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

Optimal. Leaf size=53 \[ -\frac{2 a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{b^{5/2}}+\frac{2 a \sqrt{x}}{b^2}+\frac{2 x^{3/2}}{3 b} \]

[Out]

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

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Rubi [A]  time = 0.0475495, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ -\frac{2 a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{b^{5/2}}+\frac{2 a \sqrt{x}}{b^2}+\frac{2 x^{3/2}}{3 b} \]

Antiderivative was successfully verified.

[In]  Int[x^(3/2)/(-a + b*x),x]

[Out]

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

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Rubi in Sympy [A]  time = 8.94808, size = 49, normalized size = 0.92 \[ - \frac{2 a^{\frac{3}{2}} \operatorname{atanh}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{b^{\frac{5}{2}}} + \frac{2 a \sqrt{x}}{b^{2}} + \frac{2 x^{\frac{3}{2}}}{3 b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0314575, size = 49, normalized size = 0.92 \[ \frac{2 \sqrt{x} (3 a+b x)}{3 b^2}-\frac{2 a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{b^{5/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(3/2)/(-a + b*x),x]

[Out]

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

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Maple [A]  time = 0.009, size = 43, normalized size = 0.8 \[ 2\,{\frac{1/3\,b{x}^{3/2}+a\sqrt{x}}{{b}^{2}}}-2\,{\frac{{a}^{2}}{{b}^{2}\sqrt{ab}}{\it Artanh} \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(3/2)/(b*x-a),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.228594, size = 1, normalized size = 0.02 \[ \left [\frac{3 \, a \sqrt{\frac{a}{b}} \log \left (\frac{b x - 2 \, b \sqrt{x} \sqrt{\frac{a}{b}} + a}{b x - a}\right ) + 2 \,{\left (b x + 3 \, a\right )} \sqrt{x}}{3 \, b^{2}}, -\frac{2 \,{\left (3 \, a \sqrt{-\frac{a}{b}} \arctan \left (\frac{\sqrt{x}}{\sqrt{-\frac{a}{b}}}\right ) -{\left (b x + 3 \, a\right )} \sqrt{x}\right )}}{3 \, b^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 6.04124, size = 619, normalized size = 11.68 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**(3/2)/(b*x-a),x)

[Out]

Piecewise((-6*a**(25/2)*b**5*x**(15/2)*acoth(sqrt(b)*sqrt(x)/sqrt(a))/(3*a**11*b
**(15/2)*x**(15/2) - 3*a**10*b**(17/2)*x**(17/2)) - 3*I*pi*a**(25/2)*b**5*x**(15
/2)/(3*a**11*b**(15/2)*x**(15/2) - 3*a**10*b**(17/2)*x**(17/2)) + 6*a**(23/2)*b*
*6*x**(17/2)*acoth(sqrt(b)*sqrt(x)/sqrt(a))/(3*a**11*b**(15/2)*x**(15/2) - 3*a**
10*b**(17/2)*x**(17/2)) + 3*I*pi*a**(23/2)*b**6*x**(17/2)/(3*a**11*b**(15/2)*x**
(15/2) - 3*a**10*b**(17/2)*x**(17/2)) + 6*a**12*b**(11/2)*x**8/(3*a**11*b**(15/2
)*x**(15/2) - 3*a**10*b**(17/2)*x**(17/2)) - 4*a**11*b**(13/2)*x**9/(3*a**11*b**
(15/2)*x**(15/2) - 3*a**10*b**(17/2)*x**(17/2)) - 2*a**10*b**(15/2)*x**10/(3*a**
11*b**(15/2)*x**(15/2) - 3*a**10*b**(17/2)*x**(17/2)), Abs(b*x/a) > 1), (-6*a**(
25/2)*b**5*x**(15/2)*atanh(sqrt(b)*sqrt(x)/sqrt(a))/(3*a**11*b**(15/2)*x**(15/2)
- 3*a**10*b**(17/2)*x**(17/2)) + 6*a**(23/2)*b**6*x**(17/2)*atanh(sqrt(b)*sqrt(
x)/sqrt(a))/(3*a**11*b**(15/2)*x**(15/2) - 3*a**10*b**(17/2)*x**(17/2)) + 6*a**1
2*b**(11/2)*x**8/(3*a**11*b**(15/2)*x**(15/2) - 3*a**10*b**(17/2)*x**(17/2)) - 4
*a**11*b**(13/2)*x**9/(3*a**11*b**(15/2)*x**(15/2) - 3*a**10*b**(17/2)*x**(17/2)
) - 2*a**10*b**(15/2)*x**10/(3*a**11*b**(15/2)*x**(15/2) - 3*a**10*b**(17/2)*x**
(17/2)), True))

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GIAC/XCAS [A]  time = 0.223702, size = 63, normalized size = 1.19 \[ \frac{2 \, a^{2} \arctan \left (\frac{b \sqrt{x}}{\sqrt{-a b}}\right )}{\sqrt{-a b} b^{2}} + \frac{2 \,{\left (b^{2} x^{\frac{3}{2}} + 3 \, a b \sqrt{x}\right )}}{3 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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