3.2329 \(\int \frac{\sqrt{a+b x+c x^2}}{(d+e x)^4} \, dx\)

Optimal. Leaf size=215 \[ -\frac{\left (b^2-4 a c\right ) (2 c d-b e) \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{16 \left (a e^2-b d e+c d^2\right )^{5/2}}-\frac{e \left (a+b x+c x^2\right )^{3/2}}{3 (d+e x)^3 \left (a e^2-b d e+c d^2\right )}+\frac{\sqrt{a+b x+c x^2} (2 c d-b e) (-2 a e+x (2 c d-b e)+b d)}{8 (d+e x)^2 \left (a e^2-b d e+c d^2\right )^2} \]

[Out]

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

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Rubi [A]  time = 0.426868, antiderivative size = 215, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ -\frac{\left (b^2-4 a c\right ) (2 c d-b e) \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{16 \left (a e^2-b d e+c d^2\right )^{5/2}}-\frac{e \left (a+b x+c x^2\right )^{3/2}}{3 (d+e x)^3 \left (a e^2-b d e+c d^2\right )}+\frac{\sqrt{a+b x+c x^2} (2 c d-b e) (-2 a e+x (2 c d-b e)+b d)}{8 (d+e x)^2 \left (a e^2-b d e+c d^2\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 54.5547, size = 196, normalized size = 0.91 \[ - \frac{e \left (a + b x + c x^{2}\right )^{\frac{3}{2}}}{3 \left (d + e x\right )^{3} \left (a e^{2} - b d e + c d^{2}\right )} - \frac{\left (- 4 a c + b^{2}\right ) \left (b e - 2 c d\right ) \operatorname{atanh}{\left (\frac{2 a e - b d + x \left (b e - 2 c d\right )}{2 \sqrt{a + b x + c x^{2}} \sqrt{a e^{2} - b d e + c d^{2}}} \right )}}{16 \left (a e^{2} - b d e + c d^{2}\right )^{\frac{5}{2}}} + \frac{\left (\frac{b e}{2} - c d\right ) \sqrt{a + b x + c x^{2}} \left (2 a e - b d + x \left (b e - 2 c d\right )\right )}{4 \left (d + e x\right )^{2} \left (a e^{2} - b d e + c d^{2}\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**2+b*x+a)**(1/2)/(e*x+d)**4,x)

[Out]

-e*(a + b*x + c*x**2)**(3/2)/(3*(d + e*x)**3*(a*e**2 - b*d*e + c*d**2)) - (-4*a*
c + b**2)*(b*e - 2*c*d)*atanh((2*a*e - b*d + x*(b*e - 2*c*d))/(2*sqrt(a + b*x +
c*x**2)*sqrt(a*e**2 - b*d*e + c*d**2)))/(16*(a*e**2 - b*d*e + c*d**2)**(5/2)) +
(b*e/2 - c*d)*sqrt(a + b*x + c*x**2)*(2*a*e - b*d + x*(b*e - 2*c*d))/(4*(d + e*x
)**2*(a*e**2 - b*d*e + c*d**2)**2)

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Mathematica [A]  time = 0.846676, size = 273, normalized size = 1.27 \[ \frac{-2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2} \left (-(d+e x)^2 \left (-4 c e (2 a e+b d)+3 b^2 e^2+4 c^2 d^2\right )-2 (d+e x) (2 c d-b e) \left (e (a e-b d)+c d^2\right )+8 \left (e (a e-b d)+c d^2\right )^2\right )-3 e \left (b^2-4 a c\right ) (d+e x)^3 (b e-2 c d) \log \left (2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2}+2 a e-b d+b e x-2 c d x\right )+3 e \left (b^2-4 a c\right ) (d+e x)^3 (b e-2 c d) \log (d+e x)}{48 e (d+e x)^3 \left (e (a e-b d)+c d^2\right )^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[c*d^2 + e*(-(b*d) + a*e)]*Sqrt[a + x*(b + c*x)]*(8*(c*d^2 + e*(-(b*d) +
a*e))^2 - 2*(2*c*d - b*e)*(c*d^2 + e*(-(b*d) + a*e))*(d + e*x) - (4*c^2*d^2 + 3
*b^2*e^2 - 4*c*e*(b*d + 2*a*e))*(d + e*x)^2) + 3*(b^2 - 4*a*c)*e*(-2*c*d + b*e)*
(d + e*x)^3*Log[d + e*x] - 3*(b^2 - 4*a*c)*e*(-2*c*d + b*e)*(d + e*x)^3*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)]])/(48*e*(c*d^2 + e*(-(b*d) + a*e))^(5/2)*(d + e*x)^3)

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Maple [B]  time = 0.022, size = 4844, normalized size = 22.5 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+b*x+a)^(1/2)/(e*x+d)^4,x)

[Out]

-1/4/e/(a*e^2-b*d*e+c*d^2)^2*c/((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2-b*d*
e+c*d^2)/e^2+(b*e-2*c*d)/e*(d/e+x)+2*((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*(c*(d/e+x)^
2+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))/(d/e+x))*b^2*d-1/2/e/(a*
e^2-b*d*e+c*d^2)^2*c^2/((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2-b*d*e+c*d^2)
/e^2+(b*e-2*c*d)/e*(d/e+x)+2*((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*(c*(d/e+x)^2+(b*e-2
*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))/(d/e+x))*a*d-5/4/e^2/(a*e^2-b*d*
e+c*d^2)^3/((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2-b*d*e+c*d^2)/e^2+(b*e-2*
c*d)/e*(d/e+x)+2*((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e
+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))/(d/e+x))*c^3*d^4*b-1/2*e/(a*e^2-b*d*e+c*d^2)
^3*c^2*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*x*b*d+1
/2/e/(a*e^2-b*d*e+c*d^2)^3/((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2-b*d*e+c*
d^2)/e^2+(b*e-2*c*d)/e*(d/e+x)+2*((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*(c*(d/e+x)^2+(b
*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))/(d/e+x))*a*c^3*d^3-3/4/(a*e^
2-b*d*e+c*d^2)^3/((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2-b*d*e+c*d^2)/e^2+(
b*e-2*c*d)/e*(d/e+x)+2*((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*(c*(d/e+x)^2+(b*e-2*c*d)/
e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))/(d/e+x))*a*c^2*d^2*b+1/2/e/(a*e^2-b*d*
e+c*d^2)^2*c^(3/2)*ln((1/2*(b*e-2*c*d)/e+c*(d/e+x))/c^(1/2)+(c*(d/e+x)^2+(b*e-2*
c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))*d*b-7/16/(a*e^2-b*d*e+c*d^2)^3/((
a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2-b*d*e+c*d^2)/e^2+(b*e-2*c*d)/e*(d/e+x
)+2*((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*
d*e+c*d^2)/e^2)^(1/2))/(d/e+x))*b^3*d^2*c+1/4/(a*e^2-b*d*e+c*d^2)^2*c/((a*e^2-b*
d*e+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2-b*d*e+c*d^2)/e^2+(b*e-2*c*d)/e*(d/e+x)+2*((a*
e^2-b*d*e+c*d^2)/e^2)^(1/2)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^
2)/e^2)^(1/2))/(d/e+x))*a*b-1/2/e^3/(a*e^2-b*d*e+c*d^2)^2*c^3/((a*e^2-b*d*e+c*d^
2)/e^2)^(1/2)*ln((2*(a*e^2-b*d*e+c*d^2)/e^2+(b*e-2*c*d)/e*(d/e+x)+2*((a*e^2-b*d*
e+c*d^2)/e^2)^(1/2)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^
(1/2))/(d/e+x))*d^3+1/2/e^3/(a*e^2-b*d*e+c*d^2)^3/((a*e^2-b*d*e+c*d^2)/e^2)^(1/2
)*ln((2*(a*e^2-b*d*e+c*d^2)/e^2+(b*e-2*c*d)/e*(d/e+x)+2*((a*e^2-b*d*e+c*d^2)/e^2
)^(1/2)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))/(d/e+
x))*c^4*d^5-1/8*e/(a*e^2-b*d*e+c*d^2)^3*ln((1/2*(b*e-2*c*d)/e+c*(d/e+x))/c^(1/2)
+(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))*c^(1/2)*d*b^
3+1/8*e^2/(a*e^2-b*d*e+c*d^2)^3*c*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*
e+c*d^2)/e^2)^(1/2)*x*b^2+5/8/(a*e^2-b*d*e+c*d^2)^3*ln((1/2*(b*e-2*c*d)/e+c*(d/e
+x))/c^(1/2)+(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))*
c^(3/2)*d^2*b^2+1/8*e^2/(a*e^2-b*d*e+c*d^2)^3*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)
+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*b^3+1/2/e^2/(a*e^2-b*d*e+c*d^2)^3*ln((1/2*(b*e-2
*c*d)/e+c*(d/e+x))/c^(1/2)+(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2
)/e^2)^(1/2))*c^(7/2)*d^4-1/e/(a*e^2-b*d*e+c*d^2)^3*ln((1/2*(b*e-2*c*d)/e+c*(d/e
+x))/c^(1/2)+(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))*
c^(5/2)*d^3*b-1/8/(a*e^2-b*d*e+c*d^2)^2*c^(1/2)*ln((1/2*(b*e-2*c*d)/e+c*(d/e+x))
/c^(1/2)+(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))*b^2-
1/3/e^2/(a*e^2-b*d*e+c*d^2)/(d/e+x)^3*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-
b*d*e+c*d^2)/e^2)^(3/2)+3/8*e/(a*e^2-b*d*e+c*d^2)^3/((a*e^2-b*d*e+c*d^2)/e^2)^(1
/2)*ln((2*(a*e^2-b*d*e+c*d^2)/e^2+(b*e-2*c*d)/e*(d/e+x)+2*((a*e^2-b*d*e+c*d^2)/e
^2)^(1/2)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))/(d/
e+x))*a*b^2*c*d+1/(a*e^2-b*d*e+c*d^2)^3*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)+(a*e^
2-b*d*e+c*d^2)/e^2)^(1/2)*c^2*d^2*b-1/2/(a*e^2-b*d*e+c*d^2)^3/(d/e+x)*(c*(d/e+x)
^2+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(3/2)*c^2*d^2+1/2/(a*e^2-b*d*e
+c*d^2)^3*c^3*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*
x*d^2-1/2/e^2/(a*e^2-b*d*e+c*d^2)^2*c^(5/2)*ln((1/2*(b*e-2*c*d)/e+c*(d/e+x))/c^(
1/2)+(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))*d^2+1/2/
e/(a*e^2-b*d*e+c*d^2)^2*c^2*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^
2)/e^2)^(1/2)*d+9/8/e/(a*e^2-b*d*e+c*d^2)^3/((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*ln((
2*(a*e^2-b*d*e+c*d^2)/e^2+(b*e-2*c*d)/e*(d/e+x)+2*((a*e^2-b*d*e+c*d^2)/e^2)^(1/2
)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))/(d/e+x))*b^
2*d^3*c^2-1/2*e/(a*e^2-b*d*e+c*d^2)^3*c^(3/2)*ln((1/2*(b*e-2*c*d)/e+c*(d/e+x))/c
^(1/2)+(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))*a*d*b+
3/4/e^2/(a*e^2-b*d*e+c*d^2)^2*c^2/((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2-b
*d*e+c*d^2)/e^2+(b*e-2*c*d)/e*(d/e+x)+2*((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*(c*(d/e+
x)^2+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))/(d/e+x))*b*d^2+1/8*e^
2/(a*e^2-b*d*e+c*d^2)^3*c^(1/2)*ln((1/2*(b*e-2*c*d)/e+c*(d/e+x))/c^(1/2)+(c*(d/e
+x)^2+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))*a*b^2-1/16*e^2/(a*e^
2-b*d*e+c*d^2)^3/((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2-b*d*e+c*d^2)/e^2+(
b*e-2*c*d)/e*(d/e+x)+2*((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*(c*(d/e+x)^2+(b*e-2*c*d)/
e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))/(d/e+x))*a*b^3+1/16*e/(a*e^2-b*d*e+c*d
^2)^3/((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2-b*d*e+c*d^2)/e^2+(b*e-2*c*d)/
e*(d/e+x)+2*((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)+(
a*e^2-b*d*e+c*d^2)/e^2)^(1/2))/(d/e+x))*b^4*d+1/2/(a*e^2-b*d*e+c*d^2)^3*c^(5/2)*
ln((1/2*(b*e-2*c*d)/e+c*(d/e+x))/c^(1/2)+(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)+(a*e
^2-b*d*e+c*d^2)/e^2)^(1/2))*a*d^2+1/4/(a*e^2-b*d*e+c*d^2)^2/(d/e+x)^2*(c*(d/e+x)
^2+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(3/2)*b-1/4/(a*e^2-b*d*e+c*d^2
)^2*c*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*b-1/8*e^
2/(a*e^2-b*d*e+c*d^2)^3/(d/e+x)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+
c*d^2)/e^2)^(3/2)*b^2-1/2/e/(a*e^2-b*d*e+c*d^2)^3*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/
e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*c^3*d^3-1/2/e/(a*e^2-b*d*e+c*d^2)^2/(d/e+x)^
2*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(3/2)*c*d-5/8*e/(a
*e^2-b*d*e+c*d^2)^3*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^
(1/2)*b^2*c*d+1/2*e/(a*e^2-b*d*e+c*d^2)^3/(d/e+x)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/
e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(3/2)*b*c*d

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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)/(e*x + d)^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.92362, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^2 + b*x + a)/(e*x + d)^4,x, algorithm="fricas")

[Out]

[1/96*(4*(6*b*c*d^3 + 14*a*b*d*e^2 - 8*a^2*e^3 - (3*b^2 + 20*a*c)*d^2*e + (4*c^2
*d^2*e - 4*b*c*d*e^2 + (3*b^2 - 8*a*c)*e^3)*x^2 + 2*(6*c^2*d^3 - 7*b*c*d^2*e - a
*b*e^3 + 2*(2*b^2 - 3*a*c)*d*e^2)*x)*sqrt(c*d^2 - b*d*e + a*e^2)*sqrt(c*x^2 + b*
x + a) + 3*(2*(b^2*c - 4*a*c^2)*d^4 - (b^3 - 4*a*b*c)*d^3*e + (2*(b^2*c - 4*a*c^
2)*d*e^3 - (b^3 - 4*a*b*c)*e^4)*x^3 + 3*(2*(b^2*c - 4*a*c^2)*d^2*e^2 - (b^3 - 4*
a*b*c)*d*e^3)*x^2 + 3*(2*(b^2*c - 4*a*c^2)*d^3*e - (b^3 - 4*a*b*c)*d^2*e^2)*x)*l
og(((8*a*b*d*e - 8*a^2*e^2 - (b^2 + 4*a*c)*d^2 - (8*c^2*d^2 - 8*b*c*d*e + (b^2 +
4*a*c)*e^2)*x^2 - 2*(4*b*c*d^2 + 4*a*b*e^2 - (3*b^2 + 4*a*c)*d*e)*x)*sqrt(c*d^2
- b*d*e + a*e^2) + 4*(b*c*d^3 + 3*a*b*d*e^2 - 2*a^2*e^3 - (b^2 + 2*a*c)*d^2*e +
(2*c^2*d^3 - 3*b*c*d^2*e - a*b*e^3 + (b^2 + 2*a*c)*d*e^2)*x)*sqrt(c*x^2 + b*x +
a))/(e^2*x^2 + 2*d*e*x + d^2)))/((c^2*d^7 - 2*b*c*d^6*e - 2*a*b*d^4*e^3 + a^2*d
^3*e^4 + (b^2 + 2*a*c)*d^5*e^2 + (c^2*d^4*e^3 - 2*b*c*d^3*e^4 - 2*a*b*d*e^6 + a^
2*e^7 + (b^2 + 2*a*c)*d^2*e^5)*x^3 + 3*(c^2*d^5*e^2 - 2*b*c*d^4*e^3 - 2*a*b*d^2*
e^5 + a^2*d*e^6 + (b^2 + 2*a*c)*d^3*e^4)*x^2 + 3*(c^2*d^6*e - 2*b*c*d^5*e^2 - 2*
a*b*d^3*e^4 + a^2*d^2*e^5 + (b^2 + 2*a*c)*d^4*e^3)*x)*sqrt(c*d^2 - b*d*e + a*e^2
)), 1/48*(2*(6*b*c*d^3 + 14*a*b*d*e^2 - 8*a^2*e^3 - (3*b^2 + 20*a*c)*d^2*e + (4*
c^2*d^2*e - 4*b*c*d*e^2 + (3*b^2 - 8*a*c)*e^3)*x^2 + 2*(6*c^2*d^3 - 7*b*c*d^2*e
- a*b*e^3 + 2*(2*b^2 - 3*a*c)*d*e^2)*x)*sqrt(-c*d^2 + b*d*e - a*e^2)*sqrt(c*x^2
+ b*x + a) + 3*(2*(b^2*c - 4*a*c^2)*d^4 - (b^3 - 4*a*b*c)*d^3*e + (2*(b^2*c - 4*
a*c^2)*d*e^3 - (b^3 - 4*a*b*c)*e^4)*x^3 + 3*(2*(b^2*c - 4*a*c^2)*d^2*e^2 - (b^3
- 4*a*b*c)*d*e^3)*x^2 + 3*(2*(b^2*c - 4*a*c^2)*d^3*e - (b^3 - 4*a*b*c)*d^2*e^2)*
x)*arctan(-1/2*sqrt(-c*d^2 + b*d*e - a*e^2)*(b*d - 2*a*e + (2*c*d - b*e)*x)/((c*
d^2 - b*d*e + a*e^2)*sqrt(c*x^2 + b*x + a))))/((c^2*d^7 - 2*b*c*d^6*e - 2*a*b*d^
4*e^3 + a^2*d^3*e^4 + (b^2 + 2*a*c)*d^5*e^2 + (c^2*d^4*e^3 - 2*b*c*d^3*e^4 - 2*a
*b*d*e^6 + a^2*e^7 + (b^2 + 2*a*c)*d^2*e^5)*x^3 + 3*(c^2*d^5*e^2 - 2*b*c*d^4*e^3
- 2*a*b*d^2*e^5 + a^2*d*e^6 + (b^2 + 2*a*c)*d^3*e^4)*x^2 + 3*(c^2*d^6*e - 2*b*c
*d^5*e^2 - 2*a*b*d^3*e^4 + a^2*d^2*e^5 + (b^2 + 2*a*c)*d^4*e^3)*x)*sqrt(-c*d^2 +
b*d*e - a*e^2))]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**2+b*x+a)**(1/2)/(e*x+d)**4,x)

[Out]

Integral(sqrt(a + b*x + c*x**2)/(d + e*x)**4, x)

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GIAC/XCAS [A]  time = 0.562556, size = 4, normalized size = 0.02 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^2 + b*x + a)/(e*x + d)^4,x, algorithm="giac")

[Out]

sage0*x