Integrand size = 32, antiderivative size = 136 \[ \int \frac {x \sqrt {a x+\sqrt {-b+a x}}}{\sqrt {-b+a x}} \, dx=\frac {(15-12 b+8 a x) \sqrt {a x+\sqrt {-b+a x}}}{96 a^2}+\frac {\sqrt {-b+a x} (-5+36 b+24 a x) \sqrt {a x+\sqrt {-b+a x}}}{48 a^2}+\frac {\left (5-8 b-48 b^2\right ) \log \left (1+2 \sqrt {-b+a x}-2 \sqrt {a x+\sqrt {-b+a x}}\right )}{64 a^2} \]
1/96*(8*a*x-12*b+15)*(a*x+(a*x-b)^(1/2))^(1/2)/a^2+1/48*(a*x-b)^(1/2)*(24* a*x+36*b-5)*(a*x+(a*x-b)^(1/2))^(1/2)/a^2+1/64*(-48*b^2-8*b+5)*ln(1+2*(a*x -b)^(1/2)-2*(a*x+(a*x-b)^(1/2))^(1/2))/a^2
Time = 0.23 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.96 \[ \int \frac {x \sqrt {a x+\sqrt {-b+a x}}}{\sqrt {-b+a x}} \, dx=\frac {2 \sqrt {a x+\sqrt {-b+a x}} \left (15-10 \sqrt {-b+a x}+12 b \left (-1+6 \sqrt {-b+a x}\right )+8 a \left (x+6 x \sqrt {-b+a x}\right )\right )-3 \left (-5+8 b+48 b^2\right ) \log \left (-1-2 \sqrt {-b+a x}+2 \sqrt {a x+\sqrt {-b+a x}}\right )}{192 a^2} \]
(2*Sqrt[a*x + Sqrt[-b + a*x]]*(15 - 10*Sqrt[-b + a*x] + 12*b*(-1 + 6*Sqrt[ -b + a*x]) + 8*a*(x + 6*x*Sqrt[-b + a*x])) - 3*(-5 + 8*b + 48*b^2)*Log[-1 - 2*Sqrt[-b + a*x] + 2*Sqrt[a*x + Sqrt[-b + a*x]]])/(192*a^2)
Time = 0.60 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.21, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.219, Rules used = {7267, 2192, 27, 1160, 1087, 1092, 219}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {x \sqrt {\sqrt {a x-b}+a x}}{\sqrt {a x-b}} \, dx\) |
| \(\Big \downarrow \) 7267 |
| \(\displaystyle \frac {2 \int a x \sqrt {a x+\sqrt {a x-b}}d\sqrt {a x-b}}{a^2}\) |
| \(\Big \downarrow \) 2192 |
| \(\displaystyle \frac {2 \left (\frac {1}{4} \int \frac {1}{2} \left (6 b-5 \sqrt {a x-b}\right ) \sqrt {a x+\sqrt {a x-b}}d\sqrt {a x-b}+\frac {1}{4} \sqrt {a x-b} \left (\sqrt {a x-b}+a x\right )^{3/2}\right )}{a^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 \left (\frac {1}{8} \int \left (6 b-5 \sqrt {a x-b}\right ) \sqrt {a x+\sqrt {a x-b}}d\sqrt {a x-b}+\frac {1}{4} \sqrt {a x-b} \left (\sqrt {a x-b}+a x\right )^{3/2}\right )}{a^2}\) |
| \(\Big \downarrow \) 1160 |
| \(\displaystyle \frac {2 \left (\frac {1}{8} \left (\frac {1}{2} (12 b+5) \int \sqrt {a x+\sqrt {a x-b}}d\sqrt {a x-b}-\frac {5}{3} \left (\sqrt {a x-b}+a x\right )^{3/2}\right )+\frac {1}{4} \sqrt {a x-b} \left (\sqrt {a x-b}+a x\right )^{3/2}\right )}{a^2}\) |
| \(\Big \downarrow \) 1087 |
| \(\displaystyle \frac {2 \left (\frac {1}{8} \left (\frac {1}{2} (12 b+5) \left (\frac {1}{4} \sqrt {\sqrt {a x-b}+a x} \left (2 \sqrt {a x-b}+1\right )-\frac {1}{8} (1-4 b) \int \frac {1}{\sqrt {a x+\sqrt {a x-b}}}d\sqrt {a x-b}\right )-\frac {5}{3} \left (\sqrt {a x-b}+a x\right )^{3/2}\right )+\frac {1}{4} \sqrt {a x-b} \left (\sqrt {a x-b}+a x\right )^{3/2}\right )}{a^2}\) |
| \(\Big \downarrow \) 1092 |
| \(\displaystyle \frac {2 \left (\frac {1}{8} \left (\frac {1}{2} (12 b+5) \left (\frac {1}{4} \sqrt {\sqrt {a x-b}+a x} \left (2 \sqrt {a x-b}+1\right )-\frac {1}{4} (1-4 b) \int \frac {1}{b-a x+4}d\frac {2 \sqrt {a x-b}+1}{\sqrt {a x+\sqrt {a x-b}}}\right )-\frac {5}{3} \left (\sqrt {a x-b}+a x\right )^{3/2}\right )+\frac {1}{4} \sqrt {a x-b} \left (\sqrt {a x-b}+a x\right )^{3/2}\right )}{a^2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {2 \left (\frac {1}{8} \left (\frac {1}{2} (12 b+5) \left (\frac {1}{4} \sqrt {\sqrt {a x-b}+a x} \left (2 \sqrt {a x-b}+1\right )-\frac {1}{8} (1-4 b) \text {arctanh}\left (\frac {2 \sqrt {a x-b}+1}{2 \sqrt {\sqrt {a x-b}+a x}}\right )\right )-\frac {5}{3} \left (\sqrt {a x-b}+a x\right )^{3/2}\right )+\frac {1}{4} \sqrt {a x-b} \left (\sqrt {a x-b}+a x\right )^{3/2}\right )}{a^2}\) |
(2*((Sqrt[-b + a*x]*(a*x + Sqrt[-b + a*x])^(3/2))/4 + ((-5*(a*x + Sqrt[-b + a*x])^(3/2))/3 + ((5 + 12*b)*((Sqrt[a*x + Sqrt[-b + a*x]]*(1 + 2*Sqrt[-b + a*x]))/4 - ((1 - 4*b)*ArcTanh[(1 + 2*Sqrt[-b + a*x])/(2*Sqrt[a*x + Sqrt [-b + a*x]])])/8))/2)/8))/a^2
3.20.46.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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] && (Gt Q[a, 0] || LtQ[b, 0])
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] - Simp[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] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[3*p])
Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2 Subst[I nt[1/(4*c - x^2), x], x, (b + 2*c*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a , b, c}, x]
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] + Simp[(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[p, -1]
Int[(Pq_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq, x], e = Coeff[Pq, x, Expon[Pq, x]]}, Simp[e*x^(q - 1)*((a + b*x + c*x^2)^(p + 1)/(c*(q + 2*p + 1))), x] + Simp[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]
Int[u_, x_Symbol] :> With[{lst = SubstForFractionalPowerOfLinear[u, x]}, Si mp[lst[[2]]*lst[[4]] Subst[Int[lst[[1]], x], x, lst[[3]]^(1/lst[[2]])], x ] /; !FalseQ[lst] && SubstForFractionalPowerQ[u, lst[[3]], x]]
Time = 0.16 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.34
| method | result | size |
| derivativedivides | \(\frac {\frac {3 b \left (\frac {\left (2 \sqrt {a x -b}+1\right ) \sqrt {a x +\sqrt {a x -b}}}{4}+\frac {\left (4 b -1\right ) \ln \left (\frac {1}{2}+\sqrt {a x -b}+\sqrt {a x +\sqrt {a x -b}}\right )}{8}\right )}{2}+\frac {\sqrt {a x -b}\, \left (a x +\sqrt {a x -b}\right )^{\frac {3}{2}}}{2}-\frac {5 \left (a x +\sqrt {a x -b}\right )^{\frac {3}{2}}}{12}+\frac {5 \left (2 \sqrt {a x -b}+1\right ) \sqrt {a x +\sqrt {a x -b}}}{32}+\frac {5 \left (4 b -1\right ) \ln \left (\frac {1}{2}+\sqrt {a x -b}+\sqrt {a x +\sqrt {a x -b}}\right )}{64}}{a^{2}}\) | \(182\) |
| default | \(\frac {\frac {3 b \left (\frac {\left (2 \sqrt {a x -b}+1\right ) \sqrt {a x +\sqrt {a x -b}}}{4}+\frac {\left (4 b -1\right ) \ln \left (\frac {1}{2}+\sqrt {a x -b}+\sqrt {a x +\sqrt {a x -b}}\right )}{8}\right )}{2}+\frac {\sqrt {a x -b}\, \left (a x +\sqrt {a x -b}\right )^{\frac {3}{2}}}{2}-\frac {5 \left (a x +\sqrt {a x -b}\right )^{\frac {3}{2}}}{12}+\frac {5 \left (2 \sqrt {a x -b}+1\right ) \sqrt {a x +\sqrt {a x -b}}}{32}+\frac {5 \left (4 b -1\right ) \ln \left (\frac {1}{2}+\sqrt {a x -b}+\sqrt {a x +\sqrt {a x -b}}\right )}{64}}{a^{2}}\) | \(182\) |
2/a^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)+5/64*(2*(a*x-b)^(1/2)+ 1)*(a*x+(a*x-b)^(1/2))^(1/2)+5/128*(4*b-1)*ln(1/2+(a*x-b)^(1/2)+(a*x+(a*x- b)^(1/2))^(1/2)))
Timed out. \[ \int \frac {x \sqrt {a x+\sqrt {-b+a x}}}{\sqrt {-b+a x}} \, dx=\text {Timed out} \]
Time = 0.50 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.23 \[ \int \frac {x \sqrt {a x+\sqrt {-b+a x}}}{\sqrt {-b+a x}} \, dx=\begin {cases} \frac {2 \left (\sqrt {a x + \sqrt {a x - b}} \left (\frac {a x}{24} - \frac {b}{16} + \left (\frac {5 b}{8} - \frac {5}{96}\right ) \sqrt {a x - b} + \frac {\left (a x - b\right )^{\frac {3}{2}}}{4} + \frac {5}{64}\right ) + \left (b^{2} - b \left (\frac {5 b}{8} - \frac {5}{96}\right ) + \frac {b}{96} - \frac {5}{128}\right ) \left (\begin {cases} \log {\left (2 \sqrt {a x - b} + 2 \sqrt {a x + \sqrt {a x - b}} + 1 \right )} & \text {for}\: b \neq \frac {1}{4} \\\frac {\left (\sqrt {a x - b} + \frac {1}{2}\right ) \log {\left (\sqrt {a x - b} + \frac {1}{2} \right )}}{\sqrt {\left (\sqrt {a x - b} + \frac {1}{2}\right )^{2}}} & \text {otherwise} \end {cases}\right )\right )}{a^{2}} & \text {for}\: a \neq 0 \\\frac {x^{2}}{2 \sqrt [4]{- b}} & \text {otherwise} \end {cases} \]
Piecewise((2*(sqrt(a*x + sqrt(a*x - b))*(a*x/24 - b/16 + (5*b/8 - 5/96)*sq rt(a*x - b) + (a*x - b)**(3/2)/4 + 5/64) + (b**2 - b*(5*b/8 - 5/96) + b/96 - 5/128)*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**2, Ne(a, 0)), (x**2/(2*(-b)**(1/4)), True ))
\[ \int \frac {x \sqrt {a x+\sqrt {-b+a x}}}{\sqrt {-b+a x}} \, dx=\int { \frac {\sqrt {a x + \sqrt {a x - b}} x}{\sqrt {a x - b}} \,d x } \]
Time = 0.61 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.82 \[ \int \frac {x \sqrt {a x+\sqrt {-b+a x}}}{\sqrt {-b+a x}} \, dx=-\frac {3 \, {\left (48 \, b^{2} + 8 \, b - 5\right )} \log \left ({\left | -2 \, \sqrt {a x - b} + 2 \, \sqrt {a x + \sqrt {a x - b}} - 1 \right |}\right ) - 2 \, \sqrt {a x + \sqrt {a x - b}} {\left (2 \, \sqrt {a x - b} {\left (4 \, \sqrt {a x - b} {\left (6 \, \sqrt {a x - b} + 1\right )} + 60 \, b - 5\right )} - 4 \, b + 15\right )}}{192 \, a^{2}} \]
-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)) - 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))/a^2
Timed out. \[ \int \frac {x \sqrt {a x+\sqrt {-b+a x}}}{\sqrt {-b+a x}} \, dx=\int \frac {x\,\sqrt {a\,x+\sqrt {a\,x-b}}}{\sqrt {a\,x-b}} \,d x \]