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

Optimal. Leaf size=73 \[ -\frac{2 \sqrt{x} (A c+2 b B)}{3 b c^2 \sqrt{b x+c x^2}}-\frac{2 x^{5/2} (b B-A c)}{3 b c \left (b x+c x^2\right )^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.145462, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ -\frac{2 \sqrt{x} (A c+2 b B)}{3 b c^2 \sqrt{b x+c x^2}}-\frac{2 x^{5/2} (b B-A c)}{3 b c \left (b x+c x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 10.2974, size = 63, normalized size = 0.86 \[ \frac{2 x^{\frac{5}{2}} \left (A c - B b\right )}{3 b c \left (b x + c x^{2}\right )^{\frac{3}{2}}} - \frac{4 \sqrt{x} \left (\frac{A c}{2} + B b\right )}{3 b c^{2} \sqrt{b x + c x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0456709, size = 36, normalized size = 0.49 \[ -\frac{2 x^{3/2} (c (A+3 B x)+2 b B)}{3 c^2 (x (b+c x))^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.006, size = 38, normalized size = 0.5 \[ -{\frac{ \left ( 2\,cx+2\,b \right ) \left ( 3\,Bcx+Ac+2\,Bb \right ) }{3\,{c}^{2}}{x}^{{\frac{5}{2}}} \left ( c{x}^{2}+bx \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.734896, size = 47, normalized size = 0.64 \[ -\frac{2 \,{\left (3 \, c x + 2 \, b\right )} B}{3 \,{\left (c x + b\right )}^{\frac{3}{2}} c^{2}} - \frac{2 \, A}{3 \,{\left (c x + b\right )}^{\frac{3}{2}} c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.297532, size = 76, normalized size = 1.04 \[ -\frac{2 \,{\left (3 \, B c x + 2 \, B b + A c\right )} \sqrt{c x^{2} + b x} \sqrt{x}}{3 \,{\left (c^{4} x^{3} + 2 \, b c^{3} x^{2} + b^{2} c^{2} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.275738, size = 61, normalized size = 0.84 \[ -\frac{2 \,{\left (3 \,{\left (c x + b\right )} B - B b + A c\right )}}{3 \,{\left (c x + b\right )}^{\frac{3}{2}} c^{2}} + \frac{2 \,{\left (2 \, B b + A c\right )}}{3 \, b^{\frac{3}{2}} c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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