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

Optimal. Leaf size=53 \[ \frac{2 b^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{a^{5/2}}+\frac{2 b}{a^2 \sqrt{x}}-\frac{2}{3 a x^{3/2}} \]

[Out]

-2/(3*a*x^(3/2)) + (2*b)/(a^2*Sqrt[x]) + (2*b^(3/2)*ArcTan[(Sqrt[b]*Sqrt[x])/Sqr
t[a]])/a^(5/2)

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Rubi [A]  time = 0.0457125, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \frac{2 b^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{a^{5/2}}+\frac{2 b}{a^2 \sqrt{x}}-\frac{2}{3 a x^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^(5/2)*(a + b*x)),x]

[Out]

-2/(3*a*x^(3/2)) + (2*b)/(a^2*Sqrt[x]) + (2*b^(3/2)*ArcTan[(Sqrt[b]*Sqrt[x])/Sqr
t[a]])/a^(5/2)

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Rubi in Sympy [A]  time = 8.722, size = 49, normalized size = 0.92 \[ - \frac{2}{3 a x^{\frac{3}{2}}} + \frac{2 b}{a^{2} \sqrt{x}} + \frac{2 b^{\frac{3}{2}} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{a^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/(3*a*x**(3/2)) + 2*b/(a**2*sqrt(x)) + 2*b**(3/2)*atan(sqrt(b)*sqrt(x)/sqrt(a)
)/a**(5/2)

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Mathematica [A]  time = 0.0414288, size = 50, normalized size = 0.94 \[ \frac{2 b^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{a^{5/2}}+\frac{2 (3 b x-a)}{3 a^2 x^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(-a + 3*b*x))/(3*a^2*x^(3/2)) + (2*b^(3/2)*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]])
/a^(5/2)

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Maple [A]  time = 0.013, size = 43, normalized size = 0.8 \[ -{\frac{2}{3\,a}{x}^{-{\frac{3}{2}}}}+2\,{\frac{b}{{a}^{2}\sqrt{x}}}+2\,{\frac{{b}^{2}}{{a}^{2}\sqrt{ab}}\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^(5/2)/(b*x+a),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.228294, size = 1, normalized size = 0.02 \[ \left [\frac{3 \, b x^{\frac{3}{2}} \sqrt{-\frac{b}{a}} \log \left (\frac{b x + 2 \, a \sqrt{x} \sqrt{-\frac{b}{a}} - a}{b x + a}\right ) + 6 \, b x - 2 \, a}{3 \, a^{2} x^{\frac{3}{2}}}, -\frac{2 \,{\left (3 \, b x^{\frac{3}{2}} \sqrt{\frac{b}{a}} \arctan \left (\frac{a \sqrt{\frac{b}{a}}}{b \sqrt{x}}\right ) - 3 \, b x + a\right )}}{3 \, a^{2} x^{\frac{3}{2}}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/3*(3*b*x^(3/2)*sqrt(-b/a)*log((b*x + 2*a*sqrt(x)*sqrt(-b/a) - a)/(b*x + a)) +
6*b*x - 2*a)/(a^2*x^(3/2)), -2/3*(3*b*x^(3/2)*sqrt(b/a)*arctan(a*sqrt(b/a)/(b*s
qrt(x))) - 3*b*x + a)/(a^2*x^(3/2))]

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Sympy [A]  time = 5.96103, size = 49, normalized size = 0.92 \[ - \frac{2}{3 a x^{\frac{3}{2}}} + \frac{2 b}{a^{2} \sqrt{x}} + \frac{2 b^{\frac{3}{2}} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{a^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**(5/2)/(b*x+a),x)

[Out]

-2/(3*a*x**(3/2)) + 2*b/(a**2*sqrt(x)) + 2*b**(3/2)*atan(sqrt(b)*sqrt(x)/sqrt(a)
)/a**(5/2)

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GIAC/XCAS [A]  time = 0.202737, size = 55, normalized size = 1.04 \[ \frac{2 \, b^{2} \arctan \left (\frac{b \sqrt{x}}{\sqrt{a b}}\right )}{\sqrt{a b} a^{2}} + \frac{2 \,{\left (3 \, b x - a\right )}}{3 \, a^{2} x^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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