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

Optimal. Leaf size=54 \[ -\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{a^{5/2}}+\frac{2}{a^2 \sqrt{a+b x}}+\frac{2}{3 a (a+b x)^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.0516485, antiderivative size = 54, 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 \[ -\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{a^{5/2}}+\frac{2}{a^2 \sqrt{a+b x}}+\frac{2}{3 a (a+b x)^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x*(a + b*x)^(5/2)),x]

[Out]

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

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Rubi in Sympy [A]  time = 7.36236, size = 48, normalized size = 0.89 \[ \frac{2}{3 a \left (a + b x\right )^{\frac{3}{2}}} + \frac{2}{a^{2} \sqrt{a + b x}} - \frac{2 \operatorname{atanh}{\left (\frac{\sqrt{a + b x}}{\sqrt{a}} \right )}}{a^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.103332, size = 48, normalized size = 0.89 \[ \frac{2 (4 a+3 b x)}{3 a^2 (a+b x)^{3/2}}-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{a^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(4*a + 3*b*x))/(3*a^2*(a + b*x)^(3/2)) - (2*ArcTanh[Sqrt[a + b*x]/Sqrt[a]])/a
^(5/2)

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Maple [A]  time = 0.015, size = 43, normalized size = 0.8 \[{\frac{2}{3\,a} \left ( bx+a \right ) ^{-{\frac{3}{2}}}}-2\,{\frac{1}{{a}^{5/2}}{\it Artanh} \left ({\frac{\sqrt{bx+a}}{\sqrt{a}}} \right ) }+2\,{\frac{1}{{a}^{2}\sqrt{bx+a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(b*x+a)^(5/2),x)

[Out]

2/3/a/(b*x+a)^(3/2)-2*arctanh((b*x+a)^(1/2)/a^(1/2))/a^(5/2)+2/a^2/(b*x+a)^(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)^(5/2)*x),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.254567, size = 1, normalized size = 0.02 \[ \left [\frac{3 \,{\left (b x + a\right )}^{\frac{3}{2}} \log \left (\frac{{\left (b x + 2 \, a\right )} \sqrt{a} - 2 \, \sqrt{b x + a} a}{x}\right ) + 2 \,{\left (3 \, b x + 4 \, a\right )} \sqrt{a}}{3 \,{\left (a^{2} b x + a^{3}\right )} \sqrt{b x + a} \sqrt{a}}, \frac{2 \,{\left (3 \,{\left (b x + a\right )}^{\frac{3}{2}} \arctan \left (\frac{a}{\sqrt{b x + a} \sqrt{-a}}\right ) +{\left (3 \, b x + 4 \, a\right )} \sqrt{-a}\right )}}{3 \,{\left (a^{2} b x + a^{3}\right )} \sqrt{b x + a} \sqrt{-a}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/3*(3*(b*x + a)^(3/2)*log(((b*x + 2*a)*sqrt(a) - 2*sqrt(b*x + a)*a)/x) + 2*(3*
b*x + 4*a)*sqrt(a))/((a^2*b*x + a^3)*sqrt(b*x + a)*sqrt(a)), 2/3*(3*(b*x + a)^(3
/2)*arctan(a/(sqrt(b*x + a)*sqrt(-a))) + (3*b*x + 4*a)*sqrt(-a))/((a^2*b*x + a^3
)*sqrt(b*x + a)*sqrt(-a))]

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Sympy [A]  time = 9.69957, size = 697, normalized size = 12.91 \[ \frac{8 a^{7} \sqrt{1 + \frac{b x}{a}}}{3 a^{\frac{19}{2}} + 9 a^{\frac{17}{2}} b x + 9 a^{\frac{15}{2}} b^{2} x^{2} + 3 a^{\frac{13}{2}} b^{3} x^{3}} + \frac{3 a^{7} \log{\left (\frac{b x}{a} \right )}}{3 a^{\frac{19}{2}} + 9 a^{\frac{17}{2}} b x + 9 a^{\frac{15}{2}} b^{2} x^{2} + 3 a^{\frac{13}{2}} b^{3} x^{3}} - \frac{6 a^{7} \log{\left (\sqrt{1 + \frac{b x}{a}} + 1 \right )}}{3 a^{\frac{19}{2}} + 9 a^{\frac{17}{2}} b x + 9 a^{\frac{15}{2}} b^{2} x^{2} + 3 a^{\frac{13}{2}} b^{3} x^{3}} + \frac{14 a^{6} b x \sqrt{1 + \frac{b x}{a}}}{3 a^{\frac{19}{2}} + 9 a^{\frac{17}{2}} b x + 9 a^{\frac{15}{2}} b^{2} x^{2} + 3 a^{\frac{13}{2}} b^{3} x^{3}} + \frac{9 a^{6} b x \log{\left (\frac{b x}{a} \right )}}{3 a^{\frac{19}{2}} + 9 a^{\frac{17}{2}} b x + 9 a^{\frac{15}{2}} b^{2} x^{2} + 3 a^{\frac{13}{2}} b^{3} x^{3}} - \frac{18 a^{6} b x \log{\left (\sqrt{1 + \frac{b x}{a}} + 1 \right )}}{3 a^{\frac{19}{2}} + 9 a^{\frac{17}{2}} b x + 9 a^{\frac{15}{2}} b^{2} x^{2} + 3 a^{\frac{13}{2}} b^{3} x^{3}} + \frac{6 a^{5} b^{2} x^{2} \sqrt{1 + \frac{b x}{a}}}{3 a^{\frac{19}{2}} + 9 a^{\frac{17}{2}} b x + 9 a^{\frac{15}{2}} b^{2} x^{2} + 3 a^{\frac{13}{2}} b^{3} x^{3}} + \frac{9 a^{5} b^{2} x^{2} \log{\left (\frac{b x}{a} \right )}}{3 a^{\frac{19}{2}} + 9 a^{\frac{17}{2}} b x + 9 a^{\frac{15}{2}} b^{2} x^{2} + 3 a^{\frac{13}{2}} b^{3} x^{3}} - \frac{18 a^{5} b^{2} x^{2} \log{\left (\sqrt{1 + \frac{b x}{a}} + 1 \right )}}{3 a^{\frac{19}{2}} + 9 a^{\frac{17}{2}} b x + 9 a^{\frac{15}{2}} b^{2} x^{2} + 3 a^{\frac{13}{2}} b^{3} x^{3}} + \frac{3 a^{4} b^{3} x^{3} \log{\left (\frac{b x}{a} \right )}}{3 a^{\frac{19}{2}} + 9 a^{\frac{17}{2}} b x + 9 a^{\frac{15}{2}} b^{2} x^{2} + 3 a^{\frac{13}{2}} b^{3} x^{3}} - \frac{6 a^{4} b^{3} x^{3} \log{\left (\sqrt{1 + \frac{b x}{a}} + 1 \right )}}{3 a^{\frac{19}{2}} + 9 a^{\frac{17}{2}} b x + 9 a^{\frac{15}{2}} b^{2} x^{2} + 3 a^{\frac{13}{2}} b^{3} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x/(b*x+a)**(5/2),x)

[Out]

8*a**7*sqrt(1 + b*x/a)/(3*a**(19/2) + 9*a**(17/2)*b*x + 9*a**(15/2)*b**2*x**2 +
3*a**(13/2)*b**3*x**3) + 3*a**7*log(b*x/a)/(3*a**(19/2) + 9*a**(17/2)*b*x + 9*a*
*(15/2)*b**2*x**2 + 3*a**(13/2)*b**3*x**3) - 6*a**7*log(sqrt(1 + b*x/a) + 1)/(3*
a**(19/2) + 9*a**(17/2)*b*x + 9*a**(15/2)*b**2*x**2 + 3*a**(13/2)*b**3*x**3) + 1
4*a**6*b*x*sqrt(1 + b*x/a)/(3*a**(19/2) + 9*a**(17/2)*b*x + 9*a**(15/2)*b**2*x**
2 + 3*a**(13/2)*b**3*x**3) + 9*a**6*b*x*log(b*x/a)/(3*a**(19/2) + 9*a**(17/2)*b*
x + 9*a**(15/2)*b**2*x**2 + 3*a**(13/2)*b**3*x**3) - 18*a**6*b*x*log(sqrt(1 + b*
x/a) + 1)/(3*a**(19/2) + 9*a**(17/2)*b*x + 9*a**(15/2)*b**2*x**2 + 3*a**(13/2)*b
**3*x**3) + 6*a**5*b**2*x**2*sqrt(1 + b*x/a)/(3*a**(19/2) + 9*a**(17/2)*b*x + 9*
a**(15/2)*b**2*x**2 + 3*a**(13/2)*b**3*x**3) + 9*a**5*b**2*x**2*log(b*x/a)/(3*a*
*(19/2) + 9*a**(17/2)*b*x + 9*a**(15/2)*b**2*x**2 + 3*a**(13/2)*b**3*x**3) - 18*
a**5*b**2*x**2*log(sqrt(1 + b*x/a) + 1)/(3*a**(19/2) + 9*a**(17/2)*b*x + 9*a**(1
5/2)*b**2*x**2 + 3*a**(13/2)*b**3*x**3) + 3*a**4*b**3*x**3*log(b*x/a)/(3*a**(19/
2) + 9*a**(17/2)*b*x + 9*a**(15/2)*b**2*x**2 + 3*a**(13/2)*b**3*x**3) - 6*a**4*b
**3*x**3*log(sqrt(1 + b*x/a) + 1)/(3*a**(19/2) + 9*a**(17/2)*b*x + 9*a**(15/2)*b
**2*x**2 + 3*a**(13/2)*b**3*x**3)

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GIAC/XCAS [A]  time = 0.20546, size = 61, normalized size = 1.13 \[ \frac{2 \, \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{2}} + \frac{2 \,{\left (3 \, b x + 4 \, a\right )}}{3 \,{\left (b x + a\right )}^{\frac{3}{2}} a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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