3.1613 \(\int \frac{(b+2 c x) (d+e x)^{3/2}}{a+b x+c x^2} \, dx\)

Optimal. Leaf size=422 \[ \frac{\sqrt{2} \left (-2 c^2 d \left (d \sqrt{b^2-4 a c}-4 a e\right )-2 c e \left (-b d \sqrt{b^2-4 a c}-a e \sqrt{b^2-4 a c}+2 a b e+b^2 d\right )+b^2 e^2 \left (b-\sqrt{b^2-4 a c}\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}\right )}{c^{3/2} \sqrt{b^2-4 a c} \sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}-\frac{\sqrt{2} \left (2 c^2 d \left (d \sqrt{b^2-4 a c}+4 a e\right )-2 c e \left (b d \sqrt{b^2-4 a c}+a e \sqrt{b^2-4 a c}+2 a b e+b^2 d\right )+b^2 e^2 \left (\sqrt{b^2-4 a c}+b\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}\right )}{c^{3/2} \sqrt{b^2-4 a c} \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}+\frac{2 \sqrt{d+e x} (2 c d-b e)}{c}+\frac{4}{3} (d+e x)^{3/2} \]

[Out]

(2*(2*c*d - b*e)*Sqrt[d + e*x])/c + (4*(d + e*x)^(3/2))/3 + (Sqrt[2]*(b^2*(b - S
qrt[b^2 - 4*a*c])*e^2 - 2*c^2*d*(Sqrt[b^2 - 4*a*c]*d - 4*a*e) - 2*c*e*(b^2*d - b
*Sqrt[b^2 - 4*a*c]*d + 2*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^(3/2)*Sqrt[b^2 - 4
*a*c]*Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e]) - (Sqrt[2]*(b^2*(b + Sqrt[b^2 - 4
*a*c])*e^2 + 2*c^2*d*(Sqrt[b^2 - 4*a*c]*d + 4*a*e) - 2*c*e*(b^2*d + b*Sqrt[b^2 -
4*a*c]*d + 2*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^(3/2)*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 = 4.29965, antiderivative size = 422, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ \frac{\sqrt{2} \left (-2 c^2 d \left (d \sqrt{b^2-4 a c}-4 a e\right )-2 c e \left (-b d \sqrt{b^2-4 a c}-a e \sqrt{b^2-4 a c}+2 a b e+b^2 d\right )+b^2 e^2 \left (b-\sqrt{b^2-4 a c}\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}\right )}{c^{3/2} \sqrt{b^2-4 a c} \sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}-\frac{\sqrt{2} \left (2 c^2 d \left (d \sqrt{b^2-4 a c}+4 a e\right )-2 c e \left (b d \sqrt{b^2-4 a c}+a e \sqrt{b^2-4 a c}+2 a b e+b^2 d\right )+b^2 e^2 \left (\sqrt{b^2-4 a c}+b\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}\right )}{c^{3/2} \sqrt{b^2-4 a c} \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}+\frac{2 \sqrt{d+e x} (2 c d-b e)}{c}+\frac{4}{3} (d+e x)^{3/2} \]

Antiderivative was successfully verified.

[In]  Int[((b + 2*c*x)*(d + e*x)^(3/2))/(a + b*x + c*x^2),x]

[Out]

(2*(2*c*d - b*e)*Sqrt[d + e*x])/c + (4*(d + e*x)^(3/2))/3 + (Sqrt[2]*(b^2*(b - S
qrt[b^2 - 4*a*c])*e^2 - 2*c^2*d*(Sqrt[b^2 - 4*a*c]*d - 4*a*e) - 2*c*e*(b^2*d - b
*Sqrt[b^2 - 4*a*c]*d + 2*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^(3/2)*Sqrt[b^2 - 4
*a*c]*Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e]) - (Sqrt[2]*(b^2*(b + Sqrt[b^2 - 4
*a*c])*e^2 + 2*c^2*d*(Sqrt[b^2 - 4*a*c]*d + 4*a*e) - 2*c*e*(b^2*d + b*Sqrt[b^2 -
4*a*c]*d + 2*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^(3/2)*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((2*c*x+b)*(e*x+d)**(3/2)/(c*x**2+b*x+a),x)

[Out]

Timed out

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Mathematica [A]  time = 0.770771, size = 414, normalized size = 0.98 \[ -\frac{\sqrt{2} \left (2 c^2 d \left (d \sqrt{b^2-4 a c}-4 a e\right )-2 c e \left (b d \sqrt{b^2-4 a c}+a e \sqrt{b^2-4 a c}-2 a b e+b^2 (-d)\right )+b^2 e^2 \left (\sqrt{b^2-4 a c}-b\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{e \sqrt{b^2-4 a c}-b e+2 c d}}\right )}{c^{3/2} \sqrt{b^2-4 a c} \sqrt{e \left (\sqrt{b^2-4 a c}-b\right )+2 c d}}-\frac{\sqrt{2} \left (2 c^2 d \left (d \sqrt{b^2-4 a c}+4 a e\right )-2 c e \left (b d \sqrt{b^2-4 a c}+a e \sqrt{b^2-4 a c}+2 a b e+b^2 d\right )+b^2 e^2 \left (\sqrt{b^2-4 a c}+b\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}\right )}{c^{3/2} \sqrt{b^2-4 a c} \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}+\frac{2 \sqrt{d+e x} (-3 b e+8 c d+2 c e x)}{3 c} \]

Antiderivative was successfully verified.

[In]  Integrate[((b + 2*c*x)*(d + e*x)^(3/2))/(a + b*x + c*x^2),x]

[Out]

(2*Sqrt[d + e*x]*(8*c*d - 3*b*e + 2*c*e*x))/(3*c) - (Sqrt[2]*(b^2*(-b + Sqrt[b^2
- 4*a*c])*e^2 + 2*c^2*d*(Sqrt[b^2 - 4*a*c]*d - 4*a*e) - 2*c*e*(-(b^2*d) + b*Sqr
t[b^2 - 4*a*c]*d - 2*a*b*e + a*Sqrt[b^2 - 4*a*c]*e))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sq
rt[d + e*x])/Sqrt[2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e]])/(c^(3/2)*Sqrt[b^2 - 4*a*c
]*Sqrt[2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e]) - (Sqrt[2]*(b^2*(b + Sqrt[b^2 - 4*a*
c])*e^2 + 2*c^2*d*(Sqrt[b^2 - 4*a*c]*d + 4*a*e) - 2*c*e*(b^2*d + b*Sqrt[b^2 - 4*
a*c]*d + 2*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^(3/2)*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 = 0.06, size = 1494, normalized size = 3.5 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2*c*x+b)*(e*x+d)^(3/2)/(c*x^2+b*x+a),x)

[Out]

4/3*(e*x+d)^(3/2)-2/c*b*e*(e*x+d)^(1/2)+4*(e*x+d)^(1/2)*d-4/(-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*b*e^3+8*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*e^
2*d+1/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^3*e^3-2/(-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^2*d*e^2+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^2-1/c*2^(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)*arc
tanh(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^2+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*e-2*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))*d^2-4/(-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^3+8*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*e^2*d+1/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
^3*e^3-2/(-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^2*d*e^2-2*2^(1/2)/((b*e-2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)*arcta
n(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^2+
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))*b^2*e^2-2*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*e+2*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))*d^2

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (2 \, c x + b\right )}{\left (e x + d\right )}^{\frac{3}{2}}}{c x^{2} + b x + a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*x + b)*(e*x + d)^(3/2)/(c*x^2 + b*x + a),x, algorithm="maxima")

[Out]

integrate((2*c*x + b)*(e*x + d)^(3/2)/(c*x^2 + b*x + a), x)

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Fricas [A]  time = 0.369227, size = 3927, normalized size = 9.31 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*x + b)*(e*x + d)^(3/2)/(c*x^2 + b*x + a),x, algorithm="fricas")

[Out]

1/6*(3*sqrt(2)*c*sqrt((2*c^3*d^3 - 3*b*c^2*d^2*e + 3*(b^2*c - 2*a*c^2)*d*e^2 - (
b^3 - 3*a*b*c)*e^3 + c^3*sqrt((9*(b^2*c^4 - 4*a*c^5)*d^4*e^2 - 18*(b^3*c^3 - 4*a
*b*c^4)*d^3*e^3 + 3*(5*b^4*c^2 - 22*a*b^2*c^3 + 8*a^2*c^4)*d^2*e^4 - 6*(b^5*c -
5*a*b^3*c^2 + 4*a^2*b*c^3)*d*e^5 + (b^6 - 6*a*b^4*c + 9*a^2*b^2*c^2 - 4*a^3*c^3)
*e^6)/c^6))/c^3)*log(sqrt(2)*(6*c^3*d^3 - 9*b*c^2*d^2*e + (5*b^2*c - 2*a*c^2)*d*
e^2 - (b^3 - a*b*c)*e^3 - c^3*sqrt((9*(b^2*c^4 - 4*a*c^5)*d^4*e^2 - 18*(b^3*c^3
- 4*a*b*c^4)*d^3*e^3 + 3*(5*b^4*c^2 - 22*a*b^2*c^3 + 8*a^2*c^4)*d^2*e^4 - 6*(b^5
*c - 5*a*b^3*c^2 + 4*a^2*b*c^3)*d*e^5 + (b^6 - 6*a*b^4*c + 9*a^2*b^2*c^2 - 4*a^3
*c^3)*e^6)/c^6))*sqrt((2*c^3*d^3 - 3*b*c^2*d^2*e + 3*(b^2*c - 2*a*c^2)*d*e^2 - (
b^3 - 3*a*b*c)*e^3 + c^3*sqrt((9*(b^2*c^4 - 4*a*c^5)*d^4*e^2 - 18*(b^3*c^3 - 4*a
*b*c^4)*d^3*e^3 + 3*(5*b^4*c^2 - 22*a*b^2*c^3 + 8*a^2*c^4)*d^2*e^4 - 6*(b^5*c -
5*a*b^3*c^2 + 4*a^2*b*c^3)*d*e^5 + (b^6 - 6*a*b^4*c + 9*a^2*b^2*c^2 - 4*a^3*c^3)
*e^6)/c^6))/c^3) - 4*(3*c^3*d^4 - 6*b*c^2*d^3*e + 2*(2*b^2*c + a*c^2)*d^2*e^2 -
(b^3 + 2*a*b*c)*d*e^3 + (a*b^2 - a^2*c)*e^4)*sqrt(e*x + d)) - 3*sqrt(2)*c*sqrt((
2*c^3*d^3 - 3*b*c^2*d^2*e + 3*(b^2*c - 2*a*c^2)*d*e^2 - (b^3 - 3*a*b*c)*e^3 + c^
3*sqrt((9*(b^2*c^4 - 4*a*c^5)*d^4*e^2 - 18*(b^3*c^3 - 4*a*b*c^4)*d^3*e^3 + 3*(5*
b^4*c^2 - 22*a*b^2*c^3 + 8*a^2*c^4)*d^2*e^4 - 6*(b^5*c - 5*a*b^3*c^2 + 4*a^2*b*c
^3)*d*e^5 + (b^6 - 6*a*b^4*c + 9*a^2*b^2*c^2 - 4*a^3*c^3)*e^6)/c^6))/c^3)*log(-s
qrt(2)*(6*c^3*d^3 - 9*b*c^2*d^2*e + (5*b^2*c - 2*a*c^2)*d*e^2 - (b^3 - a*b*c)*e^
3 - c^3*sqrt((9*(b^2*c^4 - 4*a*c^5)*d^4*e^2 - 18*(b^3*c^3 - 4*a*b*c^4)*d^3*e^3 +
3*(5*b^4*c^2 - 22*a*b^2*c^3 + 8*a^2*c^4)*d^2*e^4 - 6*(b^5*c - 5*a*b^3*c^2 + 4*a
^2*b*c^3)*d*e^5 + (b^6 - 6*a*b^4*c + 9*a^2*b^2*c^2 - 4*a^3*c^3)*e^6)/c^6))*sqrt(
(2*c^3*d^3 - 3*b*c^2*d^2*e + 3*(b^2*c - 2*a*c^2)*d*e^2 - (b^3 - 3*a*b*c)*e^3 + c
^3*sqrt((9*(b^2*c^4 - 4*a*c^5)*d^4*e^2 - 18*(b^3*c^3 - 4*a*b*c^4)*d^3*e^3 + 3*(5
*b^4*c^2 - 22*a*b^2*c^3 + 8*a^2*c^4)*d^2*e^4 - 6*(b^5*c - 5*a*b^3*c^2 + 4*a^2*b*
c^3)*d*e^5 + (b^6 - 6*a*b^4*c + 9*a^2*b^2*c^2 - 4*a^3*c^3)*e^6)/c^6))/c^3) - 4*(
3*c^3*d^4 - 6*b*c^2*d^3*e + 2*(2*b^2*c + a*c^2)*d^2*e^2 - (b^3 + 2*a*b*c)*d*e^3
+ (a*b^2 - a^2*c)*e^4)*sqrt(e*x + d)) + 3*sqrt(2)*c*sqrt((2*c^3*d^3 - 3*b*c^2*d^
2*e + 3*(b^2*c - 2*a*c^2)*d*e^2 - (b^3 - 3*a*b*c)*e^3 - c^3*sqrt((9*(b^2*c^4 - 4
*a*c^5)*d^4*e^2 - 18*(b^3*c^3 - 4*a*b*c^4)*d^3*e^3 + 3*(5*b^4*c^2 - 22*a*b^2*c^3
+ 8*a^2*c^4)*d^2*e^4 - 6*(b^5*c - 5*a*b^3*c^2 + 4*a^2*b*c^3)*d*e^5 + (b^6 - 6*a
*b^4*c + 9*a^2*b^2*c^2 - 4*a^3*c^3)*e^6)/c^6))/c^3)*log(sqrt(2)*(6*c^3*d^3 - 9*b
*c^2*d^2*e + (5*b^2*c - 2*a*c^2)*d*e^2 - (b^3 - a*b*c)*e^3 + c^3*sqrt((9*(b^2*c^
4 - 4*a*c^5)*d^4*e^2 - 18*(b^3*c^3 - 4*a*b*c^4)*d^3*e^3 + 3*(5*b^4*c^2 - 22*a*b^
2*c^3 + 8*a^2*c^4)*d^2*e^4 - 6*(b^5*c - 5*a*b^3*c^2 + 4*a^2*b*c^3)*d*e^5 + (b^6
- 6*a*b^4*c + 9*a^2*b^2*c^2 - 4*a^3*c^3)*e^6)/c^6))*sqrt((2*c^3*d^3 - 3*b*c^2*d^
2*e + 3*(b^2*c - 2*a*c^2)*d*e^2 - (b^3 - 3*a*b*c)*e^3 - c^3*sqrt((9*(b^2*c^4 - 4
*a*c^5)*d^4*e^2 - 18*(b^3*c^3 - 4*a*b*c^4)*d^3*e^3 + 3*(5*b^4*c^2 - 22*a*b^2*c^3
+ 8*a^2*c^4)*d^2*e^4 - 6*(b^5*c - 5*a*b^3*c^2 + 4*a^2*b*c^3)*d*e^5 + (b^6 - 6*a
*b^4*c + 9*a^2*b^2*c^2 - 4*a^3*c^3)*e^6)/c^6))/c^3) - 4*(3*c^3*d^4 - 6*b*c^2*d^3
*e + 2*(2*b^2*c + a*c^2)*d^2*e^2 - (b^3 + 2*a*b*c)*d*e^3 + (a*b^2 - a^2*c)*e^4)*
sqrt(e*x + d)) - 3*sqrt(2)*c*sqrt((2*c^3*d^3 - 3*b*c^2*d^2*e + 3*(b^2*c - 2*a*c^
2)*d*e^2 - (b^3 - 3*a*b*c)*e^3 - c^3*sqrt((9*(b^2*c^4 - 4*a*c^5)*d^4*e^2 - 18*(b
^3*c^3 - 4*a*b*c^4)*d^3*e^3 + 3*(5*b^4*c^2 - 22*a*b^2*c^3 + 8*a^2*c^4)*d^2*e^4 -
6*(b^5*c - 5*a*b^3*c^2 + 4*a^2*b*c^3)*d*e^5 + (b^6 - 6*a*b^4*c + 9*a^2*b^2*c^2
- 4*a^3*c^3)*e^6)/c^6))/c^3)*log(-sqrt(2)*(6*c^3*d^3 - 9*b*c^2*d^2*e + (5*b^2*c
- 2*a*c^2)*d*e^2 - (b^3 - a*b*c)*e^3 + c^3*sqrt((9*(b^2*c^4 - 4*a*c^5)*d^4*e^2 -
18*(b^3*c^3 - 4*a*b*c^4)*d^3*e^3 + 3*(5*b^4*c^2 - 22*a*b^2*c^3 + 8*a^2*c^4)*d^2
*e^4 - 6*(b^5*c - 5*a*b^3*c^2 + 4*a^2*b*c^3)*d*e^5 + (b^6 - 6*a*b^4*c + 9*a^2*b^
2*c^2 - 4*a^3*c^3)*e^6)/c^6))*sqrt((2*c^3*d^3 - 3*b*c^2*d^2*e + 3*(b^2*c - 2*a*c
^2)*d*e^2 - (b^3 - 3*a*b*c)*e^3 - c^3*sqrt((9*(b^2*c^4 - 4*a*c^5)*d^4*e^2 - 18*(
b^3*c^3 - 4*a*b*c^4)*d^3*e^3 + 3*(5*b^4*c^2 - 22*a*b^2*c^3 + 8*a^2*c^4)*d^2*e^4
- 6*(b^5*c - 5*a*b^3*c^2 + 4*a^2*b*c^3)*d*e^5 + (b^6 - 6*a*b^4*c + 9*a^2*b^2*c^2
- 4*a^3*c^3)*e^6)/c^6))/c^3) - 4*(3*c^3*d^4 - 6*b*c^2*d^3*e + 2*(2*b^2*c + a*c^
2)*d^2*e^2 - (b^3 + 2*a*b*c)*d*e^3 + (a*b^2 - a^2*c)*e^4)*sqrt(e*x + d)) + 4*(2*
c*e*x + 8*c*d - 3*b*e)*sqrt(e*x + d))/c

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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)**(3/2)/(c*x**2+b*x+a),x)

[Out]

Timed out

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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((2*c*x + b)*(e*x + d)^(3/2)/(c*x^2 + b*x + a),x, algorithm="giac")

[Out]

Timed out