3.861 \(\int \frac{\sqrt{a+b x+c x^2}}{(d+e x) (f+g x)^4} \, dx\)

Optimal. Leaf size=933 \[ \frac{(2 c f-b g) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{c x^2+b x+a}}\right ) e^3}{2 \sqrt{c} g (e f-d g)^4}-\frac{\sqrt{c f^2-b g f+a g^2} \tanh ^{-1}\left (\frac{b f-2 a g+(2 c f-b g) x}{2 \sqrt{c f^2-b g f+a g^2} \sqrt{c x^2+b x+a}}\right ) e^3}{g (e f-d g)^4}-\frac{\sqrt{c} \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{c x^2+b x+a}}\right ) e^2}{g (e f-d g)^3}-\frac{(2 c d-b e) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{c x^2+b x+a}}\right ) e^2}{2 \sqrt{c} (e f-d g)^4}+\frac{\sqrt{c d^2-b e d+a e^2} \tanh ^{-1}\left (\frac{b d-2 a e+(2 c d-b e) x}{2 \sqrt{c d^2-b e d+a e^2} \sqrt{c x^2+b x+a}}\right ) e^2}{(e f-d g)^4}+\frac{(2 c f-b g) \tanh ^{-1}\left (\frac{b f-2 a g+(2 c f-b g) x}{2 \sqrt{c f^2-b g f+a g^2} \sqrt{c x^2+b x+a}}\right ) e^2}{2 g (e f-d g)^3 \sqrt{c f^2-b g f+a g^2}}+\frac{\sqrt{c x^2+b x+a} e^2}{(e f-d g)^3 (f+g x)}+\frac{\left (b^2-4 a c\right ) g \tanh ^{-1}\left (\frac{b f-2 a g+(2 c f-b g) x}{2 \sqrt{c f^2-b g f+a g^2} \sqrt{c x^2+b x+a}}\right ) e}{8 (e f-d g)^2 \left (c f^2-b g f+a g^2\right )^{3/2}}-\frac{g (b f-2 a g+(2 c f-b g) x) \sqrt{c x^2+b x+a} e}{4 (e f-d g)^2 \left (c f^2-b g f+a g^2\right ) (f+g x)^2}+\frac{g^2 \left (c x^2+b x+a\right )^{3/2}}{3 (e f-d g) \left (c f^2-b g f+a g^2\right ) (f+g x)^3}+\frac{\left (b^2-4 a c\right ) g (2 c f-b g) \tanh ^{-1}\left (\frac{b f-2 a g+(2 c f-b g) x}{2 \sqrt{c f^2-b g f+a g^2} \sqrt{c x^2+b x+a}}\right )}{16 (e f-d g) \left (c f^2-b g f+a g^2\right )^{5/2}}-\frac{g (2 c f-b g) (b f-2 a g+(2 c f-b g) x) \sqrt{c x^2+b x+a}}{8 (e f-d g) \left (c f^2-b g f+a g^2\right )^2 (f+g x)^2} \]

[Out]

(e^2*Sqrt[a + b*x + c*x^2])/((e*f - d*g)^3*(f + g*x)) - (g*(2*c*f - b*g)*(b*f -
2*a*g + (2*c*f - b*g)*x)*Sqrt[a + b*x + c*x^2])/(8*(e*f - d*g)*(c*f^2 - b*f*g +
a*g^2)^2*(f + g*x)^2) - (e*g*(b*f - 2*a*g + (2*c*f - b*g)*x)*Sqrt[a + b*x + c*x^
2])/(4*(e*f - d*g)^2*(c*f^2 - b*f*g + a*g^2)*(f + g*x)^2) + (g^2*(a + b*x + c*x^
2)^(3/2))/(3*(e*f - d*g)*(c*f^2 - b*f*g + a*g^2)*(f + g*x)^3) - (e^2*(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
)^4) + (e^3*(2*c*f - b*g)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])]
)/(2*Sqrt[c]*g*(e*f - d*g)^4) - (Sqrt[c]*e^2*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt
[a + b*x + c*x^2])])/(g*(e*f - d*g)^3) + (e^2*Sqrt[c*d^2 - b*d*e + a*e^2]*ArcTan
h[(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)^4 + ((b^2 - 4*a*c)*g*(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])])/(16*(e*
f - d*g)*(c*f^2 - b*f*g + a*g^2)^(5/2)) + ((b^2 - 4*a*c)*e*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)^2*(c*f^2 - b*f*g + a*g^2)^(3/2)) + (e^2*(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)^3*Sqrt[c*f^2 - b*f*g + a*g^2]) - (e^3*Sqrt[c*f^2 - b*f*g + a*
g^2]*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])])/(g*(e*f - d*g)^4)

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Rubi [A]  time = 2.99643, antiderivative size = 933, normalized size of antiderivative = 1., number of steps used = 27, number of rules used = 9, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.31 \[ \frac{(2 c f-b g) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{c x^2+b x+a}}\right ) e^3}{2 \sqrt{c} g (e f-d g)^4}-\frac{\sqrt{c f^2-b g f+a g^2} \tanh ^{-1}\left (\frac{b f-2 a g+(2 c f-b g) x}{2 \sqrt{c f^2-b g f+a g^2} \sqrt{c x^2+b x+a}}\right ) e^3}{g (e f-d g)^4}-\frac{\sqrt{c} \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{c x^2+b x+a}}\right ) e^2}{g (e f-d g)^3}-\frac{(2 c d-b e) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{c x^2+b x+a}}\right ) e^2}{2 \sqrt{c} (e f-d g)^4}+\frac{\sqrt{c d^2-b e d+a e^2} \tanh ^{-1}\left (\frac{b d-2 a e+(2 c d-b e) x}{2 \sqrt{c d^2-b e d+a e^2} \sqrt{c x^2+b x+a}}\right ) e^2}{(e f-d g)^4}+\frac{(2 c f-b g) \tanh ^{-1}\left (\frac{b f-2 a g+(2 c f-b g) x}{2 \sqrt{c f^2-b g f+a g^2} \sqrt{c x^2+b x+a}}\right ) e^2}{2 g (e f-d g)^3 \sqrt{c f^2-b g f+a g^2}}+\frac{\sqrt{c x^2+b x+a} e^2}{(e f-d g)^3 (f+g x)}+\frac{\left (b^2-4 a c\right ) g \tanh ^{-1}\left (\frac{b f-2 a g+(2 c f-b g) x}{2 \sqrt{c f^2-b g f+a g^2} \sqrt{c x^2+b x+a}}\right ) e}{8 (e f-d g)^2 \left (c f^2-b g f+a g^2\right )^{3/2}}-\frac{g (b f-2 a g+(2 c f-b g) x) \sqrt{c x^2+b x+a} e}{4 (e f-d g)^2 \left (c f^2-b g f+a g^2\right ) (f+g x)^2}+\frac{g^2 \left (c x^2+b x+a\right )^{3/2}}{3 (e f-d g) \left (c f^2-b g f+a g^2\right ) (f+g x)^3}+\frac{\left (b^2-4 a c\right ) g (2 c f-b g) \tanh ^{-1}\left (\frac{b f-2 a g+(2 c f-b g) x}{2 \sqrt{c f^2-b g f+a g^2} \sqrt{c x^2+b x+a}}\right )}{16 (e f-d g) \left (c f^2-b g f+a g^2\right )^{5/2}}-\frac{g (2 c f-b g) (b f-2 a g+(2 c f-b g) x) \sqrt{c x^2+b x+a}}{8 (e f-d g) \left (c f^2-b g f+a g^2\right )^2 (f+g x)^2} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a + b*x + c*x^2]/((d + e*x)*(f + g*x)^4),x]

[Out]

(e^2*Sqrt[a + b*x + c*x^2])/((e*f - d*g)^3*(f + g*x)) - (g*(2*c*f - b*g)*(b*f -
2*a*g + (2*c*f - b*g)*x)*Sqrt[a + b*x + c*x^2])/(8*(e*f - d*g)*(c*f^2 - b*f*g +
a*g^2)^2*(f + g*x)^2) - (e*g*(b*f - 2*a*g + (2*c*f - b*g)*x)*Sqrt[a + b*x + c*x^
2])/(4*(e*f - d*g)^2*(c*f^2 - b*f*g + a*g^2)*(f + g*x)^2) + (g^2*(a + b*x + c*x^
2)^(3/2))/(3*(e*f - d*g)*(c*f^2 - b*f*g + a*g^2)*(f + g*x)^3) - (e^2*(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
)^4) + (e^3*(2*c*f - b*g)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])]
)/(2*Sqrt[c]*g*(e*f - d*g)^4) - (Sqrt[c]*e^2*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt
[a + b*x + c*x^2])])/(g*(e*f - d*g)^3) + (e^2*Sqrt[c*d^2 - b*d*e + a*e^2]*ArcTan
h[(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)^4 + ((b^2 - 4*a*c)*g*(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])])/(16*(e*
f - d*g)*(c*f^2 - b*f*g + a*g^2)^(5/2)) + ((b^2 - 4*a*c)*e*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)^2*(c*f^2 - b*f*g + a*g^2)^(3/2)) + (e^2*(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)^3*Sqrt[c*f^2 - b*f*g + a*g^2]) - (e^3*Sqrt[c*f^2 - b*f*g + a*
g^2]*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])])/(g*(e*f - d*g)^4)

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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)**4,x)

[Out]

Timed out

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Mathematica [A]  time = 6.93499, size = 1421, normalized size = 1.52 \[ \frac{\sqrt{c d^2-b e d+a e^2} \sqrt{a+x (b+c x)} \log (d+e x) e^2}{(e f-d g)^4 \sqrt{c x^2+b x+a}}-\frac{\sqrt{c d^2-b e d+a e^2} \sqrt{a+x (b+c x)} \log \left (-b d-2 c x d+2 a e+b e x+2 \sqrt{c d^2-b e d+a e^2} \sqrt{c x^2+b x+a}\right ) e^2}{(e f-d g)^4 \sqrt{c x^2+b x+a}}+\sqrt{a+x (b+c x)} \left (\frac{4 c e f^2+2 c d g f-5 b e g f-b d g^2+6 a e g^2}{12 (e f-d g)^2 \left (c f^2-b g f+a g^2\right ) (f+g x)^2}+\frac{8 c^2 e^2 f^4-26 b c e^2 g f^3+20 c^2 d e g f^3-4 c^2 d^2 g^2 f^2+15 b^2 e^2 g^2 f^2+44 a c e^2 g^2 f^2-26 b c d e g^2 f^2+4 b c d^2 g^3 f-42 a b e^2 g^3 f+12 b^2 d e g^3 f-4 a c d e g^3 f-3 b^2 d^2 g^4+8 a c d^2 g^4+24 a^2 e^2 g^4-6 a b d e g^4}{24 (e f-d g)^3 \left (c f^2-b g f+a g^2\right )^2 (f+g x)}-\frac{1}{3 (d g-e f) (f+g x)^3}\right )+\frac{\left (8 b c^2 e^3 f^5-16 c^3 d e^2 f^5-32 a c^2 e^3 g f^4-12 b^2 c e^3 g f^4+40 b c^2 d e^2 g f^4+5 b^3 e^3 g^2 f^3+60 a b c e^3 g^2 f^3+8 a c^2 d e^2 g^2 f^3-42 b^2 c d e^2 g^2 f^3-30 a b^2 e^3 g^3 f^2-40 a^2 c e^3 g^3 f^2+15 b^3 d e^2 g^3 f^2+20 a b c d e^2 g^3 f^2-32 a c^2 d^2 e g^3 f^2+8 b^2 c d^2 e g^3 f^2+8 a c^2 d^3 g^4 f-2 b^2 c d^3 g^4 f+40 a^2 b e^3 g^4 f-20 a b^2 d e^2 g^4 f-5 b^3 d^2 e g^4 f+20 a b c d^2 e g^4 f+b^3 d^3 g^5-4 a b c d^3 g^5-16 a^3 e^3 g^5+8 a^2 b d e^2 g^5+2 a b^2 d^2 e g^5-8 a^2 c d^2 e g^5\right ) \sqrt{a+x (b+c x)} \log (f+g x)}{16 (d g-e f)^4 \left (c f^2-b g f+a g^2\right )^{5/2} \sqrt{c x^2+b x+a}}-\frac{\left (8 b c^2 e^3 f^5-16 c^3 d e^2 f^5-32 a c^2 e^3 g f^4-12 b^2 c e^3 g f^4+40 b c^2 d e^2 g f^4+5 b^3 e^3 g^2 f^3+60 a b c e^3 g^2 f^3+8 a c^2 d e^2 g^2 f^3-42 b^2 c d e^2 g^2 f^3-30 a b^2 e^3 g^3 f^2-40 a^2 c e^3 g^3 f^2+15 b^3 d e^2 g^3 f^2+20 a b c d e^2 g^3 f^2-32 a c^2 d^2 e g^3 f^2+8 b^2 c d^2 e g^3 f^2+8 a c^2 d^3 g^4 f-2 b^2 c d^3 g^4 f+40 a^2 b e^3 g^4 f-20 a b^2 d e^2 g^4 f-5 b^3 d^2 e g^4 f+20 a b c d^2 e g^4 f+b^3 d^3 g^5-4 a b c d^3 g^5-16 a^3 e^3 g^5+8 a^2 b d e^2 g^5+2 a b^2 d^2 e g^5-8 a^2 c d^2 e g^5\right ) \sqrt{a+x (b+c x)} \log \left (-b f-2 c x f+2 a g+b g x+2 \sqrt{c f^2-b g f+a g^2} \sqrt{c x^2+b x+a}\right )}{16 (d g-e f)^4 \left (c f^2-b g f+a g^2\right )^{5/2} \sqrt{c x^2+b x+a}} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[a + b*x + c*x^2]/((d + e*x)*(f + g*x)^4),x]

[Out]

Sqrt[a + x*(b + c*x)]*(-1/(3*(-(e*f) + d*g)*(f + g*x)^3) + (4*c*e*f^2 + 2*c*d*f*
g - 5*b*e*f*g - b*d*g^2 + 6*a*e*g^2)/(12*(e*f - d*g)^2*(c*f^2 - b*f*g + a*g^2)*(
f + g*x)^2) + (8*c^2*e^2*f^4 + 20*c^2*d*e*f^3*g - 26*b*c*e^2*f^3*g - 4*c^2*d^2*f
^2*g^2 - 26*b*c*d*e*f^2*g^2 + 15*b^2*e^2*f^2*g^2 + 44*a*c*e^2*f^2*g^2 + 4*b*c*d^
2*f*g^3 + 12*b^2*d*e*f*g^3 - 4*a*c*d*e*f*g^3 - 42*a*b*e^2*f*g^3 - 3*b^2*d^2*g^4
+ 8*a*c*d^2*g^4 - 6*a*b*d*e*g^4 + 24*a^2*e^2*g^4)/(24*(e*f - d*g)^3*(c*f^2 - b*f
*g + a*g^2)^2*(f + g*x))) + (e^2*Sqrt[c*d^2 - b*d*e + a*e^2]*Sqrt[a + x*(b + c*x
)]*Log[d + e*x])/((e*f - d*g)^4*Sqrt[a + b*x + c*x^2]) + ((-16*c^3*d*e^2*f^5 + 8
*b*c^2*e^3*f^5 + 40*b*c^2*d*e^2*f^4*g - 12*b^2*c*e^3*f^4*g - 32*a*c^2*e^3*f^4*g
- 42*b^2*c*d*e^2*f^3*g^2 + 8*a*c^2*d*e^2*f^3*g^2 + 5*b^3*e^3*f^3*g^2 + 60*a*b*c*
e^3*f^3*g^2 + 8*b^2*c*d^2*e*f^2*g^3 - 32*a*c^2*d^2*e*f^2*g^3 + 15*b^3*d*e^2*f^2*
g^3 + 20*a*b*c*d*e^2*f^2*g^3 - 30*a*b^2*e^3*f^2*g^3 - 40*a^2*c*e^3*f^2*g^3 - 2*b
^2*c*d^3*f*g^4 + 8*a*c^2*d^3*f*g^4 - 5*b^3*d^2*e*f*g^4 + 20*a*b*c*d^2*e*f*g^4 -
20*a*b^2*d*e^2*f*g^4 + 40*a^2*b*e^3*f*g^4 + b^3*d^3*g^5 - 4*a*b*c*d^3*g^5 + 2*a*
b^2*d^2*e*g^5 - 8*a^2*c*d^2*e*g^5 + 8*a^2*b*d*e^2*g^5 - 16*a^3*e^3*g^5)*Sqrt[a +
x*(b + c*x)]*Log[f + g*x])/(16*(-(e*f) + d*g)^4*(c*f^2 - b*f*g + a*g^2)^(5/2)*S
qrt[a + b*x + c*x^2]) - (e^2*Sqrt[c*d^2 - b*d*e + a*e^2]*Sqrt[a + x*(b + c*x)]*L
og[-(b*d) + 2*a*e - 2*c*d*x + 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)^4*Sqrt[a + b*x + c*x^2]) - ((-16*c^3*d*e^2*f^5 + 8*b*c^
2*e^3*f^5 + 40*b*c^2*d*e^2*f^4*g - 12*b^2*c*e^3*f^4*g - 32*a*c^2*e^3*f^4*g - 42*
b^2*c*d*e^2*f^3*g^2 + 8*a*c^2*d*e^2*f^3*g^2 + 5*b^3*e^3*f^3*g^2 + 60*a*b*c*e^3*f
^3*g^2 + 8*b^2*c*d^2*e*f^2*g^3 - 32*a*c^2*d^2*e*f^2*g^3 + 15*b^3*d*e^2*f^2*g^3 +
20*a*b*c*d*e^2*f^2*g^3 - 30*a*b^2*e^3*f^2*g^3 - 40*a^2*c*e^3*f^2*g^3 - 2*b^2*c*
d^3*f*g^4 + 8*a*c^2*d^3*f*g^4 - 5*b^3*d^2*e*f*g^4 + 20*a*b*c*d^2*e*f*g^4 - 20*a*
b^2*d*e^2*f*g^4 + 40*a^2*b*e^3*f*g^4 + b^3*d^3*g^5 - 4*a*b*c*d^3*g^5 + 2*a*b^2*d
^2*e*g^5 - 8*a^2*c*d^2*e*g^5 + 8*a^2*b*d*e^2*g^5 - 16*a^3*e^3*g^5)*Sqrt[a + x*(b
+ c*x)]*Log[-(b*f) + 2*a*g - 2*c*f*x + b*g*x + 2*Sqrt[c*f^2 - b*f*g + a*g^2]*Sq
rt[a + b*x + c*x^2]])/(16*(-(e*f) + d*g)^4*(c*f^2 - b*f*g + a*g^2)^(5/2)*Sqrt[a
+ b*x + c*x^2])

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Maple [B]  time = 0.039, size = 11995, normalized size = 12.9 \[ \text{output too large to display} \]

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)^4,x)

[Out]

result too large to display

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{c x^{2} + b x + a}}{{\left (e x + d\right )}{\left (g x + f\right )}^{4}}\,{d x} \]

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)^4),x, algorithm="maxima")

[Out]

integrate(sqrt(c*x^2 + b*x + a)/((e*x + d)*(g*x + f)^4), x)

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Fricas [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)/((e*x + d)*(g*x + f)^4),x, algorithm="fricas")

[Out]

Exception raised: TypeError

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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)**4,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 14.2537, size = 4, normalized size = 0. \[ \mathit{sage}_{0} x \]

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)^4),x, algorithm="giac")

[Out]

sage0*x