3.1.56 \(\int \frac {\sqrt {a+c x^2}}{x^2 (d+e x+f x^2)} \, dx\) [56]
Optimal. Leaf size=382 \[ -\frac {\sqrt {a+c x^2}}{d x}-\frac {f \left (2 c d^2+a \left (e^2-2 d f+e \sqrt {e^2-4 d f}\right )\right ) \tanh ^{-1}\left (\frac {2 a f-c \left (e-\sqrt {e^2-4 d f}\right ) x}{\sqrt {2} \sqrt {2 a f^2+c \left (e^2-2 d f-e \sqrt {e^2-4 d f}\right )} \sqrt {a+c x^2}}\right )}{\sqrt {2} d^2 \sqrt {e^2-4 d f} \sqrt {2 a f^2+c \left (e^2-2 d f-e \sqrt {e^2-4 d f}\right )}}+\frac {f \left (2 c d^2+a \left (e^2-2 d f-e \sqrt {e^2-4 d f}\right )\right ) \tanh ^{-1}\left (\frac {2 a f-c \left (e+\sqrt {e^2-4 d f}\right ) x}{\sqrt {2} \sqrt {2 a f^2+c \left (e^2-2 d f+e \sqrt {e^2-4 d f}\right )} \sqrt {a+c x^2}}\right )}{\sqrt {2} d^2 \sqrt {e^2-4 d f} \sqrt {2 a f^2+c \left (e^2-2 d f+e \sqrt {e^2-4 d f}\right )}}+\frac {\sqrt {a} e \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{d^2} \]
[Out]
e*arctanh((c*x^2+a)^(1/2)/a^(1/2))*a^(1/2)/d^2-(c*x^2+a)^(1/2)/d/x-1/2*f*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))*(2*c*d^2+a*(e^2-2*d*f+e*
(-4*d*f+e^2)^(1/2)))/d^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*f*a
rctanh(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))*(2*c*d^2+a*(e^2-2*d*f-e*(-4*d*f+e^2)^(1/2)))/d^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 = 0.90, antiderivative size = 382, normalized size of antiderivative = 1.00, number of steps
used = 18, number of rules used = 12, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {6860, 283,
223, 212, 272, 52, 65, 214, 1034, 1094, 1048, 739} \begin {gather*} -\frac {f \left (a \left (e \sqrt {e^2-4 d f}-2 d f+e^2\right )+2 c d^2\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} d^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 {f \left (a \left (-e \sqrt {e^2-4 d f}-2 d f+e^2\right )+2 c d^2\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} d^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 {\sqrt {a} e \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{d^2}-\frac {\sqrt {a+c x^2}}{d x} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[Sqrt[a + c*x^2]/(x^2*(d + e*x + f*x^2)),x]
[Out]
-(Sqrt[a + c*x^2]/(d*x)) - (f*(2*c*d^2 + a*(e^2 - 2*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]*d^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])]) + (f*(2*c*d^2 + a*(e^2 - 2*d*f - e*S
qrt[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*Sq
rt[e^2 - 4*d*f])]*Sqrt[a + c*x^2])])/(Sqrt[2]*d^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])]) + (Sqrt[a]*e*ArcTanh[Sqrt[a + c*x^2]/Sqrt[a]])/d^2
Rule 52
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^n/(b*(
m + n + 1))), x] + Dist[n*((b*c - a*d)/(b*(m + n + 1))), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]
Rule 65
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, 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 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 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 272
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]
Rule 283
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^p/(c*(m + 1
))), x] - Dist[b*n*(p/(c^n*(m + 1))), Int[(c*x)^(m + n)*(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] &&
IGtQ[n, 0] && GtQ[p, 0] && LtQ[m, -1] && !ILtQ[(m + n*p + n + 1)/n, 0] && IntBinomialQ[a, b, c, n, m, p, x]
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 1034
Int[((g_.) + (h_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_)*((d_) + (e_.)*(x_) + (f_.)*(x_)^2)^(q_), x_Symbol] :> Simp
[h*(a + c*x^2)^p*((d + e*x + f*x^2)^(q + 1)/(2*f*(p + q + 1))), x] + Dist[1/(2*f*(p + q + 1)), Int[(a + c*x^2)
^(p - 1)*(d + e*x + f*x^2)^q*Simp[a*h*e*p - a*(h*e - 2*g*f)*(p + q + 1) - 2*h*p*(c*d - a*f)*x - (h*c*e*p + c*(
h*e - 2*g*f)*(p + q + 1))*x^2, x], x], x] /; FreeQ[{a, c, d, e, f, g, h, q}, x] && NeQ[e^2 - 4*d*f, 0] && GtQ[
p, 0] && NeQ[p + q + 1, 0]
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]
Rule 1094
Int[((A_.) + (B_.)*(x_) + (C_.)*(x_)^2)/(((a_) + (b_.)*(x_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (f_.)*(x_)^2]), x_Sym
bol] :> Dist[C/c, Int[1/Sqrt[d + f*x^2], x], x] + Dist[1/c, Int[(A*c - a*C + (B*c - b*C)*x)/((a + b*x + c*x^2)
*Sqrt[d + f*x^2]), x], x] /; FreeQ[{a, b, c, d, f, A, B, C}, x] && NeQ[b^2 - 4*a*c, 0]
Rule 6860
Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a +
b*x^n + c*x^(2*n)), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]
Rubi steps
\begin {align*} \int \frac {\sqrt {a+c x^2}}{x^2 \left (d+e x+f x^2\right )} \, dx &=\int \left (\frac {\sqrt {a+c x^2}}{d x^2}-\frac {e \sqrt {a+c x^2}}{d^2 x}+\frac {\left (e^2-d f+e f x\right ) \sqrt {a+c x^2}}{d^2 \left (d+e x+f x^2\right )}\right ) \, dx\\ &=\frac {\int \frac {\left (e^2-d f+e f x\right ) \sqrt {a+c x^2}}{d+e x+f x^2} \, dx}{d^2}+\frac {\int \frac {\sqrt {a+c x^2}}{x^2} \, dx}{d}-\frac {e \int \frac {\sqrt {a+c x^2}}{x} \, dx}{d^2}\\ &=\frac {e \sqrt {a+c x^2}}{d^2}-\frac {\sqrt {a+c x^2}}{d x}+\frac {c \int \frac {1}{\sqrt {a+c x^2}} \, dx}{d}-\frac {e \text {Subst}\left (\int \frac {\sqrt {a+c x}}{x} \, dx,x,x^2\right )}{2 d^2}+\frac {\int \frac {a f \left (e^2-d f\right )-e f (c d-a f) x-c d f^2 x^2}{\sqrt {a+c x^2} \left (d+e x+f x^2\right )} \, dx}{d^2 f}\\ &=-\frac {\sqrt {a+c x^2}}{d x}-\frac {c \int \frac {1}{\sqrt {a+c x^2}} \, dx}{d}+\frac {c \text {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {a+c x^2}}\right )}{d}-\frac {(a e) \text {Subst}\left (\int \frac {1}{x \sqrt {a+c x}} \, dx,x,x^2\right )}{2 d^2}+\frac {\int \frac {c d^2 f^2+a f^2 \left (e^2-d f\right )+\left (c d e f^2-e f^2 (c d-a f)\right ) x}{\sqrt {a+c x^2} \left (d+e x+f x^2\right )} \, dx}{d^2 f^2}\\ &=-\frac {\sqrt {a+c x^2}}{d x}+\frac {\sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{d}-\frac {c \text {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {a+c x^2}}\right )}{d}-\frac {(a e) \text {Subst}\left (\int \frac {1}{-\frac {a}{c}+\frac {x^2}{c}} \, dx,x,\sqrt {a+c x^2}\right )}{c d^2}-\frac {\left (f \left (2 c d^2+a \left (e^2-2 d f-e \sqrt {e^2-4 d f}\right )\right )\right ) \int \frac {1}{\left (e+\sqrt {e^2-4 d f}+2 f x\right ) \sqrt {a+c x^2}} \, dx}{d^2 \sqrt {e^2-4 d f}}+\frac {\left (f \left (2 c d^2+a \left (e^2-2 d f+e \sqrt {e^2-4 d f}\right )\right )\right ) \int \frac {1}{\left (e-\sqrt {e^2-4 d f}+2 f x\right ) \sqrt {a+c x^2}} \, dx}{d^2 \sqrt {e^2-4 d f}}\\ &=-\frac {\sqrt {a+c x^2}}{d x}+\frac {\sqrt {a} e \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{d^2}+\frac {\left (f \left (2 c d^2+a \left (e^2-2 d f-e \sqrt {e^2-4 d f}\right )\right )\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 )}{d^2 \sqrt {e^2-4 d f}}-\frac {\left (f \left (2 c d^2+a \left (e^2-2 d f+e \sqrt {e^2-4 d f}\right )\right )\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 )}{d^2 \sqrt {e^2-4 d f}}\\ &=-\frac {\sqrt {a+c x^2}}{d x}-\frac {f \left (2 c d^2+a \left (e^2-2 d f+e \sqrt {e^2-4 d f}\right )\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} d^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 {f \left (2 c d^2+a \left (e^2-2 d f-e \sqrt {e^2-4 d f}\right )\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} d^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 {\sqrt {a} e \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{d^2}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in
optimal.
time = 0.40, size = 336, normalized size = 0.88 \begin {gather*} -\frac {d \sqrt {a+c x^2}+2 \sqrt {a} e x \tanh ^{-1}\left (\frac {\sqrt {c} x-\sqrt {a+c x^2}}{\sqrt {a}}\right )+x \text {RootSum}\left [a^2 f+2 a \sqrt {c} e \text {$\#$1}+4 c d \text {$\#$1}^2-2 a f \text {$\#$1}^2-2 \sqrt {c} e \text {$\#$1}^3+f \text {$\#$1}^4\&,\frac {a^2 e f \log \left (-\sqrt {c} x+\sqrt {a+c x^2}-\text {$\#$1}\right )+2 c^{3/2} d^2 \log \left (-\sqrt {c} x+\sqrt {a+c x^2}-\text {$\#$1}\right ) \text {$\#$1}+2 a \sqrt {c} e^2 \log \left (-\sqrt {c} x+\sqrt {a+c x^2}-\text {$\#$1}\right ) \text {$\#$1}-2 a \sqrt {c} d f \log \left (-\sqrt {c} x+\sqrt {a+c x^2}-\text {$\#$1}\right ) \text {$\#$1}-a e f \log \left (-\sqrt {c} x+\sqrt {a+c x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{a \sqrt {c} e+4 c d \text {$\#$1}-2 a f \text {$\#$1}-3 \sqrt {c} e \text {$\#$1}^2+2 f \text {$\#$1}^3}\&\right ]}{d^2 x} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[Sqrt[a + c*x^2]/(x^2*(d + e*x + f*x^2)),x]
[Out]
-((d*Sqrt[a + c*x^2] + 2*Sqrt[a]*e*x*ArcTanh[(Sqrt[c]*x - Sqrt[a + c*x^2])/Sqrt[a]] + x*RootSum[a^2*f + 2*a*Sq
rt[c]*e*#1 + 4*c*d*#1^2 - 2*a*f*#1^2 - 2*Sqrt[c]*e*#1^3 + f*#1^4 & , (a^2*e*f*Log[-(Sqrt[c]*x) + Sqrt[a + c*x^
2] - #1] + 2*c^(3/2)*d^2*Log[-(Sqrt[c]*x) + Sqrt[a + c*x^2] - #1]*#1 + 2*a*Sqrt[c]*e^2*Log[-(Sqrt[c]*x) + Sqrt
[a + c*x^2] - #1]*#1 - 2*a*Sqrt[c]*d*f*Log[-(Sqrt[c]*x) + Sqrt[a + c*x^2] - #1]*#1 - a*e*f*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) & ])/(d^2*x))
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1414\) vs.
\(2(337)=674\).
time = 0.13, size = 1415, normalized size = 3.70
| | |
| method |
result |
size |
| | |
| default |
\(\text {Expression too large to display}\) |
\(1415\) |
| risch |
\(\text {Expression too large to display}\) |
\(2536\) |
| | |
|
|
|
|
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((c*x^2+a)^(1/2)/x^2/(f*x^2+e*x+d),x,method=_RETURNVERBOSE)
[Out]
-4*f^2/(e+(-4*d*f+e^2)^(1/2))^2/(-4*d*f+e^2)^(1/2)*(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)-1/
2*c^(1/2)*(e+(-4*d*f+e^2)^(1/2))/f*ln((-1/2*c*(e+(-4*d*f+e^2)^(1/2))/f+c*(x+1/2*(e+(-4*d*f+e^2)^(1/2))/f))/c^(
1/2)+((x+1/2*(e+(-4*d*f+e^2)^(1/2))/f)^2*c-c*(e+(-4*d*f+e^2)^(1/2))/f*(x+1/2*(e+(-4*d*f+e^2)^(1/2))/f)+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))-1/2*((-4*d*f+e^2)^(1/2)*c*e+2*a*f^2-2*c*d*f+c*e^2)/f^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)*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)))+4*f^2/(-e+(-4*d*f+e^2)^(1/2))^2/(-4*d*f+e^2)^(1/2)*(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)-1/2*c^(1/2)*(e-(-4*d*f+e^2)^(1/2))/f*ln((-1/2*c*(e-(-4*d*f+e^2)^(1/2))/f+c*(x-1/2/f*(-e+(-4*d*f+e
^2)^(1/2))))/c^(1/2)+((x-1/2/f*(-e+(-4*d*f+e^2)^(1/2)))^2*c-c*(e-(-4*d*f+e^2)^(1/2))/f*(x-1/2/f*(-e+(-4*d*f+e^
2)^(1/2)))+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))-1/2*(-(-4*d*f+e^2)^(1/2)*c*e+2*a*f^
2-2*c*d*f+c*e^2)/f^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)*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)))))-4*f/(-e+(-4*d*f+e^2)^(1/2))/(e+(-4*d*f+e^2)^(1/2))*(-1/a/x*(c*x^2+a)^(
3/2)+2*c/a*(1/2*x*(c*x^2+a)^(1/2)+1/2*a/c^(1/2)*ln(x*c^(1/2)+(c*x^2+a)^(1/2))))-16*f^2*e/(-e+(-4*d*f+e^2)^(1/2
))^2/(e+(-4*d*f+e^2)^(1/2))^2*((c*x^2+a)^(1/2)-a^(1/2)*ln((2*a+2*a^(1/2)*(c*x^2+a)^(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((c*x^2+a)^(1/2)/x^2/(f*x^2+e*x+d),x, algorithm="maxima")
[Out]
integrate(sqrt(c*x^2 + a)/((f*x^2 + x*e + d)*x^2), x)
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 2545 vs.
\(2 (346) = 692\).
time = 80.30, size = 5102, normalized size = 13.36 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((c*x^2+a)^(1/2)/x^2/(f*x^2+e*x+d),x, algorithm="fricas")
[Out]
[-1/4*(sqrt(2)*d^2*x*sqrt((2*c*d^3*f - 2*a*d^2*f^2 - a*e^4 - (c*d^2 - 4*a*d*f)*e^2 + (4*d^5*f - d^4*e^2)*sqrt(
-(a^2*e^6 + 2*(a*c*d^2 - 2*a^2*d*f)*e^4 + (c^2*d^4 - 4*a*c*d^3*f + 4*a^2*d^2*f^2)*e^2)/(4*d^9*f - d^8*e^2)))/(
4*d^5*f - d^4*e^2))*log((4*a*c*d*f*x*e^3 - 2*a^2*f*e^4 + 4*(c^2*d^3*f - 2*a*c*d^2*f^2)*x*e + sqrt(2)*((4*d^6*f
*e - d^5*e^3)*sqrt(c*x^2 + a)*sqrt(-(a^2*e^6 + 2*(a*c*d^2 - 2*a^2*d*f)*e^4 + (c^2*d^4 - 4*a*c*d^3*f + 4*a^2*d^
2*f^2)*e^2)/(4*d^9*f - d^8*e^2)) + sqrt(c*x^2 + a)*(a*d*e^5 + (c*d^3 - 6*a*d^2*f)*e^3 - 4*(c*d^4*f - 2*a*d^3*f
^2)*e))*sqrt((2*c*d^3*f - 2*a*d^2*f^2 - a*e^4 - (c*d^2 - 4*a*d*f)*e^2 + (4*d^5*f - d^4*e^2)*sqrt(-(a^2*e^6 + 2
*(a*c*d^2 - 2*a^2*d*f)*e^4 + (c^2*d^4 - 4*a*c*d^3*f + 4*a^2*d^2*f^2)*e^2)/(4*d^9*f - d^8*e^2)))/(4*d^5*f - d^4
*e^2)) - 2*(a*c*d^2*f - 2*a^2*d*f^2)*e^2 + 2*(4*a*d^5*f^2 - a*d^4*f*e^2)*sqrt(-(a^2*e^6 + 2*(a*c*d^2 - 2*a^2*d
*f)*e^4 + (c^2*d^4 - 4*a*c*d^3*f + 4*a^2*d^2*f^2)*e^2)/(4*d^9*f - d^8*e^2)))/x) - sqrt(2)*d^2*x*sqrt((2*c*d^3*
f - 2*a*d^2*f^2 - a*e^4 - (c*d^2 - 4*a*d*f)*e^2 + (4*d^5*f - d^4*e^2)*sqrt(-(a^2*e^6 + 2*(a*c*d^2 - 2*a^2*d*f)
*e^4 + (c^2*d^4 - 4*a*c*d^3*f + 4*a^2*d^2*f^2)*e^2)/(4*d^9*f - d^8*e^2)))/(4*d^5*f - d^4*e^2))*log((4*a*c*d*f*
x*e^3 - 2*a^2*f*e^4 + 4*(c^2*d^3*f - 2*a*c*d^2*f^2)*x*e - sqrt(2)*((4*d^6*f*e - d^5*e^3)*sqrt(c*x^2 + a)*sqrt(
-(a^2*e^6 + 2*(a*c*d^2 - 2*a^2*d*f)*e^4 + (c^2*d^4 - 4*a*c*d^3*f + 4*a^2*d^2*f^2)*e^2)/(4*d^9*f - d^8*e^2)) +
sqrt(c*x^2 + a)*(a*d*e^5 + (c*d^3 - 6*a*d^2*f)*e^3 - 4*(c*d^4*f - 2*a*d^3*f^2)*e))*sqrt((2*c*d^3*f - 2*a*d^2*f
^2 - a*e^4 - (c*d^2 - 4*a*d*f)*e^2 + (4*d^5*f - d^4*e^2)*sqrt(-(a^2*e^6 + 2*(a*c*d^2 - 2*a^2*d*f)*e^4 + (c^2*d
^4 - 4*a*c*d^3*f + 4*a^2*d^2*f^2)*e^2)/(4*d^9*f - d^8*e^2)))/(4*d^5*f - d^4*e^2)) - 2*(a*c*d^2*f - 2*a^2*d*f^2
)*e^2 + 2*(4*a*d^5*f^2 - a*d^4*f*e^2)*sqrt(-(a^2*e^6 + 2*(a*c*d^2 - 2*a^2*d*f)*e^4 + (c^2*d^4 - 4*a*c*d^3*f +
4*a^2*d^2*f^2)*e^2)/(4*d^9*f - d^8*e^2)))/x) - sqrt(2)*d^2*x*sqrt((2*c*d^3*f - 2*a*d^2*f^2 - a*e^4 - (c*d^2 -
4*a*d*f)*e^2 - (4*d^5*f - d^4*e^2)*sqrt(-(a^2*e^6 + 2*(a*c*d^2 - 2*a^2*d*f)*e^4 + (c^2*d^4 - 4*a*c*d^3*f + 4*a
^2*d^2*f^2)*e^2)/(4*d^9*f - d^8*e^2)))/(4*d^5*f - d^4*e^2))*log((4*a*c*d*f*x*e^3 - 2*a^2*f*e^4 + 4*(c^2*d^3*f
- 2*a*c*d^2*f^2)*x*e + sqrt(2)*((4*d^6*f*e - d^5*e^3)*sqrt(c*x^2 + a)*sqrt(-(a^2*e^6 + 2*(a*c*d^2 - 2*a^2*d*f)
*e^4 + (c^2*d^4 - 4*a*c*d^3*f + 4*a^2*d^2*f^2)*e^2)/(4*d^9*f - d^8*e^2)) - sqrt(c*x^2 + a)*(a*d*e^5 + (c*d^3 -
6*a*d^2*f)*e^3 - 4*(c*d^4*f - 2*a*d^3*f^2)*e))*sqrt((2*c*d^3*f - 2*a*d^2*f^2 - a*e^4 - (c*d^2 - 4*a*d*f)*e^2
- (4*d^5*f - d^4*e^2)*sqrt(-(a^2*e^6 + 2*(a*c*d^2 - 2*a^2*d*f)*e^4 + (c^2*d^4 - 4*a*c*d^3*f + 4*a^2*d^2*f^2)*e
^2)/(4*d^9*f - d^8*e^2)))/(4*d^5*f - d^4*e^2)) - 2*(a*c*d^2*f - 2*a^2*d*f^2)*e^2 - 2*(4*a*d^5*f^2 - a*d^4*f*e^
2)*sqrt(-(a^2*e^6 + 2*(a*c*d^2 - 2*a^2*d*f)*e^4 + (c^2*d^4 - 4*a*c*d^3*f + 4*a^2*d^2*f^2)*e^2)/(4*d^9*f - d^8*
e^2)))/x) + sqrt(2)*d^2*x*sqrt((2*c*d^3*f - 2*a*d^2*f^2 - a*e^4 - (c*d^2 - 4*a*d*f)*e^2 - (4*d^5*f - d^4*e^2)*
sqrt(-(a^2*e^6 + 2*(a*c*d^2 - 2*a^2*d*f)*e^4 + (c^2*d^4 - 4*a*c*d^3*f + 4*a^2*d^2*f^2)*e^2)/(4*d^9*f - d^8*e^2
)))/(4*d^5*f - d^4*e^2))*log((4*a*c*d*f*x*e^3 - 2*a^2*f*e^4 + 4*(c^2*d^3*f - 2*a*c*d^2*f^2)*x*e - sqrt(2)*((4*
d^6*f*e - d^5*e^3)*sqrt(c*x^2 + a)*sqrt(-(a^2*e^6 + 2*(a*c*d^2 - 2*a^2*d*f)*e^4 + (c^2*d^4 - 4*a*c*d^3*f + 4*a
^2*d^2*f^2)*e^2)/(4*d^9*f - d^8*e^2)) - sqrt(c*x^2 + a)*(a*d*e^5 + (c*d^3 - 6*a*d^2*f)*e^3 - 4*(c*d^4*f - 2*a*
d^3*f^2)*e))*sqrt((2*c*d^3*f - 2*a*d^2*f^2 - a*e^4 - (c*d^2 - 4*a*d*f)*e^2 - (4*d^5*f - d^4*e^2)*sqrt(-(a^2*e^
6 + 2*(a*c*d^2 - 2*a^2*d*f)*e^4 + (c^2*d^4 - 4*a*c*d^3*f + 4*a^2*d^2*f^2)*e^2)/(4*d^9*f - d^8*e^2)))/(4*d^5*f
- d^4*e^2)) - 2*(a*c*d^2*f - 2*a^2*d*f^2)*e^2 - 2*(4*a*d^5*f^2 - a*d^4*f*e^2)*sqrt(-(a^2*e^6 + 2*(a*c*d^2 - 2*
a^2*d*f)*e^4 + (c^2*d^4 - 4*a*c*d^3*f + 4*a^2*d^2*f^2)*e^2)/(4*d^9*f - d^8*e^2)))/x) - 2*sqrt(a)*x*e*log(-(c*x
^2 + 2*sqrt(c*x^2 + a)*sqrt(a) + 2*a)/x^2) + 4*sqrt(c*x^2 + a)*d)/(d^2*x), -1/4*(sqrt(2)*d^2*x*sqrt((2*c*d^3*f
- 2*a*d^2*f^2 - a*e^4 - (c*d^2 - 4*a*d*f)*e^2 + (4*d^5*f - d^4*e^2)*sqrt(-(a^2*e^6 + 2*(a*c*d^2 - 2*a^2*d*f)*
e^4 + (c^2*d^4 - 4*a*c*d^3*f + 4*a^2*d^2*f^2)*e^2)/(4*d^9*f - d^8*e^2)))/(4*d^5*f - d^4*e^2))*log((4*a*c*d*f*x
*e^3 - 2*a^2*f*e^4 + 4*(c^2*d^3*f - 2*a*c*d^2*f^2)*x*e + sqrt(2)*((4*d^6*f*e - d^5*e^3)*sqrt(c*x^2 + a)*sqrt(-
(a^2*e^6 + 2*(a*c*d^2 - 2*a^2*d*f)*e^4 + (c^2*d^4 - 4*a*c*d^3*f + 4*a^2*d^2*f^2)*e^2)/(4*d^9*f - d^8*e^2)) + s
qrt(c*x^2 + a)*(a*d*e^5 + (c*d^3 - 6*a*d^2*f)*e^3 - 4*(c*d^4*f - 2*a*d^3*f^2)*e))*sqrt((2*c*d^3*f - 2*a*d^2*f^
2 - a*e^4 - (c*d^2 - 4*a*d*f)*e^2 + (4*d^5*f - d^4*e^2)*sqrt(-(a^2*e^6 + 2*(a*c*d^2 - 2*a^2*d*f)*e^4 + (c^2*d^
4 - 4*a*c*d^3*f + 4*a^2*d^2*f^2)*e^2)/(4*d^9*f - d^8*e^2)))/(4*d^5*f - d^4*e^2)) - 2*(a*c*d^2*f - 2*a^2*d*f^2)
*e^2 + 2*(4*a*d^5*f^2 - a*d^4*f*e^2)*sqrt(-(a^2*e^6 + 2*(a*c*d^2 - 2*a^2*d*f)*e^4 + (c^2*d^4 - 4*a*c*d^3*f + 4
*a^2*d^2*f^2)*e^2)/(4*d^9*f - d^8*e^2)))/x) - sqrt(2)*d^2*x*sqrt((2*c*d^3*f - 2*a*d^2*f^2 - a*e^4 - (c*d^2 - 4
*a*d*f)*e^2 + (4*d^5*f - d^4*e^2)*sqrt(-(a^2*e^...
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {a + c x^{2}}}{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((c*x**2+a)**(1/2)/x**2/(f*x**2+e*x+d),x)
[Out]
Integral(sqrt(a + c*x**2)/(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((c*x^2+a)^(1/2)/x^2/(f*x^2+e*x+d),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 {c\,x^2+a}}{x^2\,\left (f\,x^2+e\,x+d\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a + c*x^2)^(1/2)/(x^2*(d + e*x + f*x^2)),x)
[Out]
int((a + c*x^2)^(1/2)/(x^2*(d + e*x + f*x^2)), x)
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