3.2331 \(\int \frac{\sqrt{a+b x+c x^2}}{(d+e x)^6} \, dx\)

Optimal. Leaf size=402 \[ -\frac{e \left (a+b x+c x^2\right )^{3/2} \left (-4 c e (8 a e+27 b d)+35 b^2 e^2+108 c^2 d^2\right )}{240 (d+e x)^3 \left (a e^2-b d e+c d^2\right )^3}+\frac{\sqrt{a+b x+c x^2} (2 c d-b e) \left (-4 c e (3 a e+4 b d)+7 b^2 e^2+16 c^2 d^2\right ) (-2 a e+x (2 c d-b e)+b d)}{128 (d+e x)^2 \left (a e^2-b d e+c d^2\right )^4}-\frac{\left (b^2-4 a c\right ) (2 c d-b e) \left (-4 c e (3 a e+4 b d)+7 b^2 e^2+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{256 \left (a e^2-b d e+c d^2\right )^{9/2}}-\frac{7 e \left (a+b x+c x^2\right )^{3/2} (2 c d-b e)}{40 (d+e x)^4 \left (a e^2-b d e+c d^2\right )^2}-\frac{e \left (a+b x+c x^2\right )^{3/2}}{5 (d+e x)^5 \left (a e^2-b d e+c d^2\right )} \]

[Out]

((2*c*d - b*e)*(16*c^2*d^2 + 7*b^2*e^2 - 4*c*e*(4*b*d + 3*a*e))*(b*d - 2*a*e + (
2*c*d - b*e)*x)*Sqrt[a + b*x + c*x^2])/(128*(c*d^2 - b*d*e + a*e^2)^4*(d + e*x)^
2) - (e*(a + b*x + c*x^2)^(3/2))/(5*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^5) - (7*e*
(2*c*d - b*e)*(a + b*x + c*x^2)^(3/2))/(40*(c*d^2 - b*d*e + a*e^2)^2*(d + e*x)^4
) - (e*(108*c^2*d^2 + 35*b^2*e^2 - 4*c*e*(27*b*d + 8*a*e))*(a + b*x + c*x^2)^(3/
2))/(240*(c*d^2 - b*d*e + a*e^2)^3*(d + e*x)^3) - ((b^2 - 4*a*c)*(2*c*d - b*e)*(
16*c^2*d^2 + 7*b^2*e^2 - 4*c*e*(4*b*d + 3*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])])/(256*(c*d^2 - b*
d*e + a*e^2)^(9/2))

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Rubi [A]  time = 1.30607, antiderivative size = 402, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273 \[ -\frac{e \left (a+b x+c x^2\right )^{3/2} \left (-4 c e (8 a e+27 b d)+35 b^2 e^2+108 c^2 d^2\right )}{240 (d+e x)^3 \left (a e^2-b d e+c d^2\right )^3}+\frac{\sqrt{a+b x+c x^2} (2 c d-b e) \left (-4 c e (3 a e+4 b d)+7 b^2 e^2+16 c^2 d^2\right ) (-2 a e+x (2 c d-b e)+b d)}{128 (d+e x)^2 \left (a e^2-b d e+c d^2\right )^4}-\frac{\left (b^2-4 a c\right ) (2 c d-b e) \left (-4 c e (3 a e+4 b d)+7 b^2 e^2+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{256 \left (a e^2-b d e+c d^2\right )^{9/2}}-\frac{7 e \left (a+b x+c x^2\right )^{3/2} (2 c d-b e)}{40 (d+e x)^4 \left (a e^2-b d e+c d^2\right )^2}-\frac{e \left (a+b x+c x^2\right )^{3/2}}{5 (d+e x)^5 \left (a e^2-b d e+c d^2\right )} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a + b*x + c*x^2]/(d + e*x)^6,x]

[Out]

((2*c*d - b*e)*(16*c^2*d^2 + 7*b^2*e^2 - 4*c*e*(4*b*d + 3*a*e))*(b*d - 2*a*e + (
2*c*d - b*e)*x)*Sqrt[a + b*x + c*x^2])/(128*(c*d^2 - b*d*e + a*e^2)^4*(d + e*x)^
2) - (e*(a + b*x + c*x^2)^(3/2))/(5*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^5) - (7*e*
(2*c*d - b*e)*(a + b*x + c*x^2)^(3/2))/(40*(c*d^2 - b*d*e + a*e^2)^2*(d + e*x)^4
) - (e*(108*c^2*d^2 + 35*b^2*e^2 - 4*c*e*(27*b*d + 8*a*e))*(a + b*x + c*x^2)^(3/
2))/(240*(c*d^2 - b*d*e + a*e^2)^3*(d + e*x)^3) - ((b^2 - 4*a*c)*(2*c*d - b*e)*(
16*c^2*d^2 + 7*b^2*e^2 - 4*c*e*(4*b*d + 3*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])])/(256*(c*d^2 - b*
d*e + a*e^2)^(9/2))

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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((c*x**2+b*x+a)**(1/2)/(e*x+d)**6,x)

[Out]

Timed out

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Mathematica [A]  time = 3.32918, size = 511, normalized size = 1.27 \[ \frac{-2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2} \left (-(d+e x)^4 \left (4 c^2 e^2 \left (64 a^2 e^2+332 a b d e+119 b^2 d^2\right )-20 b^2 c e^3 (23 a e+19 b d)-16 c^3 d^2 e (83 a e+12 b d)+105 b^4 e^4+96 c^4 d^4\right )-8 (d+e x)^2 \left (-4 c e (4 a e+3 b d)+7 b^2 e^2+12 c^2 d^2\right ) \left (e (a e-b d)+c d^2\right )^2-2 (d+e x)^3 (2 c d-b e) \left (-4 c e (29 a e+6 b d)+35 b^2 e^2+24 c^2 d^2\right ) \left (e (a e-b d)+c d^2\right )-48 (d+e x) (2 c d-b e) \left (e (a e-b d)+c d^2\right )^3+384 \left (e (a e-b d)+c d^2\right )^4\right )+15 e \left (b^2-4 a c\right ) (d+e x)^5 (b e-2 c d) \log (d+e x) \left (-4 c e (3 a e+4 b d)+7 b^2 e^2+16 c^2 d^2\right )-15 e \left (b^2-4 a c\right ) (d+e x)^5 (b e-2 c d) \left (-4 c e (3 a e+4 b d)+7 b^2 e^2+16 c^2 d^2\right ) \log \left (2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2}+2 a e-b d+b e x-2 c d x\right )}{3840 e (d+e x)^5 \left (e (a e-b d)+c d^2\right )^{9/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[a + b*x + c*x^2]/(d + e*x)^6,x]

[Out]

(-2*Sqrt[c*d^2 + e*(-(b*d) + a*e)]*Sqrt[a + x*(b + c*x)]*(384*(c*d^2 + e*(-(b*d)
+ a*e))^4 - 48*(2*c*d - b*e)*(c*d^2 + e*(-(b*d) + a*e))^3*(d + e*x) - 8*(c*d^2
+ e*(-(b*d) + a*e))^2*(12*c^2*d^2 + 7*b^2*e^2 - 4*c*e*(3*b*d + 4*a*e))*(d + e*x)
^2 - 2*(2*c*d - b*e)*(c*d^2 + e*(-(b*d) + a*e))*(24*c^2*d^2 + 35*b^2*e^2 - 4*c*e
*(6*b*d + 29*a*e))*(d + e*x)^3 - (96*c^4*d^4 + 105*b^4*e^4 - 20*b^2*c*e^3*(19*b*
d + 23*a*e) - 16*c^3*d^2*e*(12*b*d + 83*a*e) + 4*c^2*e^2*(119*b^2*d^2 + 332*a*b*
d*e + 64*a^2*e^2))*(d + e*x)^4) + 15*(b^2 - 4*a*c)*e*(-2*c*d + b*e)*(16*c^2*d^2
+ 7*b^2*e^2 - 4*c*e*(4*b*d + 3*a*e))*(d + e*x)^5*Log[d + e*x] - 15*(b^2 - 4*a*c)
*e*(-2*c*d + b*e)*(16*c^2*d^2 + 7*b^2*e^2 - 4*c*e*(4*b*d + 3*a*e))*(d + e*x)^5*L
og[-(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)]])/(3840*e*(c*d^2 + e*(-(b*d) + a*e))^(9/2)*(d + e*x)^5)

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Maple [B]  time = 0.04, size = 10791, normalized size = 26.8 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+b*x+a)^(1/2)/(e*x+d)^6,x)

[Out]

result too large to display

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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)^6,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 45.9035, 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)^6,x, algorithm="fricas")

[Out]

[1/7680*(4*(480*b*c^3*d^7 + 1488*a^3*b*d*e^6 - 384*a^4*e^7 - 720*(b^2*c^2 + 4*a*
c^3)*d^6*e + 30*(15*b^3*c + 188*a*b*c^2)*d^5*e^2 - (105*b^4 + 4360*a*b^2*c + 238
4*a^2*c^2)*d^4*e^3 + 2*(605*a*b^3 + 2284*a^2*b*c)*d^3*e^4 - 8*(263*a^2*b^2 + 196
*a^3*c)*d^2*e^5 + (96*c^4*d^4*e^3 - 192*b*c^3*d^3*e^4 + 4*(119*b^2*c^2 - 332*a*c
^3)*d^2*e^5 - 4*(95*b^3*c - 332*a*b*c^2)*d*e^6 + (105*b^4 - 460*a*b^2*c + 256*a^
2*c^2)*e^7)*x^4 + 2*(240*c^4*d^5*e^2 - 504*b*c^3*d^4*e^3 + 2*(559*b^2*c^2 - 1420
*a*c^3)*d^3*e^4 - (889*b^3*c - 2932*a*b*c^2)*d^2*e^5 + (245*b^4 - 942*a*b^2*c +
280*a^2*c^2)*d*e^6 - (35*a*b^3 - 116*a^2*b*c)*e^7)*x^3 + 2*(480*c^4*d^6*e - 1080
*b*c^3*d^5*e^2 + 2*(1049*b^2*c^2 - 2252*a*c^3)*d^4*e^3 - (1631*b^3*c - 4748*a*b*
c^2)*d^3*e^4 + 2*(224*b^4 - 679*a*b^2*c - 4*a^2*c^2)*d^2*e^5 - (161*a*b^3 - 428*
a^2*b*c)*d*e^6 + 4*(7*a^2*b^2 - 16*a^3*c)*e^7)*x^2 + 2*(480*c^4*d^7 - 1200*b*c^3
*d^6*e - 24*a^3*b*e^7 + 30*(67*b^2*c^2 - 100*a*c^3)*d^5*e^2 - (1475*b^3*c - 2996
*a*b*c^2)*d^4*e^3 + (395*b^4 - 522*a*b^2*c - 200*a^2*c^2)*d^3*e^4 - (289*a*b^3 -
292*a^2*b*c)*d^2*e^5 + 16*(8*a^2*b^2 - 5*a^3*c)*d*e^6)*x)*sqrt(c*d^2 - b*d*e +
a*e^2)*sqrt(c*x^2 + b*x + a) - 15*(32*(b^2*c^3 - 4*a*c^4)*d^8 - 48*(b^3*c^2 - 4*
a*b*c^3)*d^7*e + 6*(5*b^4*c - 24*a*b^2*c^2 + 16*a^2*c^3)*d^6*e^2 - (7*b^5 - 40*a
*b^3*c + 48*a^2*b*c^2)*d^5*e^3 + (32*(b^2*c^3 - 4*a*c^4)*d^3*e^5 - 48*(b^3*c^2 -
4*a*b*c^3)*d^2*e^6 + 6*(5*b^4*c - 24*a*b^2*c^2 + 16*a^2*c^3)*d*e^7 - (7*b^5 - 4
0*a*b^3*c + 48*a^2*b*c^2)*e^8)*x^5 + 5*(32*(b^2*c^3 - 4*a*c^4)*d^4*e^4 - 48*(b^3
*c^2 - 4*a*b*c^3)*d^3*e^5 + 6*(5*b^4*c - 24*a*b^2*c^2 + 16*a^2*c^3)*d^2*e^6 - (7
*b^5 - 40*a*b^3*c + 48*a^2*b*c^2)*d*e^7)*x^4 + 10*(32*(b^2*c^3 - 4*a*c^4)*d^5*e^
3 - 48*(b^3*c^2 - 4*a*b*c^3)*d^4*e^4 + 6*(5*b^4*c - 24*a*b^2*c^2 + 16*a^2*c^3)*d
^3*e^5 - (7*b^5 - 40*a*b^3*c + 48*a^2*b*c^2)*d^2*e^6)*x^3 + 10*(32*(b^2*c^3 - 4*
a*c^4)*d^6*e^2 - 48*(b^3*c^2 - 4*a*b*c^3)*d^5*e^3 + 6*(5*b^4*c - 24*a*b^2*c^2 +
16*a^2*c^3)*d^4*e^4 - (7*b^5 - 40*a*b^3*c + 48*a^2*b*c^2)*d^3*e^5)*x^2 + 5*(32*(
b^2*c^3 - 4*a*c^4)*d^7*e - 48*(b^3*c^2 - 4*a*b*c^3)*d^6*e^2 + 6*(5*b^4*c - 24*a*
b^2*c^2 + 16*a^2*c^3)*d^5*e^3 - (7*b^5 - 40*a*b^3*c + 48*a^2*b*c^2)*d^4*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^4*d^13 - 4*b*c^3*d^12*e - 4*a^3*b*d^6*e^7
+ a^4*d^5*e^8 + 2*(3*b^2*c^2 + 2*a*c^3)*d^11*e^2 - 4*(b^3*c + 3*a*b*c^2)*d^10*e^
3 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^9*e^4 - 4*(a*b^3 + 3*a^2*b*c)*d^8*e^5 + 2*(
3*a^2*b^2 + 2*a^3*c)*d^7*e^6 + (c^4*d^8*e^5 - 4*b*c^3*d^7*e^6 - 4*a^3*b*d*e^12 +
a^4*e^13 + 2*(3*b^2*c^2 + 2*a*c^3)*d^6*e^7 - 4*(b^3*c + 3*a*b*c^2)*d^5*e^8 + (b
^4 + 12*a*b^2*c + 6*a^2*c^2)*d^4*e^9 - 4*(a*b^3 + 3*a^2*b*c)*d^3*e^10 + 2*(3*a^2
*b^2 + 2*a^3*c)*d^2*e^11)*x^5 + 5*(c^4*d^9*e^4 - 4*b*c^3*d^8*e^5 - 4*a^3*b*d^2*e
^11 + a^4*d*e^12 + 2*(3*b^2*c^2 + 2*a*c^3)*d^7*e^6 - 4*(b^3*c + 3*a*b*c^2)*d^6*e
^7 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^5*e^8 - 4*(a*b^3 + 3*a^2*b*c)*d^4*e^9 + 2*
(3*a^2*b^2 + 2*a^3*c)*d^3*e^10)*x^4 + 10*(c^4*d^10*e^3 - 4*b*c^3*d^9*e^4 - 4*a^3
*b*d^3*e^10 + a^4*d^2*e^11 + 2*(3*b^2*c^2 + 2*a*c^3)*d^8*e^5 - 4*(b^3*c + 3*a*b*
c^2)*d^7*e^6 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^6*e^7 - 4*(a*b^3 + 3*a^2*b*c)*d^
5*e^8 + 2*(3*a^2*b^2 + 2*a^3*c)*d^4*e^9)*x^3 + 10*(c^4*d^11*e^2 - 4*b*c^3*d^10*e
^3 - 4*a^3*b*d^4*e^9 + a^4*d^3*e^10 + 2*(3*b^2*c^2 + 2*a*c^3)*d^9*e^4 - 4*(b^3*c
+ 3*a*b*c^2)*d^8*e^5 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^7*e^6 - 4*(a*b^3 + 3*a^
2*b*c)*d^6*e^7 + 2*(3*a^2*b^2 + 2*a^3*c)*d^5*e^8)*x^2 + 5*(c^4*d^12*e - 4*b*c^3*
d^11*e^2 - 4*a^3*b*d^5*e^8 + a^4*d^4*e^9 + 2*(3*b^2*c^2 + 2*a*c^3)*d^10*e^3 - 4*
(b^3*c + 3*a*b*c^2)*d^9*e^4 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^8*e^5 - 4*(a*b^3
+ 3*a^2*b*c)*d^7*e^6 + 2*(3*a^2*b^2 + 2*a^3*c)*d^6*e^7)*x)*sqrt(c*d^2 - b*d*e +
a*e^2)), 1/3840*(2*(480*b*c^3*d^7 + 1488*a^3*b*d*e^6 - 384*a^4*e^7 - 720*(b^2*c^
2 + 4*a*c^3)*d^6*e + 30*(15*b^3*c + 188*a*b*c^2)*d^5*e^2 - (105*b^4 + 4360*a*b^2
*c + 2384*a^2*c^2)*d^4*e^3 + 2*(605*a*b^3 + 2284*a^2*b*c)*d^3*e^4 - 8*(263*a^2*b
^2 + 196*a^3*c)*d^2*e^5 + (96*c^4*d^4*e^3 - 192*b*c^3*d^3*e^4 + 4*(119*b^2*c^2 -
332*a*c^3)*d^2*e^5 - 4*(95*b^3*c - 332*a*b*c^2)*d*e^6 + (105*b^4 - 460*a*b^2*c
+ 256*a^2*c^2)*e^7)*x^4 + 2*(240*c^4*d^5*e^2 - 504*b*c^3*d^4*e^3 + 2*(559*b^2*c^
2 - 1420*a*c^3)*d^3*e^4 - (889*b^3*c - 2932*a*b*c^2)*d^2*e^5 + (245*b^4 - 942*a*
b^2*c + 280*a^2*c^2)*d*e^6 - (35*a*b^3 - 116*a^2*b*c)*e^7)*x^3 + 2*(480*c^4*d^6*
e - 1080*b*c^3*d^5*e^2 + 2*(1049*b^2*c^2 - 2252*a*c^3)*d^4*e^3 - (1631*b^3*c - 4
748*a*b*c^2)*d^3*e^4 + 2*(224*b^4 - 679*a*b^2*c - 4*a^2*c^2)*d^2*e^5 - (161*a*b^
3 - 428*a^2*b*c)*d*e^6 + 4*(7*a^2*b^2 - 16*a^3*c)*e^7)*x^2 + 2*(480*c^4*d^7 - 12
00*b*c^3*d^6*e - 24*a^3*b*e^7 + 30*(67*b^2*c^2 - 100*a*c^3)*d^5*e^2 - (1475*b^3*
c - 2996*a*b*c^2)*d^4*e^3 + (395*b^4 - 522*a*b^2*c - 200*a^2*c^2)*d^3*e^4 - (289
*a*b^3 - 292*a^2*b*c)*d^2*e^5 + 16*(8*a^2*b^2 - 5*a^3*c)*d*e^6)*x)*sqrt(-c*d^2 +
b*d*e - a*e^2)*sqrt(c*x^2 + b*x + a) + 15*(32*(b^2*c^3 - 4*a*c^4)*d^8 - 48*(b^3
*c^2 - 4*a*b*c^3)*d^7*e + 6*(5*b^4*c - 24*a*b^2*c^2 + 16*a^2*c^3)*d^6*e^2 - (7*b
^5 - 40*a*b^3*c + 48*a^2*b*c^2)*d^5*e^3 + (32*(b^2*c^3 - 4*a*c^4)*d^3*e^5 - 48*(
b^3*c^2 - 4*a*b*c^3)*d^2*e^6 + 6*(5*b^4*c - 24*a*b^2*c^2 + 16*a^2*c^3)*d*e^7 - (
7*b^5 - 40*a*b^3*c + 48*a^2*b*c^2)*e^8)*x^5 + 5*(32*(b^2*c^3 - 4*a*c^4)*d^4*e^4
- 48*(b^3*c^2 - 4*a*b*c^3)*d^3*e^5 + 6*(5*b^4*c - 24*a*b^2*c^2 + 16*a^2*c^3)*d^2
*e^6 - (7*b^5 - 40*a*b^3*c + 48*a^2*b*c^2)*d*e^7)*x^4 + 10*(32*(b^2*c^3 - 4*a*c^
4)*d^5*e^3 - 48*(b^3*c^2 - 4*a*b*c^3)*d^4*e^4 + 6*(5*b^4*c - 24*a*b^2*c^2 + 16*a
^2*c^3)*d^3*e^5 - (7*b^5 - 40*a*b^3*c + 48*a^2*b*c^2)*d^2*e^6)*x^3 + 10*(32*(b^2
*c^3 - 4*a*c^4)*d^6*e^2 - 48*(b^3*c^2 - 4*a*b*c^3)*d^5*e^3 + 6*(5*b^4*c - 24*a*b
^2*c^2 + 16*a^2*c^3)*d^4*e^4 - (7*b^5 - 40*a*b^3*c + 48*a^2*b*c^2)*d^3*e^5)*x^2
+ 5*(32*(b^2*c^3 - 4*a*c^4)*d^7*e - 48*(b^3*c^2 - 4*a*b*c^3)*d^6*e^2 + 6*(5*b^4*
c - 24*a*b^2*c^2 + 16*a^2*c^3)*d^5*e^3 - (7*b^5 - 40*a*b^3*c + 48*a^2*b*c^2)*d^4
*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^4*d^13 - 4*b*c^3*d^12*e
- 4*a^3*b*d^6*e^7 + a^4*d^5*e^8 + 2*(3*b^2*c^2 + 2*a*c^3)*d^11*e^2 - 4*(b^3*c +
3*a*b*c^2)*d^10*e^3 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^9*e^4 - 4*(a*b^3 + 3*a^2*
b*c)*d^8*e^5 + 2*(3*a^2*b^2 + 2*a^3*c)*d^7*e^6 + (c^4*d^8*e^5 - 4*b*c^3*d^7*e^6
- 4*a^3*b*d*e^12 + a^4*e^13 + 2*(3*b^2*c^2 + 2*a*c^3)*d^6*e^7 - 4*(b^3*c + 3*a*b
*c^2)*d^5*e^8 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^4*e^9 - 4*(a*b^3 + 3*a^2*b*c)*d
^3*e^10 + 2*(3*a^2*b^2 + 2*a^3*c)*d^2*e^11)*x^5 + 5*(c^4*d^9*e^4 - 4*b*c^3*d^8*e
^5 - 4*a^3*b*d^2*e^11 + a^4*d*e^12 + 2*(3*b^2*c^2 + 2*a*c^3)*d^7*e^6 - 4*(b^3*c
+ 3*a*b*c^2)*d^6*e^7 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^5*e^8 - 4*(a*b^3 + 3*a^2
*b*c)*d^4*e^9 + 2*(3*a^2*b^2 + 2*a^3*c)*d^3*e^10)*x^4 + 10*(c^4*d^10*e^3 - 4*b*c
^3*d^9*e^4 - 4*a^3*b*d^3*e^10 + a^4*d^2*e^11 + 2*(3*b^2*c^2 + 2*a*c^3)*d^8*e^5 -
4*(b^3*c + 3*a*b*c^2)*d^7*e^6 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^6*e^7 - 4*(a*b
^3 + 3*a^2*b*c)*d^5*e^8 + 2*(3*a^2*b^2 + 2*a^3*c)*d^4*e^9)*x^3 + 10*(c^4*d^11*e^
2 - 4*b*c^3*d^10*e^3 - 4*a^3*b*d^4*e^9 + a^4*d^3*e^10 + 2*(3*b^2*c^2 + 2*a*c^3)*
d^9*e^4 - 4*(b^3*c + 3*a*b*c^2)*d^8*e^5 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^7*e^6
- 4*(a*b^3 + 3*a^2*b*c)*d^6*e^7 + 2*(3*a^2*b^2 + 2*a^3*c)*d^5*e^8)*x^2 + 5*(c^4
*d^12*e - 4*b*c^3*d^11*e^2 - 4*a^3*b*d^5*e^8 + a^4*d^4*e^9 + 2*(3*b^2*c^2 + 2*a*
c^3)*d^10*e^3 - 4*(b^3*c + 3*a*b*c^2)*d^9*e^4 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d
^8*e^5 - 4*(a*b^3 + 3*a^2*b*c)*d^7*e^6 + 2*(3*a^2*b^2 + 2*a^3*c)*d^6*e^7)*x)*sqr
t(-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((c*x**2+b*x+a)**(1/2)/(e*x+d)**6,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.644494, size = 4, normalized size = 0.01 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^2 + b*x + a)/(e*x + d)^6,x, algorithm="giac")

[Out]

sage0*x