3.4.70 \(\int \frac {(d+e x^2)^{3/2}}{x^3 (a+b x^2+c x^4)} \, dx\) [370]

Optimal. Leaf size=417 \[ -\frac {d \sqrt {d+e x^2}}{2 a x^2}+\frac {\sqrt {d} e \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{2 a}+\frac {\sqrt {d} (b d-2 a e) \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{a^2}-\frac {\sqrt {c} \left (b^2 d^2+b d \left (\sqrt {b^2-4 a c} d-2 a e\right )-2 a \left (c d^2+e \left (\sqrt {b^2-4 a c} d-a e\right )\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x^2}}{\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}\right )}{\sqrt {2} a^2 \sqrt {b^2-4 a c} \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}+\frac {\sqrt {c} \left (b^2 d^2-b d \left (\sqrt {b^2-4 a c} d+2 a e\right )-2 a \left (c d^2-e \left (\sqrt {b^2-4 a c} d+a e\right )\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x^2}}{\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}\right )}{\sqrt {2} a^2 \sqrt {b^2-4 a c} \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \]

[Out]

1/2*e*arctanh((e*x^2+d)^(1/2)/d^(1/2))*d^(1/2)/a+(-2*a*e+b*d)*arctanh((e*x^2+d)^(1/2)/d^(1/2))*d^(1/2)/a^2-1/2
*d*(e*x^2+d)^(1/2)/a/x^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))*c
^(1/2)*(b^2*d^2+b*d*(-2*a*e+d*(-4*a*c+b^2)^(1/2))-2*a*(c*d^2+e*(-a*e+d*(-4*a*c+b^2)^(1/2))))/a^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))*c^(1/2)*(b^2*d^2-b*d*(2*a*e+d*(-4*a*c+b^2)^(1/2))-2*a*(c*d^2-e*(a*e+d*(-4*a*c+b^2)^
(1/2))))/a^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 = 2.27, antiderivative size = 416, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {1265, 911, 1301, 205, 212, 1180, 214} \begin {gather*} -\frac {\sqrt {c} \left (b d \left (d \sqrt {b^2-4 a c}-2 a e\right )-2 a e \left (d \sqrt {b^2-4 a c}-a e\right )-2 a c d^2+b^2 d^2\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} a^2 \sqrt {b^2-4 a c} \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}+\frac {\sqrt {c} \left (-b d \left (d \sqrt {b^2-4 a c}+2 a e\right )+2 a e \left (d \sqrt {b^2-4 a c}+a e\right )-2 a c d^2+b^2 d^2\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} a^2 \sqrt {b^2-4 a c} \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}+\frac {\sqrt {d} (b d-2 a e) \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{a^2}-\frac {d \sqrt {d+e x^2}}{2 a x^2}+\frac {\sqrt {d} e \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{2 a} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 205

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^n)^(p + 1)/(a*n*(p + 1))), x] + Dist[(n*(p
+ 1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (
IntegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[
p])

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 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 911

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :
> With[{q = Denominator[m]}, Dist[q/e, Subst[Int[x^(q*(m + 1) - 1)*((e*f - d*g)/e + g*(x^q/e))^n*((c*d^2 - b*d
*e + a*e^2)/e^2 - (2*c*d - b*e)*(x^q/e^2) + c*(x^(2*q)/e^2))^p, x], x, (d + e*x)^(1/q)], 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] && IntegersQ[n,
p] && FractionQ[m]

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]

Rule 1301

Int[(((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.))/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> Int[Ex
pandIntegrand[(f*x)^m*((d + e*x^2)^q/(a + b*x^2 + c*x^4)), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b^
2 - 4*a*c, 0] && IntegerQ[q] && IntegerQ[m]

Rubi steps

\begin {align*} \int \frac {\left (d+e x^2\right )^{3/2}}{x^3 \left (a+b x^2+c x^4\right )} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {(d+e x)^{3/2}}{x^2 \left (a+b x+c x^2\right )} \, dx,x,x^2\right )\\ &=\frac {\text {Subst}\left (\int \frac {x^4}{\left (-\frac {d}{e}+\frac {x^2}{e}\right )^2 \left (\frac {c d^2-b d e+a e^2}{e^2}-\frac {(2 c d-b e) x^2}{e^2}+\frac {c x^4}{e^2}\right )} \, dx,x,\sqrt {d+e x^2}\right )}{e}\\ &=\frac {\text {Subst}\left (\int \left (\frac {d^2 e^2}{a \left (d-x^2\right )^2}-\frac {d e (-b d+2 a e)}{a^2 \left (d-x^2\right )}+\frac {e \left (-(b d-a e) \left (c d^2-b d e+a e^2\right )+c d (b d-2 a e) x^2\right )}{a^2 \left (c d^2-b d e+a e^2-(2 c d-b e) x^2+c x^4\right )}\right ) \, dx,x,\sqrt {d+e x^2}\right )}{e}\\ &=\frac {\text {Subst}\left (\int \frac {-(b d-a e) \left (c d^2-b d e+a e^2\right )+c d (b d-2 a 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 )}{a^2}+\frac {\left (d^2 e\right ) \text {Subst}\left (\int \frac {1}{\left (d-x^2\right )^2} \, dx,x,\sqrt {d+e x^2}\right )}{a}+\frac {(d (b d-2 a e)) \text {Subst}\left (\int \frac {1}{d-x^2} \, dx,x,\sqrt {d+e x^2}\right )}{a^2}\\ &=-\frac {d \sqrt {d+e x^2}}{2 a x^2}+\frac {\sqrt {d} (b d-2 a e) \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{a^2}+\frac {(d e) \text {Subst}\left (\int \frac {1}{d-x^2} \, dx,x,\sqrt {d+e x^2}\right )}{2 a}+\frac {\left (c \left (b^2 d^2-2 a c d^2+b d \left (\sqrt {b^2-4 a c} d-2 a e\right )-2 a e \left (\sqrt {b^2-4 a c} d-a e\right )\right )\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 a^2 \sqrt {b^2-4 a c}}-\frac {\left (c \left (b^2 d^2-2 a c d^2+2 a e \left (\sqrt {b^2-4 a c} d+a e\right )-b d \left (\sqrt {b^2-4 a c} d+2 a e\right )\right )\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 a^2 \sqrt {b^2-4 a c}}\\ &=-\frac {d \sqrt {d+e x^2}}{2 a x^2}+\frac {\sqrt {d} e \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{2 a}+\frac {\sqrt {d} (b d-2 a e) \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{a^2}-\frac {\sqrt {c} \left (b^2 d^2-2 a c d^2+b d \left (\sqrt {b^2-4 a c} d-2 a e\right )-2 a e \left (\sqrt {b^2-4 a c} d-a e\right )\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} a^2 \sqrt {b^2-4 a c} \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}+\frac {\sqrt {c} \left (b^2 d^2-2 a c d^2+2 a e \left (\sqrt {b^2-4 a c} d+a e\right )-b d \left (\sqrt {b^2-4 a c} d+2 a e\right )\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} a^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 = 1.55, size = 427, normalized size = 1.02 \begin {gather*} \frac {-\frac {a d \sqrt {d+e x^2}}{x^2}+\frac {\sqrt {2} \sqrt {c} \left (-i b^2 d^2+b d \left (\sqrt {-b^2+4 a c} d+2 i a e\right )-2 i a \left (-c d^2+e \left (-i \sqrt {-b^2+4 a c} d+a e\right )\right )\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 {-b^2+4 a c} \sqrt {-2 c d+\left (b-i \sqrt {-b^2+4 a c}\right ) e}}+\frac {\sqrt {2} \sqrt {c} \left (i b^2 d^2+b d \left (\sqrt {-b^2+4 a c} d-2 i a e\right )+2 i a \left (-c d^2+e \left (i \sqrt {-b^2+4 a c} d+a e\right )\right )\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 {-b^2+4 a c} \sqrt {-2 c d+\left (b+i \sqrt {-b^2+4 a c}\right ) e}}+\sqrt {d} (2 b d-3 a e) \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{2 a^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.16, size = 535, normalized size = 1.28

method	result	size
risch	\(-\frac {d \sqrt {e \,x^{2}+d}}{2 a \,x^{2}}-\frac {3 \sqrt {d}\, \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {e \,x^{2}+d}}{x}\right ) e}{2 a}+\frac {d^{\frac {3}{2}} \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {e \,x^{2}+d}}{x}\right ) b}{a^{2}}-\frac {\munderset {\textit {\_R} =\RootOf \left (c \,\textit {\_Z}^{8}+\left (4 e b -4 c d \right ) \textit {\_Z}^{6}+\left (16 a \,e^{2}-8 d e b +6 c \,d^{2}\right ) \textit {\_Z}^{4}+\left (4 d^{2} e b -4 c \,d^{3}\right ) \textit {\_Z}^{2}+d^{4} c \right )}{\sum }\frac {\left (c d \left (2 a e -b d \right ) \textit {\_R}^{6}+\left (-4 a^{2} e^{3}+8 a b d \,e^{2}-2 a \,d^{2} e c -4 b^{2} d^{2} e +3 b c \,d^{3}\right ) \textit {\_R}^{4}+d \left (4 a^{2} e^{3}-8 a b d \,e^{2}+2 a \,d^{2} e c +4 b^{2} d^{2} e -3 b c \,d^{3}\right ) \textit {\_R}^{2}-2 a c \,d^{4} e +b c \,d^{5}\right ) \ln \left (\sqrt {e \,x^{2}+d}-\sqrt {e}\, x -\textit {\_R} \right )}{\textit {\_R}^{7} c +3 \textit {\_R}^{5} b e -3 \textit {\_R}^{5} c d +8 \textit {\_R}^{3} a \,e^{2}-4 \textit {\_R}^{3} b d e +3 \textit {\_R}^{3} c \,d^{2}+\textit {\_R} b \,d^{2} e -\textit {\_R} c \,d^{3}}}{4 a^{2}}\)	\(360\)
default	\(-\frac {-\frac {b \left (\sqrt {e \,x^{2}+d}-\sqrt {e}\, x \right )^{3}}{24}+\frac {a e \left (\sqrt {e \,x^{2}+d}-\sqrt {e}\, x \right )}{2}-\frac {5 b d \left (\sqrt {e \,x^{2}+d}-\sqrt {e}\, x \right )}{8}+\frac {d \left (4 a e -5 b d \right )}{8 \sqrt {e \,x^{2}+d}-8 \sqrt {e}\, x}-\frac {b \,d^{3}}{24 \left (\sqrt {e \,x^{2}+d}-\sqrt {e}\, x \right )^{3}}-\frac {\left (\munderset {\textit {\_R} =\RootOf \left (c \,\textit {\_Z}^{8}+\left (4 e b -4 c d \right ) \textit {\_Z}^{6}+\left (16 a \,e^{2}-8 d e b +6 c \,d^{2}\right ) \textit {\_Z}^{4}+\left (4 d^{2} e b -4 c \,d^{3}\right ) \textit {\_Z}^{2}+d^{4} c \right )}{\sum }\frac {\left (c d \left (-2 a e +b d \right ) \textit {\_R}^{6}+\left (4 a^{2} e^{3}-8 a b d \,e^{2}+2 a \,d^{2} e c +4 b^{2} d^{2} e -3 b c \,d^{3}\right ) \textit {\_R}^{4}+d \left (-4 a^{2} e^{3}+8 a b d \,e^{2}-2 a \,d^{2} e c -4 b^{2} d^{2} e +3 b c \,d^{3}\right ) \textit {\_R}^{2}+2 a c \,d^{4} e -b c \,d^{5}\right ) \ln \left (\sqrt {e \,x^{2}+d}-\sqrt {e}\, x -\textit {\_R} \right )}{\textit {\_R}^{7} c +3 \textit {\_R}^{5} b e -3 \textit {\_R}^{5} c d +8 \textit {\_R}^{3} a \,e^{2}-4 \textit {\_R}^{3} b d e +3 \textit {\_R}^{3} c \,d^{2}+\textit {\_R} b \,d^{2} e -\textit {\_R} c \,d^{3}}\right )}{4}}{a^{2}}+\frac {-\frac {\left (e \,x^{2}+d \right )^{\frac {5}{2}}}{2 d \,x^{2}}+\frac {3 e \left (\frac {\left (e \,x^{2}+d \right )^{\frac {3}{2}}}{3}+d \left (\sqrt {e \,x^{2}+d}-\sqrt {d}\, \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {e \,x^{2}+d}}{x}\right )\right )\right )}{2 d}}{a}-\frac {b \left (\frac {\left (e \,x^{2}+d \right )^{\frac {3}{2}}}{3}+d \left (\sqrt {e \,x^{2}+d}-\sqrt {d}\, \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {e \,x^{2}+d}}{x}\right )\right )\right )}{a^{2}}\)	\(535\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/a^2*(-1/24*b*((e*x^2+d)^(1/2)-e^(1/2)*x)^3+1/2*a*e*((e*x^2+d)^(1/2)-e^(1/2)*x)-5/8*b*d*((e*x^2+d)^(1/2)-e^(
1/2)*x)+1/8*d*(4*a*e-5*b*d)/((e*x^2+d)^(1/2)-e^(1/2)*x)-1/24*b*d^3/((e*x^2+d)^(1/2)-e^(1/2)*x)^3-1/4*sum((c*d*
(-2*a*e+b*d)*_R^6+(4*a^2*e^3-8*a*b*d*e^2+2*a*c*d^2*e+4*b^2*d^2*e-3*b*c*d^3)*_R^4+d*(-4*a^2*e^3+8*a*b*d*e^2-2*a
*c*d^2*e-4*b^2*d^2*e+3*b*c*d^3)*_R^2+2*a*c*d^4*e-b*c*d^5)/(_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/a*(-1/2/d/x^2*(e*x^2+d)^(5/2)+3/2*e/d*(1/3*(e
*x^2+d)^(3/2)+d*((e*x^2+d)^(1/2)-d^(1/2)*ln((2*d+2*d^(1/2)*(e*x^2+d)^(1/2))/x))))-b/a^2*(1/3*(e*x^2+d)^(3/2)+d
*((e*x^2+d)^(1/2)-d^(1/2)*ln((2*d+2*d^(1/2)*(e*x^2+d)^(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((e*x^2+d)^(3/2)/x^3/(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^3), x)

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

[Out]

Timed out

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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}}}{x^{3} \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**3/(c*x**4+b*x**2+a),x)

[Out]

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

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Giac [A]
time = 5.54, size = 433, normalized size = 1.04 \begin {gather*} -\frac {{\left (2 \, b d^{2} - 3 \, a d e\right )} \arctan \left (\frac {\sqrt {x^{2} e + d}}{\sqrt {-d}}\right )}{2 \, a^{2} \sqrt {-d}} - \frac {\sqrt {-4 \, c^{2} d + 2 \, {\left (b c - \sqrt {b^{2} - 4 \, a c} c\right )} e} {\left ({\left (b^{2} - 2 \, a c + \sqrt {b^{2} - 4 \, a c} b\right )} d - {\left (a b + \sqrt {b^{2} - 4 \, a c} a\right )} e\right )} \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} \sqrt {x^{2} e + d}}{\sqrt {-\frac {2 \, a^{2} c d - a^{2} b e + \sqrt {-4 \, {\left (a^{2} c d^{2} - a^{2} b d e + a^{3} e^{2}\right )} a^{2} c + {\left (2 \, a^{2} c d - a^{2} b e\right )}^{2}}}{a^{2} c}}}\right )}{4 \, \sqrt {b^{2} - 4 \, a c} a^{2} {\left | c \right |}} + \frac {\sqrt {-4 \, c^{2} d + 2 \, {\left (b c + \sqrt {b^{2} - 4 \, a c} c\right )} e} {\left ({\left (b^{2} - 2 \, a c - \sqrt {b^{2} - 4 \, a c} b\right )} d - {\left (a b - \sqrt {b^{2} - 4 \, a c} a\right )} e\right )} \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} \sqrt {x^{2} e + d}}{\sqrt {-\frac {2 \, a^{2} c d - a^{2} b e - \sqrt {-4 \, {\left (a^{2} c d^{2} - a^{2} b d e + a^{3} e^{2}\right )} a^{2} c + {\left (2 \, a^{2} c d - a^{2} b e\right )}^{2}}}{a^{2} c}}}\right )}{4 \, \sqrt {b^{2} - 4 \, a c} a^{2} {\left | c \right |}} - \frac {\sqrt {x^{2} e + d} d}{2 \, a x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 6.10, size = 2500, normalized size = 6.00 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(d^(1/2)*atan(((d^(1/2)*(3*a*e - 2*b*d)*(((d + e*x^2)^(1/2)*(4*a^6*c^3*e^16 + 4*a^2*c^7*d^8*e^8 - 2*a^3*c^6*d^
6*e^10 + 132*a^4*c^5*d^4*e^12 - 2*a^5*c^4*d^2*e^14 + 4*b^4*c^5*d^8*e^8 + 129*a^2*b^2*c^5*d^6*e^10 - 32*a^2*b^3
*c^4*d^5*e^11 + 8*a^2*b^4*c^3*d^4*e^12 + 88*a^3*b^2*c^4*d^4*e^12 - 28*a^3*b^3*c^3*d^3*e^13 + 33*a^4*b^2*c^3*d^
2*e^14 - 16*a^5*b*c^3*d*e^15 - 8*a*b^2*c^6*d^8*e^8 - 28*a*b^3*c^5*d^7*e^9 + 8*a^2*b*c^6*d^7*e^9 - 228*a^3*b*c^
5*d^5*e^11 - 60*a^4*b*c^4*d^3*e^13))/(2*a^4) - (d^(1/2)*((56*a^4*c^6*d^6*e^9 - 44*a^5*c^5*d^4*e^11 - 100*a^6*c
^4*d^2*e^13 + 40*a^2*b^3*c^5*d^7*e^8 - 39*a^2*b^5*c^3*d^5*e^10 - 11*a^2*b^6*c^2*d^4*e^11 - 108*a^3*b^2*c^5*d^6
*e^9 + 96*a^3*b^3*c^4*d^5*e^10 + 111*a^3*b^4*c^3*d^4*e^11 + 22*a^3*b^5*c^2*d^3*e^12 - 237*a^4*b^2*c^4*d^4*e^11
- 161*a^4*b^3*c^3*d^3*e^12 - 19*a^4*b^4*c^2*d^2*e^13 + 111*a^5*b^2*c^3*d^2*e^13 - 28*a^6*b*c^3*d*e^14 - 8*a*b
^5*c^4*d^7*e^8 + 6*a*b^6*c^3*d^6*e^9 + 2*a*b^7*c^2*d^5*e^10 - 32*a^3*b*c^6*d^7*e^8 + 92*a^4*b*c^5*d^5*e^10 + 2
52*a^5*b*c^4*d^3*e^12 + 6*a^5*b^3*c^2*d*e^14)/a^4 + (d^(1/2)*(3*a*e - 2*b*d)*(((d + e*x^2)^(1/2)*(64*a^7*b*c^3
*e^13 + 352*a^7*c^4*d*e^12 - 16*a^6*b^3*c^2*e^13 - 160*a^5*c^6*d^5*e^8 + 736*a^6*c^5*d^3*e^10 + 32*a^2*b^6*c^3
*d^5*e^8 - 32*a^2*b^7*c^2*d^4*e^9 - 224*a^3*b^4*c^4*d^5*e^8 + 144*a^3*b^5*c^3*d^4*e^9 + 112*a^3*b^6*c^2*d^3*e^
10 + 432*a^4*b^2*c^5*d^5*e^8 + 144*a^4*b^3*c^4*d^4*e^9 - 716*a^4*b^4*c^3*d^3*e^10 - 132*a^4*b^5*c^2*d^2*e^11 +
936*a^5*b^2*c^4*d^3*e^10 + 860*a^5*b^3*c^3*d^2*e^11 - 896*a^5*b*c^5*d^4*e^9 + 64*a^5*b^4*c^2*d*e^12 - 1392*a^
6*b*c^4*d^2*e^11 - 336*a^6*b^2*c^3*d*e^12))/(2*a^4) + (d^(1/2)*((320*a^8*c^4*d*e^11 + 320*a^7*c^5*d^3*e^9 + 32
*a^5*b^3*c^4*d^4*e^8 - 24*a^5*b^4*c^3*d^3*e^9 - 8*a^5*b^5*c^2*d^2*e^10 + 16*a^6*b^2*c^4*d^3*e^9 + 144*a^6*b^3*
c^3*d^2*e^10 - 128*a^6*b*c^5*d^4*e^8 + 8*a^6*b^4*c^2*d*e^11 - 448*a^7*b*c^4*d^2*e^10 - 112*a^7*b^2*c^3*d*e^11)
/a^4 - (d^(1/2)*(d + e*x^2)^(1/2)*(3*a*e - 2*b*d)*(1024*a^9*c^4*e^10 + 64*a^7*b^4*c^2*e^10 - 512*a^8*b^2*c^3*e
^10 + 1536*a^8*c^5*d^2*e^8 + 128*a^6*b^4*c^3*d^2*e^8 - 896*a^7*b^2*c^4*d^2*e^8 - 1792*a^8*b*c^4*d*e^9 - 128*a^
6*b^5*c^2*d*e^9 + 960*a^7*b^3*c^3*d*e^9))/(8*a^6))*(3*a*e - 2*b*d))/(4*a^2)))/(4*a^2))*(3*a*e - 2*b*d))/(4*a^2
))*1i)/(4*a^2) + (d^(1/2)*(3*a*e - 2*b*d)*(((d + e*x^2)^(1/2)*(4*a^6*c^3*e^16 + 4*a^2*c^7*d^8*e^8 - 2*a^3*c^6*
d^6*e^10 + 132*a^4*c^5*d^4*e^12 - 2*a^5*c^4*d^2*e^14 + 4*b^4*c^5*d^8*e^8 + 129*a^2*b^2*c^5*d^6*e^10 - 32*a^2*b
^3*c^4*d^5*e^11 + 8*a^2*b^4*c^3*d^4*e^12 + 88*a^3*b^2*c^4*d^4*e^12 - 28*a^3*b^3*c^3*d^3*e^13 + 33*a^4*b^2*c^3*
d^2*e^14 - 16*a^5*b*c^3*d*e^15 - 8*a*b^2*c^6*d^8*e^8 - 28*a*b^3*c^5*d^7*e^9 + 8*a^2*b*c^6*d^7*e^9 - 228*a^3*b*
c^5*d^5*e^11 - 60*a^4*b*c^4*d^3*e^13))/(2*a^4) + (d^(1/2)*((56*a^4*c^6*d^6*e^9 - 44*a^5*c^5*d^4*e^11 - 100*a^6
*c^4*d^2*e^13 + 40*a^2*b^3*c^5*d^7*e^8 - 39*a^2*b^5*c^3*d^5*e^10 - 11*a^2*b^6*c^2*d^4*e^11 - 108*a^3*b^2*c^5*d
^6*e^9 + 96*a^3*b^3*c^4*d^5*e^10 + 111*a^3*b^4*c^3*d^4*e^11 + 22*a^3*b^5*c^2*d^3*e^12 - 237*a^4*b^2*c^4*d^4*e^
11 - 161*a^4*b^3*c^3*d^3*e^12 - 19*a^4*b^4*c^2*d^2*e^13 + 111*a^5*b^2*c^3*d^2*e^13 - 28*a^6*b*c^3*d*e^14 - 8*a
*b^5*c^4*d^7*e^8 + 6*a*b^6*c^3*d^6*e^9 + 2*a*b^7*c^2*d^5*e^10 - 32*a^3*b*c^6*d^7*e^8 + 92*a^4*b*c^5*d^5*e^10 +
252*a^5*b*c^4*d^3*e^12 + 6*a^5*b^3*c^2*d*e^14)/a^4 - (d^(1/2)*(3*a*e - 2*b*d)*(((d + e*x^2)^(1/2)*(64*a^7*b*c
^3*e^13 + 352*a^7*c^4*d*e^12 - 16*a^6*b^3*c^2*e^13 - 160*a^5*c^6*d^5*e^8 + 736*a^6*c^5*d^3*e^10 + 32*a^2*b^6*c
^3*d^5*e^8 - 32*a^2*b^7*c^2*d^4*e^9 - 224*a^3*b^4*c^4*d^5*e^8 + 144*a^3*b^5*c^3*d^4*e^9 + 112*a^3*b^6*c^2*d^3*
e^10 + 432*a^4*b^2*c^5*d^5*e^8 + 144*a^4*b^3*c^4*d^4*e^9 - 716*a^4*b^4*c^3*d^3*e^10 - 132*a^4*b^5*c^2*d^2*e^11
+ 936*a^5*b^2*c^4*d^3*e^10 + 860*a^5*b^3*c^3*d^2*e^11 - 896*a^5*b*c^5*d^4*e^9 + 64*a^5*b^4*c^2*d*e^12 - 1392*
a^6*b*c^4*d^2*e^11 - 336*a^6*b^2*c^3*d*e^12))/(2*a^4) - (d^(1/2)*((320*a^8*c^4*d*e^11 + 320*a^7*c^5*d^3*e^9 +
32*a^5*b^3*c^4*d^4*e^8 - 24*a^5*b^4*c^3*d^3*e^9 - 8*a^5*b^5*c^2*d^2*e^10 + 16*a^6*b^2*c^4*d^3*e^9 + 144*a^6*b^
3*c^3*d^2*e^10 - 128*a^6*b*c^5*d^4*e^8 + 8*a^6*b^4*c^2*d*e^11 - 448*a^7*b*c^4*d^2*e^10 - 112*a^7*b^2*c^3*d*e^1
1)/a^4 + (d^(1/2)*(d + e*x^2)^(1/2)*(3*a*e - 2*b*d)*(1024*a^9*c^4*e^10 + 64*a^7*b^4*c^2*e^10 - 512*a^8*b^2*c^3
*e^10 + 1536*a^8*c^5*d^2*e^8 + 128*a^6*b^4*c^3*d^2*e^8 - 896*a^7*b^2*c^4*d^2*e^8 - 1792*a^8*b*c^4*d*e^9 - 128*
a^6*b^5*c^2*d*e^9 + 960*a^7*b^3*c^3*d*e^9))/(8*a^6))*(3*a*e - 2*b*d))/(4*a^2)))/(4*a^2))*(3*a*e - 2*b*d))/(4*a
^2))*1i)/(4*a^2))/((3*a*c^7*d^9*e^9 + 3*a^5*c^3*d*e^17 - 2*b*c^7*d^10*e^8 + 3*a^2*c^6*d^7*e^11 + 3*a^4*c^4*d^3
*e^15 + 4*b^2*c^6*d^9*e^9 - 2*b^3*c^5*d^8*e^10 + 2*a^2*b^2*c^4*d^5*e^13 - (11*a^2*b^3*c^3*d^4*e^14)/2 + 11*a^3
*b^2*c^3*d^3*e^15 - 8*a*b*c^6*d^8*e^10 + 4*a*b^2*c^5*d^7*e^11 + a*b^4*c^3*d^5*e^13 - (3*a^2*b*c^5*d^6*e^12)/2
- 5*a^3*b*c^4*d^4*e^14 - (19*a^4*b*c^3*d^2*e^16)/2)/a^4 - (d^(1/2)*(3*a*e - 2*b*d)*(((d + e*x^2)^(1/2)*(4*a^6*
c^3*e^16 + 4*a^2*c^7*d^8*e^8 - 2*a^3*c^6*d^6*e^10 + 132*a^4*c^5*d^4*e^12 - 2*a^5*c^4*d^2*e^14 + 4*b^4*c^5*d^8*
e^8 + 129*a^2*b^2*c^5*d^6*e^10 - 32*a^2*b^3*c^4...

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