3.10.21 \(\int \frac {\sqrt {a+\frac {c}{x^2}+\frac {b}{x}}}{d+e x} \, dx\) [921]
Optimal. Leaf size=181 \[ \frac {\sqrt {a} \tanh ^{-1}\left (\frac {2 a+\frac {b}{x}}{2 \sqrt {a} \sqrt {a+\frac {c}{x^2}+\frac {b}{x}}}\right )}{e}-\frac {\sqrt {c} \tanh ^{-1}\left (\frac {b+\frac {2 c}{x}}{2 \sqrt {c} \sqrt {a+\frac {c}{x^2}+\frac {b}{x}}}\right )}{d}-\frac {\sqrt {a d^2-e (b d-c e)} \tanh ^{-1}\left (\frac {2 a d-b e+\frac {b d-2 c e}{x}}{2 \sqrt {a d^2-e (b d-c e)} \sqrt {a+\frac {c}{x^2}+\frac {b}{x}}}\right )}{d e} \]
[Out]
arctanh(1/2*(2*a+b/x)/a^(1/2)/(a+c/x^2+b/x)^(1/2))*a^(1/2)/e-arctanh(1/2*(b+2*c/x)/c^(1/2)/(a+c/x^2+b/x)^(1/2)
)*c^(1/2)/d-arctanh(1/2*(2*a*d-b*e+(b*d-2*c*e)/x)/(a*d^2-e*(b*d-c*e))^(1/2)/(a+c/x^2+b/x)^(1/2))*(a*d^2-e*(b*d
-c*e))^(1/2)/d/e
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Rubi [A]
time = 0.19, antiderivative size = 181, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 7, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {1457, 1488,
909, 738, 212, 857, 635} \begin {gather*} -\frac {\sqrt {a d^2-e (b d-c e)} \tanh ^{-1}\left (\frac {2 a d+\frac {b d-2 c e}{x}-b e}{2 \sqrt {a+\frac {b}{x}+\frac {c}{x^2}} \sqrt {a d^2-e (b d-c e)}}\right )}{d e}-\frac {\sqrt {c} \tanh ^{-1}\left (\frac {b+\frac {2 c}{x}}{2 \sqrt {c} \sqrt {a+\frac {b}{x}+\frac {c}{x^2}}}\right )}{d}+\frac {\sqrt {a} \tanh ^{-1}\left (\frac {2 a+\frac {b}{x}}{2 \sqrt {a} \sqrt {a+\frac {b}{x}+\frac {c}{x^2}}}\right )}{e} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[Sqrt[a + c/x^2 + b/x]/(d + e*x),x]
[Out]
(Sqrt[a]*ArcTanh[(2*a + b/x)/(2*Sqrt[a]*Sqrt[a + c/x^2 + b/x])])/e - (Sqrt[c]*ArcTanh[(b + (2*c)/x)/(2*Sqrt[c]
*Sqrt[a + c/x^2 + b/x])])/d - (Sqrt[a*d^2 - e*(b*d - c*e)]*ArcTanh[(2*a*d - b*e + (b*d - 2*c*e)/x)/(2*Sqrt[a*d
^2 - e*(b*d - c*e)]*Sqrt[a + c/x^2 + b/x])])/(d*e)
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 635
Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[1/(4*c - x^2), x], x, (b + 2*c*x)
/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]
Rule 738
Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[-2, Subst[Int[1/(4*c*d
^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, (2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a,
b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[2*c*d - b*e, 0]
Rule 857
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dis
t[g/e, Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] + Dist[(e*f - d*g)/e, Int[(d + e*x)^m*(a + b*x + c*x^
2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]
&& !IGtQ[m, 0]
Rule 909
Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_)/(((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))), x_Symbol] :> Dist[(c
*d^2 - b*d*e + a*e^2)/(e*(e*f - d*g)), Int[(a + b*x + c*x^2)^(p - 1)/(d + e*x), x], x] - Dist[1/(e*(e*f - d*g)
), Int[Simp[c*d*f - b*e*f + a*e*g - c*(e*f - d*g)*x, x]*((a + b*x + c*x^2)^(p - 1)/(f + g*x)), x], 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] && Fra
ctionQ[p] && GtQ[p, 0]
Rule 1457
Int[((d_) + (e_.)*(x_)^(mn_.))^(q_.)*((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.))^(p_.), x_Symbol] :> Int[((
e + d*x^n)^q*(a + b*x^n + c*x^(2*n))^p)/x^(n*q), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[n2, 2*n] && EqQ[
mn, -n] && IntegerQ[q] && (PosQ[n] || !IntegerQ[p])
Rule 1488
Int[(x_)^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.)*((d_) + (e_.)*(x_)^(n_))^(q_.), x_Symbol] :>
Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^n], x] /; FreeQ[{a,
b, c, d, e, m, n, p, q}, x] && EqQ[n2, 2*n] && IntegerQ[Simplify[(m + 1)/n]]
Rubi steps
\begin {align*} \int \frac {\sqrt {a+\frac {c}{x^2}+\frac {b}{x}}}{d+e x} \, dx &=\int \frac {\sqrt {a+\frac {c}{x^2}+\frac {b}{x}}}{\left (e+\frac {d}{x}\right ) x} \, dx\\ &=-\text {Subst}\left (\int \frac {\sqrt {a+b x+c x^2}}{x (e+d x)} \, dx,x,\frac {1}{x}\right )\\ &=\frac {\text {Subst}\left (\int \frac {a d-b e-c e x}{(e+d x) \sqrt {a+b x+c x^2}} \, dx,x,\frac {1}{x}\right )}{e}-\frac {a \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x+c x^2}} \, dx,x,\frac {1}{x}\right )}{e}\\ &=-\frac {c \text {Subst}\left (\int \frac {1}{\sqrt {a+b x+c x^2}} \, dx,x,\frac {1}{x}\right )}{d}+\frac {(2 a) \text {Subst}\left (\int \frac {1}{4 a-x^2} \, dx,x,\frac {2 a+\frac {b}{x}}{\sqrt {a+\frac {c}{x^2}+\frac {b}{x}}}\right )}{e}+\left (-b+\frac {a d}{e}+\frac {c e}{d}\right ) \text {Subst}\left (\int \frac {1}{(e+d x) \sqrt {a+b x+c x^2}} \, dx,x,\frac {1}{x}\right )\\ &=\frac {\sqrt {a} \tanh ^{-1}\left (\frac {2 a+\frac {b}{x}}{2 \sqrt {a} \sqrt {a+\frac {c}{x^2}+\frac {b}{x}}}\right )}{e}-\frac {(2 c) \text {Subst}\left (\int \frac {1}{4 c-x^2} \, dx,x,\frac {b+\frac {2 c}{x}}{\sqrt {a+\frac {c}{x^2}+\frac {b}{x}}}\right )}{d}+\left (2 \left (b-\frac {a d}{e}-\frac {c e}{d}\right )\right ) \text {Subst}\left (\int \frac {1}{4 a d^2-4 b d e+4 c e^2-x^2} \, dx,x,\frac {2 a d-b e-\frac {-b d+2 c e}{x}}{\sqrt {a+\frac {c}{x^2}+\frac {b}{x}}}\right )\\ &=\frac {\sqrt {a} \tanh ^{-1}\left (\frac {2 a+\frac {b}{x}}{2 \sqrt {a} \sqrt {a+\frac {c}{x^2}+\frac {b}{x}}}\right )}{e}-\frac {\sqrt {c} \tanh ^{-1}\left (\frac {b+\frac {2 c}{x}}{2 \sqrt {c} \sqrt {a+\frac {c}{x^2}+\frac {b}{x}}}\right )}{d}-\frac {\sqrt {a d^2-e (b d-c e)} \tanh ^{-1}\left (\frac {2 a d-b e+\frac {b d-2 c e}{x}}{2 \sqrt {a d^2-e (b d-c e)} \sqrt {a+\frac {c}{x^2}+\frac {b}{x}}}\right )}{d e}\\ \end {align*}
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Mathematica [A]
time = 0.53, size = 186, normalized size = 1.03 \begin {gather*} \frac {x \sqrt {a+\frac {c+b x}{x^2}} \left (-2 \sqrt {-a d^2+b d e-c e^2} \tan ^{-1}\left (\frac {\sqrt {a} (d+e x)-e \sqrt {c+x (b+a x)}}{\sqrt {-a d^2+b d e-c e^2}}\right )+2 \sqrt {c} e \tanh ^{-1}\left (\frac {\sqrt {a} x-\sqrt {c+x (b+a x)}}{\sqrt {c}}\right )-\sqrt {a} d \log \left (e \left (b+2 a x-2 \sqrt {a} \sqrt {c+x (b+a x)}\right )\right )\right )}{d e \sqrt {c+x (b+a x)}} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[Sqrt[a + c/x^2 + b/x]/(d + e*x),x]
[Out]
(x*Sqrt[a + (c + b*x)/x^2]*(-2*Sqrt[-(a*d^2) + b*d*e - c*e^2]*ArcTan[(Sqrt[a]*(d + e*x) - e*Sqrt[c + x*(b + a*
x)])/Sqrt[-(a*d^2) + b*d*e - c*e^2]] + 2*Sqrt[c]*e*ArcTanh[(Sqrt[a]*x - Sqrt[c + x*(b + a*x)])/Sqrt[c]] - Sqrt
[a]*d*Log[e*(b + 2*a*x - 2*Sqrt[a]*Sqrt[c + x*(b + a*x)])]))/(d*e*Sqrt[c + x*(b + a*x)])
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(396\) vs.
\(2(157)=314\).
time = 0.23, size = 397, normalized size = 2.19
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| method |
result |
size |
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| default |
\(-\frac {\sqrt {\frac {a \,x^{2}+b x +c}{x^{2}}}\, x \left (\sqrt {c}\, \ln \left (\frac {2 c +b x +2 \sqrt {c}\, \sqrt {a \,x^{2}+b x +c}}{x}\right ) \sqrt {a}\, \sqrt {\frac {a \,d^{2}-d e b +c \,e^{2}}{e^{2}}}\, e^{2}-\ln \left (\frac {2 \sqrt {a \,x^{2}+b x +c}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) \sqrt {\frac {a \,d^{2}-d e b +c \,e^{2}}{e^{2}}}\, a d e -\ln \left (\frac {2 \sqrt {\frac {a \,d^{2}-d e b +c \,e^{2}}{e^{2}}}\, \sqrt {a \,x^{2}+b x +c}\, e -2 a d x +e b x -b d +2 c e}{e x +d}\right ) a^{\frac {3}{2}} d^{2}+\ln \left (\frac {2 \sqrt {\frac {a \,d^{2}-d e b +c \,e^{2}}{e^{2}}}\, \sqrt {a \,x^{2}+b x +c}\, e -2 a d x +e b x -b d +2 c e}{e x +d}\right ) \sqrt {a}\, b d e -\ln \left (\frac {2 \sqrt {\frac {a \,d^{2}-d e b +c \,e^{2}}{e^{2}}}\, \sqrt {a \,x^{2}+b x +c}\, e -2 a d x +e b x -b d +2 c e}{e x +d}\right ) \sqrt {a}\, c \,e^{2}\right )}{\sqrt {a \,x^{2}+b x +c}\, d \,e^{2} \sqrt {a}\, \sqrt {\frac {a \,d^{2}-d e b +c \,e^{2}}{e^{2}}}}\) |
\(397\) |
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Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a+c/x^2+b/x)^(1/2)/(e*x+d),x,method=_RETURNVERBOSE)
[Out]
-((a*x^2+b*x+c)/x^2)^(1/2)*x*(c^(1/2)*ln((2*c+b*x+2*c^(1/2)*(a*x^2+b*x+c)^(1/2))/x)*a^(1/2)*((a*d^2-b*d*e+c*e^
2)/e^2)^(1/2)*e^2-ln(1/2*(2*(a*x^2+b*x+c)^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*((a*d^2-b*d*e+c*e^2)/e^2)^(1/2)*a*d*
e-ln((2*((a*d^2-b*d*e+c*e^2)/e^2)^(1/2)*(a*x^2+b*x+c)^(1/2)*e-2*a*d*x+e*b*x-b*d+2*c*e)/(e*x+d))*a^(3/2)*d^2+ln
((2*((a*d^2-b*d*e+c*e^2)/e^2)^(1/2)*(a*x^2+b*x+c)^(1/2)*e-2*a*d*x+e*b*x-b*d+2*c*e)/(e*x+d))*a^(1/2)*b*d*e-ln((
2*((a*d^2-b*d*e+c*e^2)/e^2)^(1/2)*(a*x^2+b*x+c)^(1/2)*e-2*a*d*x+e*b*x-b*d+2*c*e)/(e*x+d))*a^(1/2)*c*e^2)/(a*x^
2+b*x+c)^(1/2)/d/e^2/a^(1/2)/((a*d^2-b*d*e+c*e^2)/e^2)^(1/2)
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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((a+c/x^2+b/x)^(1/2)/(e*x+d),x, algorithm="maxima")
[Out]
integrate(sqrt(a + b/x + c/x^2)/(x*e + d), x)
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Fricas [A]
time = 41.38, size = 2407, normalized size = 13.30 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+c/x^2+b/x)^(1/2)/(e*x+d),x, algorithm="fricas")
[Out]
[1/2*(sqrt(a)*d*log(-8*a^2*x^2 - 8*a*b*x - b^2 - 4*a*c - 4*(2*a*x^2 + b*x)*sqrt(a)*sqrt((a*x^2 + b*x + c)/x^2)
) + sqrt(c)*e*log(-(8*b*c*x + (b^2 + 4*a*c)*x^2 + 8*c^2 - 4*(b*x^2 + 2*c*x)*sqrt(c)*sqrt((a*x^2 + b*x + c)/x^2
))/x^2) + sqrt(a*d^2 - b*d*e + c*e^2)*log(-(8*a^2*d^2*x^2 + 8*a*b*d^2*x + (b^2 + 4*a*c)*d^2 - 4*(2*a*d*x^2 + b
*d*x - (b*x^2 + 2*c*x)*e)*sqrt(a*d^2 - b*d*e + c*e^2)*sqrt((a*x^2 + b*x + c)/x^2) + (8*b*c*x + (b^2 + 4*a*c)*x
^2 + 8*c^2)*e^2 - 2*(4*a*b*d*x^2 + 4*b*c*d + (3*b^2 + 4*a*c)*d*x)*e)/(x^2*e^2 + 2*d*x*e + d^2)))*e^(-1)/d, -1/
2*(2*sqrt(-a)*d*arctan(1/2*(2*a*x^2 + b*x)*sqrt(-a)*sqrt((a*x^2 + b*x + c)/x^2)/(a^2*x^2 + a*b*x + a*c)) - sqr
t(c)*e*log(-(8*b*c*x + (b^2 + 4*a*c)*x^2 + 8*c^2 - 4*(b*x^2 + 2*c*x)*sqrt(c)*sqrt((a*x^2 + b*x + c)/x^2))/x^2)
- sqrt(a*d^2 - b*d*e + c*e^2)*log(-(8*a^2*d^2*x^2 + 8*a*b*d^2*x + (b^2 + 4*a*c)*d^2 - 4*(2*a*d*x^2 + b*d*x -
(b*x^2 + 2*c*x)*e)*sqrt(a*d^2 - b*d*e + c*e^2)*sqrt((a*x^2 + b*x + c)/x^2) + (8*b*c*x + (b^2 + 4*a*c)*x^2 + 8*
c^2)*e^2 - 2*(4*a*b*d*x^2 + 4*b*c*d + (3*b^2 + 4*a*c)*d*x)*e)/(x^2*e^2 + 2*d*x*e + d^2)))*e^(-1)/d, 1/2*(sqrt(
a)*d*log(-8*a^2*x^2 - 8*a*b*x - b^2 - 4*a*c - 4*(2*a*x^2 + b*x)*sqrt(a)*sqrt((a*x^2 + b*x + c)/x^2)) + sqrt(c)
*e*log(-(8*b*c*x + (b^2 + 4*a*c)*x^2 + 8*c^2 - 4*(b*x^2 + 2*c*x)*sqrt(c)*sqrt((a*x^2 + b*x + c)/x^2))/x^2) - 2
*sqrt(-a*d^2 + b*d*e - c*e^2)*arctan(-1/2*(2*a*d*x^2 + b*d*x - (b*x^2 + 2*c*x)*e)*sqrt(-a*d^2 + b*d*e - c*e^2)
*sqrt((a*x^2 + b*x + c)/x^2)/(a^2*d^2*x^2 + a*b*d^2*x + a*c*d^2 + (a*c*x^2 + b*c*x + c^2)*e^2 - (a*b*d*x^2 + b
^2*d*x + b*c*d)*e)))*e^(-1)/d, -1/2*(2*sqrt(-a)*d*arctan(1/2*(2*a*x^2 + b*x)*sqrt(-a)*sqrt((a*x^2 + b*x + c)/x
^2)/(a^2*x^2 + a*b*x + a*c)) - sqrt(c)*e*log(-(8*b*c*x + (b^2 + 4*a*c)*x^2 + 8*c^2 - 4*(b*x^2 + 2*c*x)*sqrt(c)
*sqrt((a*x^2 + b*x + c)/x^2))/x^2) + 2*sqrt(-a*d^2 + b*d*e - c*e^2)*arctan(-1/2*(2*a*d*x^2 + b*d*x - (b*x^2 +
2*c*x)*e)*sqrt(-a*d^2 + b*d*e - c*e^2)*sqrt((a*x^2 + b*x + c)/x^2)/(a^2*d^2*x^2 + a*b*d^2*x + a*c*d^2 + (a*c*x
^2 + b*c*x + c^2)*e^2 - (a*b*d*x^2 + b^2*d*x + b*c*d)*e)))*e^(-1)/d, 1/2*(2*sqrt(-c)*arctan(1/2*(b*x^2 + 2*c*x
)*sqrt(-c)*sqrt((a*x^2 + b*x + c)/x^2)/(a*c*x^2 + b*c*x + c^2))*e + sqrt(a)*d*log(-8*a^2*x^2 - 8*a*b*x - b^2 -
4*a*c - 4*(2*a*x^2 + b*x)*sqrt(a)*sqrt((a*x^2 + b*x + c)/x^2)) + sqrt(a*d^2 - b*d*e + c*e^2)*log(-(8*a^2*d^2*
x^2 + 8*a*b*d^2*x + (b^2 + 4*a*c)*d^2 - 4*(2*a*d*x^2 + b*d*x - (b*x^2 + 2*c*x)*e)*sqrt(a*d^2 - b*d*e + c*e^2)*
sqrt((a*x^2 + b*x + c)/x^2) + (8*b*c*x + (b^2 + 4*a*c)*x^2 + 8*c^2)*e^2 - 2*(4*a*b*d*x^2 + 4*b*c*d + (3*b^2 +
4*a*c)*d*x)*e)/(x^2*e^2 + 2*d*x*e + d^2)))*e^(-1)/d, -1/2*(2*sqrt(-a)*d*arctan(1/2*(2*a*x^2 + b*x)*sqrt(-a)*sq
rt((a*x^2 + b*x + c)/x^2)/(a^2*x^2 + a*b*x + a*c)) - 2*sqrt(-c)*arctan(1/2*(b*x^2 + 2*c*x)*sqrt(-c)*sqrt((a*x^
2 + b*x + c)/x^2)/(a*c*x^2 + b*c*x + c^2))*e - sqrt(a*d^2 - b*d*e + c*e^2)*log(-(8*a^2*d^2*x^2 + 8*a*b*d^2*x +
(b^2 + 4*a*c)*d^2 - 4*(2*a*d*x^2 + b*d*x - (b*x^2 + 2*c*x)*e)*sqrt(a*d^2 - b*d*e + c*e^2)*sqrt((a*x^2 + b*x +
c)/x^2) + (8*b*c*x + (b^2 + 4*a*c)*x^2 + 8*c^2)*e^2 - 2*(4*a*b*d*x^2 + 4*b*c*d + (3*b^2 + 4*a*c)*d*x)*e)/(x^2
*e^2 + 2*d*x*e + d^2)))*e^(-1)/d, 1/2*(2*sqrt(-c)*arctan(1/2*(b*x^2 + 2*c*x)*sqrt(-c)*sqrt((a*x^2 + b*x + c)/x
^2)/(a*c*x^2 + b*c*x + c^2))*e + sqrt(a)*d*log(-8*a^2*x^2 - 8*a*b*x - b^2 - 4*a*c - 4*(2*a*x^2 + b*x)*sqrt(a)*
sqrt((a*x^2 + b*x + c)/x^2)) - 2*sqrt(-a*d^2 + b*d*e - c*e^2)*arctan(-1/2*(2*a*d*x^2 + b*d*x - (b*x^2 + 2*c*x)
*e)*sqrt(-a*d^2 + b*d*e - c*e^2)*sqrt((a*x^2 + b*x + c)/x^2)/(a^2*d^2*x^2 + a*b*d^2*x + a*c*d^2 + (a*c*x^2 + b
*c*x + c^2)*e^2 - (a*b*d*x^2 + b^2*d*x + b*c*d)*e)))*e^(-1)/d, -(sqrt(-a)*d*arctan(1/2*(2*a*x^2 + b*x)*sqrt(-a
)*sqrt((a*x^2 + b*x + c)/x^2)/(a^2*x^2 + a*b*x + a*c)) - sqrt(-c)*arctan(1/2*(b*x^2 + 2*c*x)*sqrt(-c)*sqrt((a*
x^2 + b*x + c)/x^2)/(a*c*x^2 + b*c*x + c^2))*e + sqrt(-a*d^2 + b*d*e - c*e^2)*arctan(-1/2*(2*a*d*x^2 + b*d*x -
(b*x^2 + 2*c*x)*e)*sqrt(-a*d^2 + b*d*e - c*e^2)*sqrt((a*x^2 + b*x + c)/x^2)/(a^2*d^2*x^2 + a*b*d^2*x + a*c*d^
2 + (a*c*x^2 + b*c*x + c^2)*e^2 - (a*b*d*x^2 + b^2*d*x + b*c*d)*e)))*e^(-1)/d]
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {a + \frac {b}{x} + \frac {c}{x^{2}}}}{d + e x}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+c/x**2+b/x)**(1/2)/(e*x+d),x)
[Out]
Integral(sqrt(a + b/x + c/x**2)/(d + e*x), x)
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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+c/x^2+b/x)^(1/2)/(e*x+d),x, algorithm="giac")
[Out]
Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Warning, integration of abs or sign assumes constant sign by intervals (correct if the argument is real):Ch
eck [abs(sa
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\sqrt {a+\frac {b}{x}+\frac {c}{x^2}}}{d+e\,x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a + b/x + c/x^2)^(1/2)/(d + e*x),x)
[Out]
int((a + b/x + c/x^2)^(1/2)/(d + e*x), x)
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