3.1625 \(\int \frac{(b+2 c x) (d+e x)^{3/2}}{\left (a+b x+c x^2\right )^3} \, dx\)

Optimal. Leaf size=322 \[ -\frac{3 e (b+2 c x) \sqrt{d+e x}}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac{3 \sqrt{c} e \left (4 c d-e \left (2 b-\sqrt{b^2-4 a c}\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}\right )}{2 \sqrt{2} \left (b^2-4 a c\right )^{3/2} \sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}-\frac{3 \sqrt{c} e \left (4 c d-e \left (\sqrt{b^2-4 a c}+2 b\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}\right )}{2 \sqrt{2} \left (b^2-4 a c\right )^{3/2} \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}-\frac{(d+e x)^{3/2}}{2 \left (a+b x+c x^2\right )^2} \]

[Out]

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

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Rubi [A]  time = 1.70519, antiderivative size = 322, 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 \[ -\frac{3 e (b+2 c x) \sqrt{d+e x}}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac{3 \sqrt{c} e \left (4 c d-e \left (2 b-\sqrt{b^2-4 a c}\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}\right )}{2 \sqrt{2} \left (b^2-4 a c\right )^{3/2} \sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}-\frac{3 \sqrt{c} e \left (4 c d-e \left (\sqrt{b^2-4 a c}+2 b\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}\right )}{2 \sqrt{2} \left (b^2-4 a c\right )^{3/2} \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}-\frac{(d+e x)^{3/2}}{2 \left (a+b x+c x^2\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

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

[Out]

Timed out

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Mathematica [A]  time = 1.72225, size = 307, normalized size = 0.95 \[ \frac{1}{4} \left (-\frac{\sqrt{d+e x} \left (\frac{3 e (b+2 c x) (a+x (b+c x))}{b^2-4 a c}+2 (d+e x)\right )}{(a+x (b+c x))^2}+\frac{3 \sqrt{2} \sqrt{c} e \left (e \left (\sqrt{b^2-4 a c}-2 b\right )+4 c d\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{e \sqrt{b^2-4 a c}-b e+2 c d}}\right )}{\left (b^2-4 a c\right )^{3/2} \sqrt{e \left (\sqrt{b^2-4 a c}-b\right )+2 c d}}+\frac{3 \sqrt{2} \sqrt{c} e \left (e \left (\sqrt{b^2-4 a c}+2 b\right )-4 c d\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}\right )}{\left (b^2-4 a c\right )^{3/2} \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.084, size = 4025, normalized size = 12.5 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/2*e^6*c/(-e^6*(4*a*c-b^2)^3)^(1/2)/(4*a*c-b^2)*2^(1/2)/(c*(4*a*c-b^2)*(4*a*b*c
*e^3-8*a*c^2*e^2*d-b^3*e^3+2*b^2*c*d*e^2+(-e^6*(4*a*c-b^2)^3)^(1/2)))^(1/2)*arct
an(1/2*(8*a*c^2*e^2-2*b^2*c*e^2)*(e*x+d)^(1/2)*2^(1/2)/e/(c*(4*a*c-b^2)*(4*a*b*c
*e^3-8*a*c^2*e^2*d-b^3*e^3+2*b^2*c*d*e^2+(-e^6*(4*a*c-b^2)^3)^(1/2)))^(1/2))*b^5
-9/2*e^3/(c*e^2*x^2+b*e^2*x+a*e^2)^2/(4*a*c-b^2)*(e*x+d)^(3/2)*b*c*d+9/4*e^3/(c*
e^2*x^2+b*e^2*x+a*e^2)^2/(4*a*c-b^2)*(e*x+d)^(1/2)*b*c*d^2+3*e^3*c^2/(4*a*c-b^2)
*2^(1/2)/((-4*a*b*c*e^3+8*a*c^2*e^2*d+b^3*e^3-2*b^2*c*d*e^2+(-e^6*(4*a*c-b^2)^3)
^(1/2))*(4*a*c-b^2)*c)^(1/2)*arctanh(1/2*(-8*a*c^2*e^2+2*b^2*c*e^2)*(e*x+d)^(1/2
)*2^(1/2)/e/((-4*a*b*c*e^3+8*a*c^2*e^2*d+b^3*e^3-2*b^2*c*d*e^2+(-e^6*(4*a*c-b^2)
^3)^(1/2))*(4*a*c-b^2)*c)^(1/2))*a-3/4*e^3*c/(4*a*c-b^2)*2^(1/2)/((-4*a*b*c*e^3+
8*a*c^2*e^2*d+b^3*e^3-2*b^2*c*d*e^2+(-e^6*(4*a*c-b^2)^3)^(1/2))*(4*a*c-b^2)*c)^(
1/2)*arctanh(1/2*(-8*a*c^2*e^2+2*b^2*c*e^2)*(e*x+d)^(1/2)*2^(1/2)/e/((-4*a*b*c*e
^3+8*a*c^2*e^2*d+b^3*e^3-2*b^2*c*d*e^2+(-e^6*(4*a*c-b^2)^3)^(1/2))*(4*a*c-b^2)*c
)^(1/2))*b^2+3*e^3*c^2/(4*a*c-b^2)*2^(1/2)/(c*(4*a*c-b^2)*(4*a*b*c*e^3-8*a*c^2*e
^2*d-b^3*e^3+2*b^2*c*d*e^2+(-e^6*(4*a*c-b^2)^3)^(1/2)))^(1/2)*arctan(1/2*(8*a*c^
2*e^2-2*b^2*c*e^2)*(e*x+d)^(1/2)*2^(1/2)/e/(c*(4*a*c-b^2)*(4*a*b*c*e^3-8*a*c^2*e
^2*d-b^3*e^3+2*b^2*c*d*e^2+(-e^6*(4*a*c-b^2)^3)^(1/2)))^(1/2))*a-3/4*e^3*c/(4*a*
c-b^2)*2^(1/2)/(c*(4*a*c-b^2)*(4*a*b*c*e^3-8*a*c^2*e^2*d-b^3*e^3+2*b^2*c*d*e^2+(
-e^6*(4*a*c-b^2)^3)^(1/2)))^(1/2)*arctan(1/2*(8*a*c^2*e^2-2*b^2*c*e^2)*(e*x+d)^(
1/2)*2^(1/2)/e/(c*(4*a*c-b^2)*(4*a*b*c*e^3-8*a*c^2*e^2*d-b^3*e^3+2*b^2*c*d*e^2+(
-e^6*(4*a*c-b^2)^3)^(1/2)))^(1/2))*b^2-3/2*e^4/(c*e^2*x^2+b*e^2*x+a*e^2)^2/(4*a*
c-b^2)*(e*x+d)^(1/2)*a*d*c+5/4*e^4/(c*e^2*x^2+b*e^2*x+a*e^2)^2/(4*a*c-b^2)*(e*x+
d)^(3/2)*b^2+3/2*e^2/(c*e^2*x^2+b*e^2*x+a*e^2)^2*c^2/(4*a*c-b^2)*(e*x+d)^(7/2)+2
4*e^6*c^3/(-e^6*(4*a*c-b^2)^3)^(1/2)/(4*a*c-b^2)*2^(1/2)/(c*(4*a*c-b^2)*(4*a*b*c
*e^3-8*a*c^2*e^2*d-b^3*e^3+2*b^2*c*d*e^2+(-e^6*(4*a*c-b^2)^3)^(1/2)))^(1/2)*arct
an(1/2*(8*a*c^2*e^2-2*b^2*c*e^2)*(e*x+d)^(1/2)*2^(1/2)/e/(c*(4*a*c-b^2)*(4*a*b*c
*e^3-8*a*c^2*e^2*d-b^3*e^3+2*b^2*c*d*e^2+(-e^6*(4*a*c-b^2)^3)^(1/2)))^(1/2))*a^2
*b-12*e^6*c^2/(-e^6*(4*a*c-b^2)^3)^(1/2)/(4*a*c-b^2)*2^(1/2)/(c*(4*a*c-b^2)*(4*a
*b*c*e^3-8*a*c^2*e^2*d-b^3*e^3+2*b^2*c*d*e^2+(-e^6*(4*a*c-b^2)^3)^(1/2)))^(1/2)*
arctan(1/2*(8*a*c^2*e^2-2*b^2*c*e^2)*(e*x+d)^(1/2)*2^(1/2)/e/(c*(4*a*c-b^2)*(4*a
*b*c*e^3-8*a*c^2*e^2*d-b^3*e^3+2*b^2*c*d*e^2+(-e^6*(4*a*c-b^2)^3)^(1/2)))^(1/2))
*a*b^3-48*e^5*c^4/(-e^6*(4*a*c-b^2)^3)^(1/2)/(4*a*c-b^2)*2^(1/2)/(c*(4*a*c-b^2)*
(4*a*b*c*e^3-8*a*c^2*e^2*d-b^3*e^3+2*b^2*c*d*e^2+(-e^6*(4*a*c-b^2)^3)^(1/2)))^(1
/2)*arctan(1/2*(8*a*c^2*e^2-2*b^2*c*e^2)*(e*x+d)^(1/2)*2^(1/2)/e/(c*(4*a*c-b^2)*
(4*a*b*c*e^3-8*a*c^2*e^2*d-b^3*e^3+2*b^2*c*d*e^2+(-e^6*(4*a*c-b^2)^3)^(1/2)))^(1
/2))*a^2*d-1/2*e^4/(c*e^2*x^2+b*e^2*x+a*e^2)^2/(4*a*c-b^2)*(e*x+d)^(3/2)*a*c-3/4
*e^4/(c*e^2*x^2+b*e^2*x+a*e^2)^2/(4*a*c-b^2)*(e*x+d)^(1/2)*b^2*d+9/4*e^3/(c*e^2*
x^2+b*e^2*x+a*e^2)^2*c/(4*a*c-b^2)*(e*x+d)^(5/2)*b-9/2*e^2/(c*e^2*x^2+b*e^2*x+a*
e^2)^2*c^2/(4*a*c-b^2)*(e*x+d)^(5/2)*d+9/2*e^2/(c*e^2*x^2+b*e^2*x+a*e^2)^2/(4*a*
c-b^2)*(e*x+d)^(3/2)*c^2*d^2+3/4*e^5/(c*e^2*x^2+b*e^2*x+a*e^2)^2/(4*a*c-b^2)*(e*
x+d)^(1/2)*a*b-3/2*e^2/(c*e^2*x^2+b*e^2*x+a*e^2)^2/(4*a*c-b^2)*(e*x+d)^(1/2)*c^2
*d^3-3/2*e^6*c/(-e^6*(4*a*c-b^2)^3)^(1/2)/(4*a*c-b^2)*2^(1/2)/((-4*a*b*c*e^3+8*a
*c^2*e^2*d+b^3*e^3-2*b^2*c*d*e^2+(-e^6*(4*a*c-b^2)^3)^(1/2))*(4*a*c-b^2)*c)^(1/2
)*arctanh(1/2*(-8*a*c^2*e^2+2*b^2*c*e^2)*(e*x+d)^(1/2)*2^(1/2)/e/((-4*a*b*c*e^3+
8*a*c^2*e^2*d+b^3*e^3-2*b^2*c*d*e^2+(-e^6*(4*a*c-b^2)^3)^(1/2))*(4*a*c-b^2)*c)^(
1/2))*b^5-24*e^6*c^3/(-e^6*(4*a*c-b^2)^3)^(1/2)/(4*a*c-b^2)*2^(1/2)/((-4*a*b*c*e
^3+8*a*c^2*e^2*d+b^3*e^3-2*b^2*c*d*e^2+(-e^6*(4*a*c-b^2)^3)^(1/2))*(4*a*c-b^2)*c
)^(1/2)*arctanh(1/2*(-8*a*c^2*e^2+2*b^2*c*e^2)*(e*x+d)^(1/2)*2^(1/2)/e/((-4*a*b*
c*e^3+8*a*c^2*e^2*d+b^3*e^3-2*b^2*c*d*e^2+(-e^6*(4*a*c-b^2)^3)^(1/2))*(4*a*c-b^2
)*c)^(1/2))*a^2*b-3*e^5*c^2/(-e^6*(4*a*c-b^2)^3)^(1/2)/(4*a*c-b^2)*2^(1/2)/(c*(4
*a*c-b^2)*(4*a*b*c*e^3-8*a*c^2*e^2*d-b^3*e^3+2*b^2*c*d*e^2+(-e^6*(4*a*c-b^2)^3)^
(1/2)))^(1/2)*arctan(1/2*(8*a*c^2*e^2-2*b^2*c*e^2)*(e*x+d)^(1/2)*2^(1/2)/e/(c*(4
*a*c-b^2)*(4*a*b*c*e^3-8*a*c^2*e^2*d-b^3*e^3+2*b^2*c*d*e^2+(-e^6*(4*a*c-b^2)^3)^
(1/2)))^(1/2))*b^4*d+12*e^6*c^2/(-e^6*(4*a*c-b^2)^3)^(1/2)/(4*a*c-b^2)*2^(1/2)/(
(-4*a*b*c*e^3+8*a*c^2*e^2*d+b^3*e^3-2*b^2*c*d*e^2+(-e^6*(4*a*c-b^2)^3)^(1/2))*(4
*a*c-b^2)*c)^(1/2)*arctanh(1/2*(-8*a*c^2*e^2+2*b^2*c*e^2)*(e*x+d)^(1/2)*2^(1/2)/
e/((-4*a*b*c*e^3+8*a*c^2*e^2*d+b^3*e^3-2*b^2*c*d*e^2+(-e^6*(4*a*c-b^2)^3)^(1/2))
*(4*a*c-b^2)*c)^(1/2))*a*b^3+48*e^5*c^4/(-e^6*(4*a*c-b^2)^3)^(1/2)/(4*a*c-b^2)*2
^(1/2)/((-4*a*b*c*e^3+8*a*c^2*e^2*d+b^3*e^3-2*b^2*c*d*e^2+(-e^6*(4*a*c-b^2)^3)^(
1/2))*(4*a*c-b^2)*c)^(1/2)*arctanh(1/2*(-8*a*c^2*e^2+2*b^2*c*e^2)*(e*x+d)^(1/2)*
2^(1/2)/e/((-4*a*b*c*e^3+8*a*c^2*e^2*d+b^3*e^3-2*b^2*c*d*e^2+(-e^6*(4*a*c-b^2)^3
)^(1/2))*(4*a*c-b^2)*c)^(1/2))*a^2*d+3*e^5*c^2/(-e^6*(4*a*c-b^2)^3)^(1/2)/(4*a*c
-b^2)*2^(1/2)/((-4*a*b*c*e^3+8*a*c^2*e^2*d+b^3*e^3-2*b^2*c*d*e^2+(-e^6*(4*a*c-b^
2)^3)^(1/2))*(4*a*c-b^2)*c)^(1/2)*arctanh(1/2*(-8*a*c^2*e^2+2*b^2*c*e^2)*(e*x+d)
^(1/2)*2^(1/2)/e/((-4*a*b*c*e^3+8*a*c^2*e^2*d+b^3*e^3-2*b^2*c*d*e^2+(-e^6*(4*a*c
-b^2)^3)^(1/2))*(4*a*c-b^2)*c)^(1/2))*b^4*d-24*e^5*c^3/(-e^6*(4*a*c-b^2)^3)^(1/2
)/(4*a*c-b^2)*2^(1/2)/((-4*a*b*c*e^3+8*a*c^2*e^2*d+b^3*e^3-2*b^2*c*d*e^2+(-e^6*(
4*a*c-b^2)^3)^(1/2))*(4*a*c-b^2)*c)^(1/2)*arctanh(1/2*(-8*a*c^2*e^2+2*b^2*c*e^2)
*(e*x+d)^(1/2)*2^(1/2)/e/((-4*a*b*c*e^3+8*a*c^2*e^2*d+b^3*e^3-2*b^2*c*d*e^2+(-e^
6*(4*a*c-b^2)^3)^(1/2))*(4*a*c-b^2)*c)^(1/2))*a*d*b^2+24*e^5*c^3/(-e^6*(4*a*c-b^
2)^3)^(1/2)/(4*a*c-b^2)*2^(1/2)/(c*(4*a*c-b^2)*(4*a*b*c*e^3-8*a*c^2*e^2*d-b^3*e^
3+2*b^2*c*d*e^2+(-e^6*(4*a*c-b^2)^3)^(1/2)))^(1/2)*arctan(1/2*(8*a*c^2*e^2-2*b^2
*c*e^2)*(e*x+d)^(1/2)*2^(1/2)/e/(c*(4*a*c-b^2)*(4*a*b*c*e^3-8*a*c^2*e^2*d-b^3*e^
3+2*b^2*c*d*e^2+(-e^6*(4*a*c-b^2)^3)^(1/2)))^(1/2))*a*d*b^2

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

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)^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.416723, size = 9083, normalized size = 28.21 \[ \text{result too large to display} \]

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)^3,x, algorithm="fricas")

[Out]

1/8*(3*sqrt(1/2)*((b^2*c^2 - 4*a*c^3)*x^4 + a^2*b^2 - 4*a^3*c + 2*(b^3*c - 4*a*b
*c^2)*x^3 + (b^4 - 2*a*b^2*c - 8*a^2*c^2)*x^2 + 2*(a*b^3 - 4*a^2*b*c)*x)*sqrt((3
2*c^3*d^3*e^2 - 48*b*c^2*d^2*e^3 + 6*(3*b^2*c + 4*a*c^2)*d*e^4 - (b^3 + 12*a*b*c
)*e^5 + sqrt(e^10/((b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)*d^4 -
2*(b^7*c - 12*a*b^5*c^2 + 48*a^2*b^3*c^3 - 64*a^3*b*c^4)*d^3*e + (b^8 - 10*a*b^6
*c + 24*a^2*b^4*c^2 + 32*a^3*b^2*c^3 - 128*a^4*c^4)*d^2*e^2 - 2*(a*b^7 - 12*a^2*
b^5*c + 48*a^3*b^3*c^2 - 64*a^4*b*c^3)*d*e^3 + (a^2*b^6 - 12*a^3*b^4*c + 48*a^4*
b^2*c^2 - 64*a^5*c^3)*e^4))*((b^6*c - 12*a*b^4*c^2 + 48*a^2*b^2*c^3 - 64*a^3*c^4
)*d^2 - (b^7 - 12*a*b^5*c + 48*a^2*b^3*c^2 - 64*a^3*b*c^3)*d*e + (a*b^6 - 12*a^2
*b^4*c + 48*a^3*b^2*c^2 - 64*a^4*c^3)*e^2))/((b^6*c - 12*a*b^4*c^2 + 48*a^2*b^2*
c^3 - 64*a^3*c^4)*d^2 - (b^7 - 12*a*b^5*c + 48*a^2*b^3*c^2 - 64*a^3*b*c^3)*d*e +
(a*b^6 - 12*a^2*b^4*c + 48*a^3*b^2*c^2 - 64*a^4*c^3)*e^2))*log(27/2*sqrt(1/2)*(
2*(b^4*c - 8*a*b^2*c^2 + 16*a^2*c^3)*d*e^6 - (b^5 - 8*a*b^3*c + 16*a^2*b*c^2)*e^
7 - sqrt(e^10/((b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)*d^4 - 2*(b
^7*c - 12*a*b^5*c^2 + 48*a^2*b^3*c^3 - 64*a^3*b*c^4)*d^3*e + (b^8 - 10*a*b^6*c +
24*a^2*b^4*c^2 + 32*a^3*b^2*c^3 - 128*a^4*c^4)*d^2*e^2 - 2*(a*b^7 - 12*a^2*b^5*
c + 48*a^3*b^3*c^2 - 64*a^4*b*c^3)*d*e^3 + (a^2*b^6 - 12*a^3*b^4*c + 48*a^4*b^2*
c^2 - 64*a^5*c^3)*e^4))*(8*(b^6*c^3 - 12*a*b^4*c^4 + 48*a^2*b^2*c^5 - 64*a^3*c^6
)*d^4 - 16*(b^7*c^2 - 12*a*b^5*c^3 + 48*a^2*b^3*c^4 - 64*a^3*b*c^5)*d^3*e + 3*(3
*b^8*c - 32*a*b^6*c^2 + 96*a^2*b^4*c^3 - 256*a^4*c^5)*d^2*e^2 - (b^9 - 96*a^2*b^
5*c^2 + 512*a^3*b^3*c^3 - 768*a^4*b*c^4)*d*e^3 + (a*b^8 - 8*a^2*b^6*c + 128*a^4*
b^2*c^3 - 256*a^5*c^4)*e^4))*sqrt((32*c^3*d^3*e^2 - 48*b*c^2*d^2*e^3 + 6*(3*b^2*
c + 4*a*c^2)*d*e^4 - (b^3 + 12*a*b*c)*e^5 + sqrt(e^10/((b^6*c^2 - 12*a*b^4*c^3 +
48*a^2*b^2*c^4 - 64*a^3*c^5)*d^4 - 2*(b^7*c - 12*a*b^5*c^2 + 48*a^2*b^3*c^3 - 6
4*a^3*b*c^4)*d^3*e + (b^8 - 10*a*b^6*c + 24*a^2*b^4*c^2 + 32*a^3*b^2*c^3 - 128*a
^4*c^4)*d^2*e^2 - 2*(a*b^7 - 12*a^2*b^5*c + 48*a^3*b^3*c^2 - 64*a^4*b*c^3)*d*e^3
+ (a^2*b^6 - 12*a^3*b^4*c + 48*a^4*b^2*c^2 - 64*a^5*c^3)*e^4))*((b^6*c - 12*a*b
^4*c^2 + 48*a^2*b^2*c^3 - 64*a^3*c^4)*d^2 - (b^7 - 12*a*b^5*c + 48*a^2*b^3*c^2 -
64*a^3*b*c^3)*d*e + (a*b^6 - 12*a^2*b^4*c + 48*a^3*b^2*c^2 - 64*a^4*c^3)*e^2))/
((b^6*c - 12*a*b^4*c^2 + 48*a^2*b^2*c^3 - 64*a^3*c^4)*d^2 - (b^7 - 12*a*b^5*c +
48*a^2*b^3*c^2 - 64*a^3*b*c^3)*d*e + (a*b^6 - 12*a^2*b^4*c + 48*a^3*b^2*c^2 - 64
*a^4*c^3)*e^2)) + 27*(16*c^3*d^2*e^6 - 16*b*c^2*d*e^7 + (3*b^2*c + 4*a*c^2)*e^8)
*sqrt(e*x + d)) - 3*sqrt(1/2)*((b^2*c^2 - 4*a*c^3)*x^4 + a^2*b^2 - 4*a^3*c + 2*(
b^3*c - 4*a*b*c^2)*x^3 + (b^4 - 2*a*b^2*c - 8*a^2*c^2)*x^2 + 2*(a*b^3 - 4*a^2*b*
c)*x)*sqrt((32*c^3*d^3*e^2 - 48*b*c^2*d^2*e^3 + 6*(3*b^2*c + 4*a*c^2)*d*e^4 - (b
^3 + 12*a*b*c)*e^5 + sqrt(e^10/((b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^
3*c^5)*d^4 - 2*(b^7*c - 12*a*b^5*c^2 + 48*a^2*b^3*c^3 - 64*a^3*b*c^4)*d^3*e + (b
^8 - 10*a*b^6*c + 24*a^2*b^4*c^2 + 32*a^3*b^2*c^3 - 128*a^4*c^4)*d^2*e^2 - 2*(a*
b^7 - 12*a^2*b^5*c + 48*a^3*b^3*c^2 - 64*a^4*b*c^3)*d*e^3 + (a^2*b^6 - 12*a^3*b^
4*c + 48*a^4*b^2*c^2 - 64*a^5*c^3)*e^4))*((b^6*c - 12*a*b^4*c^2 + 48*a^2*b^2*c^3
- 64*a^3*c^4)*d^2 - (b^7 - 12*a*b^5*c + 48*a^2*b^3*c^2 - 64*a^3*b*c^3)*d*e + (a
*b^6 - 12*a^2*b^4*c + 48*a^3*b^2*c^2 - 64*a^4*c^3)*e^2))/((b^6*c - 12*a*b^4*c^2
+ 48*a^2*b^2*c^3 - 64*a^3*c^4)*d^2 - (b^7 - 12*a*b^5*c + 48*a^2*b^3*c^2 - 64*a^3
*b*c^3)*d*e + (a*b^6 - 12*a^2*b^4*c + 48*a^3*b^2*c^2 - 64*a^4*c^3)*e^2))*log(-27
/2*sqrt(1/2)*(2*(b^4*c - 8*a*b^2*c^2 + 16*a^2*c^3)*d*e^6 - (b^5 - 8*a*b^3*c + 16
*a^2*b*c^2)*e^7 - sqrt(e^10/((b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c
^5)*d^4 - 2*(b^7*c - 12*a*b^5*c^2 + 48*a^2*b^3*c^3 - 64*a^3*b*c^4)*d^3*e + (b^8
- 10*a*b^6*c + 24*a^2*b^4*c^2 + 32*a^3*b^2*c^3 - 128*a^4*c^4)*d^2*e^2 - 2*(a*b^7
- 12*a^2*b^5*c + 48*a^3*b^3*c^2 - 64*a^4*b*c^3)*d*e^3 + (a^2*b^6 - 12*a^3*b^4*c
+ 48*a^4*b^2*c^2 - 64*a^5*c^3)*e^4))*(8*(b^6*c^3 - 12*a*b^4*c^4 + 48*a^2*b^2*c^
5 - 64*a^3*c^6)*d^4 - 16*(b^7*c^2 - 12*a*b^5*c^3 + 48*a^2*b^3*c^4 - 64*a^3*b*c^5
)*d^3*e + 3*(3*b^8*c - 32*a*b^6*c^2 + 96*a^2*b^4*c^3 - 256*a^4*c^5)*d^2*e^2 - (b
^9 - 96*a^2*b^5*c^2 + 512*a^3*b^3*c^3 - 768*a^4*b*c^4)*d*e^3 + (a*b^8 - 8*a^2*b^
6*c + 128*a^4*b^2*c^3 - 256*a^5*c^4)*e^4))*sqrt((32*c^3*d^3*e^2 - 48*b*c^2*d^2*e
^3 + 6*(3*b^2*c + 4*a*c^2)*d*e^4 - (b^3 + 12*a*b*c)*e^5 + sqrt(e^10/((b^6*c^2 -
12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)*d^4 - 2*(b^7*c - 12*a*b^5*c^2 + 48*a
^2*b^3*c^3 - 64*a^3*b*c^4)*d^3*e + (b^8 - 10*a*b^6*c + 24*a^2*b^4*c^2 + 32*a^3*b
^2*c^3 - 128*a^4*c^4)*d^2*e^2 - 2*(a*b^7 - 12*a^2*b^5*c + 48*a^3*b^3*c^2 - 64*a^
4*b*c^3)*d*e^3 + (a^2*b^6 - 12*a^3*b^4*c + 48*a^4*b^2*c^2 - 64*a^5*c^3)*e^4))*((
b^6*c - 12*a*b^4*c^2 + 48*a^2*b^2*c^3 - 64*a^3*c^4)*d^2 - (b^7 - 12*a*b^5*c + 48
*a^2*b^3*c^2 - 64*a^3*b*c^3)*d*e + (a*b^6 - 12*a^2*b^4*c + 48*a^3*b^2*c^2 - 64*a
^4*c^3)*e^2))/((b^6*c - 12*a*b^4*c^2 + 48*a^2*b^2*c^3 - 64*a^3*c^4)*d^2 - (b^7 -
12*a*b^5*c + 48*a^2*b^3*c^2 - 64*a^3*b*c^3)*d*e + (a*b^6 - 12*a^2*b^4*c + 48*a^
3*b^2*c^2 - 64*a^4*c^3)*e^2)) + 27*(16*c^3*d^2*e^6 - 16*b*c^2*d*e^7 + (3*b^2*c +
4*a*c^2)*e^8)*sqrt(e*x + d)) + 3*sqrt(1/2)*((b^2*c^2 - 4*a*c^3)*x^4 + a^2*b^2 -
4*a^3*c + 2*(b^3*c - 4*a*b*c^2)*x^3 + (b^4 - 2*a*b^2*c - 8*a^2*c^2)*x^2 + 2*(a*
b^3 - 4*a^2*b*c)*x)*sqrt((32*c^3*d^3*e^2 - 48*b*c^2*d^2*e^3 + 6*(3*b^2*c + 4*a*c
^2)*d*e^4 - (b^3 + 12*a*b*c)*e^5 - sqrt(e^10/((b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b
^2*c^4 - 64*a^3*c^5)*d^4 - 2*(b^7*c - 12*a*b^5*c^2 + 48*a^2*b^3*c^3 - 64*a^3*b*c
^4)*d^3*e + (b^8 - 10*a*b^6*c + 24*a^2*b^4*c^2 + 32*a^3*b^2*c^3 - 128*a^4*c^4)*d
^2*e^2 - 2*(a*b^7 - 12*a^2*b^5*c + 48*a^3*b^3*c^2 - 64*a^4*b*c^3)*d*e^3 + (a^2*b
^6 - 12*a^3*b^4*c + 48*a^4*b^2*c^2 - 64*a^5*c^3)*e^4))*((b^6*c - 12*a*b^4*c^2 +
48*a^2*b^2*c^3 - 64*a^3*c^4)*d^2 - (b^7 - 12*a*b^5*c + 48*a^2*b^3*c^2 - 64*a^3*b
*c^3)*d*e + (a*b^6 - 12*a^2*b^4*c + 48*a^3*b^2*c^2 - 64*a^4*c^3)*e^2))/((b^6*c -
12*a*b^4*c^2 + 48*a^2*b^2*c^3 - 64*a^3*c^4)*d^2 - (b^7 - 12*a*b^5*c + 48*a^2*b^
3*c^2 - 64*a^3*b*c^3)*d*e + (a*b^6 - 12*a^2*b^4*c + 48*a^3*b^2*c^2 - 64*a^4*c^3)
*e^2))*log(27/2*sqrt(1/2)*(2*(b^4*c - 8*a*b^2*c^2 + 16*a^2*c^3)*d*e^6 - (b^5 - 8
*a*b^3*c + 16*a^2*b*c^2)*e^7 + sqrt(e^10/((b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c
^4 - 64*a^3*c^5)*d^4 - 2*(b^7*c - 12*a*b^5*c^2 + 48*a^2*b^3*c^3 - 64*a^3*b*c^4)*
d^3*e + (b^8 - 10*a*b^6*c + 24*a^2*b^4*c^2 + 32*a^3*b^2*c^3 - 128*a^4*c^4)*d^2*e
^2 - 2*(a*b^7 - 12*a^2*b^5*c + 48*a^3*b^3*c^2 - 64*a^4*b*c^3)*d*e^3 + (a^2*b^6 -
12*a^3*b^4*c + 48*a^4*b^2*c^2 - 64*a^5*c^3)*e^4))*(8*(b^6*c^3 - 12*a*b^4*c^4 +
48*a^2*b^2*c^5 - 64*a^3*c^6)*d^4 - 16*(b^7*c^2 - 12*a*b^5*c^3 + 48*a^2*b^3*c^4 -
64*a^3*b*c^5)*d^3*e + 3*(3*b^8*c - 32*a*b^6*c^2 + 96*a^2*b^4*c^3 - 256*a^4*c^5)
*d^2*e^2 - (b^9 - 96*a^2*b^5*c^2 + 512*a^3*b^3*c^3 - 768*a^4*b*c^4)*d*e^3 + (a*b
^8 - 8*a^2*b^6*c + 128*a^4*b^2*c^3 - 256*a^5*c^4)*e^4))*sqrt((32*c^3*d^3*e^2 - 4
8*b*c^2*d^2*e^3 + 6*(3*b^2*c + 4*a*c^2)*d*e^4 - (b^3 + 12*a*b*c)*e^5 - sqrt(e^10
/((b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)*d^4 - 2*(b^7*c - 12*a*b
^5*c^2 + 48*a^2*b^3*c^3 - 64*a^3*b*c^4)*d^3*e + (b^8 - 10*a*b^6*c + 24*a^2*b^4*c
^2 + 32*a^3*b^2*c^3 - 128*a^4*c^4)*d^2*e^2 - 2*(a*b^7 - 12*a^2*b^5*c + 48*a^3*b^
3*c^2 - 64*a^4*b*c^3)*d*e^3 + (a^2*b^6 - 12*a^3*b^4*c + 48*a^4*b^2*c^2 - 64*a^5*
c^3)*e^4))*((b^6*c - 12*a*b^4*c^2 + 48*a^2*b^2*c^3 - 64*a^3*c^4)*d^2 - (b^7 - 12
*a*b^5*c + 48*a^2*b^3*c^2 - 64*a^3*b*c^3)*d*e + (a*b^6 - 12*a^2*b^4*c + 48*a^3*b
^2*c^2 - 64*a^4*c^3)*e^2))/((b^6*c - 12*a*b^4*c^2 + 48*a^2*b^2*c^3 - 64*a^3*c^4)
*d^2 - (b^7 - 12*a*b^5*c + 48*a^2*b^3*c^2 - 64*a^3*b*c^3)*d*e + (a*b^6 - 12*a^2*
b^4*c + 48*a^3*b^2*c^2 - 64*a^4*c^3)*e^2)) + 27*(16*c^3*d^2*e^6 - 16*b*c^2*d*e^7
+ (3*b^2*c + 4*a*c^2)*e^8)*sqrt(e*x + d)) - 3*sqrt(1/2)*((b^2*c^2 - 4*a*c^3)*x^
4 + a^2*b^2 - 4*a^3*c + 2*(b^3*c - 4*a*b*c^2)*x^3 + (b^4 - 2*a*b^2*c - 8*a^2*c^2
)*x^2 + 2*(a*b^3 - 4*a^2*b*c)*x)*sqrt((32*c^3*d^3*e^2 - 48*b*c^2*d^2*e^3 + 6*(3*
b^2*c + 4*a*c^2)*d*e^4 - (b^3 + 12*a*b*c)*e^5 - sqrt(e^10/((b^6*c^2 - 12*a*b^4*c
^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)*d^4 - 2*(b^7*c - 12*a*b^5*c^2 + 48*a^2*b^3*c^3
- 64*a^3*b*c^4)*d^3*e + (b^8 - 10*a*b^6*c + 24*a^2*b^4*c^2 + 32*a^3*b^2*c^3 - 1
28*a^4*c^4)*d^2*e^2 - 2*(a*b^7 - 12*a^2*b^5*c + 48*a^3*b^3*c^2 - 64*a^4*b*c^3)*d
*e^3 + (a^2*b^6 - 12*a^3*b^4*c + 48*a^4*b^2*c^2 - 64*a^5*c^3)*e^4))*((b^6*c - 12
*a*b^4*c^2 + 48*a^2*b^2*c^3 - 64*a^3*c^4)*d^2 - (b^7 - 12*a*b^5*c + 48*a^2*b^3*c
^2 - 64*a^3*b*c^3)*d*e + (a*b^6 - 12*a^2*b^4*c + 48*a^3*b^2*c^2 - 64*a^4*c^3)*e^
2))/((b^6*c - 12*a*b^4*c^2 + 48*a^2*b^2*c^3 - 64*a^3*c^4)*d^2 - (b^7 - 12*a*b^5*
c + 48*a^2*b^3*c^2 - 64*a^3*b*c^3)*d*e + (a*b^6 - 12*a^2*b^4*c + 48*a^3*b^2*c^2
- 64*a^4*c^3)*e^2))*log(-27/2*sqrt(1/2)*(2*(b^4*c - 8*a*b^2*c^2 + 16*a^2*c^3)*d*
e^6 - (b^5 - 8*a*b^3*c + 16*a^2*b*c^2)*e^7 + sqrt(e^10/((b^6*c^2 - 12*a*b^4*c^3
+ 48*a^2*b^2*c^4 - 64*a^3*c^5)*d^4 - 2*(b^7*c - 12*a*b^5*c^2 + 48*a^2*b^3*c^3 -
64*a^3*b*c^4)*d^3*e + (b^8 - 10*a*b^6*c + 24*a^2*b^4*c^2 + 32*a^3*b^2*c^3 - 128*
a^4*c^4)*d^2*e^2 - 2*(a*b^7 - 12*a^2*b^5*c + 48*a^3*b^3*c^2 - 64*a^4*b*c^3)*d*e^
3 + (a^2*b^6 - 12*a^3*b^4*c + 48*a^4*b^2*c^2 - 64*a^5*c^3)*e^4))*(8*(b^6*c^3 - 1
2*a*b^4*c^4 + 48*a^2*b^2*c^5 - 64*a^3*c^6)*d^4 - 16*(b^7*c^2 - 12*a*b^5*c^3 + 48
*a^2*b^3*c^4 - 64*a^3*b*c^5)*d^3*e + 3*(3*b^8*c - 32*a*b^6*c^2 + 96*a^2*b^4*c^3
- 256*a^4*c^5)*d^2*e^2 - (b^9 - 96*a^2*b^5*c^2 + 512*a^3*b^3*c^3 - 768*a^4*b*c^4
)*d*e^3 + (a*b^8 - 8*a^2*b^6*c + 128*a^4*b^2*c^3 - 256*a^5*c^4)*e^4))*sqrt((32*c
^3*d^3*e^2 - 48*b*c^2*d^2*e^3 + 6*(3*b^2*c + 4*a*c^2)*d*e^4 - (b^3 + 12*a*b*c)*e
^5 - sqrt(e^10/((b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)*d^4 - 2*(
b^7*c - 12*a*b^5*c^2 + 48*a^2*b^3*c^3 - 64*a^3*b*c^4)*d^3*e + (b^8 - 10*a*b^6*c
+ 24*a^2*b^4*c^2 + 32*a^3*b^2*c^3 - 128*a^4*c^4)*d^2*e^2 - 2*(a*b^7 - 12*a^2*b^5
*c + 48*a^3*b^3*c^2 - 64*a^4*b*c^3)*d*e^3 + (a^2*b^6 - 12*a^3*b^4*c + 48*a^4*b^2
*c^2 - 64*a^5*c^3)*e^4))*((b^6*c - 12*a*b^4*c^2 + 48*a^2*b^2*c^3 - 64*a^3*c^4)*d
^2 - (b^7 - 12*a*b^5*c + 48*a^2*b^3*c^2 - 64*a^3*b*c^3)*d*e + (a*b^6 - 12*a^2*b^
4*c + 48*a^3*b^2*c^2 - 64*a^4*c^3)*e^2))/((b^6*c - 12*a*b^4*c^2 + 48*a^2*b^2*c^3
- 64*a^3*c^4)*d^2 - (b^7 - 12*a*b^5*c + 48*a^2*b^3*c^2 - 64*a^3*b*c^3)*d*e + (a
*b^6 - 12*a^2*b^4*c + 48*a^3*b^2*c^2 - 64*a^4*c^3)*e^2)) + 27*(16*c^3*d^2*e^6 -
16*b*c^2*d*e^7 + (3*b^2*c + 4*a*c^2)*e^8)*sqrt(e*x + d)) - 2*(6*c^2*e*x^3 + 9*b*
c*e*x^2 + 3*a*b*e + (5*b^2 - 2*a*c)*e*x + 2*(b^2 - 4*a*c)*d)*sqrt(e*x + d))/((b^
2*c^2 - 4*a*c^3)*x^4 + a^2*b^2 - 4*a^3*c + 2*(b^3*c - 4*a*b*c^2)*x^3 + (b^4 - 2*
a*b^2*c - 8*a^2*c^2)*x^2 + 2*(a*b^3 - 4*a^2*b*c)*x)

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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)**3,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)^3,x, algorithm="giac")

[Out]

Timed out