3.860 \(\int \frac{\sqrt{a+b x+c x^2}}{(d+e x) (f+g x)^3} \, dx\)

Optimal. Leaf size=673 \[ \frac{g \left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{-2 a g+x (2 c f-b g)+b f}{2 \sqrt{a+b x+c x^2} \sqrt{a g^2-b f g+c f^2}}\right )}{8 (e f-d g) \left (a g^2-b f g+c f^2\right )^{3/2}}+\frac{e \sqrt{a e^2-b d e+c d^2} \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{(e f-d g)^3}-\frac{e^2 \sqrt{a g^2-b f g+c f^2} \tanh ^{-1}\left (\frac{-2 a g+x (2 c f-b g)+b f}{2 \sqrt{a+b x+c x^2} \sqrt{a g^2-b f g+c f^2}}\right )}{g (e f-d g)^3}+\frac{e^2 (2 c f-b g) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{2 \sqrt{c} g (e f-d g)^3}-\frac{g \sqrt{a+b x+c x^2} (-2 a g+x (2 c f-b g)+b f)}{4 (f+g x)^2 (e f-d g) \left (a g^2-b f g+c f^2\right )}+\frac{e (2 c f-b g) \tanh ^{-1}\left (\frac{-2 a g+x (2 c f-b g)+b f}{2 \sqrt{a+b x+c x^2} \sqrt{a g^2-b f g+c f^2}}\right )}{2 g (e f-d g)^2 \sqrt{a g^2-b f g+c f^2}}+\frac{e \sqrt{a+b x+c x^2}}{(f+g x) (e f-d g)^2}-\frac{\sqrt{c} e \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{g (e f-d g)^2}-\frac{e (2 c d-b e) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{2 \sqrt{c} (e f-d g)^3} \]

[Out]

(e*Sqrt[a + b*x + c*x^2])/((e*f - d*g)^2*(f + g*x)) - (g*(b*f - 2*a*g + (2*c*f -
b*g)*x)*Sqrt[a + b*x + c*x^2])/(4*(e*f - d*g)*(c*f^2 - b*f*g + a*g^2)*(f + g*x)
^2) - (e*(2*c*d - b*e)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(
2*Sqrt[c]*(e*f - d*g)^3) + (e^2*(2*c*f - b*g)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqr
t[a + b*x + c*x^2])])/(2*Sqrt[c]*g*(e*f - d*g)^3) - (Sqrt[c]*e*ArcTanh[(b + 2*c*
x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(g*(e*f - d*g)^2) + (e*Sqrt[c*d^2 - b*d*e
+ a*e^2]*ArcTanh[(b*d - 2*a*e + (2*c*d - b*e)*x)/(2*Sqrt[c*d^2 - b*d*e + a*e^2]
*Sqrt[a + b*x + c*x^2])])/(e*f - d*g)^3 + ((b^2 - 4*a*c)*g*ArcTanh[(b*f - 2*a*g
+ (2*c*f - b*g)*x)/(2*Sqrt[c*f^2 - b*f*g + a*g^2]*Sqrt[a + b*x + c*x^2])])/(8*(e
*f - d*g)*(c*f^2 - b*f*g + a*g^2)^(3/2)) + (e*(2*c*f - b*g)*ArcTanh[(b*f - 2*a*g
+ (2*c*f - b*g)*x)/(2*Sqrt[c*f^2 - b*f*g + a*g^2]*Sqrt[a + b*x + c*x^2])])/(2*g
*(e*f - d*g)^2*Sqrt[c*f^2 - b*f*g + a*g^2]) - (e^2*Sqrt[c*f^2 - b*f*g + a*g^2]*A
rcTanh[(b*f - 2*a*g + (2*c*f - b*g)*x)/(2*Sqrt[c*f^2 - b*f*g + a*g^2]*Sqrt[a + b
*x + c*x^2])])/(g*(e*f - d*g)^3)

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Rubi [A]  time = 2.08197, antiderivative size = 673, normalized size of antiderivative = 1., number of steps used = 23, number of rules used = 8, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.276 \[ \frac{g \left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{-2 a g+x (2 c f-b g)+b f}{2 \sqrt{a+b x+c x^2} \sqrt{a g^2-b f g+c f^2}}\right )}{8 (e f-d g) \left (a g^2-b f g+c f^2\right )^{3/2}}+\frac{e \sqrt{a e^2-b d e+c d^2} \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{(e f-d g)^3}-\frac{e^2 \sqrt{a g^2-b f g+c f^2} \tanh ^{-1}\left (\frac{-2 a g+x (2 c f-b g)+b f}{2 \sqrt{a+b x+c x^2} \sqrt{a g^2-b f g+c f^2}}\right )}{g (e f-d g)^3}+\frac{e^2 (2 c f-b g) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{2 \sqrt{c} g (e f-d g)^3}-\frac{g \sqrt{a+b x+c x^2} (-2 a g+x (2 c f-b g)+b f)}{4 (f+g x)^2 (e f-d g) \left (a g^2-b f g+c f^2\right )}+\frac{e (2 c f-b g) \tanh ^{-1}\left (\frac{-2 a g+x (2 c f-b g)+b f}{2 \sqrt{a+b x+c x^2} \sqrt{a g^2-b f g+c f^2}}\right )}{2 g (e f-d g)^2 \sqrt{a g^2-b f g+c f^2}}+\frac{e \sqrt{a+b x+c x^2}}{(f+g x) (e f-d g)^2}-\frac{\sqrt{c} e \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{g (e f-d g)^2}-\frac{e (2 c d-b e) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{2 \sqrt{c} (e f-d g)^3} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a + b*x + c*x^2]/((d + e*x)*(f + g*x)^3),x]

[Out]

(e*Sqrt[a + b*x + c*x^2])/((e*f - d*g)^2*(f + g*x)) - (g*(b*f - 2*a*g + (2*c*f -
b*g)*x)*Sqrt[a + b*x + c*x^2])/(4*(e*f - d*g)*(c*f^2 - b*f*g + a*g^2)*(f + g*x)
^2) - (e*(2*c*d - b*e)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(
2*Sqrt[c]*(e*f - d*g)^3) + (e^2*(2*c*f - b*g)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqr
t[a + b*x + c*x^2])])/(2*Sqrt[c]*g*(e*f - d*g)^3) - (Sqrt[c]*e*ArcTanh[(b + 2*c*
x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(g*(e*f - d*g)^2) + (e*Sqrt[c*d^2 - b*d*e
+ a*e^2]*ArcTanh[(b*d - 2*a*e + (2*c*d - b*e)*x)/(2*Sqrt[c*d^2 - b*d*e + a*e^2]
*Sqrt[a + b*x + c*x^2])])/(e*f - d*g)^3 + ((b^2 - 4*a*c)*g*ArcTanh[(b*f - 2*a*g
+ (2*c*f - b*g)*x)/(2*Sqrt[c*f^2 - b*f*g + a*g^2]*Sqrt[a + b*x + c*x^2])])/(8*(e
*f - d*g)*(c*f^2 - b*f*g + a*g^2)^(3/2)) + (e*(2*c*f - b*g)*ArcTanh[(b*f - 2*a*g
+ (2*c*f - b*g)*x)/(2*Sqrt[c*f^2 - b*f*g + a*g^2]*Sqrt[a + b*x + c*x^2])])/(2*g
*(e*f - d*g)^2*Sqrt[c*f^2 - b*f*g + a*g^2]) - (e^2*Sqrt[c*f^2 - b*f*g + a*g^2]*A
rcTanh[(b*f - 2*a*g + (2*c*f - b*g)*x)/(2*Sqrt[c*f^2 - b*f*g + a*g^2]*Sqrt[a + b
*x + c*x^2])])/(g*(e*f - d*g)^3)

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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((c*x**2+b*x+a)**(1/2)/(e*x+d)/(g*x+f)**3,x)

[Out]

Timed out

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Mathematica [A]  time = 2.99405, size = 523, normalized size = 0.78 \[ \frac{-\frac{\log (f+g x) \left (g \left (8 a^2 e^2 g^2-4 a b e g (d g+3 e f)+b^2 \left (-d^2 g^2+6 d e f g+3 e^2 f^2\right )\right )+4 a c g \left (d^2 g^2+3 e^2 f^2\right )-4 b c e f^2 (3 d g+e f)+8 c^2 d e f^3\right )}{\left (g (a g-b f)+c f^2\right )^{3/2}}+\frac{\left (g \left (8 a^2 e^2 g^2-4 a b e g (d g+3 e f)+b^2 \left (-d^2 g^2+6 d e f g+3 e^2 f^2\right )\right )+4 a c g \left (d^2 g^2+3 e^2 f^2\right )-4 b c e f^2 (3 d g+e f)+8 c^2 d e f^3\right ) \log \left (2 \sqrt{a+x (b+c x)} \sqrt{g (a g-b f)+c f^2}+2 a g-b f+b g x-2 c f x\right )}{\left (g (a g-b f)+c f^2\right )^{3/2}}+8 e \log (d+e x) \sqrt{e (a e-b d)+c d^2}-8 e \sqrt{e (a e-b d)+c d^2} \log \left (2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2}+2 a e-b d+b e x-2 c d x\right )+\frac{2 \sqrt{a+x (b+c x)} (e f-d g) \left (\frac{(f+g x) (2 c f (d g+e f)-g (-4 a e g+b d g+3 b e f))}{g (a g-b f)+c f^2}-2 d g+2 e f\right )}{(f+g x)^2}}{8 (e f-d g)^3} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[a + b*x + c*x^2]/((d + e*x)*(f + g*x)^3),x]

[Out]

((2*(e*f - d*g)*Sqrt[a + x*(b + c*x)]*(2*e*f - 2*d*g + ((2*c*f*(e*f + d*g) - g*(
3*b*e*f + b*d*g - 4*a*e*g))*(f + g*x))/(c*f^2 + g*(-(b*f) + a*g))))/(f + g*x)^2
+ 8*e*Sqrt[c*d^2 + e*(-(b*d) + a*e)]*Log[d + e*x] - ((8*c^2*d*e*f^3 - 4*b*c*e*f^
2*(e*f + 3*d*g) + 4*a*c*g*(3*e^2*f^2 + d^2*g^2) + g*(8*a^2*e^2*g^2 - 4*a*b*e*g*(
3*e*f + d*g) + b^2*(3*e^2*f^2 + 6*d*e*f*g - d^2*g^2)))*Log[f + g*x])/(c*f^2 + g*
(-(b*f) + a*g))^(3/2) - 8*e*Sqrt[c*d^2 + e*(-(b*d) + a*e)]*Log[-(b*d) + 2*a*e -
2*c*d*x + b*e*x + 2*Sqrt[c*d^2 + e*(-(b*d) + a*e)]*Sqrt[a + x*(b + c*x)]] + ((8*
c^2*d*e*f^3 - 4*b*c*e*f^2*(e*f + 3*d*g) + 4*a*c*g*(3*e^2*f^2 + d^2*g^2) + g*(8*a
^2*e^2*g^2 - 4*a*b*e*g*(3*e*f + d*g) + b^2*(3*e^2*f^2 + 6*d*e*f*g - d^2*g^2)))*L
og[-(b*f) + 2*a*g - 2*c*f*x + b*g*x + 2*Sqrt[c*f^2 + g*(-(b*f) + a*g)]*Sqrt[a +
x*(b + c*x)]])/(c*f^2 + g*(-(b*f) + a*g))^(3/2))/(8*(e*f - d*g)^3)

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Maple [B]  time = 0.032, size = 6714, normalized size = 10. \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+b*x+a)^(1/2)/(e*x+d)/(g*x+f)^3,x)

[Out]

result too large to display

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{c x^{2} + b x + a}}{{\left (e x + d\right )}{\left (g x + f\right )}^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^2 + b*x + a)/((e*x + d)*(g*x + f)^3),x, algorithm="maxima")

[Out]

integrate(sqrt(c*x^2 + b*x + a)/((e*x + d)*(g*x + f)^3), x)

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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)/((e*x + d)*(g*x + f)^3),x, algorithm="fricas")

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**2+b*x+a)**(1/2)/(e*x+d)/(g*x+f)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 1.60089, size = 4, normalized size = 0.01 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^2 + b*x + a)/((e*x + d)*(g*x + f)^3),x, algorithm="giac")

[Out]

sage0*x