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3.1017 \(\int \frac{x^{5/2} (A+B x)}{\left (a+b x+c x^2\right )^2} \, dx\)

Optimal. Leaf size=411 \[ -\frac{\left (-\frac{20 a^2 B c^2+8 a A b c^2-19 a b^2 B c-A b^3 c+3 b^4 B}{\sqrt{b^2-4 a c}}+6 a A c^2-13 a b B c-A b^2 c+3 b^3 B\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{x}}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{\sqrt{2} c^{5/2} \left (b^2-4 a c\right ) \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\left (\frac{20 a^2 B c^2+8 a A b c^2-19 a b^2 B c-A b^3 c+3 b^4 B}{\sqrt{b^2-4 a c}}+6 a A c^2-13 a b B c-A b^2 c+3 b^3 B\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{x}}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{\sqrt{2} c^{5/2} \left (b^2-4 a c\right ) \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{\sqrt{x} \left (-10 a B c-A b c+3 b^2 B\right )}{c^2 \left (b^2-4 a c\right )}-\frac{x^{3/2} \left (x \left (-2 a B c-A b c+b^2 B\right )+a (b B-2 A c)\right )}{c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )} \]

[Out]

((3*b^2*B - A*b*c - 10*a*B*c)*Sqrt[x])/(c^2*(b^2 - 4*a*c)) - (x^(3/2)*(a*(b*B -
2*A*c) + (b^2*B - A*b*c - 2*a*B*c)*x))/(c*(b^2 - 4*a*c)*(a + b*x + c*x^2)) - ((3
*b^3*B - A*b^2*c - 13*a*b*B*c + 6*a*A*c^2 - (3*b^4*B - A*b^3*c - 19*a*b^2*B*c +
8*a*A*b*c^2 + 20*a^2*B*c^2)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[x])/
Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*c^(5/2)*(b^2 - 4*a*c)*Sqrt[b - Sqrt[b^2 -
 4*a*c]]) - ((3*b^3*B - A*b^2*c - 13*a*b*B*c + 6*a*A*c^2 + (3*b^4*B - A*b^3*c -
19*a*b^2*B*c + 8*a*A*b*c^2 + 20*a^2*B*c^2)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sq
rt[c]*Sqrt[x])/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*c^(5/2)*(b^2 - 4*a*c)*Sqrt
[b + Sqrt[b^2 - 4*a*c]])

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Rubi [A]  time = 8.71399, antiderivative size = 411, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217 \[ -\frac{\left (-\frac{20 a^2 B c^2+8 a A b c^2-19 a b^2 B c-A b^3 c+3 b^4 B}{\sqrt{b^2-4 a c}}+6 a A c^2-13 a b B c-A b^2 c+3 b^3 B\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{x}}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{\sqrt{2} c^{5/2} \left (b^2-4 a c\right ) \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\left (\frac{20 a^2 B c^2+8 a A b c^2-19 a b^2 B c-A b^3 c+3 b^4 B}{\sqrt{b^2-4 a c}}+6 a A c^2-13 a b B c-A b^2 c+3 b^3 B\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{x}}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{\sqrt{2} c^{5/2} \left (b^2-4 a c\right ) \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{\sqrt{x} \left (-10 a B c-A b c+3 b^2 B\right )}{c^2 \left (b^2-4 a c\right )}-\frac{x^{3/2} \left (x \left (-2 a B c-A b c+b^2 B\right )+a (b B-2 A c)\right )}{c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

((3*b^2*B - A*b*c - 10*a*B*c)*Sqrt[x])/(c^2*(b^2 - 4*a*c)) - (x^(3/2)*(a*(b*B -
2*A*c) + (b^2*B - A*b*c - 2*a*B*c)*x))/(c*(b^2 - 4*a*c)*(a + b*x + c*x^2)) - ((3
*b^3*B - A*b^2*c - 13*a*b*B*c + 6*a*A*c^2 - (3*b^4*B - A*b^3*c - 19*a*b^2*B*c +
8*a*A*b*c^2 + 20*a^2*B*c^2)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[x])/
Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*c^(5/2)*(b^2 - 4*a*c)*Sqrt[b - Sqrt[b^2 -
 4*a*c]]) - ((3*b^3*B - A*b^2*c - 13*a*b*B*c + 6*a*A*c^2 + (3*b^4*B - A*b^3*c -
19*a*b^2*B*c + 8*a*A*b*c^2 + 20*a^2*B*c^2)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sq
rt[c]*Sqrt[x])/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*c^(5/2)*(b^2 - 4*a*c)*Sqrt
[b + Sqrt[b^2 - 4*a*c]])

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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(x**(5/2)*(B*x+A)/(c*x**2+b*x+a)**2,x)

[Out]

Timed out

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Mathematica [A]  time = 2.70728, size = 462, normalized size = 1.12 \[ \frac{\frac{2 \sqrt{c} \sqrt{x} \left (-2 a^2 B c+a \left (-b c (A+3 B x)+2 A c^2 x+b^2 B\right )+b^2 x (b B-A c)\right )}{\left (b^2-4 a c\right ) (a+x (b+c x))}-\frac{\sqrt{2} \left (2 a c^2 \left (3 A \sqrt{b^2-4 a c}-10 a B\right )+b^2 c \left (19 a B-A \sqrt{b^2-4 a c}\right )-a b c \left (13 B \sqrt{b^2-4 a c}+8 A c\right )+b^3 \left (3 B \sqrt{b^2-4 a c}+A c\right )-3 b^4 B\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{x}}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{\left (b^2-4 a c\right )^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\sqrt{2} \left (2 a c^2 \left (3 A \sqrt{b^2-4 a c}+10 a B\right )-b^2 c \left (A \sqrt{b^2-4 a c}+19 a B\right )+a b c \left (8 A c-13 B \sqrt{b^2-4 a c}\right )+b^3 \left (3 B \sqrt{b^2-4 a c}-A c\right )+3 b^4 B\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{x}}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{\left (b^2-4 a c\right )^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}}+4 B \sqrt{c} \sqrt{x}}{2 c^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

(4*B*Sqrt[c]*Sqrt[x] + (2*Sqrt[c]*Sqrt[x]*(-2*a^2*B*c + b^2*(b*B - A*c)*x + a*(b
^2*B + 2*A*c^2*x - b*c*(A + 3*B*x))))/((b^2 - 4*a*c)*(a + x*(b + c*x))) - (Sqrt[
2]*(-3*b^4*B + b^2*c*(19*a*B - A*Sqrt[b^2 - 4*a*c]) + 2*a*c^2*(-10*a*B + 3*A*Sqr
t[b^2 - 4*a*c]) + b^3*(A*c + 3*B*Sqrt[b^2 - 4*a*c]) - a*b*c*(8*A*c + 13*B*Sqrt[b
^2 - 4*a*c]))*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[x])/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/((b
^2 - 4*a*c)^(3/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) - (Sqrt[2]*(3*b^4*B - b^2*c*(19*a
*B + A*Sqrt[b^2 - 4*a*c]) + 2*a*c^2*(10*a*B + 3*A*Sqrt[b^2 - 4*a*c]) + a*b*c*(8*
A*c - 13*B*Sqrt[b^2 - 4*a*c]) + b^3*(-(A*c) + 3*B*Sqrt[b^2 - 4*a*c]))*ArcTan[(Sq
rt[2]*Sqrt[c]*Sqrt[x])/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/((b^2 - 4*a*c)^(3/2)*Sqrt[b
 + Sqrt[b^2 - 4*a*c]]))/(2*c^(5/2))

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Maple [B]  time = 0.1, size = 4317, normalized size = 10.5 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*B*x^(1/2)/c^2-2/(c*x^2+b*x+a)/(4*a*c-b^2)*x^(3/2)*a*A+25/2/c/(4*a*c-b^2)*2^(1/
2)/((-4*a*b*c+b^3+(-(4*a*c-b^2)^3)^(1/2))*c*(4*a*c-b^2))^(1/2)*arctanh(1/2*(-8*a
*c^2+2*b^2*c)*x^(1/2)*2^(1/2)/((-4*a*b*c+b^3+(-(4*a*c-b^2)^3)^(1/2))*c*(4*a*c-b^
2))^(1/2))*B*a*b^3-3/2/c^2/(-(4*a*c-b^2)^3)^(1/2)/(4*a*c-b^2)*2^(1/2)/((-4*a*b*c
+b^3+(-(4*a*c-b^2)^3)^(1/2))*c*(4*a*c-b^2))^(1/2)*arctanh(1/2*(-8*a*c^2+2*b^2*c)
*x^(1/2)*2^(1/2)/((-4*a*b*c+b^3+(-(4*a*c-b^2)^3)^(1/2))*c*(4*a*c-b^2))^(1/2))*B*
b^8+3/c/(c*x^2+b*x+a)/(4*a*c-b^2)*x^(3/2)*a*b*B+1/c/(c*x^2+b*x+a)*a/(4*a*c-b^2)*
x^(1/2)*A*b+12*c/(4*a*c-b^2)*2^(1/2)/(c*(4*a*c-b^2)*(4*a*b*c-b^3+(-(4*a*c-b^2)^3
)^(1/2)))^(1/2)*arctan(1/2*(8*a*c^2-2*b^2*c)*x^(1/2)*2^(1/2)/(c*(4*a*c-b^2)*(4*a
*b*c-b^3+(-(4*a*c-b^2)^3)^(1/2)))^(1/2))*A*a^2+40*c/(-(4*a*c-b^2)^3)^(1/2)/(4*a*
c-b^2)*2^(1/2)/((-4*a*b*c+b^3+(-(4*a*c-b^2)^3)^(1/2))*c*(4*a*c-b^2))^(1/2)*arcta
nh(1/2*(-8*a*c^2+2*b^2*c)*x^(1/2)*2^(1/2)/((-4*a*b*c+b^3+(-(4*a*c-b^2)^3)^(1/2))
*c*(4*a*c-b^2))^(1/2))*a^2*A*b^3+232*c/(-(4*a*c-b^2)^3)^(1/2)/(4*a*c-b^2)*2^(1/2
)/((-4*a*b*c+b^3+(-(4*a*c-b^2)^3)^(1/2))*c*(4*a*c-b^2))^(1/2)*arctanh(1/2*(-8*a*
c^2+2*b^2*c)*x^(1/2)*2^(1/2)/((-4*a*b*c+b^3+(-(4*a*c-b^2)^3)^(1/2))*c*(4*a*c-b^2
))^(1/2))*B*a^3*b^2+64*c^2/(-(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-b^3+(-(4*a*c-b^2)^3)^(1/2)))^(1/2)*arctan(1/2*(8*a*c^2-2*b^2*c)*x^
(1/2)*2^(1/2)/(c*(4*a*c-b^2)*(4*a*b*c-b^3+(-(4*a*c-b^2)^3)^(1/2)))^(1/2))*a^3*A*
b-43/2/c/(-(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-b^3+
(-(4*a*c-b^2)^3)^(1/2)))^(1/2)*arctan(1/2*(8*a*c^2-2*b^2*c)*x^(1/2)*2^(1/2)/(c*(
4*a*c-b^2)*(4*a*b*c-b^3+(-(4*a*c-b^2)^3)^(1/2)))^(1/2))*B*a*b^6+43/2/c/(-(4*a*c-
b^2)^3)^(1/2)/(4*a*c-b^2)*2^(1/2)/((-4*a*b*c+b^3+(-(4*a*c-b^2)^3)^(1/2))*c*(4*a*
c-b^2))^(1/2)*arctanh(1/2*(-8*a*c^2+2*b^2*c)*x^(1/2)*2^(1/2)/((-4*a*b*c+b^3+(-(4
*a*c-b^2)^3)^(1/2))*c*(4*a*c-b^2))^(1/2))*B*a*b^6-40*c/(-(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-b^3+(-(4*a*c-b^2)^3)^(1/2)))^(1/2)*arc
tan(1/2*(8*a*c^2-2*b^2*c)*x^(1/2)*2^(1/2)/(c*(4*a*c-b^2)*(4*a*b*c-b^3+(-(4*a*c-b
^2)^3)^(1/2)))^(1/2))*a^2*A*b^3-232*c/(-(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-b^3+(-(4*a*c-b^2)^3)^(1/2)))^(1/2)*arctan(1/2*(8*a*c^2-
2*b^2*c)*x^(1/2)*2^(1/2)/(c*(4*a*c-b^2)*(4*a*b*c-b^3+(-(4*a*c-b^2)^3)^(1/2)))^(1
/2))*B*a^3*b^2-64*c^2/(-(4*a*c-b^2)^3)^(1/2)/(4*a*c-b^2)*2^(1/2)/((-4*a*b*c+b^3+
(-(4*a*c-b^2)^3)^(1/2))*c*(4*a*c-b^2))^(1/2)*arctanh(1/2*(-8*a*c^2+2*b^2*c)*x^(1
/2)*2^(1/2)/((-4*a*b*c+b^3+(-(4*a*c-b^2)^3)^(1/2))*c*(4*a*c-b^2))^(1/2))*a^3*A*b
-1/c^2/(c*x^2+b*x+a)/(4*a*c-b^2)*x^(3/2)*B*b^3+1/c/(c*x^2+b*x+a)/(4*a*c-b^2)*x^(
3/2)*A*b^2+2/c/(c*x^2+b*x+a)*a^2/(4*a*c-b^2)*x^(1/2)*B+8/(-(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-b^3+(-(4*a*c-b^2)^3)^(1/2)))^(1/2)*a
rctan(1/2*(8*a*c^2-2*b^2*c)*x^(1/2)*2^(1/2)/(c*(4*a*c-b^2)*(4*a*b*c-b^3+(-(4*a*c
-b^2)^3)^(1/2)))^(1/2))*A*a*b^5+160*c^2/(-(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-b^3+(-(4*a*c-b^2)^3)^(1/2)))^(1/2)*arctan(1/2*(8*a*c^
2-2*b^2*c)*x^(1/2)*2^(1/2)/(c*(4*a*c-b^2)*(4*a*b*c-b^3+(-(4*a*c-b^2)^3)^(1/2)))^
(1/2))*B*a^4+3/2/c^2/(-(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-b^3+(-(4*a*c-b^2)^3)^(1/2)))^(1/2)*arctan(1/2*(8*a*c^2-2*b^2*c)*x^(1/2)*
2^(1/2)/(c*(4*a*c-b^2)*(4*a*b*c-b^3+(-(4*a*c-b^2)^3)^(1/2)))^(1/2))*B*b^8+25/2/c
/(4*a*c-b^2)*2^(1/2)/(c*(4*a*c-b^2)*(4*a*b*c-b^3+(-(4*a*c-b^2)^3)^(1/2)))^(1/2)*
arctan(1/2*(8*a*c^2-2*b^2*c)*x^(1/2)*2^(1/2)/(c*(4*a*c-b^2)*(4*a*b*c-b^3+(-(4*a*
c-b^2)^3)^(1/2)))^(1/2))*B*a*b^3+1/2/c/(-(4*a*c-b^2)^3)^(1/2)/(4*a*c-b^2)*2^(1/2
)/((-4*a*b*c+b^3+(-(4*a*c-b^2)^3)^(1/2))*c*(4*a*c-b^2))^(1/2)*arctanh(1/2*(-8*a*
c^2+2*b^2*c)*x^(1/2)*2^(1/2)/((-4*a*b*c+b^3+(-(4*a*c-b^2)^3)^(1/2))*c*(4*a*c-b^2
))^(1/2))*A*b^7-3/2/c^2/(4*a*c-b^2)*2^(1/2)/(c*(4*a*c-b^2)*(4*a*b*c-b^3+(-(4*a*c
-b^2)^3)^(1/2)))^(1/2)*arctan(1/2*(8*a*c^2-2*b^2*c)*x^(1/2)*2^(1/2)/(c*(4*a*c-b^
2)*(4*a*b*c-b^3+(-(4*a*c-b^2)^3)^(1/2)))^(1/2))*B*b^5+12*c/(4*a*c-b^2)*2^(1/2)/(
(-4*a*b*c+b^3+(-(4*a*c-b^2)^3)^(1/2))*c*(4*a*c-b^2))^(1/2)*arctanh(1/2*(-8*a*c^2
+2*b^2*c)*x^(1/2)*2^(1/2)/((-4*a*b*c+b^3+(-(4*a*c-b^2)^3)^(1/2))*c*(4*a*c-b^2))^
(1/2))*A*a^2-3/2/c^2/(4*a*c-b^2)*2^(1/2)/((-4*a*b*c+b^3+(-(4*a*c-b^2)^3)^(1/2))*
c*(4*a*c-b^2))^(1/2)*arctanh(1/2*(-8*a*c^2+2*b^2*c)*x^(1/2)*2^(1/2)/((-4*a*b*c+b
^3+(-(4*a*c-b^2)^3)^(1/2))*c*(4*a*c-b^2))^(1/2))*B*b^5-26/(4*a*c-b^2)*2^(1/2)/(c
*(4*a*c-b^2)*(4*a*b*c-b^3+(-(4*a*c-b^2)^3)^(1/2)))^(1/2)*arctan(1/2*(8*a*c^2-2*b
^2*c)*x^(1/2)*2^(1/2)/(c*(4*a*c-b^2)*(4*a*b*c-b^3+(-(4*a*c-b^2)^3)^(1/2)))^(1/2)
)*B*a^2*b-5/(4*a*c-b^2)*2^(1/2)/(c*(4*a*c-b^2)*(4*a*b*c-b^3+(-(4*a*c-b^2)^3)^(1/
2)))^(1/2)*arctan(1/2*(8*a*c^2-2*b^2*c)*x^(1/2)*2^(1/2)/(c*(4*a*c-b^2)*(4*a*b*c-
b^3+(-(4*a*c-b^2)^3)^(1/2)))^(1/2))*A*a*b^2-26/(4*a*c-b^2)*2^(1/2)/((-4*a*b*c+b^
3+(-(4*a*c-b^2)^3)^(1/2))*c*(4*a*c-b^2))^(1/2)*arctanh(1/2*(-8*a*c^2+2*b^2*c)*x^
(1/2)*2^(1/2)/((-4*a*b*c+b^3+(-(4*a*c-b^2)^3)^(1/2))*c*(4*a*c-b^2))^(1/2))*B*a^2
*b+1/2/c/(4*a*c-b^2)*2^(1/2)/((-4*a*b*c+b^3+(-(4*a*c-b^2)^3)^(1/2))*c*(4*a*c-b^2
))^(1/2)*arctanh(1/2*(-8*a*c^2+2*b^2*c)*x^(1/2)*2^(1/2)/((-4*a*b*c+b^3+(-(4*a*c-
b^2)^3)^(1/2))*c*(4*a*c-b^2))^(1/2))*A*b^4-1/c^2/(c*x^2+b*x+a)*a/(4*a*c-b^2)*x^(
1/2)*b^2*B-5/(4*a*c-b^2)*2^(1/2)/((-4*a*b*c+b^3+(-(4*a*c-b^2)^3)^(1/2))*c*(4*a*c
-b^2))^(1/2)*arctanh(1/2*(-8*a*c^2+2*b^2*c)*x^(1/2)*2^(1/2)/((-4*a*b*c+b^3+(-(4*
a*c-b^2)^3)^(1/2))*c*(4*a*c-b^2))^(1/2))*A*a*b^2+1/2/c/(4*a*c-b^2)*2^(1/2)/(c*(4
*a*c-b^2)*(4*a*b*c-b^3+(-(4*a*c-b^2)^3)^(1/2)))^(1/2)*arctan(1/2*(8*a*c^2-2*b^2*
c)*x^(1/2)*2^(1/2)/(c*(4*a*c-b^2)*(4*a*b*c-b^3+(-(4*a*c-b^2)^3)^(1/2)))^(1/2))*A
*b^4-1/2/c/(-(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-b^
3+(-(4*a*c-b^2)^3)^(1/2)))^(1/2)*arctan(1/2*(8*a*c^2-2*b^2*c)*x^(1/2)*2^(1/2)/(c
*(4*a*c-b^2)*(4*a*b*c-b^3+(-(4*a*c-b^2)^3)^(1/2)))^(1/2))*A*b^7+110/(-(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-b^3+(-(4*a*c-b^2)^3)^(1/2
)))^(1/2)*arctan(1/2*(8*a*c^2-2*b^2*c)*x^(1/2)*2^(1/2)/(c*(4*a*c-b^2)*(4*a*b*c-b
^3+(-(4*a*c-b^2)^3)^(1/2)))^(1/2))*B*a^2*b^4-160*c^2/(-(4*a*c-b^2)^3)^(1/2)/(4*a
*c-b^2)*2^(1/2)/((-4*a*b*c+b^3+(-(4*a*c-b^2)^3)^(1/2))*c*(4*a*c-b^2))^(1/2)*arct
anh(1/2*(-8*a*c^2+2*b^2*c)*x^(1/2)*2^(1/2)/((-4*a*b*c+b^3+(-(4*a*c-b^2)^3)^(1/2)
)*c*(4*a*c-b^2))^(1/2))*B*a^4-8/(-(4*a*c-b^2)^3)^(1/2)/(4*a*c-b^2)*2^(1/2)/((-4*
a*b*c+b^3+(-(4*a*c-b^2)^3)^(1/2))*c*(4*a*c-b^2))^(1/2)*arctanh(1/2*(-8*a*c^2+2*b
^2*c)*x^(1/2)*2^(1/2)/((-4*a*b*c+b^3+(-(4*a*c-b^2)^3)^(1/2))*c*(4*a*c-b^2))^(1/2
))*A*a*b^5-110/(-(4*a*c-b^2)^3)^(1/2)/(4*a*c-b^2)*2^(1/2)/((-4*a*b*c+b^3+(-(4*a*
c-b^2)^3)^(1/2))*c*(4*a*c-b^2))^(1/2)*arctanh(1/2*(-8*a*c^2+2*b^2*c)*x^(1/2)*2^(
1/2)/((-4*a*b*c+b^3+(-(4*a*c-b^2)^3)^(1/2))*c*(4*a*c-b^2))^(1/2))*B*a^2*b^4

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \frac{{\left (A b c -{\left (b^{2} - 2 \, a c\right )} B\right )} x^{\frac{5}{2}} -{\left (B a b - 2 \, A a c\right )} x^{\frac{3}{2}}}{a b^{2} c - 4 \, a^{2} c^{2} +{\left (b^{2} c^{2} - 4 \, a c^{3}\right )} x^{2} +{\left (b^{3} c - 4 \, a b c^{2}\right )} x} + \int -\frac{{\left (A b c -{\left (3 \, b^{2} - 10 \, a c\right )} B\right )} x^{\frac{3}{2}} - 3 \,{\left (B a b - 2 \, A a c\right )} \sqrt{x}}{2 \,{\left (a b^{2} c - 4 \, a^{2} c^{2} +{\left (b^{2} c^{2} - 4 \, a c^{3}\right )} x^{2} +{\left (b^{3} c - 4 \, a b c^{2}\right )} x\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

((A*b*c - (b^2 - 2*a*c)*B)*x^(5/2) - (B*a*b - 2*A*a*c)*x^(3/2))/(a*b^2*c - 4*a^2
*c^2 + (b^2*c^2 - 4*a*c^3)*x^2 + (b^3*c - 4*a*b*c^2)*x) + integrate(-1/2*((A*b*c
 - (3*b^2 - 10*a*c)*B)*x^(3/2) - 3*(B*a*b - 2*A*a*c)*sqrt(x))/(a*b^2*c - 4*a^2*c
^2 + (b^2*c^2 - 4*a*c^3)*x^2 + (b^3*c - 4*a*b*c^2)*x), x)

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Fricas [A]  time = 8.51088, size = 9789, normalized size = 23.82 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*(sqrt(1/2)*(a*b^2*c^2 - 4*a^2*c^3 + (b^2*c^3 - 4*a*c^4)*x^2 + (b^3*c^2 - 4*
a*b*c^3)*x)*sqrt(-(9*B^2*b^7 + 60*(4*A*B*a^3 + A^2*a^2*b)*c^4 - 15*(28*B^2*a^3*b
 + 20*A*B*a^2*b^2 + A^2*a*b^3)*c^3 + (385*B^2*a^2*b^3 + 80*A*B*a*b^4 + A^2*b^5)*
c^2 - 3*(35*B^2*a*b^5 + 2*A*B*b^6)*c + (b^6*c^5 - 12*a*b^4*c^6 + 48*a^2*b^2*c^7
- 64*a^3*c^8)*sqrt((81*B^4*b^8 + 81*A^4*a^2*c^6 - 18*(25*A^2*B^2*a^3 + 44*A^3*B*
a^2*b + A^4*a*b^2)*c^5 + (625*B^4*a^4 + 2200*A*B^3*a^3*b + 2904*A^2*B^2*a^2*b^2
+ 196*A^3*B*a*b^3 + A^4*b^4)*c^4 - 6*(425*B^4*a^3*b^2 + 798*A*B^3*a^2*b^3 + 132*
A^2*B^2*a*b^4 + 2*A^3*B*b^5)*c^3 + 27*(113*B^4*a^2*b^4 + 52*A*B^3*a*b^5 + 2*A^2*
B^2*b^6)*c^2 - 54*(17*B^4*a*b^6 + 2*A*B^3*b^7)*c)/(b^6*c^10 - 12*a*b^4*c^11 + 48
*a^2*b^2*c^12 - 64*a^3*c^13)))/(b^6*c^5 - 12*a*b^4*c^6 + 48*a^2*b^2*c^7 - 64*a^3
*c^8))*log(sqrt(1/2)*(27*B^3*b^10 + 144*(10*A^2*B*a^4 + A^3*a^3*b)*c^6 - 8*(500*
B^3*a^5 + 930*A*B^2*a^4*b + 252*A^2*B*a^3*b^2 + 11*A^3*a^2*b^3)*c^5 + (11360*B^3
*a^4*b^2 + 7608*A*B^2*a^3*b^3 + 882*A^2*B*a^2*b^4 + 17*A^3*a*b^5)*c^4 - (8818*B^
3*a^3*b^4 + 2841*A*B^2*a^2*b^5 + 153*A^2*B*a*b^6 + A^3*b^7)*c^3 + 9*(329*B^3*a^2
*b^6 + 51*A*B^2*a*b^7 + A^2*B*b^8)*c^2 - 27*(17*B^3*a*b^8 + A*B^2*b^9)*c - (3*B*
b^9*c^5 - 768*A*a^4*c^10 + 128*(8*B*a^4*b + 5*A*a^3*b^2)*c^9 - 192*(5*B*a^3*b^3
+ A*a^2*b^4)*c^8 + 24*(14*B*a^2*b^5 + A*a*b^6)*c^7 - (52*B*a*b^7 + A*b^8)*c^6)*s
qrt((81*B^4*b^8 + 81*A^4*a^2*c^6 - 18*(25*A^2*B^2*a^3 + 44*A^3*B*a^2*b + A^4*a*b
^2)*c^5 + (625*B^4*a^4 + 2200*A*B^3*a^3*b + 2904*A^2*B^2*a^2*b^2 + 196*A^3*B*a*b
^3 + A^4*b^4)*c^4 - 6*(425*B^4*a^3*b^2 + 798*A*B^3*a^2*b^3 + 132*A^2*B^2*a*b^4 +
 2*A^3*B*b^5)*c^3 + 27*(113*B^4*a^2*b^4 + 52*A*B^3*a*b^5 + 2*A^2*B^2*b^6)*c^2 -
54*(17*B^4*a*b^6 + 2*A*B^3*b^7)*c)/(b^6*c^10 - 12*a*b^4*c^11 + 48*a^2*b^2*c^12 -
 64*a^3*c^13)))*sqrt(-(9*B^2*b^7 + 60*(4*A*B*a^3 + A^2*a^2*b)*c^4 - 15*(28*B^2*a
^3*b + 20*A*B*a^2*b^2 + A^2*a*b^3)*c^3 + (385*B^2*a^2*b^3 + 80*A*B*a*b^4 + A^2*b
^5)*c^2 - 3*(35*B^2*a*b^5 + 2*A*B*b^6)*c + (b^6*c^5 - 12*a*b^4*c^6 + 48*a^2*b^2*
c^7 - 64*a^3*c^8)*sqrt((81*B^4*b^8 + 81*A^4*a^2*c^6 - 18*(25*A^2*B^2*a^3 + 44*A^
3*B*a^2*b + A^4*a*b^2)*c^5 + (625*B^4*a^4 + 2200*A*B^3*a^3*b + 2904*A^2*B^2*a^2*
b^2 + 196*A^3*B*a*b^3 + A^4*b^4)*c^4 - 6*(425*B^4*a^3*b^2 + 798*A*B^3*a^2*b^3 +
132*A^2*B^2*a*b^4 + 2*A^3*B*b^5)*c^3 + 27*(113*B^4*a^2*b^4 + 52*A*B^3*a*b^5 + 2*
A^2*B^2*b^6)*c^2 - 54*(17*B^4*a*b^6 + 2*A*B^3*b^7)*c)/(b^6*c^10 - 12*a*b^4*c^11
+ 48*a^2*b^2*c^12 - 64*a^3*c^13)))/(b^6*c^5 - 12*a*b^4*c^6 + 48*a^2*b^2*c^7 - 64
*a^3*c^8)) + 2*(189*B^4*a^2*b^6 - 135*A*B^3*a*b^7 + 324*A^4*a^3*c^5 - 81*(28*A^3
*B*a^3*b + A^4*a^2*b^2)*c^4 - (2500*B^4*a^5 + 2500*A*B^3*a^4*b - 5016*A^2*B^2*a^
3*b^2 - 647*A^3*B*a^2*b^3 - 5*A^4*a*b^4)*c^3 + 9*(625*B^4*a^4*b^2 - 303*A*B^3*a^
3*b^3 - 186*A^2*B^2*a^2*b^4 - 5*A^3*B*a*b^5)*c^2 - 27*(73*B^4*a^3*b^4 - 49*A*B^3
*a^2*b^5 - 5*A^2*B^2*a*b^6)*c)*sqrt(x)) - sqrt(1/2)*(a*b^2*c^2 - 4*a^2*c^3 + (b^
2*c^3 - 4*a*c^4)*x^2 + (b^3*c^2 - 4*a*b*c^3)*x)*sqrt(-(9*B^2*b^7 + 60*(4*A*B*a^3
 + A^2*a^2*b)*c^4 - 15*(28*B^2*a^3*b + 20*A*B*a^2*b^2 + A^2*a*b^3)*c^3 + (385*B^
2*a^2*b^3 + 80*A*B*a*b^4 + A^2*b^5)*c^2 - 3*(35*B^2*a*b^5 + 2*A*B*b^6)*c + (b^6*
c^5 - 12*a*b^4*c^6 + 48*a^2*b^2*c^7 - 64*a^3*c^8)*sqrt((81*B^4*b^8 + 81*A^4*a^2*
c^6 - 18*(25*A^2*B^2*a^3 + 44*A^3*B*a^2*b + A^4*a*b^2)*c^5 + (625*B^4*a^4 + 2200
*A*B^3*a^3*b + 2904*A^2*B^2*a^2*b^2 + 196*A^3*B*a*b^3 + A^4*b^4)*c^4 - 6*(425*B^
4*a^3*b^2 + 798*A*B^3*a^2*b^3 + 132*A^2*B^2*a*b^4 + 2*A^3*B*b^5)*c^3 + 27*(113*B
^4*a^2*b^4 + 52*A*B^3*a*b^5 + 2*A^2*B^2*b^6)*c^2 - 54*(17*B^4*a*b^6 + 2*A*B^3*b^
7)*c)/(b^6*c^10 - 12*a*b^4*c^11 + 48*a^2*b^2*c^12 - 64*a^3*c^13)))/(b^6*c^5 - 12
*a*b^4*c^6 + 48*a^2*b^2*c^7 - 64*a^3*c^8))*log(-sqrt(1/2)*(27*B^3*b^10 + 144*(10
*A^2*B*a^4 + A^3*a^3*b)*c^6 - 8*(500*B^3*a^5 + 930*A*B^2*a^4*b + 252*A^2*B*a^3*b
^2 + 11*A^3*a^2*b^3)*c^5 + (11360*B^3*a^4*b^2 + 7608*A*B^2*a^3*b^3 + 882*A^2*B*a
^2*b^4 + 17*A^3*a*b^5)*c^4 - (8818*B^3*a^3*b^4 + 2841*A*B^2*a^2*b^5 + 153*A^2*B*
a*b^6 + A^3*b^7)*c^3 + 9*(329*B^3*a^2*b^6 + 51*A*B^2*a*b^7 + A^2*B*b^8)*c^2 - 27
*(17*B^3*a*b^8 + A*B^2*b^9)*c - (3*B*b^9*c^5 - 768*A*a^4*c^10 + 128*(8*B*a^4*b +
 5*A*a^3*b^2)*c^9 - 192*(5*B*a^3*b^3 + A*a^2*b^4)*c^8 + 24*(14*B*a^2*b^5 + A*a*b
^6)*c^7 - (52*B*a*b^7 + A*b^8)*c^6)*sqrt((81*B^4*b^8 + 81*A^4*a^2*c^6 - 18*(25*A
^2*B^2*a^3 + 44*A^3*B*a^2*b + A^4*a*b^2)*c^5 + (625*B^4*a^4 + 2200*A*B^3*a^3*b +
 2904*A^2*B^2*a^2*b^2 + 196*A^3*B*a*b^3 + A^4*b^4)*c^4 - 6*(425*B^4*a^3*b^2 + 79
8*A*B^3*a^2*b^3 + 132*A^2*B^2*a*b^4 + 2*A^3*B*b^5)*c^3 + 27*(113*B^4*a^2*b^4 + 5
2*A*B^3*a*b^5 + 2*A^2*B^2*b^6)*c^2 - 54*(17*B^4*a*b^6 + 2*A*B^3*b^7)*c)/(b^6*c^1
0 - 12*a*b^4*c^11 + 48*a^2*b^2*c^12 - 64*a^3*c^13)))*sqrt(-(9*B^2*b^7 + 60*(4*A*
B*a^3 + A^2*a^2*b)*c^4 - 15*(28*B^2*a^3*b + 20*A*B*a^2*b^2 + A^2*a*b^3)*c^3 + (3
85*B^2*a^2*b^3 + 80*A*B*a*b^4 + A^2*b^5)*c^2 - 3*(35*B^2*a*b^5 + 2*A*B*b^6)*c +
(b^6*c^5 - 12*a*b^4*c^6 + 48*a^2*b^2*c^7 - 64*a^3*c^8)*sqrt((81*B^4*b^8 + 81*A^4
*a^2*c^6 - 18*(25*A^2*B^2*a^3 + 44*A^3*B*a^2*b + A^4*a*b^2)*c^5 + (625*B^4*a^4 +
 2200*A*B^3*a^3*b + 2904*A^2*B^2*a^2*b^2 + 196*A^3*B*a*b^3 + A^4*b^4)*c^4 - 6*(4
25*B^4*a^3*b^2 + 798*A*B^3*a^2*b^3 + 132*A^2*B^2*a*b^4 + 2*A^3*B*b^5)*c^3 + 27*(
113*B^4*a^2*b^4 + 52*A*B^3*a*b^5 + 2*A^2*B^2*b^6)*c^2 - 54*(17*B^4*a*b^6 + 2*A*B
^3*b^7)*c)/(b^6*c^10 - 12*a*b^4*c^11 + 48*a^2*b^2*c^12 - 64*a^3*c^13)))/(b^6*c^5
 - 12*a*b^4*c^6 + 48*a^2*b^2*c^7 - 64*a^3*c^8)) + 2*(189*B^4*a^2*b^6 - 135*A*B^3
*a*b^7 + 324*A^4*a^3*c^5 - 81*(28*A^3*B*a^3*b + A^4*a^2*b^2)*c^4 - (2500*B^4*a^5
 + 2500*A*B^3*a^4*b - 5016*A^2*B^2*a^3*b^2 - 647*A^3*B*a^2*b^3 - 5*A^4*a*b^4)*c^
3 + 9*(625*B^4*a^4*b^2 - 303*A*B^3*a^3*b^3 - 186*A^2*B^2*a^2*b^4 - 5*A^3*B*a*b^5
)*c^2 - 27*(73*B^4*a^3*b^4 - 49*A*B^3*a^2*b^5 - 5*A^2*B^2*a*b^6)*c)*sqrt(x)) + s
qrt(1/2)*(a*b^2*c^2 - 4*a^2*c^3 + (b^2*c^3 - 4*a*c^4)*x^2 + (b^3*c^2 - 4*a*b*c^3
)*x)*sqrt(-(9*B^2*b^7 + 60*(4*A*B*a^3 + A^2*a^2*b)*c^4 - 15*(28*B^2*a^3*b + 20*A
*B*a^2*b^2 + A^2*a*b^3)*c^3 + (385*B^2*a^2*b^3 + 80*A*B*a*b^4 + A^2*b^5)*c^2 - 3
*(35*B^2*a*b^5 + 2*A*B*b^6)*c - (b^6*c^5 - 12*a*b^4*c^6 + 48*a^2*b^2*c^7 - 64*a^
3*c^8)*sqrt((81*B^4*b^8 + 81*A^4*a^2*c^6 - 18*(25*A^2*B^2*a^3 + 44*A^3*B*a^2*b +
 A^4*a*b^2)*c^5 + (625*B^4*a^4 + 2200*A*B^3*a^3*b + 2904*A^2*B^2*a^2*b^2 + 196*A
^3*B*a*b^3 + A^4*b^4)*c^4 - 6*(425*B^4*a^3*b^2 + 798*A*B^3*a^2*b^3 + 132*A^2*B^2
*a*b^4 + 2*A^3*B*b^5)*c^3 + 27*(113*B^4*a^2*b^4 + 52*A*B^3*a*b^5 + 2*A^2*B^2*b^6
)*c^2 - 54*(17*B^4*a*b^6 + 2*A*B^3*b^7)*c)/(b^6*c^10 - 12*a*b^4*c^11 + 48*a^2*b^
2*c^12 - 64*a^3*c^13)))/(b^6*c^5 - 12*a*b^4*c^6 + 48*a^2*b^2*c^7 - 64*a^3*c^8))*
log(sqrt(1/2)*(27*B^3*b^10 + 144*(10*A^2*B*a^4 + A^3*a^3*b)*c^6 - 8*(500*B^3*a^5
 + 930*A*B^2*a^4*b + 252*A^2*B*a^3*b^2 + 11*A^3*a^2*b^3)*c^5 + (11360*B^3*a^4*b^
2 + 7608*A*B^2*a^3*b^3 + 882*A^2*B*a^2*b^4 + 17*A^3*a*b^5)*c^4 - (8818*B^3*a^3*b
^4 + 2841*A*B^2*a^2*b^5 + 153*A^2*B*a*b^6 + A^3*b^7)*c^3 + 9*(329*B^3*a^2*b^6 +
51*A*B^2*a*b^7 + A^2*B*b^8)*c^2 - 27*(17*B^3*a*b^8 + A*B^2*b^9)*c + (3*B*b^9*c^5
 - 768*A*a^4*c^10 + 128*(8*B*a^4*b + 5*A*a^3*b^2)*c^9 - 192*(5*B*a^3*b^3 + A*a^2
*b^4)*c^8 + 24*(14*B*a^2*b^5 + A*a*b^6)*c^7 - (52*B*a*b^7 + A*b^8)*c^6)*sqrt((81
*B^4*b^8 + 81*A^4*a^2*c^6 - 18*(25*A^2*B^2*a^3 + 44*A^3*B*a^2*b + A^4*a*b^2)*c^5
 + (625*B^4*a^4 + 2200*A*B^3*a^3*b + 2904*A^2*B^2*a^2*b^2 + 196*A^3*B*a*b^3 + A^
4*b^4)*c^4 - 6*(425*B^4*a^3*b^2 + 798*A*B^3*a^2*b^3 + 132*A^2*B^2*a*b^4 + 2*A^3*
B*b^5)*c^3 + 27*(113*B^4*a^2*b^4 + 52*A*B^3*a*b^5 + 2*A^2*B^2*b^6)*c^2 - 54*(17*
B^4*a*b^6 + 2*A*B^3*b^7)*c)/(b^6*c^10 - 12*a*b^4*c^11 + 48*a^2*b^2*c^12 - 64*a^3
*c^13)))*sqrt(-(9*B^2*b^7 + 60*(4*A*B*a^3 + A^2*a^2*b)*c^4 - 15*(28*B^2*a^3*b +
20*A*B*a^2*b^2 + A^2*a*b^3)*c^3 + (385*B^2*a^2*b^3 + 80*A*B*a*b^4 + A^2*b^5)*c^2
 - 3*(35*B^2*a*b^5 + 2*A*B*b^6)*c - (b^6*c^5 - 12*a*b^4*c^6 + 48*a^2*b^2*c^7 - 6
4*a^3*c^8)*sqrt((81*B^4*b^8 + 81*A^4*a^2*c^6 - 18*(25*A^2*B^2*a^3 + 44*A^3*B*a^2
*b + A^4*a*b^2)*c^5 + (625*B^4*a^4 + 2200*A*B^3*a^3*b + 2904*A^2*B^2*a^2*b^2 + 1
96*A^3*B*a*b^3 + A^4*b^4)*c^4 - 6*(425*B^4*a^3*b^2 + 798*A*B^3*a^2*b^3 + 132*A^2
*B^2*a*b^4 + 2*A^3*B*b^5)*c^3 + 27*(113*B^4*a^2*b^4 + 52*A*B^3*a*b^5 + 2*A^2*B^2
*b^6)*c^2 - 54*(17*B^4*a*b^6 + 2*A*B^3*b^7)*c)/(b^6*c^10 - 12*a*b^4*c^11 + 48*a^
2*b^2*c^12 - 64*a^3*c^13)))/(b^6*c^5 - 12*a*b^4*c^6 + 48*a^2*b^2*c^7 - 64*a^3*c^
8)) + 2*(189*B^4*a^2*b^6 - 135*A*B^3*a*b^7 + 324*A^4*a^3*c^5 - 81*(28*A^3*B*a^3*
b + A^4*a^2*b^2)*c^4 - (2500*B^4*a^5 + 2500*A*B^3*a^4*b - 5016*A^2*B^2*a^3*b^2 -
 647*A^3*B*a^2*b^3 - 5*A^4*a*b^4)*c^3 + 9*(625*B^4*a^4*b^2 - 303*A*B^3*a^3*b^3 -
 186*A^2*B^2*a^2*b^4 - 5*A^3*B*a*b^5)*c^2 - 27*(73*B^4*a^3*b^4 - 49*A*B^3*a^2*b^
5 - 5*A^2*B^2*a*b^6)*c)*sqrt(x)) - sqrt(1/2)*(a*b^2*c^2 - 4*a^2*c^3 + (b^2*c^3 -
 4*a*c^4)*x^2 + (b^3*c^2 - 4*a*b*c^3)*x)*sqrt(-(9*B^2*b^7 + 60*(4*A*B*a^3 + A^2*
a^2*b)*c^4 - 15*(28*B^2*a^3*b + 20*A*B*a^2*b^2 + A^2*a*b^3)*c^3 + (385*B^2*a^2*b
^3 + 80*A*B*a*b^4 + A^2*b^5)*c^2 - 3*(35*B^2*a*b^5 + 2*A*B*b^6)*c - (b^6*c^5 - 1
2*a*b^4*c^6 + 48*a^2*b^2*c^7 - 64*a^3*c^8)*sqrt((81*B^4*b^8 + 81*A^4*a^2*c^6 - 1
8*(25*A^2*B^2*a^3 + 44*A^3*B*a^2*b + A^4*a*b^2)*c^5 + (625*B^4*a^4 + 2200*A*B^3*
a^3*b + 2904*A^2*B^2*a^2*b^2 + 196*A^3*B*a*b^3 + A^4*b^4)*c^4 - 6*(425*B^4*a^3*b
^2 + 798*A*B^3*a^2*b^3 + 132*A^2*B^2*a*b^4 + 2*A^3*B*b^5)*c^3 + 27*(113*B^4*a^2*
b^4 + 52*A*B^3*a*b^5 + 2*A^2*B^2*b^6)*c^2 - 54*(17*B^4*a*b^6 + 2*A*B^3*b^7)*c)/(
b^6*c^10 - 12*a*b^4*c^11 + 48*a^2*b^2*c^12 - 64*a^3*c^13)))/(b^6*c^5 - 12*a*b^4*
c^6 + 48*a^2*b^2*c^7 - 64*a^3*c^8))*log(-sqrt(1/2)*(27*B^3*b^10 + 144*(10*A^2*B*
a^4 + A^3*a^3*b)*c^6 - 8*(500*B^3*a^5 + 930*A*B^2*a^4*b + 252*A^2*B*a^3*b^2 + 11
*A^3*a^2*b^3)*c^5 + (11360*B^3*a^4*b^2 + 7608*A*B^2*a^3*b^3 + 882*A^2*B*a^2*b^4
+ 17*A^3*a*b^5)*c^4 - (8818*B^3*a^3*b^4 + 2841*A*B^2*a^2*b^5 + 153*A^2*B*a*b^6 +
 A^3*b^7)*c^3 + 9*(329*B^3*a^2*b^6 + 51*A*B^2*a*b^7 + A^2*B*b^8)*c^2 - 27*(17*B^
3*a*b^8 + A*B^2*b^9)*c + (3*B*b^9*c^5 - 768*A*a^4*c^10 + 128*(8*B*a^4*b + 5*A*a^
3*b^2)*c^9 - 192*(5*B*a^3*b^3 + A*a^2*b^4)*c^8 + 24*(14*B*a^2*b^5 + A*a*b^6)*c^7
 - (52*B*a*b^7 + A*b^8)*c^6)*sqrt((81*B^4*b^8 + 81*A^4*a^2*c^6 - 18*(25*A^2*B^2*
a^3 + 44*A^3*B*a^2*b + A^4*a*b^2)*c^5 + (625*B^4*a^4 + 2200*A*B^3*a^3*b + 2904*A
^2*B^2*a^2*b^2 + 196*A^3*B*a*b^3 + A^4*b^4)*c^4 - 6*(425*B^4*a^3*b^2 + 798*A*B^3
*a^2*b^3 + 132*A^2*B^2*a*b^4 + 2*A^3*B*b^5)*c^3 + 27*(113*B^4*a^2*b^4 + 52*A*B^3
*a*b^5 + 2*A^2*B^2*b^6)*c^2 - 54*(17*B^4*a*b^6 + 2*A*B^3*b^7)*c)/(b^6*c^10 - 12*
a*b^4*c^11 + 48*a^2*b^2*c^12 - 64*a^3*c^13)))*sqrt(-(9*B^2*b^7 + 60*(4*A*B*a^3 +
 A^2*a^2*b)*c^4 - 15*(28*B^2*a^3*b + 20*A*B*a^2*b^2 + A^2*a*b^3)*c^3 + (385*B^2*
a^2*b^3 + 80*A*B*a*b^4 + A^2*b^5)*c^2 - 3*(35*B^2*a*b^5 + 2*A*B*b^6)*c - (b^6*c^
5 - 12*a*b^4*c^6 + 48*a^2*b^2*c^7 - 64*a^3*c^8)*sqrt((81*B^4*b^8 + 81*A^4*a^2*c^
6 - 18*(25*A^2*B^2*a^3 + 44*A^3*B*a^2*b + A^4*a*b^2)*c^5 + (625*B^4*a^4 + 2200*A
*B^3*a^3*b + 2904*A^2*B^2*a^2*b^2 + 196*A^3*B*a*b^3 + A^4*b^4)*c^4 - 6*(425*B^4*
a^3*b^2 + 798*A*B^3*a^2*b^3 + 132*A^2*B^2*a*b^4 + 2*A^3*B*b^5)*c^3 + 27*(113*B^4
*a^2*b^4 + 52*A*B^3*a*b^5 + 2*A^2*B^2*b^6)*c^2 - 54*(17*B^4*a*b^6 + 2*A*B^3*b^7)
*c)/(b^6*c^10 - 12*a*b^4*c^11 + 48*a^2*b^2*c^12 - 64*a^3*c^13)))/(b^6*c^5 - 12*a
*b^4*c^6 + 48*a^2*b^2*c^7 - 64*a^3*c^8)) + 2*(189*B^4*a^2*b^6 - 135*A*B^3*a*b^7
+ 324*A^4*a^3*c^5 - 81*(28*A^3*B*a^3*b + A^4*a^2*b^2)*c^4 - (2500*B^4*a^5 + 2500
*A*B^3*a^4*b - 5016*A^2*B^2*a^3*b^2 - 647*A^3*B*a^2*b^3 - 5*A^4*a*b^4)*c^3 + 9*(
625*B^4*a^4*b^2 - 303*A*B^3*a^3*b^3 - 186*A^2*B^2*a^2*b^4 - 5*A^3*B*a*b^5)*c^2 -
 27*(73*B^4*a^3*b^4 - 49*A*B^3*a^2*b^5 - 5*A^2*B^2*a*b^6)*c)*sqrt(x)) - 2*(3*B*a
*b^2 + 2*(B*b^2*c - 4*B*a*c^2)*x^2 - (10*B*a^2 + A*a*b)*c + (3*B*b^3 + 2*A*a*c^2
 - (11*B*a*b + A*b^2)*c)*x)*sqrt(x))/(a*b^2*c^2 - 4*a^2*c^3 + (b^2*c^3 - 4*a*c^4
)*x^2 + (b^3*c^2 - 4*a*b*c^3)*x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{\frac{5}{2}} \left (A + B x\right )}{\left (a + b x + c x^{2}\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**(5/2)*(B*x+A)/(c*x**2+b*x+a)**2,x)

[Out]

Integral(x**(5/2)*(A + B*x)/(a + b*x + c*x**2)**2, x)

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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((B*x + A)*x^(5/2)/(c*x^2 + b*x + a)^2,x, algorithm="giac")

[Out]

Timed out

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