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

Optimal. Leaf size=100 \[ -\frac{15 a^2 \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a-b x}}\right )}{4 b^{7/2}}+\frac{15 a \sqrt{x} \sqrt{a-b x}}{4 b^3}+\frac{5 x^{3/2} \sqrt{a-b x}}{2 b^2}+\frac{2 x^{5/2}}{b \sqrt{a-b x}} \]

[Out]

(2*x^(5/2))/(b*Sqrt[a - b*x]) + (15*a*Sqrt[x]*Sqrt[a - b*x])/(4*b^3) + (5*x^(3/2
)*Sqrt[a - b*x])/(2*b^2) - (15*a^2*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a - b*x]])/(4*b
^(7/2))

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Rubi [A]  time = 0.078231, antiderivative size = 100, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{15 a^2 \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a-b x}}\right )}{4 b^{7/2}}+\frac{15 a \sqrt{x} \sqrt{a-b x}}{4 b^3}+\frac{5 x^{3/2} \sqrt{a-b x}}{2 b^2}+\frac{2 x^{5/2}}{b \sqrt{a-b x}} \]

Antiderivative was successfully verified.

[In]  Int[x^(5/2)/(a - b*x)^(3/2),x]

[Out]

(2*x^(5/2))/(b*Sqrt[a - b*x]) + (15*a*Sqrt[x]*Sqrt[a - b*x])/(4*b^3) + (5*x^(3/2
)*Sqrt[a - b*x])/(2*b^2) - (15*a^2*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a - b*x]])/(4*b
^(7/2))

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Rubi in Sympy [A]  time = 11.6016, size = 90, normalized size = 0.9 \[ - \frac{15 a^{2} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a - b x}} \right )}}{4 b^{\frac{7}{2}}} + \frac{15 a \sqrt{x} \sqrt{a - b x}}{4 b^{3}} + \frac{2 x^{\frac{5}{2}}}{b \sqrt{a - b x}} + \frac{5 x^{\frac{3}{2}} \sqrt{a - b x}}{2 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-15*a**2*atan(sqrt(b)*sqrt(x)/sqrt(a - b*x))/(4*b**(7/2)) + 15*a*sqrt(x)*sqrt(a
- b*x)/(4*b**3) + 2*x**(5/2)/(b*sqrt(a - b*x)) + 5*x**(3/2)*sqrt(a - b*x)/(2*b**
2)

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Mathematica [A]  time = 0.0926373, size = 94, normalized size = 0.94 \[ \sqrt{a-b x} \left (-\frac{2 a^2 \sqrt{x}}{b^3 (b x-a)}+\frac{7 a \sqrt{x}}{4 b^3}+\frac{x^{3/2}}{2 b^2}\right )-\frac{15 a^2 \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a-b x}}\right )}{4 b^{7/2}} \]

Antiderivative was successfully verified.

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

[Out]

Sqrt[a - b*x]*((7*a*Sqrt[x])/(4*b^3) + x^(3/2)/(2*b^2) - (2*a^2*Sqrt[x])/(b^3*(-
a + b*x))) - (15*a^2*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a - b*x]])/(4*b^(7/2))

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Maple [A]  time = 0.039, size = 127, normalized size = 1.3 \[{\frac{2\,bx+7\,a}{4\,{b}^{3}}\sqrt{x}\sqrt{-bx+a}}+{1 \left ( -{\frac{15\,{a}^{2}}{8}\arctan \left ({1\sqrt{b} \left ( x-{\frac{a}{2\,b}} \right ){\frac{1}{\sqrt{-b{x}^{2}+ax}}}} \right ){b}^{-{\frac{7}{2}}}}-2\,{\frac{{a}^{2}}{{b}^{4}}\sqrt{-b \left ( x-{\frac{a}{b}} \right ) ^{2}- \left ( x-{\frac{a}{b}} \right ) a} \left ( x-{\frac{a}{b}} \right ) ^{-1}} \right ) \sqrt{x \left ( -bx+a \right ) }{\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{-bx+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(5/2)/(-b*x+a)^(3/2),x)

[Out]

1/4*(2*b*x+7*a)/b^3*x^(1/2)*(-b*x+a)^(1/2)+(-15/8*a^2/b^(7/2)*arctan(b^(1/2)*(x-
1/2*a/b)/(-b*x^2+a*x)^(1/2))-2*a^2/b^4/(x-a/b)*(-b*(x-a/b)^2-(x-a/b)*a)^(1/2))*(
x*(-b*x+a))^(1/2)/x^(1/2)/(-b*x+a)^(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)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.222319, size = 1, normalized size = 0.01 \[ \left [\frac{15 \, \sqrt{-b x + a} a^{2} \sqrt{x} \log \left (2 \, \sqrt{-b x + a} b \sqrt{x} -{\left (2 \, b x - a\right )} \sqrt{-b}\right ) - 2 \,{\left (2 \, b^{2} x^{3} + 5 \, a b x^{2} - 15 \, a^{2} x\right )} \sqrt{-b}}{8 \, \sqrt{-b x + a} \sqrt{-b} b^{3} \sqrt{x}}, \frac{15 \, \sqrt{-b x + a} a^{2} \sqrt{x} \arctan \left (\frac{\sqrt{-b x + a}}{\sqrt{b} \sqrt{x}}\right ) -{\left (2 \, b^{2} x^{3} + 5 \, a b x^{2} - 15 \, a^{2} x\right )} \sqrt{b}}{4 \, \sqrt{-b x + a} b^{\frac{7}{2}} \sqrt{x}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/8*(15*sqrt(-b*x + a)*a^2*sqrt(x)*log(2*sqrt(-b*x + a)*b*sqrt(x) - (2*b*x - a)
*sqrt(-b)) - 2*(2*b^2*x^3 + 5*a*b*x^2 - 15*a^2*x)*sqrt(-b))/(sqrt(-b*x + a)*sqrt
(-b)*b^3*sqrt(x)), 1/4*(15*sqrt(-b*x + a)*a^2*sqrt(x)*arctan(sqrt(-b*x + a)/(sqr
t(b)*sqrt(x))) - (2*b^2*x^3 + 5*a*b*x^2 - 15*a^2*x)*sqrt(b))/(sqrt(-b*x + a)*b^(
7/2)*sqrt(x))]

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Sympy [A]  time = 85.3056, size = 224, normalized size = 2.24 \[ \begin{cases} - \frac{15 i a^{\frac{3}{2}} \sqrt{x}}{4 b^{3} \sqrt{-1 + \frac{b x}{a}}} + \frac{5 i \sqrt{a} x^{\frac{3}{2}}}{4 b^{2} \sqrt{-1 + \frac{b x}{a}}} + \frac{15 i a^{2} \operatorname{acosh}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{4 b^{\frac{7}{2}}} + \frac{i x^{\frac{5}{2}}}{2 \sqrt{a} b \sqrt{-1 + \frac{b x}{a}}} & \text{for}\: \left |{\frac{b x}{a}}\right | > 1 \\\frac{15 a^{\frac{3}{2}} \sqrt{x}}{4 b^{3} \sqrt{1 - \frac{b x}{a}}} - \frac{5 \sqrt{a} x^{\frac{3}{2}}}{4 b^{2} \sqrt{1 - \frac{b x}{a}}} - \frac{15 a^{2} \operatorname{asin}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{4 b^{\frac{7}{2}}} - \frac{x^{\frac{5}{2}}}{2 \sqrt{a} b \sqrt{1 - \frac{b x}{a}}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((-15*I*a**(3/2)*sqrt(x)/(4*b**3*sqrt(-1 + b*x/a)) + 5*I*sqrt(a)*x**(3/
2)/(4*b**2*sqrt(-1 + b*x/a)) + 15*I*a**2*acosh(sqrt(b)*sqrt(x)/sqrt(a))/(4*b**(7
/2)) + I*x**(5/2)/(2*sqrt(a)*b*sqrt(-1 + b*x/a)), Abs(b*x/a) > 1), (15*a**(3/2)*
sqrt(x)/(4*b**3*sqrt(1 - b*x/a)) - 5*sqrt(a)*x**(3/2)/(4*b**2*sqrt(1 - b*x/a)) -
15*a**2*asin(sqrt(b)*sqrt(x)/sqrt(a))/(4*b**(7/2)) - x**(5/2)/(2*sqrt(a)*b*sqrt
(1 - b*x/a)), True))

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GIAC/XCAS [A]  time = 0.228643, size = 208, normalized size = 2.08 \[ \frac{{\left (2 \, \sqrt{{\left (b x - a\right )} b + a b} \sqrt{-b x + a}{\left (\frac{2 \,{\left (b x - a\right )}}{b^{3}} + \frac{9 \, a}{b^{3}}\right )} - \frac{32 \, a^{3} \sqrt{-b}}{{\left ({\left (\sqrt{-b x + a} \sqrt{-b} - \sqrt{{\left (b x - a\right )} b + a b}\right )}^{2} - a b\right )} b^{2}} + \frac{15 \, a^{2} \sqrt{-b}{\rm ln}\left ({\left (\sqrt{-b x + a} \sqrt{-b} - \sqrt{{\left (b x - a\right )} b + a b}\right )}^{2}\right )}{b^{3}}\right )}{\left | b \right |}}{8 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*(2*sqrt((b*x - a)*b + a*b)*sqrt(-b*x + a)*(2*(b*x - a)/b^3 + 9*a/b^3) - 32*a
^3*sqrt(-b)/(((sqrt(-b*x + a)*sqrt(-b) - sqrt((b*x - a)*b + a*b))^2 - a*b)*b^2)
+ 15*a^2*sqrt(-b)*ln((sqrt(-b*x + a)*sqrt(-b) - sqrt((b*x - a)*b + a*b))^2)/b^3)
*abs(b)/b^2