^
2*e^2 - 4*a*b^2*c*e^4 - 2*b^3*c*d*e^3 + 6*a*b*c^2*d*e^3))/c)*((8*a^2*c^3*d - b^5*e - b^2*e*(-(4*a*c - b^2)^3)^
(1/2) + b^4*c*d + 7*a*b^3*c*e + a*c*e*(-(4*a*c - b^2)^3)^(1/2) + b*c*d*(-(4*a*c - b^2)^3)^(1/2) - 6*a*b^2*c^2*
d - 12*a^2*b*c^2*e)/(8*(16*a^2*c^5 + b^4*c^3 - 8*a*b^2*c^4)))^(1/2)*1i - (((16*a^2*c^3*e^4 - 4*a*b^2*c^2*e^4 +
16*a*c^4*d^2*e^2 + 4*b^3*c^2*d*e^3 - 4*b^2*c^3*d^2*e^2 - 16*a*b*c^3*d*e^3)/c + (2*(d + e*x^2)^(1/2)*((8*a^2*c
^3*d - b^5*e - b^2*e*(-(4*a*c - b^2)^3)^(1/2) + b^4*c*d + 7*a*b^3*c*e + a*c*e*(-(4*a*c - b^2)^3)^(1/2) + b*c*d
*(-(4*a*c - b^2)^3)^(1/2) - 6*a*b^2*c^2*d - 12*a^2*b*c^2*e)/(8*(16*a^2*c^5 + b^4*c^3 - 8*a*b^2*c^4)))^(1/2)*(4
*b^3*c^3*e^3 - 8*b^2*c^4*d*e^2 - 16*a*b*c^4*e^3 + 32*a*c^5*d*e^2))/c)*((8*a^2*c^3*d - b^5*e - b^2*e*(-(4*a*c -
b^2)^3)^(1/2) + b^4*c*d + 7*a*b^3*c*e + a*c*e*(-(4*a*c - b^2)^3)^(1/2) + b*c*d*(-(4*a*c - b^2)^3)^(1/2) - 6*a
*b^2*c^2*d - 12*a^2*b*c^2*e)/(8*(16*a^2*c^5 + b^4*c^3 - 8*a*b^2*c^4)))^(1/2) + (2*(d + e*x^2)^(1/2)*(b^4*e^4 +
2*a^2*c^2*e^4 - 2*a*c^3*d^2*e^2 + b^2*c^2*d^2*e^2 - 4*a*b^2*c*e^4 - 2*b^3*c*d*e^3 + 6*a*b*c^2*d*e^3))/c)*((8*
a^2*c^3*d - b^5*e - b^2*e*(-(4*a*c - b^2)^3)^(1/2) + b^4*c*d + 7*a*b^3*c*e + a*c*e*(-(4*a*c - b^2)^3)^(1/2) +
b*c*d*(-(4*a*c - b^2)^3)^(1/2) - 6*a*b^2*c^2*d - 12*a^2*b*c^2*e)/(8*(16*a^2*c^5 + b^4*c^3 - 8*a*b^2*c^4)))^(1/
2)*1i)/((((16*a^2*c^3*e^4 - 4*a*b^2*c^2*e^4 + 16*a*c^4*d^2*e^2 + 4*b^3*c^2*d*e^3 - 4*b^2*c^3*d^2*e^2 - 16*a*b*
c^3*d*e^3)/c - (2*(d + e*x^2)^(1/2)*((8*a^2*c^3*d - b^5*e - b^2*e*(-(4*a*c - b^2)^3)^(1/2) + b^4*c*d + 7*a*b^3
*c*e + a*c*e*(-(4*a*c - b^2)^3)^(1/2) + b*c*d*(-(4*a*c - b^2)^3)^(1/2) - 6*a*b^2*c^2*d - 12*a^2*b*c^2*e)/(8*(1
6*a^2*c^5 + b^4*c^3 - 8*a*b^2*c^4)))^(1/2)*(4*b^3*c^3*e^3 - 8*b^2*c^4*d*e^2 - 16*a*b*c^4*e^3 + 32*a*c^5*d*e^2)
)/c)*((8*a^2*c^3*d - b^5*e - b^2*e*(-(4*a*c - b^2)^3)^(1/2) + b^4*c*d + 7*a*b^3*c*e + a*c*e*(-(4*a*c - b^2)^3)
^(1/2) + b*c*d*(-(4*a*c - b^2)^3)^(1/2) - 6*a*b^2*c^2*d - 12*a^2*b*c^2*e)/(8*(16*a^2*c^5 + b^4*c^3 - 8*a*b^2*c
^4)))^(1/2) - (2*(d + e*x^2)^(1/2)*(b^4*e^4 + 2*a^2*c^2*e^4 - 2*a*c^3*d^2*e^2 + b^2*c^2*d^2*e^2 - 4*a*b^2*c*e^
4 - 2*b^3*c*d*e^3 + 6*a*b*c^2*d*e^3))/c)*((8*a^2*c^3*d - b^5*e - b^2*e*(-(4*a*c - b^2)^3)^(1/2) + b^4*c*d + 7*
a*b^3*c*e + a*c*e*(-(4*a*c - b^2)^3)^(1/2) + b*c*d*(-(4*a*c - b^2)^3)^(1/2) - 6*a*b^2*c^2*d - 12*a^2*b*c^2*e)/
(8*(16*a^2*c^5 + b^4*c^3 - 8*a*b^2*c^4)))^(1/2) - (2*(a*c^2*d^3*e^2 - a^2*b*e^5 + a*b^2*d*e^4 + a^2*c*d*e^4 -
2*a*b*c*d^2*e^3))/c + (((16*a^2*c^3*e^4 - 4*a*b^2*c^2*e^4 + 16*a*c^4*d^2*e^2 + 4*b^3*c^2*d*e^3 - 4*b^2*c^3*d^2
*e^2 - 16*a*b*c^3*d*e^3)/c + (2*(d + e*x^2)^(1/2)*((8*a^2*c^3*d - b^5*e - b^2*e*(-(4*a*c - b^2)^3)^(1/2) + b^4
*c*d + 7*a*b^3*c*e + a*c*e*(-(4*a*c - b^2)^3)^(1/2) + b*c*d*(-(4*a*c - b^2)^3)^(1/2) - 6*a*b^2*c^2*d - 12*a^2*
b*c^2*e)/(8*(16*a^2*c^5 + b^4*c^3 - 8*a*b^2*c^4)))^(1/2)*(4*b^3*c^3*e^3 - 8*b^2*c^4*d*e^2 - 16*a*b*c^4*e^3 + 3
2*a*c^5*d*e^2))/c)*((8*a^2*c^3*d - b^5*e - b^2*e*(-(4*a*c - b^2)^3)^(1/2) + b^4*c*d + 7*a*b^3*c*e + a*c*e*(-(4
*a*c - b^2)^3)^(1/2) + b*c*d*(-(4*a*c - b^2)^3)^(1/2) - 6*a*b^2*c^2*d - 12*a^2*b*c^2*e)/(8*(16*a^2*c^5 + b^4*c
^3 - 8*a*b^2*c^4)))^(1/2) + (2*(d + e*x^2)^(1/2)*(b^4*e^4 + 2*a^2*c^2*e^4 - 2*a*c^3*d^2*e^2 + b^2*c^2*d^2*e^2
- 4*a*b^2*c*e^4 - 2*b^3*c*d*e^3 + 6*a*b*c^2*d*e^3))/c)*((8*a^2*c^3*d - b^5*e - b^2*e*(-(4*a*c - b^2)^3)^(1/2)
+ b^4*c*d + 7*a*b^3*c*e + a*c*e*(-(4*a*c - b^2)^3)^(1/2) + b*c*d*(-(4*a*c - b^2)^3)^(1/2) - 6*a*b^2*c^2*d - 12
*a^2*b*c^2*e)/(8*(16*a^2*c^5 + b^4*c^3 - 8*a*b^2*c^4)))^(1/2)))*((8*a^2*c^3*d - b^5*e - b^2*e*(-(4*a*c - b^2)^
3)^(1/2) + b^4*c*d + 7*a*b^3*c*e + a*c*e*(-(4*a*c - b^2)^3)^(1/2) + b*c*d*(-(4*a*c - b^2)^3)^(1/2) - 6*a*b^2*c
^2*d - 12*a^2*b*c^2*e)/(8*(16*a^2*c^5 + b^4*c^3 - 8*a*b^2*c^4)))^(1/2)*2i - atan(((((16*a^2*c^3*e^4 - 4*a*b^2*
c^2*e^4 + 16*a*c^4*d^2*e^2 + 4*b^3*c^2*d*e^3 - 4*b^2*c^3*d^2*e^2 - 16*a*b*c^3*d*e^3)/c - (2*(d + e*x^2)^(1/2)*
(-(b^5*e - 8*a^2*c^3*d - b^2*e*(-(4*a*c - b^2)^3)^(1/2) - b^4*c*d - 7*a*b^3*c*e + a*c*e*(-(4*a*c - b^2)^3)^(1/
2) + b*c*d*(-(4*a*c - b^2)^3)^(1/2) + 6*a*b^2*c^2*d + 12*a^2*b*c^2*e)/(8*(16*a^2*c^5 + b^4*c^3 - 8*a*b^2*c^4))
)^(1/2)*(4*b^3*c^3*e^3 - 8*b^2*c^4*d*e^2 - 16*a*b*c^4*e^3 + 32*a*c^5*d*e^2))/c)*(-(b^5*e - 8*a^2*c^3*d - b^2*e
*(-(4*a*c - b^2)^3)^(1/2) - b^4*c*d - 7*a*b^3*c...

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