3.1556 \(\int \frac{(b+2 c x) \sqrt{a+b x+c x^2}}{(d+e x)^6} \, dx\)

Optimal. Leaf size=430 \[ -\frac{e \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2} \left (-4 c e (a e+6 b d)+7 b^2 e^2+24 c^2 d^2\right ) (-2 a e+x (2 c d-b e)+b d)}{128 (d+e x)^2 \left (a e^2-b d e+c d^2\right )^4}+\frac{\left (a+b x+c x^2\right )^{3/2} (2 c d-b e) \left (-4 c e (33 a e+2 b d)+35 b^2 e^2+8 c^2 d^2\right )}{240 (d+e x)^3 \left (a e^2-b d e+c d^2\right )^3}+\frac{\left (a+b x+c x^2\right )^{3/2} \left (-4 c e (5 a e+2 b d)+7 b^2 e^2+8 c^2 d^2\right )}{40 (d+e x)^4 \left (a e^2-b d e+c d^2\right )^2}+\frac{e \left (b^2-4 a c\right )^2 \left (-4 c e (a e+6 b d)+7 b^2 e^2+24 c^2 d^2\right ) \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{256 \left (a e^2-b d e+c d^2\right )^{9/2}}+\frac{\left (a+b x+c x^2\right )^{3/2} (2 c d-b e)}{5 (d+e x)^5 \left (a e^2-b d e+c d^2\right )} \]

[Out]

-((b^2 - 4*a*c)*e*(24*c^2*d^2 + 7*b^2*e^2 - 4*c*e*(6*b*d + 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) + ((2*c*d - b*e)*(a + b*x + c*x^2)^(3/2))/(5*(c*d^2 - b*d*e + a*e^2)*(d + e*
x)^5) + ((8*c^2*d^2 + 7*b^2*e^2 - 4*c*e*(2*b*d + 5*a*e))*(a + b*x + c*x^2)^(3/2)
)/(40*(c*d^2 - b*d*e + a*e^2)^2*(d + e*x)^4) + ((2*c*d - b*e)*(8*c^2*d^2 + 35*b^
2*e^2 - 4*c*e*(2*b*d + 33*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*e*(24*c^2*d^2 + 7*b^2*e^2 - 4*c*e*(6*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])])/(256*(c*d^2 - b*d*e + a*e^2)^(9/2))

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Rubi [A]  time = 1.65614, antiderivative size = 430, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179 \[ -\frac{e \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2} \left (-4 c e (a e+6 b d)+7 b^2 e^2+24 c^2 d^2\right ) (-2 a e+x (2 c d-b e)+b d)}{128 (d+e x)^2 \left (a e^2-b d e+c d^2\right )^4}+\frac{\left (a+b x+c x^2\right )^{3/2} (2 c d-b e) \left (-4 c e (33 a e+2 b d)+35 b^2 e^2+8 c^2 d^2\right )}{240 (d+e x)^3 \left (a e^2-b d e+c d^2\right )^3}+\frac{\left (a+b x+c x^2\right )^{3/2} \left (-4 c e (5 a e+2 b d)+7 b^2 e^2+8 c^2 d^2\right )}{40 (d+e x)^4 \left (a e^2-b d e+c d^2\right )^2}+\frac{e \left (b^2-4 a c\right )^2 \left (-4 c e (a e+6 b d)+7 b^2 e^2+24 c^2 d^2\right ) \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{256 \left (a e^2-b d e+c d^2\right )^{9/2}}+\frac{\left (a+b x+c x^2\right )^{3/2} (2 c d-b e)}{5 (d+e x)^5 \left (a e^2-b d e+c d^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

-((b^2 - 4*a*c)*e*(24*c^2*d^2 + 7*b^2*e^2 - 4*c*e*(6*b*d + 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) + ((2*c*d - b*e)*(a + b*x + c*x^2)^(3/2))/(5*(c*d^2 - b*d*e + a*e^2)*(d + e*
x)^5) + ((8*c^2*d^2 + 7*b^2*e^2 - 4*c*e*(2*b*d + 5*a*e))*(a + b*x + c*x^2)^(3/2)
)/(40*(c*d^2 - b*d*e + a*e^2)^2*(d + e*x)^4) + ((2*c*d - b*e)*(8*c^2*d^2 + 35*b^
2*e^2 - 4*c*e*(2*b*d + 33*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*e*(24*c^2*d^2 + 7*b^2*e^2 - 4*c*e*(6*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])])/(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((2*c*x+b)*(c*x**2+b*x+a)**(1/2)/(e*x+d)**6,x)

[Out]

Timed out

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Mathematica [A]  time = 4.68456, size = 578, normalized size = 1.34 \[ \frac{1}{256} \left (\frac{2 \sqrt{a+x (b+c x)} \left ((d+e x)^4 (2 c d-b e) \left (-16 c^2 e^2 \left (81 a^2 e^2+28 a b d e+b^2 d^2\right )+40 b^2 c e^3 (19 a e+2 b d)-64 c^3 d^2 e (2 b d-7 a e)-105 b^4 e^4+64 c^4 d^4\right )-48 (d+e x) \left (4 c e (5 a e-6 b d)+b^2 e^2+24 c^2 d^2\right ) \left (e (a e-b d)+c d^2\right )^3+8 (d+e x)^2 (2 c d-b e) \left (4 c e (9 a e-2 b d)-7 b^2 e^2+8 c^2 d^2\right ) \left (e (a e-b d)+c d^2\right )^2+2 (d+e x)^3 \left (8 b^2 c e^3 (27 a e+8 b d)-128 c^3 d^2 e (b d-3 a e)-48 a c^2 e^3 (5 a e+8 b d)-35 b^4 e^4+64 c^4 d^4\right ) \left (e (a e-b d)+c d^2\right )+384 (2 c d-b e) \left (e (a e-b d)+c d^2\right )^4\right )}{15 e^2 (d+e x)^5 \left (e (a e-b d)+c d^2\right )^4}+\frac{e \left (b^2-4 a c\right )^2 \log (d+e x) \left (-4 c e (a e+6 b d)+7 b^2 e^2+24 c^2 d^2\right )}{\left (e (a e-b d)+c d^2\right )^{9/2}}-\frac{e \left (b^2-4 a c\right )^2 \left (-4 c e (a e+6 b d)+7 b^2 e^2+24 c^2 d^2\right ) \log \left (2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2}+2 a e-b d+b e x-2 c d x\right )}{\left (e (a e-b d)+c d^2\right )^{9/2}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

((2*Sqrt[a + x*(b + c*x)]*(384*(2*c*d - b*e)*(c*d^2 + e*(-(b*d) + a*e))^4 - 48*(
c*d^2 + e*(-(b*d) + a*e))^3*(24*c^2*d^2 + b^2*e^2 + 4*c*e*(-6*b*d + 5*a*e))*(d +
e*x) + 8*(2*c*d - b*e)*(c*d^2 + e*(-(b*d) + a*e))^2*(8*c^2*d^2 - 7*b^2*e^2 + 4*
c*e*(-2*b*d + 9*a*e))*(d + e*x)^2 + 2*(c*d^2 + e*(-(b*d) + a*e))*(64*c^4*d^4 - 3
5*b^4*e^4 - 128*c^3*d^2*e*(b*d - 3*a*e) - 48*a*c^2*e^3*(8*b*d + 5*a*e) + 8*b^2*c
*e^3*(8*b*d + 27*a*e))*(d + e*x)^3 + (2*c*d - b*e)*(64*c^4*d^4 - 105*b^4*e^4 - 6
4*c^3*d^2*e*(2*b*d - 7*a*e) + 40*b^2*c*e^3*(2*b*d + 19*a*e) - 16*c^2*e^2*(b^2*d^
2 + 28*a*b*d*e + 81*a^2*e^2))*(d + e*x)^4))/(15*e^2*(c*d^2 + e*(-(b*d) + a*e))^4
*(d + e*x)^5) + ((b^2 - 4*a*c)^2*e*(24*c^2*d^2 + 7*b^2*e^2 - 4*c*e*(6*b*d + a*e)
)*Log[d + e*x])/(c*d^2 + e*(-(b*d) + a*e))^(9/2) - ((b^2 - 4*a*c)^2*e*(24*c^2*d^
2 + 7*b^2*e^2 - 4*c*e*(6*b*d + a*e))*Log[-(b*d) + 2*a*e - 2*c*d*x + b*e*x + 2*Sq
rt[c*d^2 + e*(-(b*d) + a*e)]*Sqrt[a + x*(b + c*x)]])/(c*d^2 + e*(-(b*d) + a*e))^
(9/2))/256

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Maple [B]  time = 0.035, size = 15192, normalized size = 35.3 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2*c*x+b)*(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)*(2*c*x + b)/(e*x + d)^6,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 42.6219, 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)*(2*c*x + b)/(e*x + d)^6,x, algorithm="fricas")

[Out]

[1/7680*(4*(1280*a*c^4*d^7 - 384*a^4*b*e^7 - 40*(9*b^3*c^2 + 68*a*b*c^3)*d^6*e +
8*(45*b^4*c + 630*a*b^2*c^2 - 344*a^2*c^3)*d^5*e^2 - 5*(21*b^5 + 824*a*b^3*c -
48*a^2*b*c^2)*d^4*e^3 + 2*(605*a*b^4 + 1848*a^2*b^2*c - 432*a^3*c^2)*d^3*e^4 - 8
*(263*a^2*b^3 + 108*a^3*b*c)*d^2*e^5 + 48*(31*a^3*b^2 - 4*a^4*c)*d*e^6 + (128*c^
5*d^5*e^2 - 320*b*c^4*d^4*e^3 + 32*(3*b^2*c^3 + 28*a*c^4)*d^3*e^4 + 16*(11*b^3*c
^2 - 84*a*b*c^3)*d^2*e^5 - 2*(145*b^4*c - 984*a*b^2*c^2 + 1296*a^2*c^3)*d*e^6 +
(105*b^5 - 760*a*b^3*c + 1296*a^2*b*c^2)*e^7)*x^4 + 2*(320*c^5*d^6*e - 832*b*c^4
*d^5*e^2 + 320*(b^2*c^3 + 7*a*c^4)*d^4*e^3 + 32*(13*b^3*c^2 - 112*a*b*c^3)*d^3*e
^4 - 7*(97*b^4*c - 648*a*b^2*c^2 + 720*a^2*c^3)*d^2*e^5 + (245*b^5 - 1672*a*b^3*
c + 2448*a^2*b*c^2)*d*e^6 - (35*a*b^4 - 216*a^2*b^2*c + 240*a^3*c^2)*e^7)*x^3 +
2*(640*c^5*d^7 - 1760*b*c^4*d^6*e + 8*(113*b^2*c^3 + 556*a*c^4)*d^5*e^2 + 20*(35
*b^3*c^2 - 388*a*b*c^3)*d^4*e^3 - (1247*b^4*c - 8424*a*b^2*c^2 + 6704*a^2*c^3)*d
^3*e^4 + 8*(56*b^5 - 347*a*b^3*c + 312*a^2*b*c^2)*d^2*e^5 - (161*a*b^4 - 912*a^2
*b^2*c + 432*a^3*c^2)*d*e^6 + 4*(7*a^2*b^3 - 36*a^3*b*c)*e^7)*x^2 + 2*(640*b*c^4
*d^7 - 40*(61*b^2*c^3 - 44*a*c^4)*d^6*e + 64*(45*b^3*c^2 - 13*a*b*c^3)*d^5*e^2 -
5*(337*b^4*c - 96*a*b^2*c^2 + 1328*a^2*c^3)*d^4*e^3 + (395*b^5 + 296*a*b^3*c +
6576*a^2*b*c^2)*d^3*e^4 - (289*a*b^4 + 2064*a^2*b^2*c + 2160*a^3*c^2)*d^2*e^5 +
64*(2*a^2*b^3 + 27*a^3*b*c)*d*e^6 - 24*(a^3*b^2 + 20*a^4*c)*e^7)*x)*sqrt(c*d^2 -
b*d*e + a*e^2)*sqrt(c*x^2 + b*x + a) - 15*(24*(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c
^4)*d^7*e - 24*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*d^6*e^2 + (7*b^6 - 60*a*b^4*
c + 144*a^2*b^2*c^2 - 64*a^3*c^3)*d^5*e^3 + (24*(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*
c^4)*d^2*e^6 - 24*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*d*e^7 + (7*b^6 - 60*a*b^4
*c + 144*a^2*b^2*c^2 - 64*a^3*c^3)*e^8)*x^5 + 5*(24*(b^4*c^2 - 8*a*b^2*c^3 + 16*
a^2*c^4)*d^3*e^5 - 24*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*d^2*e^6 + (7*b^6 - 60
*a*b^4*c + 144*a^2*b^2*c^2 - 64*a^3*c^3)*d*e^7)*x^4 + 10*(24*(b^4*c^2 - 8*a*b^2*
c^3 + 16*a^2*c^4)*d^4*e^4 - 24*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*d^3*e^5 + (7
*b^6 - 60*a*b^4*c + 144*a^2*b^2*c^2 - 64*a^3*c^3)*d^2*e^6)*x^3 + 10*(24*(b^4*c^2
- 8*a*b^2*c^3 + 16*a^2*c^4)*d^5*e^3 - 24*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*d
^4*e^4 + (7*b^6 - 60*a*b^4*c + 144*a^2*b^2*c^2 - 64*a^3*c^3)*d^3*e^5)*x^2 + 5*(2
4*(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*d^6*e^2 - 24*(b^5*c - 8*a*b^3*c^2 + 16*a^
2*b*c^3)*d^5*e^3 + (7*b^6 - 60*a*b^4*c + 144*a^2*b^2*c^2 - 64*a^3*c^3)*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^1
2 + 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^1
0*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*(1280*a*c^4*d^7 - 384*a^4*b*e^7 - 40*(9*b^3*c^2 + 68*a*b*c
^3)*d^6*e + 8*(45*b^4*c + 630*a*b^2*c^2 - 344*a^2*c^3)*d^5*e^2 - 5*(21*b^5 + 824
*a*b^3*c - 48*a^2*b*c^2)*d^4*e^3 + 2*(605*a*b^4 + 1848*a^2*b^2*c - 432*a^3*c^2)*
d^3*e^4 - 8*(263*a^2*b^3 + 108*a^3*b*c)*d^2*e^5 + 48*(31*a^3*b^2 - 4*a^4*c)*d*e^
6 + (128*c^5*d^5*e^2 - 320*b*c^4*d^4*e^3 + 32*(3*b^2*c^3 + 28*a*c^4)*d^3*e^4 + 1
6*(11*b^3*c^2 - 84*a*b*c^3)*d^2*e^5 - 2*(145*b^4*c - 984*a*b^2*c^2 + 1296*a^2*c^
3)*d*e^6 + (105*b^5 - 760*a*b^3*c + 1296*a^2*b*c^2)*e^7)*x^4 + 2*(320*c^5*d^6*e
- 832*b*c^4*d^5*e^2 + 320*(b^2*c^3 + 7*a*c^4)*d^4*e^3 + 32*(13*b^3*c^2 - 112*a*b
*c^3)*d^3*e^4 - 7*(97*b^4*c - 648*a*b^2*c^2 + 720*a^2*c^3)*d^2*e^5 + (245*b^5 -
1672*a*b^3*c + 2448*a^2*b*c^2)*d*e^6 - (35*a*b^4 - 216*a^2*b^2*c + 240*a^3*c^2)*
e^7)*x^3 + 2*(640*c^5*d^7 - 1760*b*c^4*d^6*e + 8*(113*b^2*c^3 + 556*a*c^4)*d^5*e
^2 + 20*(35*b^3*c^2 - 388*a*b*c^3)*d^4*e^3 - (1247*b^4*c - 8424*a*b^2*c^2 + 6704
*a^2*c^3)*d^3*e^4 + 8*(56*b^5 - 347*a*b^3*c + 312*a^2*b*c^2)*d^2*e^5 - (161*a*b^
4 - 912*a^2*b^2*c + 432*a^3*c^2)*d*e^6 + 4*(7*a^2*b^3 - 36*a^3*b*c)*e^7)*x^2 + 2
*(640*b*c^4*d^7 - 40*(61*b^2*c^3 - 44*a*c^4)*d^6*e + 64*(45*b^3*c^2 - 13*a*b*c^3
)*d^5*e^2 - 5*(337*b^4*c - 96*a*b^2*c^2 + 1328*a^2*c^3)*d^4*e^3 + (395*b^5 + 296
*a*b^3*c + 6576*a^2*b*c^2)*d^3*e^4 - (289*a*b^4 + 2064*a^2*b^2*c + 2160*a^3*c^2)
*d^2*e^5 + 64*(2*a^2*b^3 + 27*a^3*b*c)*d*e^6 - 24*(a^3*b^2 + 20*a^4*c)*e^7)*x)*s
qrt(-c*d^2 + b*d*e - a*e^2)*sqrt(c*x^2 + b*x + a) - 15*(24*(b^4*c^2 - 8*a*b^2*c^
3 + 16*a^2*c^4)*d^7*e - 24*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*d^6*e^2 + (7*b^6
- 60*a*b^4*c + 144*a^2*b^2*c^2 - 64*a^3*c^3)*d^5*e^3 + (24*(b^4*c^2 - 8*a*b^2*c
^3 + 16*a^2*c^4)*d^2*e^6 - 24*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*d*e^7 + (7*b^
6 - 60*a*b^4*c + 144*a^2*b^2*c^2 - 64*a^3*c^3)*e^8)*x^5 + 5*(24*(b^4*c^2 - 8*a*b
^2*c^3 + 16*a^2*c^4)*d^3*e^5 - 24*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*d^2*e^6 +
(7*b^6 - 60*a*b^4*c + 144*a^2*b^2*c^2 - 64*a^3*c^3)*d*e^7)*x^4 + 10*(24*(b^4*c^
2 - 8*a*b^2*c^3 + 16*a^2*c^4)*d^4*e^4 - 24*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*
d^3*e^5 + (7*b^6 - 60*a*b^4*c + 144*a^2*b^2*c^2 - 64*a^3*c^3)*d^2*e^6)*x^3 + 10*
(24*(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*d^5*e^3 - 24*(b^5*c - 8*a*b^3*c^2 + 16*
a^2*b*c^3)*d^4*e^4 + (7*b^6 - 60*a*b^4*c + 144*a^2*b^2*c^2 - 64*a^3*c^3)*d^3*e^5
)*x^2 + 5*(24*(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*d^6*e^2 - 24*(b^5*c - 8*a*b^3
*c^2 + 16*a^2*b*c^3)*d^5*e^3 + (7*b^6 - 60*a*b^4*c + 144*a^2*b^2*c^2 - 64*a^3*c^
3)*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)*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)*(c*x**2+b*x+a)**(1/2)/(e*x+d)**6,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.697268, 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)*(2*c*x + b)/(e*x + d)^6,x, algorithm="giac")

[Out]

sage0*x