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

Optimal. Leaf size=32 \[ \frac{2 a}{3 b^2 (a+b x)^{3/2}}-\frac{2}{b^2 \sqrt{a+b x}} \]

[Out]

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

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Rubi [A]  time = 0.0249004, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{2 a}{3 b^2 (a+b x)^{3/2}}-\frac{2}{b^2 \sqrt{a+b x}} \]

Antiderivative was successfully verified.

[In]  Int[x/(a + b*x)^(5/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 4.98279, size = 29, normalized size = 0.91 \[ \frac{2 a}{3 b^{2} \left (a + b x\right )^{\frac{3}{2}}} - \frac{2}{b^{2} \sqrt{a + b x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0175843, size = 24, normalized size = 0.75 \[ -\frac{2 (2 a+3 b x)}{3 b^2 (a+b x)^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[x/(a + b*x)^(5/2),x]

[Out]

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

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Maple [A]  time = 0.004, size = 21, normalized size = 0.7 \[ -{\frac{6\,bx+4\,a}{3\,{b}^{2}} \left ( bx+a \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(b*x+a)^(5/2),x)

[Out]

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

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Maxima [A]  time = 1.34625, size = 35, normalized size = 1.09 \[ -\frac{2}{\sqrt{b x + a} b^{2}} + \frac{2 \, a}{3 \,{\left (b x + a\right )}^{\frac{3}{2}} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.213969, size = 41, normalized size = 1.28 \[ -\frac{2 \,{\left (3 \, b x + 2 \, a\right )}}{3 \,{\left (b^{3} x + a b^{2}\right )} \sqrt{b x + a}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 3.43473, size = 80, normalized size = 2.5 \[ \begin{cases} - \frac{4 a}{3 a b^{2} \sqrt{a + b x} + 3 b^{3} x \sqrt{a + b x}} - \frac{6 b x}{3 a b^{2} \sqrt{a + b x} + 3 b^{3} x \sqrt{a + b x}} & \text{for}\: b \neq 0 \\\frac{x^{2}}{2 a^{\frac{5}{2}}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(b*x+a)**(5/2),x)

[Out]

Piecewise((-4*a/(3*a*b**2*sqrt(a + b*x) + 3*b**3*x*sqrt(a + b*x)) - 6*b*x/(3*a*b
**2*sqrt(a + b*x) + 3*b**3*x*sqrt(a + b*x)), Ne(b, 0)), (x**2/(2*a**(5/2)), True
))

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GIAC/XCAS [A]  time = 0.202811, size = 27, normalized size = 0.84 \[ -\frac{2 \,{\left (3 \, b x + 2 \, a\right )}}{3 \,{\left (b x + a\right )}^{\frac{3}{2}} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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