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

Optimal. Leaf size=146 \[ \frac{3 a^5 \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a-b x}}\right )}{128 b^{5/2}}-\frac{3 a^4 \sqrt{x} \sqrt{a-b x}}{128 b^2}-\frac{a^3 x^{3/2} \sqrt{a-b x}}{64 b}+\frac{1}{16} a^2 x^{5/2} \sqrt{a-b x}+\frac{1}{8} a x^{5/2} (a-b x)^{3/2}+\frac{1}{5} x^{5/2} (a-b x)^{5/2} \]

[Out]

(-3*a^4*Sqrt[x]*Sqrt[a - b*x])/(128*b^2) - (a^3*x^(3/2)*Sqrt[a - b*x])/(64*b) +
(a^2*x^(5/2)*Sqrt[a - b*x])/16 + (a*x^(5/2)*(a - b*x)^(3/2))/8 + (x^(5/2)*(a - b
*x)^(5/2))/5 + (3*a^5*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a - b*x]])/(128*b^(5/2))

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Rubi [A]  time = 0.121534, antiderivative size = 146, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188 \[ \frac{3 a^5 \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a-b x}}\right )}{128 b^{5/2}}-\frac{3 a^4 \sqrt{x} \sqrt{a-b x}}{128 b^2}-\frac{a^3 x^{3/2} \sqrt{a-b x}}{64 b}+\frac{1}{16} a^2 x^{5/2} \sqrt{a-b x}+\frac{1}{8} a x^{5/2} (a-b x)^{3/2}+\frac{1}{5} x^{5/2} (a-b x)^{5/2} \]

Antiderivative was successfully verified.

[In]  Int[x^(3/2)*(a - b*x)^(5/2),x]

[Out]

(-3*a^4*Sqrt[x]*Sqrt[a - b*x])/(128*b^2) - (a^3*x^(3/2)*Sqrt[a - b*x])/(64*b) +
(a^2*x^(5/2)*Sqrt[a - b*x])/16 + (a*x^(5/2)*(a - b*x)^(3/2))/8 + (x^(5/2)*(a - b
*x)^(5/2))/5 + (3*a^5*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a - b*x]])/(128*b^(5/2))

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Rubi in Sympy [A]  time = 20.7348, size = 138, normalized size = 0.95 \[ \frac{3 a^{5} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a - b x}} \right )}}{128 b^{\frac{5}{2}}} + \frac{3 a^{4} \sqrt{x} \sqrt{a - b x}}{128 b^{2}} + \frac{a^{3} \sqrt{x} \left (a - b x\right )^{\frac{3}{2}}}{64 b^{2}} + \frac{a^{2} \sqrt{x} \left (a - b x\right )^{\frac{5}{2}}}{80 b^{2}} - \frac{3 a \sqrt{x} \left (a - b x\right )^{\frac{7}{2}}}{40 b^{2}} - \frac{x^{\frac{3}{2}} \left (a - b x\right )^{\frac{7}{2}}}{5 b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*a**5*atan(sqrt(b)*sqrt(x)/sqrt(a - b*x))/(128*b**(5/2)) + 3*a**4*sqrt(x)*sqrt(
a - b*x)/(128*b**2) + a**3*sqrt(x)*(a - b*x)**(3/2)/(64*b**2) + a**2*sqrt(x)*(a
- b*x)**(5/2)/(80*b**2) - 3*a*sqrt(x)*(a - b*x)**(7/2)/(40*b**2) - x**(3/2)*(a -
b*x)**(7/2)/(5*b)

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Mathematica [A]  time = 0.083481, size = 99, normalized size = 0.68 \[ \frac{15 a^5 \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a-b x}}\right )+\sqrt{b} \sqrt{x} \sqrt{a-b x} \left (-15 a^4-10 a^3 b x+248 a^2 b^2 x^2-336 a b^3 x^3+128 b^4 x^4\right )}{640 b^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[b]*Sqrt[x]*Sqrt[a - b*x]*(-15*a^4 - 10*a^3*b*x + 248*a^2*b^2*x^2 - 336*a*b
^3*x^3 + 128*b^4*x^4) + 15*a^5*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a - b*x]])/(640*b^(
5/2))

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Maple [A]  time = 0.009, size = 146, normalized size = 1. \[ -{\frac{1}{5\,b}{x}^{{\frac{3}{2}}} \left ( -bx+a \right ) ^{{\frac{7}{2}}}}-{\frac{3\,a}{40\,{b}^{2}}\sqrt{x} \left ( -bx+a \right ) ^{{\frac{7}{2}}}}+{\frac{{a}^{2}}{80\,{b}^{2}} \left ( -bx+a \right ) ^{{\frac{5}{2}}}\sqrt{x}}+{\frac{{a}^{3}}{64\,{b}^{2}} \left ( -bx+a \right ) ^{{\frac{3}{2}}}\sqrt{x}}+{\frac{3\,{a}^{4}}{128\,{b}^{2}}\sqrt{x}\sqrt{-bx+a}}+{\frac{3\,{a}^{5}}{256}\sqrt{x \left ( -bx+a \right ) }\arctan \left ({1\sqrt{b} \left ( x-{\frac{a}{2\,b}} \right ){\frac{1}{\sqrt{-b{x}^{2}+ax}}}} \right ){b}^{-{\frac{5}{2}}}{\frac{1}{\sqrt{-bx+a}}}{\frac{1}{\sqrt{x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(3/2)*(-b*x+a)^(5/2),x)

[Out]

-1/5/b*x^(3/2)*(-b*x+a)^(7/2)-3/40*a/b^2*x^(1/2)*(-b*x+a)^(7/2)+1/80*a^2/b^2*(-b
*x+a)^(5/2)*x^(1/2)+1/64*a^3/b^2*(-b*x+a)^(3/2)*x^(1/2)+3/128*a^4*x^(1/2)*(-b*x+
a)^(1/2)/b^2+3/256*a^5/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)^(5/2)*x^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.223451, size = 1, normalized size = 0.01 \[ \left [\frac{15 \, a^{5} \log \left (-2 \, \sqrt{-b x + a} b \sqrt{x} -{\left (2 \, b x - a\right )} \sqrt{-b}\right ) + 2 \,{\left (128 \, b^{4} x^{4} - 336 \, a b^{3} x^{3} + 248 \, a^{2} b^{2} x^{2} - 10 \, a^{3} b x - 15 \, a^{4}\right )} \sqrt{-b x + a} \sqrt{-b} \sqrt{x}}{1280 \, \sqrt{-b} b^{2}}, -\frac{15 \, a^{5} \arctan \left (\frac{\sqrt{-b x + a}}{\sqrt{b} \sqrt{x}}\right ) -{\left (128 \, b^{4} x^{4} - 336 \, a b^{3} x^{3} + 248 \, a^{2} b^{2} x^{2} - 10 \, a^{3} b x - 15 \, a^{4}\right )} \sqrt{-b x + a} \sqrt{b} \sqrt{x}}{640 \, b^{\frac{5}{2}}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/1280*(15*a^5*log(-2*sqrt(-b*x + a)*b*sqrt(x) - (2*b*x - a)*sqrt(-b)) + 2*(128
*b^4*x^4 - 336*a*b^3*x^3 + 248*a^2*b^2*x^2 - 10*a^3*b*x - 15*a^4)*sqrt(-b*x + a)
*sqrt(-b)*sqrt(x))/(sqrt(-b)*b^2), -1/640*(15*a^5*arctan(sqrt(-b*x + a)/(sqrt(b)
*sqrt(x))) - (128*b^4*x^4 - 336*a*b^3*x^3 + 248*a^2*b^2*x^2 - 10*a^3*b*x - 15*a^
4)*sqrt(-b*x + a)*sqrt(b)*sqrt(x))/b^(5/2)]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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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)^(5/2)*x^(3/2),x, algorithm="giac")

[Out]

Timed out