3.2477 \(\int \frac{A+B x}{(d+e x)^2 \left (a+b x+c x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=334 \[ \frac{e \sqrt{a+b x+c x^2} \left (2 b \left (a B e^2+2 A c d e+B c d^2\right )-4 c \left (-2 a A e^2+3 a B d e+A c d^2\right )+b^2 e (B d-3 A e)\right )}{\left (b^2-4 a c\right ) (d+e x) \left (a e^2-b d e+c d^2\right )^2}+\frac{2 \left (-A \left (2 a c e+b^2 (-e)+b c d\right )+c x (-2 a B e+A b e-2 A c d+b B d)+a B (2 c d-b e)\right )}{\left (b^2-4 a c\right ) (d+e x) \sqrt{a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}+\frac{e \left (3 A e (2 c d-b e)-B \left (4 c d^2-e (2 a e+b d)\right )\right ) \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{2 \left (a e^2-b d e+c d^2\right )^{5/2}} \]

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

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Rubi [A]  time = 1.08483, antiderivative size = 332, normalized size of antiderivative = 0.99, number of steps used = 4, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148 \[ \frac{e \sqrt{a+b x+c x^2} \left (2 b \left (a B e^2+2 A c d e+B c d^2\right )-4 c \left (-2 a A e^2+3 a B d e+A c d^2\right )+b^2 e (B d-3 A e)\right )}{\left (b^2-4 a c\right ) (d+e x) \left (a e^2-b d e+c d^2\right )^2}+\frac{2 \left (-A \left (2 a c e+b^2 (-e)+b c d\right )+c x (-2 a B e+A b e-2 A c d+b B d)+a B (2 c d-b e)\right )}{\left (b^2-4 a c\right ) (d+e x) \sqrt{a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}-\frac{e \left (-B e (2 a e+b d)-3 A e (2 c d-b e)+4 B c d^2\right ) \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{2 \left (a e^2-b d e+c d^2\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x)/((d + e*x)^2*(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)*Sqrt[a + b*x +
c*x^2]) + (e*(b^2*e*(B*d - 3*A*e) - 4*c*(A*c*d^2 + 3*a*B*d*e - 2*a*A*e^2) + 2*b*
(B*c*d^2 + 2*A*c*d*e + a*B*e^2))*Sqrt[a + b*x + c*x^2])/((b^2 - 4*a*c)*(c*d^2 -
b*d*e + a*e^2)^2*(d + e*x)) - (e*(4*B*c*d^2 - B*e*(b*d + 2*a*e) - 3*A*e*(2*c*d -
b*e))*ArcTanh[(b*d - 2*a*e + (2*c*d - b*e)*x)/(2*Sqrt[c*d^2 - b*d*e + a*e^2]*Sq
rt[a + b*x + c*x^2])])/(2*(c*d^2 - b*d*e + a*e^2)^(5/2))

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

[Out]

Timed out

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Mathematica [A]  time = 3.49139, size = 388, normalized size = 1.16 \[ \frac{1}{2} \left (-\frac{4 \left (B \left (2 a^2 c e^2-a \left (b^2 e^2+b c e (e x-2 d)+2 c^2 d (d-2 e x)\right )-b c^2 d^2 x\right )+A \left (b c \left (c d (d-2 e x)-3 a e^2\right )+2 c^2 \left (a e (2 d-e x)+c d^2 x\right )+b^3 e^2+b^2 c e (e x-2 d)\right )\right )}{\left (b^2-4 a c\right ) \sqrt{a+x (b+c x)} \left (e (a e-b d)+c d^2\right )^2}-\frac{2 e^2 \sqrt{a+x (b+c x)} (A e-B d)}{(d+e x) \left (e (a e-b d)+c d^2\right )^2}+\frac{e \log (d+e x) \left (B e (2 a e+b d)-3 A e (b e-2 c d)-4 B c d^2\right )}{\left (e (a e-b d)+c d^2\right )^{5/2}}-\frac{e \left (B e (2 a e+b d)-3 A e (b e-2 c d)-4 B c d^2\right ) \log \left (2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2}+2 a e-b d+b e x-2 c d x\right )}{\left (e (a e-b d)+c d^2\right )^{5/2}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.027, size = 3090, normalized size = 9.3 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

6*B/e/(a*e^2-b*d*e+c*d^2)/(4*a*c-b^2)/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-
b*d*e+c*d^2)/e^2)^(1/2)*b*c*d+12/(a*e^2-b*d*e+c*d^2)^2/(4*a*c-b^2)/((x+d/e)^2*c+
(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*x*b*c^2*d^2*B+3*e^2/(a*e^2-
b*d*e+c*d^2)^2/(4*a*c-b^2)/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2
)/e^2)^(1/2)*x*b^2*c*A-12/e/(a*e^2-b*d*e+c*d^2)^2/(4*a*c-b^2)/((x+d/e)^2*c+(b*e-
2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*x*c^3*d^3*B-6*e/(a*e^2-b*d*e+c*d
^2)^2/(4*a*c-b^2)/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1
/2)*b^2*c*d*A-6/e/(a*e^2-b*d*e+c*d^2)^2/(4*a*c-b^2)/((x+d/e)^2*c+(b*e-2*c*d)/e*(
x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*b*c^2*d^3*B-3*e/(a*e^2-b*d*e+c*d^2)^2/(4*a
*c-b^2)/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*x*b^2*
c*B*d-12*e/(a*e^2-b*d*e+c*d^2)^2/(4*a*c-b^2)/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+
(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*x*b*c^2*d*A+3/2*e^2/(a*e^2-b*d*e+c*d^2)^2/((a*e^2
-b*d*e+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2-b*d*e+c*d^2)/e^2+(b*e-2*c*d)/e*(x+d/e)+2*(
(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c
*d^2)/e^2)^(1/2))/(x+d/e))*b*A+3/2*e/(a*e^2-b*d*e+c*d^2)^2/((x+d/e)^2*c+(b*e-2*c
*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*b*B*d+3/(a*e^2-b*d*e+c*d^2)^2/((a*e
^2-b*d*e+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2-b*d*e+c*d^2)/e^2+(b*e-2*c*d)/e*(x+d/e)+2
*((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e
+c*d^2)/e^2)^(1/2))/(x+d/e))*c*d^2*B-8*c^2/(a*e^2-b*d*e+c*d^2)/(4*a*c-b^2)/((x+d
/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*x*A-4*c/(a*e^2-b*d*
e+c*d^2)/(4*a*c-b^2)/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)
^(1/2)*b*A+1/e/(a*e^2-b*d*e+c*d^2)/(x+d/e)/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a
*e^2-b*d*e+c*d^2)/e^2)^(1/2)*B*d+3*e/(a*e^2-b*d*e+c*d^2)^2/((x+d/e)^2*c+(b*e-2*c
*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*c*d*A+3/2*e^2/(a*e^2-b*d*e+c*d^2)^2
/(4*a*c-b^2)/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*b
^3*A-1/(a*e^2-b*d*e+c*d^2)/(x+d/e)/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d
*e+c*d^2)/e^2)^(1/2)*A-B/(a*e^2-b*d*e+c*d^2)/((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*ln(
(2*(a*e^2-b*d*e+c*d^2)/e^2+(b*e-2*c*d)/e*(x+d/e)+2*((a*e^2-b*d*e+c*d^2)/e^2)^(1/
2)*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))/(x+d/e))+1
2*B/e/(a*e^2-b*d*e+c*d^2)/(4*a*c-b^2)/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-
b*d*e+c*d^2)/e^2)^(1/2)*x*c^2*d+B/(a*e^2-b*d*e+c*d^2)/((x+d/e)^2*c+(b*e-2*c*d)/e
*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)-B/(a*e^2-b*d*e+c*d^2)/(4*a*c-b^2)/((x+d/
e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*b^2-3/2*e^2/(a*e^2-b
*d*e+c*d^2)^2/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*
b*A-3/(a*e^2-b*d*e+c*d^2)^2/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^
2)/e^2)^(1/2)*c*d^2*B-2*B/(a*e^2-b*d*e+c*d^2)/(4*a*c-b^2)/((x+d/e)^2*c+(b*e-2*c*
d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*x*b*c+12/(a*e^2-b*d*e+c*d^2)^2/(4*a*
c-b^2)/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*x*c^3*d
^2*A-3/2*e/(a*e^2-b*d*e+c*d^2)^2/(4*a*c-b^2)/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+
(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*b^3*B*d-3/2*e/(a*e^2-b*d*e+c*d^2)^2/((a*e^2-b*d*e
+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2-b*d*e+c*d^2)/e^2+(b*e-2*c*d)/e*(x+d/e)+2*((a*e^2
-b*d*e+c*d^2)/e^2)^(1/2)*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/
e^2)^(1/2))/(x+d/e))*b*B*d+6/(a*e^2-b*d*e+c*d^2)^2/(4*a*c-b^2)/((x+d/e)^2*c+(b*e
-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*b^2*c*d^2*B+6/(a*e^2-b*d*e+c*d^
2)^2/(4*a*c-b^2)/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/
2)*b*c^2*d^2*A-3*e/(a*e^2-b*d*e+c*d^2)^2/((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*ln((2*(
a*e^2-b*d*e+c*d^2)/e^2+(b*e-2*c*d)/e*(x+d/e)+2*((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*(
(x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))/(x+d/e))*c*d*A

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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)^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 5.82579, 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)^2),x, algorithm="fricas")

[Out]

[1/4*(4*(2*(2*B*a - A*b)*c^2*d^3 - 4*(2*A*a*c^2 + (B*a*b - A*b^2)*c)*d^2*e + (3*
B*a*b^2 - 2*A*b^3 - 2*(4*B*a^2 - 3*A*a*b)*c)*d*e^2 - (A*a*b^2 - 4*A*a^2*c)*e^3 +
(2*(B*b*c^2 - 2*A*c^3)*d^2*e + (B*b^2*c - 4*(3*B*a - A*b)*c^2)*d*e^2 + (8*A*a*c
^2 + (2*B*a*b - 3*A*b^2)*c)*e^3)*x^2 - (2*(2*B*a - A*b)*c^2*d^2*e - 2*(B*b*c^2 -
2*A*c^3)*d^3 - (B*b^3 - 4*A*a*c^2 - 2*(3*B*a*b - A*b^2)*c)*d*e^2 - (2*B*a*b^2 -
3*A*b^3 - 2*(2*B*a^2 - 5*A*a*b)*c)*e^3)*x)*sqrt(c*d^2 - b*d*e + a*e^2)*sqrt(c*x
^2 + b*x + a) + (4*(B*a*b^2*c - 4*B*a^2*c^2)*d^3*e - (B*a*b^3 - 24*A*a^2*c^2 - 2
*(2*B*a^2*b - 3*A*a*b^2)*c)*d^2*e^2 - (2*B*a^2*b^2 - 3*A*a*b^3 - 4*(2*B*a^3 - 3*
A*a^2*b)*c)*d*e^3 + (4*(B*b^2*c^2 - 4*B*a*c^3)*d^2*e^2 - (B*b^3*c - 24*A*a*c^3 -
2*(2*B*a*b - 3*A*b^2)*c^2)*d*e^3 + (4*(2*B*a^2 - 3*A*a*b)*c^2 - (2*B*a*b^2 - 3*
A*b^3)*c)*e^4)*x^3 + (4*(B*b^2*c^2 - 4*B*a*c^3)*d^3*e + 3*(B*b^3*c + 8*A*a*c^3 -
2*(2*B*a*b + A*b^2)*c^2)*d^2*e^2 - (B*b^4 - 4*(2*B*a^2 + 3*A*a*b)*c^2 - (2*B*a*
b^2 - 3*A*b^3)*c)*d*e^3 - (2*B*a*b^3 - 3*A*b^4 - 4*(2*B*a^2*b - 3*A*a*b^2)*c)*e^
4)*x^2 + (4*(B*b^3*c - 4*B*a*b*c^2)*d^3*e - (B*b^4 + 8*(2*B*a^2 - 3*A*a*b)*c^2 -
2*(4*B*a*b^2 - 3*A*b^3)*c)*d^2*e^2 - 3*(B*a*b^3 - A*b^4 - 8*A*a^2*c^2 - 2*(2*B*
a^2*b - 3*A*a*b^2)*c)*d*e^3 - (2*B*a^2*b^2 - 3*A*a*b^3 - 4*(2*B*a^3 - 3*A*a^2*b)
*c)*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)))/(((a*b^2*c^2 - 4*a^2*c^3)*d^5 - 2*
(a*b^3*c - 4*a^2*b*c^2)*d^4*e + (a*b^4 - 2*a^2*b^2*c - 8*a^3*c^2)*d^3*e^2 - 2*(a
^2*b^3 - 4*a^3*b*c)*d^2*e^3 + (a^3*b^2 - 4*a^4*c)*d*e^4 + ((b^2*c^3 - 4*a*c^4)*d
^4*e - 2*(b^3*c^2 - 4*a*b*c^3)*d^3*e^2 + (b^4*c - 2*a*b^2*c^2 - 8*a^2*c^3)*d^2*e
^3 - 2*(a*b^3*c - 4*a^2*b*c^2)*d*e^4 + (a^2*b^2*c - 4*a^3*c^2)*e^5)*x^3 + ((b^2*
c^3 - 4*a*c^4)*d^5 - (b^3*c^2 - 4*a*b*c^3)*d^4*e - (b^4*c - 6*a*b^2*c^2 + 8*a^2*
c^3)*d^3*e^2 + (b^5 - 4*a*b^3*c)*d^2*e^3 - (2*a*b^4 - 9*a^2*b^2*c + 4*a^3*c^2)*d
*e^4 + (a^2*b^3 - 4*a^3*b*c)*e^5)*x^2 + ((b^3*c^2 - 4*a*b*c^3)*d^5 - (2*b^4*c -
9*a*b^2*c^2 + 4*a^2*c^3)*d^4*e + (b^5 - 4*a*b^3*c)*d^3*e^2 - (a*b^4 - 6*a^2*b^2*
c + 8*a^3*c^2)*d^2*e^3 - (a^2*b^3 - 4*a^3*b*c)*d*e^4 + (a^3*b^2 - 4*a^4*c)*e^5)*
x)*sqrt(c*d^2 - b*d*e + a*e^2)), 1/2*(2*(2*(2*B*a - A*b)*c^2*d^3 - 4*(2*A*a*c^2
+ (B*a*b - A*b^2)*c)*d^2*e + (3*B*a*b^2 - 2*A*b^3 - 2*(4*B*a^2 - 3*A*a*b)*c)*d*e
^2 - (A*a*b^2 - 4*A*a^2*c)*e^3 + (2*(B*b*c^2 - 2*A*c^3)*d^2*e + (B*b^2*c - 4*(3*
B*a - A*b)*c^2)*d*e^2 + (8*A*a*c^2 + (2*B*a*b - 3*A*b^2)*c)*e^3)*x^2 - (2*(2*B*a
- A*b)*c^2*d^2*e - 2*(B*b*c^2 - 2*A*c^3)*d^3 - (B*b^3 - 4*A*a*c^2 - 2*(3*B*a*b
- A*b^2)*c)*d*e^2 - (2*B*a*b^2 - 3*A*b^3 - 2*(2*B*a^2 - 5*A*a*b)*c)*e^3)*x)*sqrt
(-c*d^2 + b*d*e - a*e^2)*sqrt(c*x^2 + b*x + a) + (4*(B*a*b^2*c - 4*B*a^2*c^2)*d^
3*e - (B*a*b^3 - 24*A*a^2*c^2 - 2*(2*B*a^2*b - 3*A*a*b^2)*c)*d^2*e^2 - (2*B*a^2*
b^2 - 3*A*a*b^3 - 4*(2*B*a^3 - 3*A*a^2*b)*c)*d*e^3 + (4*(B*b^2*c^2 - 4*B*a*c^3)*
d^2*e^2 - (B*b^3*c - 24*A*a*c^3 - 2*(2*B*a*b - 3*A*b^2)*c^2)*d*e^3 + (4*(2*B*a^2
- 3*A*a*b)*c^2 - (2*B*a*b^2 - 3*A*b^3)*c)*e^4)*x^3 + (4*(B*b^2*c^2 - 4*B*a*c^3)
*d^3*e + 3*(B*b^3*c + 8*A*a*c^3 - 2*(2*B*a*b + A*b^2)*c^2)*d^2*e^2 - (B*b^4 - 4*
(2*B*a^2 + 3*A*a*b)*c^2 - (2*B*a*b^2 - 3*A*b^3)*c)*d*e^3 - (2*B*a*b^3 - 3*A*b^4
- 4*(2*B*a^2*b - 3*A*a*b^2)*c)*e^4)*x^2 + (4*(B*b^3*c - 4*B*a*b*c^2)*d^3*e - (B*
b^4 + 8*(2*B*a^2 - 3*A*a*b)*c^2 - 2*(4*B*a*b^2 - 3*A*b^3)*c)*d^2*e^2 - 3*(B*a*b^
3 - A*b^4 - 8*A*a^2*c^2 - 2*(2*B*a^2*b - 3*A*a*b^2)*c)*d*e^3 - (2*B*a^2*b^2 - 3*
A*a*b^3 - 4*(2*B*a^3 - 3*A*a^2*b)*c)*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))))/(((a*b^2*c^2 - 4*a^2*c^3)*d^5 - 2*(a*b^3*c - 4*a^2*b*c^2)*d^4*e + (a*b^4
- 2*a^2*b^2*c - 8*a^3*c^2)*d^3*e^2 - 2*(a^2*b^3 - 4*a^3*b*c)*d^2*e^3 + (a^3*b^2
- 4*a^4*c)*d*e^4 + ((b^2*c^3 - 4*a*c^4)*d^4*e - 2*(b^3*c^2 - 4*a*b*c^3)*d^3*e^2
+ (b^4*c - 2*a*b^2*c^2 - 8*a^2*c^3)*d^2*e^3 - 2*(a*b^3*c - 4*a^2*b*c^2)*d*e^4 +
(a^2*b^2*c - 4*a^3*c^2)*e^5)*x^3 + ((b^2*c^3 - 4*a*c^4)*d^5 - (b^3*c^2 - 4*a*b*
c^3)*d^4*e - (b^4*c - 6*a*b^2*c^2 + 8*a^2*c^3)*d^3*e^2 + (b^5 - 4*a*b^3*c)*d^2*e
^3 - (2*a*b^4 - 9*a^2*b^2*c + 4*a^3*c^2)*d*e^4 + (a^2*b^3 - 4*a^3*b*c)*e^5)*x^2
+ ((b^3*c^2 - 4*a*b*c^3)*d^5 - (2*b^4*c - 9*a*b^2*c^2 + 4*a^2*c^3)*d^4*e + (b^5
- 4*a*b^3*c)*d^3*e^2 - (a*b^4 - 6*a^2*b^2*c + 8*a^3*c^2)*d^2*e^3 - (a^2*b^3 - 4*
a^3*b*c)*d*e^4 + (a^3*b^2 - 4*a^4*c)*e^5)*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)**2/(c*x**2+b*x+a)**(3/2),x)

[Out]

Timed out

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

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)^2),x, algorithm="giac")

[Out]

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