3.1.66 \(\int \frac {x}{\sqrt {a+c x^2} (d+e x+f x^2)} \, dx\) [66]
Optimal. Leaf size=294 \[ \frac {\left (e-\sqrt {e^2-4 d f}\right ) \tanh ^{-1}\left (\frac {2 a f-c \left (e-\sqrt {e^2-4 d f}\right ) x}{\sqrt {2} \sqrt {2 a f^2+c \left (e^2-2 d f-e \sqrt {e^2-4 d f}\right )} \sqrt {a+c x^2}}\right )}{\sqrt {2} \sqrt {e^2-4 d f} \sqrt {2 a f^2+c \left (e^2-2 d f-e \sqrt {e^2-4 d f}\right )}}-\frac {\left (e+\sqrt {e^2-4 d f}\right ) \tanh ^{-1}\left (\frac {2 a f-c \left (e+\sqrt {e^2-4 d f}\right ) x}{\sqrt {2} \sqrt {2 a f^2+c \left (e^2-2 d f+e \sqrt {e^2-4 d f}\right )} \sqrt {a+c x^2}}\right )}{\sqrt {2} \sqrt {e^2-4 d f} \sqrt {2 a f^2+c \left (e^2-2 d f+e \sqrt {e^2-4 d f}\right )}} \]
[Out]
1/2*arctanh(1/2*(2*a*f-c*x*(e-(-4*d*f+e^2)^(1/2)))*2^(1/2)/(c*x^2+a)^(1/2)/(2*a*f^2+c*(e^2-2*d*f-e*(-4*d*f+e^2
)^(1/2)))^(1/2))*(e-(-4*d*f+e^2)^(1/2))*2^(1/2)/(-4*d*f+e^2)^(1/2)/(2*a*f^2+c*(e^2-2*d*f-e*(-4*d*f+e^2)^(1/2))
)^(1/2)-1/2*arctanh(1/2*(2*a*f-c*x*(e+(-4*d*f+e^2)^(1/2)))*2^(1/2)/(c*x^2+a)^(1/2)/(2*a*f^2+c*(e^2-2*d*f+e*(-4
*d*f+e^2)^(1/2)))^(1/2))*(e+(-4*d*f+e^2)^(1/2))*2^(1/2)/(-4*d*f+e^2)^(1/2)/(2*a*f^2+c*(e^2-2*d*f+e*(-4*d*f+e^2
)^(1/2)))^(1/2)
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Rubi [A]
time = 0.17, antiderivative size = 294, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {1048, 739, 212}
\begin {gather*} \frac {\left (e-\sqrt {e^2-4 d f}\right ) \tanh ^{-1}\left (\frac {2 a f-c x \left (e-\sqrt {e^2-4 d f}\right )}{\sqrt {2} \sqrt {a+c x^2} \sqrt {2 a f^2+c \left (-e \sqrt {e^2-4 d f}-2 d f+e^2\right )}}\right )}{\sqrt {2} \sqrt {e^2-4 d f} \sqrt {2 a f^2+c \left (-e \sqrt {e^2-4 d f}-2 d f+e^2\right )}}-\frac {\left (\sqrt {e^2-4 d f}+e\right ) \tanh ^{-1}\left (\frac {2 a f-c x \left (\sqrt {e^2-4 d f}+e\right )}{\sqrt {2} \sqrt {a+c x^2} \sqrt {2 a f^2+c \left (e \sqrt {e^2-4 d f}-2 d f+e^2\right )}}\right )}{\sqrt {2} \sqrt {e^2-4 d f} \sqrt {2 a f^2+c \left (e \sqrt {e^2-4 d f}-2 d f+e^2\right )}} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[x/(Sqrt[a + c*x^2]*(d + e*x + f*x^2)),x]
[Out]
((e - Sqrt[e^2 - 4*d*f])*ArcTanh[(2*a*f - c*(e - Sqrt[e^2 - 4*d*f])*x)/(Sqrt[2]*Sqrt[2*a*f^2 + c*(e^2 - 2*d*f
- e*Sqrt[e^2 - 4*d*f])]*Sqrt[a + c*x^2])])/(Sqrt[2]*Sqrt[e^2 - 4*d*f]*Sqrt[2*a*f^2 + c*(e^2 - 2*d*f - e*Sqrt[e
^2 - 4*d*f])]) - ((e + Sqrt[e^2 - 4*d*f])*ArcTanh[(2*a*f - c*(e + Sqrt[e^2 - 4*d*f])*x)/(Sqrt[2]*Sqrt[2*a*f^2
+ c*(e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f])]*Sqrt[a + c*x^2])])/(Sqrt[2]*Sqrt[e^2 - 4*d*f]*Sqrt[2*a*f^2 + c*(e^2 -
2*d*f + e*Sqrt[e^2 - 4*d*f])])
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 739
Int[1/(((d_) + (e_.)*(x_))*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> -Subst[Int[1/(c*d^2 + a*e^2 - x^2), x], x,
(a*e - c*d*x)/Sqrt[a + c*x^2]] /; FreeQ[{a, c, d, e}, x]
Rule 1048
Int[((g_.) + (h_.)*(x_))/(((a_) + (b_.)*(x_) + (c_.)*(x_)^2)*Sqrt[(d_) + (f_.)*(x_)^2]), x_Symbol] :> With[{q
= Rt[b^2 - 4*a*c, 2]}, Dist[(2*c*g - h*(b - q))/q, Int[1/((b - q + 2*c*x)*Sqrt[d + f*x^2]), x], x] - Dist[(2*c
*g - h*(b + q))/q, Int[1/((b + q + 2*c*x)*Sqrt[d + f*x^2]), x], x]] /; FreeQ[{a, b, c, d, f, g, h}, x] && NeQ[
b^2 - 4*a*c, 0] && PosQ[b^2 - 4*a*c]
Rubi steps
\begin {align*} \int \frac {x}{\sqrt {a+c x^2} \left (d+e x+f x^2\right )} \, dx &=-\left (\left (-1-\frac {e}{\sqrt {e^2-4 d f}}\right ) \int \frac {1}{\left (e+\sqrt {e^2-4 d f}+2 f x\right ) \sqrt {a+c x^2}} \, dx\right )+\left (1-\frac {e}{\sqrt {e^2-4 d f}}\right ) \int \frac {1}{\left (e-\sqrt {e^2-4 d f}+2 f x\right ) \sqrt {a+c x^2}} \, dx\\ &=\left (-1+\frac {e}{\sqrt {e^2-4 d f}}\right ) \text {Subst}\left (\int \frac {1}{4 a f^2+c \left (e-\sqrt {e^2-4 d f}\right )^2-x^2} \, dx,x,\frac {2 a f-c \left (e-\sqrt {e^2-4 d f}\right ) x}{\sqrt {a+c x^2}}\right )-\left (1+\frac {e}{\sqrt {e^2-4 d f}}\right ) \text {Subst}\left (\int \frac {1}{4 a f^2+c \left (e+\sqrt {e^2-4 d f}\right )^2-x^2} \, dx,x,\frac {2 a f-c \left (e+\sqrt {e^2-4 d f}\right ) x}{\sqrt {a+c x^2}}\right )\\ &=-\frac {\left (1-\frac {e}{\sqrt {e^2-4 d f}}\right ) \tanh ^{-1}\left (\frac {2 a f-c \left (e-\sqrt {e^2-4 d f}\right ) x}{\sqrt {2} \sqrt {2 a f^2+c \left (e^2-2 d f-e \sqrt {e^2-4 d f}\right )} \sqrt {a+c x^2}}\right )}{\sqrt {2} \sqrt {2 a f^2+c \left (e^2-2 d f-e \sqrt {e^2-4 d f}\right )}}-\frac {\left (1+\frac {e}{\sqrt {e^2-4 d f}}\right ) \tanh ^{-1}\left (\frac {2 a f-c \left (e+\sqrt {e^2-4 d f}\right ) x}{\sqrt {2} \sqrt {2 a f^2+c \left (e^2-2 d f+e \sqrt {e^2-4 d f}\right )} \sqrt {a+c x^2}}\right )}{\sqrt {2} \sqrt {2 a f^2+c \left (e^2-2 d f+e \sqrt {e^2-4 d f}\right )}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in
optimal.
time = 0.31, size = 156, normalized size = 0.53 \begin {gather*} \text {RootSum}\left [a^2 f+2 a \sqrt {c} e \text {$\#$1}+4 c d \text {$\#$1}^2-2 a f \text {$\#$1}^2-2 \sqrt {c} e \text {$\#$1}^3+f \text {$\#$1}^4\&,\frac {-a \log \left (-\sqrt {c} x+\sqrt {a+c x^2}-\text {$\#$1}\right )+\log \left (-\sqrt {c} x+\sqrt {a+c x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{a \sqrt {c} e+4 c d \text {$\#$1}-2 a f \text {$\#$1}-3 \sqrt {c} e \text {$\#$1}^2+2 f \text {$\#$1}^3}\&\right ] \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[x/(Sqrt[a + c*x^2]*(d + e*x + f*x^2)),x]
[Out]
RootSum[a^2*f + 2*a*Sqrt[c]*e*#1 + 4*c*d*#1^2 - 2*a*f*#1^2 - 2*Sqrt[c]*e*#1^3 + f*#1^4 & , (-(a*Log[-(Sqrt[c]*
x) + Sqrt[a + c*x^2] - #1]) + Log[-(Sqrt[c]*x) + Sqrt[a + c*x^2] - #1]*#1^2)/(a*Sqrt[c]*e + 4*c*d*#1 - 2*a*f*#
1 - 3*Sqrt[c]*e*#1^2 + 2*f*#1^3) & ]
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(621\) vs.
\(2(257)=514\).
time = 0.12, size = 622, normalized size = 2.12
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method |
result |
size |
| | |
default |
\(-\frac {\left (e +\sqrt {-4 d f +e^{2}}\right ) \sqrt {2}\, \ln \left (\frac {\frac {\sqrt {-4 d f +e^{2}}\, c e +2 a \,f^{2}-2 c d f +c \,e^{2}}{f^{2}}-\frac {c \left (e +\sqrt {-4 d f +e^{2}}\right ) \left (x +\frac {e +\sqrt {-4 d f +e^{2}}}{2 f}\right )}{f}+\frac {\sqrt {2}\, \sqrt {\frac {\sqrt {-4 d f +e^{2}}\, c e +2 a \,f^{2}-2 c d f +c \,e^{2}}{f^{2}}}\, \sqrt {4 \left (x +\frac {e +\sqrt {-4 d f +e^{2}}}{2 f}\right )^{2} c -\frac {4 c \left (e +\sqrt {-4 d f +e^{2}}\right ) \left (x +\frac {e +\sqrt {-4 d f +e^{2}}}{2 f}\right )}{f}+\frac {2 \sqrt {-4 d f +e^{2}}\, c e +4 a \,f^{2}-4 c d f +2 c \,e^{2}}{f^{2}}}}{2}}{x +\frac {e +\sqrt {-4 d f +e^{2}}}{2 f}}\right )}{2 \sqrt {-4 d f +e^{2}}\, f \sqrt {\frac {\sqrt {-4 d f +e^{2}}\, c e +2 a \,f^{2}-2 c d f +c \,e^{2}}{f^{2}}}}-\frac {\left (-e +\sqrt {-4 d f +e^{2}}\right ) \sqrt {2}\, \ln \left (\frac {\frac {-\sqrt {-4 d f +e^{2}}\, c e +2 a \,f^{2}-2 c d f +c \,e^{2}}{f^{2}}-\frac {c \left (e -\sqrt {-4 d f +e^{2}}\right ) \left (x -\frac {-e +\sqrt {-4 d f +e^{2}}}{2 f}\right )}{f}+\frac {\sqrt {2}\, \sqrt {\frac {-\sqrt {-4 d f +e^{2}}\, c e +2 a \,f^{2}-2 c d f +c \,e^{2}}{f^{2}}}\, \sqrt {4 \left (x -\frac {-e +\sqrt {-4 d f +e^{2}}}{2 f}\right )^{2} c -\frac {4 c \left (e -\sqrt {-4 d f +e^{2}}\right ) \left (x -\frac {-e +\sqrt {-4 d f +e^{2}}}{2 f}\right )}{f}+\frac {-2 \sqrt {-4 d f +e^{2}}\, c e +4 a \,f^{2}-4 c d f +2 c \,e^{2}}{f^{2}}}}{2}}{x -\frac {-e +\sqrt {-4 d f +e^{2}}}{2 f}}\right )}{2 \sqrt {-4 d f +e^{2}}\, f \sqrt {\frac {-\sqrt {-4 d f +e^{2}}\, c e +2 a \,f^{2}-2 c d f +c \,e^{2}}{f^{2}}}}\) |
\(622\) |
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Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x/(f*x^2+e*x+d)/(c*x^2+a)^(1/2),x,method=_RETURNVERBOSE)
[Out]
-1/2*(e+(-4*d*f+e^2)^(1/2))/(-4*d*f+e^2)^(1/2)/f*2^(1/2)/(((-4*d*f+e^2)^(1/2)*c*e+2*a*f^2-2*c*d*f+c*e^2)/f^2)^
(1/2)*ln((((-4*d*f+e^2)^(1/2)*c*e+2*a*f^2-2*c*d*f+c*e^2)/f^2-c*(e+(-4*d*f+e^2)^(1/2))/f*(x+1/2*(e+(-4*d*f+e^2)
^(1/2))/f)+1/2*2^(1/2)*(((-4*d*f+e^2)^(1/2)*c*e+2*a*f^2-2*c*d*f+c*e^2)/f^2)^(1/2)*(4*(x+1/2*(e+(-4*d*f+e^2)^(1
/2))/f)^2*c-4*c*(e+(-4*d*f+e^2)^(1/2))/f*(x+1/2*(e+(-4*d*f+e^2)^(1/2))/f)+2*((-4*d*f+e^2)^(1/2)*c*e+2*a*f^2-2*
c*d*f+c*e^2)/f^2)^(1/2))/(x+1/2*(e+(-4*d*f+e^2)^(1/2))/f))-1/2*(-e+(-4*d*f+e^2)^(1/2))/(-4*d*f+e^2)^(1/2)/f*2^
(1/2)/((-(-4*d*f+e^2)^(1/2)*c*e+2*a*f^2-2*c*d*f+c*e^2)/f^2)^(1/2)*ln(((-(-4*d*f+e^2)^(1/2)*c*e+2*a*f^2-2*c*d*f
+c*e^2)/f^2-c*(e-(-4*d*f+e^2)^(1/2))/f*(x-1/2/f*(-e+(-4*d*f+e^2)^(1/2)))+1/2*2^(1/2)*((-(-4*d*f+e^2)^(1/2)*c*e
+2*a*f^2-2*c*d*f+c*e^2)/f^2)^(1/2)*(4*(x-1/2/f*(-e+(-4*d*f+e^2)^(1/2)))^2*c-4*c*(e-(-4*d*f+e^2)^(1/2))/f*(x-1/
2/f*(-e+(-4*d*f+e^2)^(1/2)))+2*(-(-4*d*f+e^2)^(1/2)*c*e+2*a*f^2-2*c*d*f+c*e^2)/f^2)^(1/2))/(x-1/2/f*(-e+(-4*d*
f+e^2)^(1/2))))
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x/(f*x^2+e*x+d)/(c*x^2+a)^(1/2),x, algorithm="maxima")
[Out]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*d*f-%e^2>0)', see `assume?`
for more det
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 4789 vs.
\(2 (261) = 522\).
time = 0.96, size = 4789, normalized size = 16.29 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x/(f*x^2+e*x+d)/(c*x^2+a)^(1/2),x, algorithm="fricas")
[Out]
1/4*sqrt(2)*sqrt(-(2*c*d^2 - 2*a*d*f + a*e^2 + (4*c^2*d^3*f - 8*a*c*d^2*f^2 + 4*a^2*d*f^3 - a*c*e^4 - (c^2*d^2
- 6*a*c*d*f + a^2*f^2)*e^2)*sqrt(-a^2*e^2/(4*c^4*d^5*f - 16*a*c^3*d^4*f^2 + 24*a^2*c^2*d^3*f^3 - 16*a^3*c*d^2
*f^4 + 4*a^4*d*f^5 - a^2*c^2*e^6 - 2*(a*c^3*d^2 - 4*a^2*c^2*d*f + a^3*c*f^2)*e^4 - (c^4*d^4 - 12*a*c^3*d^3*f +
22*a^2*c^2*d^2*f^2 - 12*a^3*c*d*f^3 + a^4*f^4)*e^2)))/(4*c^2*d^3*f - 8*a*c*d^2*f^2 + 4*a^2*d*f^3 - a*c*e^4 -
(c^2*d^2 - 6*a*c*d*f + a^2*f^2)*e^2))*log((4*a*c*d^2*x*e - 2*a^2*d*e^2 + sqrt(2)*(4*a^2*d*f*e^2 - a^2*e^4 + (8
*c^3*d^5*f - 24*a*c^2*d^4*f^2 + 24*a^2*c*d^3*f^3 - 8*a^3*d^2*f^4 - a^2*c*e^6 - (3*a*c^2*d^2 - 8*a^2*c*d*f + a^
3*f^2)*e^4 - 2*(c^3*d^4 - 9*a*c^2*d^3*f + 11*a^2*c*d^2*f^2 - 3*a^3*d*f^3)*e^2)*sqrt(-a^2*e^2/(4*c^4*d^5*f - 16
*a*c^3*d^4*f^2 + 24*a^2*c^2*d^3*f^3 - 16*a^3*c*d^2*f^4 + 4*a^4*d*f^5 - a^2*c^2*e^6 - 2*(a*c^3*d^2 - 4*a^2*c^2*
d*f + a^3*c*f^2)*e^4 - (c^4*d^4 - 12*a*c^3*d^3*f + 22*a^2*c^2*d^2*f^2 - 12*a^3*c*d*f^3 + a^4*f^4)*e^2)))*sqrt(
c*x^2 + a)*sqrt(-(2*c*d^2 - 2*a*d*f + a*e^2 + (4*c^2*d^3*f - 8*a*c*d^2*f^2 + 4*a^2*d*f^3 - a*c*e^4 - (c^2*d^2
- 6*a*c*d*f + a^2*f^2)*e^2)*sqrt(-a^2*e^2/(4*c^4*d^5*f - 16*a*c^3*d^4*f^2 + 24*a^2*c^2*d^3*f^3 - 16*a^3*c*d^2*
f^4 + 4*a^4*d*f^5 - a^2*c^2*e^6 - 2*(a*c^3*d^2 - 4*a^2*c^2*d*f + a^3*c*f^2)*e^4 - (c^4*d^4 - 12*a*c^3*d^3*f +
22*a^2*c^2*d^2*f^2 - 12*a^3*c*d*f^3 + a^4*f^4)*e^2)))/(4*c^2*d^3*f - 8*a*c*d^2*f^2 + 4*a^2*d*f^3 - a*c*e^4 - (
c^2*d^2 - 6*a*c*d*f + a^2*f^2)*e^2)) + 2*(4*a*c^2*d^4*f - 8*a^2*c*d^3*f^2 + 4*a^3*d^2*f^3 - a^2*c*d*e^4 - (a*c
^2*d^3 - 6*a^2*c*d^2*f + a^3*d*f^2)*e^2)*sqrt(-a^2*e^2/(4*c^4*d^5*f - 16*a*c^3*d^4*f^2 + 24*a^2*c^2*d^3*f^3 -
16*a^3*c*d^2*f^4 + 4*a^4*d*f^5 - a^2*c^2*e^6 - 2*(a*c^3*d^2 - 4*a^2*c^2*d*f + a^3*c*f^2)*e^4 - (c^4*d^4 - 12*a
*c^3*d^3*f + 22*a^2*c^2*d^2*f^2 - 12*a^3*c*d*f^3 + a^4*f^4)*e^2)))/x) - 1/4*sqrt(2)*sqrt(-(2*c*d^2 - 2*a*d*f +
a*e^2 + (4*c^2*d^3*f - 8*a*c*d^2*f^2 + 4*a^2*d*f^3 - a*c*e^4 - (c^2*d^2 - 6*a*c*d*f + a^2*f^2)*e^2)*sqrt(-a^2
*e^2/(4*c^4*d^5*f - 16*a*c^3*d^4*f^2 + 24*a^2*c^2*d^3*f^3 - 16*a^3*c*d^2*f^4 + 4*a^4*d*f^5 - a^2*c^2*e^6 - 2*(
a*c^3*d^2 - 4*a^2*c^2*d*f + a^3*c*f^2)*e^4 - (c^4*d^4 - 12*a*c^3*d^3*f + 22*a^2*c^2*d^2*f^2 - 12*a^3*c*d*f^3 +
a^4*f^4)*e^2)))/(4*c^2*d^3*f - 8*a*c*d^2*f^2 + 4*a^2*d*f^3 - a*c*e^4 - (c^2*d^2 - 6*a*c*d*f + a^2*f^2)*e^2))*
log((4*a*c*d^2*x*e - 2*a^2*d*e^2 - sqrt(2)*(4*a^2*d*f*e^2 - a^2*e^4 + (8*c^3*d^5*f - 24*a*c^2*d^4*f^2 + 24*a^2
*c*d^3*f^3 - 8*a^3*d^2*f^4 - a^2*c*e^6 - (3*a*c^2*d^2 - 8*a^2*c*d*f + a^3*f^2)*e^4 - 2*(c^3*d^4 - 9*a*c^2*d^3*
f + 11*a^2*c*d^2*f^2 - 3*a^3*d*f^3)*e^2)*sqrt(-a^2*e^2/(4*c^4*d^5*f - 16*a*c^3*d^4*f^2 + 24*a^2*c^2*d^3*f^3 -
16*a^3*c*d^2*f^4 + 4*a^4*d*f^5 - a^2*c^2*e^6 - 2*(a*c^3*d^2 - 4*a^2*c^2*d*f + a^3*c*f^2)*e^4 - (c^4*d^4 - 12*a
*c^3*d^3*f + 22*a^2*c^2*d^2*f^2 - 12*a^3*c*d*f^3 + a^4*f^4)*e^2)))*sqrt(c*x^2 + a)*sqrt(-(2*c*d^2 - 2*a*d*f +
a*e^2 + (4*c^2*d^3*f - 8*a*c*d^2*f^2 + 4*a^2*d*f^3 - a*c*e^4 - (c^2*d^2 - 6*a*c*d*f + a^2*f^2)*e^2)*sqrt(-a^2*
e^2/(4*c^4*d^5*f - 16*a*c^3*d^4*f^2 + 24*a^2*c^2*d^3*f^3 - 16*a^3*c*d^2*f^4 + 4*a^4*d*f^5 - a^2*c^2*e^6 - 2*(a
*c^3*d^2 - 4*a^2*c^2*d*f + a^3*c*f^2)*e^4 - (c^4*d^4 - 12*a*c^3*d^3*f + 22*a^2*c^2*d^2*f^2 - 12*a^3*c*d*f^3 +
a^4*f^4)*e^2)))/(4*c^2*d^3*f - 8*a*c*d^2*f^2 + 4*a^2*d*f^3 - a*c*e^4 - (c^2*d^2 - 6*a*c*d*f + a^2*f^2)*e^2)) +
2*(4*a*c^2*d^4*f - 8*a^2*c*d^3*f^2 + 4*a^3*d^2*f^3 - a^2*c*d*e^4 - (a*c^2*d^3 - 6*a^2*c*d^2*f + a^3*d*f^2)*e^
2)*sqrt(-a^2*e^2/(4*c^4*d^5*f - 16*a*c^3*d^4*f^2 + 24*a^2*c^2*d^3*f^3 - 16*a^3*c*d^2*f^4 + 4*a^4*d*f^5 - a^2*c
^2*e^6 - 2*(a*c^3*d^2 - 4*a^2*c^2*d*f + a^3*c*f^2)*e^4 - (c^4*d^4 - 12*a*c^3*d^3*f + 22*a^2*c^2*d^2*f^2 - 12*a
^3*c*d*f^3 + a^4*f^4)*e^2)))/x) + 1/4*sqrt(2)*sqrt(-(2*c*d^2 - 2*a*d*f + a*e^2 - (4*c^2*d^3*f - 8*a*c*d^2*f^2
+ 4*a^2*d*f^3 - a*c*e^4 - (c^2*d^2 - 6*a*c*d*f + a^2*f^2)*e^2)*sqrt(-a^2*e^2/(4*c^4*d^5*f - 16*a*c^3*d^4*f^2 +
24*a^2*c^2*d^3*f^3 - 16*a^3*c*d^2*f^4 + 4*a^4*d*f^5 - a^2*c^2*e^6 - 2*(a*c^3*d^2 - 4*a^2*c^2*d*f + a^3*c*f^2)
*e^4 - (c^4*d^4 - 12*a*c^3*d^3*f + 22*a^2*c^2*d^2*f^2 - 12*a^3*c*d*f^3 + a^4*f^4)*e^2)))/(4*c^2*d^3*f - 8*a*c*
d^2*f^2 + 4*a^2*d*f^3 - a*c*e^4 - (c^2*d^2 - 6*a*c*d*f + a^2*f^2)*e^2))*log((4*a*c*d^2*x*e - 2*a^2*d*e^2 + sqr
t(2)*(4*a^2*d*f*e^2 - a^2*e^4 - (8*c^3*d^5*f - 24*a*c^2*d^4*f^2 + 24*a^2*c*d^3*f^3 - 8*a^3*d^2*f^4 - a^2*c*e^6
- (3*a*c^2*d^2 - 8*a^2*c*d*f + a^3*f^2)*e^4 - 2*(c^3*d^4 - 9*a*c^2*d^3*f + 11*a^2*c*d^2*f^2 - 3*a^3*d*f^3)*e^
2)*sqrt(-a^2*e^2/(4*c^4*d^5*f - 16*a*c^3*d^4*f^2 + 24*a^2*c^2*d^3*f^3 - 16*a^3*c*d^2*f^4 + 4*a^4*d*f^5 - a^2*c
^2*e^6 - 2*(a*c^3*d^2 - 4*a^2*c^2*d*f + a^3*c*f^2)*e^4 - (c^4*d^4 - 12*a*c^3*d^3*f + 22*a^2*c^2*d^2*f^2 - 12*a
^3*c*d*f^3 + a^4*f^4)*e^2)))*sqrt(c*x^2 + a)*sqrt(-(2*c*d^2 - 2*a*d*f + a*e^2 - (4*c^2*d^3*f - 8*a*c*d^2*f^2 +
4*a^2*d*f^3 - a*c*e^4 - (c^2*d^2 - 6*a*c*d*f + a^2*f^2)*e^2)*sqrt(-a^2*e^2/(4*c^4*d^5*f - 16*a*c^3*d^4*f^2 +
24*a^2*c^2*d^3*f^3 - 16*a^3*c*d^2*f^4 + 4*a^4*d*f^5 - a^2*c^2*e^6 - 2*(a*c^3*d^2 - 4*a^2*c^2*d*f + a^3*c*f^2)*
e^4 - (c^4*d^4 - 12*a*c^3*d^3*f + 22*a^2*c^2*d^...
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x}{\sqrt {a + c x^{2}} \left (d + e x + f x^{2}\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x/(f*x**2+e*x+d)/(c*x**2+a)**(1/2),x)
[Out]
Integral(x/(sqrt(a + c*x**2)*(d + e*x + f*x**2)), 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(x/(f*x^2+e*x+d)/(c*x^2+a)^(1/2),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 {x}{\sqrt {c\,x^2+a}\,\left (f\,x^2+e\,x+d\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x/((a + c*x^2)^(1/2)*(d + e*x + f*x^2)),x)
[Out]
int(x/((a + c*x^2)^(1/2)*(d + e*x + f*x^2)), x)
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