∖int A+Bx_________________ ∖left(a+bx+cx2∖right)∘ _______ d+ex+fx2 ∖,dx

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3.21 \(\int \frac{A+B x}{\left (a+b x+c x^2\right ) \sqrt{d+e x+f x^2}} \, dx\)

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]*Sq

rt[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 = 6.3233, 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 \[ \frac{\left (-B \sqrt{b^2-4 a c}-2 A c+b B\right ) \tanh ^{-1}\left (\frac{2 x \left (c e-f \left (b-\sqrt{b^2-4 a c}\right )\right )-e \left (b-\sqrt{b^2-4 a c}\right )+4 c d}{2 \sqrt{2} \sqrt{d+e x+f x^2} \sqrt{\sqrt{b^2-4 a c} (c e-b f)-2 a c f+b^2 f-b c e+2 c^2 d}}\right )}{\sqrt{2} \sqrt{b^2-4 a c} \sqrt{\sqrt{b^2-4 a c} (c e-b f)-2 a c f+b^2 f-b c e+2 c^2 d}}+\frac{\left (2 A c-B \left (\sqrt{b^2-4 a c}+b\right )\right ) \tanh ^{-1}\left (\frac{2 x \left (c e-f \left (\sqrt{b^2-4 a c}+b\right )\right )-e \left (\sqrt{b^2-4 a c}+b\right )+4 c d}{2 \sqrt{2} \sqrt{d+e x+f x^2} \sqrt{-\sqrt{b^2-4 a c} (c e-b f)-2 a c f+b^2 f-b c e+2 c^2 d}}\right )}{\sqrt{2} \sqrt{b^2-4 a c} \sqrt{-\sqrt{b^2-4 a c} (c e-b f)-2 a c f+b^2 f-b c e+2 c^2 d}} \]

Warning: Unable to verify antiderivative.

[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]*Sq

rt[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 in Sympy [A] time = 141.626, size = 394, normalized size = 0.95 \[ - \frac{\sqrt{2} \left (2 A c - B \left (b - \sqrt{- 4 a c + b^{2}}\right )\right ) \operatorname{atanh}{\left (\frac{\sqrt{2} \left (- b e + 4 c d + e \sqrt{- 4 a c + b^{2}} + x \left (2 c e - 2 f \left (b - \sqrt{- 4 a c + b^{2}}\right )\right )\right )}{4 \sqrt{d + e x + f x^{2}} \sqrt{- 2 a c f + b^{2} f - b c e + 2 c^{2} d - \sqrt{- 4 a c + b^{2}} \left (b f - c e\right )}} \right )}}{2 \sqrt{- 4 a c + b^{2}} \sqrt{- 2 a c f + b^{2} f - b c e + 2 c^{2} d - \sqrt{- 4 a c + b^{2}} \left (b f - c e\right )}} + \frac{\sqrt{2} \left (2 A c - B \left (b + \sqrt{- 4 a c + b^{2}}\right )\right ) \operatorname{atanh}{\left (\frac{\sqrt{2} \left (4 c d - e \left (b + \sqrt{- 4 a c + b^{2}}\right ) + x \left (2 c e - 2 f \left (b + \sqrt{- 4 a c + b^{2}}\right )\right )\right )}{4 \sqrt{d + e x + f x^{2}} \sqrt{- 2 a c f + b^{2} f - b c e + 2 c^{2} d - \sqrt{- 4 a c + b^{2}} \left (- b f + c e\right )}} \right )}}{2 \sqrt{- 4 a c + b^{2}} \sqrt{- 2 a c f + b^{2} f - b c e + 2 c^{2} d - \sqrt{- 4 a c + b^{2}} \left (- b f + c e\right )}} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In] rubi_integrate((B*x+A)/(c*x**2+b*x+a)/(f*x**2+e*x+d)**(1/2),x)

[Out]

-sqrt(2)*(2*A*c - B*(b - sqrt(-4*a*c + b**2)))*atanh(sqrt(2)*(-b*e + 4*c*d + e*s

qrt(-4*a*c + b**2) + x*(2*c*e - 2*f*(b - sqrt(-4*a*c + b**2))))/(4*sqrt(d + e*x

+ f*x**2)*sqrt(-2*a*c*f + b**2*f - b*c*e + 2*c**2*d - sqrt(-4*a*c + b**2)*(b*f -

c*e))))/(2*sqrt(-4*a*c + b**2)*sqrt(-2*a*c*f + b**2*f - b*c*e + 2*c**2*d - sqrt

(-4*a*c + b**2)*(b*f - c*e))) + sqrt(2)*(2*A*c - B*(b + sqrt(-4*a*c + b**2)))*at

anh(sqrt(2)*(4*c*d - e*(b + sqrt(-4*a*c + b**2)) + x*(2*c*e - 2*f*(b + sqrt(-4*a

*c + b**2))))/(4*sqrt(d + e*x + f*x**2)*sqrt(-2*a*c*f + b**2*f - b*c*e + 2*c**2*

d - sqrt(-4*a*c + b**2)*(-b*f + c*e))))/(2*sqrt(-4*a*c + b**2)*sqrt(-2*a*c*f + b

**2*f - b*c*e + 2*c**2*d - sqrt(-4*a*c + b**2)*(-b*f + c*e)))

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Mathematica [A] time = 1.7273, size = 696, normalized size = 1.67 \[ \frac{\frac{\left (B \sqrt{b^2-4 a c}+2 A c-b B\right ) \log \left (\sqrt{b^2-4 a c}-b-2 c x\right )}{\sqrt{c \left (e \sqrt{b^2-4 a c}-2 a f-b e\right )+b f \left (b-\sqrt{b^2-4 a c}\right )+2 c^2 d}}+\frac{\left (B \sqrt{b^2-4 a c}-2 A c+b B\right ) \log \left (\sqrt{b^2-4 a c}+b+2 c x\right )}{\sqrt{-c \left (e \sqrt{b^2-4 a c}+2 a f+b e\right )+b f \left (\sqrt{b^2-4 a c}+b\right )+2 c^2 d}}-\frac{\left (B \sqrt{b^2-4 a c}-2 A c+b B\right ) \log \left (2 \sqrt{2} \sqrt{b^2-4 a c} \sqrt{d+x (e+f x)} \sqrt{-c \left (e \sqrt{b^2-4 a c}+2 a f+b e\right )+b f \left (\sqrt{b^2-4 a c}+b\right )+2 c^2 d}+2 c \left (2 d \sqrt{b^2-4 a c}+e x \sqrt{b^2-4 a c}+2 a e+4 a f x\right )-b \sqrt{b^2-4 a c} (e+2 f x)+b^2 (-(e+2 f x))\right )}{\sqrt{-c \left (e \sqrt{b^2-4 a c}+2 a f+b e\right )+b f \left (\sqrt{b^2-4 a c}+b\right )+2 c^2 d}}-\frac{\left (B \sqrt{b^2-4 a c}+2 A c-b B\right ) \log \left (2 \sqrt{2} \sqrt{b^2-4 a c} \sqrt{d+x (e+f x)} \sqrt{c \left (e \sqrt{b^2-4 a c}-2 a f-b e\right )+b f \left (b-\sqrt{b^2-4 a c}\right )+2 c^2 d}+2 c \left (2 d \sqrt{b^2-4 a c}+e x \sqrt{b^2-4 a c}-2 a (e+2 f x)\right )-b \sqrt{b^2-4 a c} (e+2 f x)+b^2 (e+2 f x)\right )}{\sqrt{c \left (e \sqrt{b^2-4 a c}-2 a f-b e\right )+b f \left (b-\sqrt{b^2-4 a c}\right )+2 c^2 d}}}{\sqrt{2} \sqrt{b^2-4 a c}} \]

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])*Log[-b + Sqrt[b^2 - 4*a*c] - 2*c*x])/Sq

rt[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])*Log[b + Sqrt[b^2 - 4*a*c] + 2*c*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])*Log[-(b^2*(e + 2*f*x)) - b*Sqrt[b^2 -

4*a*c]*(e + 2*f*x) + 2*c*(2*Sqrt[b^2 - 4*a*c]*d + 2*a*e + Sqrt[b^2 - 4*a*c]*e*x

+ 4*a*f*x) + 2*Sqrt[2]*Sqrt[b^2 - 4*a*c]*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])*Log[b^2*(e + 2*f*x) - b*Sqrt[b^2 - 4*a*c]*(e

+ 2*f*x) + 2*Sqrt[2]*Sqrt[b^2 - 4*a*c]*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)] + 2*c*(2*Sq

rt[b^2 - 4*a*c]*d + Sqrt[b^2 - 4*a*c]*e*x - 2*a*(e + 2*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 = 0.079, size = 2269, normalized size = 5.5 \[ \text{result too large to display} \]

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*(

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-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

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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In] integrate((B*x + A)/((c*x^2 + b*x + a)*sqrt(f*x^2 + e*x + d)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In] integrate((B*x + A)/((c*x^2 + b*x + a)*sqrt(f*x^2 + e*x + d)),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{A + B x}{\left (a + b x + c x^{2}\right ) \sqrt{d + e x + f x^{2}}}\, dx \]

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/XCAS [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: RuntimeError} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In] integrate((B*x + A)/((c*x^2 + b*x + a)*sqrt(f*x^2 + e*x + d)),x, algorithm="giac")

[Out]

Exception raised: RuntimeError

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