3.1012 \(\int \frac{A+B x}{\sqrt{x} (a+b x+c x^2)} \, dx\)

Optimal. Leaf size=180 \[ \frac{\sqrt{2} \left (B-\frac{b B-2 A c}{\sqrt{b^2-4 a c}}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{x}}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{\sqrt{c} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\sqrt{2} \left (\frac{b B-2 A c}{\sqrt{b^2-4 a c}}+B\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{x}}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{\sqrt{c} \sqrt{\sqrt{b^2-4 a c}+b}} \]

[Out]

(Sqrt[2]*(B - (b*B - 2*A*c)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[x])/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/
(Sqrt[c]*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (Sqrt[2]*(B + (b*B - 2*A*c)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]
*Sqrt[x])/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(Sqrt[c]*Sqrt[b + Sqrt[b^2 - 4*a*c]])

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Rubi [A]  time = 0.285088, antiderivative size = 180, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {826, 1166, 205} \[ \frac{\sqrt{2} \left (B-\frac{b B-2 A c}{\sqrt{b^2-4 a c}}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{x}}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{\sqrt{c} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\sqrt{2} \left (\frac{b B-2 A c}{\sqrt{b^2-4 a c}}+B\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{x}}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{\sqrt{c} \sqrt{\sqrt{b^2-4 a c}+b}} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*x)/(Sqrt[x]*(a + b*x + c*x^2)),x]

[Out]

(Sqrt[2]*(B - (b*B - 2*A*c)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[x])/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/
(Sqrt[c]*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (Sqrt[2]*(B + (b*B - 2*A*c)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]
*Sqrt[x])/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(Sqrt[c]*Sqrt[b + Sqrt[b^2 - 4*a*c]])

Rule 826

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 1166

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 205

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

Rubi steps

\begin{align*} \int \frac{A+B x}{\sqrt{x} \left (a+b x+c x^2\right )} \, dx &=2 \operatorname{Subst}\left (\int \frac{A+B x^2}{a+b x^2+c x^4} \, dx,x,\sqrt{x}\right )\\ &=\left (B-\frac{b B-2 A c}{\sqrt{b^2-4 a c}}\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{b}{2}-\frac{1}{2} \sqrt{b^2-4 a c}+c x^2} \, dx,x,\sqrt{x}\right )+\left (B+\frac{b B-2 A c}{\sqrt{b^2-4 a c}}\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{b}{2}+\frac{1}{2} \sqrt{b^2-4 a c}+c x^2} \, dx,x,\sqrt{x}\right )\\ &=\frac{\sqrt{2} \left (B-\frac{b B-2 A c}{\sqrt{b^2-4 a c}}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{x}}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{\sqrt{c} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\sqrt{2} \left (B+\frac{b B-2 A c}{\sqrt{b^2-4 a c}}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{x}}{\sqrt{b+\sqrt{b^2-4 a c}}}\right )}{\sqrt{c} \sqrt{b+\sqrt{b^2-4 a c}}}\\ \end{align*}

Mathematica [A]  time = 0.281208, size = 181, normalized size = 1.01 \[ \frac{\sqrt{2} \left (\frac{\left (B \sqrt{b^2-4 a c}+2 A c-b B\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{x}}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{\sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\left (B \sqrt{b^2-4 a c}-2 A c+b B\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{x}}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{\sqrt{c} \sqrt{b^2-4 a c}} \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x)/(Sqrt[x]*(a + b*x + c*x^2)),x]

[Out]

(Sqrt[2]*(((-(b*B) + 2*A*c + B*Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[x])/Sqrt[b - Sqrt[b^2 - 4*a*c]]
])/Sqrt[b - Sqrt[b^2 - 4*a*c]] + ((b*B - 2*A*c + B*Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[x])/Sqrt[b
+ Sqrt[b^2 - 4*a*c]]])/Sqrt[b + Sqrt[b^2 - 4*a*c]]))/(Sqrt[c]*Sqrt[b^2 - 4*a*c])

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Maple [B]  time = 0.022, size = 337, normalized size = 1.9 \begin{align*} -2\,{\frac{c\sqrt{2}A}{\sqrt{-4\,ac+{b}^{2}}\sqrt{ \left ( -b+\sqrt{-4\,ac+{b}^{2}} \right ) c}}{\it Artanh} \left ({\frac{\sqrt{x}c\sqrt{2}}{\sqrt{ \left ( -b+\sqrt{-4\,ac+{b}^{2}} \right ) c}}} \right ) }-{\sqrt{2}B{\it Artanh} \left ({c\sqrt{2}\sqrt{x}{\frac{1}{\sqrt{ \left ( -b+\sqrt{-4\,ac+{b}^{2}} \right ) c}}}} \right ){\frac{1}{\sqrt{ \left ( -b+\sqrt{-4\,ac+{b}^{2}} \right ) c}}}}+{\sqrt{2}bB{\it Artanh} \left ({c\sqrt{2}\sqrt{x}{\frac{1}{\sqrt{ \left ( -b+\sqrt{-4\,ac+{b}^{2}} \right ) c}}}} \right ){\frac{1}{\sqrt{-4\,ac+{b}^{2}}}}{\frac{1}{\sqrt{ \left ( -b+\sqrt{-4\,ac+{b}^{2}} \right ) c}}}}-2\,{\frac{c\sqrt{2}A}{\sqrt{-4\,ac+{b}^{2}}\sqrt{ \left ( b+\sqrt{-4\,ac+{b}^{2}} \right ) c}}\arctan \left ({\frac{\sqrt{x}c\sqrt{2}}{\sqrt{ \left ( b+\sqrt{-4\,ac+{b}^{2}} \right ) c}}} \right ) }+{\sqrt{2}B\arctan \left ({c\sqrt{2}\sqrt{x}{\frac{1}{\sqrt{ \left ( b+\sqrt{-4\,ac+{b}^{2}} \right ) c}}}} \right ){\frac{1}{\sqrt{ \left ( b+\sqrt{-4\,ac+{b}^{2}} \right ) c}}}}+{\sqrt{2}bB\arctan \left ({c\sqrt{2}\sqrt{x}{\frac{1}{\sqrt{ \left ( b+\sqrt{-4\,ac+{b}^{2}} \right ) c}}}} \right ){\frac{1}{\sqrt{-4\,ac+{b}^{2}}}}{\frac{1}{\sqrt{ \left ( b+\sqrt{-4\,ac+{b}^{2}} \right ) c}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)/(c*x^2+b*x+a)/x^(1/2),x)

[Out]

-2*c/(-4*a*c+b^2)^(1/2)*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctanh(x^(1/2)*c*2^(1/2)/((-b+(-4*a*c+b^2)^
(1/2))*c)^(1/2))*A-2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctanh(x^(1/2)*c*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2)
)*c)^(1/2))*B+1/(-4*a*c+b^2)^(1/2)*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctanh(x^(1/2)*c*2^(1/2)/((-b+(-
4*a*c+b^2)^(1/2))*c)^(1/2))*b*B-2*c/(-4*a*c+b^2)^(1/2)*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(x^(1/2)
*c*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*A+2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(x^(1/2)*c*2^(1/
2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*B+1/(-4*a*c+b^2)^(1/2)*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(x^
(1/2)*c*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*b*B

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{2 \, A \sqrt{x}}{a} - \int \frac{A c x^{\frac{3}{2}} -{\left (B a - A b\right )} \sqrt{x}}{a c x^{2} + a b x + a^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/(c*x^2+b*x+a)/x^(1/2),x, algorithm="maxima")

[Out]

2*A*sqrt(x)/a - integrate((A*c*x^(3/2) - (B*a - A*b)*sqrt(x))/(a*c*x^2 + a*b*x + a^2), x)

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Fricas [B]  time = 2.51236, size = 3085, normalized size = 17.14 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/(c*x^2+b*x+a)/x^(1/2),x, algorithm="fricas")

[Out]

1/2*sqrt(2)*sqrt(-(B^2*a*b - (4*A*B*a - A^2*b)*c + (a*b^2*c - 4*a^2*c^2)*sqrt((B^4*a^2 - 2*A^2*B^2*a*c + A^4*c
^2)/(a^2*b^2*c^2 - 4*a^3*c^3)))/(a*b^2*c - 4*a^2*c^2))*log(sqrt(2)*(A*B^2*a*b^2 + 4*A^3*a*c^2 - (4*A*B^2*a^2 +
A^3*b^2)*c + (4*(2*B*a^3 - A*a^2*b)*c^2 - (2*B*a^2*b^2 - A*a*b^3)*c)*sqrt((B^4*a^2 - 2*A^2*B^2*a*c + A^4*c^2)
/(a^2*b^2*c^2 - 4*a^3*c^3)))*sqrt(-(B^2*a*b - (4*A*B*a - A^2*b)*c + (a*b^2*c - 4*a^2*c^2)*sqrt((B^4*a^2 - 2*A^
2*B^2*a*c + A^4*c^2)/(a^2*b^2*c^2 - 4*a^3*c^3)))/(a*b^2*c - 4*a^2*c^2)) - 4*(B^4*a^2 - A*B^3*a*b + A^3*B*b*c -
A^4*c^2)*sqrt(x)) - 1/2*sqrt(2)*sqrt(-(B^2*a*b - (4*A*B*a - A^2*b)*c + (a*b^2*c - 4*a^2*c^2)*sqrt((B^4*a^2 -
2*A^2*B^2*a*c + A^4*c^2)/(a^2*b^2*c^2 - 4*a^3*c^3)))/(a*b^2*c - 4*a^2*c^2))*log(-sqrt(2)*(A*B^2*a*b^2 + 4*A^3*
a*c^2 - (4*A*B^2*a^2 + A^3*b^2)*c + (4*(2*B*a^3 - A*a^2*b)*c^2 - (2*B*a^2*b^2 - A*a*b^3)*c)*sqrt((B^4*a^2 - 2*
A^2*B^2*a*c + A^4*c^2)/(a^2*b^2*c^2 - 4*a^3*c^3)))*sqrt(-(B^2*a*b - (4*A*B*a - A^2*b)*c + (a*b^2*c - 4*a^2*c^2
)*sqrt((B^4*a^2 - 2*A^2*B^2*a*c + A^4*c^2)/(a^2*b^2*c^2 - 4*a^3*c^3)))/(a*b^2*c - 4*a^2*c^2)) - 4*(B^4*a^2 - A
*B^3*a*b + A^3*B*b*c - A^4*c^2)*sqrt(x)) + 1/2*sqrt(2)*sqrt(-(B^2*a*b - (4*A*B*a - A^2*b)*c - (a*b^2*c - 4*a^2
*c^2)*sqrt((B^4*a^2 - 2*A^2*B^2*a*c + A^4*c^2)/(a^2*b^2*c^2 - 4*a^3*c^3)))/(a*b^2*c - 4*a^2*c^2))*log(sqrt(2)*
(A*B^2*a*b^2 + 4*A^3*a*c^2 - (4*A*B^2*a^2 + A^3*b^2)*c - (4*(2*B*a^3 - A*a^2*b)*c^2 - (2*B*a^2*b^2 - A*a*b^3)*
c)*sqrt((B^4*a^2 - 2*A^2*B^2*a*c + A^4*c^2)/(a^2*b^2*c^2 - 4*a^3*c^3)))*sqrt(-(B^2*a*b - (4*A*B*a - A^2*b)*c -
(a*b^2*c - 4*a^2*c^2)*sqrt((B^4*a^2 - 2*A^2*B^2*a*c + A^4*c^2)/(a^2*b^2*c^2 - 4*a^3*c^3)))/(a*b^2*c - 4*a^2*c
^2)) - 4*(B^4*a^2 - A*B^3*a*b + A^3*B*b*c - A^4*c^2)*sqrt(x)) - 1/2*sqrt(2)*sqrt(-(B^2*a*b - (4*A*B*a - A^2*b)
*c - (a*b^2*c - 4*a^2*c^2)*sqrt((B^4*a^2 - 2*A^2*B^2*a*c + A^4*c^2)/(a^2*b^2*c^2 - 4*a^3*c^3)))/(a*b^2*c - 4*a
^2*c^2))*log(-sqrt(2)*(A*B^2*a*b^2 + 4*A^3*a*c^2 - (4*A*B^2*a^2 + A^3*b^2)*c - (4*(2*B*a^3 - A*a^2*b)*c^2 - (2
*B*a^2*b^2 - A*a*b^3)*c)*sqrt((B^4*a^2 - 2*A^2*B^2*a*c + A^4*c^2)/(a^2*b^2*c^2 - 4*a^3*c^3)))*sqrt(-(B^2*a*b -
(4*A*B*a - A^2*b)*c - (a*b^2*c - 4*a^2*c^2)*sqrt((B^4*a^2 - 2*A^2*B^2*a*c + A^4*c^2)/(a^2*b^2*c^2 - 4*a^3*c^3
)))/(a*b^2*c - 4*a^2*c^2)) - 4*(B^4*a^2 - A*B^3*a*b + A^3*B*b*c - A^4*c^2)*sqrt(x))

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Sympy [B]  time = 35.9341, size = 1950, normalized size = 10.83 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/(c*x**2+b*x+a)/x**(1/2),x)

[Out]

Piecewise((-2*A/(b*sqrt(x)) + I*A*log(-I*sqrt(b)*sqrt(1/c) + sqrt(x))/(b**(3/2)*sqrt(1/c)) - I*A*log(I*sqrt(b)
*sqrt(1/c) + sqrt(x))/(b**(3/2)*sqrt(1/c)) - I*B*log(-I*sqrt(b)*sqrt(1/c) + sqrt(x))/(sqrt(b)*c*sqrt(1/c)) + I
*B*log(I*sqrt(b)*sqrt(1/c) + sqrt(x))/(sqrt(b)*c*sqrt(1/c)), Eq(a, 0)), (8*I*A*sqrt(b)*c**2*sqrt(x)*sqrt(1/c)/
(2*I*b**(5/2)*c*sqrt(1/c) + 4*I*b**(3/2)*c**2*x*sqrt(1/c)) + 2*sqrt(2)*A*b*c*log(-sqrt(2)*I*sqrt(b)*sqrt(1/c)/
2 + sqrt(x))/(2*I*b**(5/2)*c*sqrt(1/c) + 4*I*b**(3/2)*c**2*x*sqrt(1/c)) - 2*sqrt(2)*A*b*c*log(sqrt(2)*I*sqrt(b
)*sqrt(1/c)/2 + sqrt(x))/(2*I*b**(5/2)*c*sqrt(1/c) + 4*I*b**(3/2)*c**2*x*sqrt(1/c)) + 4*sqrt(2)*A*c**2*x*log(-
sqrt(2)*I*sqrt(b)*sqrt(1/c)/2 + sqrt(x))/(2*I*b**(5/2)*c*sqrt(1/c) + 4*I*b**(3/2)*c**2*x*sqrt(1/c)) - 4*sqrt(2
)*A*c**2*x*log(sqrt(2)*I*sqrt(b)*sqrt(1/c)/2 + sqrt(x))/(2*I*b**(5/2)*c*sqrt(1/c) + 4*I*b**(3/2)*c**2*x*sqrt(1
/c)) - 4*I*B*b**(3/2)*c*sqrt(x)*sqrt(1/c)/(2*I*b**(5/2)*c*sqrt(1/c) + 4*I*b**(3/2)*c**2*x*sqrt(1/c)) + sqrt(2)
*B*b**2*log(-sqrt(2)*I*sqrt(b)*sqrt(1/c)/2 + sqrt(x))/(2*I*b**(5/2)*c*sqrt(1/c) + 4*I*b**(3/2)*c**2*x*sqrt(1/c
)) - sqrt(2)*B*b**2*log(sqrt(2)*I*sqrt(b)*sqrt(1/c)/2 + sqrt(x))/(2*I*b**(5/2)*c*sqrt(1/c) + 4*I*b**(3/2)*c**2
*x*sqrt(1/c)) + 2*sqrt(2)*B*b*c*x*log(-sqrt(2)*I*sqrt(b)*sqrt(1/c)/2 + sqrt(x))/(2*I*b**(5/2)*c*sqrt(1/c) + 4*
I*b**(3/2)*c**2*x*sqrt(1/c)) - 2*sqrt(2)*B*b*c*x*log(sqrt(2)*I*sqrt(b)*sqrt(1/c)/2 + sqrt(x))/(2*I*b**(5/2)*c*
sqrt(1/c) + 4*I*b**(3/2)*c**2*x*sqrt(1/c)), Eq(a, b**2/(4*c))), (-I*A*log(-I*sqrt(a)*sqrt(1/b) + sqrt(x))/(sqr
t(a)*b*sqrt(1/b)) + I*A*log(I*sqrt(a)*sqrt(1/b) + sqrt(x))/(sqrt(a)*b*sqrt(1/b)) + I*B*sqrt(a)*log(-I*sqrt(a)*
sqrt(1/b) + sqrt(x))/(b**2*sqrt(1/b)) - I*B*sqrt(a)*log(I*sqrt(a)*sqrt(1/b) + sqrt(x))/(b**2*sqrt(1/b)) + 2*B*
sqrt(x)/b, Eq(c, 0)), (sqrt(2)*A*b*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)*log(sqrt(x) - sqrt(2)*sqrt(-b/c - sqrt(-
4*a*c + b**2)/c)/2)/(4*a*sqrt(-4*a*c + b**2)) - sqrt(2)*A*b*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)*log(sqrt(x) + s
qrt(2)*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)/2)/(4*a*sqrt(-4*a*c + b**2)) - sqrt(2)*A*b*sqrt(-b/c + sqrt(-4*a*c +
b**2)/c)*log(sqrt(x) - sqrt(2)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c)/2)/(4*a*sqrt(-4*a*c + b**2)) + sqrt(2)*A*b*
sqrt(-b/c + sqrt(-4*a*c + b**2)/c)*log(sqrt(x) + sqrt(2)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c)/2)/(4*a*sqrt(-4*a*
c + b**2)) - sqrt(2)*A*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)*log(sqrt(x) - sqrt(2)*sqrt(-b/c - sqrt(-4*a*c + b**2
)/c)/2)/(4*a) + sqrt(2)*A*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)*log(sqrt(x) + sqrt(2)*sqrt(-b/c - sqrt(-4*a*c + b
**2)/c)/2)/(4*a) - sqrt(2)*A*sqrt(-b/c + sqrt(-4*a*c + b**2)/c)*log(sqrt(x) - sqrt(2)*sqrt(-b/c + sqrt(-4*a*c
+ b**2)/c)/2)/(4*a) + sqrt(2)*A*sqrt(-b/c + sqrt(-4*a*c + b**2)/c)*log(sqrt(x) + sqrt(2)*sqrt(-b/c + sqrt(-4*a
*c + b**2)/c)/2)/(4*a) - sqrt(2)*B*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)*log(sqrt(x) - sqrt(2)*sqrt(-b/c - sqrt(-
4*a*c + b**2)/c)/2)/(2*sqrt(-4*a*c + b**2)) + sqrt(2)*B*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)*log(sqrt(x) + sqrt(
2)*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)/2)/(2*sqrt(-4*a*c + b**2)) + sqrt(2)*B*sqrt(-b/c + sqrt(-4*a*c + b**2)/c
)*log(sqrt(x) - sqrt(2)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c)/2)/(2*sqrt(-4*a*c + b**2)) - sqrt(2)*B*sqrt(-b/c +
sqrt(-4*a*c + b**2)/c)*log(sqrt(x) + sqrt(2)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c)/2)/(2*sqrt(-4*a*c + b**2)), Tr
ue))

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Giac [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/(c*x^2+b*x+a)/x^(1/2),x, algorithm="giac")

[Out]

Timed out