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

Optimal. Leaf size=153 \[ \frac{\sqrt{a+b x+c x^2} \left (-8 a B c-2 A b c+3 b^2 B\right )}{c^2 \left (b^2-4 a c\right )}-\frac{2 x \left (x \left (-2 a B c-A b c+b^2 B\right )+a (b B-2 A c)\right )}{c \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}}-\frac{(3 b B-2 A c) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{2 c^{5/2}} \]

[Out]

(-2*x*(a*(b*B - 2*A*c) + (b^2*B - A*b*c - 2*a*B*c)*x))/(c*(b^2 - 4*a*c)*Sqrt[a +
b*x + c*x^2]) + ((3*b^2*B - 2*A*b*c - 8*a*B*c)*Sqrt[a + b*x + c*x^2])/(c^2*(b^2
- 4*a*c)) - ((3*b*B - 2*A*c)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^
2])])/(2*c^(5/2))

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Rubi [A]  time = 0.296527, antiderivative size = 153, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174 \[ \frac{\sqrt{a+b x+c x^2} \left (-8 a B c-2 A b c+3 b^2 B\right )}{c^2 \left (b^2-4 a c\right )}-\frac{2 x \left (x \left (-2 a B c-A b c+b^2 B\right )+a (b B-2 A c)\right )}{c \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}}-\frac{(3 b B-2 A c) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{2 c^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*x*(a*(b*B - 2*A*c) + (b^2*B - A*b*c - 2*a*B*c)*x))/(c*(b^2 - 4*a*c)*Sqrt[a +
b*x + c*x^2]) + ((3*b^2*B - 2*A*b*c - 8*a*B*c)*Sqrt[a + b*x + c*x^2])/(c^2*(b^2
- 4*a*c)) - ((3*b*B - 2*A*c)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^
2])])/(2*c^(5/2))

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Rubi in Sympy [A]  time = 34.885, size = 146, normalized size = 0.95 \[ \frac{2 x \left (a \left (2 A c - B b\right ) - x \left (- A b c - 2 B a c + B b^{2}\right )\right )}{c \left (- 4 a c + b^{2}\right ) \sqrt{a + b x + c x^{2}}} + \frac{2 \left (\frac{3 B b^{2}}{2} - c \left (A b + 4 B a\right )\right ) \sqrt{a + b x + c x^{2}}}{c^{2} \left (- 4 a c + b^{2}\right )} + \frac{\left (2 A c - 3 B b\right ) \operatorname{atanh}{\left (\frac{b + 2 c x}{2 \sqrt{c} \sqrt{a + b x + c x^{2}}} \right )}}{2 c^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*x*(a*(2*A*c - B*b) - x*(-A*b*c - 2*B*a*c + B*b**2))/(c*(-4*a*c + b**2)*sqrt(a
+ b*x + c*x**2)) + 2*(3*B*b**2/2 - c*(A*b + 4*B*a))*sqrt(a + b*x + c*x**2)/(c**2
*(-4*a*c + b**2)) + (2*A*c - 3*B*b)*atanh((b + 2*c*x)/(2*sqrt(c)*sqrt(a + b*x +
c*x**2)))/(2*c**(5/2))

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Mathematica [A]  time = 0.334812, size = 146, normalized size = 0.95 \[ \frac{\frac{2 \sqrt{c} \left (8 a^2 B c+a \left (2 b c (A+5 B x)+4 c^2 x (B x-A)-3 b^2 B\right )-b^2 x (-2 A c+3 b B+B c x)\right )}{\sqrt{a+x (b+c x)}}+\left (b^2-4 a c\right ) (3 b B-2 A c) \log \left (2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right )}{2 c^{5/2} \left (4 a c-b^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

((2*Sqrt[c]*(8*a^2*B*c - b^2*x*(3*b*B - 2*A*c + B*c*x) + a*(-3*b^2*B + 4*c^2*x*(
-A + B*x) + 2*b*c*(A + 5*B*x))))/Sqrt[a + x*(b + c*x)] + (b^2 - 4*a*c)*(3*b*B -
2*A*c)*Log[b + 2*c*x + 2*Sqrt[c]*Sqrt[a + x*(b + c*x)]])/(2*c^(5/2)*(-b^2 + 4*a*
c))

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Maple [B]  time = 0.011, size = 382, normalized size = 2.5 \[ -{\frac{Ax}{c}{\frac{1}{\sqrt{c{x}^{2}+bx+a}}}}+{\frac{Ab}{2\,{c}^{2}}{\frac{1}{\sqrt{c{x}^{2}+bx+a}}}}+{\frac{Ax{b}^{2}}{c \left ( 4\,ac-{b}^{2} \right ) }{\frac{1}{\sqrt{c{x}^{2}+bx+a}}}}+{\frac{A{b}^{3}}{2\,{c}^{2} \left ( 4\,ac-{b}^{2} \right ) }{\frac{1}{\sqrt{c{x}^{2}+bx+a}}}}+{A\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{3}{2}}}}+{\frac{{x}^{2}B}{c}{\frac{1}{\sqrt{c{x}^{2}+bx+a}}}}+{\frac{3\,xBb}{2\,{c}^{2}}{\frac{1}{\sqrt{c{x}^{2}+bx+a}}}}-{\frac{3\,{b}^{2}B}{4\,{c}^{3}}{\frac{1}{\sqrt{c{x}^{2}+bx+a}}}}-{\frac{3\,Bx{b}^{3}}{2\,{c}^{2} \left ( 4\,ac-{b}^{2} \right ) }{\frac{1}{\sqrt{c{x}^{2}+bx+a}}}}-{\frac{3\,{b}^{4}B}{4\,{c}^{3} \left ( 4\,ac-{b}^{2} \right ) }{\frac{1}{\sqrt{c{x}^{2}+bx+a}}}}-{\frac{3\,Bb}{2}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{5}{2}}}}+2\,{\frac{Ba}{{c}^{2}\sqrt{c{x}^{2}+bx+a}}}+4\,{\frac{Bxab}{c \left ( 4\,ac-{b}^{2} \right ) \sqrt{c{x}^{2}+bx+a}}}+2\,{\frac{a{b}^{2}B}{{c}^{2} \left ( 4\,ac-{b}^{2} \right ) \sqrt{c{x}^{2}+bx+a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.435641, size = 1, normalized size = 0.01 \[ \left [\frac{4 \,{\left (3 \, B a b^{2} +{\left (B b^{2} c - 4 \, B a c^{2}\right )} x^{2} - 2 \,{\left (4 \, B a^{2} + A a b\right )} c +{\left (3 \, B b^{3} + 4 \, A a c^{2} - 2 \,{\left (5 \, B a b + A b^{2}\right )} c\right )} x\right )} \sqrt{c x^{2} + b x + a} \sqrt{c} -{\left (3 \, B a b^{3} + 8 \, A a^{2} c^{2} +{\left (3 \, B b^{3} c + 8 \, A a c^{3} - 2 \,{\left (6 \, B a b + A b^{2}\right )} c^{2}\right )} x^{2} - 2 \,{\left (6 \, B a^{2} b + A a b^{2}\right )} c +{\left (3 \, B b^{4} + 8 \, A a b c^{2} - 2 \,{\left (6 \, B a b^{2} + A b^{3}\right )} c\right )} x\right )} \log \left (-4 \,{\left (2 \, c^{2} x + b c\right )} \sqrt{c x^{2} + b x + a} -{\left (8 \, c^{2} x^{2} + 8 \, b c x + b^{2} + 4 \, a c\right )} \sqrt{c}\right )}{4 \,{\left (a b^{2} c^{2} - 4 \, a^{2} c^{3} +{\left (b^{2} c^{3} - 4 \, a c^{4}\right )} x^{2} +{\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} x\right )} \sqrt{c}}, \frac{2 \,{\left (3 \, B a b^{2} +{\left (B b^{2} c - 4 \, B a c^{2}\right )} x^{2} - 2 \,{\left (4 \, B a^{2} + A a b\right )} c +{\left (3 \, B b^{3} + 4 \, A a c^{2} - 2 \,{\left (5 \, B a b + A b^{2}\right )} c\right )} x\right )} \sqrt{c x^{2} + b x + a} \sqrt{-c} -{\left (3 \, B a b^{3} + 8 \, A a^{2} c^{2} +{\left (3 \, B b^{3} c + 8 \, A a c^{3} - 2 \,{\left (6 \, B a b + A b^{2}\right )} c^{2}\right )} x^{2} - 2 \,{\left (6 \, B a^{2} b + A a b^{2}\right )} c +{\left (3 \, B b^{4} + 8 \, A a b c^{2} - 2 \,{\left (6 \, B a b^{2} + A b^{3}\right )} c\right )} x\right )} \arctan \left (\frac{{\left (2 \, c x + b\right )} \sqrt{-c}}{2 \, \sqrt{c x^{2} + b x + a} c}\right )}{2 \,{\left (a b^{2} c^{2} - 4 \, a^{2} c^{3} +{\left (b^{2} c^{3} - 4 \, a c^{4}\right )} x^{2} +{\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} x\right )} \sqrt{-c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/4*(4*(3*B*a*b^2 + (B*b^2*c - 4*B*a*c^2)*x^2 - 2*(4*B*a^2 + A*a*b)*c + (3*B*b^
3 + 4*A*a*c^2 - 2*(5*B*a*b + A*b^2)*c)*x)*sqrt(c*x^2 + b*x + a)*sqrt(c) - (3*B*a
*b^3 + 8*A*a^2*c^2 + (3*B*b^3*c + 8*A*a*c^3 - 2*(6*B*a*b + A*b^2)*c^2)*x^2 - 2*(
6*B*a^2*b + A*a*b^2)*c + (3*B*b^4 + 8*A*a*b*c^2 - 2*(6*B*a*b^2 + A*b^3)*c)*x)*lo
g(-4*(2*c^2*x + b*c)*sqrt(c*x^2 + b*x + a) - (8*c^2*x^2 + 8*b*c*x + b^2 + 4*a*c)
*sqrt(c)))/((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(c)), 1/2*(2*(3*B*a*b^2 + (B*b^2*c - 4*B*a*c^2)*x^2 - 2*(4*B*a^2 + A
*a*b)*c + (3*B*b^3 + 4*A*a*c^2 - 2*(5*B*a*b + A*b^2)*c)*x)*sqrt(c*x^2 + b*x + a)
*sqrt(-c) - (3*B*a*b^3 + 8*A*a^2*c^2 + (3*B*b^3*c + 8*A*a*c^3 - 2*(6*B*a*b + A*b
^2)*c^2)*x^2 - 2*(6*B*a^2*b + A*a*b^2)*c + (3*B*b^4 + 8*A*a*b*c^2 - 2*(6*B*a*b^2
+ A*b^3)*c)*x)*arctan(1/2*(2*c*x + b)*sqrt(-c)/(sqrt(c*x^2 + b*x + a)*c)))/((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(-c
))]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.29585, size = 239, normalized size = 1.56 \[ \frac{{\left (\frac{{\left (B b^{2} c - 4 \, B a c^{2}\right )} x}{b^{2} c^{2} - 4 \, a c^{3}} + \frac{3 \, B b^{3} - 10 \, B a b c - 2 \, A b^{2} c + 4 \, A a c^{2}}{b^{2} c^{2} - 4 \, a c^{3}}\right )} x + \frac{3 \, B a b^{2} - 8 \, B a^{2} c - 2 \, A a b c}{b^{2} c^{2} - 4 \, a c^{3}}}{\sqrt{c x^{2} + b x + a}} + \frac{{\left (3 \, B b - 2 \, A c\right )}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} \sqrt{c} - b \right |}\right )}{2 \, c^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(((B*b^2*c - 4*B*a*c^2)*x/(b^2*c^2 - 4*a*c^3) + (3*B*b^3 - 10*B*a*b*c - 2*A*b^2*
c + 4*A*a*c^2)/(b^2*c^2 - 4*a*c^3))*x + (3*B*a*b^2 - 8*B*a^2*c - 2*A*a*b*c)/(b^2
*c^2 - 4*a*c^3))/sqrt(c*x^2 + b*x + a) + 1/2*(3*B*b - 2*A*c)*ln(abs(-2*(sqrt(c)*
x - sqrt(c*x^2 + b*x + a))*sqrt(c) - b))/c^(5/2)