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

Optimal. Leaf size=124 \[ \frac{3 a^4 \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a-b x}}\right )}{64 b^{5/2}}-\frac{3 a^3 \sqrt{x} \sqrt{a-b x}}{64 b^2}-\frac{a^2 x^{3/2} \sqrt{a-b x}}{32 b}+\frac{1}{8} a x^{5/2} \sqrt{a-b x}+\frac{1}{4} x^{5/2} (a-b x)^{3/2} \]

[Out]

(-3*a^3*Sqrt[x]*Sqrt[a - b*x])/(64*b^2) - (a^2*x^(3/2)*Sqrt[a - b*x])/(32*b) + (
a*x^(5/2)*Sqrt[a - b*x])/8 + (x^(5/2)*(a - b*x)^(3/2))/4 + (3*a^4*ArcTan[(Sqrt[b
]*Sqrt[x])/Sqrt[a - b*x]])/(64*b^(5/2))

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Rubi [A]  time = 0.0966688, antiderivative size = 124, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188 \[ \frac{3 a^4 \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a-b x}}\right )}{64 b^{5/2}}-\frac{3 a^3 \sqrt{x} \sqrt{a-b x}}{64 b^2}-\frac{a^2 x^{3/2} \sqrt{a-b x}}{32 b}+\frac{1}{8} a x^{5/2} \sqrt{a-b x}+\frac{1}{4} x^{5/2} (a-b x)^{3/2} \]

Antiderivative was successfully verified.

[In]  Int[x^(3/2)*(a - b*x)^(3/2),x]

[Out]

(-3*a^3*Sqrt[x]*Sqrt[a - b*x])/(64*b^2) - (a^2*x^(3/2)*Sqrt[a - b*x])/(32*b) + (
a*x^(5/2)*Sqrt[a - b*x])/8 + (x^(5/2)*(a - b*x)^(3/2))/4 + (3*a^4*ArcTan[(Sqrt[b
]*Sqrt[x])/Sqrt[a - b*x]])/(64*b^(5/2))

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Rubi in Sympy [A]  time = 14.9515, size = 114, normalized size = 0.92 \[ - \frac{3 a^{4} \operatorname{atan}{\left (\frac{\sqrt{a - b x}}{\sqrt{b} \sqrt{x}} \right )}}{64 b^{\frac{5}{2}}} + \frac{3 a^{3} \sqrt{x} \sqrt{a - b x}}{64 b^{2}} + \frac{a^{2} \sqrt{x} \left (a - b x\right )^{\frac{3}{2}}}{32 b^{2}} - \frac{a \sqrt{x} \left (a - b x\right )^{\frac{5}{2}}}{8 b^{2}} - \frac{x^{\frac{3}{2}} \left (a - b x\right )^{\frac{5}{2}}}{4 b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3*a**4*atan(sqrt(a - b*x)/(sqrt(b)*sqrt(x)))/(64*b**(5/2)) + 3*a**3*sqrt(x)*sqr
t(a - b*x)/(64*b**2) + a**2*sqrt(x)*(a - b*x)**(3/2)/(32*b**2) - a*sqrt(x)*(a -
b*x)**(5/2)/(8*b**2) - x**(3/2)*(a - b*x)**(5/2)/(4*b)

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Mathematica [A]  time = 0.0763476, size = 89, normalized size = 0.72 \[ \frac{3 a^4 \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a-b x}}\right )-\sqrt{b} \sqrt{x} \sqrt{a-b x} \left (3 a^3+2 a^2 b x-24 a b^2 x^2+16 b^3 x^3\right )}{64 b^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

(-(Sqrt[b]*Sqrt[x]*Sqrt[a - b*x]*(3*a^3 + 2*a^2*b*x - 24*a*b^2*x^2 + 16*b^3*x^3)
) + 3*a^4*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a - b*x]])/(64*b^(5/2))

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Maple [A]  time = 0.009, size = 127, normalized size = 1. \[ -{\frac{1}{4\,b}{x}^{{\frac{3}{2}}} \left ( -bx+a \right ) ^{{\frac{5}{2}}}}-{\frac{a}{8\,{b}^{2}} \left ( -bx+a \right ) ^{{\frac{5}{2}}}\sqrt{x}}+{\frac{{a}^{2}}{32\,{b}^{2}} \left ( -bx+a \right ) ^{{\frac{3}{2}}}\sqrt{x}}+{\frac{3\,{a}^{3}}{64\,{b}^{2}}\sqrt{x}\sqrt{-bx+a}}+{\frac{3\,{a}^{4}}{128}\sqrt{x \left ( -bx+a \right ) }\arctan \left ({1\sqrt{b} \left ( x-{\frac{a}{2\,b}} \right ){\frac{1}{\sqrt{-b{x}^{2}+ax}}}} \right ){b}^{-{\frac{5}{2}}}{\frac{1}{\sqrt{-bx+a}}}{\frac{1}{\sqrt{x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(3/2)*(-b*x+a)^(3/2),x)

[Out]

-1/4/b*x^(3/2)*(-b*x+a)^(5/2)-1/8*a/b^2*x^(1/2)*(-b*x+a)^(5/2)+1/32*a^2/b^2*(-b*
x+a)^(3/2)*x^(1/2)+3/64*a^3*x^(1/2)*(-b*x+a)^(1/2)/b^2+3/128*a^4/b^(5/2)*(x*(-b*
x+a))^(1/2)/(-b*x+a)^(1/2)/x^(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)*x^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.232363, size = 1, normalized size = 0.01 \[ \left [\frac{3 \, a^{4} \log \left (-2 \, \sqrt{-b x + a} b \sqrt{x} -{\left (2 \, b x - a\right )} \sqrt{-b}\right ) - 2 \,{\left (16 \, b^{3} x^{3} - 24 \, a b^{2} x^{2} + 2 \, a^{2} b x + 3 \, a^{3}\right )} \sqrt{-b x + a} \sqrt{-b} \sqrt{x}}{128 \, \sqrt{-b} b^{2}}, -\frac{3 \, a^{4} \arctan \left (\frac{\sqrt{-b x + a}}{\sqrt{b} \sqrt{x}}\right ) +{\left (16 \, b^{3} x^{3} - 24 \, a b^{2} x^{2} + 2 \, a^{2} b x + 3 \, a^{3}\right )} \sqrt{-b x + a} \sqrt{b} \sqrt{x}}{64 \, b^{\frac{5}{2}}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/128*(3*a^4*log(-2*sqrt(-b*x + a)*b*sqrt(x) - (2*b*x - a)*sqrt(-b)) - 2*(16*b^
3*x^3 - 24*a*b^2*x^2 + 2*a^2*b*x + 3*a^3)*sqrt(-b*x + a)*sqrt(-b)*sqrt(x))/(sqrt
(-b)*b^2), -1/64*(3*a^4*arctan(sqrt(-b*x + a)/(sqrt(b)*sqrt(x))) + (16*b^3*x^3 -
24*a*b^2*x^2 + 2*a^2*b*x + 3*a^3)*sqrt(-b*x + a)*sqrt(b)*sqrt(x))/b^(5/2)]

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Sympy [A]  time = 54.6287, size = 323, normalized size = 2.6 \[ \begin{cases} \frac{3 i a^{\frac{7}{2}} \sqrt{x}}{64 b^{2} \sqrt{-1 + \frac{b x}{a}}} - \frac{i a^{\frac{5}{2}} x^{\frac{3}{2}}}{64 b \sqrt{-1 + \frac{b x}{a}}} - \frac{13 i a^{\frac{3}{2}} x^{\frac{5}{2}}}{32 \sqrt{-1 + \frac{b x}{a}}} + \frac{5 i \sqrt{a} b x^{\frac{7}{2}}}{8 \sqrt{-1 + \frac{b x}{a}}} - \frac{3 i a^{4} \operatorname{acosh}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{64 b^{\frac{5}{2}}} - \frac{i b^{2} x^{\frac{9}{2}}}{4 \sqrt{a} \sqrt{-1 + \frac{b x}{a}}} & \text{for}\: \left |{\frac{b x}{a}}\right | > 1 \\- \frac{3 a^{\frac{7}{2}} \sqrt{x}}{64 b^{2} \sqrt{1 - \frac{b x}{a}}} + \frac{a^{\frac{5}{2}} x^{\frac{3}{2}}}{64 b \sqrt{1 - \frac{b x}{a}}} + \frac{13 a^{\frac{3}{2}} x^{\frac{5}{2}}}{32 \sqrt{1 - \frac{b x}{a}}} - \frac{5 \sqrt{a} b x^{\frac{7}{2}}}{8 \sqrt{1 - \frac{b x}{a}}} + \frac{3 a^{4} \operatorname{asin}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{64 b^{\frac{5}{2}}} + \frac{b^{2} x^{\frac{9}{2}}}{4 \sqrt{a} \sqrt{1 - \frac{b x}{a}}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((3*I*a**(7/2)*sqrt(x)/(64*b**2*sqrt(-1 + b*x/a)) - I*a**(5/2)*x**(3/2)
/(64*b*sqrt(-1 + b*x/a)) - 13*I*a**(3/2)*x**(5/2)/(32*sqrt(-1 + b*x/a)) + 5*I*sq
rt(a)*b*x**(7/2)/(8*sqrt(-1 + b*x/a)) - 3*I*a**4*acosh(sqrt(b)*sqrt(x)/sqrt(a))/
(64*b**(5/2)) - I*b**2*x**(9/2)/(4*sqrt(a)*sqrt(-1 + b*x/a)), Abs(b*x/a) > 1), (
-3*a**(7/2)*sqrt(x)/(64*b**2*sqrt(1 - b*x/a)) + a**(5/2)*x**(3/2)/(64*b*sqrt(1 -
b*x/a)) + 13*a**(3/2)*x**(5/2)/(32*sqrt(1 - b*x/a)) - 5*sqrt(a)*b*x**(7/2)/(8*s
qrt(1 - b*x/a)) + 3*a**4*asin(sqrt(b)*sqrt(x)/sqrt(a))/(64*b**(5/2)) + b**2*x**(
9/2)/(4*sqrt(a)*sqrt(1 - b*x/a)), True))

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GIAC/XCAS [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out