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

Optimal. Leaf size=74 \[ \frac{3 a^2 \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a-b x}}\right )}{4 \sqrt{b}}+\frac{3}{4} a \sqrt{x} \sqrt{a-b x}+\frac{1}{2} \sqrt{x} (a-b x)^{3/2} \]

[Out]

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

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Rubi [A]  time = 0.0505026, antiderivative size = 74, 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 \[ \frac{3 a^2 \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a-b x}}\right )}{4 \sqrt{b}}+\frac{3}{4} a \sqrt{x} \sqrt{a-b x}+\frac{1}{2} \sqrt{x} (a-b x)^{3/2} \]

Antiderivative was successfully verified.

[In]  Int[(a - b*x)^(3/2)/Sqrt[x],x]

[Out]

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

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Rubi in Sympy [A]  time = 7.83581, size = 65, normalized size = 0.88 \[ \frac{3 a^{2} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a - b x}} \right )}}{4 \sqrt{b}} + \frac{3 a \sqrt{x} \sqrt{a - b x}}{4} + \frac{\sqrt{x} \left (a - b x\right )^{\frac{3}{2}}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((-b*x+a)**(3/2)/x**(1/2),x)

[Out]

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

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Mathematica [A]  time = 0.0549427, size = 61, normalized size = 0.82 \[ \frac{1}{4} \left (\frac{3 a^2 \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a-b x}}\right )}{\sqrt{b}}+\sqrt{x} (5 a-2 b x) \sqrt{a-b x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.009, size = 83, normalized size = 1.1 \[{\frac{1}{2} \left ( -bx+a \right ) ^{{\frac{3}{2}}}\sqrt{x}}+{\frac{3\,a}{4}\sqrt{x}\sqrt{-bx+a}}+{\frac{3\,{a}^{2}}{8}\sqrt{x \left ( -bx+a \right ) }\arctan \left ({1\sqrt{b} \left ( x-{\frac{a}{2\,b}} \right ){\frac{1}{\sqrt{-b{x}^{2}+ax}}}} \right ){\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{-bx+a}}}{\frac{1}{\sqrt{b}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-b*x+a)^(3/2)/x^(1/2),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.220834, size = 1, normalized size = 0.01 \[ \left [\frac{3 \, a^{2} \log \left (-2 \, \sqrt{-b x + a} b \sqrt{x} -{\left (2 \, b x - a\right )} \sqrt{-b}\right ) - 2 \,{\left (2 \, b x - 5 \, a\right )} \sqrt{-b x + a} \sqrt{-b} \sqrt{x}}{8 \, \sqrt{-b}}, -\frac{3 \, a^{2} \arctan \left (\frac{\sqrt{-b x + a}}{\sqrt{b} \sqrt{x}}\right ) +{\left (2 \, b x - 5 \, a\right )} \sqrt{-b x + a} \sqrt{b} \sqrt{x}}{4 \, \sqrt{b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 14.0811, size = 190, normalized size = 2.57 \[ \begin{cases} - \frac{5 i a^{\frac{3}{2}} \sqrt{x}}{4 \sqrt{-1 + \frac{b x}{a}}} + \frac{7 i \sqrt{a} b x^{\frac{3}{2}}}{4 \sqrt{-1 + \frac{b x}{a}}} - \frac{3 i a^{2} \operatorname{acosh}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{4 \sqrt{b}} - \frac{i b^{2} x^{\frac{5}{2}}}{2 \sqrt{a} \sqrt{-1 + \frac{b x}{a}}} & \text{for}\: \left |{\frac{b x}{a}}\right | > 1 \\\frac{5 a^{\frac{3}{2}} \sqrt{x} \sqrt{1 - \frac{b x}{a}}}{4} - \frac{\sqrt{a} b x^{\frac{3}{2}} \sqrt{1 - \frac{b x}{a}}}{2} + \frac{3 a^{2} \operatorname{asin}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{4 \sqrt{b}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-b*x+a)**(3/2)/x**(1/2),x)

[Out]

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

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: NotImplementedError