3.1018 \(\int \frac{x^{3/2} (A+B x)}{\left (a+b x+c x^2\right )^2} \, dx\)

Optimal. Leaf size=321 \[ -\frac{\sqrt{x} \left (x \left (-2 a B c-A b c+b^2 B\right )+a (b B-2 A c)\right )}{c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac{\left (-\frac{4 a A c^2-8 a b B c+A b^2 c+b^3 B}{\sqrt{b^2-4 a c}}-6 a B c+A b c+b^2 B\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{x}}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{\sqrt{2} c^{3/2} \left (b^2-4 a c\right ) \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\left (\frac{4 a A c^2-8 a b B c+A b^2 c+b^3 B}{\sqrt{b^2-4 a c}}-6 a B c+A b c+b^2 B\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{x}}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{\sqrt{2} c^{3/2} \left (b^2-4 a c\right ) \sqrt{\sqrt{b^2-4 a c}+b}} \]

[Out]

-((Sqrt[x]*(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))) + ((b^2*B + A*b*c - 6*a*B*c - (b^3*B + A*b^2*c - 8*a*b*B*c + 4*
a*A*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^(3/2)*(b^2 - 4*a*c)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + ((b^2*B
+ A*b*c - 6*a*B*c + (b^3*B + A*b^2*c - 8*a*b*B*c + 4*a*A*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^(3/2
)*(b^2 - 4*a*c)*Sqrt[b + Sqrt[b^2 - 4*a*c]])

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Rubi [A]  time = 2.95573, antiderivative size = 321, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174 \[ -\frac{\sqrt{x} \left (x \left (-2 a B c-A b c+b^2 B\right )+a (b B-2 A c)\right )}{c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac{\left (-\frac{4 a A c^2-8 a b B c+A b^2 c+b^3 B}{\sqrt{b^2-4 a c}}-6 a B c+A b c+b^2 B\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{x}}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{\sqrt{2} c^{3/2} \left (b^2-4 a c\right ) \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\left (\frac{4 a A c^2-8 a b B c+A b^2 c+b^3 B}{\sqrt{b^2-4 a c}}-6 a B c+A b c+b^2 B\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{x}}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{\sqrt{2} c^{3/2} \left (b^2-4 a c\right ) \sqrt{\sqrt{b^2-4 a c}+b}} \]

Antiderivative was successfully verified.

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

[Out]

-((Sqrt[x]*(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))) + ((b^2*B + A*b*c - 6*a*B*c - (b^3*B + A*b^2*c - 8*a*b*B*c + 4*
a*A*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^(3/2)*(b^2 - 4*a*c)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + ((b^2*B
+ A*b*c - 6*a*B*c + (b^3*B + A*b^2*c - 8*a*b*B*c + 4*a*A*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^(3/2
)*(b^2 - 4*a*c)*Sqrt[b + Sqrt[b^2 - 4*a*c]])

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Rubi in Sympy [A]  time = 132.278, size = 318, normalized size = 0.99 \[ \frac{\sqrt{x} \left (a \left (2 A c - B b\right ) - x \left (- A b c - 2 B a c + B b^{2}\right )\right )}{c \left (- 4 a c + b^{2}\right ) \left (a + b x + c x^{2}\right )} + \frac{\sqrt{2} \left (2 a c \left (2 A c - B b\right ) + b \left (B b^{2} + c \left (A b - 6 B a\right )\right ) + \left (B b^{2} + c \left (A b - 6 B a\right )\right ) \sqrt{- 4 a c + b^{2}}\right ) \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt{c} \sqrt{x}}{\sqrt{b + \sqrt{- 4 a c + b^{2}}}} \right )}}{2 c^{\frac{3}{2}} \sqrt{b + \sqrt{- 4 a c + b^{2}}} \left (- 4 a c + b^{2}\right )^{\frac{3}{2}}} - \frac{\sqrt{2} \left (2 a c \left (2 A c - B b\right ) + b \left (B b^{2} + c \left (A b - 6 B a\right )\right ) - \left (B b^{2} + c \left (A b - 6 B a\right )\right ) \sqrt{- 4 a c + b^{2}}\right ) \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt{c} \sqrt{x}}{\sqrt{b - \sqrt{- 4 a c + b^{2}}}} \right )}}{2 c^{\frac{3}{2}} \sqrt{b - \sqrt{- 4 a c + b^{2}}} \left (- 4 a c + b^{2}\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(3/2)*(B*x+A)/(c*x**2+b*x+a)**2,x)

[Out]

sqrt(x)*(a*(2*A*c - B*b) - x*(-A*b*c - 2*B*a*c + B*b**2))/(c*(-4*a*c + b**2)*(a
+ b*x + c*x**2)) + sqrt(2)*(2*a*c*(2*A*c - B*b) + b*(B*b**2 + c*(A*b - 6*B*a)) +
(B*b**2 + c*(A*b - 6*B*a))*sqrt(-4*a*c + b**2))*atan(sqrt(2)*sqrt(c)*sqrt(x)/sq
rt(b + sqrt(-4*a*c + b**2)))/(2*c**(3/2)*sqrt(b + sqrt(-4*a*c + b**2))*(-4*a*c +
b**2)**(3/2)) - sqrt(2)*(2*a*c*(2*A*c - B*b) + b*(B*b**2 + c*(A*b - 6*B*a)) - (
B*b**2 + c*(A*b - 6*B*a))*sqrt(-4*a*c + b**2))*atan(sqrt(2)*sqrt(c)*sqrt(x)/sqrt
(b - sqrt(-4*a*c + b**2)))/(2*c**(3/2)*sqrt(b - sqrt(-4*a*c + b**2))*(-4*a*c + b
**2)**(3/2))

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Mathematica [A]  time = 1.8487, size = 366, normalized size = 1.14 \[ \frac{\frac{2 \sqrt{c} \sqrt{x} (2 a c (A+B x)-a b B+b x (A c-b B))}{\left (b^2-4 a c\right ) (a+x (b+c x))}+\frac{\sqrt{2} \left (b^2 \left (B \sqrt{b^2-4 a c}-A c\right )+b c \left (A \sqrt{b^2-4 a c}+8 a B\right )-2 a c \left (3 B \sqrt{b^2-4 a c}+2 A c\right )+b^3 (-B)\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{x}}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{\left (b^2-4 a c\right )^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\sqrt{2} \left (b^2 \left (B \sqrt{b^2-4 a c}+A c\right )+b \left (A c \sqrt{b^2-4 a c}-8 a B c\right )+2 a c \left (2 A c-3 B \sqrt{b^2-4 a c}\right )+b^3 B\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{x}}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{\left (b^2-4 a c\right )^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}}}{2 c^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

((2*Sqrt[c]*Sqrt[x]*(-(a*b*B) + b*(-(b*B) + A*c)*x + 2*a*c*(A + B*x)))/((b^2 - 4
*a*c)*(a + x*(b + c*x))) + (Sqrt[2]*(-(b^3*B) + b*c*(8*a*B + A*Sqrt[b^2 - 4*a*c]
) + b^2*(-(A*c) + B*Sqrt[b^2 - 4*a*c]) - 2*a*c*(2*A*c + 3*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]*(b^3*B + 2*a*c*(2*A*c - 3*B*Sqrt[b^2
- 4*a*c]) + b^2*(A*c + B*Sqrt[b^2 - 4*a*c]) + b*(-8*a*B*c + A*c*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]]))/(2*c^(3/2))

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Maple [B]  time = 0.14, size = 4063, normalized size = 12.7 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-32*c^4/(-c^2*(4*a*c-b^2)^3)^(1/2)/(4*a*c-b^2)*2^(1/2)/((4*a*c-b^2)*(4*a*b*c^2-b
^3*c+(-c^2*(4*a*c-b^2)^3)^(1/2)))^(1/2)*arctan(1/2*(8*a*c^3-2*b^2*c^2)*x^(1/2)*2
^(1/2)/c/((4*a*c-b^2)*(4*a*b*c^2-b^3*c+(-c^2*(4*a*c-b^2)^3)^(1/2)))^(1/2))*A*a^3
-1/2*c/(-c^2*(4*a*c-b^2)^3)^(1/2)/(4*a*c-b^2)*2^(1/2)/((4*a*c-b^2)*(4*a*b*c^2-b^
3*c+(-c^2*(4*a*c-b^2)^3)^(1/2)))^(1/2)*arctan(1/2*(8*a*c^3-2*b^2*c^2)*x^(1/2)*2^
(1/2)/c/((4*a*c-b^2)*(4*a*b*c^2-b^3*c+(-c^2*(4*a*c-b^2)^3)^(1/2)))^(1/2))*A*b^6-
2*c/(4*a*c-b^2)*2^(1/2)/((4*a*c-b^2)*(4*a*b*c^2-b^3*c+(-c^2*(4*a*c-b^2)^3)^(1/2)
))^(1/2)*arctan(1/2*(8*a*c^3-2*b^2*c^2)*x^(1/2)*2^(1/2)/c/((4*a*c-b^2)*(4*a*b*c^
2-b^3*c+(-c^2*(4*a*c-b^2)^3)^(1/2)))^(1/2))*A*b*a+32*c^4/(-c^2*(4*a*c-b^2)^3)^(1
/2)/(4*a*c-b^2)*2^(1/2)/((-4*a*b*c^2+b^3*c+(-c^2*(4*a*c-b^2)^3)^(1/2))*(4*a*c-b^
2))^(1/2)*arctanh(1/2*(-8*a*c^3+2*b^2*c^2)*x^(1/2)*2^(1/2)/c/((-4*a*b*c^2+b^3*c+
(-c^2*(4*a*c-b^2)^3)^(1/2))*(4*a*c-b^2))^(1/2))*A*a^3+1/2*c/(-c^2*(4*a*c-b^2)^3)
^(1/2)/(4*a*c-b^2)*2^(1/2)/((-4*a*b*c^2+b^3*c+(-c^2*(4*a*c-b^2)^3)^(1/2))*(4*a*c
-b^2))^(1/2)*arctanh(1/2*(-8*a*c^3+2*b^2*c^2)*x^(1/2)*2^(1/2)/c/((-4*a*b*c^2+b^3
*c+(-c^2*(4*a*c-b^2)^3)^(1/2))*(4*a*c-b^2))^(1/2))*A*b^6+1/2/(4*a*c-b^2)*2^(1/2)
/((4*a*c-b^2)*(4*a*b*c^2-b^3*c+(-c^2*(4*a*c-b^2)^3)^(1/2)))^(1/2)*arctan(1/2*(8*
a*c^3-2*b^2*c^2)*x^(1/2)*2^(1/2)/c/((4*a*c-b^2)*(4*a*b*c^2-b^3*c+(-c^2*(4*a*c-b^
2)^3)^(1/2)))^(1/2))*A*b^3+1/2/(4*a*c-b^2)*2^(1/2)/((-4*a*b*c^2+b^3*c+(-c^2*(4*a
*c-b^2)^3)^(1/2))*(4*a*c-b^2))^(1/2)*arctanh(1/2*(-8*a*c^3+2*b^2*c^2)*x^(1/2)*2^
(1/2)/c/((-4*a*b*c^2+b^3*c+(-c^2*(4*a*c-b^2)^3)^(1/2))*(4*a*c-b^2))^(1/2))*A*b^3
+64*c^3/(-c^2*(4*a*c-b^2)^3)^(1/2)/(4*a*c-b^2)*2^(1/2)/((4*a*c-b^2)*(4*a*b*c^2-b
^3*c+(-c^2*(4*a*c-b^2)^3)^(1/2)))^(1/2)*arctan(1/2*(8*a*c^3-2*b^2*c^2)*x^(1/2)*2
^(1/2)/c/((4*a*c-b^2)*(4*a*b*c^2-b^3*c+(-c^2*(4*a*c-b^2)^3)^(1/2)))^(1/2))*B*a^3
*b+40*c^2/(-c^2*(4*a*c-b^2)^3)^(1/2)/(4*a*c-b^2)*2^(1/2)/((-4*a*b*c^2+b^3*c+(-c^
2*(4*a*c-b^2)^3)^(1/2))*(4*a*c-b^2))^(1/2)*arctanh(1/2*(-8*a*c^3+2*b^2*c^2)*x^(1
/2)*2^(1/2)/c/((-4*a*b*c^2+b^3*c+(-c^2*(4*a*c-b^2)^3)^(1/2))*(4*a*c-b^2))^(1/2))
*B*a^2*b^3-40*c^2/(-c^2*(4*a*c-b^2)^3)^(1/2)/(4*a*c-b^2)*2^(1/2)/((4*a*c-b^2)*(4
*a*b*c^2-b^3*c+(-c^2*(4*a*c-b^2)^3)^(1/2)))^(1/2)*arctan(1/2*(8*a*c^3-2*b^2*c^2)
*x^(1/2)*2^(1/2)/c/((4*a*c-b^2)*(4*a*b*c^2-b^3*c+(-c^2*(4*a*c-b^2)^3)^(1/2)))^(1
/2))*B*a^2*b^3+8*c/(-c^2*(4*a*c-b^2)^3)^(1/2)/(4*a*c-b^2)*2^(1/2)/((4*a*c-b^2)*(
4*a*b*c^2-b^3*c+(-c^2*(4*a*c-b^2)^3)^(1/2)))^(1/2)*arctan(1/2*(8*a*c^3-2*b^2*c^2
)*x^(1/2)*2^(1/2)/c/((4*a*c-b^2)*(4*a*b*c^2-b^3*c+(-c^2*(4*a*c-b^2)^3)^(1/2)))^(
1/2))*B*a*b^5-8*c^3/(-c^2*(4*a*c-b^2)^3)^(1/2)/(4*a*c-b^2)*2^(1/2)/((-4*a*b*c^2+
b^3*c+(-c^2*(4*a*c-b^2)^3)^(1/2))*(4*a*c-b^2))^(1/2)*arctanh(1/2*(-8*a*c^3+2*b^2
*c^2)*x^(1/2)*2^(1/2)/c/((-4*a*b*c^2+b^3*c+(-c^2*(4*a*c-b^2)^3)^(1/2))*(4*a*c-b^
2))^(1/2))*A*a^2*b^2-2*c^2/(-c^2*(4*a*c-b^2)^3)^(1/2)/(4*a*c-b^2)*2^(1/2)/((-4*a
*b*c^2+b^3*c+(-c^2*(4*a*c-b^2)^3)^(1/2))*(4*a*c-b^2))^(1/2)*arctanh(1/2*(-8*a*c^
3+2*b^2*c^2)*x^(1/2)*2^(1/2)/c/((-4*a*b*c^2+b^3*c+(-c^2*(4*a*c-b^2)^3)^(1/2))*(4
*a*c-b^2))^(1/2))*A*b^4*a-64*c^3/(-c^2*(4*a*c-b^2)^3)^(1/2)/(4*a*c-b^2)*2^(1/2)/
((-4*a*b*c^2+b^3*c+(-c^2*(4*a*c-b^2)^3)^(1/2))*(4*a*c-b^2))^(1/2)*arctanh(1/2*(-
8*a*c^3+2*b^2*c^2)*x^(1/2)*2^(1/2)/c/((-4*a*b*c^2+b^3*c+(-c^2*(4*a*c-b^2)^3)^(1/
2))*(4*a*c-b^2))^(1/2))*B*a^3*b-8*c/(-c^2*(4*a*c-b^2)^3)^(1/2)/(4*a*c-b^2)*2^(1/
2)/((-4*a*b*c^2+b^3*c+(-c^2*(4*a*c-b^2)^3)^(1/2))*(4*a*c-b^2))^(1/2)*arctanh(1/2
*(-8*a*c^3+2*b^2*c^2)*x^(1/2)*2^(1/2)/c/((-4*a*b*c^2+b^3*c+(-c^2*(4*a*c-b^2)^3)^
(1/2))*(4*a*c-b^2))^(1/2))*B*a*b^5+8*c^3/(-c^2*(4*a*c-b^2)^3)^(1/2)/(4*a*c-b^2)*
2^(1/2)/((4*a*c-b^2)*(4*a*b*c^2-b^3*c+(-c^2*(4*a*c-b^2)^3)^(1/2)))^(1/2)*arctan(
1/2*(8*a*c^3-2*b^2*c^2)*x^(1/2)*2^(1/2)/c/((4*a*c-b^2)*(4*a*b*c^2-b^3*c+(-c^2*(4
*a*c-b^2)^3)^(1/2)))^(1/2))*A*a^2*b^2+2*c^2/(-c^2*(4*a*c-b^2)^3)^(1/2)/(4*a*c-b^
2)*2^(1/2)/((4*a*c-b^2)*(4*a*b*c^2-b^3*c+(-c^2*(4*a*c-b^2)^3)^(1/2)))^(1/2)*arct
an(1/2*(8*a*c^3-2*b^2*c^2)*x^(1/2)*2^(1/2)/c/((4*a*c-b^2)*(4*a*b*c^2-b^3*c+(-c^2
*(4*a*c-b^2)^3)^(1/2)))^(1/2))*A*b^4*a+1/2/(-c^2*(4*a*c-b^2)^3)^(1/2)/(4*a*c-b^2
)*2^(1/2)/((-4*a*b*c^2+b^3*c+(-c^2*(4*a*c-b^2)^3)^(1/2))*(4*a*c-b^2))^(1/2)*arct
anh(1/2*(-8*a*c^3+2*b^2*c^2)*x^(1/2)*2^(1/2)/c/((-4*a*b*c^2+b^3*c+(-c^2*(4*a*c-b
^2)^3)^(1/2))*(4*a*c-b^2))^(1/2))*B*b^7-5/(4*a*c-b^2)*2^(1/2)/((-4*a*b*c^2+b^3*c
+(-c^2*(4*a*c-b^2)^3)^(1/2))*(4*a*c-b^2))^(1/2)*arctanh(1/2*(-8*a*c^3+2*b^2*c^2)
*x^(1/2)*2^(1/2)/c/((-4*a*b*c^2+b^3*c+(-c^2*(4*a*c-b^2)^3)^(1/2))*(4*a*c-b^2))^(
1/2))*B*a*b^2-1/2/(-c^2*(4*a*c-b^2)^3)^(1/2)/(4*a*c-b^2)*2^(1/2)/((4*a*c-b^2)*(4
*a*b*c^2-b^3*c+(-c^2*(4*a*c-b^2)^3)^(1/2)))^(1/2)*arctan(1/2*(8*a*c^3-2*b^2*c^2)
*x^(1/2)*2^(1/2)/c/((4*a*c-b^2)*(4*a*b*c^2-b^3*c+(-c^2*(4*a*c-b^2)^3)^(1/2)))^(1
/2))*B*b^7-5/(4*a*c-b^2)*2^(1/2)/((4*a*c-b^2)*(4*a*b*c^2-b^3*c+(-c^2*(4*a*c-b^2)
^3)^(1/2)))^(1/2)*arctan(1/2*(8*a*c^3-2*b^2*c^2)*x^(1/2)*2^(1/2)/c/((4*a*c-b^2)*
(4*a*b*c^2-b^3*c+(-c^2*(4*a*c-b^2)^3)^(1/2)))^(1/2))*B*a*b^2+12*c/(4*a*c-b^2)*2^
(1/2)/((4*a*c-b^2)*(4*a*b*c^2-b^3*c+(-c^2*(4*a*c-b^2)^3)^(1/2)))^(1/2)*arctan(1/
2*(8*a*c^3-2*b^2*c^2)*x^(1/2)*2^(1/2)/c/((4*a*c-b^2)*(4*a*b*c^2-b^3*c+(-c^2*(4*a
*c-b^2)^3)^(1/2)))^(1/2))*B*a^2+1/2/c/(4*a*c-b^2)*2^(1/2)/((4*a*c-b^2)*(4*a*b*c^
2-b^3*c+(-c^2*(4*a*c-b^2)^3)^(1/2)))^(1/2)*arctan(1/2*(8*a*c^3-2*b^2*c^2)*x^(1/2
)*2^(1/2)/c/((4*a*c-b^2)*(4*a*b*c^2-b^3*c+(-c^2*(4*a*c-b^2)^3)^(1/2)))^(1/2))*B*
b^4+12*c/(4*a*c-b^2)*2^(1/2)/((-4*a*b*c^2+b^3*c+(-c^2*(4*a*c-b^2)^3)^(1/2))*(4*a
*c-b^2))^(1/2)*arctanh(1/2*(-8*a*c^3+2*b^2*c^2)*x^(1/2)*2^(1/2)/c/((-4*a*b*c^2+b
^3*c+(-c^2*(4*a*c-b^2)^3)^(1/2))*(4*a*c-b^2))^(1/2))*B*a^2+1/2/c/(4*a*c-b^2)*2^(
1/2)/((-4*a*b*c^2+b^3*c+(-c^2*(4*a*c-b^2)^3)^(1/2))*(4*a*c-b^2))^(1/2)*arctanh(1
/2*(-8*a*c^3+2*b^2*c^2)*x^(1/2)*2^(1/2)/c/((-4*a*b*c^2+b^3*c+(-c^2*(4*a*c-b^2)^3
)^(1/2))*(4*a*c-b^2))^(1/2))*B*b^4-2*c/(4*a*c-b^2)*2^(1/2)/((-4*a*b*c^2+b^3*c+(-
c^2*(4*a*c-b^2)^3)^(1/2))*(4*a*c-b^2))^(1/2)*arctanh(1/2*(-8*a*c^3+2*b^2*c^2)*x^
(1/2)*2^(1/2)/c/((-4*a*b*c^2+b^3*c+(-c^2*(4*a*c-b^2)^3)^(1/2))*(4*a*c-b^2))^(1/2
))*A*b*a+2*(-1/2*(A*b*c+2*B*a*c-B*b^2)/c/(4*a*c-b^2)*x^(3/2)-1/2*a*(2*A*c-B*b)/c
/(4*a*c-b^2)*x^(1/2))/(c*x^2+b*x+a)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 2.46501, size = 6282, normalized size = 19.57 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*(sqrt(1/2)*(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)*sqrt(-(B^2*b^5 - 12*(4*A*B*a^2 - A^2*a*b)*c^3 + (60*B^2*a^2*b - 12*A*B*a
*b^2 + A^2*b^3)*c^2 - (15*B^2*a*b^3 - 2*A*B*b^4)*c + (b^6*c^3 - 12*a*b^4*c^4 + 4
8*a^2*b^2*c^5 - 64*a^3*c^6)*sqrt((B^4*b^4 + A^4*c^4 - 2*(9*A^2*B^2*a - 2*A^3*B*b
)*c^3 + 3*(27*B^4*a^2 - 12*A*B^3*a*b + 2*A^2*B^2*b^2)*c^2 - 2*(9*B^4*a*b^2 - 2*A
*B^3*b^3)*c)/(b^6*c^6 - 12*a*b^4*c^7 + 48*a^2*b^2*c^8 - 64*a^3*c^9)))/(b^6*c^3 -
12*a*b^4*c^4 + 48*a^2*b^2*c^5 - 64*a^3*c^6))*log(sqrt(1/2)*(B^3*b^7 - 17*B^3*a*
b^5*c - 32*A^3*a^2*c^5 + 16*(18*A*B^2*a^3 - 3*A^2*B*a^2*b + A^3*a*b^2)*c^4 - 2*(
72*B^3*a^3*b + 72*A*B^2*a^2*b^2 - 12*A^2*B*a*b^3 + A^3*b^4)*c^3 + (88*B^3*a^2*b^
3 + 18*A*B^2*a*b^4 - 3*A^2*B*b^5)*c^2 - (B*b^8*c^3 + 256*(3*B*a^4 - A*a^3*b)*c^7
- 64*(10*B*a^3*b^2 - 3*A*a^2*b^3)*c^6 + 48*(4*B*a^2*b^4 - A*a*b^5)*c^5 - 4*(6*B
*a*b^6 - A*b^7)*c^4)*sqrt((B^4*b^4 + A^4*c^4 - 2*(9*A^2*B^2*a - 2*A^3*B*b)*c^3 +
3*(27*B^4*a^2 - 12*A*B^3*a*b + 2*A^2*B^2*b^2)*c^2 - 2*(9*B^4*a*b^2 - 2*A*B^3*b^
3)*c)/(b^6*c^6 - 12*a*b^4*c^7 + 48*a^2*b^2*c^8 - 64*a^3*c^9)))*sqrt(-(B^2*b^5 -
12*(4*A*B*a^2 - A^2*a*b)*c^3 + (60*B^2*a^2*b - 12*A*B*a*b^2 + A^2*b^3)*c^2 - (15
*B^2*a*b^3 - 2*A*B*b^4)*c + (b^6*c^3 - 12*a*b^4*c^4 + 48*a^2*b^2*c^5 - 64*a^3*c^
6)*sqrt((B^4*b^4 + A^4*c^4 - 2*(9*A^2*B^2*a - 2*A^3*B*b)*c^3 + 3*(27*B^4*a^2 - 1
2*A*B^3*a*b + 2*A^2*B^2*b^2)*c^2 - 2*(9*B^4*a*b^2 - 2*A*B^3*b^3)*c)/(b^6*c^6 - 1
2*a*b^4*c^7 + 48*a^2*b^2*c^8 - 64*a^3*c^9)))/(b^6*c^3 - 12*a*b^4*c^4 + 48*a^2*b^
2*c^5 - 64*a^3*c^6)) - 2*(5*B^4*a*b^4 - 3*A*B^3*b^5 - 4*A^4*a*c^4 + (20*A^3*B*a*
b - 3*A^4*b^2)*c^3 + 3*(108*B^4*a^3 - 108*A*B^3*a^2*b + 28*A^2*B^2*a*b^2 - 3*A^3
*B*b^3)*c^2 - (81*B^4*a^2*b^2 - 65*A*B^3*a*b^3 + 9*A^2*B^2*b^4)*c)*sqrt(x)) - sq
rt(1/2)*(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)*
sqrt(-(B^2*b^5 - 12*(4*A*B*a^2 - A^2*a*b)*c^3 + (60*B^2*a^2*b - 12*A*B*a*b^2 + A
^2*b^3)*c^2 - (15*B^2*a*b^3 - 2*A*B*b^4)*c + (b^6*c^3 - 12*a*b^4*c^4 + 48*a^2*b^
2*c^5 - 64*a^3*c^6)*sqrt((B^4*b^4 + A^4*c^4 - 2*(9*A^2*B^2*a - 2*A^3*B*b)*c^3 +
3*(27*B^4*a^2 - 12*A*B^3*a*b + 2*A^2*B^2*b^2)*c^2 - 2*(9*B^4*a*b^2 - 2*A*B^3*b^3
)*c)/(b^6*c^6 - 12*a*b^4*c^7 + 48*a^2*b^2*c^8 - 64*a^3*c^9)))/(b^6*c^3 - 12*a*b^
4*c^4 + 48*a^2*b^2*c^5 - 64*a^3*c^6))*log(-sqrt(1/2)*(B^3*b^7 - 17*B^3*a*b^5*c -
32*A^3*a^2*c^5 + 16*(18*A*B^2*a^3 - 3*A^2*B*a^2*b + A^3*a*b^2)*c^4 - 2*(72*B^3*
a^3*b + 72*A*B^2*a^2*b^2 - 12*A^2*B*a*b^3 + A^3*b^4)*c^3 + (88*B^3*a^2*b^3 + 18*
A*B^2*a*b^4 - 3*A^2*B*b^5)*c^2 - (B*b^8*c^3 + 256*(3*B*a^4 - A*a^3*b)*c^7 - 64*(
10*B*a^3*b^2 - 3*A*a^2*b^3)*c^6 + 48*(4*B*a^2*b^4 - A*a*b^5)*c^5 - 4*(6*B*a*b^6
- A*b^7)*c^4)*sqrt((B^4*b^4 + A^4*c^4 - 2*(9*A^2*B^2*a - 2*A^3*B*b)*c^3 + 3*(27*
B^4*a^2 - 12*A*B^3*a*b + 2*A^2*B^2*b^2)*c^2 - 2*(9*B^4*a*b^2 - 2*A*B^3*b^3)*c)/(
b^6*c^6 - 12*a*b^4*c^7 + 48*a^2*b^2*c^8 - 64*a^3*c^9)))*sqrt(-(B^2*b^5 - 12*(4*A
*B*a^2 - A^2*a*b)*c^3 + (60*B^2*a^2*b - 12*A*B*a*b^2 + A^2*b^3)*c^2 - (15*B^2*a*
b^3 - 2*A*B*b^4)*c + (b^6*c^3 - 12*a*b^4*c^4 + 48*a^2*b^2*c^5 - 64*a^3*c^6)*sqrt
((B^4*b^4 + A^4*c^4 - 2*(9*A^2*B^2*a - 2*A^3*B*b)*c^3 + 3*(27*B^4*a^2 - 12*A*B^3
*a*b + 2*A^2*B^2*b^2)*c^2 - 2*(9*B^4*a*b^2 - 2*A*B^3*b^3)*c)/(b^6*c^6 - 12*a*b^4
*c^7 + 48*a^2*b^2*c^8 - 64*a^3*c^9)))/(b^6*c^3 - 12*a*b^4*c^4 + 48*a^2*b^2*c^5 -
64*a^3*c^6)) - 2*(5*B^4*a*b^4 - 3*A*B^3*b^5 - 4*A^4*a*c^4 + (20*A^3*B*a*b - 3*A
^4*b^2)*c^3 + 3*(108*B^4*a^3 - 108*A*B^3*a^2*b + 28*A^2*B^2*a*b^2 - 3*A^3*B*b^3)
*c^2 - (81*B^4*a^2*b^2 - 65*A*B^3*a*b^3 + 9*A^2*B^2*b^4)*c)*sqrt(x)) + sqrt(1/2)
*(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)*sqrt(-(
B^2*b^5 - 12*(4*A*B*a^2 - A^2*a*b)*c^3 + (60*B^2*a^2*b - 12*A*B*a*b^2 + A^2*b^3)
*c^2 - (15*B^2*a*b^3 - 2*A*B*b^4)*c - (b^6*c^3 - 12*a*b^4*c^4 + 48*a^2*b^2*c^5 -
64*a^3*c^6)*sqrt((B^4*b^4 + A^4*c^4 - 2*(9*A^2*B^2*a - 2*A^3*B*b)*c^3 + 3*(27*B
^4*a^2 - 12*A*B^3*a*b + 2*A^2*B^2*b^2)*c^2 - 2*(9*B^4*a*b^2 - 2*A*B^3*b^3)*c)/(b
^6*c^6 - 12*a*b^4*c^7 + 48*a^2*b^2*c^8 - 64*a^3*c^9)))/(b^6*c^3 - 12*a*b^4*c^4 +
48*a^2*b^2*c^5 - 64*a^3*c^6))*log(sqrt(1/2)*(B^3*b^7 - 17*B^3*a*b^5*c - 32*A^3*
a^2*c^5 + 16*(18*A*B^2*a^3 - 3*A^2*B*a^2*b + A^3*a*b^2)*c^4 - 2*(72*B^3*a^3*b +
72*A*B^2*a^2*b^2 - 12*A^2*B*a*b^3 + A^3*b^4)*c^3 + (88*B^3*a^2*b^3 + 18*A*B^2*a*
b^4 - 3*A^2*B*b^5)*c^2 + (B*b^8*c^3 + 256*(3*B*a^4 - A*a^3*b)*c^7 - 64*(10*B*a^3
*b^2 - 3*A*a^2*b^3)*c^6 + 48*(4*B*a^2*b^4 - A*a*b^5)*c^5 - 4*(6*B*a*b^6 - A*b^7)
*c^4)*sqrt((B^4*b^4 + A^4*c^4 - 2*(9*A^2*B^2*a - 2*A^3*B*b)*c^3 + 3*(27*B^4*a^2
- 12*A*B^3*a*b + 2*A^2*B^2*b^2)*c^2 - 2*(9*B^4*a*b^2 - 2*A*B^3*b^3)*c)/(b^6*c^6
- 12*a*b^4*c^7 + 48*a^2*b^2*c^8 - 64*a^3*c^9)))*sqrt(-(B^2*b^5 - 12*(4*A*B*a^2 -
A^2*a*b)*c^3 + (60*B^2*a^2*b - 12*A*B*a*b^2 + A^2*b^3)*c^2 - (15*B^2*a*b^3 - 2*
A*B*b^4)*c - (b^6*c^3 - 12*a*b^4*c^4 + 48*a^2*b^2*c^5 - 64*a^3*c^6)*sqrt((B^4*b^
4 + A^4*c^4 - 2*(9*A^2*B^2*a - 2*A^3*B*b)*c^3 + 3*(27*B^4*a^2 - 12*A*B^3*a*b + 2
*A^2*B^2*b^2)*c^2 - 2*(9*B^4*a*b^2 - 2*A*B^3*b^3)*c)/(b^6*c^6 - 12*a*b^4*c^7 + 4
8*a^2*b^2*c^8 - 64*a^3*c^9)))/(b^6*c^3 - 12*a*b^4*c^4 + 48*a^2*b^2*c^5 - 64*a^3*
c^6)) - 2*(5*B^4*a*b^4 - 3*A*B^3*b^5 - 4*A^4*a*c^4 + (20*A^3*B*a*b - 3*A^4*b^2)*
c^3 + 3*(108*B^4*a^3 - 108*A*B^3*a^2*b + 28*A^2*B^2*a*b^2 - 3*A^3*B*b^3)*c^2 - (
81*B^4*a^2*b^2 - 65*A*B^3*a*b^3 + 9*A^2*B^2*b^4)*c)*sqrt(x)) - sqrt(1/2)*(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)*sqrt(-(B^2*b^5
- 12*(4*A*B*a^2 - A^2*a*b)*c^3 + (60*B^2*a^2*b - 12*A*B*a*b^2 + A^2*b^3)*c^2 - (
15*B^2*a*b^3 - 2*A*B*b^4)*c - (b^6*c^3 - 12*a*b^4*c^4 + 48*a^2*b^2*c^5 - 64*a^3*
c^6)*sqrt((B^4*b^4 + A^4*c^4 - 2*(9*A^2*B^2*a - 2*A^3*B*b)*c^3 + 3*(27*B^4*a^2 -
12*A*B^3*a*b + 2*A^2*B^2*b^2)*c^2 - 2*(9*B^4*a*b^2 - 2*A*B^3*b^3)*c)/(b^6*c^6 -
12*a*b^4*c^7 + 48*a^2*b^2*c^8 - 64*a^3*c^9)))/(b^6*c^3 - 12*a*b^4*c^4 + 48*a^2*
b^2*c^5 - 64*a^3*c^6))*log(-sqrt(1/2)*(B^3*b^7 - 17*B^3*a*b^5*c - 32*A^3*a^2*c^5
+ 16*(18*A*B^2*a^3 - 3*A^2*B*a^2*b + A^3*a*b^2)*c^4 - 2*(72*B^3*a^3*b + 72*A*B^
2*a^2*b^2 - 12*A^2*B*a*b^3 + A^3*b^4)*c^3 + (88*B^3*a^2*b^3 + 18*A*B^2*a*b^4 - 3
*A^2*B*b^5)*c^2 + (B*b^8*c^3 + 256*(3*B*a^4 - A*a^3*b)*c^7 - 64*(10*B*a^3*b^2 -
3*A*a^2*b^3)*c^6 + 48*(4*B*a^2*b^4 - A*a*b^5)*c^5 - 4*(6*B*a*b^6 - A*b^7)*c^4)*s
qrt((B^4*b^4 + A^4*c^4 - 2*(9*A^2*B^2*a - 2*A^3*B*b)*c^3 + 3*(27*B^4*a^2 - 12*A*
B^3*a*b + 2*A^2*B^2*b^2)*c^2 - 2*(9*B^4*a*b^2 - 2*A*B^3*b^3)*c)/(b^6*c^6 - 12*a*
b^4*c^7 + 48*a^2*b^2*c^8 - 64*a^3*c^9)))*sqrt(-(B^2*b^5 - 12*(4*A*B*a^2 - A^2*a*
b)*c^3 + (60*B^2*a^2*b - 12*A*B*a*b^2 + A^2*b^3)*c^2 - (15*B^2*a*b^3 - 2*A*B*b^4
)*c - (b^6*c^3 - 12*a*b^4*c^4 + 48*a^2*b^2*c^5 - 64*a^3*c^6)*sqrt((B^4*b^4 + A^4
*c^4 - 2*(9*A^2*B^2*a - 2*A^3*B*b)*c^3 + 3*(27*B^4*a^2 - 12*A*B^3*a*b + 2*A^2*B^
2*b^2)*c^2 - 2*(9*B^4*a*b^2 - 2*A*B^3*b^3)*c)/(b^6*c^6 - 12*a*b^4*c^7 + 48*a^2*b
^2*c^8 - 64*a^3*c^9)))/(b^6*c^3 - 12*a*b^4*c^4 + 48*a^2*b^2*c^5 - 64*a^3*c^6)) -
2*(5*B^4*a*b^4 - 3*A*B^3*b^5 - 4*A^4*a*c^4 + (20*A^3*B*a*b - 3*A^4*b^2)*c^3 + 3
*(108*B^4*a^3 - 108*A*B^3*a^2*b + 28*A^2*B^2*a*b^2 - 3*A^3*B*b^3)*c^2 - (81*B^4*
a^2*b^2 - 65*A*B^3*a*b^3 + 9*A^2*B^2*b^4)*c)*sqrt(x)) + 2*(B*a*b - 2*A*a*c + (B*
b^2 - (2*B*a + A*b)*c)*x)*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)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**(3/2)*(B*x+A)/(c*x**2+b*x+a)**2,x)

[Out]

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

[Out]

Timed out