3.1580 \(\int \frac{b+2 c x}{(d+e x)^4 \sqrt{a+b x+c x^2}} \, dx\)

Optimal. Leaf size=328 \[ \frac{\sqrt{a+b x+c x^2} (2 c d-b e) \left (-4 c e (13 a e+2 b d)+15 b^2 e^2+8 c^2 d^2\right )}{24 (d+e x) \left (a e^2-b d e+c d^2\right )^3}+\frac{\sqrt{a+b x+c x^2} \left (-4 c e (3 a e+2 b d)+5 b^2 e^2+8 c^2 d^2\right )}{12 (d+e x)^2 \left (a e^2-b d e+c d^2\right )^2}-\frac{e \left (b^2-4 a c\right ) \left (-4 c e (a e+4 b d)+5 b^2 e^2+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{16 \left (a e^2-b d e+c d^2\right )^{7/2}}+\frac{\sqrt{a+b x+c x^2} (2 c d-b e)}{3 (d+e x)^3 \left (a e^2-b d e+c d^2\right )} \]

[Out]

((2*c*d - b*e)*Sqrt[a + b*x + c*x^2])/(3*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^3) +
((8*c^2*d^2 + 5*b^2*e^2 - 4*c*e*(2*b*d + 3*a*e))*Sqrt[a + b*x + c*x^2])/(12*(c*d
^2 - b*d*e + a*e^2)^2*(d + e*x)^2) + ((2*c*d - b*e)*(8*c^2*d^2 + 15*b^2*e^2 - 4*
c*e*(2*b*d + 13*a*e))*Sqrt[a + b*x + c*x^2])/(24*(c*d^2 - b*d*e + a*e^2)^3*(d +
e*x)) - ((b^2 - 4*a*c)*e*(16*c^2*d^2 + 5*b^2*e^2 - 4*c*e*(4*b*d + a*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)^(7/2))

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Rubi [A]  time = 1.0653, antiderivative size = 328, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ \frac{\sqrt{a+b x+c x^2} (2 c d-b e) \left (-4 c e (13 a e+2 b d)+15 b^2 e^2+8 c^2 d^2\right )}{24 (d+e x) \left (a e^2-b d e+c d^2\right )^3}+\frac{\sqrt{a+b x+c x^2} \left (-4 c e (3 a e+2 b d)+5 b^2 e^2+8 c^2 d^2\right )}{12 (d+e x)^2 \left (a e^2-b d e+c d^2\right )^2}-\frac{e \left (b^2-4 a c\right ) \left (-4 c e (a e+4 b d)+5 b^2 e^2+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{16 \left (a e^2-b d e+c d^2\right )^{7/2}}+\frac{\sqrt{a+b x+c x^2} (2 c d-b e)}{3 (d+e x)^3 \left (a e^2-b d e+c d^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

((2*c*d - b*e)*Sqrt[a + b*x + c*x^2])/(3*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^3) +
((8*c^2*d^2 + 5*b^2*e^2 - 4*c*e*(2*b*d + 3*a*e))*Sqrt[a + b*x + c*x^2])/(12*(c*d
^2 - b*d*e + a*e^2)^2*(d + e*x)^2) + ((2*c*d - b*e)*(8*c^2*d^2 + 15*b^2*e^2 - 4*
c*e*(2*b*d + 13*a*e))*Sqrt[a + b*x + c*x^2])/(24*(c*d^2 - b*d*e + a*e^2)^3*(d +
e*x)) - ((b^2 - 4*a*c)*e*(16*c^2*d^2 + 5*b^2*e^2 - 4*c*e*(4*b*d + a*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)^(7/2))

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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((2*c*x+b)/(e*x+d)**4/(c*x**2+b*x+a)**(1/2),x)

[Out]

Timed out

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Mathematica [A]  time = 0.801842, size = 351, normalized size = 1.07 \[ \frac{2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2} \left (2 (d+e x) \left (-4 c e (3 a e+2 b d)+5 b^2 e^2+8 c^2 d^2\right ) \left (e (a e-b d)+c d^2\right )+(d+e x)^2 (2 c d-b e) \left (-4 c e (13 a e+2 b d)+15 b^2 e^2+8 c^2 d^2\right )+8 (2 c d-b e) \left (e (a e-b d)+c d^2\right )^2\right )-3 e \left (b^2-4 a c\right ) (d+e x)^3 \log (d+e x) \left (-4 c e (a e+4 b d)+5 b^2 e^2+16 c^2 d^2\right )+3 e \left (b^2-4 a c\right ) (d+e x)^3 \left (-4 c e (a e+4 b d)+5 b^2 e^2+16 c^2 d^2\right ) \log \left (2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2}+2 a e-b d+b e x-2 c d x\right )}{48 (d+e x)^3 \left (e (a e-b d)+c d^2\right )^{7/2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[c*d^2 + e*(-(b*d) + a*e)]*Sqrt[a + x*(b + c*x)]*(8*(2*c*d - b*e)*(c*d^2
+ e*(-(b*d) + a*e))^2 + 2*(c*d^2 + e*(-(b*d) + a*e))*(8*c^2*d^2 + 5*b^2*e^2 - 4*
c*e*(2*b*d + 3*a*e))*(d + e*x) + (2*c*d - b*e)*(8*c^2*d^2 + 15*b^2*e^2 - 4*c*e*(
2*b*d + 13*a*e))*(d + e*x)^2) - 3*(b^2 - 4*a*c)*e*(16*c^2*d^2 + 5*b^2*e^2 - 4*c*
e*(4*b*d + a*e))*(d + e*x)^3*Log[d + e*x] + 3*(b^2 - 4*a*c)*e*(16*c^2*d^2 + 5*b^
2*e^2 - 4*c*e*(4*b*d + a*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*(c*d^2 + e*(-(b*d)
+ a*e))^(7/2)*(d + e*x)^3)

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Maple [B]  time = 0.021, size = 2659, normalized size = 8.1 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

c^2/e^2/(a*e^2-b*d*e+c*d^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))-c/e^2/(a*e^2-b*d*e
+c*d^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)^(1
/2)+5/16*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))*b^4-3/2/(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+13/6/(a*e^2-b*d*e+c*d
^2)^2*c/(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)^(1/2
)*b+15/4*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)^(1/2)*b^2*c*d-5/2*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))*b^3*c*d-5/3/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)^(1/2)*b*c*d-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)^(1/2)*b
-5/8*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)^(1/2)*b^3-13/3/e/(a*e^2-b*d*e+c*d^2)^2*c^2/(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)^(1/2)*d+5/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^4*d^4+2/3/e^3/(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)^(1/2)*c*d+5/3/
e^2/(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)^(1/2)*c^2*d^2+5/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)^(1/2)*c^3*d^3+15/2/(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))*b^2*c^2*d^2-6/e^2/(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^2-15/2/(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)^(1/2)*b*c^2*d^2-10/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*c^3*d^3+6/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))*b*d+5/12/(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)^(1/2)*b^2

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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((2*c*x + b)/(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 = 4.8255, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/96*(4*(48*c^3*d^5 - 96*b*c^2*d^4*e - 8*a^2*b*e^5 + 16*(6*b^2*c - 5*a*c^2)*d^3
*e^2 - 3*(11*b^3 - 4*a*b*c)*d^2*e^3 + 2*(13*a*b^2 - 4*a^2*c)*d*e^4 + (16*c^3*d^3
*e^2 - 24*b*c^2*d^2*e^3 + 2*(19*b^2*c - 52*a*c^2)*d*e^4 - (15*b^3 - 52*a*b*c)*e^
5)*x^2 + 2*(24*c^3*d^4*e - 40*b*c^2*d^3*e^2 + 3*(17*b^2*c - 36*a*c^2)*d^2*e^3 -
4*(5*b^3 - 14*a*b*c)*d*e^4 + (5*a*b^2 - 12*a^2*c)*e^5)*x)*sqrt(c*d^2 - b*d*e + a
*e^2)*sqrt(c*x^2 + b*x + a) + 3*(16*(b^2*c^2 - 4*a*c^3)*d^5*e - 16*(b^3*c - 4*a*
b*c^2)*d^4*e^2 + (5*b^4 - 24*a*b^2*c + 16*a^2*c^2)*d^3*e^3 + (16*(b^2*c^2 - 4*a*
c^3)*d^2*e^4 - 16*(b^3*c - 4*a*b*c^2)*d*e^5 + (5*b^4 - 24*a*b^2*c + 16*a^2*c^2)*
e^6)*x^3 + 3*(16*(b^2*c^2 - 4*a*c^3)*d^3*e^3 - 16*(b^3*c - 4*a*b*c^2)*d^2*e^4 +
(5*b^4 - 24*a*b^2*c + 16*a^2*c^2)*d*e^5)*x^2 + 3*(16*(b^2*c^2 - 4*a*c^3)*d^4*e^2
- 16*(b^3*c - 4*a*b*c^2)*d^3*e^3 + (5*b^4 - 24*a*b^2*c + 16*a^2*c^2)*d^2*e^4)*x
)*log(((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^3*d^9 - 3*b*c^2*d^8*e - 3*a^2*b*d^4*e^5
+ a^3*d^3*e^6 + 3*(b^2*c + a*c^2)*d^7*e^2 - (b^3 + 6*a*b*c)*d^6*e^3 + 3*(a*b^2 +
a^2*c)*d^5*e^4 + (c^3*d^6*e^3 - 3*b*c^2*d^5*e^4 - 3*a^2*b*d*e^8 + a^3*e^9 + 3*(
b^2*c + a*c^2)*d^4*e^5 - (b^3 + 6*a*b*c)*d^3*e^6 + 3*(a*b^2 + a^2*c)*d^2*e^7)*x^
3 + 3*(c^3*d^7*e^2 - 3*b*c^2*d^6*e^3 - 3*a^2*b*d^2*e^7 + a^3*d*e^8 + 3*(b^2*c +
a*c^2)*d^5*e^4 - (b^3 + 6*a*b*c)*d^4*e^5 + 3*(a*b^2 + a^2*c)*d^3*e^6)*x^2 + 3*(c
^3*d^8*e - 3*b*c^2*d^7*e^2 - 3*a^2*b*d^3*e^6 + a^3*d^2*e^7 + 3*(b^2*c + a*c^2)*d
^6*e^3 - (b^3 + 6*a*b*c)*d^5*e^4 + 3*(a*b^2 + a^2*c)*d^4*e^5)*x)*sqrt(c*d^2 - b*
d*e + a*e^2)), 1/48*(2*(48*c^3*d^5 - 96*b*c^2*d^4*e - 8*a^2*b*e^5 + 16*(6*b^2*c
- 5*a*c^2)*d^3*e^2 - 3*(11*b^3 - 4*a*b*c)*d^2*e^3 + 2*(13*a*b^2 - 4*a^2*c)*d*e^4
+ (16*c^3*d^3*e^2 - 24*b*c^2*d^2*e^3 + 2*(19*b^2*c - 52*a*c^2)*d*e^4 - (15*b^3
- 52*a*b*c)*e^5)*x^2 + 2*(24*c^3*d^4*e - 40*b*c^2*d^3*e^2 + 3*(17*b^2*c - 36*a*c
^2)*d^2*e^3 - 4*(5*b^3 - 14*a*b*c)*d*e^4 + (5*a*b^2 - 12*a^2*c)*e^5)*x)*sqrt(-c*
d^2 + b*d*e - a*e^2)*sqrt(c*x^2 + b*x + a) + 3*(16*(b^2*c^2 - 4*a*c^3)*d^5*e - 1
6*(b^3*c - 4*a*b*c^2)*d^4*e^2 + (5*b^4 - 24*a*b^2*c + 16*a^2*c^2)*d^3*e^3 + (16*
(b^2*c^2 - 4*a*c^3)*d^2*e^4 - 16*(b^3*c - 4*a*b*c^2)*d*e^5 + (5*b^4 - 24*a*b^2*c
+ 16*a^2*c^2)*e^6)*x^3 + 3*(16*(b^2*c^2 - 4*a*c^3)*d^3*e^3 - 16*(b^3*c - 4*a*b*
c^2)*d^2*e^4 + (5*b^4 - 24*a*b^2*c + 16*a^2*c^2)*d*e^5)*x^2 + 3*(16*(b^2*c^2 - 4
*a*c^3)*d^4*e^2 - 16*(b^3*c - 4*a*b*c^2)*d^3*e^3 + (5*b^4 - 24*a*b^2*c + 16*a^2*
c^2)*d^2*e^4)*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^3*d^9 - 3*b*c^2*
d^8*e - 3*a^2*b*d^4*e^5 + a^3*d^3*e^6 + 3*(b^2*c + a*c^2)*d^7*e^2 - (b^3 + 6*a*b
*c)*d^6*e^3 + 3*(a*b^2 + a^2*c)*d^5*e^4 + (c^3*d^6*e^3 - 3*b*c^2*d^5*e^4 - 3*a^2
*b*d*e^8 + a^3*e^9 + 3*(b^2*c + a*c^2)*d^4*e^5 - (b^3 + 6*a*b*c)*d^3*e^6 + 3*(a*
b^2 + a^2*c)*d^2*e^7)*x^3 + 3*(c^3*d^7*e^2 - 3*b*c^2*d^6*e^3 - 3*a^2*b*d^2*e^7 +
a^3*d*e^8 + 3*(b^2*c + a*c^2)*d^5*e^4 - (b^3 + 6*a*b*c)*d^4*e^5 + 3*(a*b^2 + a^
2*c)*d^3*e^6)*x^2 + 3*(c^3*d^8*e - 3*b*c^2*d^7*e^2 - 3*a^2*b*d^3*e^6 + a^3*d^2*e
^7 + 3*(b^2*c + a*c^2)*d^6*e^3 - (b^3 + 6*a*b*c)*d^5*e^4 + 3*(a*b^2 + a^2*c)*d^4
*e^5)*x)*sqrt(-c*d^2 + b*d*e - a*e^2))]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.651104, size = 4, normalized size = 0.01 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sage0*x