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

Optimal. Leaf size=171 \[ \frac{5 a^6 \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a-b x}}\right )}{512 b^{7/2}}-\frac{5 a^5 \sqrt{x} \sqrt{a-b x}}{512 b^3}-\frac{5 a^4 x^{3/2} \sqrt{a-b x}}{768 b^2}-\frac{a^3 x^{5/2} \sqrt{a-b x}}{192 b}+\frac{1}{32} a^2 x^{7/2} \sqrt{a-b x}+\frac{1}{12} a x^{7/2} (a-b x)^{3/2}+\frac{1}{6} x^{7/2} (a-b x)^{5/2} \]

[Out]

(-5*a^5*Sqrt[x]*Sqrt[a - b*x])/(512*b^3) - (5*a^4*x^(3/2)*Sqrt[a - b*x])/(768*b^
2) - (a^3*x^(5/2)*Sqrt[a - b*x])/(192*b) + (a^2*x^(7/2)*Sqrt[a - b*x])/32 + (a*x
^(7/2)*(a - b*x)^(3/2))/12 + (x^(7/2)*(a - b*x)^(5/2))/6 + (5*a^6*ArcTan[(Sqrt[b
]*Sqrt[x])/Sqrt[a - b*x]])/(512*b^(7/2))

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Rubi [A]  time = 0.153295, antiderivative size = 171, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188 \[ \frac{5 a^6 \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a-b x}}\right )}{512 b^{7/2}}-\frac{5 a^5 \sqrt{x} \sqrt{a-b x}}{512 b^3}-\frac{5 a^4 x^{3/2} \sqrt{a-b x}}{768 b^2}-\frac{a^3 x^{5/2} \sqrt{a-b x}}{192 b}+\frac{1}{32} a^2 x^{7/2} \sqrt{a-b x}+\frac{1}{12} a x^{7/2} (a-b x)^{3/2}+\frac{1}{6} x^{7/2} (a-b x)^{5/2} \]

Antiderivative was successfully verified.

[In]  Int[x^(5/2)*(a - b*x)^(5/2),x]

[Out]

(-5*a^5*Sqrt[x]*Sqrt[a - b*x])/(512*b^3) - (5*a^4*x^(3/2)*Sqrt[a - b*x])/(768*b^
2) - (a^3*x^(5/2)*Sqrt[a - b*x])/(192*b) + (a^2*x^(7/2)*Sqrt[a - b*x])/32 + (a*x
^(7/2)*(a - b*x)^(3/2))/12 + (x^(7/2)*(a - b*x)^(5/2))/6 + (5*a^6*ArcTan[(Sqrt[b
]*Sqrt[x])/Sqrt[a - b*x]])/(512*b^(7/2))

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Rubi in Sympy [A]  time = 24.6945, size = 160, normalized size = 0.94 \[ \frac{5 a^{6} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a - b x}} \right )}}{512 b^{\frac{7}{2}}} + \frac{5 a^{5} \sqrt{x} \sqrt{a - b x}}{512 b^{3}} + \frac{5 a^{4} \sqrt{x} \left (a - b x\right )^{\frac{3}{2}}}{768 b^{3}} + \frac{a^{3} \sqrt{x} \left (a - b x\right )^{\frac{5}{2}}}{192 b^{3}} - \frac{a^{2} \sqrt{x} \left (a - b x\right )^{\frac{7}{2}}}{32 b^{3}} - \frac{a x^{\frac{3}{2}} \left (a - b x\right )^{\frac{7}{2}}}{12 b^{2}} - \frac{x^{\frac{5}{2}} \left (a - b x\right )^{\frac{7}{2}}}{6 b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

5*a**6*atan(sqrt(b)*sqrt(x)/sqrt(a - b*x))/(512*b**(7/2)) + 5*a**5*sqrt(x)*sqrt(
a - b*x)/(512*b**3) + 5*a**4*sqrt(x)*(a - b*x)**(3/2)/(768*b**3) + a**3*sqrt(x)*
(a - b*x)**(5/2)/(192*b**3) - a**2*sqrt(x)*(a - b*x)**(7/2)/(32*b**3) - a*x**(3/
2)*(a - b*x)**(7/2)/(12*b**2) - x**(5/2)*(a - b*x)**(7/2)/(6*b)

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Mathematica [A]  time = 0.102836, size = 110, normalized size = 0.64 \[ \frac{15 a^6 \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a-b x}}\right )+\sqrt{b} \sqrt{x} \sqrt{a-b x} \left (-15 a^5-10 a^4 b x-8 a^3 b^2 x^2+432 a^2 b^3 x^3-640 a b^4 x^4+256 b^5 x^5\right )}{1536 b^{7/2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[b]*Sqrt[x]*Sqrt[a - b*x]*(-15*a^5 - 10*a^4*b*x - 8*a^3*b^2*x^2 + 432*a^2*b
^3*x^3 - 640*a*b^4*x^4 + 256*b^5*x^5) + 15*a^6*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a -
b*x]])/(1536*b^(7/2))

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Maple [A]  time = 0.008, size = 165, normalized size = 1. \[ -{\frac{1}{6\,b}{x}^{{\frac{5}{2}}} \left ( -bx+a \right ) ^{{\frac{7}{2}}}}-{\frac{a}{12\,{b}^{2}}{x}^{{\frac{3}{2}}} \left ( -bx+a \right ) ^{{\frac{7}{2}}}}-{\frac{{a}^{2}}{32\,{b}^{3}}\sqrt{x} \left ( -bx+a \right ) ^{{\frac{7}{2}}}}+{\frac{{a}^{3}}{192\,{b}^{3}} \left ( -bx+a \right ) ^{{\frac{5}{2}}}\sqrt{x}}+{\frac{5\,{a}^{4}}{768\,{b}^{3}} \left ( -bx+a \right ) ^{{\frac{3}{2}}}\sqrt{x}}+{\frac{5\,{a}^{5}}{512\,{b}^{3}}\sqrt{x}\sqrt{-bx+a}}+{\frac{5\,{a}^{6}}{1024}\sqrt{x \left ( -bx+a \right ) }\arctan \left ({1\sqrt{b} \left ( x-{\frac{a}{2\,b}} \right ){\frac{1}{\sqrt{-b{x}^{2}+ax}}}} \right ){b}^{-{\frac{7}{2}}}{\frac{1}{\sqrt{-bx+a}}}{\frac{1}{\sqrt{x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(5/2)*(-b*x+a)^(5/2),x)

[Out]

-1/6/b*x^(5/2)*(-b*x+a)^(7/2)-1/12*a/b^2*x^(3/2)*(-b*x+a)^(7/2)-1/32*a^2/b^3*x^(
1/2)*(-b*x+a)^(7/2)+1/192*a^3/b^3*(-b*x+a)^(5/2)*x^(1/2)+5/768*a^4/b^3*(-b*x+a)^
(3/2)*x^(1/2)+5/512*a^5*x^(1/2)*(-b*x+a)^(1/2)/b^3+5/1024*a^6/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)^(5/2)*x^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.224598, size = 1, normalized size = 0.01 \[ \left [\frac{15 \, a^{6} \log \left (-2 \, \sqrt{-b x + a} b \sqrt{x} -{\left (2 \, b x - a\right )} \sqrt{-b}\right ) + 2 \,{\left (256 \, b^{5} x^{5} - 640 \, a b^{4} x^{4} + 432 \, a^{2} b^{3} x^{3} - 8 \, a^{3} b^{2} x^{2} - 10 \, a^{4} b x - 15 \, a^{5}\right )} \sqrt{-b x + a} \sqrt{-b} \sqrt{x}}{3072 \, \sqrt{-b} b^{3}}, -\frac{15 \, a^{6} \arctan \left (\frac{\sqrt{-b x + a}}{\sqrt{b} \sqrt{x}}\right ) -{\left (256 \, b^{5} x^{5} - 640 \, a b^{4} x^{4} + 432 \, a^{2} b^{3} x^{3} - 8 \, a^{3} b^{2} x^{2} - 10 \, a^{4} b x - 15 \, a^{5}\right )} \sqrt{-b x + a} \sqrt{b} \sqrt{x}}{1536 \, b^{\frac{7}{2}}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/3072*(15*a^6*log(-2*sqrt(-b*x + a)*b*sqrt(x) - (2*b*x - a)*sqrt(-b)) + 2*(256
*b^5*x^5 - 640*a*b^4*x^4 + 432*a^2*b^3*x^3 - 8*a^3*b^2*x^2 - 10*a^4*b*x - 15*a^5
)*sqrt(-b*x + a)*sqrt(-b)*sqrt(x))/(sqrt(-b)*b^3), -1/1536*(15*a^6*arctan(sqrt(-
b*x + a)/(sqrt(b)*sqrt(x))) - (256*b^5*x^5 - 640*a*b^4*x^4 + 432*a^2*b^3*x^3 - 8
*a^3*b^2*x^2 - 10*a^4*b*x - 15*a^5)*sqrt(-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)**(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^(5/2),x, algorithm="giac")

[Out]

Timed out