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

Optimal. Leaf size=70 \[ \frac{5 a^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{b^{7/2}}-\frac{5 a \sqrt{x}}{b^3}-\frac{x^{5/2}}{b (a+b x)}+\frac{5 x^{3/2}}{3 b^2} \]

[Out]

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

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Rubi [A]  time = 0.0594221, antiderivative size = 70, 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 \[ \frac{5 a^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{b^{7/2}}-\frac{5 a \sqrt{x}}{b^3}-\frac{x^{5/2}}{b (a+b x)}+\frac{5 x^{3/2}}{3 b^2} \]

Antiderivative was successfully verified.

[In]  Int[x^(5/2)/(a + b*x)^2,x]

[Out]

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

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Rubi in Sympy [A]  time = 11.9204, size = 63, normalized size = 0.9 \[ \frac{5 a^{\frac{3}{2}} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{b^{\frac{7}{2}}} - \frac{5 a \sqrt{x}}{b^{3}} - \frac{x^{\frac{5}{2}}}{b \left (a + b x\right )} + \frac{5 x^{\frac{3}{2}}}{3 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0666243, size = 68, normalized size = 0.97 \[ \frac{5 a^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{b^{7/2}}+\frac{\sqrt{x} \left (-15 a^2-10 a b x+2 b^2 x^2\right )}{3 b^3 (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.018, size = 61, normalized size = 0.9 \[{\frac{2}{3\,{b}^{2}}{x}^{{\frac{3}{2}}}}-4\,{\frac{a\sqrt{x}}{{b}^{3}}}-{\frac{{a}^{2}}{{b}^{3} \left ( bx+a \right ) }\sqrt{x}}+5\,{\frac{{a}^{2}}{{b}^{3}\sqrt{ab}}\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(5/2)/(b*x+a)^2,x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.227829, size = 1, normalized size = 0.01 \[ \left [\frac{15 \,{\left (a b x + a^{2}\right )} \sqrt{-\frac{a}{b}} \log \left (\frac{b x + 2 \, b \sqrt{x} \sqrt{-\frac{a}{b}} - a}{b x + a}\right ) + 2 \,{\left (2 \, b^{2} x^{2} - 10 \, a b x - 15 \, a^{2}\right )} \sqrt{x}}{6 \,{\left (b^{4} x + a b^{3}\right )}}, \frac{15 \,{\left (a b x + a^{2}\right )} \sqrt{\frac{a}{b}} \arctan \left (\frac{\sqrt{x}}{\sqrt{\frac{a}{b}}}\right ) +{\left (2 \, b^{2} x^{2} - 10 \, a b x - 15 \, a^{2}\right )} \sqrt{x}}{3 \,{\left (b^{4} x + a b^{3}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/6*(15*(a*b*x + a^2)*sqrt(-a/b)*log((b*x + 2*b*sqrt(x)*sqrt(-a/b) - a)/(b*x +
a)) + 2*(2*b^2*x^2 - 10*a*b*x - 15*a^2)*sqrt(x))/(b^4*x + a*b^3), 1/3*(15*(a*b*x
+ a^2)*sqrt(a/b)*arctan(sqrt(x)/sqrt(a/b)) + (2*b^2*x^2 - 10*a*b*x - 15*a^2)*sq
rt(x))/(b^4*x + a*b^3)]

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Sympy [A]  time = 12.3633, size = 257, normalized size = 3.67 \[ \frac{15 a^{\frac{61}{2}} b^{17} x^{\frac{41}{2}} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{3 a^{29} b^{\frac{41}{2}} x^{\frac{41}{2}} + 3 a^{28} b^{\frac{43}{2}} x^{\frac{43}{2}}} + \frac{15 a^{\frac{59}{2}} b^{18} x^{\frac{43}{2}} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{3 a^{29} b^{\frac{41}{2}} x^{\frac{41}{2}} + 3 a^{28} b^{\frac{43}{2}} x^{\frac{43}{2}}} - \frac{15 a^{30} b^{\frac{35}{2}} x^{21}}{3 a^{29} b^{\frac{41}{2}} x^{\frac{41}{2}} + 3 a^{28} b^{\frac{43}{2}} x^{\frac{43}{2}}} - \frac{10 a^{29} b^{\frac{37}{2}} x^{22}}{3 a^{29} b^{\frac{41}{2}} x^{\frac{41}{2}} + 3 a^{28} b^{\frac{43}{2}} x^{\frac{43}{2}}} + \frac{2 a^{28} b^{\frac{39}{2}} x^{23}}{3 a^{29} b^{\frac{41}{2}} x^{\frac{41}{2}} + 3 a^{28} b^{\frac{43}{2}} x^{\frac{43}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

15*a**(61/2)*b**17*x**(41/2)*atan(sqrt(b)*sqrt(x)/sqrt(a))/(3*a**29*b**(41/2)*x*
*(41/2) + 3*a**28*b**(43/2)*x**(43/2)) + 15*a**(59/2)*b**18*x**(43/2)*atan(sqrt(
b)*sqrt(x)/sqrt(a))/(3*a**29*b**(41/2)*x**(41/2) + 3*a**28*b**(43/2)*x**(43/2))
- 15*a**30*b**(35/2)*x**21/(3*a**29*b**(41/2)*x**(41/2) + 3*a**28*b**(43/2)*x**(
43/2)) - 10*a**29*b**(37/2)*x**22/(3*a**29*b**(41/2)*x**(41/2) + 3*a**28*b**(43/
2)*x**(43/2)) + 2*a**28*b**(39/2)*x**23/(3*a**29*b**(41/2)*x**(41/2) + 3*a**28*b
**(43/2)*x**(43/2))

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GIAC/XCAS [A]  time = 0.205941, size = 88, normalized size = 1.26 \[ \frac{5 \, a^{2} \arctan \left (\frac{b \sqrt{x}}{\sqrt{a b}}\right )}{\sqrt{a b} b^{3}} - \frac{a^{2} \sqrt{x}}{{\left (b x + a\right )} b^{3}} + \frac{2 \,{\left (b^{4} x^{\frac{3}{2}} - 6 \, a b^{3} \sqrt{x}\right )}}{3 \, b^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

5*a^2*arctan(b*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*b^3) - a^2*sqrt(x)/((b*x + a)*b^3)
+ 2/3*(b^4*x^(3/2) - 6*a*b^3*sqrt(x))/b^6