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

Optimal. Leaf size=617 \[ \frac{\left (2 f (A f (c d-a f)-B d (c e-b f))-\left (e-\sqrt{e^2-4 d f}\right ) \left (B \left (f (b e-a f)-c \left (e^2-d f\right )\right )+A f (c e-b f)\right )\right ) \tanh ^{-1}\left (\frac{4 a f+2 x \left (b f-c \left (e-\sqrt{e^2-4 d f}\right )\right )-b \left (e-\sqrt{e^2-4 d f}\right )}{2 \sqrt{2} \sqrt{a+b x+c x^2} \sqrt{2 a f^2-\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}\right )}{\sqrt{2} f^2 \sqrt{e^2-4 d f} \sqrt{2 a f^2-\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}-\frac{\left (2 f (A f (c d-a f)-B d (c e-b f))-\left (\sqrt{e^2-4 d f}+e\right ) \left (B \left (f (b e-a f)-c \left (e^2-d f\right )\right )+A f (c e-b f)\right )\right ) \tanh ^{-1}\left (\frac{4 a f+2 x \left (b f-c \left (\sqrt{e^2-4 d f}+e\right )\right )-b \left (\sqrt{e^2-4 d f}+e\right )}{2 \sqrt{2} \sqrt{a+b x+c x^2} \sqrt{2 a f^2+\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}\right )}{\sqrt{2} f^2 \sqrt{e^2-4 d f} \sqrt{2 a f^2+\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}-\frac{\tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right ) (-2 A c f-b B f+2 B c e)}{2 \sqrt{c} f^2}+\frac{B \sqrt{a+b x+c x^2}}{f} \]

[Out]

(B*Sqrt[a + b*x + c*x^2])/f - ((2*B*c*e - b*B*f - 2*A*c*f)*ArcTanh[(b + 2*c*x)/(
2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(2*Sqrt[c]*f^2) + ((2*f*(A*f*(c*d - a*f) - B*
d*(c*e - b*f)) - (e - Sqrt[e^2 - 4*d*f])*(A*f*(c*e - b*f) + B*(f*(b*e - a*f) - c
*(e^2 - d*f))))*ArcTanh[(4*a*f - b*(e - Sqrt[e^2 - 4*d*f]) + 2*(b*f - c*(e - Sqr
t[e^2 - 4*d*f]))*x)/(2*Sqrt[2]*Sqrt[c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 - (c*e - b
*f)*Sqrt[e^2 - 4*d*f]]*Sqrt[a + b*x + c*x^2])])/(Sqrt[2]*f^2*Sqrt[e^2 - 4*d*f]*S
qrt[c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 - (c*e - b*f)*Sqrt[e^2 - 4*d*f]]) - ((2*f*
(A*f*(c*d - a*f) - B*d*(c*e - b*f)) - (e + Sqrt[e^2 - 4*d*f])*(A*f*(c*e - b*f) +
B*(f*(b*e - a*f) - c*(e^2 - d*f))))*ArcTanh[(4*a*f - b*(e + Sqrt[e^2 - 4*d*f])
+ 2*(b*f - c*(e + Sqrt[e^2 - 4*d*f]))*x)/(2*Sqrt[2]*Sqrt[c*e^2 - 2*c*d*f - b*e*f
+ 2*a*f^2 + (c*e - b*f)*Sqrt[e^2 - 4*d*f]]*Sqrt[a + b*x + c*x^2])])/(Sqrt[2]*f^
2*Sqrt[e^2 - 4*d*f]*Sqrt[c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 + (c*e - b*f)*Sqrt[e^
2 - 4*d*f]])

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Rubi [A]  time = 18.5359, antiderivative size = 615, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188 \[ \frac{\left (2 f (A f (c d-a f)-B d (c e-b f))-\left (e-\sqrt{e^2-4 d f}\right ) \left (A f (c e-b f)-B \left (a f^2-b e f-c d f+c e^2\right )\right )\right ) \tanh ^{-1}\left (\frac{4 a f+2 x \left (b f-c \left (e-\sqrt{e^2-4 d f}\right )\right )-b \left (e-\sqrt{e^2-4 d f}\right )}{2 \sqrt{2} \sqrt{a+b x+c x^2} \sqrt{2 a f^2-\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}\right )}{\sqrt{2} f^2 \sqrt{e^2-4 d f} \sqrt{2 a f^2-\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}-\frac{\left (2 f (A f (c d-a f)-B d (c e-b f))-\left (\sqrt{e^2-4 d f}+e\right ) \left (B f (b e-a f)+A f (c e-b f)-B c \left (e^2-d f\right )\right )\right ) \tanh ^{-1}\left (\frac{4 a f+2 x \left (b f-c \left (\sqrt{e^2-4 d f}+e\right )\right )-b \left (\sqrt{e^2-4 d f}+e\right )}{2 \sqrt{2} \sqrt{a+b x+c x^2} \sqrt{2 a f^2+\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}\right )}{\sqrt{2} f^2 \sqrt{e^2-4 d f} \sqrt{2 a f^2+\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}-\frac{\tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right ) (-2 A c f-b B f+2 B c e)}{2 \sqrt{c} f^2}+\frac{B \sqrt{a+b x+c x^2}}{f} \]

Warning: Unable to verify antiderivative.

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

[Out]

(B*Sqrt[a + b*x + c*x^2])/f - ((2*B*c*e - b*B*f - 2*A*c*f)*ArcTanh[(b + 2*c*x)/(
2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(2*Sqrt[c]*f^2) + ((2*f*(A*f*(c*d - a*f) - B*
d*(c*e - b*f)) - (e - Sqrt[e^2 - 4*d*f])*(A*f*(c*e - b*f) - B*(c*e^2 - c*d*f - b
*e*f + a*f^2)))*ArcTanh[(4*a*f - b*(e - Sqrt[e^2 - 4*d*f]) + 2*(b*f - c*(e - Sqr
t[e^2 - 4*d*f]))*x)/(2*Sqrt[2]*Sqrt[c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 - (c*e - b
*f)*Sqrt[e^2 - 4*d*f]]*Sqrt[a + b*x + c*x^2])])/(Sqrt[2]*f^2*Sqrt[e^2 - 4*d*f]*S
qrt[c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 - (c*e - b*f)*Sqrt[e^2 - 4*d*f]]) - ((2*f*
(A*f*(c*d - a*f) - B*d*(c*e - b*f)) - (e + Sqrt[e^2 - 4*d*f])*(B*f*(b*e - a*f) +
A*f*(c*e - b*f) - B*c*(e^2 - d*f)))*ArcTanh[(4*a*f - b*(e + Sqrt[e^2 - 4*d*f])
+ 2*(b*f - c*(e + Sqrt[e^2 - 4*d*f]))*x)/(2*Sqrt[2]*Sqrt[c*e^2 - 2*c*d*f - b*e*f
+ 2*a*f^2 + (c*e - b*f)*Sqrt[e^2 - 4*d*f]]*Sqrt[a + b*x + c*x^2])])/(Sqrt[2]*f^
2*Sqrt[e^2 - 4*d*f]*Sqrt[c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 + (c*e - b*f)*Sqrt[e^
2 - 4*d*f]])

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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Mathematica [B]  time = 6.24853, size = 1629, normalized size = 2.64 \[ \frac{\sqrt{a+x (b+c x)} B}{f}+\frac{\left (-B c e^3+b B f e^2+A c f e^2+B c \sqrt{e^2-4 d f} e^2-A b f^2 e-a B f^2 e+3 B c d f e-b B f \sqrt{e^2-4 d f} e-A c f \sqrt{e^2-4 d f} e+2 a A f^3-2 b B d f^2-2 A c d f^2+A b f^2 \sqrt{e^2-4 d f}+a B f^2 \sqrt{e^2-4 d f}-B c d f \sqrt{e^2-4 d f}\right ) \sqrt{a+x (b+c x)} \log \left (-e-2 f x+\sqrt{e^2-4 d f}\right )}{\sqrt{2} f^2 \sqrt{e^2-4 d f} \sqrt{c e^2-b f e-c \sqrt{e^2-4 d f} e+2 a f^2-2 c d f+b f \sqrt{e^2-4 d f}} \sqrt{c x^2+b x+a}}+\frac{\left (B c e^3-b B f e^2-A c f e^2+B c \sqrt{e^2-4 d f} e^2+A b f^2 e+a B f^2 e-3 B c d f e-b B f \sqrt{e^2-4 d f} e-A c f \sqrt{e^2-4 d f} e-2 a A f^3+2 b B d f^2+2 A c d f^2+A b f^2 \sqrt{e^2-4 d f}+a B f^2 \sqrt{e^2-4 d f}-B c d f \sqrt{e^2-4 d f}\right ) \sqrt{a+x (b+c x)} \log \left (e+2 f x+\sqrt{e^2-4 d f}\right )}{\sqrt{2} f^2 \sqrt{e^2-4 d f} \sqrt{c e^2-b f e+c \sqrt{e^2-4 d f} e+2 a f^2-2 c d f-b f \sqrt{e^2-4 d f}} \sqrt{c x^2+b x+a}}-\frac{(2 B c e-b B f-2 A c f) \sqrt{a+x (b+c x)} \log \left (b+2 c x+2 \sqrt{c} \sqrt{c x^2+b x+a}\right )}{2 \sqrt{c} f^2 \sqrt{c x^2+b x+a}}-\frac{\left (B c e^3-b B f e^2-A c f e^2+B c \sqrt{e^2-4 d f} e^2+A b f^2 e+a B f^2 e-3 B c d f e-b B f \sqrt{e^2-4 d f} e-A c f \sqrt{e^2-4 d f} e-2 a A f^3+2 b B d f^2+2 A c d f^2+A b f^2 \sqrt{e^2-4 d f}+a B f^2 \sqrt{e^2-4 d f}-B c d f \sqrt{e^2-4 d f}\right ) \sqrt{a+x (b+c x)} \log \left (-b e^2-2 c x e^2-2 c \sqrt{e^2-4 d f} x e-b \sqrt{e^2-4 d f} e+4 b d f+8 c d f x+2 b f \sqrt{e^2-4 d f} x+4 a f \sqrt{e^2-4 d f}+2 \sqrt{2} \sqrt{e^2-4 d f} \sqrt{c e^2-b f e+c \sqrt{e^2-4 d f} e+2 a f^2-2 c d f-b f \sqrt{e^2-4 d f}} \sqrt{c x^2+b x+a}\right )}{\sqrt{2} f^2 \sqrt{e^2-4 d f} \sqrt{c e^2-b f e+c \sqrt{e^2-4 d f} e+2 a f^2-2 c d f-b f \sqrt{e^2-4 d f}} \sqrt{c x^2+b x+a}}-\frac{\left (-B c e^3+b B f e^2+A c f e^2+B c \sqrt{e^2-4 d f} e^2-A b f^2 e-a B f^2 e+3 B c d f e-b B f \sqrt{e^2-4 d f} e-A c f \sqrt{e^2-4 d f} e+2 a A f^3-2 b B d f^2-2 A c d f^2+A b f^2 \sqrt{e^2-4 d f}+a B f^2 \sqrt{e^2-4 d f}-B c d f \sqrt{e^2-4 d f}\right ) \sqrt{a+x (b+c x)} \log \left (b e^2+2 c x e^2-2 c \sqrt{e^2-4 d f} x e-b \sqrt{e^2-4 d f} e-4 b d f-8 c d f x+2 b f \sqrt{e^2-4 d f} x+4 a f \sqrt{e^2-4 d f}+2 \sqrt{2} \sqrt{e^2-4 d f} \sqrt{c e^2-b f e-c \sqrt{e^2-4 d f} e+2 a f^2-2 c d f+b f \sqrt{e^2-4 d f}} \sqrt{c x^2+b x+a}\right )}{\sqrt{2} f^2 \sqrt{e^2-4 d f} \sqrt{c e^2-b f e-c \sqrt{e^2-4 d f} e+2 a f^2-2 c d f+b f \sqrt{e^2-4 d f}} \sqrt{c x^2+b x+a}} \]

Antiderivative was successfully verified.

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

[Out]

(B*Sqrt[a + x*(b + c*x)])/f + ((-(B*c*e^3) + 3*B*c*d*e*f + b*B*e^2*f + A*c*e^2*f
- 2*b*B*d*f^2 - 2*A*c*d*f^2 - A*b*e*f^2 - a*B*e*f^2 + 2*a*A*f^3 + B*c*e^2*Sqrt[
e^2 - 4*d*f] - B*c*d*f*Sqrt[e^2 - 4*d*f] - b*B*e*f*Sqrt[e^2 - 4*d*f] - A*c*e*f*S
qrt[e^2 - 4*d*f] + A*b*f^2*Sqrt[e^2 - 4*d*f] + a*B*f^2*Sqrt[e^2 - 4*d*f])*Sqrt[a
+ x*(b + c*x)]*Log[-e + Sqrt[e^2 - 4*d*f] - 2*f*x])/(Sqrt[2]*f^2*Sqrt[e^2 - 4*d
*f]*Sqrt[c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 - c*e*Sqrt[e^2 - 4*d*f] + b*f*Sqrt[e^
2 - 4*d*f]]*Sqrt[a + b*x + c*x^2]) + ((B*c*e^3 - 3*B*c*d*e*f - b*B*e^2*f - A*c*e
^2*f + 2*b*B*d*f^2 + 2*A*c*d*f^2 + A*b*e*f^2 + a*B*e*f^2 - 2*a*A*f^3 + B*c*e^2*S
qrt[e^2 - 4*d*f] - B*c*d*f*Sqrt[e^2 - 4*d*f] - b*B*e*f*Sqrt[e^2 - 4*d*f] - A*c*e
*f*Sqrt[e^2 - 4*d*f] + A*b*f^2*Sqrt[e^2 - 4*d*f] + a*B*f^2*Sqrt[e^2 - 4*d*f])*Sq
rt[a + x*(b + c*x)]*Log[e + Sqrt[e^2 - 4*d*f] + 2*f*x])/(Sqrt[2]*f^2*Sqrt[e^2 -
4*d*f]*Sqrt[c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 + c*e*Sqrt[e^2 - 4*d*f] - b*f*Sqrt
[e^2 - 4*d*f]]*Sqrt[a + b*x + c*x^2]) - ((2*B*c*e - b*B*f - 2*A*c*f)*Sqrt[a + x*
(b + c*x)]*Log[b + 2*c*x + 2*Sqrt[c]*Sqrt[a + b*x + c*x^2]])/(2*Sqrt[c]*f^2*Sqrt
[a + b*x + c*x^2]) - ((B*c*e^3 - 3*B*c*d*e*f - b*B*e^2*f - A*c*e^2*f + 2*b*B*d*f
^2 + 2*A*c*d*f^2 + A*b*e*f^2 + a*B*e*f^2 - 2*a*A*f^3 + B*c*e^2*Sqrt[e^2 - 4*d*f]
- B*c*d*f*Sqrt[e^2 - 4*d*f] - b*B*e*f*Sqrt[e^2 - 4*d*f] - A*c*e*f*Sqrt[e^2 - 4*
d*f] + A*b*f^2*Sqrt[e^2 - 4*d*f] + a*B*f^2*Sqrt[e^2 - 4*d*f])*Sqrt[a + x*(b + c*
x)]*Log[-(b*e^2) + 4*b*d*f - b*e*Sqrt[e^2 - 4*d*f] + 4*a*f*Sqrt[e^2 - 4*d*f] - 2
*c*e^2*x + 8*c*d*f*x - 2*c*e*Sqrt[e^2 - 4*d*f]*x + 2*b*f*Sqrt[e^2 - 4*d*f]*x + 2
*Sqrt[2]*Sqrt[e^2 - 4*d*f]*Sqrt[c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 + c*e*Sqrt[e^2
- 4*d*f] - b*f*Sqrt[e^2 - 4*d*f]]*Sqrt[a + b*x + c*x^2]])/(Sqrt[2]*f^2*Sqrt[e^2
- 4*d*f]*Sqrt[c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 + c*e*Sqrt[e^2 - 4*d*f] - b*f*S
qrt[e^2 - 4*d*f]]*Sqrt[a + b*x + c*x^2]) - ((-(B*c*e^3) + 3*B*c*d*e*f + b*B*e^2*
f + A*c*e^2*f - 2*b*B*d*f^2 - 2*A*c*d*f^2 - A*b*e*f^2 - a*B*e*f^2 + 2*a*A*f^3 +
B*c*e^2*Sqrt[e^2 - 4*d*f] - B*c*d*f*Sqrt[e^2 - 4*d*f] - b*B*e*f*Sqrt[e^2 - 4*d*f
] - A*c*e*f*Sqrt[e^2 - 4*d*f] + A*b*f^2*Sqrt[e^2 - 4*d*f] + a*B*f^2*Sqrt[e^2 - 4
*d*f])*Sqrt[a + x*(b + c*x)]*Log[b*e^2 - 4*b*d*f - b*e*Sqrt[e^2 - 4*d*f] + 4*a*f
*Sqrt[e^2 - 4*d*f] + 2*c*e^2*x - 8*c*d*f*x - 2*c*e*Sqrt[e^2 - 4*d*f]*x + 2*b*f*S
qrt[e^2 - 4*d*f]*x + 2*Sqrt[2]*Sqrt[e^2 - 4*d*f]*Sqrt[c*e^2 - 2*c*d*f - b*e*f +
2*a*f^2 - c*e*Sqrt[e^2 - 4*d*f] + b*f*Sqrt[e^2 - 4*d*f]]*Sqrt[a + b*x + c*x^2]])
/(Sqrt[2]*f^2*Sqrt[e^2 - 4*d*f]*Sqrt[c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 - c*e*Sqr
t[e^2 - 4*d*f] + b*f*Sqrt[e^2 - 4*d*f]]*Sqrt[a + b*x + c*x^2])

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Maple [B]  time = 0.026, size = 16209, normalized size = 26.3 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

result too large to display

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((A + B*x)*sqrt(a + b*x + c*x**2)/(d + e*x + f*x**2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: TypeError