3.1594 \(\int \frac{b+2 c x}{(d+e x)^2 \left (a+b x+c x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=423 \[ \frac{e^2 \sqrt{a+b x+c x^2} (2 c d-b e) \left (-4 c e (13 a e+2 b d)+15 b^2 e^2+8 c^2 d^2\right )}{3 \left (b^2-4 a c\right ) (d+e x) \left (a e^2-b d e+c d^2\right )^3}-\frac{e^3 \left (-4 c e (a e+4 b d)+5 b^2 e^2+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{2 \left (a e^2-b d e+c d^2\right )^{7/2}}-\frac{2 \left (\left (b^2-4 a c\right ) (c d-b e)-c e x \left (b^2-4 a c\right )\right )}{3 \left (b^2-4 a c\right ) (d+e x) \left (a+b x+c x^2\right )^{3/2} \left (a e^2-b d e+c d^2\right )}-\frac{2 e \left (-c x \left (-4 c e (3 a e+2 b d)+5 b^2 e^2+8 c^2 d^2\right )-4 b c \left (c d^2-4 a e^2\right )-20 a c^2 d e-5 b^3 e^2+9 b^2 c d e\right )}{3 \left (b^2-4 a c\right ) (d+e x) \sqrt{a+b x+c x^2} \left (a e^2-b d e+c d^2\right )^2} \]

[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)*(d + e*x)*(a + b*x + c*x^2)^(3/2)) - (2*e*(9*b^2*c*d*e - 20*a*c^
2*d*e - 5*b^3*e^2 - 4*b*c*(c*d^2 - 4*a*e^2) - c*(8*c^2*d^2 + 5*b^2*e^2 - 4*c*e*(
2*b*d + 3*a*e))*x))/(3*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)^2*(d + e*x)*Sqrt[a
+ b*x + c*x^2]) + (e^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])/(3*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)^3*(d + e*
x)) - (e^3*(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])])/(2*(c*
d^2 - b*d*e + a*e^2)^(7/2))

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Rubi [A]  time = 1.45835, antiderivative size = 423, 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 \[ \frac{e^2 \sqrt{a+b x+c x^2} (2 c d-b e) \left (-4 c e (13 a e+2 b d)+15 b^2 e^2+8 c^2 d^2\right )}{3 \left (b^2-4 a c\right ) (d+e x) \left (a e^2-b d e+c d^2\right )^3}-\frac{e^3 \left (-4 c e (a e+4 b d)+5 b^2 e^2+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{2 \left (a e^2-b d e+c d^2\right )^{7/2}}-\frac{2 e \left (-c x \left (-4 c e (3 a e+2 b d)+5 b^2 e^2+8 c^2 d^2\right )-4 b c \left (c d^2-4 a e^2\right )-20 a c^2 d e-5 b^3 e^2+9 b^2 c d e\right )}{3 \left (b^2-4 a c\right ) (d+e x) \sqrt{a+b x+c x^2} \left (a e^2-b d e+c d^2\right )^2}-\frac{2 (-b e+c d-c e x)}{3 (d+e x) \left (a+b x+c x^2\right )^{3/2} \left (a e^2-b d e+c d^2\right )} \]

Antiderivative was successfully verified.

[In]  Int[(b + 2*c*x)/((d + e*x)^2*(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)*(d + e*x)*(a + b*x + c*x^2)^
(3/2)) - (2*e*(9*b^2*c*d*e - 20*a*c^2*d*e - 5*b^3*e^2 - 4*b*c*(c*d^2 - 4*a*e^2)
- c*(8*c^2*d^2 + 5*b^2*e^2 - 4*c*e*(2*b*d + 3*a*e))*x))/(3*(b^2 - 4*a*c)*(c*d^2
- b*d*e + a*e^2)^2*(d + e*x)*Sqrt[a + b*x + c*x^2]) + (e^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])/(3*(b^2 - 4*a*
c)*(c*d^2 - b*d*e + a*e^2)^3*(d + e*x)) - (e^3*(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])])/(2*(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)**2/(c*x**2+b*x+a)**(5/2),x)

[Out]

Timed out

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Mathematica [A]  time = 4.26549, size = 454, normalized size = 1.07 \[ \frac{1}{2} \left (2 \sqrt{a+x (b+c x)} \left (\frac{2 e \left (4 c^2 \left (3 a^2 e^3+a c d e (10 e x-9 d)-2 c^2 d^3 x\right )+b^2 c e \left (c d (15 d-16 e x)-25 a e^2\right )-4 b c^2 \left (a e^2 (5 e x-14 d)+c d^2 (d-3 e x)\right )+6 b^4 e^3+b^3 c e^2 (6 e x-17 d)\right )}{3 \left (b^2-4 a c\right ) (a+x (b+c x)) \left (e (b d-a e)-c d^2\right )^3}-\frac{2 \left (c e (-a e-2 b d+b e x)+b^2 e^2+c^2 d (d-2 e x)\right )}{3 (a+x (b+c x))^2 \left (e (a e-b d)+c d^2\right )^2}-\frac{e^4 (b e-2 c d)}{(d+e x) \left (e (a e-b d)+c d^2\right )^3}\right )+\frac{e^3 \log (d+e x) \left (4 c e (a e+4 b d)-5 b^2 e^2-16 c^2 d^2\right )}{\left (e (a e-b d)+c d^2\right )^{7/2}}+\frac{e^3 \left (-4 c e (a e+4 b d)+5 b^2 e^2+16 c^2 d^2\right ) \log \left (2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2}+2 a e-b d+b e x-2 c d x\right )}{\left (e (a e-b d)+c d^2\right )^{7/2}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[a + x*(b + c*x)]*(-((e^4*(-2*c*d + b*e))/((c*d^2 + e*(-(b*d) + a*e))^3*(
d + e*x))) - (2*(b^2*e^2 + c^2*d*(d - 2*e*x) + c*e*(-2*b*d - a*e + b*e*x)))/(3*(
c*d^2 + e*(-(b*d) + a*e))^2*(a + x*(b + c*x))^2) + (2*e*(6*b^4*e^3 + b^3*c*e^2*(
-17*d + 6*e*x) + b^2*c*e*(-25*a*e^2 + c*d*(15*d - 16*e*x)) - 4*b*c^2*(c*d^2*(d -
3*e*x) + a*e^2*(-14*d + 5*e*x)) + 4*c^2*(3*a^2*e^3 - 2*c^2*d^3*x + a*c*d*e*(-9*
d + 10*e*x))))/(3*(b^2 - 4*a*c)*(-(c*d^2) + e*(b*d - a*e))^3*(a + x*(b + c*x))))
+ (e^3*(-16*c^2*d^2 - 5*b^2*e^2 + 4*c*e*(4*b*d + a*e))*Log[d + e*x])/(c*d^2 + e
*(-(b*d) + a*e))^(7/2) + (e^3*(16*c^2*d^2 + 5*b^2*e^2 - 4*c*e*(4*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)]])/(c*d^2 + e*(-(b*d) + a*e))^(7/2))/2

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Maple [B]  time = 0.03, size = 4855, normalized size = 11.5 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-20/3/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)^(3/2)*b*c^3*d^3+20/3/e/(a*e^2-b*d*e+c*d^2)*c^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*d+320/3/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+160/3/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-10*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*d*b-40*e/(a*e^2-b*d*e+c*d^2)^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^3*d+5/3*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)^(3/2)*c*x*b^3-40/3/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)^(3/2)*c^4*x*d^
3-5*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)^(3/2)*b^3*c*d+160/(a*e^2-b*d*e+c*d^2)^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*b-4*c^2*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+8*c^3*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*d+4*c^2*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*d+2/3*c/(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)-10/3/(a*e^2-b*d*e+c*d^2)*c/(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+30*e^2/(a*e^2-b*d*e+c*d^2)^3/
(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^2*d^2-20*e/(a*e^2-b*d*e+c*d^2)^3/(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^3*d^3+40/3/e/(a*e^2-b*d*e+c*d^2)*c^3/(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)*x*d+20
/(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)^(3/2)*c^3*x*d^2*b-80/3/(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+5/6*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)^(3/2)*b^4-10*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)*c^2*d^2+5/2*e^4/(a*e^2-b*d*e+c*d^2)^3/(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^4+5/2*e^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))*b^2+60*e^2/(a*e^2-b*d*e
+c*d^2)^3/(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*b-80*e/(a*e^2-b*d*e+c*d^2)^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^2*d-10*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)^(3
/2)*c^2*x*b^2*d-320/3/e/(a*e^2-b*d*e+c*d^2)^2*c^5/(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^3+40/3*e^2/(a*e^2-b*d*e+c*
d^2)^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^3-160/3/e/(a*e^2-b*d*e+c*d^2)^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)*b*d^3+5*e^4/(a*e^2-b*d*e+c*d^
2)^3/(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^3*c-40*e/(a*e^2-b*d*e+c*d^2)^3/(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^4*d^3-15*e^3/(a*e^2-b*d*e+c*d^2)^3/(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*c*d
+80/(a*e^2-b*d*e+c*d^2)^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^2*d^2-20/3/(a*e^2-b*d*e+c*d^2)*c^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)*x*b-160/3/(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+10/(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)^(3/2)*b^2*c^2*d^2+20/3*e^2/(a*e^2-b*d*e+
c*d^2)^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^4+2/e/(a*e^2-b*d*e+c*d^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)^(3/2)*c*d+10/3*e/(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)^(3/2)*b*c*d+10*e^3/(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*c*d+1
0*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^2*d^2-2*c*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^2-30*e^3/(a*e^2-b*d*e+c*d^2)^3/(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^2*d-2*c*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))-5/6*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)^(3/2)*b^2-5/2*e^4/(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^2-1/(a*e^2-b*d*e+c*d^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)^(3/2)*b-10/3/(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)^(3/2)*c^2*d^2+2*c*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)

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 5.8964, 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)^2),x, algorithm="fricas")

[Out]

[-1/12*(4*(2*(b^2*c^3 - 4*a*c^4)*d^5 - 2*(3*b^3*c^2 - 8*a*b*c^3)*d^4*e + 6*(b^4*
c + a*b^2*c^2 - 12*a^2*c^3)*d^3*e^2 - 2*(b^5 + 14*a*b^3*c - 60*a^2*b*c^2)*d^2*e^
3 + 2*(7*a*b^4 - 33*a^2*b^2*c + 28*a^3*c^2)*d*e^4 + 3*(a^2*b^3 - 4*a^3*b*c)*e^5
- (16*c^5*d^3*e^2 - 24*b*c^4*d^2*e^3 + 2*(19*b^2*c^3 - 52*a*c^4)*d*e^4 - (15*b^3
*c^2 - 52*a*b*c^3)*e^5)*x^4 - 2*(8*c^5*d^4*e - (11*b^2*c^3 + 4*a*c^4)*d^2*e^3 +
(33*b^3*c^2 - 100*a*b*c^3)*d*e^4 - 3*(5*b^4*c - 19*a*b^2*c^2 + 4*a^2*c^3)*e^5)*x
^3 - 3*(8*b*c^4*d^4*e - 2*(7*b^2*c^3 - 12*a*c^4)*d^3*e^2 + 10*(b^3*c^2 - 4*a*b*c
^3)*d^2*e^3 + 2*(3*b^4*c - a*b^2*c^2 - 28*a^2*c^3)*d*e^4 - (5*b^5 - 14*a*b^3*c -
16*a^2*b*c^2)*e^5)*x^2 - 2*((5*b^2*c^3 + 4*a*c^4)*d^4*e - (15*b^3*c^2 - 28*a*b*
c^3)*d^3*e^2 + 3*(5*b^4*c - 15*a*b^2*c^2 - 4*a^2*c^3)*d^2*e^3 - (5*b^5 - 38*a*b^
3*c + 72*a^2*b*c^2)*d*e^4 - 2*(5*a*b^4 - 21*a^2*b^2*c + 8*a^3*c^2)*e^5)*x)*sqrt(
c*d^2 - b*d*e + a*e^2)*sqrt(c*x^2 + b*x + a) + 3*(16*(a^2*b^2*c^2 - 4*a^3*c^3)*d
^3*e^3 - 16*(a^2*b^3*c - 4*a^3*b*c^2)*d^2*e^4 + (5*a^2*b^4 - 24*a^3*b^2*c + 16*a
^4*c^2)*d*e^5 + (16*(b^2*c^4 - 4*a*c^5)*d^2*e^4 - 16*(b^3*c^3 - 4*a*b*c^4)*d*e^5
+ (5*b^4*c^2 - 24*a*b^2*c^3 + 16*a^2*c^4)*e^6)*x^5 + (16*(b^2*c^4 - 4*a*c^5)*d^
3*e^3 + 16*(b^3*c^3 - 4*a*b*c^4)*d^2*e^4 - (27*b^4*c^2 - 104*a*b^2*c^3 - 16*a^2*
c^4)*d*e^5 + 2*(5*b^5*c - 24*a*b^3*c^2 + 16*a^2*b*c^3)*e^6)*x^4 + (32*(b^3*c^3 -
4*a*b*c^4)*d^3*e^3 - 16*(b^4*c^2 - 6*a*b^2*c^3 + 8*a^2*c^4)*d^2*e^4 - 2*(3*b^5*
c + 8*a*b^3*c^2 - 80*a^2*b*c^3)*d*e^5 + (5*b^6 - 14*a*b^4*c - 32*a^2*b^2*c^2 + 3
2*a^3*c^3)*e^6)*x^3 + (16*(b^4*c^2 - 2*a*b^2*c^3 - 8*a^2*c^4)*d^3*e^3 - 16*(b^5*
c - 4*a*b^3*c^2)*d^2*e^4 + (5*b^6 - 46*a*b^4*c + 96*a^2*b^2*c^2 + 32*a^3*c^3)*d*
e^5 + 2*(5*a*b^5 - 24*a^2*b^3*c + 16*a^3*b*c^2)*e^6)*x^2 + (32*(a*b^3*c^2 - 4*a^
2*b*c^3)*d^3*e^3 - 16*(2*a*b^4*c - 9*a^2*b^2*c^2 + 4*a^3*c^3)*d^2*e^4 + 2*(5*a*b
^5 - 32*a^2*b^3*c + 48*a^3*b*c^2)*d*e^5 + (5*a^2*b^4 - 24*a^3*b^2*c + 16*a^4*c^2
)*e^6)*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^3 - 4*a^3*c^4)*d^7 - 3*
(a^2*b^3*c^2 - 4*a^3*b*c^3)*d^6*e + 3*(a^2*b^4*c - 3*a^3*b^2*c^2 - 4*a^4*c^3)*d^
5*e^2 - (a^2*b^5 + 2*a^3*b^3*c - 24*a^4*b*c^2)*d^4*e^3 + 3*(a^3*b^4 - 3*a^4*b^2*
c - 4*a^5*c^2)*d^3*e^4 - 3*(a^4*b^3 - 4*a^5*b*c)*d^2*e^5 + (a^5*b^2 - 4*a^6*c)*d
*e^6 + ((b^2*c^5 - 4*a*c^6)*d^6*e - 3*(b^3*c^4 - 4*a*b*c^5)*d^5*e^2 + 3*(b^4*c^3
- 3*a*b^2*c^4 - 4*a^2*c^5)*d^4*e^3 - (b^5*c^2 + 2*a*b^3*c^3 - 24*a^2*b*c^4)*d^3
*e^4 + 3*(a*b^4*c^2 - 3*a^2*b^2*c^3 - 4*a^3*c^4)*d^2*e^5 - 3*(a^2*b^3*c^2 - 4*a^
3*b*c^3)*d*e^6 + (a^3*b^2*c^2 - 4*a^4*c^3)*e^7)*x^5 + ((b^2*c^5 - 4*a*c^6)*d^7 -
(b^3*c^4 - 4*a*b*c^5)*d^6*e - 3*(b^4*c^3 - 5*a*b^2*c^4 + 4*a^2*c^5)*d^5*e^2 + 5
*(b^5*c^2 - 4*a*b^3*c^3)*d^4*e^3 - (2*b^6*c + a*b^4*c^2 - 39*a^2*b^2*c^3 + 12*a^
3*c^4)*d^3*e^4 + 3*(2*a*b^5*c - 7*a^2*b^3*c^2 - 4*a^3*b*c^3)*d^2*e^5 - (6*a^2*b^
4*c - 25*a^3*b^2*c^2 + 4*a^4*c^3)*d*e^6 + 2*(a^3*b^3*c - 4*a^4*b*c^2)*e^7)*x^4 +
(2*(b^3*c^4 - 4*a*b*c^5)*d^7 - (5*b^4*c^3 - 22*a*b^2*c^4 + 8*a^2*c^5)*d^6*e + 3
*(b^5*c^2 - 4*a*b^3*c^3)*d^5*e^2 + (b^6*c - 7*a*b^4*c^2 + 18*a^2*b^2*c^3 - 24*a^
3*c^4)*d^4*e^3 - (b^7 - 2*a*b^5*c - 2*a^2*b^3*c^2 - 24*a^3*b*c^3)*d^3*e^4 + 3*(a
*b^6 - 3*a^2*b^4*c - 2*a^3*b^2*c^2 - 8*a^4*c^3)*d^2*e^5 - (3*a^2*b^5 - 8*a^3*b^3
*c - 16*a^4*b*c^2)*d*e^6 + (a^3*b^4 - 2*a^4*b^2*c - 8*a^5*c^2)*e^7)*x^3 + ((b^4*
c^3 - 2*a*b^2*c^4 - 8*a^2*c^5)*d^7 - (3*b^5*c^2 - 8*a*b^3*c^3 - 16*a^2*b*c^4)*d^
6*e + 3*(b^6*c - 3*a*b^4*c^2 - 2*a^2*b^2*c^3 - 8*a^3*c^4)*d^5*e^2 - (b^7 - 2*a*b
^5*c - 2*a^2*b^3*c^2 - 24*a^3*b*c^3)*d^4*e^3 + (a*b^6 - 7*a^2*b^4*c + 18*a^3*b^2
*c^2 - 24*a^4*c^3)*d^3*e^4 + 3*(a^2*b^5 - 4*a^3*b^3*c)*d^2*e^5 - (5*a^3*b^4 - 22
*a^4*b^2*c + 8*a^5*c^2)*d*e^6 + 2*(a^4*b^3 - 4*a^5*b*c)*e^7)*x^2 + (2*(a*b^3*c^3
- 4*a^2*b*c^4)*d^7 - (6*a*b^4*c^2 - 25*a^2*b^2*c^3 + 4*a^3*c^4)*d^6*e + 3*(2*a*
b^5*c - 7*a^2*b^3*c^2 - 4*a^3*b*c^3)*d^5*e^2 - (2*a*b^6 + a^2*b^4*c - 39*a^3*b^2
*c^2 + 12*a^4*c^3)*d^4*e^3 + 5*(a^2*b^5 - 4*a^3*b^3*c)*d^3*e^4 - 3*(a^3*b^4 - 5*
a^4*b^2*c + 4*a^5*c^2)*d^2*e^5 - (a^4*b^3 - 4*a^5*b*c)*d*e^6 + (a^5*b^2 - 4*a^6*
c)*e^7)*x)*sqrt(c*d^2 - b*d*e + a*e^2)), -1/6*(2*(2*(b^2*c^3 - 4*a*c^4)*d^5 - 2*
(3*b^3*c^2 - 8*a*b*c^3)*d^4*e + 6*(b^4*c + a*b^2*c^2 - 12*a^2*c^3)*d^3*e^2 - 2*(
b^5 + 14*a*b^3*c - 60*a^2*b*c^2)*d^2*e^3 + 2*(7*a*b^4 - 33*a^2*b^2*c + 28*a^3*c^
2)*d*e^4 + 3*(a^2*b^3 - 4*a^3*b*c)*e^5 - (16*c^5*d^3*e^2 - 24*b*c^4*d^2*e^3 + 2*
(19*b^2*c^3 - 52*a*c^4)*d*e^4 - (15*b^3*c^2 - 52*a*b*c^3)*e^5)*x^4 - 2*(8*c^5*d^
4*e - (11*b^2*c^3 + 4*a*c^4)*d^2*e^3 + (33*b^3*c^2 - 100*a*b*c^3)*d*e^4 - 3*(5*b
^4*c - 19*a*b^2*c^2 + 4*a^2*c^3)*e^5)*x^3 - 3*(8*b*c^4*d^4*e - 2*(7*b^2*c^3 - 12
*a*c^4)*d^3*e^2 + 10*(b^3*c^2 - 4*a*b*c^3)*d^2*e^3 + 2*(3*b^4*c - a*b^2*c^2 - 28
*a^2*c^3)*d*e^4 - (5*b^5 - 14*a*b^3*c - 16*a^2*b*c^2)*e^5)*x^2 - 2*((5*b^2*c^3 +
4*a*c^4)*d^4*e - (15*b^3*c^2 - 28*a*b*c^3)*d^3*e^2 + 3*(5*b^4*c - 15*a*b^2*c^2
- 4*a^2*c^3)*d^2*e^3 - (5*b^5 - 38*a*b^3*c + 72*a^2*b*c^2)*d*e^4 - 2*(5*a*b^4 -
21*a^2*b^2*c + 8*a^3*c^2)*e^5)*x)*sqrt(-c*d^2 + b*d*e - a*e^2)*sqrt(c*x^2 + b*x
+ a) - 3*(16*(a^2*b^2*c^2 - 4*a^3*c^3)*d^3*e^3 - 16*(a^2*b^3*c - 4*a^3*b*c^2)*d^
2*e^4 + (5*a^2*b^4 - 24*a^3*b^2*c + 16*a^4*c^2)*d*e^5 + (16*(b^2*c^4 - 4*a*c^5)*
d^2*e^4 - 16*(b^3*c^3 - 4*a*b*c^4)*d*e^5 + (5*b^4*c^2 - 24*a*b^2*c^3 + 16*a^2*c^
4)*e^6)*x^5 + (16*(b^2*c^4 - 4*a*c^5)*d^3*e^3 + 16*(b^3*c^3 - 4*a*b*c^4)*d^2*e^4
- (27*b^4*c^2 - 104*a*b^2*c^3 - 16*a^2*c^4)*d*e^5 + 2*(5*b^5*c - 24*a*b^3*c^2 +
16*a^2*b*c^3)*e^6)*x^4 + (32*(b^3*c^3 - 4*a*b*c^4)*d^3*e^3 - 16*(b^4*c^2 - 6*a*
b^2*c^3 + 8*a^2*c^4)*d^2*e^4 - 2*(3*b^5*c + 8*a*b^3*c^2 - 80*a^2*b*c^3)*d*e^5 +
(5*b^6 - 14*a*b^4*c - 32*a^2*b^2*c^2 + 32*a^3*c^3)*e^6)*x^3 + (16*(b^4*c^2 - 2*a
*b^2*c^3 - 8*a^2*c^4)*d^3*e^3 - 16*(b^5*c - 4*a*b^3*c^2)*d^2*e^4 + (5*b^6 - 46*a
*b^4*c + 96*a^2*b^2*c^2 + 32*a^3*c^3)*d*e^5 + 2*(5*a*b^5 - 24*a^2*b^3*c + 16*a^3
*b*c^2)*e^6)*x^2 + (32*(a*b^3*c^2 - 4*a^2*b*c^3)*d^3*e^3 - 16*(2*a*b^4*c - 9*a^2
*b^2*c^2 + 4*a^3*c^3)*d^2*e^4 + 2*(5*a*b^5 - 32*a^2*b^3*c + 48*a^3*b*c^2)*d*e^5
+ (5*a^2*b^4 - 24*a^3*b^2*c + 16*a^4*c^2)*e^6)*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^3 - 4*a^3*c^4)*d^7 - 3*(a^2*b^3*c^2 - 4*a^3*b*c^3)*d^6
*e + 3*(a^2*b^4*c - 3*a^3*b^2*c^2 - 4*a^4*c^3)*d^5*e^2 - (a^2*b^5 + 2*a^3*b^3*c
- 24*a^4*b*c^2)*d^4*e^3 + 3*(a^3*b^4 - 3*a^4*b^2*c - 4*a^5*c^2)*d^3*e^4 - 3*(a^4
*b^3 - 4*a^5*b*c)*d^2*e^5 + (a^5*b^2 - 4*a^6*c)*d*e^6 + ((b^2*c^5 - 4*a*c^6)*d^6
*e - 3*(b^3*c^4 - 4*a*b*c^5)*d^5*e^2 + 3*(b^4*c^3 - 3*a*b^2*c^4 - 4*a^2*c^5)*d^4
*e^3 - (b^5*c^2 + 2*a*b^3*c^3 - 24*a^2*b*c^4)*d^3*e^4 + 3*(a*b^4*c^2 - 3*a^2*b^2
*c^3 - 4*a^3*c^4)*d^2*e^5 - 3*(a^2*b^3*c^2 - 4*a^3*b*c^3)*d*e^6 + (a^3*b^2*c^2 -
4*a^4*c^3)*e^7)*x^5 + ((b^2*c^5 - 4*a*c^6)*d^7 - (b^3*c^4 - 4*a*b*c^5)*d^6*e -
3*(b^4*c^3 - 5*a*b^2*c^4 + 4*a^2*c^5)*d^5*e^2 + 5*(b^5*c^2 - 4*a*b^3*c^3)*d^4*e^
3 - (2*b^6*c + a*b^4*c^2 - 39*a^2*b^2*c^3 + 12*a^3*c^4)*d^3*e^4 + 3*(2*a*b^5*c -
7*a^2*b^3*c^2 - 4*a^3*b*c^3)*d^2*e^5 - (6*a^2*b^4*c - 25*a^3*b^2*c^2 + 4*a^4*c^
3)*d*e^6 + 2*(a^3*b^3*c - 4*a^4*b*c^2)*e^7)*x^4 + (2*(b^3*c^4 - 4*a*b*c^5)*d^7 -
(5*b^4*c^3 - 22*a*b^2*c^4 + 8*a^2*c^5)*d^6*e + 3*(b^5*c^2 - 4*a*b^3*c^3)*d^5*e^
2 + (b^6*c - 7*a*b^4*c^2 + 18*a^2*b^2*c^3 - 24*a^3*c^4)*d^4*e^3 - (b^7 - 2*a*b^5
*c - 2*a^2*b^3*c^2 - 24*a^3*b*c^3)*d^3*e^4 + 3*(a*b^6 - 3*a^2*b^4*c - 2*a^3*b^2*
c^2 - 8*a^4*c^3)*d^2*e^5 - (3*a^2*b^5 - 8*a^3*b^3*c - 16*a^4*b*c^2)*d*e^6 + (a^3
*b^4 - 2*a^4*b^2*c - 8*a^5*c^2)*e^7)*x^3 + ((b^4*c^3 - 2*a*b^2*c^4 - 8*a^2*c^5)*
d^7 - (3*b^5*c^2 - 8*a*b^3*c^3 - 16*a^2*b*c^4)*d^6*e + 3*(b^6*c - 3*a*b^4*c^2 -
2*a^2*b^2*c^3 - 8*a^3*c^4)*d^5*e^2 - (b^7 - 2*a*b^5*c - 2*a^2*b^3*c^2 - 24*a^3*b
*c^3)*d^4*e^3 + (a*b^6 - 7*a^2*b^4*c + 18*a^3*b^2*c^2 - 24*a^4*c^3)*d^3*e^4 + 3*
(a^2*b^5 - 4*a^3*b^3*c)*d^2*e^5 - (5*a^3*b^4 - 22*a^4*b^2*c + 8*a^5*c^2)*d*e^6 +
2*(a^4*b^3 - 4*a^5*b*c)*e^7)*x^2 + (2*(a*b^3*c^3 - 4*a^2*b*c^4)*d^7 - (6*a*b^4*
c^2 - 25*a^2*b^2*c^3 + 4*a^3*c^4)*d^6*e + 3*(2*a*b^5*c - 7*a^2*b^3*c^2 - 4*a^3*b
*c^3)*d^5*e^2 - (2*a*b^6 + a^2*b^4*c - 39*a^3*b^2*c^2 + 12*a^4*c^3)*d^4*e^3 + 5*
(a^2*b^5 - 4*a^3*b^3*c)*d^3*e^4 - 3*(a^3*b^4 - 5*a^4*b^2*c + 4*a^5*c^2)*d^2*e^5
- (a^4*b^3 - 4*a^5*b*c)*d*e^6 + (a^5*b^2 - 4*a^6*c)*e^7)*x)*sqrt(-c*d^2 + b*d*e
- a*e^2))]

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

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{2 \, c x + b}{{\left (c x^{2} + b x + a\right )}^{\frac{5}{2}}{\left (e x + d\right )}^{2}}\,{d x} \]

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)^2),x, algorithm="giac")

[Out]

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