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

Optimal. Leaf size=94 \[ \frac{8 (2 a+b x) (A b-2 a B)}{3 \left (b^2-4 a c\right )^2 \sqrt{a+b x+c x^2}}-\frac{2 x^2 (-2 a B-x (b B-2 A c)+A b)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.141898, antiderivative size = 94, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087 \[ \frac{8 (2 a+b x) (A b-2 a B)}{3 \left (b^2-4 a c\right )^2 \sqrt{a+b x+c x^2}}-\frac{2 x^2 (-2 a B-x (b B-2 A c)+A b)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 13.6376, size = 90, normalized size = 0.96 \[ - \frac{2 x^{2} \left (A b - 2 B a + x \left (2 A c - B b\right )\right )}{3 \left (- 4 a c + b^{2}\right ) \left (a + b x + c x^{2}\right )^{\frac{3}{2}}} + \frac{4 \left (4 a + 2 b x\right ) \left (A b - 2 B a\right )}{3 \left (- 4 a c + b^{2}\right )^{2} \sqrt{a + b x + c x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.204526, size = 110, normalized size = 1.17 \[ \frac{2 \left (-16 a^3 B+8 a^2 (A b-3 B x (b+c x))+2 a x \left (A \left (6 b^2+6 b c x+4 c^2 x^2\right )-3 b B x (b+2 c x)\right )+b^2 x^2 (3 A b+2 A c x+b B x)\right )}{3 \left (b^2-4 a c\right )^2 (a+x (b+c x))^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.013, size = 141, normalized size = 1.5 \[{\frac{16\,aA{c}^{2}{x}^{3}+4\,A{x}^{3}{b}^{2}c-24\,B{x}^{3}abc+2\,B{b}^{3}{x}^{3}+24\,A{x}^{2}abc+6\,A{b}^{3}{x}^{2}-48\,{a}^{2}Bc{x}^{2}-12\,B{x}^{2}a{b}^{2}+24\,Axa{b}^{2}-48\,Bx{a}^{2}b+16\,A{a}^{2}b-32\,B{a}^{3}}{48\,{a}^{2}{c}^{2}-24\,ac{b}^{2}+3\,{b}^{4}} \left ( c{x}^{2}+bx+a \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3/(c*x^2+b*x+a)^(3/2)*(8*A*a*c^2*x^3+2*A*b^2*c*x^3-12*B*a*b*c*x^3+B*b^3*x^3+12
*A*a*b*c*x^2+3*A*b^3*x^2-24*B*a^2*c*x^2-6*B*a*b^2*x^2+12*A*a*b^2*x-24*B*a^2*b*x+
8*A*a^2*b-16*B*a^3)/(16*a^2*c^2-8*a*b^2*c+b^4)

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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)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.41801, size = 335, normalized size = 3.56 \[ -\frac{2 \,{\left (16 \, B a^{3} - 8 \, A a^{2} b -{\left (B b^{3} + 8 \, A a c^{2} - 2 \,{\left (6 \, B a b - A b^{2}\right )} c\right )} x^{3} + 3 \,{\left (2 \, B a b^{2} - A b^{3} + 4 \,{\left (2 \, B a^{2} - A a b\right )} c\right )} x^{2} + 12 \,{\left (2 \, B a^{2} b - A a b^{2}\right )} x\right )} \sqrt{c x^{2} + b x + a}}{3 \,{\left (a^{2} b^{4} - 8 \, a^{3} b^{2} c + 16 \, a^{4} c^{2} +{\left (b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} x^{4} + 2 \,{\left (b^{5} c - 8 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} x^{3} +{\left (b^{6} - 6 \, a b^{4} c + 32 \, a^{3} c^{3}\right )} x^{2} + 2 \,{\left (a b^{5} - 8 \, a^{2} b^{3} c + 16 \, a^{3} b c^{2}\right )} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/3*(16*B*a^3 - 8*A*a^2*b - (B*b^3 + 8*A*a*c^2 - 2*(6*B*a*b - A*b^2)*c)*x^3 + 3
*(2*B*a*b^2 - A*b^3 + 4*(2*B*a^2 - A*a*b)*c)*x^2 + 12*(2*B*a^2*b - A*a*b^2)*x)*s
qrt(c*x^2 + b*x + a)/(a^2*b^4 - 8*a^3*b^2*c + 16*a^4*c^2 + (b^4*c^2 - 8*a*b^2*c^
3 + 16*a^2*c^4)*x^4 + 2*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*x^3 + (b^6 - 6*a*b^
4*c + 32*a^3*c^3)*x^2 + 2*(a*b^5 - 8*a^2*b^3*c + 16*a^3*b*c^2)*x)

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.28312, size = 296, normalized size = 3.15 \[ \frac{{\left ({\left (\frac{{\left (B b^{3} - 12 \, B a b c + 2 \, A b^{2} c + 8 \, A a c^{2}\right )} x}{b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}} - \frac{3 \,{\left (2 \, B a b^{2} - A b^{3} + 8 \, B a^{2} c - 4 \, A a b c\right )}}{b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}}\right )} x - \frac{12 \,{\left (2 \, B a^{2} b - A a b^{2}\right )}}{b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}}\right )} x - \frac{8 \,{\left (2 \, B a^{3} - A a^{2} b\right )}}{b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}}}{3 \,{\left (c x^{2} + b x + a\right )}^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*((((B*b^3 - 12*B*a*b*c + 2*A*b^2*c + 8*A*a*c^2)*x/(b^4*c^2 - 8*a*b^2*c^3 + 1
6*a^2*c^4) - 3*(2*B*a*b^2 - A*b^3 + 8*B*a^2*c - 4*A*a*b*c)/(b^4*c^2 - 8*a*b^2*c^
3 + 16*a^2*c^4))*x - 12*(2*B*a^2*b - A*a*b^2)/(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^
4))*x - 8*(2*B*a^3 - A*a^2*b)/(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4))/(c*x^2 + b*x
+ a)^(3/2)