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

Optimal. Leaf size=69 \[ \frac{5 b^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{a^{7/2}}+\frac{5 b}{a^3 \sqrt{x}}-\frac{5}{3 a^2 x^{3/2}}+\frac{1}{a x^{3/2} (a+b x)} \]

[Out]

-5/(3*a^2*x^(3/2)) + (5*b)/(a^3*Sqrt[x]) + 1/(a*x^(3/2)*(a + b*x)) + (5*b^(3/2)*
ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/a^(7/2)

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Rubi [A]  time = 0.0580196, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308 \[ \frac{5 b^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{a^{7/2}}+\frac{5 b}{a^3 \sqrt{x}}-\frac{5}{3 a^2 x^{3/2}}+\frac{1}{a x^{3/2} (a+b x)} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^(5/2)*(a + b*x)^2),x]

[Out]

-5/(3*a^2*x^(3/2)) + (5*b)/(a^3*Sqrt[x]) + 1/(a*x^(3/2)*(a + b*x)) + (5*b^(3/2)*
ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/a^(7/2)

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Rubi in Sympy [A]  time = 11.9465, size = 65, normalized size = 0.94 \[ \frac{1}{a x^{\frac{3}{2}} \left (a + b x\right )} - \frac{5}{3 a^{2} x^{\frac{3}{2}}} + \frac{5 b}{a^{3} \sqrt{x}} + \frac{5 b^{\frac{3}{2}} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{a^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/(a*x**(3/2)*(a + b*x)) - 5/(3*a**2*x**(3/2)) + 5*b/(a**3*sqrt(x)) + 5*b**(3/2)
*atan(sqrt(b)*sqrt(x)/sqrt(a))/a**(7/2)

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Mathematica [A]  time = 0.0654992, size = 68, normalized size = 0.99 \[ \frac{5 b^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{a^{7/2}}+\frac{-2 a^2+10 a b x+15 b^2 x^2}{3 a^3 x^{3/2} (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

(-2*a^2 + 10*a*b*x + 15*b^2*x^2)/(3*a^3*x^(3/2)*(a + b*x)) + (5*b^(3/2)*ArcTan[(
Sqrt[b]*Sqrt[x])/Sqrt[a]])/a^(7/2)

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Maple [A]  time = 0.022, size = 60, normalized size = 0.9 \[ -{\frac{2}{3\,{a}^{2}}{x}^{-{\frac{3}{2}}}}+4\,{\frac{b}{{a}^{3}\sqrt{x}}}+{\frac{{b}^{2}}{{a}^{3} \left ( bx+a \right ) }\sqrt{x}}+5\,{\frac{{b}^{2}}{{a}^{3}\sqrt{ab}}\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^(5/2)/(b*x+a)^2,x)

[Out]

-2/3/a^2/x^(3/2)+4*b/a^3/x^(1/2)+1/a^3*b^2*x^(1/2)/(b*x+a)+5/a^3*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)^2*x^(5/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.25695, size = 1, normalized size = 0.01 \[ \left [\frac{30 \, b^{2} x^{2} + 20 \, a b x + 15 \,{\left (b^{2} x^{2} + a b x\right )} \sqrt{x} \sqrt{-\frac{b}{a}} \log \left (\frac{b x + 2 \, a \sqrt{x} \sqrt{-\frac{b}{a}} - a}{b x + a}\right ) - 4 \, a^{2}}{6 \,{\left (a^{3} b x^{2} + a^{4} x\right )} \sqrt{x}}, \frac{15 \, b^{2} x^{2} + 10 \, a b x - 15 \,{\left (b^{2} x^{2} + a b x\right )} \sqrt{x} \sqrt{\frac{b}{a}} \arctan \left (\frac{a \sqrt{\frac{b}{a}}}{b \sqrt{x}}\right ) - 2 \, a^{2}}{3 \,{\left (a^{3} b x^{2} + a^{4} x\right )} \sqrt{x}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/6*(30*b^2*x^2 + 20*a*b*x + 15*(b^2*x^2 + a*b*x)*sqrt(x)*sqrt(-b/a)*log((b*x +
2*a*sqrt(x)*sqrt(-b/a) - a)/(b*x + a)) - 4*a^2)/((a^3*b*x^2 + a^4*x)*sqrt(x)),
1/3*(15*b^2*x^2 + 10*a*b*x - 15*(b^2*x^2 + a*b*x)*sqrt(x)*sqrt(b/a)*arctan(a*sqr
t(b/a)/(b*sqrt(x))) - 2*a^2)/((a^3*b*x^2 + a^4*x)*sqrt(x))]

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Sympy [A]  time = 11.6904, size = 991, normalized size = 14.36 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

15*a**(27/2)*b**2*x**2*atan(sqrt(b)*sqrt(x)/sqrt(a))/(3*a**17*sqrt(b)*x**2 + 12*
a**16*b**(3/2)*x**3 + 18*a**15*b**(5/2)*x**4 + 12*a**14*b**(7/2)*x**5 + 3*a**13*
b**(9/2)*x**6) + 60*a**(25/2)*b**3*x**3*atan(sqrt(b)*sqrt(x)/sqrt(a))/(3*a**17*s
qrt(b)*x**2 + 12*a**16*b**(3/2)*x**3 + 18*a**15*b**(5/2)*x**4 + 12*a**14*b**(7/2
)*x**5 + 3*a**13*b**(9/2)*x**6) + 90*a**(23/2)*b**4*x**4*atan(sqrt(b)*sqrt(x)/sq
rt(a))/(3*a**17*sqrt(b)*x**2 + 12*a**16*b**(3/2)*x**3 + 18*a**15*b**(5/2)*x**4 +
12*a**14*b**(7/2)*x**5 + 3*a**13*b**(9/2)*x**6) + 60*a**(21/2)*b**5*x**5*atan(s
qrt(b)*sqrt(x)/sqrt(a))/(3*a**17*sqrt(b)*x**2 + 12*a**16*b**(3/2)*x**3 + 18*a**1
5*b**(5/2)*x**4 + 12*a**14*b**(7/2)*x**5 + 3*a**13*b**(9/2)*x**6) + 15*a**(19/2)
*b**6*x**6*atan(sqrt(b)*sqrt(x)/sqrt(a))/(3*a**17*sqrt(b)*x**2 + 12*a**16*b**(3/
2)*x**3 + 18*a**15*b**(5/2)*x**4 + 12*a**14*b**(7/2)*x**5 + 3*a**13*b**(9/2)*x**
6) - 2*a**15*sqrt(b)*sqrt(x)/(3*a**17*sqrt(b)*x**2 + 12*a**16*b**(3/2)*x**3 + 18
*a**15*b**(5/2)*x**4 + 12*a**14*b**(7/2)*x**5 + 3*a**13*b**(9/2)*x**6) + 4*a**14
*b**(3/2)*x**(3/2)/(3*a**17*sqrt(b)*x**2 + 12*a**16*b**(3/2)*x**3 + 18*a**15*b**
(5/2)*x**4 + 12*a**14*b**(7/2)*x**5 + 3*a**13*b**(9/2)*x**6) + 39*a**13*b**(5/2)
*x**(5/2)/(3*a**17*sqrt(b)*x**2 + 12*a**16*b**(3/2)*x**3 + 18*a**15*b**(5/2)*x**
4 + 12*a**14*b**(7/2)*x**5 + 3*a**13*b**(9/2)*x**6) + 73*a**12*b**(7/2)*x**(7/2)
/(3*a**17*sqrt(b)*x**2 + 12*a**16*b**(3/2)*x**3 + 18*a**15*b**(5/2)*x**4 + 12*a*
*14*b**(7/2)*x**5 + 3*a**13*b**(9/2)*x**6) + 55*a**11*b**(9/2)*x**(9/2)/(3*a**17
*sqrt(b)*x**2 + 12*a**16*b**(3/2)*x**3 + 18*a**15*b**(5/2)*x**4 + 12*a**14*b**(7
/2)*x**5 + 3*a**13*b**(9/2)*x**6) + 15*a**10*b**(11/2)*x**(11/2)/(3*a**17*sqrt(b
)*x**2 + 12*a**16*b**(3/2)*x**3 + 18*a**15*b**(5/2)*x**4 + 12*a**14*b**(7/2)*x**
5 + 3*a**13*b**(9/2)*x**6)

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GIAC/XCAS [A]  time = 0.202231, size = 78, normalized size = 1.13 \[ \frac{5 \, b^{2} \arctan \left (\frac{b \sqrt{x}}{\sqrt{a b}}\right )}{\sqrt{a b} a^{3}} + \frac{b^{2} \sqrt{x}}{{\left (b x + a\right )} a^{3}} + \frac{2 \,{\left (6 \, b x - a\right )}}{3 \, a^{3} x^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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