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

Optimal. Leaf size=67 \[ \frac{2 x (2 A c+b B)}{3 b^2 c \sqrt{b x+c x^2}}-\frac{2 x^2 (b B-A c)}{3 b c \left (b x+c x^2\right )^{3/2}} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.158917, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{2 x (2 A c+b B)}{3 b^2 c \sqrt{b x+c x^2}}-\frac{2 x^2 (b B-A c)}{3 b c \left (b x+c x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 10.3367, size = 58, normalized size = 0.87 \[ \frac{2 x^{2} \left (A c - B b\right )}{3 b c \left (b x + c x^{2}\right )^{\frac{3}{2}}} + \frac{4 x \left (A c + \frac{B b}{2}\right )}{3 b^{2} c \sqrt{b x + c x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*x**2*(A*c - B*b)/(3*b*c*(b*x + c*x**2)**(3/2)) + 4*x*(A*c + B*b/2)/(3*b**2*c*s
qrt(b*x + c*x**2))

_______________________________________________________________________________________

Mathematica [A]  time = 0.0497503, size = 35, normalized size = 0.52 \[ \frac{2 x^2 (3 A b+2 A c x+b B x)}{3 b^2 (x (b+c x))^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*x^2*(3*A*b + b*B*x + 2*A*c*x))/(3*b^2*(x*(b + c*x))^(3/2))

_______________________________________________________________________________________

Maple [A]  time = 0.007, size = 39, normalized size = 0.6 \[{\frac{2\,{x}^{3} \left ( cx+b \right ) \left ( 2\,Acx+xBb+3\,Ab \right ) }{3\,{b}^{2}} \left ( c{x}^{2}+bx \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*x^3*(c*x+b)*(2*A*c*x+B*b*x+3*A*b)/b^2/(c*x^2+b*x)^(5/2)

_______________________________________________________________________________________

Maxima [A]  time = 0.682031, size = 181, normalized size = 2.7 \[ -\frac{B x^{2}}{{\left (c x^{2} + b x\right )}^{\frac{3}{2}} c} + \frac{4 \, A x}{3 \, \sqrt{c x^{2} + b x} b^{2}} - \frac{B b x}{3 \,{\left (c x^{2} + b x\right )}^{\frac{3}{2}} c^{2}} - \frac{2 \, A x}{3 \,{\left (c x^{2} + b x\right )}^{\frac{3}{2}} c} + \frac{2 \, B x}{3 \, \sqrt{c x^{2} + b x} b c} + \frac{B}{3 \, \sqrt{c x^{2} + b x} c^{2}} + \frac{2 \, A}{3 \, \sqrt{c x^{2} + b x} b c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.305115, size = 58, normalized size = 0.87 \[ \frac{2 \,{\left (3 \, A b x +{\left (B b + 2 \, A c\right )} x^{2}\right )}}{3 \,{\left (b^{2} c x + b^{3}\right )} \sqrt{c x^{2} + b x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*(3*A*b*x + (B*b + 2*A*c)*x^2)/((b^2*c*x + b^3)*sqrt(c*x^2 + b*x))

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.283243, size = 161, normalized size = 2.4 \[ \frac{2 \,{\left (3 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{2} B c + 3 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} B b \sqrt{c} + 3 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} A c^{\frac{3}{2}} + B b^{2} + 2 \, A b c\right )}}{3 \,{\left ({\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} \sqrt{c} + b\right )}^{3} c^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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