3.582 \(\int \frac{1}{x^{5/2} (a+b x)^{3/2}} \, dx\)

Optimal. Leaf size=63 \[ \frac{16 b \sqrt{a+b x}}{3 a^3 \sqrt{x}}-\frac{8 \sqrt{a+b x}}{3 a^2 x^{3/2}}+\frac{2}{a x^{3/2} \sqrt{a+b x}} \]

[Out]

2/(a*x^(3/2)*Sqrt[a + b*x]) - (8*Sqrt[a + b*x])/(3*a^2*x^(3/2)) + (16*b*Sqrt[a +
b*x])/(3*a^3*Sqrt[x])

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Rubi [A]  time = 0.0416455, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{16 b \sqrt{a+b x}}{3 a^3 \sqrt{x}}-\frac{8 \sqrt{a+b x}}{3 a^2 x^{3/2}}+\frac{2}{a x^{3/2} \sqrt{a+b x}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^(5/2)*(a + b*x)^(3/2)),x]

[Out]

2/(a*x^(3/2)*Sqrt[a + b*x]) - (8*Sqrt[a + b*x])/(3*a^2*x^(3/2)) + (16*b*Sqrt[a +
b*x])/(3*a^3*Sqrt[x])

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Rubi in Sympy [A]  time = 6.26241, size = 58, normalized size = 0.92 \[ \frac{2}{a x^{\frac{3}{2}} \sqrt{a + b x}} - \frac{8 \sqrt{a + b x}}{3 a^{2} x^{\frac{3}{2}}} + \frac{16 b \sqrt{a + b x}}{3 a^{3} \sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**(5/2)/(b*x+a)**(3/2),x)

[Out]

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

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Mathematica [A]  time = 0.0280075, size = 38, normalized size = 0.6 \[ -\frac{2 \left (a^2-4 a b x-8 b^2 x^2\right )}{3 a^3 x^{3/2} \sqrt{a+b x}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^(5/2)*(a + b*x)^(3/2)),x]

[Out]

(-2*(a^2 - 4*a*b*x - 8*b^2*x^2))/(3*a^3*x^(3/2)*Sqrt[a + b*x])

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Maple [A]  time = 0.006, size = 33, normalized size = 0.5 \[ -{\frac{-16\,{b}^{2}{x}^{2}-8\,abx+2\,{a}^{2}}{3\,{a}^{3}}{x}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{bx+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^(5/2)/(b*x+a)^(3/2),x)

[Out]

-2/3*(-8*b^2*x^2-4*a*b*x+a^2)/x^(3/2)/(b*x+a)^(1/2)/a^3

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Maxima [A]  time = 1.34735, size = 68, normalized size = 1.08 \[ \frac{2 \, b^{2} \sqrt{x}}{\sqrt{b x + a} a^{3}} + \frac{2 \,{\left (\frac{6 \, \sqrt{b x + a} b}{\sqrt{x}} - \frac{{\left (b x + a\right )}^{\frac{3}{2}}}{x^{\frac{3}{2}}}\right )}}{3 \, a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^(3/2)*x^(5/2)),x, algorithm="maxima")

[Out]

2*b^2*sqrt(x)/(sqrt(b*x + a)*a^3) + 2/3*(6*sqrt(b*x + a)*b/sqrt(x) - (b*x + a)^(
3/2)/x^(3/2))/a^3

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Fricas [A]  time = 0.209284, size = 46, normalized size = 0.73 \[ \frac{2 \,{\left (8 \, b^{2} x^{2} + 4 \, a b x - a^{2}\right )}}{3 \, \sqrt{b x + a} a^{3} x^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^(3/2)*x^(5/2)),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 103.614, size = 219, normalized size = 3.48 \[ - \frac{2 a^{3} b^{\frac{9}{2}} \sqrt{\frac{a}{b x} + 1}}{3 a^{5} b^{4} x + 6 a^{4} b^{5} x^{2} + 3 a^{3} b^{6} x^{3}} + \frac{6 a^{2} b^{\frac{11}{2}} x \sqrt{\frac{a}{b x} + 1}}{3 a^{5} b^{4} x + 6 a^{4} b^{5} x^{2} + 3 a^{3} b^{6} x^{3}} + \frac{24 a b^{\frac{13}{2}} x^{2} \sqrt{\frac{a}{b x} + 1}}{3 a^{5} b^{4} x + 6 a^{4} b^{5} x^{2} + 3 a^{3} b^{6} x^{3}} + \frac{16 b^{\frac{15}{2}} x^{3} \sqrt{\frac{a}{b x} + 1}}{3 a^{5} b^{4} x + 6 a^{4} b^{5} x^{2} + 3 a^{3} b^{6} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**(5/2)/(b*x+a)**(3/2),x)

[Out]

-2*a**3*b**(9/2)*sqrt(a/(b*x) + 1)/(3*a**5*b**4*x + 6*a**4*b**5*x**2 + 3*a**3*b*
*6*x**3) + 6*a**2*b**(11/2)*x*sqrt(a/(b*x) + 1)/(3*a**5*b**4*x + 6*a**4*b**5*x**
2 + 3*a**3*b**6*x**3) + 24*a*b**(13/2)*x**2*sqrt(a/(b*x) + 1)/(3*a**5*b**4*x + 6
*a**4*b**5*x**2 + 3*a**3*b**6*x**3) + 16*b**(15/2)*x**3*sqrt(a/(b*x) + 1)/(3*a**
5*b**4*x + 6*a**4*b**5*x**2 + 3*a**3*b**6*x**3)

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GIAC/XCAS [A]  time = 0.216189, size = 126, normalized size = 2. \[ -\frac{\sqrt{b x + a}{\left (\frac{5 \,{\left (b x + a\right )}{\left | b \right |}}{b^{2}} - \frac{6 \, a{\left | b \right |}}{b^{2}}\right )}}{24 \,{\left ({\left (b x + a\right )} b - a b\right )}^{\frac{3}{2}}} + \frac{4 \, b^{\frac{7}{2}}}{{\left ({\left (\sqrt{b x + a} \sqrt{b} - \sqrt{{\left (b x + a\right )} b - a b}\right )}^{2} + a b\right )} a^{2}{\left | b \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^(3/2)*x^(5/2)),x, algorithm="giac")

[Out]

-1/24*sqrt(b*x + a)*(5*(b*x + a)*abs(b)/b^2 - 6*a*abs(b)/b^2)/((b*x + a)*b - a*b
)^(3/2) + 4*b^(7/2)/(((sqrt(b*x + a)*sqrt(b) - sqrt((b*x + a)*b - a*b))^2 + a*b)
*a^2*abs(b))