3.2478 \(\int \frac{A+B x}{(d+e x)^3 \left (a+b x+c x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=545 \[ \frac{3 e \left (A e \left (-4 c e (a e+4 b d)+5 b^2 e^2+16 c^2 d^2\right )-B \left (-4 c d e (3 a e+b d)+b e^2 (4 a e+b d)+8 c^2 d^3\right )\right ) \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{8 \left (a e^2-b d e+c d^2\right )^{7/2}}+\frac{e \sqrt{a+b x+c x^2} \left (4 b \left (a B e^2+2 A c d e+B c d^2\right )-4 c \left (-3 a A e^2+5 a B d e+2 A c d^2\right )+b^2 e (B d-5 A e)\right )}{2 \left (b^2-4 a c\right ) (d+e x)^2 \left (a e^2-b d e+c d^2\right )^2}+\frac{2 \left (-A \left (2 a c e+b^2 (-e)+b c d\right )+c x (-2 a B e+A b e-2 A c d+b B d)+a B (2 c d-b e)\right )}{\left (b^2-4 a c\right ) (d+e x)^2 \sqrt{a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}-\frac{e \sqrt{a+b x+c x^2} \left (-2 b^2 e \left (-6 a B e^2-19 A c d e+5 B c d^2\right )-4 b c \left (-13 a A e^3+9 a B d e^2+6 A c d^2 e+2 B c d^3\right )+8 c \left (A c d \left (2 c d^2-13 a e^2\right )+a B e \left (11 c d^2-4 a e^2\right )\right )+3 b^3 e^2 (B d-5 A e)\right )}{4 \left (b^2-4 a c\right ) (d+e x) \left (a e^2-b d e+c d^2\right )^3} \]

[Out]

(2*(a*B*(2*c*d - b*e) - A*(b*c*d - b^2*e + 2*a*c*e) + c*(b*B*d - 2*A*c*d + A*b*e
- 2*a*B*e)*x))/((b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^2*Sqrt[a + b*x
+ c*x^2]) + (e*(b^2*e*(B*d - 5*A*e) - 4*c*(2*A*c*d^2 + 5*a*B*d*e - 3*a*A*e^2) +
4*b*(B*c*d^2 + 2*A*c*d*e + a*B*e^2))*Sqrt[a + b*x + c*x^2])/(2*(b^2 - 4*a*c)*(c*
d^2 - b*d*e + a*e^2)^2*(d + e*x)^2) - (e*(3*b^3*e^2*(B*d - 5*A*e) - 2*b^2*e*(5*B
*c*d^2 - 19*A*c*d*e - 6*a*B*e^2) - 4*b*c*(2*B*c*d^3 + 6*A*c*d^2*e + 9*a*B*d*e^2
- 13*a*A*e^3) + 8*c*(A*c*d*(2*c*d^2 - 13*a*e^2) + a*B*e*(11*c*d^2 - 4*a*e^2)))*S
qrt[a + b*x + c*x^2])/(4*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)^3*(d + e*x)) + (3
*e*(A*e*(16*c^2*d^2 + 5*b^2*e^2 - 4*c*e*(4*b*d + a*e)) - B*(8*c^2*d^3 - 4*c*d*e*
(b*d + 3*a*e) + b*e^2*(b*d + 4*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])])/(8*(c*d^2 - b*d*e + a*e^2)
^(7/2))

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Rubi [A]  time = 2.69613, antiderivative size = 545, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185 \[ \frac{3 e \left (A e \left (-4 c e (a e+4 b d)+5 b^2 e^2+16 c^2 d^2\right )-B \left (-4 c d e (3 a e+b d)+b e^2 (4 a e+b d)+8 c^2 d^3\right )\right ) \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{8 \left (a e^2-b d e+c d^2\right )^{7/2}}+\frac{e \sqrt{a+b x+c x^2} \left (4 b \left (a B e^2+2 A c d e+B c d^2\right )-4 c \left (-3 a A e^2+5 a B d e+2 A c d^2\right )+b^2 e (B d-5 A e)\right )}{2 \left (b^2-4 a c\right ) (d+e x)^2 \left (a e^2-b d e+c d^2\right )^2}+\frac{2 \left (-A \left (2 a c e+b^2 (-e)+b c d\right )+c x (-2 a B e+A b e-2 A c d+b B d)+a B (2 c d-b e)\right )}{\left (b^2-4 a c\right ) (d+e x)^2 \sqrt{a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}-\frac{e \sqrt{a+b x+c x^2} \left (-2 b^2 e \left (-6 a B e^2-19 A c d e+5 B c d^2\right )-4 b c \left (-13 a A e^3+9 a B d e^2+6 A c d^2 e+2 B c d^3\right )+8 c \left (A c d \left (2 c d^2-13 a e^2\right )+a B e \left (11 c d^2-4 a e^2\right )\right )+3 b^3 e^2 (B d-5 A e)\right )}{4 \left (b^2-4 a c\right ) (d+e x) \left (a e^2-b d e+c d^2\right )^3} \]

Antiderivative was successfully verified.

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

[Out]

(2*(a*B*(2*c*d - b*e) - A*(b*c*d - b^2*e + 2*a*c*e) + c*(b*B*d - 2*A*c*d + A*b*e
- 2*a*B*e)*x))/((b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^2*Sqrt[a + b*x
+ c*x^2]) + (e*(b^2*e*(B*d - 5*A*e) - 4*c*(2*A*c*d^2 + 5*a*B*d*e - 3*a*A*e^2) +
4*b*(B*c*d^2 + 2*A*c*d*e + a*B*e^2))*Sqrt[a + b*x + c*x^2])/(2*(b^2 - 4*a*c)*(c*
d^2 - b*d*e + a*e^2)^2*(d + e*x)^2) - (e*(3*b^3*e^2*(B*d - 5*A*e) - 2*b^2*e*(5*B
*c*d^2 - 19*A*c*d*e - 6*a*B*e^2) - 4*b*c*(2*B*c*d^3 + 6*A*c*d^2*e + 9*a*B*d*e^2
- 13*a*A*e^3) + 8*c*(A*c*d*(2*c*d^2 - 13*a*e^2) + a*B*e*(11*c*d^2 - 4*a*e^2)))*S
qrt[a + b*x + c*x^2])/(4*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)^3*(d + e*x)) + (3
*e*(A*e*(16*c^2*d^2 + 5*b^2*e^2 - 4*c*e*(4*b*d + a*e)) - B*(8*c^2*d^3 - 4*c*d*e*
(b*d + 3*a*e) + b*e^2*(b*d + 4*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])])/(8*(c*d^2 - b*d*e + a*e^2)
^(7/2))

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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((B*x+A)/(e*x+d)**3/(c*x**2+b*x+a)**(3/2),x)

[Out]

Timed out

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Mathematica [A]  time = 7.33089, size = 792, normalized size = 1.45 \[ \frac{\left (a+b x+c x^2\right )^2 \left (-\frac{2 \left (2 a^2 A c^2 e^3+3 a^2 b B c e^3-6 a^2 B c^2 d e^2+2 a^2 B c^2 e^3 x-4 a A b^2 c e^3+9 a A b c^2 d e^2-3 a A b c^2 e^3 x-6 a A c^3 d^2 e+6 a A c^3 d e^2 x-a b^3 B e^3+3 a b^2 B c d e^2-a b^2 B c e^3 x-3 a b B c^2 d^2 e+3 a b B c^2 d e^2 x+2 a B c^3 d^3-6 a B c^3 d^2 e x+A b^4 e^3-3 A b^3 c d e^2+A b^3 c e^3 x+3 A b^2 c^2 d^2 e-3 A b^2 c^2 d e^2 x-A b c^3 d^3+3 A b c^3 d^2 e x-2 A c^4 d^3 x+b B c^3 d^3 x\right )}{\left (4 a c-b^2\right ) \left (a+b x+c x^2\right ) \left (a e^2-b d e+c d^2\right )^3}-\frac{e^2 \left (4 a B e^2-7 A b e^2+14 A c d e+3 b B d e-10 B c d^2\right )}{4 (d+e x) \left (a e^2-b d e+c d^2\right )^3}-\frac{e^2 (A e-B d)}{2 (d+e x)^2 \left (a e^2-b d e+c d^2\right )^2}\right )}{(a+x (b+c x))^{3/2}}-\frac{3 e \left (a+b x+c x^2\right )^{3/2} \log (d+e x) \left (4 a A c e^3+4 a b B e^3-12 a B c d e^2-5 A b^2 e^3+16 A b c d e^2-16 A c^2 d^2 e+b^2 B d e^2-4 b B c d^2 e+8 B c^2 d^3\right )}{8 (a+x (b+c x))^{3/2} \left (a e^2-b d e+c d^2\right )^{7/2}}+\frac{3 e \left (a+b x+c x^2\right )^{3/2} \left (4 a A c e^3+4 a b B e^3-12 a B c d e^2-5 A b^2 e^3+16 A b c d e^2-16 A c^2 d^2 e+b^2 B d e^2-4 b B c d^2 e+8 B c^2 d^3\right ) \log \left (2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}+2 a e-b d+b e x-2 c d x\right )}{8 (a+x (b+c x))^{3/2} \left (a e^2-b d e+c d^2\right )^{7/2}} \]

Antiderivative was successfully verified.

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

[Out]

((a + b*x + c*x^2)^2*(-(e^2*(-(B*d) + A*e))/(2*(c*d^2 - b*d*e + a*e^2)^2*(d + e*
x)^2) - (e^2*(-10*B*c*d^2 + 3*b*B*d*e + 14*A*c*d*e - 7*A*b*e^2 + 4*a*B*e^2))/(4*
(c*d^2 - b*d*e + a*e^2)^3*(d + e*x)) - (2*(-(A*b*c^3*d^3) + 2*a*B*c^3*d^3 + 3*A*
b^2*c^2*d^2*e - 3*a*b*B*c^2*d^2*e - 6*a*A*c^3*d^2*e - 3*A*b^3*c*d*e^2 + 3*a*b^2*
B*c*d*e^2 + 9*a*A*b*c^2*d*e^2 - 6*a^2*B*c^2*d*e^2 + A*b^4*e^3 - a*b^3*B*e^3 - 4*
a*A*b^2*c*e^3 + 3*a^2*b*B*c*e^3 + 2*a^2*A*c^2*e^3 + b*B*c^3*d^3*x - 2*A*c^4*d^3*
x + 3*A*b*c^3*d^2*e*x - 6*a*B*c^3*d^2*e*x - 3*A*b^2*c^2*d*e^2*x + 3*a*b*B*c^2*d*
e^2*x + 6*a*A*c^3*d*e^2*x + A*b^3*c*e^3*x - a*b^2*B*c*e^3*x - 3*a*A*b*c^2*e^3*x
+ 2*a^2*B*c^2*e^3*x))/((-b^2 + 4*a*c)*(c*d^2 - b*d*e + a*e^2)^3*(a + b*x + c*x^2
))))/(a + x*(b + c*x))^(3/2) - (3*e*(8*B*c^2*d^3 - 4*b*B*c*d^2*e - 16*A*c^2*d^2*
e + b^2*B*d*e^2 + 16*A*b*c*d*e^2 - 12*a*B*c*d*e^2 - 5*A*b^2*e^3 + 4*a*b*B*e^3 +
4*a*A*c*e^3)*(a + b*x + c*x^2)^(3/2)*Log[d + e*x])/(8*(c*d^2 - b*d*e + a*e^2)^(7
/2)*(a + x*(b + c*x))^(3/2)) + (3*e*(8*B*c^2*d^3 - 4*b*B*c*d^2*e - 16*A*c^2*d^2*
e + b^2*B*d*e^2 + 16*A*b*c*d*e^2 - 12*a*B*c*d*e^2 - 5*A*b^2*e^3 + 4*a*b*B*e^3 +
4*a*A*c*e^3)*(a + b*x + c*x^2)^(3/2)*Log[-(b*d) + 2*a*e - 2*c*d*x + b*e*x + 2*Sq
rt[c*d^2 - b*d*e + a*e^2]*Sqrt[a + b*x + c*x^2]])/(8*(c*d^2 - b*d*e + a*e^2)^(7/
2)*(a + x*(b + c*x))^(3/2))

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Maple [B]  time = 0.033, size = 5528, normalized size = 10.1 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 27.0668, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/16*(4*(8*(2*B*a - A*b)*c^3*d^5 - 24*(2*A*a*c^3 + (B*a*b - A*b^2)*c^2)*d^4*e -
12*(2*(4*B*a^2 - 3*A*a*b)*c^2 - (3*B*a*b^2 - 2*A*b^3)*c)*d^3*e^2 - (13*B*a*b^3
- 8*A*b^4 - 80*A*a^2*c^2 - 4*(11*B*a^2*b - 12*A*a*b^2)*c)*d^2*e^3 - (2*B*a^2*b^2
- 9*A*a*b^3 - 4*(2*B*a^3 - 9*A*a^2*b)*c)*d*e^4 - 2*(A*a^2*b^2 - 4*A*a^3*c)*e^5
+ (8*(B*b*c^3 - 2*A*c^4)*d^3*e^2 + 2*(5*B*b^2*c^2 - 4*(11*B*a - 3*A*b)*c^3)*d^2*
e^3 - (3*B*b^3*c - 104*A*a*c^3 - 2*(18*B*a*b - 19*A*b^2)*c^2)*d*e^4 + (4*(8*B*a^
2 - 13*A*a*b)*c^2 - 3*(4*B*a*b^2 - 5*A*b^3)*c)*e^5)*x^3 + (16*(B*b*c^3 - 2*A*c^4
)*d^4*e + 4*(3*B*b^2*c^2 - 2*(16*B*a - 5*A*b)*c^3)*d^3*e^2 + (5*B*b^3*c + 112*A*
a*c^3 + 4*(B*a*b - 10*A*b^2)*c^2)*d^2*e^3 - (3*B*b^4 + 4*(2*B*a^2 - 11*A*a*b)*c^
2 - (18*B*a*b^2 - 13*A*b^3)*c)*d*e^4 - (12*B*a*b^3 - 15*A*b^4 - 24*A*a^2*c^2 - 2
*(20*B*a^2*b - 31*A*a*b^2)*c)*e^5)*x^2 - (8*(2*B*a - A*b)*c^3*d^4*e - 8*(B*b*c^3
- 2*A*c^4)*d^5 - 12*(B*b^3*c - 4*A*a*c^3 - 2*(3*B*a*b - A*b^2)*c^2)*d^3*e^2 + (
5*B*b^4 + 8*(15*B*a^2 - 23*A*a*b)*c^2 - 14*(5*B*a*b^2 - 4*A*b^3)*c)*d^2*e^3 + (2
1*B*a*b^3 - 25*A*b^4 - 88*A*a^2*c^2 - 2*(34*B*a^2*b - 57*A*a*b^2)*c)*d*e^4 + (4*
B*a^2*b^2 - 5*A*a*b^3 - 4*(4*B*a^3 - 5*A*a^2*b)*c)*e^5)*x)*sqrt(c*d^2 - b*d*e +
a*e^2)*sqrt(c*x^2 + b*x + a) + 3*(8*(B*a*b^2*c^2 - 4*B*a^2*c^3)*d^5*e - 4*(B*a*b
^3*c - 16*A*a^2*c^3 - 4*(B*a^2*b - A*a*b^2)*c^2)*d^4*e^2 + (B*a*b^4 + 16*(3*B*a^
3 - 4*A*a^2*b)*c^2 - 16*(B*a^2*b^2 - A*a*b^3)*c)*d^3*e^3 + (4*B*a^2*b^3 - 5*A*a*
b^4 - 16*A*a^3*c^2 - 8*(2*B*a^3*b - 3*A*a^2*b^2)*c)*d^2*e^4 + (8*(B*b^2*c^3 - 4*
B*a*c^4)*d^3*e^3 - 4*(B*b^3*c^2 - 16*A*a*c^4 - 4*(B*a*b - A*b^2)*c^3)*d^2*e^4 +
(B*b^4*c + 16*(3*B*a^2 - 4*A*a*b)*c^3 - 16*(B*a*b^2 - A*b^3)*c^2)*d*e^5 - (16*A*
a^2*c^3 + 8*(2*B*a^2*b - 3*A*a*b^2)*c^2 - (4*B*a*b^3 - 5*A*b^4)*c)*e^6)*x^4 + (1
6*(B*b^2*c^3 - 4*B*a*c^4)*d^4*e^2 - 32*(A*b^2*c^3 - 4*A*a*c^4)*d^3*e^3 - 2*(B*b^
4*c - 16*(3*B*a^2 - 2*A*a*b)*c^3 + 8*(B*a*b^2 - A*b^3)*c^2)*d^2*e^4 + (B*b^5 - 3
2*A*a^2*c^3 + 16*(B*a^2*b - A*a*b^2)*c^2 - 2*(4*B*a*b^3 - 3*A*b^4)*c)*d*e^5 + (4
*B*a*b^4 - 5*A*b^5 - 16*A*a^2*b*c^2 - 8*(2*B*a^2*b^2 - 3*A*a*b^3)*c)*e^6)*x^3 +
(8*(B*b^2*c^3 - 4*B*a*c^4)*d^5*e + 4*(3*B*b^3*c^2 + 16*A*a*c^4 - 4*(3*B*a*b + A*
b^2)*c^3)*d^4*e^2 - (7*B*b^4*c - 16*(B*a^2 + 4*A*a*b)*c^3 - 8*(3*B*a*b^2 - 2*A*b
^3)*c^2)*d^3*e^3 + (2*B*b^5 + 48*A*a^2*c^3 + 24*(4*B*a^2*b - 5*A*a*b^2)*c^2 - (3
2*B*a*b^3 - 27*A*b^4)*c)*d^2*e^4 + (9*B*a*b^4 - 10*A*b^5 + 48*(B*a^3 - 2*A*a^2*b
)*c^2 - 16*(3*B*a^2*b^2 - 4*A*a*b^3)*c)*d*e^5 + (4*B*a^2*b^3 - 5*A*a*b^4 - 16*A*
a^3*c^2 - 8*(2*B*a^3*b - 3*A*a^2*b^2)*c)*e^6)*x^2 + (8*(B*b^3*c^2 - 4*B*a*b*c^3)
*d^5*e - 4*(B*b^4*c + 16*(B*a^2 - A*a*b)*c^3 - 4*(2*B*a*b^2 - A*b^3)*c^2)*d^4*e^
2 + (B*b^5 + 128*A*a^2*c^3 + 16*(5*B*a^2*b - 6*A*a*b^2)*c^2 - 8*(3*B*a*b^3 - 2*A
*b^4)*c)*d^3*e^3 + (6*B*a*b^4 - 5*A*b^5 + 48*(2*B*a^3 - 3*A*a^2*b)*c^2 - 8*(6*B*
a^2*b^2 - 7*A*a*b^3)*c)*d^2*e^4 + 2*(4*B*a^2*b^3 - 5*A*a*b^4 - 16*A*a^3*c^2 - 8*
(2*B*a^3*b - 3*A*a^2*b^2)*c)*d*e^5)*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)))/(((a*b^
2*c^3 - 4*a^2*c^4)*d^8 - 3*(a*b^3*c^2 - 4*a^2*b*c^3)*d^7*e + 3*(a*b^4*c - 3*a^2*
b^2*c^2 - 4*a^3*c^3)*d^6*e^2 - (a*b^5 + 2*a^2*b^3*c - 24*a^3*b*c^2)*d^5*e^3 + 3*
(a^2*b^4 - 3*a^3*b^2*c - 4*a^4*c^2)*d^4*e^4 - 3*(a^3*b^3 - 4*a^4*b*c)*d^3*e^5 +
(a^4*b^2 - 4*a^5*c)*d^2*e^6 + ((b^2*c^4 - 4*a*c^5)*d^6*e^2 - 3*(b^3*c^3 - 4*a*b*
c^4)*d^5*e^3 + 3*(b^4*c^2 - 3*a*b^2*c^3 - 4*a^2*c^4)*d^4*e^4 - (b^5*c + 2*a*b^3*
c^2 - 24*a^2*b*c^3)*d^3*e^5 + 3*(a*b^4*c - 3*a^2*b^2*c^2 - 4*a^3*c^3)*d^2*e^6 -
3*(a^2*b^3*c - 4*a^3*b*c^2)*d*e^7 + (a^3*b^2*c - 4*a^4*c^2)*e^8)*x^4 + (2*(b^2*c
^4 - 4*a*c^5)*d^7*e - 5*(b^3*c^3 - 4*a*b*c^4)*d^6*e^2 + 3*(b^4*c^2 - 2*a*b^2*c^3
- 8*a^2*c^4)*d^5*e^3 + (b^5*c - 13*a*b^3*c^2 + 36*a^2*b*c^3)*d^4*e^4 - (b^6 - 4
*a*b^4*c - 6*a^2*b^2*c^2 + 24*a^3*c^3)*d^3*e^5 + 3*(a*b^5 - 5*a^2*b^3*c + 4*a^3*
b*c^2)*d^2*e^6 - (3*a^2*b^4 - 14*a^3*b^2*c + 8*a^4*c^2)*d*e^7 + (a^3*b^3 - 4*a^4
*b*c)*e^8)*x^3 + ((b^2*c^4 - 4*a*c^5)*d^8 - (b^3*c^3 - 4*a*b*c^4)*d^7*e - (3*b^4
*c^2 - 16*a*b^2*c^3 + 16*a^2*c^4)*d^6*e^2 + (5*b^5*c - 23*a*b^3*c^2 + 12*a^2*b*c
^3)*d^5*e^3 - 2*(b^6 - a*b^4*c - 15*a^2*b^2*c^2 + 12*a^3*c^3)*d^4*e^4 + (5*a*b^5
- 23*a^2*b^3*c + 12*a^3*b*c^2)*d^3*e^5 - (3*a^2*b^4 - 16*a^3*b^2*c + 16*a^4*c^2
)*d^2*e^6 - (a^3*b^3 - 4*a^4*b*c)*d*e^7 + (a^4*b^2 - 4*a^5*c)*e^8)*x^2 + ((b^3*c
^3 - 4*a*b*c^4)*d^8 - (3*b^4*c^2 - 14*a*b^2*c^3 + 8*a^2*c^4)*d^7*e + 3*(b^5*c -
5*a*b^3*c^2 + 4*a^2*b*c^3)*d^6*e^2 - (b^6 - 4*a*b^4*c - 6*a^2*b^2*c^2 + 24*a^3*c
^3)*d^5*e^3 + (a*b^5 - 13*a^2*b^3*c + 36*a^3*b*c^2)*d^4*e^4 + 3*(a^2*b^4 - 2*a^3
*b^2*c - 8*a^4*c^2)*d^3*e^5 - 5*(a^3*b^3 - 4*a^4*b*c)*d^2*e^6 + 2*(a^4*b^2 - 4*a
^5*c)*d*e^7)*x)*sqrt(c*d^2 - b*d*e + a*e^2)), 1/8*(2*(8*(2*B*a - A*b)*c^3*d^5 -
24*(2*A*a*c^3 + (B*a*b - A*b^2)*c^2)*d^4*e - 12*(2*(4*B*a^2 - 3*A*a*b)*c^2 - (3*
B*a*b^2 - 2*A*b^3)*c)*d^3*e^2 - (13*B*a*b^3 - 8*A*b^4 - 80*A*a^2*c^2 - 4*(11*B*a
^2*b - 12*A*a*b^2)*c)*d^2*e^3 - (2*B*a^2*b^2 - 9*A*a*b^3 - 4*(2*B*a^3 - 9*A*a^2*
b)*c)*d*e^4 - 2*(A*a^2*b^2 - 4*A*a^3*c)*e^5 + (8*(B*b*c^3 - 2*A*c^4)*d^3*e^2 + 2
*(5*B*b^2*c^2 - 4*(11*B*a - 3*A*b)*c^3)*d^2*e^3 - (3*B*b^3*c - 104*A*a*c^3 - 2*(
18*B*a*b - 19*A*b^2)*c^2)*d*e^4 + (4*(8*B*a^2 - 13*A*a*b)*c^2 - 3*(4*B*a*b^2 - 5
*A*b^3)*c)*e^5)*x^3 + (16*(B*b*c^3 - 2*A*c^4)*d^4*e + 4*(3*B*b^2*c^2 - 2*(16*B*a
- 5*A*b)*c^3)*d^3*e^2 + (5*B*b^3*c + 112*A*a*c^3 + 4*(B*a*b - 10*A*b^2)*c^2)*d^
2*e^3 - (3*B*b^4 + 4*(2*B*a^2 - 11*A*a*b)*c^2 - (18*B*a*b^2 - 13*A*b^3)*c)*d*e^4
- (12*B*a*b^3 - 15*A*b^4 - 24*A*a^2*c^2 - 2*(20*B*a^2*b - 31*A*a*b^2)*c)*e^5)*x
^2 - (8*(2*B*a - A*b)*c^3*d^4*e - 8*(B*b*c^3 - 2*A*c^4)*d^5 - 12*(B*b^3*c - 4*A*
a*c^3 - 2*(3*B*a*b - A*b^2)*c^2)*d^3*e^2 + (5*B*b^4 + 8*(15*B*a^2 - 23*A*a*b)*c^
2 - 14*(5*B*a*b^2 - 4*A*b^3)*c)*d^2*e^3 + (21*B*a*b^3 - 25*A*b^4 - 88*A*a^2*c^2
- 2*(34*B*a^2*b - 57*A*a*b^2)*c)*d*e^4 + (4*B*a^2*b^2 - 5*A*a*b^3 - 4*(4*B*a^3 -
5*A*a^2*b)*c)*e^5)*x)*sqrt(-c*d^2 + b*d*e - a*e^2)*sqrt(c*x^2 + b*x + a) + 3*(8
*(B*a*b^2*c^2 - 4*B*a^2*c^3)*d^5*e - 4*(B*a*b^3*c - 16*A*a^2*c^3 - 4*(B*a^2*b -
A*a*b^2)*c^2)*d^4*e^2 + (B*a*b^4 + 16*(3*B*a^3 - 4*A*a^2*b)*c^2 - 16*(B*a^2*b^2
- A*a*b^3)*c)*d^3*e^3 + (4*B*a^2*b^3 - 5*A*a*b^4 - 16*A*a^3*c^2 - 8*(2*B*a^3*b -
3*A*a^2*b^2)*c)*d^2*e^4 + (8*(B*b^2*c^3 - 4*B*a*c^4)*d^3*e^3 - 4*(B*b^3*c^2 - 1
6*A*a*c^4 - 4*(B*a*b - A*b^2)*c^3)*d^2*e^4 + (B*b^4*c + 16*(3*B*a^2 - 4*A*a*b)*c
^3 - 16*(B*a*b^2 - A*b^3)*c^2)*d*e^5 - (16*A*a^2*c^3 + 8*(2*B*a^2*b - 3*A*a*b^2)
*c^2 - (4*B*a*b^3 - 5*A*b^4)*c)*e^6)*x^4 + (16*(B*b^2*c^3 - 4*B*a*c^4)*d^4*e^2 -
32*(A*b^2*c^3 - 4*A*a*c^4)*d^3*e^3 - 2*(B*b^4*c - 16*(3*B*a^2 - 2*A*a*b)*c^3 +
8*(B*a*b^2 - A*b^3)*c^2)*d^2*e^4 + (B*b^5 - 32*A*a^2*c^3 + 16*(B*a^2*b - A*a*b^2
)*c^2 - 2*(4*B*a*b^3 - 3*A*b^4)*c)*d*e^5 + (4*B*a*b^4 - 5*A*b^5 - 16*A*a^2*b*c^2
- 8*(2*B*a^2*b^2 - 3*A*a*b^3)*c)*e^6)*x^3 + (8*(B*b^2*c^3 - 4*B*a*c^4)*d^5*e +
4*(3*B*b^3*c^2 + 16*A*a*c^4 - 4*(3*B*a*b + A*b^2)*c^3)*d^4*e^2 - (7*B*b^4*c - 16
*(B*a^2 + 4*A*a*b)*c^3 - 8*(3*B*a*b^2 - 2*A*b^3)*c^2)*d^3*e^3 + (2*B*b^5 + 48*A*
a^2*c^3 + 24*(4*B*a^2*b - 5*A*a*b^2)*c^2 - (32*B*a*b^3 - 27*A*b^4)*c)*d^2*e^4 +
(9*B*a*b^4 - 10*A*b^5 + 48*(B*a^3 - 2*A*a^2*b)*c^2 - 16*(3*B*a^2*b^2 - 4*A*a*b^3
)*c)*d*e^5 + (4*B*a^2*b^3 - 5*A*a*b^4 - 16*A*a^3*c^2 - 8*(2*B*a^3*b - 3*A*a^2*b^
2)*c)*e^6)*x^2 + (8*(B*b^3*c^2 - 4*B*a*b*c^3)*d^5*e - 4*(B*b^4*c + 16*(B*a^2 - A
*a*b)*c^3 - 4*(2*B*a*b^2 - A*b^3)*c^2)*d^4*e^2 + (B*b^5 + 128*A*a^2*c^3 + 16*(5*
B*a^2*b - 6*A*a*b^2)*c^2 - 8*(3*B*a*b^3 - 2*A*b^4)*c)*d^3*e^3 + (6*B*a*b^4 - 5*A
*b^5 + 48*(2*B*a^3 - 3*A*a^2*b)*c^2 - 8*(6*B*a^2*b^2 - 7*A*a*b^3)*c)*d^2*e^4 + 2
*(4*B*a^2*b^3 - 5*A*a*b^4 - 16*A*a^3*c^2 - 8*(2*B*a^3*b - 3*A*a^2*b^2)*c)*d*e^5)
*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))))/(((a*b^2*c^3 - 4*a^2*c^4)*d^8 - 3
*(a*b^3*c^2 - 4*a^2*b*c^3)*d^7*e + 3*(a*b^4*c - 3*a^2*b^2*c^2 - 4*a^3*c^3)*d^6*e
^2 - (a*b^5 + 2*a^2*b^3*c - 24*a^3*b*c^2)*d^5*e^3 + 3*(a^2*b^4 - 3*a^3*b^2*c - 4
*a^4*c^2)*d^4*e^4 - 3*(a^3*b^3 - 4*a^4*b*c)*d^3*e^5 + (a^4*b^2 - 4*a^5*c)*d^2*e^
6 + ((b^2*c^4 - 4*a*c^5)*d^6*e^2 - 3*(b^3*c^3 - 4*a*b*c^4)*d^5*e^3 + 3*(b^4*c^2
- 3*a*b^2*c^3 - 4*a^2*c^4)*d^4*e^4 - (b^5*c + 2*a*b^3*c^2 - 24*a^2*b*c^3)*d^3*e^
5 + 3*(a*b^4*c - 3*a^2*b^2*c^2 - 4*a^3*c^3)*d^2*e^6 - 3*(a^2*b^3*c - 4*a^3*b*c^2
)*d*e^7 + (a^3*b^2*c - 4*a^4*c^2)*e^8)*x^4 + (2*(b^2*c^4 - 4*a*c^5)*d^7*e - 5*(b
^3*c^3 - 4*a*b*c^4)*d^6*e^2 + 3*(b^4*c^2 - 2*a*b^2*c^3 - 8*a^2*c^4)*d^5*e^3 + (b
^5*c - 13*a*b^3*c^2 + 36*a^2*b*c^3)*d^4*e^4 - (b^6 - 4*a*b^4*c - 6*a^2*b^2*c^2 +
24*a^3*c^3)*d^3*e^5 + 3*(a*b^5 - 5*a^2*b^3*c + 4*a^3*b*c^2)*d^2*e^6 - (3*a^2*b^
4 - 14*a^3*b^2*c + 8*a^4*c^2)*d*e^7 + (a^3*b^3 - 4*a^4*b*c)*e^8)*x^3 + ((b^2*c^4
- 4*a*c^5)*d^8 - (b^3*c^3 - 4*a*b*c^4)*d^7*e - (3*b^4*c^2 - 16*a*b^2*c^3 + 16*a
^2*c^4)*d^6*e^2 + (5*b^5*c - 23*a*b^3*c^2 + 12*a^2*b*c^3)*d^5*e^3 - 2*(b^6 - a*b
^4*c - 15*a^2*b^2*c^2 + 12*a^3*c^3)*d^4*e^4 + (5*a*b^5 - 23*a^2*b^3*c + 12*a^3*b
*c^2)*d^3*e^5 - (3*a^2*b^4 - 16*a^3*b^2*c + 16*a^4*c^2)*d^2*e^6 - (a^3*b^3 - 4*a
^4*b*c)*d*e^7 + (a^4*b^2 - 4*a^5*c)*e^8)*x^2 + ((b^3*c^3 - 4*a*b*c^4)*d^8 - (3*b
^4*c^2 - 14*a*b^2*c^3 + 8*a^2*c^4)*d^7*e + 3*(b^5*c - 5*a*b^3*c^2 + 4*a^2*b*c^3)
*d^6*e^2 - (b^6 - 4*a*b^4*c - 6*a^2*b^2*c^2 + 24*a^3*c^3)*d^5*e^3 + (a*b^5 - 13*
a^2*b^3*c + 36*a^3*b*c^2)*d^4*e^4 + 3*(a^2*b^4 - 2*a^3*b^2*c - 8*a^4*c^2)*d^3*e^
5 - 5*(a^3*b^3 - 4*a^4*b*c)*d^2*e^6 + 2*(a^4*b^2 - 4*a^5*c)*d*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((B*x+A)/(e*x+d)**3/(c*x**2+b*x+a)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: TypeError