3.1019 \(\int \frac{\sqrt{x} (A+B x)}{\left (a+b x+c x^2\right )^2} \, dx\)

Optimal. Leaf size=276 \[ -\frac{\sqrt{x} (-2 a B-x (b B-2 A c)+A b)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac{\left (-\frac{4 a B c-4 A b c+b^2 B}{\sqrt{b^2-4 a c}}-2 A c+b B\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{x}}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{\sqrt{2} \sqrt{c} \left (b^2-4 a c\right ) \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\left (\frac{4 a B c-4 A b c+b^2 B}{\sqrt{b^2-4 a c}}-2 A c+b B\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{x}}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{\sqrt{2} \sqrt{c} \left (b^2-4 a c\right ) \sqrt{\sqrt{b^2-4 a c}+b}} \]

[Out]

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

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Rubi [A]  time = 1.84965, antiderivative size = 276, 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 \[ -\frac{\sqrt{x} (-2 a B-x (b B-2 A c)+A b)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac{\left (-\frac{4 a B c-4 A b c+b^2 B}{\sqrt{b^2-4 a c}}-2 A c+b B\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{x}}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{\sqrt{2} \sqrt{c} \left (b^2-4 a c\right ) \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\left (\frac{4 a B c-4 A b c+b^2 B}{\sqrt{b^2-4 a c}}-2 A c+b B\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{x}}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{\sqrt{2} \sqrt{c} \left (b^2-4 a c\right ) \sqrt{\sqrt{b^2-4 a c}+b}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 159.808, size = 274, normalized size = 0.99 \[ - \frac{\sqrt{x} \left (A b - 2 B a + x \left (2 A c - B b\right )\right )}{\left (- 4 a c + b^{2}\right ) \left (a + b x + c x^{2}\right )} - \frac{\sqrt{2} \left (b \left (2 A c - B b\right ) + 2 c \left (A b - 2 B a\right ) + \left (2 A c - B b\right ) \sqrt{- 4 a c + b^{2}}\right ) \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt{c} \sqrt{x}}{\sqrt{b + \sqrt{- 4 a c + b^{2}}}} \right )}}{2 \sqrt{c} \sqrt{b + \sqrt{- 4 a c + b^{2}}} \left (- 4 a c + b^{2}\right )^{\frac{3}{2}}} - \frac{\sqrt{2} \left (- 4 A b c + 4 B a c + B b^{2} + \left (2 A c - B b\right ) \sqrt{- 4 a c + b^{2}}\right ) \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt{c} \sqrt{x}}{\sqrt{b - \sqrt{- 4 a c + b^{2}}}} \right )}}{2 \sqrt{c} \sqrt{b - \sqrt{- 4 a c + b^{2}}} \left (- 4 a c + b^{2}\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*x**(1/2)/(c*x**2+b*x+a)**2,x)

[Out]

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

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Mathematica [A]  time = 0.74336, size = 298, normalized size = 1.08 \[ \frac{\sqrt{x} (B (2 a+b x)-A (b+2 c x))}{\left (b^2-4 a c\right ) (a+x (b+c x))}+\frac{\left (-2 A c \sqrt{b^2-4 a c}+b B \sqrt{b^2-4 a c}-4 a B c+4 A b c+b^2 (-B)\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{x}}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{\sqrt{2} \sqrt{c} \left (b^2-4 a c\right )^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\left (-2 A c \sqrt{b^2-4 a c}+b B \sqrt{b^2-4 a c}+4 a B c-4 A b c+b^2 B\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{x}}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{\sqrt{2} \sqrt{c} \left (b^2-4 a c\right )^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.07, size = 3042, normalized size = 11. \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*x^(1/2)/(c*x^2+b*x+a)^2,x)

[Out]

2*(1/2*(2*A*c-B*b)/(4*a*c-b^2)*x^(3/2)+1/2*(A*b-2*B*a)/(4*a*c-b^2)*x^(1/2))/(c*x
^2+b*x+a)+32*c^3/(-(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^2*A*b-16*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*A*b^3+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^5-32*c^3/(-(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*B+8*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^2*B*b^2+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^4*B*a-1/2/(-(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)*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))*b^6*B+4/(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*c^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^2-2*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*b*a+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^3-32/(-(4*a*c-b^2)^3)^(1/2)*c^3/(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^2*A*b+16/(-(4*a*
c-b^2)^3)^(1/2)*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))*a*A*b^3-2*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^5+32/(-(4*a*c-b^2)^3)^(1/2)*c^3/(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*B-8/(-(4*a*c-b^2)^3)^(1/2)*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))*a^
2*B*b^2-2*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^4*B*a+1/2/(-(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^6*B+4/(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*c^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^2-2*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*
b*a+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^3

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 1.34455, size = 4674, normalized size = 16.93 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*(sqrt(1/2)*(a*b^2 - 4*a^2*c + (b^2*c - 4*a*c^2)*x^2 + (b^3 - 4*a*b*c)*x)*sq
rt(-(B^2*a*b^3 - 4*(4*A*B*a^2 - 3*A^2*a*b)*c^2 + (12*B^2*a^2*b - 12*A*B*a*b^2 +
A^2*b^3)*c + (a*b^6*c - 12*a^2*b^4*c^2 + 48*a^3*b^2*c^3 - 64*a^4*c^4)*sqrt((B^4*
a^2 - 2*A^2*B^2*a*c + A^4*c^2)/(a^2*b^6*c^2 - 12*a^3*b^4*c^3 + 48*a^4*b^2*c^4 -
64*a^5*c^5)))/(a*b^6*c - 12*a^2*b^4*c^2 + 48*a^3*b^2*c^3 - 64*a^4*c^4))*log(sqrt
(1/2)*(2*B^3*a^2*b^4 - A*B^2*a*b^5 - 16*(2*A^2*B*a^3 - A^3*a^2*b)*c^3 + 8*(4*B^3
*a^4 - 2*A*B^2*a^3*b + 2*A^2*B*a^2*b^2 - A^3*a*b^3)*c^2 - (16*B^3*a^3*b^2 - 8*A*
B^2*a^2*b^3 + 2*A^2*B*a*b^4 - A^3*b^5)*c + (192*B*a^4*b^3*c^3 + 256*A*a^5*c^5 -
128*(2*B*a^5*b + A*a^4*b^2)*c^4 - 8*(6*B*a^3*b^5 - A*a^2*b^6)*c^2 + (4*B*a^2*b^7
- A*a*b^8)*c)*sqrt((B^4*a^2 - 2*A^2*B^2*a*c + A^4*c^2)/(a^2*b^6*c^2 - 12*a^3*b^
4*c^3 + 48*a^4*b^2*c^4 - 64*a^5*c^5)))*sqrt(-(B^2*a*b^3 - 4*(4*A*B*a^2 - 3*A^2*a
*b)*c^2 + (12*B^2*a^2*b - 12*A*B*a*b^2 + A^2*b^3)*c + (a*b^6*c - 12*a^2*b^4*c^2
+ 48*a^3*b^2*c^3 - 64*a^4*c^4)*sqrt((B^4*a^2 - 2*A^2*B^2*a*c + A^4*c^2)/(a^2*b^6
*c^2 - 12*a^3*b^4*c^3 + 48*a^4*b^2*c^4 - 64*a^5*c^5)))/(a*b^6*c - 12*a^2*b^4*c^2
+ 48*a^3*b^2*c^3 - 64*a^4*c^4)) - 2*(3*B^4*a^2*b^2 - A*B^3*a*b^3 - 4*A^4*a*c^3
+ 3*(4*A^3*B*a*b - A^4*b^2)*c^2 + (4*B^4*a^3 - 12*A*B^3*a^2*b + A^3*B*b^3)*c)*sq
rt(x)) - sqrt(1/2)*(a*b^2 - 4*a^2*c + (b^2*c - 4*a*c^2)*x^2 + (b^3 - 4*a*b*c)*x)
*sqrt(-(B^2*a*b^3 - 4*(4*A*B*a^2 - 3*A^2*a*b)*c^2 + (12*B^2*a^2*b - 12*A*B*a*b^2
+ A^2*b^3)*c + (a*b^6*c - 12*a^2*b^4*c^2 + 48*a^3*b^2*c^3 - 64*a^4*c^4)*sqrt((B
^4*a^2 - 2*A^2*B^2*a*c + A^4*c^2)/(a^2*b^6*c^2 - 12*a^3*b^4*c^3 + 48*a^4*b^2*c^4
- 64*a^5*c^5)))/(a*b^6*c - 12*a^2*b^4*c^2 + 48*a^3*b^2*c^3 - 64*a^4*c^4))*log(-
sqrt(1/2)*(2*B^3*a^2*b^4 - A*B^2*a*b^5 - 16*(2*A^2*B*a^3 - A^3*a^2*b)*c^3 + 8*(4
*B^3*a^4 - 2*A*B^2*a^3*b + 2*A^2*B*a^2*b^2 - A^3*a*b^3)*c^2 - (16*B^3*a^3*b^2 -
8*A*B^2*a^2*b^3 + 2*A^2*B*a*b^4 - A^3*b^5)*c + (192*B*a^4*b^3*c^3 + 256*A*a^5*c^
5 - 128*(2*B*a^5*b + A*a^4*b^2)*c^4 - 8*(6*B*a^3*b^5 - A*a^2*b^6)*c^2 + (4*B*a^2
*b^7 - A*a*b^8)*c)*sqrt((B^4*a^2 - 2*A^2*B^2*a*c + A^4*c^2)/(a^2*b^6*c^2 - 12*a^
3*b^4*c^3 + 48*a^4*b^2*c^4 - 64*a^5*c^5)))*sqrt(-(B^2*a*b^3 - 4*(4*A*B*a^2 - 3*A
^2*a*b)*c^2 + (12*B^2*a^2*b - 12*A*B*a*b^2 + A^2*b^3)*c + (a*b^6*c - 12*a^2*b^4*
c^2 + 48*a^3*b^2*c^3 - 64*a^4*c^4)*sqrt((B^4*a^2 - 2*A^2*B^2*a*c + A^4*c^2)/(a^2
*b^6*c^2 - 12*a^3*b^4*c^3 + 48*a^4*b^2*c^4 - 64*a^5*c^5)))/(a*b^6*c - 12*a^2*b^4
*c^2 + 48*a^3*b^2*c^3 - 64*a^4*c^4)) - 2*(3*B^4*a^2*b^2 - A*B^3*a*b^3 - 4*A^4*a*
c^3 + 3*(4*A^3*B*a*b - A^4*b^2)*c^2 + (4*B^4*a^3 - 12*A*B^3*a^2*b + A^3*B*b^3)*c
)*sqrt(x)) + sqrt(1/2)*(a*b^2 - 4*a^2*c + (b^2*c - 4*a*c^2)*x^2 + (b^3 - 4*a*b*c
)*x)*sqrt(-(B^2*a*b^3 - 4*(4*A*B*a^2 - 3*A^2*a*b)*c^2 + (12*B^2*a^2*b - 12*A*B*a
*b^2 + A^2*b^3)*c - (a*b^6*c - 12*a^2*b^4*c^2 + 48*a^3*b^2*c^3 - 64*a^4*c^4)*sqr
t((B^4*a^2 - 2*A^2*B^2*a*c + A^4*c^2)/(a^2*b^6*c^2 - 12*a^3*b^4*c^3 + 48*a^4*b^2
*c^4 - 64*a^5*c^5)))/(a*b^6*c - 12*a^2*b^4*c^2 + 48*a^3*b^2*c^3 - 64*a^4*c^4))*l
og(sqrt(1/2)*(2*B^3*a^2*b^4 - A*B^2*a*b^5 - 16*(2*A^2*B*a^3 - A^3*a^2*b)*c^3 + 8
*(4*B^3*a^4 - 2*A*B^2*a^3*b + 2*A^2*B*a^2*b^2 - A^3*a*b^3)*c^2 - (16*B^3*a^3*b^2
- 8*A*B^2*a^2*b^3 + 2*A^2*B*a*b^4 - A^3*b^5)*c - (192*B*a^4*b^3*c^3 + 256*A*a^5
*c^5 - 128*(2*B*a^5*b + A*a^4*b^2)*c^4 - 8*(6*B*a^3*b^5 - A*a^2*b^6)*c^2 + (4*B*
a^2*b^7 - A*a*b^8)*c)*sqrt((B^4*a^2 - 2*A^2*B^2*a*c + A^4*c^2)/(a^2*b^6*c^2 - 12
*a^3*b^4*c^3 + 48*a^4*b^2*c^4 - 64*a^5*c^5)))*sqrt(-(B^2*a*b^3 - 4*(4*A*B*a^2 -
3*A^2*a*b)*c^2 + (12*B^2*a^2*b - 12*A*B*a*b^2 + A^2*b^3)*c - (a*b^6*c - 12*a^2*b
^4*c^2 + 48*a^3*b^2*c^3 - 64*a^4*c^4)*sqrt((B^4*a^2 - 2*A^2*B^2*a*c + A^4*c^2)/(
a^2*b^6*c^2 - 12*a^3*b^4*c^3 + 48*a^4*b^2*c^4 - 64*a^5*c^5)))/(a*b^6*c - 12*a^2*
b^4*c^2 + 48*a^3*b^2*c^3 - 64*a^4*c^4)) - 2*(3*B^4*a^2*b^2 - A*B^3*a*b^3 - 4*A^4
*a*c^3 + 3*(4*A^3*B*a*b - A^4*b^2)*c^2 + (4*B^4*a^3 - 12*A*B^3*a^2*b + A^3*B*b^3
)*c)*sqrt(x)) - sqrt(1/2)*(a*b^2 - 4*a^2*c + (b^2*c - 4*a*c^2)*x^2 + (b^3 - 4*a*
b*c)*x)*sqrt(-(B^2*a*b^3 - 4*(4*A*B*a^2 - 3*A^2*a*b)*c^2 + (12*B^2*a^2*b - 12*A*
B*a*b^2 + A^2*b^3)*c - (a*b^6*c - 12*a^2*b^4*c^2 + 48*a^3*b^2*c^3 - 64*a^4*c^4)*
sqrt((B^4*a^2 - 2*A^2*B^2*a*c + A^4*c^2)/(a^2*b^6*c^2 - 12*a^3*b^4*c^3 + 48*a^4*
b^2*c^4 - 64*a^5*c^5)))/(a*b^6*c - 12*a^2*b^4*c^2 + 48*a^3*b^2*c^3 - 64*a^4*c^4)
)*log(-sqrt(1/2)*(2*B^3*a^2*b^4 - A*B^2*a*b^5 - 16*(2*A^2*B*a^3 - A^3*a^2*b)*c^3
+ 8*(4*B^3*a^4 - 2*A*B^2*a^3*b + 2*A^2*B*a^2*b^2 - A^3*a*b^3)*c^2 - (16*B^3*a^3
*b^2 - 8*A*B^2*a^2*b^3 + 2*A^2*B*a*b^4 - A^3*b^5)*c - (192*B*a^4*b^3*c^3 + 256*A
*a^5*c^5 - 128*(2*B*a^5*b + A*a^4*b^2)*c^4 - 8*(6*B*a^3*b^5 - A*a^2*b^6)*c^2 + (
4*B*a^2*b^7 - A*a*b^8)*c)*sqrt((B^4*a^2 - 2*A^2*B^2*a*c + A^4*c^2)/(a^2*b^6*c^2
- 12*a^3*b^4*c^3 + 48*a^4*b^2*c^4 - 64*a^5*c^5)))*sqrt(-(B^2*a*b^3 - 4*(4*A*B*a^
2 - 3*A^2*a*b)*c^2 + (12*B^2*a^2*b - 12*A*B*a*b^2 + A^2*b^3)*c - (a*b^6*c - 12*a
^2*b^4*c^2 + 48*a^3*b^2*c^3 - 64*a^4*c^4)*sqrt((B^4*a^2 - 2*A^2*B^2*a*c + A^4*c^
2)/(a^2*b^6*c^2 - 12*a^3*b^4*c^3 + 48*a^4*b^2*c^4 - 64*a^5*c^5)))/(a*b^6*c - 12*
a^2*b^4*c^2 + 48*a^3*b^2*c^3 - 64*a^4*c^4)) - 2*(3*B^4*a^2*b^2 - A*B^3*a*b^3 - 4
*A^4*a*c^3 + 3*(4*A^3*B*a*b - A^4*b^2)*c^2 + (4*B^4*a^3 - 12*A*B^3*a^2*b + A^3*B
*b^3)*c)*sqrt(x)) - 2*(2*B*a - A*b + (B*b - 2*A*c)*x)*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)

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Sympy [A]  time = 73.0415, size = 1807, normalized size = 6.55 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)*x**(1/2)/(c*x**2+b*x+a)**2,x)

[Out]

2*A*b*sqrt(x)/(8*a**2*c - 2*a*b**2 + 8*a*b*c*x + 8*a*c**2*x**2 - 2*b**3*x - 2*b*
*2*c*x**2) + 4*A*c*x**(3/2)/(8*a**2*c - 2*a*b**2 + 8*a*b*c*x + 8*a*c**2*x**2 - 2
*b**3*x - 2*b**2*c*x**2) + 2*A*RootSum(_t**4*(1048576*a**7*c**6 - 1572864*a**6*b
**2*c**5 + 983040*a**5*b**4*c**4 - 327680*a**4*b**6*c**3 + 61440*a**3*b**8*c**2
- 6144*a**2*b**10*c + 256*a*b**12) + _t**2*(-12288*a**4*b*c**4 + 8192*a**3*b**3*
c**3 - 1536*a**2*b**5*c**2 + 16*b**9) + 16*a**2*c**3 + 24*a*b**2*c**2 + 9*b**4*c
, Lambda(_t, _t*log(16384*_t**3*a**5*c**4/(4*a*c**2 + 3*b**2*c) - 8192*_t**3*a**
4*b**2*c**3/(4*a*c**2 + 3*b**2*c) + 512*_t**3*a**2*b**6*c/(4*a*c**2 + 3*b**2*c)
- 64*_t**3*a*b**8/(4*a*c**2 + 3*b**2*c) - 128*_t*a**2*b*c**2/(4*a*c**2 + 3*b**2*
c) - 16*_t*a*b**3*c/(4*a*c**2 + 3*b**2*c) - 4*_t*b**5/(4*a*c**2 + 3*b**2*c) + sq
rt(x)))) - 4*B*a**2*sqrt(x)/(8*a**3*c - 2*a**2*b**2 + 8*a**2*b*c*x + 8*a**2*c**2
*x**2 - 2*a*b**3*x - 2*a*b**2*c*x**2) + 2*B*a*b**2*sqrt(x)/(8*a**3*c**2 - 2*a**2
*b**2*c + 8*a**2*b*c**2*x + 8*a**2*c**3*x**2 - 2*a*b**3*c*x - 2*a*b**2*c**2*x**2
) + 2*B*a*b*x**(3/2)/(8*a**3*c - 2*a**2*b**2 + 8*a**2*b*c*x + 8*a**2*c**2*x**2 -
2*a*b**3*x - 2*a*b**2*c*x**2) - 2*B*a*RootSum(_t**4*(1048576*a**9*c**6 - 157286
4*a**8*b**2*c**5 + 983040*a**7*b**4*c**4 - 327680*a**6*b**6*c**3 + 61440*a**5*b*
*8*c**2 - 6144*a**4*b**10*c + 256*a**3*b**12) + _t**2*(-61440*a**5*b*c**5 + 6144
0*a**4*b**3*c**4 - 24064*a**3*b**5*c**3 + 4608*a**2*b**7*c**2 - 432*a*b**9*c + 1
6*b**11) + 1296*a**2*c**5 - 360*a*b**2*c**4 + 25*b**4*c**3, Lambda(_t, _t*log(32
768*_t**3*a**7*b*c**4/(324*a**2*c**4 - 81*a*b**2*c**3 + 5*b**4*c**2) - 28672*_t*
*3*a**6*b**3*c**3/(324*a**2*c**4 - 81*a*b**2*c**3 + 5*b**4*c**2) + 9216*_t**3*a*
*5*b**5*c**2/(324*a**2*c**4 - 81*a*b**2*c**3 + 5*b**4*c**2) - 1280*_t**3*a**4*b*
*7*c/(324*a**2*c**4 - 81*a*b**2*c**3 + 5*b**4*c**2) + 64*_t**3*a**3*b**9/(324*a*
*2*c**4 - 81*a*b**2*c**3 + 5*b**4*c**2) + 1728*_t*a**4*c**4/(324*a**2*c**4 - 81*
a*b**2*c**3 + 5*b**4*c**2) - 2304*_t*a**3*b**2*c**3/(324*a**2*c**4 - 81*a*b**2*c
**3 + 5*b**4*c**2) + 740*_t*a**2*b**4*c**2/(324*a**2*c**4 - 81*a*b**2*c**3 + 5*b
**4*c**2) - 92*_t*a*b**6*c/(324*a**2*c**4 - 81*a*b**2*c**3 + 5*b**4*c**2) + 4*_t
*b**8/(324*a**2*c**4 - 81*a*b**2*c**3 + 5*b**4*c**2) + sqrt(x))))/c - 2*B*b**2*s
qrt(x)/(8*a**2*c**2 - 2*a*b**2*c + 8*a*b*c**2*x + 8*a*c**3*x**2 - 2*b**3*c*x - 2
*b**2*c**2*x**2) - 4*B*b*x**(3/2)/(8*a**2*c - 2*a*b**2 + 8*a*b*c*x + 8*a*c**2*x*
*2 - 2*b**3*x - 2*b**2*c*x**2) - 2*B*b*RootSum(_t**4*(1048576*a**7*c**6 - 157286
4*a**6*b**2*c**5 + 983040*a**5*b**4*c**4 - 327680*a**4*b**6*c**3 + 61440*a**3*b*
*8*c**2 - 6144*a**2*b**10*c + 256*a*b**12) + _t**2*(-12288*a**4*b*c**4 + 8192*a*
*3*b**3*c**3 - 1536*a**2*b**5*c**2 + 16*b**9) + 16*a**2*c**3 + 24*a*b**2*c**2 +
9*b**4*c, Lambda(_t, _t*log(16384*_t**3*a**5*c**4/(4*a*c**2 + 3*b**2*c) - 8192*_
t**3*a**4*b**2*c**3/(4*a*c**2 + 3*b**2*c) + 512*_t**3*a**2*b**6*c/(4*a*c**2 + 3*
b**2*c) - 64*_t**3*a*b**8/(4*a*c**2 + 3*b**2*c) - 128*_t*a**2*b*c**2/(4*a*c**2 +
3*b**2*c) - 16*_t*a*b**3*c/(4*a*c**2 + 3*b**2*c) - 4*_t*b**5/(4*a*c**2 + 3*b**2
*c) + sqrt(x))))/c + 2*B*RootSum(_t**4*(256*a**3*c**2 - 128*a**2*b**2*c + 16*a*b
**4) + _t**2*(-16*a*b*c + 4*b**3) + c, Lambda(_t, _t*log(32*_t**3*a**2*b - 8*_t*
*3*a*b**3/c + 4*_t*a - 2*_t*b**2/c + sqrt(x))))/c

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

[Out]

Exception raised: TypeError