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

Optimal. Leaf size=149 \[ \frac{3 a^5 \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a-b x}}\right )}{128 b^{7/2}}-\frac{3 a^4 \sqrt{x} \sqrt{a-b x}}{128 b^3}-\frac{a^3 x^{3/2} \sqrt{a-b x}}{64 b^2}-\frac{a^2 x^{5/2} \sqrt{a-b x}}{80 b}+\frac{3}{40} a x^{7/2} \sqrt{a-b x}+\frac{1}{5} x^{7/2} (a-b x)^{3/2} \]

[Out]

(-3*a^4*Sqrt[x]*Sqrt[a - b*x])/(128*b^3) - (a^3*x^(3/2)*Sqrt[a - b*x])/(64*b^2)
- (a^2*x^(5/2)*Sqrt[a - b*x])/(80*b) + (3*a*x^(7/2)*Sqrt[a - b*x])/40 + (x^(7/2)
*(a - b*x)^(3/2))/5 + (3*a^5*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a - b*x]])/(128*b^(7/
2))

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Rubi [A]  time = 0.125709, antiderivative size = 149, 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 \[ \frac{3 a^5 \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a-b x}}\right )}{128 b^{7/2}}-\frac{3 a^4 \sqrt{x} \sqrt{a-b x}}{128 b^3}-\frac{a^3 x^{3/2} \sqrt{a-b x}}{64 b^2}-\frac{a^2 x^{5/2} \sqrt{a-b x}}{80 b}+\frac{3}{40} a x^{7/2} \sqrt{a-b x}+\frac{1}{5} x^{7/2} (a-b x)^{3/2} \]

Antiderivative was successfully verified.

[In]  Int[x^(5/2)*(a - b*x)^(3/2),x]

[Out]

(-3*a^4*Sqrt[x]*Sqrt[a - b*x])/(128*b^3) - (a^3*x^(3/2)*Sqrt[a - b*x])/(64*b^2)
- (a^2*x^(5/2)*Sqrt[a - b*x])/(80*b) + (3*a*x^(7/2)*Sqrt[a - b*x])/40 + (x^(7/2)
*(a - b*x)^(3/2))/5 + (3*a^5*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a - b*x]])/(128*b^(7/
2))

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Rubi in Sympy [A]  time = 20.1527, size = 136, normalized size = 0.91 \[ \frac{3 a^{5} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a - b x}} \right )}}{128 b^{\frac{7}{2}}} + \frac{3 a^{4} \sqrt{x} \sqrt{a - b x}}{128 b^{3}} + \frac{a^{3} \sqrt{x} \left (a - b x\right )^{\frac{3}{2}}}{64 b^{3}} - \frac{a^{2} \sqrt{x} \left (a - b x\right )^{\frac{5}{2}}}{16 b^{3}} - \frac{a x^{\frac{3}{2}} \left (a - b x\right )^{\frac{5}{2}}}{8 b^{2}} - \frac{x^{\frac{5}{2}} \left (a - b x\right )^{\frac{5}{2}}}{5 b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*a**5*atan(sqrt(b)*sqrt(x)/sqrt(a - b*x))/(128*b**(7/2)) + 3*a**4*sqrt(x)*sqrt(
a - b*x)/(128*b**3) + a**3*sqrt(x)*(a - b*x)**(3/2)/(64*b**3) - a**2*sqrt(x)*(a
- b*x)**(5/2)/(16*b**3) - a*x**(3/2)*(a - b*x)**(5/2)/(8*b**2) - x**(5/2)*(a - b
*x)**(5/2)/(5*b)

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Mathematica [A]  time = 0.0879275, size = 100, normalized size = 0.67 \[ \frac{15 a^5 \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a-b x}}\right )-\sqrt{b} \sqrt{x} \sqrt{a-b x} \left (15 a^4+10 a^3 b x+8 a^2 b^2 x^2-176 a b^3 x^3+128 b^4 x^4\right )}{640 b^{7/2}} \]

Antiderivative was successfully verified.

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

[Out]

(-(Sqrt[b]*Sqrt[x]*Sqrt[a - b*x]*(15*a^4 + 10*a^3*b*x + 8*a^2*b^2*x^2 - 176*a*b^
3*x^3 + 128*b^4*x^4)) + 15*a^5*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a - b*x]])/(640*b^(
7/2))

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Maple [A]  time = 0.01, size = 146, normalized size = 1. \[ -{\frac{1}{5\,b}{x}^{{\frac{5}{2}}} \left ( -bx+a \right ) ^{{\frac{5}{2}}}}-{\frac{a}{8\,{b}^{2}}{x}^{{\frac{3}{2}}} \left ( -bx+a \right ) ^{{\frac{5}{2}}}}-{\frac{{a}^{2}}{16\,{b}^{3}} \left ( -bx+a \right ) ^{{\frac{5}{2}}}\sqrt{x}}+{\frac{{a}^{3}}{64\,{b}^{3}} \left ( -bx+a \right ) ^{{\frac{3}{2}}}\sqrt{x}}+{\frac{3\,{a}^{4}}{128\,{b}^{3}}\sqrt{x}\sqrt{-bx+a}}+{\frac{3\,{a}^{5}}{256}\sqrt{x \left ( -bx+a \right ) }\arctan \left ({1\sqrt{b} \left ( x-{\frac{a}{2\,b}} \right ){\frac{1}{\sqrt{-b{x}^{2}+ax}}}} \right ){b}^{-{\frac{7}{2}}}{\frac{1}{\sqrt{-bx+a}}}{\frac{1}{\sqrt{x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(5/2)*(-b*x+a)^(3/2),x)

[Out]

-1/5/b*x^(5/2)*(-b*x+a)^(5/2)-1/8*a/b^2*x^(3/2)*(-b*x+a)^(5/2)-1/16*a^2/b^3*x^(1
/2)*(-b*x+a)^(5/2)+1/64*a^3/b^3*(-b*x+a)^(3/2)*x^(1/2)+3/128*a^4*x^(1/2)*(-b*x+a
)^(1/2)/b^3+3/256*a^5/b^(7/2)*(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^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.223228, size = 1, normalized size = 0.01 \[ \left [\frac{15 \, a^{5} \log \left (-2 \, \sqrt{-b x + a} b \sqrt{x} -{\left (2 \, b x - a\right )} \sqrt{-b}\right ) - 2 \,{\left (128 \, b^{4} x^{4} - 176 \, a b^{3} x^{3} + 8 \, a^{2} b^{2} x^{2} + 10 \, a^{3} b x + 15 \, a^{4}\right )} \sqrt{-b x + a} \sqrt{-b} \sqrt{x}}{1280 \, \sqrt{-b} b^{3}}, -\frac{15 \, a^{5} \arctan \left (\frac{\sqrt{-b x + a}}{\sqrt{b} \sqrt{x}}\right ) +{\left (128 \, b^{4} x^{4} - 176 \, a b^{3} x^{3} + 8 \, a^{2} b^{2} x^{2} + 10 \, a^{3} b x + 15 \, a^{4}\right )} \sqrt{-b x + a} \sqrt{b} \sqrt{x}}{640 \, b^{\frac{7}{2}}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-b*x + a)^(3/2)*x^(5/2),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 - 176*a*b^3*x^3 + 8*a^2*b^2*x^2 + 10*a^3*b*x + 15*a^4)*sqrt(-b*x + a)*s
qrt(-b)*sqrt(x))/(sqrt(-b)*b^3), -1/640*(15*a^5*arctan(sqrt(-b*x + a)/(sqrt(b)*s
qrt(x))) + (128*b^4*x^4 - 176*a*b^3*x^3 + 8*a^2*b^2*x^2 + 10*a^3*b*x + 15*a^4)*s
qrt(-b*x + a)*sqrt(b)*sqrt(x))/b^(7/2)]

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

[Out]

Timed out