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

Optimal. Leaf size=64 \[ -\frac{16 \sqrt{a+b x}}{3 a^3 \sqrt{x}}+\frac{8}{3 a^2 \sqrt{x} \sqrt{a+b x}}+\frac{2}{3 a \sqrt{x} (a+b x)^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.0450158, antiderivative size = 64, 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 \[ -\frac{16 \sqrt{a+b x}}{3 a^3 \sqrt{x}}+\frac{8}{3 a^2 \sqrt{x} \sqrt{a+b x}}+\frac{2}{3 a \sqrt{x} (a+b x)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 6.45407, size = 58, normalized size = 0.91 \[ \frac{2}{3 a \sqrt{x} \left (a + b x\right )^{\frac{3}{2}}} + \frac{8}{3 a^{2} \sqrt{x} \sqrt{a + b x}} - \frac{16 \sqrt{a + b x}}{3 a^{3} \sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0285809, size = 40, normalized size = 0.62 \[ -\frac{2 \left (3 a^2+12 a b x+8 b^2 x^2\right )}{3 a^3 \sqrt{x} (a+b x)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.006, size = 35, normalized size = 0.6 \[ -{\frac{16\,{b}^{2}{x}^{2}+24\,abx+6\,{a}^{2}}{3\,{a}^{3}} \left ( bx+a \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^(3/2)/(b*x+a)^(5/2),x)

[Out]

-2/3*(8*b^2*x^2+12*a*b*x+3*a^2)/x^(1/2)/(b*x+a)^(3/2)/a^3

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Maxima [A]  time = 1.35515, size = 62, normalized size = 0.97 \[ \frac{2 \,{\left (b^{2} - \frac{6 \,{\left (b x + a\right )} b}{x}\right )} x^{\frac{3}{2}}}{3 \,{\left (b x + a\right )}^{\frac{3}{2}} a^{3}} - \frac{2 \, \sqrt{b x + a}}{a^{3} \sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.236679, size = 58, normalized size = 0.91 \[ -\frac{2 \,{\left (8 \, b^{2} x^{2} + 12 \, a b x + 3 \, a^{2}\right )}}{3 \,{\left (a^{3} b x + a^{4}\right )} \sqrt{b x + a} \sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

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Sympy [A]  time = 104.053, size = 153, normalized size = 2.39 \[ - \frac{6 a^{2} b^{\frac{9}{2}} \sqrt{\frac{a}{b x} + 1}}{3 a^{5} b^{4} + 6 a^{4} b^{5} x + 3 a^{3} b^{6} x^{2}} - \frac{24 a b^{\frac{11}{2}} x \sqrt{\frac{a}{b x} + 1}}{3 a^{5} b^{4} + 6 a^{4} b^{5} x + 3 a^{3} b^{6} x^{2}} - \frac{16 b^{\frac{13}{2}} x^{2} \sqrt{\frac{a}{b x} + 1}}{3 a^{5} b^{4} + 6 a^{4} b^{5} x + 3 a^{3} b^{6} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.223933, size = 215, normalized size = 3.36 \[ -\frac{2 \, \sqrt{b x + a} b^{2}}{\sqrt{{\left (b x + a\right )} b - a b} a^{3}{\left | b \right |}} - \frac{4 \,{\left (3 \,{\left (\sqrt{b x + a} \sqrt{b} - \sqrt{{\left (b x + a\right )} b - a b}\right )}^{4} b^{\frac{5}{2}} + 12 \, a{\left (\sqrt{b x + a} \sqrt{b} - \sqrt{{\left (b x + a\right )} b - a b}\right )}^{2} b^{\frac{7}{2}} + 5 \, a^{2} b^{\frac{9}{2}}\right )}}{3 \,{\left ({\left (\sqrt{b x + a} \sqrt{b} - \sqrt{{\left (b x + a\right )} b - a b}\right )}^{2} + a b\right )}^{3} a^{2}{\left | b \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*sqrt(b*x + a)*b^2/(sqrt((b*x + a)*b - a*b)*a^3*abs(b)) - 4/3*(3*(sqrt(b*x + a
)*sqrt(b) - sqrt((b*x + a)*b - a*b))^4*b^(5/2) + 12*a*(sqrt(b*x + a)*sqrt(b) - s
qrt((b*x + a)*b - a*b))^2*b^(7/2) + 5*a^2*b^(9/2))/(((sqrt(b*x + a)*sqrt(b) - sq
rt((b*x + a)*b - a*b))^2 + a*b)^3*a^2*abs(b))