3.603 \(\int \frac{x^{3/2}}{(a-b x)^{5/2}} \, dx\)

Optimal. Leaf size=72 \[ \frac{2 \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a-b x}}\right )}{b^{5/2}}-\frac{2 \sqrt{x}}{b^2 \sqrt{a-b x}}+\frac{2 x^{3/2}}{3 b (a-b x)^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.0545072, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188 \[ \frac{2 \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a-b x}}\right )}{b^{5/2}}-\frac{2 \sqrt{x}}{b^2 \sqrt{a-b x}}+\frac{2 x^{3/2}}{3 b (a-b x)^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[x^(3/2)/(a - b*x)^(5/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 9.13329, size = 63, normalized size = 0.88 \[ \frac{2 x^{\frac{3}{2}}}{3 b \left (a - b x\right )^{\frac{3}{2}}} - \frac{2 \sqrt{x}}{b^{2} \sqrt{a - b x}} + \frac{2 \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a - b x}} \right )}}{b^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(3/2)/(-b*x+a)**(5/2),x)

[Out]

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

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Mathematica [A]  time = 0.0988373, size = 60, normalized size = 0.83 \[ \frac{2 \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a-b x}}\right )}{b^{5/2}}+\frac{2 \sqrt{x} (4 b x-3 a)}{3 b^2 (a-b x)^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(3/2)/(a - b*x)^(5/2),x]

[Out]

(2*Sqrt[x]*(-3*a + 4*b*x))/(3*b^2*(a - b*x)^(3/2)) + (2*ArcTan[(Sqrt[b]*Sqrt[x])
/Sqrt[a - b*x]])/b^(5/2)

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Maple [F]  time = 0.037, size = 0, normalized size = 0. \[ \int{1{x}^{{\frac{3}{2}}} \left ( -bx+a \right ) ^{-{\frac{5}{2}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(3/2)/(-b*x+a)^(5/2),x)

[Out]

int(x^(3/2)/(-b*x+a)^(5/2),x)

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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)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.226536, size = 1, normalized size = 0.01 \[ \left [\frac{3 \,{\left (b x - a\right )} \sqrt{-b x + a} \sqrt{x} \log \left (-2 \, \sqrt{-b x + a} b \sqrt{x} -{\left (2 \, b x - a\right )} \sqrt{-b}\right ) - 2 \,{\left (4 \, b x^{2} - 3 \, a x\right )} \sqrt{-b}}{3 \,{\left (b^{3} x - a b^{2}\right )} \sqrt{-b x + a} \sqrt{-b} \sqrt{x}}, -\frac{2 \,{\left (3 \,{\left (b x - a\right )} \sqrt{-b x + a} \sqrt{x} \arctan \left (\frac{\sqrt{-b x + a}}{\sqrt{b} \sqrt{x}}\right ) +{\left (4 \, b x^{2} - 3 \, a x\right )} \sqrt{b}\right )}}{3 \,{\left (b^{3} x - a b^{2}\right )} \sqrt{-b x + a} \sqrt{b} \sqrt{x}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(3/2)/(-b*x + a)^(5/2),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 31.8005, size = 833, normalized size = 11.57 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**(3/2)/(-b*x+a)**(5/2),x)

[Out]

Piecewise((-6*I*a**(39/2)*b**11*x**(27/2)*sqrt(-1 + b*x/a)*acosh(sqrt(b)*sqrt(x)
/sqrt(a))/(3*a**(39/2)*b**(27/2)*x**(27/2)*sqrt(-1 + b*x/a) - 3*a**(37/2)*b**(29
/2)*x**(29/2)*sqrt(-1 + b*x/a)) + 3*pi*a**(39/2)*b**11*x**(27/2)*sqrt(-1 + b*x/a
)/(3*a**(39/2)*b**(27/2)*x**(27/2)*sqrt(-1 + b*x/a) - 3*a**(37/2)*b**(29/2)*x**(
29/2)*sqrt(-1 + b*x/a)) + 6*I*a**(37/2)*b**12*x**(29/2)*sqrt(-1 + b*x/a)*acosh(s
qrt(b)*sqrt(x)/sqrt(a))/(3*a**(39/2)*b**(27/2)*x**(27/2)*sqrt(-1 + b*x/a) - 3*a*
*(37/2)*b**(29/2)*x**(29/2)*sqrt(-1 + b*x/a)) - 3*pi*a**(37/2)*b**12*x**(29/2)*s
qrt(-1 + b*x/a)/(3*a**(39/2)*b**(27/2)*x**(27/2)*sqrt(-1 + b*x/a) - 3*a**(37/2)*
b**(29/2)*x**(29/2)*sqrt(-1 + b*x/a)) + 6*I*a**19*b**(23/2)*x**14/(3*a**(39/2)*b
**(27/2)*x**(27/2)*sqrt(-1 + b*x/a) - 3*a**(37/2)*b**(29/2)*x**(29/2)*sqrt(-1 +
b*x/a)) - 8*I*a**18*b**(25/2)*x**15/(3*a**(39/2)*b**(27/2)*x**(27/2)*sqrt(-1 + b
*x/a) - 3*a**(37/2)*b**(29/2)*x**(29/2)*sqrt(-1 + b*x/a)), Abs(b*x/a) > 1), (6*a
**(39/2)*b**11*x**(27/2)*sqrt(1 - b*x/a)*asin(sqrt(b)*sqrt(x)/sqrt(a))/(3*a**(39
/2)*b**(27/2)*x**(27/2)*sqrt(1 - b*x/a) - 3*a**(37/2)*b**(29/2)*x**(29/2)*sqrt(1
- b*x/a)) - 6*a**(37/2)*b**12*x**(29/2)*sqrt(1 - b*x/a)*asin(sqrt(b)*sqrt(x)/sq
rt(a))/(3*a**(39/2)*b**(27/2)*x**(27/2)*sqrt(1 - b*x/a) - 3*a**(37/2)*b**(29/2)*
x**(29/2)*sqrt(1 - b*x/a)) - 6*a**19*b**(23/2)*x**14/(3*a**(39/2)*b**(27/2)*x**(
27/2)*sqrt(1 - b*x/a) - 3*a**(37/2)*b**(29/2)*x**(29/2)*sqrt(1 - b*x/a)) + 8*a**
18*b**(25/2)*x**15/(3*a**(39/2)*b**(27/2)*x**(27/2)*sqrt(1 - b*x/a) - 3*a**(37/2
)*b**(29/2)*x**(29/2)*sqrt(1 - b*x/a)), True))

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GIAC/XCAS [A]  time = 0.230126, size = 266, normalized size = 3.69 \[ -\frac{{\left (\frac{3 \, \sqrt{-b}{\rm ln}\left ({\left (\sqrt{-b x + a} \sqrt{-b} - \sqrt{{\left (b x - a\right )} b + a b}\right )}^{2}\right )}{b} - \frac{8 \,{\left (3 \, a{\left (\sqrt{-b x + a} \sqrt{-b} - \sqrt{{\left (b x - a\right )} b + a b}\right )}^{4} \sqrt{-b} - 3 \, a^{2}{\left (\sqrt{-b x + a} \sqrt{-b} - \sqrt{{\left (b x - a\right )} b + a b}\right )}^{2} \sqrt{-b} b + 2 \, a^{3} \sqrt{-b} b^{2}\right )}}{{\left ({\left (\sqrt{-b x + a} \sqrt{-b} - \sqrt{{\left (b x - a\right )} b + a b}\right )}^{2} - a b\right )}^{3}}\right )}{\left | b \right |}}{3 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(3/2)/(-b*x + a)^(5/2),x, algorithm="giac")

[Out]

-1/3*(3*sqrt(-b)*ln((sqrt(-b*x + a)*sqrt(-b) - sqrt((b*x - a)*b + a*b))^2)/b - 8
*(3*a*(sqrt(-b*x + a)*sqrt(-b) - sqrt((b*x - a)*b + a*b))^4*sqrt(-b) - 3*a^2*(sq
rt(-b*x + a)*sqrt(-b) - sqrt((b*x - a)*b + a*b))^2*sqrt(-b)*b + 2*a^3*sqrt(-b)*b
^2)/((sqrt(-b*x + a)*sqrt(-b) - sqrt((b*x - a)*b + a*b))^2 - a*b)^3)*abs(b)/b^3