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

Optimal. Leaf size=121 \[ \frac{5 a^4 \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a-b x}}\right )}{64 b^{3/2}}-\frac{5 a^3 \sqrt{x} \sqrt{a-b x}}{64 b}+\frac{5}{32} a^2 x^{3/2} \sqrt{a-b x}+\frac{5}{24} a x^{3/2} (a-b x)^{3/2}+\frac{1}{4} x^{3/2} (a-b x)^{5/2} \]

[Out]

(-5*a^3*Sqrt[x]*Sqrt[a - b*x])/(64*b) + (5*a^2*x^(3/2)*Sqrt[a - b*x])/32 + (5*a*
x^(3/2)*(a - b*x)^(3/2))/24 + (x^(3/2)*(a - b*x)^(5/2))/4 + (5*a^4*ArcTan[(Sqrt[
b]*Sqrt[x])/Sqrt[a - b*x]])/(64*b^(3/2))

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Rubi [A]  time = 0.0914399, antiderivative size = 121, 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 \[ \frac{5 a^4 \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a-b x}}\right )}{64 b^{3/2}}-\frac{5 a^3 \sqrt{x} \sqrt{a-b x}}{64 b}+\frac{5}{32} a^2 x^{3/2} \sqrt{a-b x}+\frac{5}{24} a x^{3/2} (a-b x)^{3/2}+\frac{1}{4} x^{3/2} (a-b x)^{5/2} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[x]*(a - b*x)^(5/2),x]

[Out]

(-5*a^3*Sqrt[x]*Sqrt[a - b*x])/(64*b) + (5*a^2*x^(3/2)*Sqrt[a - b*x])/32 + (5*a*
x^(3/2)*(a - b*x)^(3/2))/24 + (x^(3/2)*(a - b*x)^(5/2))/4 + (5*a^4*ArcTan[(Sqrt[
b]*Sqrt[x])/Sqrt[a - b*x]])/(64*b^(3/2))

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Rubi in Sympy [A]  time = 15.7889, size = 110, normalized size = 0.91 \[ \frac{5 a^{4} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a - b x}} \right )}}{64 b^{\frac{3}{2}}} + \frac{5 a^{3} \sqrt{x} \sqrt{a - b x}}{64 b} + \frac{5 a^{2} \sqrt{x} \left (a - b x\right )^{\frac{3}{2}}}{96 b} + \frac{a \sqrt{x} \left (a - b x\right )^{\frac{5}{2}}}{24 b} - \frac{\sqrt{x} \left (a - b x\right )^{\frac{7}{2}}}{4 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((-b*x+a)**(5/2)*x**(1/2),x)

[Out]

5*a**4*atan(sqrt(b)*sqrt(x)/sqrt(a - b*x))/(64*b**(3/2)) + 5*a**3*sqrt(x)*sqrt(a
- b*x)/(64*b) + 5*a**2*sqrt(x)*(a - b*x)**(3/2)/(96*b) + a*sqrt(x)*(a - b*x)**(
5/2)/(24*b) - sqrt(x)*(a - b*x)**(7/2)/(4*b)

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Mathematica [A]  time = 0.0699371, size = 88, normalized size = 0.73 \[ \frac{15 a^4 \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a-b x}}\right )+\sqrt{b} \sqrt{x} \sqrt{a-b x} \left (-15 a^3+118 a^2 b x-136 a b^2 x^2+48 b^3 x^3\right )}{192 b^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[b]*Sqrt[x]*Sqrt[a - b*x]*(-15*a^3 + 118*a^2*b*x - 136*a*b^2*x^2 + 48*b^3*x
^3) + 15*a^4*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a - b*x]])/(192*b^(3/2))

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Maple [A]  time = 0.009, size = 118, normalized size = 1. \[{\frac{1}{4}{x}^{{\frac{3}{2}}} \left ( -bx+a \right ) ^{{\frac{5}{2}}}}+{\frac{5\,a}{24}{x}^{{\frac{3}{2}}} \left ( -bx+a \right ) ^{{\frac{3}{2}}}}+{\frac{5\,{a}^{2}}{32}{x}^{{\frac{3}{2}}}\sqrt{-bx+a}}-{\frac{5\,{a}^{3}}{64\,b}\sqrt{x}\sqrt{-bx+a}}+{\frac{5\,{a}^{4}}{128}\sqrt{x \left ( -bx+a \right ) }\arctan \left ({1\sqrt{b} \left ( x-{\frac{a}{2\,b}} \right ){\frac{1}{\sqrt{-b{x}^{2}+ax}}}} \right ){b}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{-bx+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-b*x+a)^(5/2)*x^(1/2),x)

[Out]

1/4*x^(3/2)*(-b*x+a)^(5/2)+5/24*a*x^(3/2)*(-b*x+a)^(3/2)+5/32*a^2*x^(3/2)*(-b*x+
a)^(1/2)-5/64*a^3*x^(1/2)*(-b*x+a)^(1/2)/b+5/128*a^4/b^(3/2)*(x*(-b*x+a))^(1/2)/
x^(1/2)/(-b*x+a)^(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)*sqrt(x),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.224621, size = 1, normalized size = 0.01 \[ \left [\frac{15 \, a^{4} \log \left (-2 \, \sqrt{-b x + a} b \sqrt{x} -{\left (2 \, b x - a\right )} \sqrt{-b}\right ) + 2 \,{\left (48 \, b^{3} x^{3} - 136 \, a b^{2} x^{2} + 118 \, a^{2} b x - 15 \, a^{3}\right )} \sqrt{-b x + a} \sqrt{-b} \sqrt{x}}{384 \, \sqrt{-b} b}, -\frac{15 \, a^{4} \arctan \left (\frac{\sqrt{-b x + a}}{\sqrt{b} \sqrt{x}}\right ) -{\left (48 \, b^{3} x^{3} - 136 \, a b^{2} x^{2} + 118 \, a^{2} b x - 15 \, a^{3}\right )} \sqrt{-b x + a} \sqrt{b} \sqrt{x}}{192 \, b^{\frac{3}{2}}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/384*(15*a^4*log(-2*sqrt(-b*x + a)*b*sqrt(x) - (2*b*x - a)*sqrt(-b)) + 2*(48*b
^3*x^3 - 136*a*b^2*x^2 + 118*a^2*b*x - 15*a^3)*sqrt(-b*x + a)*sqrt(-b)*sqrt(x))/
(sqrt(-b)*b), -1/192*(15*a^4*arctan(sqrt(-b*x + a)/(sqrt(b)*sqrt(x))) - (48*b^3*
x^3 - 136*a*b^2*x^2 + 118*a^2*b*x - 15*a^3)*sqrt(-b*x + a)*sqrt(b)*sqrt(x))/b^(3
/2)]

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Sympy [A]  time = 101.263, size = 326, normalized size = 2.69 \[ \begin{cases} \frac{5 i a^{\frac{7}{2}} \sqrt{x}}{64 b \sqrt{-1 + \frac{b x}{a}}} - \frac{133 i a^{\frac{5}{2}} x^{\frac{3}{2}}}{192 \sqrt{-1 + \frac{b x}{a}}} + \frac{127 i a^{\frac{3}{2}} b x^{\frac{5}{2}}}{96 \sqrt{-1 + \frac{b x}{a}}} - \frac{23 i \sqrt{a} b^{2} x^{\frac{7}{2}}}{24 \sqrt{-1 + \frac{b x}{a}}} - \frac{5 i a^{4} \operatorname{acosh}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{64 b^{\frac{3}{2}}} + \frac{i b^{3} x^{\frac{9}{2}}}{4 \sqrt{a} \sqrt{-1 + \frac{b x}{a}}} & \text{for}\: \left |{\frac{b x}{a}}\right | > 1 \\- \frac{5 a^{\frac{7}{2}} \sqrt{x}}{64 b \sqrt{1 - \frac{b x}{a}}} + \frac{133 a^{\frac{5}{2}} x^{\frac{3}{2}}}{192 \sqrt{1 - \frac{b x}{a}}} - \frac{127 a^{\frac{3}{2}} b x^{\frac{5}{2}}}{96 \sqrt{1 - \frac{b x}{a}}} + \frac{23 \sqrt{a} b^{2} x^{\frac{7}{2}}}{24 \sqrt{1 - \frac{b x}{a}}} + \frac{5 a^{4} \operatorname{asin}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{64 b^{\frac{3}{2}}} - \frac{b^{3} x^{\frac{9}{2}}}{4 \sqrt{a} \sqrt{1 - \frac{b x}{a}}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-b*x+a)**(5/2)*x**(1/2),x)

[Out]

Piecewise((5*I*a**(7/2)*sqrt(x)/(64*b*sqrt(-1 + b*x/a)) - 133*I*a**(5/2)*x**(3/2
)/(192*sqrt(-1 + b*x/a)) + 127*I*a**(3/2)*b*x**(5/2)/(96*sqrt(-1 + b*x/a)) - 23*
I*sqrt(a)*b**2*x**(7/2)/(24*sqrt(-1 + b*x/a)) - 5*I*a**4*acosh(sqrt(b)*sqrt(x)/s
qrt(a))/(64*b**(3/2)) + I*b**3*x**(9/2)/(4*sqrt(a)*sqrt(-1 + b*x/a)), Abs(b*x/a)
> 1), (-5*a**(7/2)*sqrt(x)/(64*b*sqrt(1 - b*x/a)) + 133*a**(5/2)*x**(3/2)/(192*
sqrt(1 - b*x/a)) - 127*a**(3/2)*b*x**(5/2)/(96*sqrt(1 - b*x/a)) + 23*sqrt(a)*b**
2*x**(7/2)/(24*sqrt(1 - b*x/a)) + 5*a**4*asin(sqrt(b)*sqrt(x)/sqrt(a))/(64*b**(3
/2)) - b**3*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)^(5/2)*sqrt(x),x, algorithm="giac")

[Out]

Timed out