3.22 \(\int \frac{x^3 (1+a x)}{\sqrt{a x} \sqrt{1-a x}} \, dx\)

Optimal. Leaf size=111 \[ -\frac{\sqrt{1-a x} (a x)^{7/2}}{4 a^4}-\frac{5 \sqrt{1-a x} (a x)^{5/2}}{8 a^4}-\frac{25 \sqrt{1-a x} (a x)^{3/2}}{32 a^4}-\frac{75 \sqrt{1-a x} \sqrt{a x}}{64 a^4}-\frac{75 \sin ^{-1}(1-2 a x)}{128 a^4} \]

[Out]

(-75*Sqrt[a*x]*Sqrt[1 - a*x])/(64*a^4) - (25*(a*x)^(3/2)*Sqrt[1 - a*x])/(32*a^4)
- (5*(a*x)^(5/2)*Sqrt[1 - a*x])/(8*a^4) - ((a*x)^(7/2)*Sqrt[1 - a*x])/(4*a^4) -
(75*ArcSin[1 - 2*a*x])/(128*a^4)

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Rubi [A]  time = 0.158344, antiderivative size = 111, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ -\frac{\sqrt{1-a x} (a x)^{7/2}}{4 a^4}-\frac{5 \sqrt{1-a x} (a x)^{5/2}}{8 a^4}-\frac{25 \sqrt{1-a x} (a x)^{3/2}}{32 a^4}-\frac{75 \sqrt{1-a x} \sqrt{a x}}{64 a^4}-\frac{75 \sin ^{-1}(1-2 a x)}{128 a^4} \]

Antiderivative was successfully verified.

[In]  Int[(x^3*(1 + a*x))/(Sqrt[a*x]*Sqrt[1 - a*x]),x]

[Out]

(-75*Sqrt[a*x]*Sqrt[1 - a*x])/(64*a^4) - (25*(a*x)^(3/2)*Sqrt[1 - a*x])/(32*a^4)
- (5*(a*x)^(5/2)*Sqrt[1 - a*x])/(8*a^4) - ((a*x)^(7/2)*Sqrt[1 - a*x])/(4*a^4) -
(75*ArcSin[1 - 2*a*x])/(128*a^4)

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Rubi in Sympy [A]  time = 17.4243, size = 100, normalized size = 0.9 \[ - \frac{\left (a x\right )^{\frac{7}{2}} \sqrt{- a x + 1}}{4 a^{4}} - \frac{5 \left (a x\right )^{\frac{5}{2}} \sqrt{- a x + 1}}{8 a^{4}} - \frac{25 \left (a x\right )^{\frac{3}{2}} \sqrt{- a x + 1}}{32 a^{4}} - \frac{75 \sqrt{a x} \sqrt{- a x + 1}}{64 a^{4}} + \frac{75 \operatorname{asin}{\left (2 a x - 1 \right )}}{128 a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3*(a*x+1)/(a*x)**(1/2)/(-a*x+1)**(1/2),x)

[Out]

-(a*x)**(7/2)*sqrt(-a*x + 1)/(4*a**4) - 5*(a*x)**(5/2)*sqrt(-a*x + 1)/(8*a**4) -
25*(a*x)**(3/2)*sqrt(-a*x + 1)/(32*a**4) - 75*sqrt(a*x)*sqrt(-a*x + 1)/(64*a**4
) + 75*asin(2*a*x - 1)/(128*a**4)

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Mathematica [A]  time = 0.0932974, size = 89, normalized size = 0.8 \[ \frac{\sqrt{a} x \left (16 a^4 x^4+24 a^3 x^3+10 a^2 x^2+25 a x-75\right )+75 \sqrt{x} \sqrt{1-a x} \sin ^{-1}\left (\sqrt{a} \sqrt{x}\right )}{64 a^{7/2} \sqrt{-a x (a x-1)}} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^3*(1 + a*x))/(Sqrt[a*x]*Sqrt[1 - a*x]),x]

[Out]

(Sqrt[a]*x*(-75 + 25*a*x + 10*a^2*x^2 + 24*a^3*x^3 + 16*a^4*x^4) + 75*Sqrt[x]*Sq
rt[1 - a*x]*ArcSin[Sqrt[a]*Sqrt[x]])/(64*a^(7/2)*Sqrt[-(a*x*(-1 + a*x))])

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Maple [C]  time = 0.056, size = 132, normalized size = 1.2 \[ -{\frac{x{\it csgn} \left ( a \right ) }{128\,{a}^{3}}\sqrt{-ax+1} \left ( 32\,{\it csgn} \left ( a \right ){x}^{3}{a}^{3}\sqrt{-x \left ( ax-1 \right ) a}+80\,{\it csgn} \left ( a \right ){a}^{2}{x}^{2}\sqrt{-x \left ( ax-1 \right ) a}+100\,{\it csgn} \left ( a \right ) \sqrt{-x \left ( ax-1 \right ) a}xa+150\,{\it csgn} \left ( a \right ) \sqrt{-x \left ( ax-1 \right ) a}-75\,\arctan \left ( 1/2\,{\frac{ \left ( 2\,ax-1 \right ){\it csgn} \left ( a \right ) }{\sqrt{-x \left ( ax-1 \right ) a}}} \right ) \right ){\frac{1}{\sqrt{ax}}}{\frac{1}{\sqrt{-x \left ( ax-1 \right ) a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3*(a*x+1)/(a*x)^(1/2)/(-a*x+1)^(1/2),x)

[Out]

-1/128*(-a*x+1)^(1/2)*x*(32*csgn(a)*x^3*a^3*(-x*(a*x-1)*a)^(1/2)+80*csgn(a)*a^2*
x^2*(-x*(a*x-1)*a)^(1/2)+100*csgn(a)*(-x*(a*x-1)*a)^(1/2)*x*a+150*csgn(a)*(-x*(a
*x-1)*a)^(1/2)-75*arctan(1/2*(2*a*x-1)*csgn(a)/(-x*(a*x-1)*a)^(1/2)))*csgn(a)/a^
3/(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^3/(sqrt(a*x)*sqrt(-a*x + 1)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.225729, size = 88, normalized size = 0.79 \[ -\frac{{\left (16 \, a^{3} x^{3} + 40 \, a^{2} x^{2} + 50 \, a x + 75\right )} \sqrt{a x} \sqrt{-a x + 1} + 75 \, \arctan \left (\frac{\sqrt{a x} \sqrt{-a x + 1}}{a x}\right )}{64 \, a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a*x + 1)*x^3/(sqrt(a*x)*sqrt(-a*x + 1)),x, algorithm="fricas")

[Out]

-1/64*((16*a^3*x^3 + 40*a^2*x^2 + 50*a*x + 75)*sqrt(a*x)*sqrt(-a*x + 1) + 75*arc
tan(sqrt(a*x)*sqrt(-a*x + 1)/(a*x)))/a^4

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Sympy [A]  time = 69.7642, size = 484, normalized size = 4.36 \[ a \left (\begin{cases} - \frac{35 i \operatorname{acosh}{\left (\sqrt{a} \sqrt{x} \right )}}{64 a^{5}} - \frac{i x^{\frac{9}{2}}}{4 \sqrt{a} \sqrt{a x - 1}} - \frac{i x^{\frac{7}{2}}}{24 a^{\frac{3}{2}} \sqrt{a x - 1}} - \frac{7 i x^{\frac{5}{2}}}{96 a^{\frac{5}{2}} \sqrt{a x - 1}} - \frac{35 i x^{\frac{3}{2}}}{192 a^{\frac{7}{2}} \sqrt{a x - 1}} + \frac{35 i \sqrt{x}}{64 a^{\frac{9}{2}} \sqrt{a x - 1}} & \text{for}\: \left |{a x}\right | > 1 \\\frac{35 \operatorname{asin}{\left (\sqrt{a} \sqrt{x} \right )}}{64 a^{5}} + \frac{x^{\frac{9}{2}}}{4 \sqrt{a} \sqrt{- a x + 1}} + \frac{x^{\frac{7}{2}}}{24 a^{\frac{3}{2}} \sqrt{- a x + 1}} + \frac{7 x^{\frac{5}{2}}}{96 a^{\frac{5}{2}} \sqrt{- a x + 1}} + \frac{35 x^{\frac{3}{2}}}{192 a^{\frac{7}{2}} \sqrt{- a x + 1}} - \frac{35 \sqrt{x}}{64 a^{\frac{9}{2}} \sqrt{- a x + 1}} & \text{otherwise} \end{cases}\right ) + \begin{cases} - \frac{5 i \operatorname{acosh}{\left (\sqrt{a} \sqrt{x} \right )}}{8 a^{4}} - \frac{i x^{\frac{7}{2}}}{3 \sqrt{a} \sqrt{a x - 1}} - \frac{i x^{\frac{5}{2}}}{12 a^{\frac{3}{2}} \sqrt{a x - 1}} - \frac{5 i x^{\frac{3}{2}}}{24 a^{\frac{5}{2}} \sqrt{a x - 1}} + \frac{5 i \sqrt{x}}{8 a^{\frac{7}{2}} \sqrt{a x - 1}} & \text{for}\: \left |{a x}\right | > 1 \\\frac{5 \operatorname{asin}{\left (\sqrt{a} \sqrt{x} \right )}}{8 a^{4}} + \frac{x^{\frac{7}{2}}}{3 \sqrt{a} \sqrt{- a x + 1}} + \frac{x^{\frac{5}{2}}}{12 a^{\frac{3}{2}} \sqrt{- a x + 1}} + \frac{5 x^{\frac{3}{2}}}{24 a^{\frac{5}{2}} \sqrt{- a x + 1}} - \frac{5 \sqrt{x}}{8 a^{\frac{7}{2}} \sqrt{- a x + 1}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3*(a*x+1)/(a*x)**(1/2)/(-a*x+1)**(1/2),x)

[Out]

a*Piecewise((-35*I*acosh(sqrt(a)*sqrt(x))/(64*a**5) - I*x**(9/2)/(4*sqrt(a)*sqrt
(a*x - 1)) - I*x**(7/2)/(24*a**(3/2)*sqrt(a*x - 1)) - 7*I*x**(5/2)/(96*a**(5/2)*
sqrt(a*x - 1)) - 35*I*x**(3/2)/(192*a**(7/2)*sqrt(a*x - 1)) + 35*I*sqrt(x)/(64*a
**(9/2)*sqrt(a*x - 1)), Abs(a*x) > 1), (35*asin(sqrt(a)*sqrt(x))/(64*a**5) + x**
(9/2)/(4*sqrt(a)*sqrt(-a*x + 1)) + x**(7/2)/(24*a**(3/2)*sqrt(-a*x + 1)) + 7*x**
(5/2)/(96*a**(5/2)*sqrt(-a*x + 1)) + 35*x**(3/2)/(192*a**(7/2)*sqrt(-a*x + 1)) -
35*sqrt(x)/(64*a**(9/2)*sqrt(-a*x + 1)), True)) + 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)*sqrt(a*x - 1)) + 5*I*sqrt(x)/(8*a
**(7/2)*sqrt(a*x - 1)), Abs(a*x) > 1), (5*asin(sqrt(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))

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GIAC/XCAS [A]  time = 0.22885, size = 85, normalized size = 0.77 \[ -\frac{{\left (2 \,{\left (4 \, a x{\left (\frac{2 \, x}{a^{2}} + \frac{5}{a^{3}}\right )} + \frac{25}{a^{3}}\right )} a x + \frac{75}{a^{3}}\right )} \sqrt{a x} \sqrt{-a x + 1} - \frac{75 \, \arcsin \left (\sqrt{a x}\right )}{a^{3}}}{64 \, a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a*x + 1)*x^3/(sqrt(a*x)*sqrt(-a*x + 1)),x, algorithm="giac")

[Out]

-1/64*((2*(4*a*x*(2*x/a^2 + 5/a^3) + 25/a^3)*a*x + 75/a^3)*sqrt(a*x)*sqrt(-a*x +
1) - 75*arcsin(sqrt(a*x))/a^3)/a