\(\int \frac {(-1+a x) \sqrt {a x+\sqrt {-b+a x}}}{\sqrt {-b+a x}} \, dx\) [1947]

Optimal result

Integrand size = 36, antiderivative size = 136 \[ \int \frac {(-1+a x) \sqrt {a x+\sqrt {-b+a x}}}{\sqrt {-b+a x}} \, dx=\frac {(-33-12 b+8 a x) \sqrt {a x+\sqrt {-b+a x}}}{96 a}+\frac {\sqrt {-b+a x} (-53+36 b+24 a x) \sqrt {a x+\sqrt {-b+a x}}}{48 a}+\frac {\left (-11+56 b-48 b^2\right ) \log \left (1+2 \sqrt {-b+a x}-2 \sqrt {a x+\sqrt {-b+a x}}\right )}{64 a} \]

[Out]

1/96*(8*a*x-12*b-33)*(a*x+(a*x-b)^(1/2))^(1/2)/a+1/48*(a*x-b)^(1/2)*(24*a*x+36*b-53)*(a*x+(a*x-b)^(1/2))^(1/2)
/a+1/64*(-48*b^2+56*b-11)*ln(1+2*(a*x-b)^(1/2)-2*(a*x+(a*x-b)^(1/2))^(1/2))/a

Rubi [A] (verified)

Time = 0.27 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.22, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.139, Rules used = {1675, 654, 626, 635, 212} \[ \int \frac {(-1+a x) \sqrt {a x+\sqrt {-b+a x}}}{\sqrt {-b+a x}} \, dx=\frac {(11-12 b) (1-4 b) \text {arctanh}\left (\frac {2 \sqrt {a x-b}+1}{2 \sqrt {\sqrt {a x-b}+a x}}\right )}{64 a}+\frac {\sqrt {a x-b} \left (\sqrt {a x-b}+a x\right )^{3/2}}{2 a}-\frac {5 \left (\sqrt {a x-b}+a x\right )^{3/2}}{12 a}-\frac {(11-12 b) \left (2 \sqrt {a x-b}+1\right ) \sqrt {\sqrt {a x-b}+a x}}{32 a} \]

[In]

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

[Out]

(-5*(a*x + Sqrt[-b + a*x])^(3/2))/(12*a) + (Sqrt[-b + a*x]*(a*x + Sqrt[-b + a*x])^(3/2))/(2*a) - ((11 - 12*b)*
Sqrt[a*x + Sqrt[-b + a*x]]*(1 + 2*Sqrt[-b + a*x]))/(32*a) + ((11 - 12*b)*(1 - 4*b)*ArcTanh[(1 + 2*Sqrt[-b + a*
x])/(2*Sqrt[a*x + Sqrt[-b + a*x]])])/(64*a)

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 626

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(b + 2*c*x)*((a + b*x + c*x^2)^p/(2*c*(2*p + 1
))), x] - Dist[p*((b^2 - 4*a*c)/(2*c*(2*p + 1))), Int[(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x]
&& NeQ[b^2 - 4*a*c, 0] && GtQ[p, 0] && IntegerQ[4*p]

Rule 635

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[1/(4*c - x^2), x], x, (b + 2*c*x)
/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 654

Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[e*((a + b*x + c*x^2)^(p +
1)/(2*c*(p + 1))), x] + Dist[(2*c*d - b*e)/(2*c), Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}
, x] && NeQ[2*c*d - b*e, 0] && NeQ[p, -1]

Rule 1675

Int[(Pq_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq, x], e = Coeff[Pq, x, Expo
n[Pq, x]]}, Simp[e*x^(q - 1)*((a + b*x + c*x^2)^(p + 1)/(c*(q + 2*p + 1))), x] + Dist[1/(c*(q + 2*p + 1)), Int
[(a + b*x + c*x^2)^p*ExpandToSum[c*(q + 2*p + 1)*Pq - a*e*(q - 1)*x^(q - 2) - b*e*(q + p)*x^(q - 1) - c*e*(q +
2*p + 1)*x^q, x], x], x]] /; FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] &&  !LeQ[p, -1]

Rubi steps \begin{align*} \text {integral}& = \frac {2 \text {Subst}\left (\int \left (-1+b+x^2\right ) \sqrt {b+x+x^2} \, dx,x,\sqrt {-b+a x}\right )}{a} \\ & = \frac {\sqrt {-b+a x} \left (a x+\sqrt {-b+a x}\right )^{3/2}}{2 a}+\frac {\text {Subst}\left (\int \left (-4+3 b-\frac {5 x}{2}\right ) \sqrt {b+x+x^2} \, dx,x,\sqrt {-b+a x}\right )}{2 a} \\ & = -\frac {5 \left (a x+\sqrt {-b+a x}\right )^{3/2}}{12 a}+\frac {\sqrt {-b+a x} \left (a x+\sqrt {-b+a x}\right )^{3/2}}{2 a}-\frac {(11-12 b) \text {Subst}\left (\int \sqrt {b+x+x^2} \, dx,x,\sqrt {-b+a x}\right )}{8 a} \\ & = -\frac {5 \left (a x+\sqrt {-b+a x}\right )^{3/2}}{12 a}+\frac {\sqrt {-b+a x} \left (a x+\sqrt {-b+a x}\right )^{3/2}}{2 a}-\frac {(11-12 b) \sqrt {a x+\sqrt {-b+a x}} \left (1+2 \sqrt {-b+a x}\right )}{32 a}+\frac {((11-12 b) (1-4 b)) \text {Subst}\left (\int \frac {1}{\sqrt {b+x+x^2}} \, dx,x,\sqrt {-b+a x}\right )}{64 a} \\ & = -\frac {5 \left (a x+\sqrt {-b+a x}\right )^{3/2}}{12 a}+\frac {\sqrt {-b+a x} \left (a x+\sqrt {-b+a x}\right )^{3/2}}{2 a}-\frac {(11-12 b) \sqrt {a x+\sqrt {-b+a x}} \left (1+2 \sqrt {-b+a x}\right )}{32 a}+\frac {((11-12 b) (1-4 b)) \text {Subst}\left (\int \frac {1}{4-x^2} \, dx,x,\frac {1+2 \sqrt {-b+a x}}{\sqrt {a x+\sqrt {-b+a x}}}\right )}{32 a} \\ & = -\frac {5 \left (a x+\sqrt {-b+a x}\right )^{3/2}}{12 a}+\frac {\sqrt {-b+a x} \left (a x+\sqrt {-b+a x}\right )^{3/2}}{2 a}-\frac {(11-12 b) \sqrt {a x+\sqrt {-b+a x}} \left (1+2 \sqrt {-b+a x}\right )}{32 a}+\frac {(11-12 b) (1-4 b) \text {arctanh}\left (\frac {1+2 \sqrt {-b+a x}}{2 \sqrt {a x+\sqrt {-b+a x}}}\right )}{64 a} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.26 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.97 \[ \int \frac {(-1+a x) \sqrt {a x+\sqrt {-b+a x}}}{\sqrt {-b+a x}} \, dx=\frac {2 \sqrt {a x+\sqrt {-b+a x}} \left (-33-106 \sqrt {-b+a x}+12 b \left (-1+6 \sqrt {-b+a x}\right )+8 a \left (x+6 x \sqrt {-b+a x}\right )\right )-3 \left (11-56 b+48 b^2\right ) \log \left (a \left (-1-2 \sqrt {-b+a x}+2 \sqrt {a x+\sqrt {-b+a x}}\right )\right )}{192 a} \]

[In]

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

[Out]

(2*Sqrt[a*x + Sqrt[-b + a*x]]*(-33 - 106*Sqrt[-b + a*x] + 12*b*(-1 + 6*Sqrt[-b + a*x]) + 8*a*(x + 6*x*Sqrt[-b
+ a*x])) - 3*(11 - 56*b + 48*b^2)*Log[a*(-1 - 2*Sqrt[-b + a*x] + 2*Sqrt[a*x + Sqrt[-b + a*x]])])/(192*a)

Maple [A] (verified)

Time = 0.26 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.34

method	result	size
derivativedivides	\(\frac {-\frac {11 \left (2 \sqrt {a x -b}+1\right ) \sqrt {a x +\sqrt {a x -b}}}{32}-\frac {11 \left (4 b -1\right ) \ln \left (\frac {1}{2}+\sqrt {a x -b}+\sqrt {a x +\sqrt {a x -b}}\right )}{64}+\frac {3 b \left (\frac {\left (2 \sqrt {a x -b}+1\right ) \sqrt {a x +\sqrt {a x -b}}}{4}+\frac {\left (4 b -1\right ) \ln \left (\frac {1}{2}+\sqrt {a x -b}+\sqrt {a x +\sqrt {a x -b}}\right )}{8}\right )}{2}+\frac {\sqrt {a x -b}\, \left (a x +\sqrt {a x -b}\right )^{\frac {3}{2}}}{2}-\frac {5 \left (a x +\sqrt {a x -b}\right )^{\frac {3}{2}}}{12}}{a}\)	\(182\)
default	\(\frac {-\frac {11 \left (2 \sqrt {a x -b}+1\right ) \sqrt {a x +\sqrt {a x -b}}}{32}-\frac {11 \left (4 b -1\right ) \ln \left (\frac {1}{2}+\sqrt {a x -b}+\sqrt {a x +\sqrt {a x -b}}\right )}{64}+\frac {3 b \left (\frac {\left (2 \sqrt {a x -b}+1\right ) \sqrt {a x +\sqrt {a x -b}}}{4}+\frac {\left (4 b -1\right ) \ln \left (\frac {1}{2}+\sqrt {a x -b}+\sqrt {a x +\sqrt {a x -b}}\right )}{8}\right )}{2}+\frac {\sqrt {a x -b}\, \left (a x +\sqrt {a x -b}\right )^{\frac {3}{2}}}{2}-\frac {5 \left (a x +\sqrt {a x -b}\right )^{\frac {3}{2}}}{12}}{a}\)	\(182\)

[In]

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

[Out]

2/a*(-11/64*(2*(a*x-b)^(1/2)+1)*(a*x+(a*x-b)^(1/2))^(1/2)-11/128*(4*b-1)*ln(1/2+(a*x-b)^(1/2)+(a*x+(a*x-b)^(1/
2))^(1/2))+3/4*b*(1/4*(2*(a*x-b)^(1/2)+1)*(a*x+(a*x-b)^(1/2))^(1/2)+1/8*(4*b-1)*ln(1/2+(a*x-b)^(1/2)+(a*x+(a*x
-b)^(1/2))^(1/2)))+1/4*(a*x-b)^(1/2)*(a*x+(a*x-b)^(1/2))^(3/2)-5/24*(a*x+(a*x-b)^(1/2))^(3/2))

Fricas [F(-1)]

Timed out. \[ \int \frac {(-1+a x) \sqrt {a x+\sqrt {-b+a x}}}{\sqrt {-b+a x}} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Sympy [A] (verification not implemented)

Time = 0.60 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.21 \[ \int \frac {(-1+a x) \sqrt {a x+\sqrt {-b+a x}}}{\sqrt {-b+a x}} \, dx=\begin {cases} \frac {2 \left (\sqrt {a x + \sqrt {a x - b}} \left (\frac {a x}{24} - \frac {b}{16} + \left (\frac {5 b}{8} - \frac {53}{96}\right ) \sqrt {a x - b} + \frac {\left (a x - b\right )^{\frac {3}{2}}}{4} - \frac {11}{64}\right ) + \left (b^{2} - b \left (\frac {5 b}{8} - \frac {53}{96}\right ) - \frac {95 b}{96} + \frac {11}{128}\right ) \left (\begin {cases} \log {\left (2 \sqrt {a x - b} + 2 \sqrt {a x + \sqrt {a x - b}} + 1 \right )} & \text {for}\: b \neq \frac {1}{4} \\\frac {\left (\sqrt {a x - b} + \frac {1}{2}\right ) \log {\left (\sqrt {a x - b} + \frac {1}{2} \right )}}{\sqrt {\left (\sqrt {a x - b} + \frac {1}{2}\right )^{2}}} & \text {otherwise} \end {cases}\right )\right )}{a} & \text {for}\: a \neq 0 \\- \frac {x}{\sqrt [4]{- b}} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((2*(sqrt(a*x + sqrt(a*x - b))*(a*x/24 - b/16 + (5*b/8 - 53/96)*sqrt(a*x - b) + (a*x - b)**(3/2)/4 -
11/64) + (b**2 - b*(5*b/8 - 53/96) - 95*b/96 + 11/128)*Piecewise((log(2*sqrt(a*x - b) + 2*sqrt(a*x + sqrt(a*x
- b)) + 1), Ne(b, 1/4)), ((sqrt(a*x - b) + 1/2)*log(sqrt(a*x - b) + 1/2)/sqrt((sqrt(a*x - b) + 1/2)**2), True)
))/a, Ne(a, 0)), (-x/(-b)**(1/4), True))

Maxima [F]

\[ \int \frac {(-1+a x) \sqrt {a x+\sqrt {-b+a x}}}{\sqrt {-b+a x}} \, dx=\int { \frac {\sqrt {a x + \sqrt {a x - b}} {\left (a x - 1\right )}}{\sqrt {a x - b}} \,d x } \]

[In]

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

[Out]

integrate(sqrt(a*x + sqrt(a*x - b))*(a*x - 1)/sqrt(a*x - b), x)

Giac [A] (verification not implemented)

none

Time = 0.93 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.33 \[ \int \frac {(-1+a x) \sqrt {a x+\sqrt {-b+a x}}}{\sqrt {-b+a x}} \, dx=-\frac {3 \, {\left (48 \, b^{2} + 8 \, b - 5\right )} \log \left ({\left | -2 \, \sqrt {a x - b} + 2 \, \sqrt {a x + \sqrt {a x - b}} - 1 \right |}\right ) - 48 \, {\left (4 \, b - 1\right )} \log \left ({\left | -2 \, \sqrt {a x - b} + 2 \, \sqrt {a x + \sqrt {a x - b}} - 1 \right |}\right ) - 2 \, \sqrt {a x + \sqrt {a x - b}} {\left (2 \, \sqrt {a x - b} {\left (4 \, \sqrt {a x - b} {\left (6 \, \sqrt {a x - b} + 1\right )} + 60 \, b - 5\right )} - 4 \, b + 15\right )} + 96 \, \sqrt {a x + \sqrt {a x - b}} {\left (2 \, \sqrt {a x - b} + 1\right )}}{192 \, a} \]

[In]

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

[Out]

-1/192*(3*(48*b^2 + 8*b - 5)*log(abs(-2*sqrt(a*x - b) + 2*sqrt(a*x + sqrt(a*x - b)) - 1)) - 48*(4*b - 1)*log(a
bs(-2*sqrt(a*x - b) + 2*sqrt(a*x + sqrt(a*x - b)) - 1)) - 2*sqrt(a*x + sqrt(a*x - b))*(2*sqrt(a*x - b)*(4*sqrt
(a*x - b)*(6*sqrt(a*x - b) + 1) + 60*b - 5) - 4*b + 15) + 96*sqrt(a*x + sqrt(a*x - b))*(2*sqrt(a*x - b) + 1))/
a

Mupad [F(-1)]

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

[In]

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

[Out]

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