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

Optimal. Leaf size=70 \[ \frac{2 (b+2 c x) (b B-4 A c)}{3 b^3 c \sqrt{b x+c x^2}}-\frac{2 x (b B-A c)}{3 b c \left (b x+c x^2\right )^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.107399, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ \frac{2 (b+2 c x) (b B-4 A c)}{3 b^3 c \sqrt{b x+c x^2}}-\frac{2 x (b B-A c)}{3 b c \left (b x+c x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 8.12253, size = 61, normalized size = 0.87 \[ \frac{2 x \left (A c - B b\right )}{3 b c \left (b x + c x^{2}\right )^{\frac{3}{2}}} - \frac{\left (2 b + 4 c x\right ) \left (4 A c - B b\right )}{3 b^{3} c \sqrt{b x + c x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0796265, size = 55, normalized size = 0.79 \[ \frac{x \left (2 b B x (3 b+2 c x)-2 A \left (3 b^2+12 b c x+8 c^2 x^2\right )\right )}{3 b^3 (x (b+c x))^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(x*(2*b*B*x*(3*b + 2*c*x) - 2*A*(3*b^2 + 12*b*c*x + 8*c^2*x^2)))/(3*b^3*(x*(b +
c*x))^(3/2))

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Maple [A]  time = 0.008, size = 62, normalized size = 0.9 \[ -{\frac{2\,{x}^{2} \left ( cx+b \right ) \left ( 8\,A{c}^{2}{x}^{2}-2\,B{x}^{2}bc+12\,Abcx-3\,{b}^{2}Bx+3\,{b}^{2}A \right ) }{3\,{b}^{3}} \left ( c{x}^{2}+bx \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/3*x^2*(c*x+b)*(8*A*c^2*x^2-2*B*b*c*x^2+12*A*b*c*x-3*B*b^2*x+3*A*b^2)/b^3/(c*x
^2+b*x)^(5/2)

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Maxima [A]  time = 0.685086, size = 150, normalized size = 2.14 \[ \frac{4 \, B x}{3 \, \sqrt{c x^{2} + b x} b^{2}} + \frac{2 \, A x}{3 \,{\left (c x^{2} + b x\right )}^{\frac{3}{2}} b} - \frac{2 \, B x}{3 \,{\left (c x^{2} + b x\right )}^{\frac{3}{2}} c} - \frac{16 \, A c x}{3 \, \sqrt{c x^{2} + b x} b^{3}} - \frac{8 \, A}{3 \, \sqrt{c x^{2} + b x} b^{2}} + \frac{2 \, B}{3 \, \sqrt{c x^{2} + b x} b c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.303694, size = 84, normalized size = 1.2 \[ -\frac{2 \,{\left (3 \, A b^{2} - 2 \,{\left (B b c - 4 \, A c^{2}\right )} x^{2} - 3 \,{\left (B b^{2} - 4 \, A b c\right )} x\right )}}{3 \,{\left (b^{3} c x + b^{4}\right )} \sqrt{c x^{2} + b x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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