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 = 0.0649047, 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 \[ -\frac{\sqrt{1-a x} (a x+1)^{3/2}}{2 a}-\frac{3 \sqrt{1-a x} \sqrt{a x+1}}{2 a}+\frac{3 \sin ^{-1}(a x)}{2 a} \]

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)

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Rubi in Sympy [A]  time = 10.0375, size = 51, normalized size = 0.8 \[ - \frac{\sqrt{- a x + 1} \left (a x + 1\right )^{\frac{3}{2}}}{2 a} - \frac{3 \sqrt{- a x + 1} \sqrt{a x + 1}}{2 a} + \frac{3 \operatorname{asin}{\left (a x \right )}}{2 a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0541555, size = 47, normalized size = 0.73 \[ \frac{6 \sin ^{-1}\left (\frac{\sqrt{a x+1}}{\sqrt{2}}\right )-(a x+4) \sqrt{1-a^2 x^2}}{2 a} \]

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 = 0.017, size = 98, normalized size = 1.5 \[ -{\frac{1}{2\,a} \left ( ax+1 \right ) ^{{\frac{3}{2}}}\sqrt{-ax+1}}-{\frac{3}{2\,a}\sqrt{-ax+1}\sqrt{ax+1}}+{\frac{3}{2}\sqrt{ \left ( ax+1 \right ) \left ( -ax+1 \right ) }\arctan \left ({x\sqrt{{a}^{2}}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ){\frac{1}{\sqrt{-ax+1}}}{\frac{1}{\sqrt{ax+1}}}{\frac{1}{\sqrt{{a}^{2}}}}} \]

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 = 1.47995, size = 69, normalized size = 1.08 \[ -\frac{1}{2} \, \sqrt{-a^{2} x^{2} + 1} x + \frac{3 \, \arcsin \left (\frac{a^{2} x}{\sqrt{a^{2}}}\right )}{2 \, \sqrt{a^{2}}} - \frac{2 \, \sqrt{-a^{2} x^{2} + 1}}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a*x + 1)^(3/2)/sqrt(-a*x + 1),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 = 0.211831, size = 196, normalized size = 3.06 \[ \frac{2 \, a^{3} x^{3} + 4 \, a^{2} x^{2} -{\left (a^{3} x^{3} + 4 \, a^{2} x^{2} - 2 \, a x\right )} \sqrt{a x + 1} \sqrt{-a x + 1} - 2 \, a x - 6 \,{\left (a^{2} x^{2} + 2 \, \sqrt{a x + 1} \sqrt{-a x + 1} - 2\right )} \arctan \left (\frac{\sqrt{a x + 1} \sqrt{-a x + 1} - 1}{a x}\right )}{2 \,{\left (a^{3} x^{2} + 2 \, \sqrt{a x + 1} \sqrt{-a x + 1} a - 2 \, a\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(2*a^3*x^3 + 4*a^2*x^2 - (a^3*x^3 + 4*a^2*x^2 - 2*a*x)*sqrt(a*x + 1)*sqrt(-a
*x + 1) - 2*a*x - 6*(a^2*x^2 + 2*sqrt(a*x + 1)*sqrt(-a*x + 1) - 2)*arctan((sqrt(
a*x + 1)*sqrt(-a*x + 1) - 1)/(a*x)))/(a^3*x^2 + 2*sqrt(a*x + 1)*sqrt(-a*x + 1)*a
- 2*a)

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Sympy [A]  time = 33.8873, size = 73, normalized size = 1.14 \[ \begin{cases} \frac{2 \left (\begin{cases} - \frac{a x \sqrt{- a x + 1} \sqrt{a x + 1}}{4} - \sqrt{- a x + 1} \sqrt{a x + 1} + \frac{3 \operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{a x + 1}}{2} \right )}}{2} & \text{for}\: - a x + 1 \leq 2 \wedge - a x + 1 > 0 \end{cases}\right )}{a} & \text{for}\: a \neq 0 \\x & \text{otherwise} \end{cases} \]

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)*sqr
t(a*x + 1) + 3*asin(sqrt(2)*sqrt(a*x + 1)/2)/2, (-a*x + 1 <= 2) & (-a*x + 1 > 0)
))/a, Ne(a, 0)), (x, True))

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GIAC/XCAS [A]  time = 0.214039, size = 57, normalized size = 0.89 \[ -\frac{{\left (a x + 4\right )} \sqrt{a x + 1} \sqrt{-a x + 1} - 6 \, \arcsin \left (\frac{1}{2} \, \sqrt{2} \sqrt{a x + 1}\right )}{2 \, a} \]

Verification of antiderivative is not currently implemented for this CAS.

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