3.2278 \(\int \frac{(d+e x)^{5/2}}{a+b x+c x^2} \, dx\)

Optimal. Leaf size=459 \[ -\frac{\sqrt{2} \left (-3 c^2 d e \left (-d \sqrt{b^2-4 a c}+2 a e+b d\right )+c e^2 \left (-3 b d \sqrt{b^2-4 a c}-a e \sqrt{b^2-4 a c}+3 a b e+3 b^2 d\right )-b^2 e^3 \left (b-\sqrt{b^2-4 a c}\right )+2 c^3 d^3\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}\right )}{c^{5/2} \sqrt{b^2-4 a c} \sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}+\frac{\sqrt{2} \left (-3 c^2 d e \left (d \sqrt{b^2-4 a c}+2 a e+b d\right )+c e^2 \left (3 b \left (d \sqrt{b^2-4 a c}+a e\right )+a e \sqrt{b^2-4 a c}+3 b^2 d\right )-b^2 e^3 \left (\sqrt{b^2-4 a c}+b\right )+2 c^3 d^3\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}\right )}{c^{5/2} \sqrt{b^2-4 a c} \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}+\frac{2 e \sqrt{d+e x} (2 c d-b e)}{c^2}+\frac{2 e (d+e x)^{3/2}}{3 c} \]

[Out]

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

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Rubi [A]  time = 8.79726, antiderivative size = 459, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ -\frac{\sqrt{2} \left (-3 c^2 d e \left (-d \sqrt{b^2-4 a c}+2 a e+b d\right )+c e^2 \left (-3 b d \sqrt{b^2-4 a c}-a e \sqrt{b^2-4 a c}+3 a b e+3 b^2 d\right )-b^2 e^3 \left (b-\sqrt{b^2-4 a c}\right )+2 c^3 d^3\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}\right )}{c^{5/2} \sqrt{b^2-4 a c} \sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}+\frac{\sqrt{2} \left (-3 c^2 d e \left (d \sqrt{b^2-4 a c}+2 a e+b d\right )+c e^2 \left (3 b \left (d \sqrt{b^2-4 a c}+a e\right )+a e \sqrt{b^2-4 a c}+3 b^2 d\right )-b^2 e^3 \left (\sqrt{b^2-4 a c}+b\right )+2 c^3 d^3\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}\right )}{c^{5/2} \sqrt{b^2-4 a c} \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}+\frac{2 e \sqrt{d+e x} (2 c d-b e)}{c^2}+\frac{2 e (d+e x)^{3/2}}{3 c} \]

Antiderivative was successfully verified.

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

[Out]

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

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

[Out]

Timed out

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Mathematica [A]  time = 0.943113, size = 445, normalized size = 0.97 \[ -\frac{\sqrt{2} \left (3 c^2 d e \left (d \sqrt{b^2-4 a c}-2 a e-b d\right )+c e^2 \left (-3 b d \sqrt{b^2-4 a c}-a e \sqrt{b^2-4 a c}+3 a b e+3 b^2 d\right )+b^2 e^3 \left (\sqrt{b^2-4 a c}-b\right )+2 c^3 d^3\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{e \sqrt{b^2-4 a c}-b e+2 c d}}\right )}{c^{5/2} \sqrt{b^2-4 a c} \sqrt{e \left (\sqrt{b^2-4 a c}-b\right )+2 c d}}+\frac{\sqrt{2} \left (-3 c^2 d e \left (d \sqrt{b^2-4 a c}+2 a e+b d\right )+c e^2 \left (3 b \left (d \sqrt{b^2-4 a c}+a e\right )+a e \sqrt{b^2-4 a c}+3 b^2 d\right )-b^2 e^3 \left (\sqrt{b^2-4 a c}+b\right )+2 c^3 d^3\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}\right )}{c^{5/2} \sqrt{b^2-4 a c} \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}+\frac{2 e \sqrt{d+e x} (-3 b e+7 c d+c e x)}{3 c^2} \]

Antiderivative was successfully verified.

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

[Out]

(2*e*Sqrt[d + e*x]*(7*c*d - 3*b*e + c*e*x))/(3*c^2) - (Sqrt[2]*(2*c^3*d^3 + b^2*
(-b + Sqrt[b^2 - 4*a*c])*e^3 + 3*c^2*d*e*(-(b*d) + Sqrt[b^2 - 4*a*c]*d - 2*a*e)
+ c*e^2*(3*b^2*d - 3*b*Sqrt[b^2 - 4*a*c]*d + 3*a*b*e - a*Sqrt[b^2 - 4*a*c]*e))*A
rcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e]])
/(c^(5/2)*Sqrt[b^2 - 4*a*c]*Sqrt[2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e]) + (Sqrt[2]
*(2*c^3*d^3 - b^2*(b + Sqrt[b^2 - 4*a*c])*e^3 - 3*c^2*d*e*(b*d + Sqrt[b^2 - 4*a*
c]*d + 2*a*e) + c*e^2*(3*b^2*d + a*Sqrt[b^2 - 4*a*c]*e + 3*b*(Sqrt[b^2 - 4*a*c]*
d + a*e)))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - (b + Sqrt[b^2 -
4*a*c])*e]])/(c^(5/2)*Sqrt[b^2 - 4*a*c]*Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e])

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Maple [B]  time = 0.104, size = 1929, normalized size = 4.2 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*e*(e*x+d)^(3/2)/c-2/c^2*b*e^2*(e*x+d)^(1/2)+4*d*e*(e*x+d)^(1/2)/c-3/c/(-e^2*
(4*a*c-b^2))^(1/2)*2^(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)*arcta
nh(c*(e*x+d)^(1/2)*2^(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2))*a*b*
e^4+6/(-e^2*(4*a*c-b^2))^(1/2)*2^(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)
^(1/2)*arctanh(c*(e*x+d)^(1/2)*2^(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)
^(1/2))*a*e^3*d+1/c^2/(-e^2*(4*a*c-b^2))^(1/2)*2^(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c
-b^2))^(1/2))*c)^(1/2)*arctanh(c*(e*x+d)^(1/2)*2^(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c
-b^2))^(1/2))*c)^(1/2))*b^3*e^4-3/c/(-e^2*(4*a*c-b^2))^(1/2)*2^(1/2)/((-b*e+2*c*
d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)*arctanh(c*(e*x+d)^(1/2)*2^(1/2)/((-b*e+2*c*
d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2))*b^2*d*e^3+3/(-e^2*(4*a*c-b^2))^(1/2)*2^(1/
2)/((-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)*arctanh(c*(e*x+d)^(1/2)*2^(1/
2)/((-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2))*b*d^2*e^2-2*e*c/(-e^2*(4*a*c
-b^2))^(1/2)*2^(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)*arctanh(c*(
e*x+d)^(1/2)*2^(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2))*d^3+1/c*2^
(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)*arctanh(c*(e*x+d)^(1/2)*2^
(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2))*a*e^3-1/c^2*2^(1/2)/((-b*
e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)*arctanh(c*(e*x+d)^(1/2)*2^(1/2)/((-b*
e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2))*b^2*e^3+3/c*2^(1/2)/((-b*e+2*c*d+(-e
^2*(4*a*c-b^2))^(1/2))*c)^(1/2)*arctanh(c*(e*x+d)^(1/2)*2^(1/2)/((-b*e+2*c*d+(-e
^2*(4*a*c-b^2))^(1/2))*c)^(1/2))*b*d*e^2-3*e*2^(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c-b
^2))^(1/2))*c)^(1/2)*arctanh(c*(e*x+d)^(1/2)*2^(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c-b
^2))^(1/2))*c)^(1/2))*d^2-3/c/(-e^2*(4*a*c-b^2))^(1/2)*2^(1/2)/((b*e-2*c*d+(-e^2
*(4*a*c-b^2))^(1/2))*c)^(1/2)*arctan(c*(e*x+d)^(1/2)*2^(1/2)/((b*e-2*c*d+(-e^2*(
4*a*c-b^2))^(1/2))*c)^(1/2))*a*b*e^4+6/(-e^2*(4*a*c-b^2))^(1/2)*2^(1/2)/((b*e-2*
c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)*arctan(c*(e*x+d)^(1/2)*2^(1/2)/((b*e-2*c*
d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2))*a*e^3*d+1/c^2/(-e^2*(4*a*c-b^2))^(1/2)*2^(
1/2)/((b*e-2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)*arctan(c*(e*x+d)^(1/2)*2^(1/
2)/((b*e-2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2))*b^3*e^4-3/c/(-e^2*(4*a*c-b^2)
)^(1/2)*2^(1/2)/((b*e-2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)*arctan(c*(e*x+d)^
(1/2)*2^(1/2)/((b*e-2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2))*b^2*d*e^3+3/(-e^2*
(4*a*c-b^2))^(1/2)*2^(1/2)/((b*e-2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)*arctan
(c*(e*x+d)^(1/2)*2^(1/2)/((b*e-2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2))*b*d^2*e
^2-2*e*c/(-e^2*(4*a*c-b^2))^(1/2)*2^(1/2)/((b*e-2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*
c)^(1/2)*arctan(c*(e*x+d)^(1/2)*2^(1/2)/((b*e-2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)
^(1/2))*d^3-1/c*2^(1/2)/((b*e-2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)*arctan(c*
(e*x+d)^(1/2)*2^(1/2)/((b*e-2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2))*a*e^3+1/c^
2*2^(1/2)/((b*e-2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)*arctan(c*(e*x+d)^(1/2)*
2^(1/2)/((b*e-2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2))*b^2*e^3-3/c*2^(1/2)/((b*
e-2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)*arctan(c*(e*x+d)^(1/2)*2^(1/2)/((b*e-
2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2))*b*d*e^2+3*e*2^(1/2)/((b*e-2*c*d+(-e^2*
(4*a*c-b^2))^(1/2))*c)^(1/2)*arctan(c*(e*x+d)^(1/2)*2^(1/2)/((b*e-2*c*d+(-e^2*(4
*a*c-b^2))^(1/2))*c)^(1/2))*d^2

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 1.85495, size = 9046, normalized size = 19.71 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*(3*sqrt(2)*c^2*sqrt((2*c^5*d^5 - 5*b*c^4*d^4*e + 10*(b^2*c^3 - 2*a*c^4)*d^3*
e^2 - 10*(b^3*c^2 - 3*a*b*c^3)*d^2*e^3 + 5*(b^4*c - 4*a*b^2*c^2 + 2*a^2*c^3)*d*e
^4 - (b^5 - 5*a*b^3*c + 5*a^2*b*c^2)*e^5 + (b^2*c^5 - 4*a*c^6)*sqrt((25*c^8*d^8*
e^2 - 100*b*c^7*d^7*e^3 + 100*(2*b^2*c^6 - a*c^7)*d^6*e^4 - 50*(5*b^3*c^5 - 6*a*
b*c^6)*d^5*e^5 + 10*(21*b^4*c^4 - 43*a*b^2*c^5 + 11*a^2*c^6)*d^4*e^6 - 20*(6*b^5
*c^3 - 18*a*b^3*c^4 + 11*a^2*b*c^5)*d^3*e^7 + 5*(9*b^6*c^2 - 36*a*b^4*c^3 + 36*a
^2*b^2*c^4 - 4*a^3*c^5)*d^2*e^8 - 10*(b^7*c - 5*a*b^5*c^2 + 7*a^2*b^3*c^3 - 2*a^
3*b*c^4)*d*e^9 + (b^8 - 6*a*b^6*c + 11*a^2*b^4*c^2 - 6*a^3*b^2*c^3 + a^4*c^4)*e^
10)/(b^2*c^10 - 4*a*c^11)))/(b^2*c^5 - 4*a*c^6))*log(sqrt(2)*(10*(b^2*c^5 - 4*a*
c^6)*d^5*e^2 - 25*(b^3*c^4 - 4*a*b*c^5)*d^4*e^3 + 10*(3*b^4*c^3 - 14*a*b^2*c^4 +
8*a^2*c^5)*d^3*e^4 - 10*(2*b^5*c^2 - 11*a*b^3*c^3 + 12*a^2*b*c^4)*d^2*e^5 + (7*
b^6*c - 44*a*b^4*c^2 + 66*a^2*b^2*c^3 - 8*a^3*c^4)*d*e^6 - (b^7 - 7*a*b^5*c + 13
*a^2*b^3*c^2 - 4*a^3*b*c^3)*e^7 - (2*(b^2*c^7 - 4*a*c^8)*d^2 - 2*(b^3*c^6 - 4*a*
b*c^7)*d*e + (b^4*c^5 - 6*a*b^2*c^6 + 8*a^2*c^7)*e^2)*sqrt((25*c^8*d^8*e^2 - 100
*b*c^7*d^7*e^3 + 100*(2*b^2*c^6 - a*c^7)*d^6*e^4 - 50*(5*b^3*c^5 - 6*a*b*c^6)*d^
5*e^5 + 10*(21*b^4*c^4 - 43*a*b^2*c^5 + 11*a^2*c^6)*d^4*e^6 - 20*(6*b^5*c^3 - 18
*a*b^3*c^4 + 11*a^2*b*c^5)*d^3*e^7 + 5*(9*b^6*c^2 - 36*a*b^4*c^3 + 36*a^2*b^2*c^
4 - 4*a^3*c^5)*d^2*e^8 - 10*(b^7*c - 5*a*b^5*c^2 + 7*a^2*b^3*c^3 - 2*a^3*b*c^4)*
d*e^9 + (b^8 - 6*a*b^6*c + 11*a^2*b^4*c^2 - 6*a^3*b^2*c^3 + a^4*c^4)*e^10)/(b^2*
c^10 - 4*a*c^11)))*sqrt((2*c^5*d^5 - 5*b*c^4*d^4*e + 10*(b^2*c^3 - 2*a*c^4)*d^3*
e^2 - 10*(b^3*c^2 - 3*a*b*c^3)*d^2*e^3 + 5*(b^4*c - 4*a*b^2*c^2 + 2*a^2*c^3)*d*e
^4 - (b^5 - 5*a*b^3*c + 5*a^2*b*c^2)*e^5 + (b^2*c^5 - 4*a*c^6)*sqrt((25*c^8*d^8*
e^2 - 100*b*c^7*d^7*e^3 + 100*(2*b^2*c^6 - a*c^7)*d^6*e^4 - 50*(5*b^3*c^5 - 6*a*
b*c^6)*d^5*e^5 + 10*(21*b^4*c^4 - 43*a*b^2*c^5 + 11*a^2*c^6)*d^4*e^6 - 20*(6*b^5
*c^3 - 18*a*b^3*c^4 + 11*a^2*b*c^5)*d^3*e^7 + 5*(9*b^6*c^2 - 36*a*b^4*c^3 + 36*a
^2*b^2*c^4 - 4*a^3*c^5)*d^2*e^8 - 10*(b^7*c - 5*a*b^5*c^2 + 7*a^2*b^3*c^3 - 2*a^
3*b*c^4)*d*e^9 + (b^8 - 6*a*b^6*c + 11*a^2*b^4*c^2 - 6*a^3*b^2*c^3 + a^4*c^4)*e^
10)/(b^2*c^10 - 4*a*c^11)))/(b^2*c^5 - 4*a*c^6)) + 4*(5*c^6*d^8*e - 20*b*c^5*d^7
*e^2 + 35*b^2*c^4*d^6*e^3 - 35*b^3*c^3*d^5*e^4 + 7*(3*b^4*c^2 + a*b^2*c^3 - 2*a^
2*c^4)*d^4*e^5 - 7*(b^5*c + 2*a*b^3*c^2 - 4*a^2*b*c^3)*d^3*e^6 + (b^6 + 9*a*b^4*
c - 15*a^2*b^2*c^2 - 8*a^3*c^3)*d^2*e^7 - (2*a*b^5 - a^2*b^3*c - 8*a^3*b*c^2)*d*
e^8 + (a^2*b^4 - 3*a^3*b^2*c + a^4*c^2)*e^9)*sqrt(e*x + d)) - 3*sqrt(2)*c^2*sqrt
((2*c^5*d^5 - 5*b*c^4*d^4*e + 10*(b^2*c^3 - 2*a*c^4)*d^3*e^2 - 10*(b^3*c^2 - 3*a
*b*c^3)*d^2*e^3 + 5*(b^4*c - 4*a*b^2*c^2 + 2*a^2*c^3)*d*e^4 - (b^5 - 5*a*b^3*c +
5*a^2*b*c^2)*e^5 + (b^2*c^5 - 4*a*c^6)*sqrt((25*c^8*d^8*e^2 - 100*b*c^7*d^7*e^3
+ 100*(2*b^2*c^6 - a*c^7)*d^6*e^4 - 50*(5*b^3*c^5 - 6*a*b*c^6)*d^5*e^5 + 10*(21
*b^4*c^4 - 43*a*b^2*c^5 + 11*a^2*c^6)*d^4*e^6 - 20*(6*b^5*c^3 - 18*a*b^3*c^4 + 1
1*a^2*b*c^5)*d^3*e^7 + 5*(9*b^6*c^2 - 36*a*b^4*c^3 + 36*a^2*b^2*c^4 - 4*a^3*c^5)
*d^2*e^8 - 10*(b^7*c - 5*a*b^5*c^2 + 7*a^2*b^3*c^3 - 2*a^3*b*c^4)*d*e^9 + (b^8 -
6*a*b^6*c + 11*a^2*b^4*c^2 - 6*a^3*b^2*c^3 + a^4*c^4)*e^10)/(b^2*c^10 - 4*a*c^1
1)))/(b^2*c^5 - 4*a*c^6))*log(-sqrt(2)*(10*(b^2*c^5 - 4*a*c^6)*d^5*e^2 - 25*(b^3
*c^4 - 4*a*b*c^5)*d^4*e^3 + 10*(3*b^4*c^3 - 14*a*b^2*c^4 + 8*a^2*c^5)*d^3*e^4 -
10*(2*b^5*c^2 - 11*a*b^3*c^3 + 12*a^2*b*c^4)*d^2*e^5 + (7*b^6*c - 44*a*b^4*c^2 +
66*a^2*b^2*c^3 - 8*a^3*c^4)*d*e^6 - (b^7 - 7*a*b^5*c + 13*a^2*b^3*c^2 - 4*a^3*b
*c^3)*e^7 - (2*(b^2*c^7 - 4*a*c^8)*d^2 - 2*(b^3*c^6 - 4*a*b*c^7)*d*e + (b^4*c^5
- 6*a*b^2*c^6 + 8*a^2*c^7)*e^2)*sqrt((25*c^8*d^8*e^2 - 100*b*c^7*d^7*e^3 + 100*(
2*b^2*c^6 - a*c^7)*d^6*e^4 - 50*(5*b^3*c^5 - 6*a*b*c^6)*d^5*e^5 + 10*(21*b^4*c^4
- 43*a*b^2*c^5 + 11*a^2*c^6)*d^4*e^6 - 20*(6*b^5*c^3 - 18*a*b^3*c^4 + 11*a^2*b*
c^5)*d^3*e^7 + 5*(9*b^6*c^2 - 36*a*b^4*c^3 + 36*a^2*b^2*c^4 - 4*a^3*c^5)*d^2*e^8
- 10*(b^7*c - 5*a*b^5*c^2 + 7*a^2*b^3*c^3 - 2*a^3*b*c^4)*d*e^9 + (b^8 - 6*a*b^6
*c + 11*a^2*b^4*c^2 - 6*a^3*b^2*c^3 + a^4*c^4)*e^10)/(b^2*c^10 - 4*a*c^11)))*sqr
t((2*c^5*d^5 - 5*b*c^4*d^4*e + 10*(b^2*c^3 - 2*a*c^4)*d^3*e^2 - 10*(b^3*c^2 - 3*
a*b*c^3)*d^2*e^3 + 5*(b^4*c - 4*a*b^2*c^2 + 2*a^2*c^3)*d*e^4 - (b^5 - 5*a*b^3*c
+ 5*a^2*b*c^2)*e^5 + (b^2*c^5 - 4*a*c^6)*sqrt((25*c^8*d^8*e^2 - 100*b*c^7*d^7*e^
3 + 100*(2*b^2*c^6 - a*c^7)*d^6*e^4 - 50*(5*b^3*c^5 - 6*a*b*c^6)*d^5*e^5 + 10*(2
1*b^4*c^4 - 43*a*b^2*c^5 + 11*a^2*c^6)*d^4*e^6 - 20*(6*b^5*c^3 - 18*a*b^3*c^4 +
11*a^2*b*c^5)*d^3*e^7 + 5*(9*b^6*c^2 - 36*a*b^4*c^3 + 36*a^2*b^2*c^4 - 4*a^3*c^5
)*d^2*e^8 - 10*(b^7*c - 5*a*b^5*c^2 + 7*a^2*b^3*c^3 - 2*a^3*b*c^4)*d*e^9 + (b^8
- 6*a*b^6*c + 11*a^2*b^4*c^2 - 6*a^3*b^2*c^3 + a^4*c^4)*e^10)/(b^2*c^10 - 4*a*c^
11)))/(b^2*c^5 - 4*a*c^6)) + 4*(5*c^6*d^8*e - 20*b*c^5*d^7*e^2 + 35*b^2*c^4*d^6*
e^3 - 35*b^3*c^3*d^5*e^4 + 7*(3*b^4*c^2 + a*b^2*c^3 - 2*a^2*c^4)*d^4*e^5 - 7*(b^
5*c + 2*a*b^3*c^2 - 4*a^2*b*c^3)*d^3*e^6 + (b^6 + 9*a*b^4*c - 15*a^2*b^2*c^2 - 8
*a^3*c^3)*d^2*e^7 - (2*a*b^5 - a^2*b^3*c - 8*a^3*b*c^2)*d*e^8 + (a^2*b^4 - 3*a^3
*b^2*c + a^4*c^2)*e^9)*sqrt(e*x + d)) + 3*sqrt(2)*c^2*sqrt((2*c^5*d^5 - 5*b*c^4*
d^4*e + 10*(b^2*c^3 - 2*a*c^4)*d^3*e^2 - 10*(b^3*c^2 - 3*a*b*c^3)*d^2*e^3 + 5*(b
^4*c - 4*a*b^2*c^2 + 2*a^2*c^3)*d*e^4 - (b^5 - 5*a*b^3*c + 5*a^2*b*c^2)*e^5 - (b
^2*c^5 - 4*a*c^6)*sqrt((25*c^8*d^8*e^2 - 100*b*c^7*d^7*e^3 + 100*(2*b^2*c^6 - a*
c^7)*d^6*e^4 - 50*(5*b^3*c^5 - 6*a*b*c^6)*d^5*e^5 + 10*(21*b^4*c^4 - 43*a*b^2*c^
5 + 11*a^2*c^6)*d^4*e^6 - 20*(6*b^5*c^3 - 18*a*b^3*c^4 + 11*a^2*b*c^5)*d^3*e^7 +
5*(9*b^6*c^2 - 36*a*b^4*c^3 + 36*a^2*b^2*c^4 - 4*a^3*c^5)*d^2*e^8 - 10*(b^7*c -
5*a*b^5*c^2 + 7*a^2*b^3*c^3 - 2*a^3*b*c^4)*d*e^9 + (b^8 - 6*a*b^6*c + 11*a^2*b^
4*c^2 - 6*a^3*b^2*c^3 + a^4*c^4)*e^10)/(b^2*c^10 - 4*a*c^11)))/(b^2*c^5 - 4*a*c^
6))*log(sqrt(2)*(10*(b^2*c^5 - 4*a*c^6)*d^5*e^2 - 25*(b^3*c^4 - 4*a*b*c^5)*d^4*e
^3 + 10*(3*b^4*c^3 - 14*a*b^2*c^4 + 8*a^2*c^5)*d^3*e^4 - 10*(2*b^5*c^2 - 11*a*b^
3*c^3 + 12*a^2*b*c^4)*d^2*e^5 + (7*b^6*c - 44*a*b^4*c^2 + 66*a^2*b^2*c^3 - 8*a^3
*c^4)*d*e^6 - (b^7 - 7*a*b^5*c + 13*a^2*b^3*c^2 - 4*a^3*b*c^3)*e^7 + (2*(b^2*c^7
- 4*a*c^8)*d^2 - 2*(b^3*c^6 - 4*a*b*c^7)*d*e + (b^4*c^5 - 6*a*b^2*c^6 + 8*a^2*c
^7)*e^2)*sqrt((25*c^8*d^8*e^2 - 100*b*c^7*d^7*e^3 + 100*(2*b^2*c^6 - a*c^7)*d^6*
e^4 - 50*(5*b^3*c^5 - 6*a*b*c^6)*d^5*e^5 + 10*(21*b^4*c^4 - 43*a*b^2*c^5 + 11*a^
2*c^6)*d^4*e^6 - 20*(6*b^5*c^3 - 18*a*b^3*c^4 + 11*a^2*b*c^5)*d^3*e^7 + 5*(9*b^6
*c^2 - 36*a*b^4*c^3 + 36*a^2*b^2*c^4 - 4*a^3*c^5)*d^2*e^8 - 10*(b^7*c - 5*a*b^5*
c^2 + 7*a^2*b^3*c^3 - 2*a^3*b*c^4)*d*e^9 + (b^8 - 6*a*b^6*c + 11*a^2*b^4*c^2 - 6
*a^3*b^2*c^3 + a^4*c^4)*e^10)/(b^2*c^10 - 4*a*c^11)))*sqrt((2*c^5*d^5 - 5*b*c^4*
d^4*e + 10*(b^2*c^3 - 2*a*c^4)*d^3*e^2 - 10*(b^3*c^2 - 3*a*b*c^3)*d^2*e^3 + 5*(b
^4*c - 4*a*b^2*c^2 + 2*a^2*c^3)*d*e^4 - (b^5 - 5*a*b^3*c + 5*a^2*b*c^2)*e^5 - (b
^2*c^5 - 4*a*c^6)*sqrt((25*c^8*d^8*e^2 - 100*b*c^7*d^7*e^3 + 100*(2*b^2*c^6 - a*
c^7)*d^6*e^4 - 50*(5*b^3*c^5 - 6*a*b*c^6)*d^5*e^5 + 10*(21*b^4*c^4 - 43*a*b^2*c^
5 + 11*a^2*c^6)*d^4*e^6 - 20*(6*b^5*c^3 - 18*a*b^3*c^4 + 11*a^2*b*c^5)*d^3*e^7 +
5*(9*b^6*c^2 - 36*a*b^4*c^3 + 36*a^2*b^2*c^4 - 4*a^3*c^5)*d^2*e^8 - 10*(b^7*c -
5*a*b^5*c^2 + 7*a^2*b^3*c^3 - 2*a^3*b*c^4)*d*e^9 + (b^8 - 6*a*b^6*c + 11*a^2*b^
4*c^2 - 6*a^3*b^2*c^3 + a^4*c^4)*e^10)/(b^2*c^10 - 4*a*c^11)))/(b^2*c^5 - 4*a*c^
6)) + 4*(5*c^6*d^8*e - 20*b*c^5*d^7*e^2 + 35*b^2*c^4*d^6*e^3 - 35*b^3*c^3*d^5*e^
4 + 7*(3*b^4*c^2 + a*b^2*c^3 - 2*a^2*c^4)*d^4*e^5 - 7*(b^5*c + 2*a*b^3*c^2 - 4*a
^2*b*c^3)*d^3*e^6 + (b^6 + 9*a*b^4*c - 15*a^2*b^2*c^2 - 8*a^3*c^3)*d^2*e^7 - (2*
a*b^5 - a^2*b^3*c - 8*a^3*b*c^2)*d*e^8 + (a^2*b^4 - 3*a^3*b^2*c + a^4*c^2)*e^9)*
sqrt(e*x + d)) - 3*sqrt(2)*c^2*sqrt((2*c^5*d^5 - 5*b*c^4*d^4*e + 10*(b^2*c^3 - 2
*a*c^4)*d^3*e^2 - 10*(b^3*c^2 - 3*a*b*c^3)*d^2*e^3 + 5*(b^4*c - 4*a*b^2*c^2 + 2*
a^2*c^3)*d*e^4 - (b^5 - 5*a*b^3*c + 5*a^2*b*c^2)*e^5 - (b^2*c^5 - 4*a*c^6)*sqrt(
(25*c^8*d^8*e^2 - 100*b*c^7*d^7*e^3 + 100*(2*b^2*c^6 - a*c^7)*d^6*e^4 - 50*(5*b^
3*c^5 - 6*a*b*c^6)*d^5*e^5 + 10*(21*b^4*c^4 - 43*a*b^2*c^5 + 11*a^2*c^6)*d^4*e^6
- 20*(6*b^5*c^3 - 18*a*b^3*c^4 + 11*a^2*b*c^5)*d^3*e^7 + 5*(9*b^6*c^2 - 36*a*b^
4*c^3 + 36*a^2*b^2*c^4 - 4*a^3*c^5)*d^2*e^8 - 10*(b^7*c - 5*a*b^5*c^2 + 7*a^2*b^
3*c^3 - 2*a^3*b*c^4)*d*e^9 + (b^8 - 6*a*b^6*c + 11*a^2*b^4*c^2 - 6*a^3*b^2*c^3 +
a^4*c^4)*e^10)/(b^2*c^10 - 4*a*c^11)))/(b^2*c^5 - 4*a*c^6))*log(-sqrt(2)*(10*(b
^2*c^5 - 4*a*c^6)*d^5*e^2 - 25*(b^3*c^4 - 4*a*b*c^5)*d^4*e^3 + 10*(3*b^4*c^3 - 1
4*a*b^2*c^4 + 8*a^2*c^5)*d^3*e^4 - 10*(2*b^5*c^2 - 11*a*b^3*c^3 + 12*a^2*b*c^4)*
d^2*e^5 + (7*b^6*c - 44*a*b^4*c^2 + 66*a^2*b^2*c^3 - 8*a^3*c^4)*d*e^6 - (b^7 - 7
*a*b^5*c + 13*a^2*b^3*c^2 - 4*a^3*b*c^3)*e^7 + (2*(b^2*c^7 - 4*a*c^8)*d^2 - 2*(b
^3*c^6 - 4*a*b*c^7)*d*e + (b^4*c^5 - 6*a*b^2*c^6 + 8*a^2*c^7)*e^2)*sqrt((25*c^8*
d^8*e^2 - 100*b*c^7*d^7*e^3 + 100*(2*b^2*c^6 - a*c^7)*d^6*e^4 - 50*(5*b^3*c^5 -
6*a*b*c^6)*d^5*e^5 + 10*(21*b^4*c^4 - 43*a*b^2*c^5 + 11*a^2*c^6)*d^4*e^6 - 20*(6
*b^5*c^3 - 18*a*b^3*c^4 + 11*a^2*b*c^5)*d^3*e^7 + 5*(9*b^6*c^2 - 36*a*b^4*c^3 +
36*a^2*b^2*c^4 - 4*a^3*c^5)*d^2*e^8 - 10*(b^7*c - 5*a*b^5*c^2 + 7*a^2*b^3*c^3 -
2*a^3*b*c^4)*d*e^9 + (b^8 - 6*a*b^6*c + 11*a^2*b^4*c^2 - 6*a^3*b^2*c^3 + a^4*c^4
)*e^10)/(b^2*c^10 - 4*a*c^11)))*sqrt((2*c^5*d^5 - 5*b*c^4*d^4*e + 10*(b^2*c^3 -
2*a*c^4)*d^3*e^2 - 10*(b^3*c^2 - 3*a*b*c^3)*d^2*e^3 + 5*(b^4*c - 4*a*b^2*c^2 + 2
*a^2*c^3)*d*e^4 - (b^5 - 5*a*b^3*c + 5*a^2*b*c^2)*e^5 - (b^2*c^5 - 4*a*c^6)*sqrt
((25*c^8*d^8*e^2 - 100*b*c^7*d^7*e^3 + 100*(2*b^2*c^6 - a*c^7)*d^6*e^4 - 50*(5*b
^3*c^5 - 6*a*b*c^6)*d^5*e^5 + 10*(21*b^4*c^4 - 43*a*b^2*c^5 + 11*a^2*c^6)*d^4*e^
6 - 20*(6*b^5*c^3 - 18*a*b^3*c^4 + 11*a^2*b*c^5)*d^3*e^7 + 5*(9*b^6*c^2 - 36*a*b
^4*c^3 + 36*a^2*b^2*c^4 - 4*a^3*c^5)*d^2*e^8 - 10*(b^7*c - 5*a*b^5*c^2 + 7*a^2*b
^3*c^3 - 2*a^3*b*c^4)*d*e^9 + (b^8 - 6*a*b^6*c + 11*a^2*b^4*c^2 - 6*a^3*b^2*c^3
+ a^4*c^4)*e^10)/(b^2*c^10 - 4*a*c^11)))/(b^2*c^5 - 4*a*c^6)) + 4*(5*c^6*d^8*e -
20*b*c^5*d^7*e^2 + 35*b^2*c^4*d^6*e^3 - 35*b^3*c^3*d^5*e^4 + 7*(3*b^4*c^2 + a*b
^2*c^3 - 2*a^2*c^4)*d^4*e^5 - 7*(b^5*c + 2*a*b^3*c^2 - 4*a^2*b*c^3)*d^3*e^6 + (b
^6 + 9*a*b^4*c - 15*a^2*b^2*c^2 - 8*a^3*c^3)*d^2*e^7 - (2*a*b^5 - a^2*b^3*c - 8*
a^3*b*c^2)*d*e^8 + (a^2*b^4 - 3*a^3*b^2*c + a^4*c^2)*e^9)*sqrt(e*x + d)) + 4*(c*
e^2*x + 7*c*d*e - 3*b*e^2)*sqrt(e*x + d))/c^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((e*x+d)**(5/2)/(c*x**2+b*x+a),x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out