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 = 0.0448675, 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, Rules used = {16, 80, 50, 53, 619, 216} \[ -\frac{\sqrt{1-a x} (a x)^{7/2}}{4 a^4}-\frac{5 \sqrt{1-a x} (a x)^{5/2}}{8 a^4}-\frac{25 \sqrt{1-a x} (a x)^{3/2}}{32 a^4}-\frac{75 \sqrt{1-a x} \sqrt{a x}}{64 a^4}-\frac{75 \sin ^{-1}(1-2 a x)}{128 a^4} \]

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)

Rule 16

Int[(u_.)*(v_)^(m_.)*((b_)*(v_))^(n_), x_Symbol] :> Dist[1/b^m, Int[u*(b*v)^(m + n), x], x] /; FreeQ[{b, n}, x
] && IntegerQ[m]

Rule 80

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(c + d*x)
^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 2)), x] + Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(
d*f*(n + p + 2)), Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2,
0]

Rule 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 53

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]), x_Symbol] :> Int[1/Sqrt[a*c - b*(a - c)*x - b^2*x^2]
, x] /; FreeQ[{a, b, c, d}, x] && EqQ[b + d, 0] && GtQ[a + c, 0]

Rule 619

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[1/(2*c*((-4*c)/(b^2 - 4*a*c))^p), Subst[Int[Si
mp[1 - x^2/(b^2 - 4*a*c), x]^p, x], x, b + 2*c*x], x] /; FreeQ[{a, b, c, p}, x] && GtQ[4*a - b^2/c, 0]

Rule 216

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[(Rt[-b, 2]*x)/Sqrt[a]]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[b]

Rubi steps

\begin{align*} \int \frac{x^3 (1+a x)}{\sqrt{a x} \sqrt{1-a x}} \, dx &=\frac{\int \frac{(a x)^{5/2} (1+a x)}{\sqrt{1-a x}} \, dx}{a^3}\\ &=-\frac{(a x)^{7/2} \sqrt{1-a x}}{4 a^4}+\frac{15 \int \frac{(a x)^{5/2}}{\sqrt{1-a x}} \, dx}{8 a^3}\\ &=-\frac{5 (a x)^{5/2} \sqrt{1-a x}}{8 a^4}-\frac{(a x)^{7/2} \sqrt{1-a x}}{4 a^4}+\frac{25 \int \frac{(a x)^{3/2}}{\sqrt{1-a x}} \, dx}{16 a^3}\\ &=-\frac{25 (a x)^{3/2} \sqrt{1-a x}}{32 a^4}-\frac{5 (a x)^{5/2} \sqrt{1-a x}}{8 a^4}-\frac{(a x)^{7/2} \sqrt{1-a x}}{4 a^4}+\frac{75 \int \frac{\sqrt{a x}}{\sqrt{1-a x}} \, dx}{64 a^3}\\ &=-\frac{75 \sqrt{a x} \sqrt{1-a x}}{64 a^4}-\frac{25 (a x)^{3/2} \sqrt{1-a x}}{32 a^4}-\frac{5 (a x)^{5/2} \sqrt{1-a x}}{8 a^4}-\frac{(a x)^{7/2} \sqrt{1-a x}}{4 a^4}+\frac{75 \int \frac{1}{\sqrt{a x} \sqrt{1-a x}} \, dx}{128 a^3}\\ &=-\frac{75 \sqrt{a x} \sqrt{1-a x}}{64 a^4}-\frac{25 (a x)^{3/2} \sqrt{1-a x}}{32 a^4}-\frac{5 (a x)^{5/2} \sqrt{1-a x}}{8 a^4}-\frac{(a x)^{7/2} \sqrt{1-a x}}{4 a^4}+\frac{75 \int \frac{1}{\sqrt{a x-a^2 x^2}} \, dx}{128 a^3}\\ &=-\frac{75 \sqrt{a x} \sqrt{1-a x}}{64 a^4}-\frac{25 (a x)^{3/2} \sqrt{1-a x}}{32 a^4}-\frac{5 (a x)^{5/2} \sqrt{1-a x}}{8 a^4}-\frac{(a x)^{7/2} \sqrt{1-a x}}{4 a^4}-\frac{75 \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-\frac{x^2}{a^2}}} \, dx,x,a-2 a^2 x\right )}{128 a^5}\\ &=-\frac{75 \sqrt{a x} \sqrt{1-a x}}{64 a^4}-\frac{25 (a x)^{3/2} \sqrt{1-a x}}{32 a^4}-\frac{5 (a x)^{5/2} \sqrt{1-a x}}{8 a^4}-\frac{(a x)^{7/2} \sqrt{1-a x}}{4 a^4}-\frac{75 \sin ^{-1}(1-2 a x)}{128 a^4}\\ \end{align*}

Mathematica [A]  time = 0.0846198, 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]*Sqrt[1 - a*x]*ArcSin[Sqrt[a]*Sqr
t[x]])/(64*a^(7/2)*Sqrt[-(a*x*(-1 + a*x))])

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Maple [C]  time = 0.03, size = 132, normalized size = 1.2 \begin{align*} -{\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\,{a}^{2}{x}^{2}\sqrt{-x \left ( ax-1 \right ) a}{\it csgn} \left ( a \right ) +100\,\sqrt{-x \left ( ax-1 \right ) a}{\it csgn} \left ( a \right ) xa+150\,\sqrt{-x \left ( ax-1 \right ) a}{\it csgn} \left ( a \right ) -75\,\arctan \left ( 1/2\,{\frac{{\it csgn} \left ( a \right ) \left ( 2\,ax-1 \right ) }{\sqrt{-x \left ( ax-1 \right ) a}}} \right ) \right ){\frac{1}{\sqrt{ax}}}{\frac{1}{\sqrt{-x \left ( ax-1 \right ) a}}}} \end{align*}

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*a^2*x^2*(-x*(a*x-1)*a)^(1/2)*csgn(a)+100*(
-x*(a*x-1)*a)^(1/2)*csgn(a)*x*a+150*(-x*(a*x-1)*a)^(1/2)*csgn(a)-75*arctan(1/2*csgn(a)*(2*a*x-1)/(-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(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

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, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.26349, size = 165, normalized size = 1.49 \begin{align*} -\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}} \end{align*}

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, 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*arctan(sqrt(a*x)*sqrt(-a*x + 1)/(
a*x)))/a^4

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Sympy [C]  time = 25.7371, size = 484, normalized size = 4.36 \begin{align*} 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} \end{align*}

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*s
qrt(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*aco
sh(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(sqr
t(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 [A]  time = 1.77946, size = 85, normalized size = 0.77 \begin{align*} -\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} \end{align*}

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, 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