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

Optimal. Leaf size=56 \[ -\frac{3 \sqrt{b} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{a^{5/2}}-\frac{3}{a^2 \sqrt{x}}+\frac{1}{a \sqrt{x} (a+b x)} \]

[Out]

-3/(a^2*Sqrt[x]) + 1/(a*Sqrt[x]*(a + b*x)) - (3*Sqrt[b]*ArcTan[(Sqrt[b]*Sqrt[x])
/Sqrt[a]])/a^(5/2)

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Rubi [A]  time = 0.0456107, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308 \[ -\frac{3 \sqrt{b} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{a^{5/2}}-\frac{3}{a^2 \sqrt{x}}+\frac{1}{a \sqrt{x} (a+b x)} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^(3/2)*(a + b*x)^2),x]

[Out]

-3/(a^2*Sqrt[x]) + 1/(a*Sqrt[x]*(a + b*x)) - (3*Sqrt[b]*ArcTan[(Sqrt[b]*Sqrt[x])
/Sqrt[a]])/a^(5/2)

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Rubi in Sympy [A]  time = 9.12143, size = 51, normalized size = 0.91 \[ \frac{1}{a \sqrt{x} \left (a + b x\right )} - \frac{3}{a^{2} \sqrt{x}} - \frac{3 \sqrt{b} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{a^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0538791, size = 54, normalized size = 0.96 \[ \frac{-2 a-3 b x}{a^2 \sqrt{x} (a+b x)}-\frac{3 \sqrt{b} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{a^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.019, size = 48, normalized size = 0.9 \[ -2\,{\frac{1}{{a}^{2}\sqrt{x}}}-{\frac{b}{{a}^{2} \left ( bx+a \right ) }\sqrt{x}}-3\,{\frac{b}{{a}^{2}\sqrt{ab}}\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^(3/2)/(b*x+a)^2,x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.265427, size = 1, normalized size = 0.02 \[ \left [\frac{3 \,{\left (b x + a\right )} \sqrt{x} \sqrt{-\frac{b}{a}} \log \left (\frac{b x - 2 \, a \sqrt{x} \sqrt{-\frac{b}{a}} - a}{b x + a}\right ) - 6 \, b x - 4 \, a}{2 \,{\left (a^{2} b x + a^{3}\right )} \sqrt{x}}, \frac{3 \,{\left (b x + a\right )} \sqrt{x} \sqrt{\frac{b}{a}} \arctan \left (\frac{a \sqrt{\frac{b}{a}}}{b \sqrt{x}}\right ) - 3 \, b x - 2 \, a}{{\left (a^{2} b x + a^{3}\right )} \sqrt{x}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/2*(3*(b*x + a)*sqrt(x)*sqrt(-b/a)*log((b*x - 2*a*sqrt(x)*sqrt(-b/a) - a)/(b*x
+ a)) - 6*b*x - 4*a)/((a^2*b*x + a^3)*sqrt(x)), (3*(b*x + a)*sqrt(x)*sqrt(b/a)*
arctan(a*sqrt(b/a)/(b*sqrt(x))) - 3*b*x - 2*a)/((a^2*b*x + a^3)*sqrt(x))]

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Sympy [A]  time = 6.06974, size = 595, normalized size = 10.62 \[ - \frac{2 a^{\frac{15}{2}}}{a^{\frac{19}{2}} \sqrt{x} + 3 a^{\frac{17}{2}} b x^{\frac{3}{2}} + 3 a^{\frac{15}{2}} b^{2} x^{\frac{5}{2}} + a^{\frac{13}{2}} b^{3} x^{\frac{7}{2}}} - \frac{7 a^{\frac{13}{2}} b x}{a^{\frac{19}{2}} \sqrt{x} + 3 a^{\frac{17}{2}} b x^{\frac{3}{2}} + 3 a^{\frac{15}{2}} b^{2} x^{\frac{5}{2}} + a^{\frac{13}{2}} b^{3} x^{\frac{7}{2}}} - \frac{8 a^{\frac{11}{2}} b^{2} x^{2}}{a^{\frac{19}{2}} \sqrt{x} + 3 a^{\frac{17}{2}} b x^{\frac{3}{2}} + 3 a^{\frac{15}{2}} b^{2} x^{\frac{5}{2}} + a^{\frac{13}{2}} b^{3} x^{\frac{7}{2}}} - \frac{3 a^{\frac{9}{2}} b^{3} x^{3}}{a^{\frac{19}{2}} \sqrt{x} + 3 a^{\frac{17}{2}} b x^{\frac{3}{2}} + 3 a^{\frac{15}{2}} b^{2} x^{\frac{5}{2}} + a^{\frac{13}{2}} b^{3} x^{\frac{7}{2}}} - \frac{3 a^{7} \sqrt{b} \sqrt{x} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{a^{\frac{19}{2}} \sqrt{x} + 3 a^{\frac{17}{2}} b x^{\frac{3}{2}} + 3 a^{\frac{15}{2}} b^{2} x^{\frac{5}{2}} + a^{\frac{13}{2}} b^{3} x^{\frac{7}{2}}} - \frac{9 a^{6} b^{\frac{3}{2}} x^{\frac{3}{2}} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{a^{\frac{19}{2}} \sqrt{x} + 3 a^{\frac{17}{2}} b x^{\frac{3}{2}} + 3 a^{\frac{15}{2}} b^{2} x^{\frac{5}{2}} + a^{\frac{13}{2}} b^{3} x^{\frac{7}{2}}} - \frac{9 a^{5} b^{\frac{5}{2}} x^{\frac{5}{2}} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{a^{\frac{19}{2}} \sqrt{x} + 3 a^{\frac{17}{2}} b x^{\frac{3}{2}} + 3 a^{\frac{15}{2}} b^{2} x^{\frac{5}{2}} + a^{\frac{13}{2}} b^{3} x^{\frac{7}{2}}} - \frac{3 a^{4} b^{\frac{7}{2}} x^{\frac{7}{2}} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{a^{\frac{19}{2}} \sqrt{x} + 3 a^{\frac{17}{2}} b x^{\frac{3}{2}} + 3 a^{\frac{15}{2}} b^{2} x^{\frac{5}{2}} + a^{\frac{13}{2}} b^{3} x^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*a**(15/2)/(a**(19/2)*sqrt(x) + 3*a**(17/2)*b*x**(3/2) + 3*a**(15/2)*b**2*x**(
5/2) + a**(13/2)*b**3*x**(7/2)) - 7*a**(13/2)*b*x/(a**(19/2)*sqrt(x) + 3*a**(17/
2)*b*x**(3/2) + 3*a**(15/2)*b**2*x**(5/2) + a**(13/2)*b**3*x**(7/2)) - 8*a**(11/
2)*b**2*x**2/(a**(19/2)*sqrt(x) + 3*a**(17/2)*b*x**(3/2) + 3*a**(15/2)*b**2*x**(
5/2) + a**(13/2)*b**3*x**(7/2)) - 3*a**(9/2)*b**3*x**3/(a**(19/2)*sqrt(x) + 3*a*
*(17/2)*b*x**(3/2) + 3*a**(15/2)*b**2*x**(5/2) + a**(13/2)*b**3*x**(7/2)) - 3*a*
*7*sqrt(b)*sqrt(x)*atan(sqrt(b)*sqrt(x)/sqrt(a))/(a**(19/2)*sqrt(x) + 3*a**(17/2
)*b*x**(3/2) + 3*a**(15/2)*b**2*x**(5/2) + a**(13/2)*b**3*x**(7/2)) - 9*a**6*b**
(3/2)*x**(3/2)*atan(sqrt(b)*sqrt(x)/sqrt(a))/(a**(19/2)*sqrt(x) + 3*a**(17/2)*b*
x**(3/2) + 3*a**(15/2)*b**2*x**(5/2) + a**(13/2)*b**3*x**(7/2)) - 9*a**5*b**(5/2
)*x**(5/2)*atan(sqrt(b)*sqrt(x)/sqrt(a))/(a**(19/2)*sqrt(x) + 3*a**(17/2)*b*x**(
3/2) + 3*a**(15/2)*b**2*x**(5/2) + a**(13/2)*b**3*x**(7/2)) - 3*a**4*b**(7/2)*x*
*(7/2)*atan(sqrt(b)*sqrt(x)/sqrt(a))/(a**(19/2)*sqrt(x) + 3*a**(17/2)*b*x**(3/2)
+ 3*a**(15/2)*b**2*x**(5/2) + a**(13/2)*b**3*x**(7/2))

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GIAC/XCAS [A]  time = 0.203329, size = 66, normalized size = 1.18 \[ -\frac{3 \, b \arctan \left (\frac{b \sqrt{x}}{\sqrt{a b}}\right )}{\sqrt{a b} a^{2}} - \frac{3 \, b x + 2 \, a}{{\left (b x^{\frac{3}{2}} + a \sqrt{x}\right )} a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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