3.21 \(\int \frac{A+B x}{(a+b x+c x^2) \sqrt{d+e x+f x^2}} \, dx\)
Optimal. Leaf size=416 \[ \frac{\left (-B \sqrt{b^2-4 a c}-2 A c+b B\right ) \tanh ^{-1}\left (\frac{2 x \left (c e-f \left (b-\sqrt{b^2-4 a c}\right )\right )-e \left (b-\sqrt{b^2-4 a c}\right )+4 c d}{2 \sqrt{2} \sqrt{d+e x+f x^2} \sqrt{\sqrt{b^2-4 a c} (c e-b f)-2 a c f+b^2 f-b c e+2 c^2 d}}\right )}{\sqrt{2} \sqrt{b^2-4 a c} \sqrt{\sqrt{b^2-4 a c} (c e-b f)-2 a c f+b^2 f-b c e+2 c^2 d}}+\frac{\left (2 A c-B \left (\sqrt{b^2-4 a c}+b\right )\right ) \tanh ^{-1}\left (\frac{2 x \left (c e-f \left (\sqrt{b^2-4 a c}+b\right )\right )-e \left (\sqrt{b^2-4 a c}+b\right )+4 c d}{2 \sqrt{2} \sqrt{d+e x+f x^2} \sqrt{-\sqrt{b^2-4 a c} (c e-b f)-2 a c f+b^2 f-b c e+2 c^2 d}}\right )}{\sqrt{2} \sqrt{b^2-4 a c} \sqrt{-\sqrt{b^2-4 a c} (c e-b f)-2 a c f+b^2 f-b c e+2 c^2 d}} \]
[Out]
((b*B - 2*A*c - B*Sqrt[b^2 - 4*a*c])*ArcTanh[(4*c*d - (b - Sqrt[b^2 - 4*a*c])*e + 2*(c*e - (b - Sqrt[b^2 - 4*a
*c])*f)*x)/(2*Sqrt[2]*Sqrt[2*c^2*d - b*c*e + b^2*f - 2*a*c*f + Sqrt[b^2 - 4*a*c]*(c*e - b*f)]*Sqrt[d + e*x + f
*x^2])])/(Sqrt[2]*Sqrt[b^2 - 4*a*c]*Sqrt[2*c^2*d - b*c*e + b^2*f - 2*a*c*f + Sqrt[b^2 - 4*a*c]*(c*e - b*f)]) +
((2*A*c - B*(b + Sqrt[b^2 - 4*a*c]))*ArcTanh[(4*c*d - (b + Sqrt[b^2 - 4*a*c])*e + 2*(c*e - (b + Sqrt[b^2 - 4*
a*c])*f)*x)/(2*Sqrt[2]*Sqrt[2*c^2*d - b*c*e + b^2*f - 2*a*c*f - Sqrt[b^2 - 4*a*c]*(c*e - b*f)]*Sqrt[d + e*x +
f*x^2])])/(Sqrt[2]*Sqrt[b^2 - 4*a*c]*Sqrt[2*c^2*d - b*c*e + b^2*f - 2*a*c*f - Sqrt[b^2 - 4*a*c]*(c*e - b*f)])
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Rubi [A] time = 2.70211, antiderivative size = 416, normalized size of antiderivative = 1.,
number of steps used = 5, number of rules used = 3, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.094, Rules used =
{1032, 724, 206} \[ \frac{\left (-B \sqrt{b^2-4 a c}-2 A c+b B\right ) \tanh ^{-1}\left (\frac{2 x \left (c e-f \left (b-\sqrt{b^2-4 a c}\right )\right )-e \left (b-\sqrt{b^2-4 a c}\right )+4 c d}{2 \sqrt{2} \sqrt{d+e x+f x^2} \sqrt{\sqrt{b^2-4 a c} (c e-b f)-2 a c f+b^2 f-b c e+2 c^2 d}}\right )}{\sqrt{2} \sqrt{b^2-4 a c} \sqrt{\sqrt{b^2-4 a c} (c e-b f)-2 a c f+b^2 f-b c e+2 c^2 d}}+\frac{\left (2 A c-B \left (\sqrt{b^2-4 a c}+b\right )\right ) \tanh ^{-1}\left (\frac{2 x \left (c e-f \left (\sqrt{b^2-4 a c}+b\right )\right )-e \left (\sqrt{b^2-4 a c}+b\right )+4 c d}{2 \sqrt{2} \sqrt{d+e x+f x^2} \sqrt{-\sqrt{b^2-4 a c} (c e-b f)-2 a c f+b^2 f-b c e+2 c^2 d}}\right )}{\sqrt{2} \sqrt{b^2-4 a c} \sqrt{-\sqrt{b^2-4 a c} (c e-b f)-2 a c f+b^2 f-b c e+2 c^2 d}} \]
Antiderivative was successfully verified.
[In]
Int[(A + B*x)/((a + b*x + c*x^2)*Sqrt[d + e*x + f*x^2]),x]
[Out]
((b*B - 2*A*c - B*Sqrt[b^2 - 4*a*c])*ArcTanh[(4*c*d - (b - Sqrt[b^2 - 4*a*c])*e + 2*(c*e - (b - Sqrt[b^2 - 4*a
*c])*f)*x)/(2*Sqrt[2]*Sqrt[2*c^2*d - b*c*e + b^2*f - 2*a*c*f + Sqrt[b^2 - 4*a*c]*(c*e - b*f)]*Sqrt[d + e*x + f
*x^2])])/(Sqrt[2]*Sqrt[b^2 - 4*a*c]*Sqrt[2*c^2*d - b*c*e + b^2*f - 2*a*c*f + Sqrt[b^2 - 4*a*c]*(c*e - b*f)]) +
((2*A*c - B*(b + Sqrt[b^2 - 4*a*c]))*ArcTanh[(4*c*d - (b + Sqrt[b^2 - 4*a*c])*e + 2*(c*e - (b + Sqrt[b^2 - 4*
a*c])*f)*x)/(2*Sqrt[2]*Sqrt[2*c^2*d - b*c*e + b^2*f - 2*a*c*f - Sqrt[b^2 - 4*a*c]*(c*e - b*f)]*Sqrt[d + e*x +
f*x^2])])/(Sqrt[2]*Sqrt[b^2 - 4*a*c]*Sqrt[2*c^2*d - b*c*e + b^2*f - 2*a*c*f - Sqrt[b^2 - 4*a*c]*(c*e - b*f)])
Rule 1032
Int[((g_.) + (h_.)*(x_))/(((a_) + (b_.)*(x_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + (f_.)*(x_)^2]), x_Symbo
l] :> 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 + e*x + f*x^2])
, x], x] - Dist[(2*c*g - h*(b + q))/q, Int[1/((b + q + 2*c*x)*Sqrt[d + e*x + f*x^2]), x], x]] /; FreeQ[{a, b,
c, d, e, f, g, h}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0] && PosQ[b^2 - 4*a*c]
Rule 724
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 206
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])
Rubi steps
\begin{align*} \int \frac{A+B x}{\left (a+b x+c x^2\right ) \sqrt{d+e x+f x^2}} \, dx &=\frac{\left (2 A c-B \left (b-\sqrt{b^2-4 a c}\right )\right ) \int \frac{1}{\left (b-\sqrt{b^2-4 a c}+2 c x\right ) \sqrt{d+e x+f x^2}} \, dx}{\sqrt{b^2-4 a c}}-\frac{\left (2 A c-B \left (b+\sqrt{b^2-4 a c}\right )\right ) \int \frac{1}{\left (b+\sqrt{b^2-4 a c}+2 c x\right ) \sqrt{d+e x+f x^2}} \, dx}{\sqrt{b^2-4 a c}}\\ &=\frac{\left (2 \left (b B-2 A c-B \sqrt{b^2-4 a c}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{16 c^2 d-8 c \left (b-\sqrt{b^2-4 a c}\right ) e+4 \left (b-\sqrt{b^2-4 a c}\right )^2 f-x^2} \, dx,x,\frac{4 c d-\left (b-\sqrt{b^2-4 a c}\right ) e-\left (-2 c e+2 \left (b-\sqrt{b^2-4 a c}\right ) f\right ) x}{\sqrt{d+e x+f x^2}}\right )}{\sqrt{b^2-4 a c}}+\frac{\left (2 \left (2 A c-B \left (b+\sqrt{b^2-4 a c}\right )\right )\right ) \operatorname{Subst}\left (\int \frac{1}{16 c^2 d-8 c \left (b+\sqrt{b^2-4 a c}\right ) e+4 \left (b+\sqrt{b^2-4 a c}\right )^2 f-x^2} \, dx,x,\frac{4 c d-\left (b+\sqrt{b^2-4 a c}\right ) e-\left (-2 c e+2 \left (b+\sqrt{b^2-4 a c}\right ) f\right ) x}{\sqrt{d+e x+f x^2}}\right )}{\sqrt{b^2-4 a c}}\\ &=\frac{\left (b B-2 A c-B \sqrt{b^2-4 a c}\right ) \tanh ^{-1}\left (\frac{4 c d-\left (b-\sqrt{b^2-4 a c}\right ) e+2 \left (c e-\left (b-\sqrt{b^2-4 a c}\right ) f\right ) x}{2 \sqrt{2} \sqrt{2 c^2 d-b c e+b^2 f-2 a c f+\sqrt{b^2-4 a c} (c e-b f)} \sqrt{d+e x+f x^2}}\right )}{\sqrt{2} \sqrt{b^2-4 a c} \sqrt{2 c^2 d-b c e+b^2 f-2 a c f+\sqrt{b^2-4 a c} (c e-b f)}}+\frac{\left (2 A c-B \left (b+\sqrt{b^2-4 a c}\right )\right ) \tanh ^{-1}\left (\frac{4 c d-\left (b+\sqrt{b^2-4 a c}\right ) e+2 \left (c e-\left (b+\sqrt{b^2-4 a c}\right ) f\right ) x}{2 \sqrt{2} \sqrt{2 c^2 d-b c e+b^2 f-2 a c f-\sqrt{b^2-4 a c} (c e-b f)} \sqrt{d+e x+f x^2}}\right )}{\sqrt{2} \sqrt{b^2-4 a c} \sqrt{2 c^2 d-b c e+b^2 f-2 a c f-\sqrt{b^2-4 a c} (c e-b f)}}\\ \end{align*}
Mathematica [A] time = 4.23536, size = 393, normalized size = 0.94 \[ \frac{-\frac{\left (B \sqrt{b^2-4 a c}+2 A c-b B\right ) \tanh ^{-1}\left (\frac{\left (\sqrt{b^2-4 a c}-b\right ) (e+2 f x)+2 c (2 d+e x)}{2 \sqrt{2} \sqrt{d+x (e+f x)} \sqrt{c \left (e \sqrt{b^2-4 a c}-2 a f-b e\right )+b f \left (b-\sqrt{b^2-4 a c}\right )+2 c^2 d}}\right )}{\sqrt{c \left (e \sqrt{b^2-4 a c}-2 a f-b e\right )+b f \left (b-\sqrt{b^2-4 a c}\right )+2 c^2 d}}-\frac{\left (B \sqrt{b^2-4 a c}-2 A c+b B\right ) \tanh ^{-1}\left (\frac{2 c (2 d+e x)-\left (\sqrt{b^2-4 a c}+b\right ) (e+2 f x)}{2 \sqrt{d+x (e+f x)} \sqrt{-2 c \left (e \sqrt{b^2-4 a c}+2 a f+b e\right )+2 b f \left (\sqrt{b^2-4 a c}+b\right )+4 c^2 d}}\right )}{\sqrt{-c \left (e \sqrt{b^2-4 a c}+2 a f+b e\right )+b f \left (\sqrt{b^2-4 a c}+b\right )+2 c^2 d}}}{\sqrt{2} \sqrt{b^2-4 a c}} \]
Antiderivative was successfully verified.
[In]
Integrate[(A + B*x)/((a + b*x + c*x^2)*Sqrt[d + e*x + f*x^2]),x]
[Out]
(-(((-(b*B) + 2*A*c + B*Sqrt[b^2 - 4*a*c])*ArcTanh[(2*c*(2*d + e*x) + (-b + Sqrt[b^2 - 4*a*c])*(e + 2*f*x))/(2
*Sqrt[2]*Sqrt[2*c^2*d + b*(b - Sqrt[b^2 - 4*a*c])*f + c*(-(b*e) + Sqrt[b^2 - 4*a*c]*e - 2*a*f)]*Sqrt[d + x*(e
+ f*x)])])/Sqrt[2*c^2*d + b*(b - Sqrt[b^2 - 4*a*c])*f + c*(-(b*e) + Sqrt[b^2 - 4*a*c]*e - 2*a*f)]) - ((b*B - 2
*A*c + B*Sqrt[b^2 - 4*a*c])*ArcTanh[(2*c*(2*d + e*x) - (b + Sqrt[b^2 - 4*a*c])*(e + 2*f*x))/(2*Sqrt[4*c^2*d +
2*b*(b + Sqrt[b^2 - 4*a*c])*f - 2*c*(b*e + Sqrt[b^2 - 4*a*c]*e + 2*a*f)]*Sqrt[d + x*(e + f*x)])])/Sqrt[2*c^2*d
+ b*(b + Sqrt[b^2 - 4*a*c])*f - c*(b*e + Sqrt[b^2 - 4*a*c]*e + 2*a*f)])/(Sqrt[2]*Sqrt[b^2 - 4*a*c])
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Maple [B] time = 0.431, size = 2269, normalized size = 5.5 \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)/(c*x^2+b*x+a)/(f*x^2+e*x+d)^(1/2),x)
[Out]
-2/(-4*a*c+b^2)^(1/2)/(-2*(b*f*(-4*a*c+b^2)^(1/2)-(-4*a*c+b^2)^(1/2)*c*e+2*a*c*f-b^2*f+b*c*e-2*c^2*d)/c^2)^(1/
2)*ln((-(b*f*(-4*a*c+b^2)^(1/2)-(-4*a*c+b^2)^(1/2)*c*e+2*a*c*f-b^2*f+b*c*e-2*c^2*d)/c^2-(-f*(-4*a*c+b^2)^(1/2)
+b*f-c*e)/c*(x-1/2/c*(-b+(-4*a*c+b^2)^(1/2)))+1/2*(-2*(b*f*(-4*a*c+b^2)^(1/2)-(-4*a*c+b^2)^(1/2)*c*e+2*a*c*f-b
^2*f+b*c*e-2*c^2*d)/c^2)^(1/2)*(4*(x-1/2/c*(-b+(-4*a*c+b^2)^(1/2)))^2*f-4*(-f*(-4*a*c+b^2)^(1/2)+b*f-c*e)/c*(x
-1/2/c*(-b+(-4*a*c+b^2)^(1/2)))-2*(b*f*(-4*a*c+b^2)^(1/2)-(-4*a*c+b^2)^(1/2)*c*e+2*a*c*f-b^2*f+b*c*e-2*c^2*d)/
c^2)^(1/2))/(x-1/2/c*(-b+(-4*a*c+b^2)^(1/2))))*A-1/c/(-2*(b*f*(-4*a*c+b^2)^(1/2)-(-4*a*c+b^2)^(1/2)*c*e+2*a*c*
f-b^2*f+b*c*e-2*c^2*d)/c^2)^(1/2)*ln((-(b*f*(-4*a*c+b^2)^(1/2)-(-4*a*c+b^2)^(1/2)*c*e+2*a*c*f-b^2*f+b*c*e-2*c^
2*d)/c^2-(-f*(-4*a*c+b^2)^(1/2)+b*f-c*e)/c*(x-1/2/c*(-b+(-4*a*c+b^2)^(1/2)))+1/2*(-2*(b*f*(-4*a*c+b^2)^(1/2)-(
-4*a*c+b^2)^(1/2)*c*e+2*a*c*f-b^2*f+b*c*e-2*c^2*d)/c^2)^(1/2)*(4*(x-1/2/c*(-b+(-4*a*c+b^2)^(1/2)))^2*f-4*(-f*(
-4*a*c+b^2)^(1/2)+b*f-c*e)/c*(x-1/2/c*(-b+(-4*a*c+b^2)^(1/2)))-2*(b*f*(-4*a*c+b^2)^(1/2)-(-4*a*c+b^2)^(1/2)*c*
e+2*a*c*f-b^2*f+b*c*e-2*c^2*d)/c^2)^(1/2))/(x-1/2/c*(-b+(-4*a*c+b^2)^(1/2))))*B+1/(-4*a*c+b^2)^(1/2)/c/(-2*(b*
f*(-4*a*c+b^2)^(1/2)-(-4*a*c+b^2)^(1/2)*c*e+2*a*c*f-b^2*f+b*c*e-2*c^2*d)/c^2)^(1/2)*ln((-(b*f*(-4*a*c+b^2)^(1/
2)-(-4*a*c+b^2)^(1/2)*c*e+2*a*c*f-b^2*f+b*c*e-2*c^2*d)/c^2-(-f*(-4*a*c+b^2)^(1/2)+b*f-c*e)/c*(x-1/2/c*(-b+(-4*
a*c+b^2)^(1/2)))+1/2*(-2*(b*f*(-4*a*c+b^2)^(1/2)-(-4*a*c+b^2)^(1/2)*c*e+2*a*c*f-b^2*f+b*c*e-2*c^2*d)/c^2)^(1/2
)*(4*(x-1/2/c*(-b+(-4*a*c+b^2)^(1/2)))^2*f-4*(-f*(-4*a*c+b^2)^(1/2)+b*f-c*e)/c*(x-1/2/c*(-b+(-4*a*c+b^2)^(1/2)
))-2*(b*f*(-4*a*c+b^2)^(1/2)-(-4*a*c+b^2)^(1/2)*c*e+2*a*c*f-b^2*f+b*c*e-2*c^2*d)/c^2)^(1/2))/(x-1/2/c*(-b+(-4*
a*c+b^2)^(1/2))))*B*b+2/(-4*a*c+b^2)^(1/2)/(-2*(-b*f*(-4*a*c+b^2)^(1/2)+(-4*a*c+b^2)^(1/2)*c*e+2*a*c*f-b^2*f+b
*c*e-2*c^2*d)/c^2)^(1/2)*ln((-(-b*f*(-4*a*c+b^2)^(1/2)+(-4*a*c+b^2)^(1/2)*c*e+2*a*c*f-b^2*f+b*c*e-2*c^2*d)/c^2
-(f*(-4*a*c+b^2)^(1/2)+b*f-c*e)/c*(x+1/2*(b+(-4*a*c+b^2)^(1/2))/c)+1/2*(-2*(-b*f*(-4*a*c+b^2)^(1/2)+(-4*a*c+b^
2)^(1/2)*c*e+2*a*c*f-b^2*f+b*c*e-2*c^2*d)/c^2)^(1/2)*(4*(x+1/2*(b+(-4*a*c+b^2)^(1/2))/c)^2*f-4*(f*(-4*a*c+b^2)
^(1/2)+b*f-c*e)/c*(x+1/2*(b+(-4*a*c+b^2)^(1/2))/c)-2*(-b*f*(-4*a*c+b^2)^(1/2)+(-4*a*c+b^2)^(1/2)*c*e+2*a*c*f-b
^2*f+b*c*e-2*c^2*d)/c^2)^(1/2))/(x+1/2*(b+(-4*a*c+b^2)^(1/2))/c))*A-1/c/(-2*(-b*f*(-4*a*c+b^2)^(1/2)+(-4*a*c+b
^2)^(1/2)*c*e+2*a*c*f-b^2*f+b*c*e-2*c^2*d)/c^2)^(1/2)*ln((-(-b*f*(-4*a*c+b^2)^(1/2)+(-4*a*c+b^2)^(1/2)*c*e+2*a
*c*f-b^2*f+b*c*e-2*c^2*d)/c^2-(f*(-4*a*c+b^2)^(1/2)+b*f-c*e)/c*(x+1/2*(b+(-4*a*c+b^2)^(1/2))/c)+1/2*(-2*(-b*f*
(-4*a*c+b^2)^(1/2)+(-4*a*c+b^2)^(1/2)*c*e+2*a*c*f-b^2*f+b*c*e-2*c^2*d)/c^2)^(1/2)*(4*(x+1/2*(b+(-4*a*c+b^2)^(1
/2))/c)^2*f-4*(f*(-4*a*c+b^2)^(1/2)+b*f-c*e)/c*(x+1/2*(b+(-4*a*c+b^2)^(1/2))/c)-2*(-b*f*(-4*a*c+b^2)^(1/2)+(-4
*a*c+b^2)^(1/2)*c*e+2*a*c*f-b^2*f+b*c*e-2*c^2*d)/c^2)^(1/2))/(x+1/2*(b+(-4*a*c+b^2)^(1/2))/c))*B-1/(-4*a*c+b^2
)^(1/2)/c/(-2*(-b*f*(-4*a*c+b^2)^(1/2)+(-4*a*c+b^2)^(1/2)*c*e+2*a*c*f-b^2*f+b*c*e-2*c^2*d)/c^2)^(1/2)*ln((-(-b
*f*(-4*a*c+b^2)^(1/2)+(-4*a*c+b^2)^(1/2)*c*e+2*a*c*f-b^2*f+b*c*e-2*c^2*d)/c^2-(f*(-4*a*c+b^2)^(1/2)+b*f-c*e)/c
*(x+1/2*(b+(-4*a*c+b^2)^(1/2))/c)+1/2*(-2*(-b*f*(-4*a*c+b^2)^(1/2)+(-4*a*c+b^2)^(1/2)*c*e+2*a*c*f-b^2*f+b*c*e-
2*c^2*d)/c^2)^(1/2)*(4*(x+1/2*(b+(-4*a*c+b^2)^(1/2))/c)^2*f-4*(f*(-4*a*c+b^2)^(1/2)+b*f-c*e)/c*(x+1/2*(b+(-4*a
*c+b^2)^(1/2))/c)-2*(-b*f*(-4*a*c+b^2)^(1/2)+(-4*a*c+b^2)^(1/2)*c*e+2*a*c*f-b^2*f+b*c*e-2*c^2*d)/c^2)^(1/2))/(
x+1/2*(b+(-4*a*c+b^2)^(1/2))/c))*B*b
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((B*x+A)/(c*x^2+b*x+a)/(f*x^2+e*x+d)^(1/2),x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [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)/(f*x^2+e*x+d)^(1/2),x, algorithm="fricas")
[Out]
Timed out
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{A + B x}{\left (a + b x + c x^{2}\right ) \sqrt{d + e x + f x^{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((B*x+A)/(c*x**2+b*x+a)/(f*x**2+e*x+d)**(1/2),x)
[Out]
Integral((A + B*x)/((a + b*x + c*x**2)*sqrt(d + e*x + f*x**2)), x)
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((B*x+A)/(c*x^2+b*x+a)/(f*x^2+e*x+d)^(1/2),x, algorithm="giac")
[Out]
Exception raised: TypeError