3.24 \(\int \frac{x (1+a x)}{\sqrt{a x} \sqrt{1-a x}} \, dx\)

Optimal. Leaf size=63 \[ -\frac{\sqrt{1-a x} (a x)^{3/2}}{2 a^2}-\frac{7 \sqrt{1-a x} \sqrt{a x}}{4 a^2}-\frac{7 \sin ^{-1}(1-2 a x)}{8 a^2} \]

[Out]

(-7*Sqrt[a*x]*Sqrt[1 - a*x])/(4*a^2) - ((a*x)^(3/2)*Sqrt[1 - a*x])/(2*a^2) - (7*ArcSin[1 - 2*a*x])/(8*a^2)

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Rubi [A]  time = 0.0232313, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {16, 80, 50, 53, 619, 216} \[ -\frac{\sqrt{1-a x} (a x)^{3/2}}{2 a^2}-\frac{7 \sqrt{1-a x} \sqrt{a x}}{4 a^2}-\frac{7 \sin ^{-1}(1-2 a x)}{8 a^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-7*Sqrt[a*x]*Sqrt[1 - a*x])/(4*a^2) - ((a*x)^(3/2)*Sqrt[1 - a*x])/(2*a^2) - (7*ArcSin[1 - 2*a*x])/(8*a^2)

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 (1+a x)}{\sqrt{a x} \sqrt{1-a x}} \, dx &=\frac{\int \frac{\sqrt{a x} (1+a x)}{\sqrt{1-a x}} \, dx}{a}\\ &=-\frac{(a x)^{3/2} \sqrt{1-a x}}{2 a^2}+\frac{7 \int \frac{\sqrt{a x}}{\sqrt{1-a x}} \, dx}{4 a}\\ &=-\frac{7 \sqrt{a x} \sqrt{1-a x}}{4 a^2}-\frac{(a x)^{3/2} \sqrt{1-a x}}{2 a^2}+\frac{7 \int \frac{1}{\sqrt{a x} \sqrt{1-a x}} \, dx}{8 a}\\ &=-\frac{7 \sqrt{a x} \sqrt{1-a x}}{4 a^2}-\frac{(a x)^{3/2} \sqrt{1-a x}}{2 a^2}+\frac{7 \int \frac{1}{\sqrt{a x-a^2 x^2}} \, dx}{8 a}\\ &=-\frac{7 \sqrt{a x} \sqrt{1-a x}}{4 a^2}-\frac{(a x)^{3/2} \sqrt{1-a x}}{2 a^2}-\frac{7 \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-\frac{x^2}{a^2}}} \, dx,x,a-2 a^2 x\right )}{8 a^3}\\ &=-\frac{7 \sqrt{a x} \sqrt{1-a x}}{4 a^2}-\frac{(a x)^{3/2} \sqrt{1-a x}}{2 a^2}-\frac{7 \sin ^{-1}(1-2 a x)}{8 a^2}\\ \end{align*}

Mathematica [A]  time = 0.0267346, size = 73, normalized size = 1.16 \[ \frac{\sqrt{a} x \left (2 a^2 x^2+5 a x-7\right )+7 \sqrt{x} \sqrt{1-a x} \sin ^{-1}\left (\sqrt{a} \sqrt{x}\right )}{4 a^{3/2} \sqrt{-a x (a x-1)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[a]*x*(-7 + 5*a*x + 2*a^2*x^2) + 7*Sqrt[x]*Sqrt[1 - a*x]*ArcSin[Sqrt[a]*Sqrt[x]])/(4*a^(3/2)*Sqrt[-(a*x*(
-1 + a*x))])

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Maple [C]  time = 0.011, size = 90, normalized size = 1.4 \begin{align*} -{\frac{x{\it csgn} \left ( a \right ) }{8\,a}\sqrt{-ax+1} \left ( 4\,\sqrt{-x \left ( ax-1 \right ) a}{\it csgn} \left ( a \right ) xa+14\,\sqrt{-x \left ( ax-1 \right ) a}{\it csgn} \left ( a \right ) -7\,\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*(a*x+1)/(a*x)^(1/2)/(-a*x+1)^(1/2),x)

[Out]

-1/8*(-a*x+1)^(1/2)*x/a*(4*(-x*(a*x-1)*a)^(1/2)*csgn(a)*x*a+14*(-x*(a*x-1)*a)^(1/2)*csgn(a)-7*arctan(1/2*csgn(
a)*(2*a*x-1)/(-x*(a*x-1)*a)^(1/2)))*csgn(a)/(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*(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 = 1.31869, size = 124, normalized size = 1.97 \begin{align*} -\frac{{\left (2 \, a x + 7\right )} \sqrt{a x} \sqrt{-a x + 1} + 7 \, \arctan \left (\frac{\sqrt{a x} \sqrt{-a x + 1}}{a x}\right )}{4 \, a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*((2*a*x + 7)*sqrt(a*x)*sqrt(-a*x + 1) + 7*arctan(sqrt(a*x)*sqrt(-a*x + 1)/(a*x)))/a^2

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Sympy [C]  time = 14.3845, size = 269, normalized size = 4.27 \begin{align*} a \left (\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}\right ) + \begin{cases} - \frac{i \operatorname{acosh}{\left (\sqrt{a} \sqrt{x} \right )}}{a^{2}} - \frac{i \sqrt{x} \sqrt{a x - 1}}{a^{\frac{3}{2}}} & \text{for}\: \left |{a x}\right | > 1 \\\frac{\operatorname{asin}{\left (\sqrt{a} \sqrt{x} \right )}}{a^{2}} + \frac{x^{\frac{3}{2}}}{\sqrt{a} \sqrt{- a x + 1}} - \frac{\sqrt{x}}{a^{\frac{3}{2}} \sqrt{- a x + 1}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*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)) + Piecewise((-I*acosh(sqrt(a)*sqrt(x))/a**2 - I*sqrt(x)*sqrt(a*x - 1)/a**(3/2), Abs(a*x) > 1),
(asin(sqrt(a)*sqrt(x))/a**2 + x**(3/2)/(sqrt(a)*sqrt(-a*x + 1)) - sqrt(x)/(a**(3/2)*sqrt(-a*x + 1)), True))

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Giac [A]  time = 2.49052, size = 54, normalized size = 0.86 \begin{align*} -\frac{\sqrt{a x} \sqrt{-a x + 1}{\left (2 \, x + \frac{7}{a}\right )} - \frac{7 \, \arcsin \left (\sqrt{a x}\right )}{a}}{4 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(sqrt(a*x)*sqrt(-a*x + 1)*(2*x + 7/a) - 7*arcsin(sqrt(a*x))/a)/a