3.1104 \(\int \frac{(1+a x)^{3/2}}{\sqrt{1-a x}} \, dx\)

Optimal. Leaf size=64 \[ -\frac{\sqrt{1-a x} (a x+1)^{3/2}}{2 a}-\frac{3 \sqrt{1-a x} \sqrt{a x+1}}{2 a}+\frac{3 \sin ^{-1}(a x)}{2 a} \]

[Out]

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

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Rubi [A]  time = 0.0117152, antiderivative size = 64, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {50, 41, 216} \[ -\frac{\sqrt{1-a x} (a x+1)^{3/2}}{2 a}-\frac{3 \sqrt{1-a x} \sqrt{a x+1}}{2 a}+\frac{3 \sin ^{-1}(a x)}{2 a} \]

Antiderivative was successfully verified.

[In]

Int[(1 + a*x)^(3/2)/Sqrt[1 - a*x],x]

[Out]

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

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 41

Int[((a_) + (b_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(m_.), x_Symbol] :> Int[(a*c + b*d*x^2)^m, x] /; FreeQ[{a, b
, c, d, m}, x] && EqQ[b*c + a*d, 0] && (IntegerQ[m] || (GtQ[a, 0] && GtQ[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{(1+a x)^{3/2}}{\sqrt{1-a x}} \, dx &=-\frac{\sqrt{1-a x} (1+a x)^{3/2}}{2 a}+\frac{3}{2} \int \frac{\sqrt{1+a x}}{\sqrt{1-a x}} \, dx\\ &=-\frac{3 \sqrt{1-a x} \sqrt{1+a x}}{2 a}-\frac{\sqrt{1-a x} (1+a x)^{3/2}}{2 a}+\frac{3}{2} \int \frac{1}{\sqrt{1-a x} \sqrt{1+a x}} \, dx\\ &=-\frac{3 \sqrt{1-a x} \sqrt{1+a x}}{2 a}-\frac{\sqrt{1-a x} (1+a x)^{3/2}}{2 a}+\frac{3}{2} \int \frac{1}{\sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{3 \sqrt{1-a x} \sqrt{1+a x}}{2 a}-\frac{\sqrt{1-a x} (1+a x)^{3/2}}{2 a}+\frac{3 \sin ^{-1}(a x)}{2 a}\\ \end{align*}

Mathematica [A]  time = 0.0401253, size = 47, normalized size = 0.73 \[ -\frac{\sqrt{1-a^2 x^2} (a x+4)+6 \sin ^{-1}\left (\frac{\sqrt{1-a x}}{\sqrt{2}}\right )}{2 a} \]

Antiderivative was successfully verified.

[In]

Integrate[(1 + a*x)^(3/2)/Sqrt[1 - a*x],x]

[Out]

-((4 + a*x)*Sqrt[1 - a^2*x^2] + 6*ArcSin[Sqrt[1 - a*x]/Sqrt[2]])/(2*a)

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Maple [A]  time = 0.01, size = 98, normalized size = 1.5 \begin{align*} -{\frac{1}{2\,a} \left ( ax+1 \right ) ^{{\frac{3}{2}}}\sqrt{-ax+1}}-{\frac{3}{2\,a}\sqrt{-ax+1}\sqrt{ax+1}}+{\frac{3}{2}\sqrt{ \left ( ax+1 \right ) \left ( -ax+1 \right ) }\arctan \left ({x\sqrt{{a}^{2}}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ){\frac{1}{\sqrt{-ax+1}}}{\frac{1}{\sqrt{ax+1}}}{\frac{1}{\sqrt{{a}^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(a*x+1)^(3/2)*(-a*x+1)^(1/2)/a-3/2*(-a*x+1)^(1/2)*(a*x+1)^(1/2)/a+3/2*((a*x+1)*(-a*x+1))^(1/2)/(a*x+1)^(1
/2)/(-a*x+1)^(1/2)/(a^2)^(1/2)*arctan((a^2)^(1/2)*x/(-a^2*x^2+1)^(1/2))

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Maxima [A]  time = 1.47734, size = 69, normalized size = 1.08 \begin{align*} -\frac{1}{2} \, \sqrt{-a^{2} x^{2} + 1} x + \frac{3 \, \arcsin \left (\frac{a^{2} x}{\sqrt{a^{2}}}\right )}{2 \, \sqrt{a^{2}}} - \frac{2 \, \sqrt{-a^{2} x^{2} + 1}}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^(3/2)/(-a*x+1)^(1/2),x, algorithm="maxima")

[Out]

-1/2*sqrt(-a^2*x^2 + 1)*x + 3/2*arcsin(a^2*x/sqrt(a^2))/sqrt(a^2) - 2*sqrt(-a^2*x^2 + 1)/a

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Fricas [A]  time = 1.64875, size = 138, normalized size = 2.16 \begin{align*} -\frac{{\left (a x + 4\right )} \sqrt{a x + 1} \sqrt{-a x + 1} + 6 \, \arctan \left (\frac{\sqrt{a x + 1} \sqrt{-a x + 1} - 1}{a x}\right )}{2 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^(3/2)/(-a*x+1)^(1/2),x, algorithm="fricas")

[Out]

-1/2*((a*x + 4)*sqrt(a*x + 1)*sqrt(-a*x + 1) + 6*arctan((sqrt(a*x + 1)*sqrt(-a*x + 1) - 1)/(a*x)))/a

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Sympy [A]  time = 24.9349, size = 75, normalized size = 1.17 \begin{align*} \begin{cases} \frac{2 \left (\begin{cases} - \frac{a x \sqrt{- a x + 1} \sqrt{a x + 1}}{4} - \sqrt{- a x + 1} \sqrt{a x + 1} + \frac{3 \operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{a x + 1}}{2} \right )}}{2} & \text{for}\: a x - 1 \geq -2 \wedge a x - 1 < 0 \end{cases}\right )}{a} & \text{for}\: a \neq 0 \\x & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((2*Piecewise((-a*x*sqrt(-a*x + 1)*sqrt(a*x + 1)/4 - sqrt(-a*x + 1)*sqrt(a*x + 1) + 3*asin(sqrt(2)*sq
rt(a*x + 1)/2)/2, (a*x - 1 >= -2) & (a*x - 1 < 0)))/a, Ne(a, 0)), (x, True))

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Giac [A]  time = 1.0726, size = 57, normalized size = 0.89 \begin{align*} -\frac{{\left (a x + 4\right )} \sqrt{a x + 1} \sqrt{-a x + 1} - 6 \, \arcsin \left (\frac{1}{2} \, \sqrt{2} \sqrt{a x + 1}\right )}{2 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^(3/2)/(-a*x+1)^(1/2),x, algorithm="giac")

[Out]

-1/2*((a*x + 4)*sqrt(a*x + 1)*sqrt(-a*x + 1) - 6*arcsin(1/2*sqrt(2)*sqrt(a*x + 1)))/a