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

Optimal. Leaf size=57 \[ -\frac{3 \sqrt{a} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{b^{5/2}}-\frac{x^{3/2}}{b (a+b x)}+\frac{3 \sqrt{x}}{b^2} \]

[Out]

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

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Rubi [A]  time = 0.0461128, antiderivative size = 57, 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 \[ -\frac{3 \sqrt{a} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{b^{5/2}}-\frac{x^{3/2}}{b (a+b x)}+\frac{3 \sqrt{x}}{b^2} \]

Antiderivative was successfully verified.

[In]  Int[x^(3/2)/(a + b*x)^2,x]

[Out]

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

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Rubi in Sympy [A]  time = 9.13102, size = 49, normalized size = 0.86 \[ - \frac{3 \sqrt{a} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{b^{\frac{5}{2}}} - \frac{x^{\frac{3}{2}}}{b \left (a + b x\right )} + \frac{3 \sqrt{x}}{b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0516494, size = 54, normalized size = 0.95 \[ \frac{\sqrt{x} (3 a+2 b x)}{b^2 (a+b x)}-\frac{3 \sqrt{a} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{b^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.016, size = 47, normalized size = 0.8 \[ 2\,{\frac{\sqrt{x}}{{b}^{2}}}+{\frac{a}{{b}^{2} \left ( bx+a \right ) }\sqrt{x}}-3\,{\frac{a}{{b}^{2}\sqrt{ab}}\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(3/2)/(b*x+a)^2,x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.234073, size = 1, normalized size = 0.02 \[ \left [\frac{3 \,{\left (b x + a\right )} \sqrt{-\frac{a}{b}} \log \left (\frac{b x - 2 \, b \sqrt{x} \sqrt{-\frac{a}{b}} - a}{b x + a}\right ) + 2 \,{\left (2 \, b x + 3 \, a\right )} \sqrt{x}}{2 \,{\left (b^{3} x + a b^{2}\right )}}, -\frac{3 \,{\left (b x + a\right )} \sqrt{\frac{a}{b}} \arctan \left (\frac{\sqrt{x}}{\sqrt{\frac{a}{b}}}\right ) -{\left (2 \, b x + 3 \, a\right )} \sqrt{x}}{b^{3} x + a b^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 5.86463, size = 199, normalized size = 3.49 \[ - \frac{3 a^{\frac{17}{2}} b^{4} x^{\frac{13}{2}} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{a^{8} b^{\frac{13}{2}} x^{\frac{13}{2}} + a^{7} b^{\frac{15}{2}} x^{\frac{15}{2}}} - \frac{3 a^{\frac{15}{2}} b^{5} x^{\frac{15}{2}} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{a^{8} b^{\frac{13}{2}} x^{\frac{13}{2}} + a^{7} b^{\frac{15}{2}} x^{\frac{15}{2}}} + \frac{3 a^{8} b^{\frac{9}{2}} x^{7}}{a^{8} b^{\frac{13}{2}} x^{\frac{13}{2}} + a^{7} b^{\frac{15}{2}} x^{\frac{15}{2}}} + \frac{2 a^{7} b^{\frac{11}{2}} x^{8}}{a^{8} b^{\frac{13}{2}} x^{\frac{13}{2}} + a^{7} b^{\frac{15}{2}} x^{\frac{15}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.205131, size = 62, normalized size = 1.09 \[ -\frac{3 \, a \arctan \left (\frac{b \sqrt{x}}{\sqrt{a b}}\right )}{\sqrt{a b} b^{2}} + \frac{a \sqrt{x}}{{\left (b x + a\right )} b^{2}} + \frac{2 \, \sqrt{x}}{b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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