3.65 \(\int \frac{x}{\left (a x+b x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=48 \[ \frac{2 x}{3 a \left (a x+b x^2\right )^{3/2}}-\frac{8 (a+2 b x)}{3 a^3 \sqrt{a x+b x^2}} \]

[Out]

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

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Rubi [A]  time = 0.0439084, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{2 x}{3 a \left (a x+b x^2\right )^{3/2}}-\frac{8 (a+2 b x)}{3 a^3 \sqrt{a x+b x^2}} \]

Antiderivative was successfully verified.

[In]  Int[x/(a*x + b*x^2)^(5/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 4.68897, size = 42, normalized size = 0.88 \[ \frac{2 x}{3 a \left (a x + b x^{2}\right )^{\frac{3}{2}}} - \frac{4 \left (2 a + 4 b x\right )}{3 a^{3} \sqrt{a x + b x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0357549, size = 38, normalized size = 0.79 \[ -\frac{2 x \left (3 a^2+12 a b x+8 b^2 x^2\right )}{3 a^3 (x (a+b x))^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*x*(3*a^2 + 12*a*b*x + 8*b^2*x^2))/(3*a^3*(x*(a + b*x))^(3/2))

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Maple [A]  time = 0.007, size = 44, normalized size = 0.9 \[ -{\frac{2\,{x}^{2} \left ( bx+a \right ) \left ( 8\,{b}^{2}{x}^{2}+12\,abx+3\,{a}^{2} \right ) }{3\,{a}^{3}} \left ( b{x}^{2}+ax \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(b*x^2+a*x)^(5/2),x)

[Out]

-2/3*x^2*(b*x+a)*(8*b^2*x^2+12*a*b*x+3*a^2)/a^3/(b*x^2+a*x)^(5/2)

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Maxima [A]  time = 0.695022, size = 70, normalized size = 1.46 \[ \frac{2 \, x}{3 \,{\left (b x^{2} + a x\right )}^{\frac{3}{2}} a} - \frac{16 \, b x}{3 \, \sqrt{b x^{2} + a x} a^{3}} - \frac{8}{3 \, \sqrt{b x^{2} + a x} a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*x/((b*x^2 + a*x)^(3/2)*a) - 16/3*b*x/(sqrt(b*x^2 + a*x)*a^3) - 8/3/(sqrt(b*x
^2 + a*x)*a^2)

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Fricas [A]  time = 0.218623, size = 59, normalized size = 1.23 \[ -\frac{2 \,{\left (8 \, b^{2} x^{2} + 12 \, a b x + 3 \, a^{2}\right )}}{3 \,{\left (a^{3} b x + a^{4}\right )} \sqrt{b x^{2} + a x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/3*(8*b^2*x^2 + 12*a*b*x + 3*a^2)/((a^3*b*x + a^4)*sqrt(b*x^2 + a*x))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(b*x**2+a*x)**(5/2),x)

[Out]

Integral(x/(x*(a + b*x))**(5/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x/(b*x^2 + a*x)^(5/2), x)