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

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

________________________________________________________________________________________

Rubi [A]  time = 0.01, antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {50, 41, 216} \begin {gather*} -\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} \end {gather*}

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 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 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 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 = 0.04, size = 47, normalized size = 0.73 \begin {gather*} -\frac {\sqrt {1-a^2 x^2} (a x+4)+6 \sin ^{-1}\left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )}{2 a} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.08, size = 86, normalized size = 1.34 \begin {gather*} -\frac {\sqrt {1-a x} \left (\frac {3 (1-a x)}{a x+1}+5\right )}{a \sqrt {a x+1} \left (\frac {1-a x}{a x+1}+1\right )^2}-\frac {3 \tan ^{-1}\left (\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right )}{a} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-((Sqrt[1 - a*x]*(5 + (3*(1 - a*x))/(1 + a*x)))/(a*Sqrt[1 + a*x]*(1 + (1 - a*x)/(1 + a*x))^2)) - (3*ArcTan[Sqr
t[1 - a*x]/Sqrt[1 + a*x]])/a

________________________________________________________________________________________

fricas [A]  time = 1.27, size = 55, normalized size = 0.86 \begin {gather*} -\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 {gather*}

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

________________________________________________________________________________________

giac [A]  time = 0.70, size = 42, normalized size = 0.66 \begin {gather*} -\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 {gather*}

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

________________________________________________________________________________________

maple [A]  time = 0.01, size = 98, normalized size = 1.53 \begin {gather*} \frac {3 \sqrt {\left (a x +1\right ) \left (-a x +1\right )}\, \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 \sqrt {a x +1}\, \sqrt {-a x +1}\, \sqrt {a^{2}}}-\frac {\left (a x +1\right )^{\frac {3}{2}} \sqrt {-a x +1}}{2 a}-\frac {3 \sqrt {-a x +1}\, \sqrt {a x +1}}{2 a} \end {gather*}

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))

________________________________________________________________________________________

maxima [A]  time = 2.99, size = 42, normalized size = 0.66 \begin {gather*} -\frac {1}{2} \, \sqrt {-a^{2} x^{2} + 1} x + \frac {3 \, \arcsin \left (a x\right )}{2 \, a} - \frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{a} \end {gather*}

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*x)/a - 2*sqrt(-a^2*x^2 + 1)/a

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {{\left (a\,x+1\right )}^{3/2}}{\sqrt {1-a\,x}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [A]  time = 33.75, size = 75, normalized size = 1.17 \begin {gather*} \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 {gather*}

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))

________________________________________________________________________________________