3.1011 \(\int \frac{\sqrt{x} (A+B x)}{a+b x+c x^2} \, dx\)

Optimal. Leaf size=221 \[ -\frac{\sqrt{2} \left (-\frac{-2 a B c-A b c+b^2 B}{\sqrt{b^2-4 a c}}-A c+b B\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{x}}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{c^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\sqrt{2} \left (\frac{-2 a B c-A b c+b^2 B}{\sqrt{b^2-4 a c}}-A c+b B\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{x}}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{c^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{2 B \sqrt{x}}{c} \]

[Out]

(2*B*Sqrt[x])/c - (Sqrt[2]*(b*B - A*c - (b^2*B - A*b*c - 2*a*B*c)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*S
qrt[x])/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(c^(3/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) - (Sqrt[2]*(b*B - A*c + (b^2*B - A
*b*c - 2*a*B*c)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[x])/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(c^(3/2)*Sqr
t[b + Sqrt[b^2 - 4*a*c]])

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Rubi [A]  time = 0.811903, antiderivative size = 221, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used = {824, 826, 1166, 205} \[ -\frac{\sqrt{2} \left (-\frac{-2 a B c-A b c+b^2 B}{\sqrt{b^2-4 a c}}-A c+b B\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{x}}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{c^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\sqrt{2} \left (\frac{-2 a B c-A b c+b^2 B}{\sqrt{b^2-4 a c}}-A c+b B\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{x}}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{c^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{2 B \sqrt{x}}{c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*B*Sqrt[x])/c - (Sqrt[2]*(b*B - A*c - (b^2*B - A*b*c - 2*a*B*c)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*S
qrt[x])/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(c^(3/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) - (Sqrt[2]*(b*B - A*c + (b^2*B - A
*b*c - 2*a*B*c)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[x])/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(c^(3/2)*Sqr
t[b + Sqrt[b^2 - 4*a*c]])

Rule 824

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(g
*(d + e*x)^m)/(c*m), x] + Dist[1/c, Int[((d + e*x)^(m - 1)*Simp[c*d*f - a*e*g + (g*c*d - b*e*g + c*e*f)*x, x])
/(a + b*x + c*x^2), 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] && FractionQ[m] && GtQ[m, 0]

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{\sqrt{x} (A+B x)}{a+b x+c x^2} \, dx &=\frac{2 B \sqrt{x}}{c}+\frac{\int \frac{-a B-(b B-A c) x}{\sqrt{x} \left (a+b x+c x^2\right )} \, dx}{c}\\ &=\frac{2 B \sqrt{x}}{c}+\frac{2 \operatorname{Subst}\left (\int \frac{-a B+(-b B+A c) x^2}{a+b x^2+c x^4} \, dx,x,\sqrt{x}\right )}{c}\\ &=\frac{2 B \sqrt{x}}{c}-\frac{\left (b B-A c+\frac{b^2 B-A b c-2 a B 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 )}{c}+\frac{\left (-b B+A c+\frac{b^2 B-A b c-2 a B 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 )}{c}\\ &=\frac{2 B \sqrt{x}}{c}-\frac{\sqrt{2} \left (b B-A c-\frac{b^2 B-A b c-2 a B 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 )}{c^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\sqrt{2} \left (b B-A c+\frac{b^2 B-A b c-2 a B 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 )}{c^{3/2} \sqrt{b+\sqrt{b^2-4 a c}}}\\ \end{align*}

Mathematica [A]  time = 0.235985, size = 264, normalized size = 1.19 \[ -\frac{\sqrt{2} \left (-A c \sqrt{b^2-4 a c}+b B \sqrt{b^2-4 a c}+2 a B c+A b c+b^2 (-B)\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{x}}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{c^{3/2} \sqrt{b^2-4 a c} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\sqrt{2} \left (-A c \sqrt{b^2-4 a c}+b B \sqrt{b^2-4 a c}-2 a B c-A b c+b^2 B\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{x}}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{c^{3/2} \sqrt{b^2-4 a c} \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{2 B \sqrt{x}}{c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*B*Sqrt[x])/c - (Sqrt[2]*(-(b^2*B) + A*b*c + 2*a*B*c + b*B*Sqrt[b^2 - 4*a*c] - A*c*Sqrt[b^2 - 4*a*c])*ArcTan
[(Sqrt[2]*Sqrt[c]*Sqrt[x])/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(c^(3/2)*Sqrt[b^2 - 4*a*c]*Sqrt[b - Sqrt[b^2 - 4*a*c]
]) - (Sqrt[2]*(b^2*B - A*b*c - 2*a*B*c + b*B*Sqrt[b^2 - 4*a*c] - A*c*Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c
]*Sqrt[x])/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(c^(3/2)*Sqrt[b^2 - 4*a*c]*Sqrt[b + Sqrt[b^2 - 4*a*c]])

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Maple [B]  time = 0.029, size = 581, normalized size = 2.6 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 3.96366, size = 5285, normalized size = 23.91 \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)*x^(1/2)/(c*x^2+b*x+a),x, algorithm="fricas")

[Out]

1/2*(sqrt(2)*c*sqrt(-(B^2*b^3 + (4*A*B*a + A^2*b)*c^2 - (3*B^2*a*b + 2*A*B*b^2)*c + (b^2*c^3 - 4*a*c^4)*sqrt((
B^4*b^4 + A^4*c^4 - 2*(A^2*B^2*a + 2*A^3*B*b)*c^3 + (B^4*a^2 + 4*A*B^3*a*b + 6*A^2*B^2*b^2)*c^2 - 2*(B^4*a*b^2
+ 2*A*B^3*b^3)*c)/(b^2*c^6 - 4*a*c^7)))/(b^2*c^3 - 4*a*c^4))*log(sqrt(2)*(B^3*b^4 - 4*A^2*B*a*c^3 + (4*B^3*a^
2 + 8*A*B^2*a*b + A^2*B*b^2)*c^2 - (5*B^3*a*b^2 + 2*A*B^2*b^3)*c - (B*b^3*c^3 + 8*A*a*c^5 - 2*(2*B*a*b + A*b^2
)*c^4)*sqrt((B^4*b^4 + A^4*c^4 - 2*(A^2*B^2*a + 2*A^3*B*b)*c^3 + (B^4*a^2 + 4*A*B^3*a*b + 6*A^2*B^2*b^2)*c^2 -
2*(B^4*a*b^2 + 2*A*B^3*b^3)*c)/(b^2*c^6 - 4*a*c^7)))*sqrt(-(B^2*b^3 + (4*A*B*a + A^2*b)*c^2 - (3*B^2*a*b + 2*
A*B*b^2)*c + (b^2*c^3 - 4*a*c^4)*sqrt((B^4*b^4 + A^4*c^4 - 2*(A^2*B^2*a + 2*A^3*B*b)*c^3 + (B^4*a^2 + 4*A*B^3*
a*b + 6*A^2*B^2*b^2)*c^2 - 2*(B^4*a*b^2 + 2*A*B^3*b^3)*c)/(b^2*c^6 - 4*a*c^7)))/(b^2*c^3 - 4*a*c^4)) + 4*(B^4*
a*b^2 - A*B^3*b^3 - 3*A^3*B*b*c^2 + A^4*c^3 - (B^4*a^2 + A*B^3*a*b - 3*A^2*B^2*b^2)*c)*sqrt(x)) - sqrt(2)*c*sq
rt(-(B^2*b^3 + (4*A*B*a + A^2*b)*c^2 - (3*B^2*a*b + 2*A*B*b^2)*c + (b^2*c^3 - 4*a*c^4)*sqrt((B^4*b^4 + A^4*c^4
- 2*(A^2*B^2*a + 2*A^3*B*b)*c^3 + (B^4*a^2 + 4*A*B^3*a*b + 6*A^2*B^2*b^2)*c^2 - 2*(B^4*a*b^2 + 2*A*B^3*b^3)*c
)/(b^2*c^6 - 4*a*c^7)))/(b^2*c^3 - 4*a*c^4))*log(-sqrt(2)*(B^3*b^4 - 4*A^2*B*a*c^3 + (4*B^3*a^2 + 8*A*B^2*a*b
+ A^2*B*b^2)*c^2 - (5*B^3*a*b^2 + 2*A*B^2*b^3)*c - (B*b^3*c^3 + 8*A*a*c^5 - 2*(2*B*a*b + A*b^2)*c^4)*sqrt((B^4
*b^4 + A^4*c^4 - 2*(A^2*B^2*a + 2*A^3*B*b)*c^3 + (B^4*a^2 + 4*A*B^3*a*b + 6*A^2*B^2*b^2)*c^2 - 2*(B^4*a*b^2 +
2*A*B^3*b^3)*c)/(b^2*c^6 - 4*a*c^7)))*sqrt(-(B^2*b^3 + (4*A*B*a + A^2*b)*c^2 - (3*B^2*a*b + 2*A*B*b^2)*c + (b^
2*c^3 - 4*a*c^4)*sqrt((B^4*b^4 + A^4*c^4 - 2*(A^2*B^2*a + 2*A^3*B*b)*c^3 + (B^4*a^2 + 4*A*B^3*a*b + 6*A^2*B^2*
b^2)*c^2 - 2*(B^4*a*b^2 + 2*A*B^3*b^3)*c)/(b^2*c^6 - 4*a*c^7)))/(b^2*c^3 - 4*a*c^4)) + 4*(B^4*a*b^2 - A*B^3*b^
3 - 3*A^3*B*b*c^2 + A^4*c^3 - (B^4*a^2 + A*B^3*a*b - 3*A^2*B^2*b^2)*c)*sqrt(x)) + sqrt(2)*c*sqrt(-(B^2*b^3 + (
4*A*B*a + A^2*b)*c^2 - (3*B^2*a*b + 2*A*B*b^2)*c - (b^2*c^3 - 4*a*c^4)*sqrt((B^4*b^4 + A^4*c^4 - 2*(A^2*B^2*a
+ 2*A^3*B*b)*c^3 + (B^4*a^2 + 4*A*B^3*a*b + 6*A^2*B^2*b^2)*c^2 - 2*(B^4*a*b^2 + 2*A*B^3*b^3)*c)/(b^2*c^6 - 4*a
*c^7)))/(b^2*c^3 - 4*a*c^4))*log(sqrt(2)*(B^3*b^4 - 4*A^2*B*a*c^3 + (4*B^3*a^2 + 8*A*B^2*a*b + A^2*B*b^2)*c^2
- (5*B^3*a*b^2 + 2*A*B^2*b^3)*c + (B*b^3*c^3 + 8*A*a*c^5 - 2*(2*B*a*b + A*b^2)*c^4)*sqrt((B^4*b^4 + A^4*c^4 -
2*(A^2*B^2*a + 2*A^3*B*b)*c^3 + (B^4*a^2 + 4*A*B^3*a*b + 6*A^2*B^2*b^2)*c^2 - 2*(B^4*a*b^2 + 2*A*B^3*b^3)*c)/(
b^2*c^6 - 4*a*c^7)))*sqrt(-(B^2*b^3 + (4*A*B*a + A^2*b)*c^2 - (3*B^2*a*b + 2*A*B*b^2)*c - (b^2*c^3 - 4*a*c^4)*
sqrt((B^4*b^4 + A^4*c^4 - 2*(A^2*B^2*a + 2*A^3*B*b)*c^3 + (B^4*a^2 + 4*A*B^3*a*b + 6*A^2*B^2*b^2)*c^2 - 2*(B^4
*a*b^2 + 2*A*B^3*b^3)*c)/(b^2*c^6 - 4*a*c^7)))/(b^2*c^3 - 4*a*c^4)) + 4*(B^4*a*b^2 - A*B^3*b^3 - 3*A^3*B*b*c^2
+ A^4*c^3 - (B^4*a^2 + A*B^3*a*b - 3*A^2*B^2*b^2)*c)*sqrt(x)) - sqrt(2)*c*sqrt(-(B^2*b^3 + (4*A*B*a + A^2*b)*
c^2 - (3*B^2*a*b + 2*A*B*b^2)*c - (b^2*c^3 - 4*a*c^4)*sqrt((B^4*b^4 + A^4*c^4 - 2*(A^2*B^2*a + 2*A^3*B*b)*c^3
+ (B^4*a^2 + 4*A*B^3*a*b + 6*A^2*B^2*b^2)*c^2 - 2*(B^4*a*b^2 + 2*A*B^3*b^3)*c)/(b^2*c^6 - 4*a*c^7)))/(b^2*c^3
- 4*a*c^4))*log(-sqrt(2)*(B^3*b^4 - 4*A^2*B*a*c^3 + (4*B^3*a^2 + 8*A*B^2*a*b + A^2*B*b^2)*c^2 - (5*B^3*a*b^2 +
2*A*B^2*b^3)*c + (B*b^3*c^3 + 8*A*a*c^5 - 2*(2*B*a*b + A*b^2)*c^4)*sqrt((B^4*b^4 + A^4*c^4 - 2*(A^2*B^2*a + 2
*A^3*B*b)*c^3 + (B^4*a^2 + 4*A*B^3*a*b + 6*A^2*B^2*b^2)*c^2 - 2*(B^4*a*b^2 + 2*A*B^3*b^3)*c)/(b^2*c^6 - 4*a*c^
7)))*sqrt(-(B^2*b^3 + (4*A*B*a + A^2*b)*c^2 - (3*B^2*a*b + 2*A*B*b^2)*c - (b^2*c^3 - 4*a*c^4)*sqrt((B^4*b^4 +
A^4*c^4 - 2*(A^2*B^2*a + 2*A^3*B*b)*c^3 + (B^4*a^2 + 4*A*B^3*a*b + 6*A^2*B^2*b^2)*c^2 - 2*(B^4*a*b^2 + 2*A*B^3
*b^3)*c)/(b^2*c^6 - 4*a*c^7)))/(b^2*c^3 - 4*a*c^4)) + 4*(B^4*a*b^2 - A*B^3*b^3 - 3*A^3*B*b*c^2 + A^4*c^3 - (B^
4*a^2 + A*B^3*a*b - 3*A^2*B^2*b^2)*c)*sqrt(x)) + 4*B*sqrt(x))/c

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Sympy [B]  time = 21.7764, size = 2443, normalized size = 11.05 \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)*x**(1/2)/(c*x**2+b*x+a),x)

[Out]

Piecewise((-8*I*A*sqrt(b)*c**2*sqrt(x)*sqrt(1/c)/(4*I*b**(3/2)*c**2*sqrt(1/c) + 8*I*sqrt(b)*c**3*x*sqrt(1/c))
+ 2*sqrt(2)*A*b*c*log(-sqrt(2)*I*sqrt(b)*sqrt(1/c)/2 + sqrt(x))/(4*I*b**(3/2)*c**2*sqrt(1/c) + 8*I*sqrt(b)*c**
3*x*sqrt(1/c)) - 2*sqrt(2)*A*b*c*log(sqrt(2)*I*sqrt(b)*sqrt(1/c)/2 + sqrt(x))/(4*I*b**(3/2)*c**2*sqrt(1/c) + 8
*I*sqrt(b)*c**3*x*sqrt(1/c)) + 4*sqrt(2)*A*c**2*x*log(-sqrt(2)*I*sqrt(b)*sqrt(1/c)/2 + sqrt(x))/(4*I*b**(3/2)*
c**2*sqrt(1/c) + 8*I*sqrt(b)*c**3*x*sqrt(1/c)) - 4*sqrt(2)*A*c**2*x*log(sqrt(2)*I*sqrt(b)*sqrt(1/c)/2 + sqrt(x
))/(4*I*b**(3/2)*c**2*sqrt(1/c) + 8*I*sqrt(b)*c**3*x*sqrt(1/c)) + 12*I*B*b**(3/2)*c*sqrt(x)*sqrt(1/c)/(4*I*b**
(3/2)*c**2*sqrt(1/c) + 8*I*sqrt(b)*c**3*x*sqrt(1/c)) + 16*I*B*sqrt(b)*c**2*x**(3/2)*sqrt(1/c)/(4*I*b**(3/2)*c*
*2*sqrt(1/c) + 8*I*sqrt(b)*c**3*x*sqrt(1/c)) - 3*sqrt(2)*B*b**2*log(-sqrt(2)*I*sqrt(b)*sqrt(1/c)/2 + sqrt(x))/
(4*I*b**(3/2)*c**2*sqrt(1/c) + 8*I*sqrt(b)*c**3*x*sqrt(1/c)) + 3*sqrt(2)*B*b**2*log(sqrt(2)*I*sqrt(b)*sqrt(1/c
)/2 + sqrt(x))/(4*I*b**(3/2)*c**2*sqrt(1/c) + 8*I*sqrt(b)*c**3*x*sqrt(1/c)) - 6*sqrt(2)*B*b*c*x*log(-sqrt(2)*I
*sqrt(b)*sqrt(1/c)/2 + sqrt(x))/(4*I*b**(3/2)*c**2*sqrt(1/c) + 8*I*sqrt(b)*c**3*x*sqrt(1/c)) + 6*sqrt(2)*B*b*c
*x*log(sqrt(2)*I*sqrt(b)*sqrt(1/c)/2 + sqrt(x))/(4*I*b**(3/2)*c**2*sqrt(1/c) + 8*I*sqrt(b)*c**3*x*sqrt(1/c)),
Eq(a, b**2/(4*c))), (I*A*sqrt(a)*log(-I*sqrt(a)*sqrt(1/b) + sqrt(x))/(b**2*sqrt(1/b)) - I*A*sqrt(a)*log(I*sqrt
(a)*sqrt(1/b) + sqrt(x))/(b**2*sqrt(1/b)) + 2*A*sqrt(x)/b - I*B*a**(3/2)*log(-I*sqrt(a)*sqrt(1/b) + sqrt(x))/(
b**3*sqrt(1/b)) + I*B*a**(3/2)*log(I*sqrt(a)*sqrt(1/b) + sqrt(x))/(b**3*sqrt(1/b)) - 2*B*a*sqrt(x)/b**2 + 2*B*
x**(3/2)/(3*b), Eq(c, 0)), (2*sqrt(2)*A*c*sqrt(-4*a*c + b**2)*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)/(16*a*c**2 - 4*b**2*c) - 2*sqrt(2)*A*c*sqrt(-4*a*c + b**2)*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)/(16*a*c**2 - 4*b**2
*c) - 2*sqrt(2)*A*c*sqrt(-4*a*c + b**2)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c)*log(sqrt(x) - sqrt(2)*sqrt(-b/c + s
qrt(-4*a*c + b**2)/c)/2)/(16*a*c**2 - 4*b**2*c) + 2*sqrt(2)*A*c*sqrt(-4*a*c + b**2)*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)/(16*a*c**2 - 4*b**2*c) + 32*B*a*c*sqrt(x)
/(16*a*c**2 - 4*b**2*c) + 4*sqrt(2)*B*a*c*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)/(16*a*c**2 - 4*b**2*c) - 4*sqrt(2)*B*a*c*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)*log(sqr
t(x) + sqrt(2)*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)/2)/(16*a*c**2 - 4*b**2*c) + 4*sqrt(2)*B*a*c*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)/(16*a*c**2 - 4*b**2*c) - 4*sqrt
(2)*B*a*c*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)/(16*a
*c**2 - 4*b**2*c) - 8*B*b**2*sqrt(x)/(16*a*c**2 - 4*b**2*c) - sqrt(2)*B*b**2*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)/(16*a*c**2 - 4*b**2*c) + sqrt(2)*B*b**2*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)/(16*a*c**2 - 4*b**2*c)
- sqrt(2)*B*b**2*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
)/(16*a*c**2 - 4*b**2*c) + sqrt(2)*B*b**2*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)/(16*a*c**2 - 4*b**2*c) - sqrt(2)*B*b*sqrt(-4*a*c + b**2)*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)/(16*a*c**2 - 4*b**2*c) + sqrt(2)*B*b*sqrt
(-4*a*c + b**2)*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)
/(16*a*c**2 - 4*b**2*c) + sqrt(2)*B*b*sqrt(-4*a*c + b**2)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c)*log(sqrt(x) - sqr
t(2)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c)/2)/(16*a*c**2 - 4*b**2*c) - sqrt(2)*B*b*sqrt(-4*a*c + b**2)*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)/(16*a*c**2 - 4*b**2*c), T
rue))

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: NotImplementedError