3.1593 \(\int \frac{b+2 c x}{(d+e x) \left (a+b x+c x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=291 \[ -\frac{2 \left (\left (b^2-4 a c\right ) (c d-b e)-c e x \left (b^2-4 a c\right )\right )}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2} \left (a e^2-b d e+c d^2\right )}-\frac{2 e \left (-c x \left (-4 c e (2 a e+b d)+3 b^2 e^2+4 c^2 d^2\right )-2 b c \left (c d^2-5 a e^2\right )-12 a c^2 d e-3 b^3 e^2+5 b^2 c d e\right )}{3 \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2} \left (a e^2-b d e+c d^2\right )^2}-\frac{e^3 (2 c d-b e) \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{\left (a e^2-b d e+c d^2\right )^{5/2}} \]

[Out]

(-2*((b^2 - 4*a*c)*(c*d - b*e) - c*(b^2 - 4*a*c)*e*x))/(3*(b^2 - 4*a*c)*(c*d^2 -
b*d*e + a*e^2)*(a + b*x + c*x^2)^(3/2)) - (2*e*(5*b^2*c*d*e - 12*a*c^2*d*e - 3*
b^3*e^2 - 2*b*c*(c*d^2 - 5*a*e^2) - c*(4*c^2*d^2 + 3*b^2*e^2 - 4*c*e*(b*d + 2*a*
e))*x))/(3*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)^2*Sqrt[a + b*x + c*x^2]) - (e^3
*(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])])/(c*d^2 - b*d*e + a*e^2)^(5/2)

_______________________________________________________________________________________

Rubi [A]  time = 0.782848, antiderivative size = 291, 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 \[ -\frac{2 e \left (-c x \left (-4 c e (2 a e+b d)+3 b^2 e^2+4 c^2 d^2\right )-2 b c \left (c d^2-5 a e^2\right )-12 a c^2 d e-3 b^3 e^2+5 b^2 c d e\right )}{3 \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2} \left (a e^2-b d e+c d^2\right )^2}-\frac{2 (-b e+c d-c e x)}{3 \left (a+b x+c x^2\right )^{3/2} \left (a e^2-b d e+c d^2\right )}-\frac{e^3 (2 c d-b e) \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{\left (a e^2-b d e+c d^2\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]  Int[(b + 2*c*x)/((d + e*x)*(a + b*x + c*x^2)^(5/2)),x]

[Out]

(-2*(c*d - b*e - c*e*x))/(3*(c*d^2 - b*d*e + a*e^2)*(a + b*x + c*x^2)^(3/2)) - (
2*e*(5*b^2*c*d*e - 12*a*c^2*d*e - 3*b^3*e^2 - 2*b*c*(c*d^2 - 5*a*e^2) - c*(4*c^2
*d^2 + 3*b^2*e^2 - 4*c*e*(b*d + 2*a*e))*x))/(3*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*
e^2)^2*Sqrt[a + b*x + c*x^2]) - (e^3*(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])])/(c*d^2 - b*d*
e + a*e^2)^(5/2)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 127.062, size = 257, normalized size = 0.88 \[ - \frac{e^{3} \left (b e - 2 c d\right ) \operatorname{atanh}{\left (\frac{2 a e - b d + x \left (b e - 2 c d\right )}{2 \sqrt{a + b x + c x^{2}} \sqrt{a e^{2} - b d e + c d^{2}}} \right )}}{\left (a e^{2} - b d e + c d^{2}\right )^{\frac{5}{2}}} + \frac{4 e \left (- 2 a c e \left (b e - 2 c d\right ) + \frac{c x \left (- 8 a c e^{2} + 3 b^{2} e^{2} - 4 b c d e + 4 c^{2} d^{2}\right )}{2} + \left (\frac{3 b e}{2} - c d\right ) \left (- 2 a c e + b^{2} e - b c d\right )\right )}{3 \left (- 4 a c + b^{2}\right ) \sqrt{a + b x + c x^{2}} \left (a e^{2} - b d e + c d^{2}\right )^{2}} + \frac{2 \left (b e - c d + c e x\right )}{3 \left (a + b x + c x^{2}\right )^{\frac{3}{2}} \left (a e^{2} - b d e + c d^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((2*c*x+b)/(e*x+d)/(c*x**2+b*x+a)**(5/2),x)

[Out]

-e**3*(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)))/(a*e**2 - b*d*e + c*d**2)**(5/2) + 4*e*(-2*a
*c*e*(b*e - 2*c*d) + c*x*(-8*a*c*e**2 + 3*b**2*e**2 - 4*b*c*d*e + 4*c**2*d**2)/2
+ (3*b*e/2 - c*d)*(-2*a*c*e + b**2*e - b*c*d))/(3*(-4*a*c + b**2)*sqrt(a + b*x
+ c*x**2)*(a*e**2 - b*d*e + c*d**2)**2) + 2*(b*e - c*d + c*e*x)/(3*(a + b*x + c*
x**2)**(3/2)*(a*e**2 - b*d*e + c*d**2))

_______________________________________________________________________________________

Mathematica [A]  time = 1.19075, size = 285, normalized size = 0.98 \[ \frac{2 e \left (2 b c \left (c d (d-2 e x)-5 a e^2\right )+4 c^2 \left (a e (3 d-2 e x)+c d^2 x\right )+3 b^3 e^2+b^2 c e (3 e x-5 d)\right )}{3 \left (b^2-4 a c\right ) \sqrt{a+x (b+c x)} \left (e (a e-b d)+c d^2\right )^2}+\frac{e^3 (b e-2 c d) \log (d+e x)}{\left (e (a e-b d)+c d^2\right )^{5/2}}+\frac{e^3 (2 c d-b e) \log \left (2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2}+2 a e-b d+b e x-2 c d x\right )}{\left (e (a e-b d)+c d^2\right )^{5/2}}+\frac{2 (b e-c d+c e x)}{3 (a+x (b+c x))^{3/2} \left (e (a e-b d)+c d^2\right )} \]

Antiderivative was successfully verified.

[In]  Integrate[(b + 2*c*x)/((d + e*x)*(a + b*x + c*x^2)^(5/2)),x]

[Out]

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

_______________________________________________________________________________________

Maple [B]  time = 0.018, size = 2451, normalized size = 8.4 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2*c*x+b)/(e*x+d)/(c*x^2+b*x+a)^(5/2),x)

[Out]

-e^3/(a*e^2-b*d*e+c*d^2)^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-2/3*e/(a*e^2-b*d*
e+c*d^2)/(4*a*c-b^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*x*b^2-4/3/e/(a*e^2-b*d*e+c*d^2)/(4*a*c-b^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*c^2*d^2-16/3*e/(a*e^2-b*d*e+c*d^2)*c^2/
(4*a*c-b^2)^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^2-4*e/(a*e^2-b*d*e+c*d^2)^2/(4*a*c-b^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^2*d^2-32/3/e/(a*e^2-b*d*e+c*d^2)*c^3/(4*a*c-b^
2)^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*d^2-2*e
^3/(a*e^2-b*d*e+c*d^2)^2/(4*a*c-b^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^2*c-8*e/(a*e^2-b*d*e+c*d^2)^2/(4*a*c-b^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*c^3*d^2+4*e^2/(a*e^2-b*
d*e+c*d^2)^2/(4*a*c-b^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*c*d-8/3/e/(a*e^2-b*d*e+c*d^2)/(4*a*c-b^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^3*x*d^2-64/3/e/(a*e^2-b*d*e+c*d^2)*
c^4/(4*a*c-b^2)^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*d^2+8/3/(a*e^2-b*d*e+c*d^2)/(4*a*c-b^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^2*x*b*d+64/3/(a*e^2-b*d*e+c*d^2)*c^3/(4*a*c-b^
2)^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+64/
3*c^3/e/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*x+32/3*c^2/e/(4*a*c-b^2)^2/(c*x^2+b*x+
a)^(1/2)*b+4/3/(a*e^2-b*d*e+c*d^2)/(4*a*c-b^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^2*c*d+32/3/(a*e^2-b*d*e+c*d^2)*c^2/(4*a*c-b^2
)^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*d-8/3*
e/(a*e^2-b*d*e+c*d^2)*c/(4*a*c-b^2)^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^3+2*e^2/(a*e^2-b*d*e+c*d^2)^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*d-2*e^2/(a*e^2-b*d*e+c*d^2)^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*d-e^3/(a*e^2-b*d*e+c*d^2)^2/(4*a*c-b^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^3+8/3*c^2/e/(4*a*c-b^2
)/(c*x^2+b*x+a)^(3/2)*x+4/3*c/e/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*b-1/3*e/(a*e^2-b
*d*e+c*d^2)/(4*a*c-b^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^3+1/3*e/(a*e^2-b*d*e+c*d^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+e^3/(a*e^2-b*d*e+c*d^2)^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-2/3/(a*e^2-b*d*e+c*d^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+8*e^2/(a*e^2-b*d*e+c*d^2
)^2/(4*a*c-b^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*c^2*d

_______________________________________________________________________________________

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)/((c*x^2 + b*x + a)^(5/2)*(e*x + d)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 1.41375, 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)/((c*x^2 + b*x + a)^(5/2)*(e*x + d)),x, algorithm="fricas")

[Out]

[-1/6*(4*((b^2*c^2 - 4*a*c^3)*d^3 - 2*(b^3*c - 3*a*b*c^2)*d^2*e + (b^4 + 2*a*b^2
*c - 16*a^2*c^2)*d*e^2 - 2*(2*a*b^3 - 7*a^2*b*c)*e^3 - (4*c^4*d^2*e - 4*b*c^3*d*
e^2 + (3*b^2*c^2 - 8*a*c^3)*e^3)*x^3 - 3*(2*b*c^3*d^2*e - (3*b^2*c^2 - 4*a*c^3)*
d*e^2 + 2*(b^3*c - 3*a*b*c^2)*e^3)*x^2 - 3*(b^2*c^2*d^2*e - 2*(b^3*c - 2*a*b*c^2
)*d*e^2 + (b^4 - 2*a*b^2*c - 4*a^2*c^2)*e^3)*x)*sqrt(c*d^2 - b*d*e + a*e^2)*sqrt
(c*x^2 + b*x + a) + 3*(2*(a^2*b^2*c - 4*a^3*c^2)*d*e^3 - (a^2*b^3 - 4*a^3*b*c)*e
^4 + (2*(b^2*c^3 - 4*a*c^4)*d*e^3 - (b^3*c^2 - 4*a*b*c^3)*e^4)*x^4 + 2*(2*(b^3*c
^2 - 4*a*b*c^3)*d*e^3 - (b^4*c - 4*a*b^2*c^2)*e^4)*x^3 + (2*(b^4*c - 2*a*b^2*c^2
- 8*a^2*c^3)*d*e^3 - (b^5 - 2*a*b^3*c - 8*a^2*b*c^2)*e^4)*x^2 + 2*(2*(a*b^3*c -
4*a^2*b*c^2)*d*e^3 - (a*b^4 - 4*a^2*b^2*c)*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)))/(((a^2*b^2*c^2 - 4*a^3*c^3)*d^4 - 2*(a^2*b^3*c - 4*a^3*b*c^2)*d^3*e + (a^2
*b^4 - 2*a^3*b^2*c - 8*a^4*c^2)*d^2*e^2 - 2*(a^3*b^3 - 4*a^4*b*c)*d*e^3 + (a^4*b
^2 - 4*a^5*c)*e^4 + ((b^2*c^4 - 4*a*c^5)*d^4 - 2*(b^3*c^3 - 4*a*b*c^4)*d^3*e + (
b^4*c^2 - 2*a*b^2*c^3 - 8*a^2*c^4)*d^2*e^2 - 2*(a*b^3*c^2 - 4*a^2*b*c^3)*d*e^3 +
(a^2*b^2*c^2 - 4*a^3*c^3)*e^4)*x^4 + 2*((b^3*c^3 - 4*a*b*c^4)*d^4 - 2*(b^4*c^2
- 4*a*b^2*c^3)*d^3*e + (b^5*c - 2*a*b^3*c^2 - 8*a^2*b*c^3)*d^2*e^2 - 2*(a*b^4*c
- 4*a^2*b^2*c^2)*d*e^3 + (a^2*b^3*c - 4*a^3*b*c^2)*e^4)*x^3 + ((b^4*c^2 - 2*a*b^
2*c^3 - 8*a^2*c^4)*d^4 - 2*(b^5*c - 2*a*b^3*c^2 - 8*a^2*b*c^3)*d^3*e + (b^6 - 12
*a^2*b^2*c^2 - 16*a^3*c^3)*d^2*e^2 - 2*(a*b^5 - 2*a^2*b^3*c - 8*a^3*b*c^2)*d*e^3
+ (a^2*b^4 - 2*a^3*b^2*c - 8*a^4*c^2)*e^4)*x^2 + 2*((a*b^3*c^2 - 4*a^2*b*c^3)*d
^4 - 2*(a*b^4*c - 4*a^2*b^2*c^2)*d^3*e + (a*b^5 - 2*a^2*b^3*c - 8*a^3*b*c^2)*d^2
*e^2 - 2*(a^2*b^4 - 4*a^3*b^2*c)*d*e^3 + (a^3*b^3 - 4*a^4*b*c)*e^4)*x)*sqrt(c*d^
2 - b*d*e + a*e^2)), -1/3*(2*((b^2*c^2 - 4*a*c^3)*d^3 - 2*(b^3*c - 3*a*b*c^2)*d^
2*e + (b^4 + 2*a*b^2*c - 16*a^2*c^2)*d*e^2 - 2*(2*a*b^3 - 7*a^2*b*c)*e^3 - (4*c^
4*d^2*e - 4*b*c^3*d*e^2 + (3*b^2*c^2 - 8*a*c^3)*e^3)*x^3 - 3*(2*b*c^3*d^2*e - (3
*b^2*c^2 - 4*a*c^3)*d*e^2 + 2*(b^3*c - 3*a*b*c^2)*e^3)*x^2 - 3*(b^2*c^2*d^2*e -
2*(b^3*c - 2*a*b*c^2)*d*e^2 + (b^4 - 2*a*b^2*c - 4*a^2*c^2)*e^3)*x)*sqrt(-c*d^2
+ b*d*e - a*e^2)*sqrt(c*x^2 + b*x + a) - 3*(2*(a^2*b^2*c - 4*a^3*c^2)*d*e^3 - (a
^2*b^3 - 4*a^3*b*c)*e^4 + (2*(b^2*c^3 - 4*a*c^4)*d*e^3 - (b^3*c^2 - 4*a*b*c^3)*e
^4)*x^4 + 2*(2*(b^3*c^2 - 4*a*b*c^3)*d*e^3 - (b^4*c - 4*a*b^2*c^2)*e^4)*x^3 + (2
*(b^4*c - 2*a*b^2*c^2 - 8*a^2*c^3)*d*e^3 - (b^5 - 2*a*b^3*c - 8*a^2*b*c^2)*e^4)*
x^2 + 2*(2*(a*b^3*c - 4*a^2*b*c^2)*d*e^3 - (a*b^4 - 4*a^2*b^2*c)*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))))/(((a^2*b^2*c^2 - 4*a^3*c^3)*d^4 - 2*(a^2*b^3
*c - 4*a^3*b*c^2)*d^3*e + (a^2*b^4 - 2*a^3*b^2*c - 8*a^4*c^2)*d^2*e^2 - 2*(a^3*b
^3 - 4*a^4*b*c)*d*e^3 + (a^4*b^2 - 4*a^5*c)*e^4 + ((b^2*c^4 - 4*a*c^5)*d^4 - 2*(
b^3*c^3 - 4*a*b*c^4)*d^3*e + (b^4*c^2 - 2*a*b^2*c^3 - 8*a^2*c^4)*d^2*e^2 - 2*(a*
b^3*c^2 - 4*a^2*b*c^3)*d*e^3 + (a^2*b^2*c^2 - 4*a^3*c^3)*e^4)*x^4 + 2*((b^3*c^3
- 4*a*b*c^4)*d^4 - 2*(b^4*c^2 - 4*a*b^2*c^3)*d^3*e + (b^5*c - 2*a*b^3*c^2 - 8*a^
2*b*c^3)*d^2*e^2 - 2*(a*b^4*c - 4*a^2*b^2*c^2)*d*e^3 + (a^2*b^3*c - 4*a^3*b*c^2)
*e^4)*x^3 + ((b^4*c^2 - 2*a*b^2*c^3 - 8*a^2*c^4)*d^4 - 2*(b^5*c - 2*a*b^3*c^2 -
8*a^2*b*c^3)*d^3*e + (b^6 - 12*a^2*b^2*c^2 - 16*a^3*c^3)*d^2*e^2 - 2*(a*b^5 - 2*
a^2*b^3*c - 8*a^3*b*c^2)*d*e^3 + (a^2*b^4 - 2*a^3*b^2*c - 8*a^4*c^2)*e^4)*x^2 +
2*((a*b^3*c^2 - 4*a^2*b*c^3)*d^4 - 2*(a*b^4*c - 4*a^2*b^2*c^2)*d^3*e + (a*b^5 -
2*a^2*b^3*c - 8*a^3*b*c^2)*d^2*e^2 - 2*(a^2*b^4 - 4*a^3*b^2*c)*d*e^3 + (a^3*b^3
- 4*a^4*b*c)*e^4)*x)*sqrt(-c*d^2 + b*d*e - a*e^2))]

_______________________________________________________________________________________

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((2*c*x+b)/(e*x+d)/(c*x**2+b*x+a)**(5/2),x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.358359, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*x + b)/((c*x^2 + b*x + a)^(5/2)*(e*x + d)),x, algorithm="giac")

[Out]

Done