\(\int \frac {(1+a x)^{3/2}}{\sqrt {1-a x}} \, dx\) [1104]

Optimal result

Integrand size = 20, antiderivative size = 64 \[ \int \frac {(1+a x)^{3/2}}{\sqrt {1-a x}} \, dx=-\frac {3 \sqrt {1-a x} \sqrt {1+a x}}{2 a}-\frac {\sqrt {1-a x} (1+a x)^{3/2}}{2 a}+\frac {3 \arcsin (a x)}{2 a} \]

[Out]

3/2*arcsin(a*x)/a-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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {52, 41, 222} \[ \int \frac {(1+a x)^{3/2}}{\sqrt {1-a x}} \, dx=\frac {3 \arcsin (a x)}{2 a}-\frac {\sqrt {1-a x} (a x+1)^{3/2}}{2 a}-\frac {3 \sqrt {1-a x} \sqrt {a x+1}}{2 a} \]

[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 52

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 222

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*} \text {integral}& = -\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] (verified)

Time = 0.11 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.86 \[ \int \frac {(1+a x)^{3/2}}{\sqrt {1-a x}} \, dx=\frac {-\left ((4+a x) \sqrt {1-a^2 x^2}\right )+6 \arctan \left (\frac {\sqrt {1-a^2 x^2}}{1-a x}\right )}{2 a} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.20 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.53

method	result	size
default	\(-\frac {\left (a x +1\right )^{\frac {3}{2}} \sqrt {-a x +1}}{2 a}-\frac {3 \sqrt {-a x +1}\, \sqrt {a x +1}}{2 a}+\frac {3 \sqrt {\left (a x +1\right ) \left (-a x +1\right )}\, \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 \sqrt {a x +1}\, \sqrt {-a x +1}\, \sqrt {a^{2}}}\)	\(98\)
risch	\(\frac {\left (a x +4\right ) \sqrt {a x +1}\, \left (a x -1\right ) \sqrt {\left (a x +1\right ) \left (-a x +1\right )}}{2 a \sqrt {-\left (a x +1\right ) \left (a x -1\right )}\, \sqrt {-a x +1}}+\frac {3 \sqrt {\left (a x +1\right ) \left (-a x +1\right )}\, \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 \sqrt {a x +1}\, \sqrt {-a x +1}\, \sqrt {a^{2}}}\)	\(116\)

[In]

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

[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))

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.86 \[ \int \frac {(1+a x)^{3/2}}{\sqrt {1-a x}} \, dx=-\frac {{\left (a x + 4\right )} \sqrt {a x + 1} \sqrt {-a x + 1} + 6 \, \arctan \left (\frac {\sqrt {a x + 1} \sqrt {-a x + 1} - 1}{a x}\right )}{2 \, a} \]

[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

Sympy [F]

\[ \int \frac {(1+a x)^{3/2}}{\sqrt {1-a x}} \, dx=\int \frac {\left (a x + 1\right )^{\frac {3}{2}}}{\sqrt {- a x + 1}}\, dx \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.66 \[ \int \frac {(1+a x)^{3/2}}{\sqrt {1-a x}} \, dx=-\frac {1}{2} \, \sqrt {-a^{2} x^{2} + 1} x + \frac {3 \, \arcsin \left (a x\right )}{2 \, a} - \frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{a} \]

[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

Giac [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.66 \[ \int \frac {(1+a x)^{3/2}}{\sqrt {1-a x}} \, dx=-\frac {{\left (a x + 4\right )} \sqrt {a x + 1} \sqrt {-a x + 1} - 6 \, \arcsin \left (\frac {1}{2} \, \sqrt {2} \sqrt {a x + 1}\right )}{2 \, a} \]

[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

Mupad [F(-1)]

Timed out. \[ \int \frac {(1+a x)^{3/2}}{\sqrt {1-a x}} \, dx=\int \frac {{\left (a\,x+1\right )}^{3/2}}{\sqrt {1-a\,x}} \,d x \]

[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)