3.23 \(\int \frac{x^2 (1+a x)}{\sqrt{a x} \sqrt{1-a x}} \, dx\)

Optimal. Leaf size=87 \[ -\frac{\sqrt{1-a x} (a x)^{5/2}}{3 a^3}-\frac{11 \sqrt{1-a x} (a x)^{3/2}}{12 a^3}-\frac{11 \sqrt{1-a x} \sqrt{a x}}{8 a^3}-\frac{11 \sin ^{-1}(1-2 a x)}{16 a^3} \]

[Out]

(-11*Sqrt[a*x]*Sqrt[1 - a*x])/(8*a^3) - (11*(a*x)^(3/2)*Sqrt[1 - a*x])/(12*a^3)
- ((a*x)^(5/2)*Sqrt[1 - a*x])/(3*a^3) - (11*ArcSin[1 - 2*a*x])/(16*a^3)

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Rubi [A]  time = 0.123107, antiderivative size = 87, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ -\frac{\sqrt{1-a x} (a x)^{5/2}}{3 a^3}-\frac{11 \sqrt{1-a x} (a x)^{3/2}}{12 a^3}-\frac{11 \sqrt{1-a x} \sqrt{a x}}{8 a^3}-\frac{11 \sin ^{-1}(1-2 a x)}{16 a^3} \]

Antiderivative was successfully verified.

[In]  Int[(x^2*(1 + a*x))/(Sqrt[a*x]*Sqrt[1 - a*x]),x]

[Out]

(-11*Sqrt[a*x]*Sqrt[1 - a*x])/(8*a^3) - (11*(a*x)^(3/2)*Sqrt[1 - a*x])/(12*a^3)
- ((a*x)^(5/2)*Sqrt[1 - a*x])/(3*a^3) - (11*ArcSin[1 - 2*a*x])/(16*a^3)

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Rubi in Sympy [A]  time = 14.1193, size = 78, normalized size = 0.9 \[ - \frac{\left (a x\right )^{\frac{5}{2}} \sqrt{- a x + 1}}{3 a^{3}} - \frac{11 \left (a x\right )^{\frac{3}{2}} \sqrt{- a x + 1}}{12 a^{3}} - \frac{11 \sqrt{a x} \sqrt{- a x + 1}}{8 a^{3}} + \frac{11 \operatorname{asin}{\left (2 a x - 1 \right )}}{16 a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(a*x)**(5/2)*sqrt(-a*x + 1)/(3*a**3) - 11*(a*x)**(3/2)*sqrt(-a*x + 1)/(12*a**3)
- 11*sqrt(a*x)*sqrt(-a*x + 1)/(8*a**3) + 11*asin(2*a*x - 1)/(16*a**3)

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Mathematica [A]  time = 0.0750891, size = 81, normalized size = 0.93 \[ \frac{\sqrt{a} x \left (8 a^3 x^3+14 a^2 x^2+11 a x-33\right )+33 \sqrt{x} \sqrt{1-a x} \sin ^{-1}\left (\sqrt{a} \sqrt{x}\right )}{24 a^{5/2} \sqrt{-a x (a x-1)}} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^2*(1 + a*x))/(Sqrt[a*x]*Sqrt[1 - a*x]),x]

[Out]

(Sqrt[a]*x*(-33 + 11*a*x + 14*a^2*x^2 + 8*a^3*x^3) + 33*Sqrt[x]*Sqrt[1 - a*x]*Ar
cSin[Sqrt[a]*Sqrt[x]])/(24*a^(5/2)*Sqrt[-(a*x*(-1 + a*x))])

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Maple [C]  time = 0.021, size = 111, normalized size = 1.3 \[{\frac{x{\it csgn} \left ( a \right ) }{48\,{a}^{2}}\sqrt{-ax+1} \left ( -16\,{\it csgn} \left ( a \right ){a}^{2}{x}^{2}\sqrt{-x \left ( ax-1 \right ) a}-44\,{\it csgn} \left ( a \right ) \sqrt{-x \left ( ax-1 \right ) a}xa-66\,{\it csgn} \left ( a \right ) \sqrt{-x \left ( ax-1 \right ) a}+33\,\arctan \left ( 1/2\,{\frac{ \left ( 2\,ax-1 \right ){\it csgn} \left ( a \right ) }{\sqrt{-x \left ( ax-1 \right ) a}}} \right ) \right ){\frac{1}{\sqrt{ax}}}{\frac{1}{\sqrt{-x \left ( ax-1 \right ) a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2*(a*x+1)/(a*x)^(1/2)/(-a*x+1)^(1/2),x)

[Out]

1/48*(-a*x+1)^(1/2)*x*(-16*csgn(a)*a^2*x^2*(-x*(a*x-1)*a)^(1/2)-44*csgn(a)*(-x*(
a*x-1)*a)^(1/2)*x*a-66*csgn(a)*(-x*(a*x-1)*a)^(1/2)+33*arctan(1/2*(2*a*x-1)*csgn
(a)/(-x*(a*x-1)*a)^(1/2)))*csgn(a)/a^2/(a*x)^(1/2)/(-x*(a*x-1)*a)^(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((a*x + 1)*x^2/(sqrt(a*x)*sqrt(-a*x + 1)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.228744, size = 77, normalized size = 0.89 \[ -\frac{{\left (8 \, a^{2} x^{2} + 22 \, a x + 33\right )} \sqrt{a x} \sqrt{-a x + 1} + 33 \, \arctan \left (\frac{\sqrt{a x} \sqrt{-a x + 1}}{a x}\right )}{24 \, a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a*x + 1)*x^2/(sqrt(a*x)*sqrt(-a*x + 1)),x, algorithm="fricas")

[Out]

-1/24*((8*a^2*x^2 + 22*a*x + 33)*sqrt(a*x)*sqrt(-a*x + 1) + 33*arctan(sqrt(a*x)*
sqrt(-a*x + 1)/(a*x)))/a^3

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Sympy [A]  time = 48.1127, size = 393, normalized size = 4.52 \[ a \left (\begin{cases} - \frac{5 i \operatorname{acosh}{\left (\sqrt{a} \sqrt{x} \right )}}{8 a^{4}} - \frac{i x^{\frac{7}{2}}}{3 \sqrt{a} \sqrt{a x - 1}} - \frac{i x^{\frac{5}{2}}}{12 a^{\frac{3}{2}} \sqrt{a x - 1}} - \frac{5 i x^{\frac{3}{2}}}{24 a^{\frac{5}{2}} \sqrt{a x - 1}} + \frac{5 i \sqrt{x}}{8 a^{\frac{7}{2}} \sqrt{a x - 1}} & \text{for}\: \left |{a x}\right | > 1 \\\frac{5 \operatorname{asin}{\left (\sqrt{a} \sqrt{x} \right )}}{8 a^{4}} + \frac{x^{\frac{7}{2}}}{3 \sqrt{a} \sqrt{- a x + 1}} + \frac{x^{\frac{5}{2}}}{12 a^{\frac{3}{2}} \sqrt{- a x + 1}} + \frac{5 x^{\frac{3}{2}}}{24 a^{\frac{5}{2}} \sqrt{- a x + 1}} - \frac{5 \sqrt{x}}{8 a^{\frac{7}{2}} \sqrt{- a x + 1}} & \text{otherwise} \end{cases}\right ) + \begin{cases} - \frac{3 i \operatorname{acosh}{\left (\sqrt{a} \sqrt{x} \right )}}{4 a^{3}} - \frac{i x^{\frac{5}{2}}}{2 \sqrt{a} \sqrt{a x - 1}} - \frac{i x^{\frac{3}{2}}}{4 a^{\frac{3}{2}} \sqrt{a x - 1}} + \frac{3 i \sqrt{x}}{4 a^{\frac{5}{2}} \sqrt{a x - 1}} & \text{for}\: \left |{a x}\right | > 1 \\\frac{3 \operatorname{asin}{\left (\sqrt{a} \sqrt{x} \right )}}{4 a^{3}} + \frac{x^{\frac{5}{2}}}{2 \sqrt{a} \sqrt{- a x + 1}} + \frac{x^{\frac{3}{2}}}{4 a^{\frac{3}{2}} \sqrt{- a x + 1}} - \frac{3 \sqrt{x}}{4 a^{\frac{5}{2}} \sqrt{- a x + 1}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a*Piecewise((-5*I*acosh(sqrt(a)*sqrt(x))/(8*a**4) - I*x**(7/2)/(3*sqrt(a)*sqrt(a
*x - 1)) - I*x**(5/2)/(12*a**(3/2)*sqrt(a*x - 1)) - 5*I*x**(3/2)/(24*a**(5/2)*sq
rt(a*x - 1)) + 5*I*sqrt(x)/(8*a**(7/2)*sqrt(a*x - 1)), Abs(a*x) > 1), (5*asin(sq
rt(a)*sqrt(x))/(8*a**4) + x**(7/2)/(3*sqrt(a)*sqrt(-a*x + 1)) + x**(5/2)/(12*a**
(3/2)*sqrt(-a*x + 1)) + 5*x**(3/2)/(24*a**(5/2)*sqrt(-a*x + 1)) - 5*sqrt(x)/(8*a
**(7/2)*sqrt(-a*x + 1)), True)) + Piecewise((-3*I*acosh(sqrt(a)*sqrt(x))/(4*a**3
) - I*x**(5/2)/(2*sqrt(a)*sqrt(a*x - 1)) - I*x**(3/2)/(4*a**(3/2)*sqrt(a*x - 1))
+ 3*I*sqrt(x)/(4*a**(5/2)*sqrt(a*x - 1)), Abs(a*x) > 1), (3*asin(sqrt(a)*sqrt(x
))/(4*a**3) + x**(5/2)/(2*sqrt(a)*sqrt(-a*x + 1)) + x**(3/2)/(4*a**(3/2)*sqrt(-a
*x + 1)) - 3*sqrt(x)/(4*a**(5/2)*sqrt(-a*x + 1)), True))

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GIAC/XCAS [A]  time = 0.223501, size = 72, normalized size = 0.83 \[ -\frac{{\left (2 \, a x{\left (\frac{4 \, x}{a} + \frac{11}{a^{2}}\right )} + \frac{33}{a^{2}}\right )} \sqrt{a x} \sqrt{-a x + 1} - \frac{33 \, \arcsin \left (\sqrt{a x}\right )}{a^{2}}}{24 \, a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a*x + 1)*x^2/(sqrt(a*x)*sqrt(-a*x + 1)),x, algorithm="giac")

[Out]

-1/24*((2*a*x*(4*x/a + 11/a^2) + 33/a^2)*sqrt(a*x)*sqrt(-a*x + 1) - 33*arcsin(sq
rt(a*x))/a^2)/a