\(\int \frac {x^2 \sqrt {a x+\sqrt {-b+a x}}}{\sqrt {-b+a x}} \, dx\) [2313]

Optimal result

Integrand size = 34, antiderivative size = 177 \[ \int \frac {x^2 \sqrt {a x+\sqrt {-b+a x}}}{\sqrt {-b+a x}} \, dx=\frac {\left (315-648 b-400 b^2+168 a x-160 a b x+128 a^2 x^2\right ) \sqrt {a x+\sqrt {-b+a x}}}{3840 a^3}+\frac {\sqrt {-b+a x} \left (-105+120 b+1200 b^2-72 a x+800 a b x+640 a^2 x^2\right ) \sqrt {a x+\sqrt {-b+a x}}}{1920 a^3}+\frac {\left (21-60 b-16 b^2-320 b^3\right ) \log \left (1+2 \sqrt {-b+a x}-2 \sqrt {a x+\sqrt {-b+a x}}\right )}{512 a^3} \]

[Out]

1/3840*(128*a^2*x^2-160*a*b*x+168*a*x-400*b^2-648*b+315)*(a*x+(a*x-b)^(1/2))^(1/2)/a^3+1/1920*(a*x-b)^(1/2)*(6
40*a^2*x^2+800*a*b*x-72*a*x+1200*b^2+120*b-105)*(a*x+(a*x-b)^(1/2))^(1/2)/a^3+1/512*(-320*b^3-16*b^2-60*b+21)*
ln(1+2*(a*x-b)^(1/2)-2*(a*x+(a*x-b)^(1/2))^(1/2))/a^3

Rubi [A] (verified)

Time = 0.54 (sec) , antiderivative size = 255, normalized size of antiderivative = 1.44, number of steps used = 8, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.147, Rules used = {1675, 654, 626, 635, 212} \[ \int \frac {x^2 \sqrt {a x+\sqrt {-b+a x}}}{\sqrt {-b+a x}} \, dx=-\frac {(1-4 b) \left (80 b^2+24 b+21\right ) \text {arctanh}\left (\frac {2 \sqrt {a x-b}+1}{2 \sqrt {\sqrt {a x-b}+a x}}\right )}{512 a^3}+\frac {\left (80 b^2+24 b+21\right ) \left (2 \sqrt {a x-b}+1\right ) \sqrt {\sqrt {a x-b}+a x}}{256 a^3}+\frac {(a x-b)^{3/2} \left (\sqrt {a x-b}+a x\right )^{3/2}}{3 a^3}-\frac {(68 b+35) \left (\sqrt {a x-b}+a x\right )^{3/2}}{160 a^3}+\frac {3 (b-a x) \left (\sqrt {a x-b}+a x\right )^{3/2}}{10 a^3}+\frac {3 (20 b+7) \sqrt {a x-b} \left (\sqrt {a x-b}+a x\right )^{3/2}}{80 a^3} \]

[In]

Int[(x^2*Sqrt[a*x + Sqrt[-b + a*x]])/Sqrt[-b + a*x],x]

[Out]

-1/160*((35 + 68*b)*(a*x + Sqrt[-b + a*x])^(3/2))/a^3 + (3*(b - a*x)*(a*x + Sqrt[-b + a*x])^(3/2))/(10*a^3) +
(3*(7 + 20*b)*Sqrt[-b + a*x]*(a*x + Sqrt[-b + a*x])^(3/2))/(80*a^3) + ((-b + a*x)^(3/2)*(a*x + Sqrt[-b + a*x])
^(3/2))/(3*a^3) + ((21 + 24*b + 80*b^2)*Sqrt[a*x + Sqrt[-b + a*x]]*(1 + 2*Sqrt[-b + a*x]))/(256*a^3) - ((1 - 4
*b)*(21 + 24*b + 80*b^2)*ArcTanh[(1 + 2*Sqrt[-b + a*x])/(2*Sqrt[a*x + Sqrt[-b + a*x]])])/(512*a^3)

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 (b+x^2\right )^2 \sqrt {b+x+x^2} \, dx,x,\sqrt {-b+a x}\right )}{a^3} \\ & = \frac {(-b+a x)^{3/2} \left (a x+\sqrt {-b+a x}\right )^{3/2}}{3 a^3}+\frac {\text {Subst}\left (\int \sqrt {b+x+x^2} \left (6 b^2+9 b x^2-\frac {9 x^3}{2}\right ) \, dx,x,\sqrt {-b+a x}\right )}{3 a^3} \\ & = \frac {3 (b-a x) \left (a x+\sqrt {-b+a x}\right )^{3/2}}{10 a^3}+\frac {(-b+a x)^{3/2} \left (a x+\sqrt {-b+a x}\right )^{3/2}}{3 a^3}+\frac {\text {Subst}\left (\int \sqrt {b+x+x^2} \left (30 b^2+9 b x+\frac {9}{4} (7+20 b) x^2\right ) \, dx,x,\sqrt {-b+a x}\right )}{15 a^3} \\ & = \frac {3 (b-a x) \left (a x+\sqrt {-b+a x}\right )^{3/2}}{10 a^3}+\frac {3 (7+20 b) \sqrt {-b+a x} \left (a x+\sqrt {-b+a x}\right )^{3/2}}{80 a^3}+\frac {(-b+a x)^{3/2} \left (a x+\sqrt {-b+a x}\right )^{3/2}}{3 a^3}+\frac {\text {Subst}\left (\int \left (-\frac {3}{4} (21-100 b) b-\frac {9}{8} (35+68 b) x\right ) \sqrt {b+x+x^2} \, dx,x,\sqrt {-b+a x}\right )}{60 a^3} \\ & = -\frac {(35+68 b) \left (a x+\sqrt {-b+a x}\right )^{3/2}}{160 a^3}+\frac {3 (b-a x) \left (a x+\sqrt {-b+a x}\right )^{3/2}}{10 a^3}+\frac {3 (7+20 b) \sqrt {-b+a x} \left (a x+\sqrt {-b+a x}\right )^{3/2}}{80 a^3}+\frac {(-b+a x)^{3/2} \left (a x+\sqrt {-b+a x}\right )^{3/2}}{3 a^3}+\frac {\left (21+24 b+80 b^2\right ) \text {Subst}\left (\int \sqrt {b+x+x^2} \, dx,x,\sqrt {-b+a x}\right )}{64 a^3} \\ & = -\frac {(35+68 b) \left (a x+\sqrt {-b+a x}\right )^{3/2}}{160 a^3}+\frac {3 (b-a x) \left (a x+\sqrt {-b+a x}\right )^{3/2}}{10 a^3}+\frac {3 (7+20 b) \sqrt {-b+a x} \left (a x+\sqrt {-b+a x}\right )^{3/2}}{80 a^3}+\frac {(-b+a x)^{3/2} \left (a x+\sqrt {-b+a x}\right )^{3/2}}{3 a^3}+\frac {\left (21+24 b+80 b^2\right ) \sqrt {a x+\sqrt {-b+a x}} \left (1+2 \sqrt {-b+a x}\right )}{256 a^3}-\frac {\left ((1-4 b) \left (21+24 b+80 b^2\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {b+x+x^2}} \, dx,x,\sqrt {-b+a x}\right )}{512 a^3} \\ & = -\frac {(35+68 b) \left (a x+\sqrt {-b+a x}\right )^{3/2}}{160 a^3}+\frac {3 (b-a x) \left (a x+\sqrt {-b+a x}\right )^{3/2}}{10 a^3}+\frac {3 (7+20 b) \sqrt {-b+a x} \left (a x+\sqrt {-b+a x}\right )^{3/2}}{80 a^3}+\frac {(-b+a x)^{3/2} \left (a x+\sqrt {-b+a x}\right )^{3/2}}{3 a^3}+\frac {\left (21+24 b+80 b^2\right ) \sqrt {a x+\sqrt {-b+a x}} \left (1+2 \sqrt {-b+a x}\right )}{256 a^3}-\frac {\left ((1-4 b) \left (21+24 b+80 b^2\right )\right ) \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 )}{256 a^3} \\ & = -\frac {(35+68 b) \left (a x+\sqrt {-b+a x}\right )^{3/2}}{160 a^3}+\frac {3 (b-a x) \left (a x+\sqrt {-b+a x}\right )^{3/2}}{10 a^3}+\frac {3 (7+20 b) \sqrt {-b+a x} \left (a x+\sqrt {-b+a x}\right )^{3/2}}{80 a^3}+\frac {(-b+a x)^{3/2} \left (a x+\sqrt {-b+a x}\right )^{3/2}}{3 a^3}+\frac {\left (21+24 b+80 b^2\right ) \sqrt {a x+\sqrt {-b+a x}} \left (1+2 \sqrt {-b+a x}\right )}{256 a^3}-\frac {(1-4 b) \left (21+24 b+80 b^2\right ) \text {arctanh}\left (\frac {1+2 \sqrt {-b+a x}}{2 \sqrt {a x+\sqrt {-b+a x}}}\right )}{512 a^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.39 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.11 \[ \int \frac {x^2 \sqrt {a x+\sqrt {-b+a x}}}{\sqrt {-b+a x}} \, dx=\frac {2 \sqrt {a x+\sqrt {-b+a x}} \left (315-210 \sqrt {-b+a x}-24 a x \left (-7+6 \sqrt {-b+a x}\right )+400 b^2 \left (-1+6 \sqrt {-b+a x}\right )+128 a^2 x^2 \left (1+10 \sqrt {-b+a x}\right )+8 b \left (-81+30 \sqrt {-b+a x}+20 a x \left (-1+10 \sqrt {-b+a x}\right )\right )\right )-15 \left (-21+60 b+16 b^2+320 b^3\right ) \log \left (-1-2 \sqrt {-b+a x}+2 \sqrt {a x+\sqrt {-b+a x}}\right )}{7680 a^3} \]

[In]

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

[Out]

(2*Sqrt[a*x + Sqrt[-b + a*x]]*(315 - 210*Sqrt[-b + a*x] - 24*a*x*(-7 + 6*Sqrt[-b + a*x]) + 400*b^2*(-1 + 6*Sqr
t[-b + a*x]) + 128*a^2*x^2*(1 + 10*Sqrt[-b + a*x]) + 8*b*(-81 + 30*Sqrt[-b + a*x] + 20*a*x*(-1 + 10*Sqrt[-b +
a*x]))) - 15*(-21 + 60*b + 16*b^2 + 320*b^3)*Log[-1 - 2*Sqrt[-b + a*x] + 2*Sqrt[a*x + Sqrt[-b + a*x]]])/(7680*
a^3)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(564\) vs. \(2(155)=310\).

Time = 0.10 (sec) , antiderivative size = 565, normalized size of antiderivative = 3.19

method	result	size
derivativedivides	\(\frac {2 b^{2} \left (\frac {\left (2 \sqrt {a x -b}+1\right ) \sqrt {a x +\sqrt {a x -b}}}{4}+\frac {\left (4 b -1\right ) \ln \left (\frac {1}{2}+\sqrt {a x -b}+\sqrt {a x +\sqrt {a x -b}}\right )}{8}\right )+\frac {\left (a x -b \right )^{\frac {3}{2}} \left (a x +\sqrt {a x -b}\right )^{\frac {3}{2}}}{3}-\frac {3 \left (a x -b \right ) \left (a x +\sqrt {a x -b}\right )^{\frac {3}{2}}}{10}+\frac {21 \sqrt {a x -b}\, \left (a x +\sqrt {a x -b}\right )^{\frac {3}{2}}}{80}-\frac {7 \left (a x +\sqrt {a x -b}\right )^{\frac {3}{2}}}{32}+\frac {21 \left (2 \sqrt {a x -b}+1\right ) \sqrt {a x +\sqrt {a x -b}}}{256}+\frac {21 \left (4 b -1\right ) \ln \left (\frac {1}{2}+\sqrt {a x -b}+\sqrt {a x +\sqrt {a x -b}}\right )}{512}-\frac {21 b \left (\frac {\left (2 \sqrt {a x -b}+1\right ) \sqrt {a x +\sqrt {a x -b}}}{4}+\frac {\left (4 b -1\right ) \ln \left (\frac {1}{2}+\sqrt {a x -b}+\sqrt {a x +\sqrt {a x -b}}\right )}{8}\right )}{80}+\frac {3 b \left (\frac {\left (a x +\sqrt {a x -b}\right )^{\frac {3}{2}}}{3}-\frac {\left (2 \sqrt {a x -b}+1\right ) \sqrt {a x +\sqrt {a x -b}}}{8}-\frac {\left (4 b -1\right ) \ln \left (\frac {1}{2}+\sqrt {a x -b}+\sqrt {a x +\sqrt {a x -b}}\right )}{16}\right )}{5}+3 b \left (\frac {\sqrt {a x -b}\, \left (a x +\sqrt {a x -b}\right )^{\frac {3}{2}}}{4}-\frac {5 \left (a x +\sqrt {a x -b}\right )^{\frac {3}{2}}}{24}+\frac {5 \left (2 \sqrt {a x -b}+1\right ) \sqrt {a x +\sqrt {a x -b}}}{64}+\frac {5 \left (4 b -1\right ) \ln \left (\frac {1}{2}+\sqrt {a x -b}+\sqrt {a x +\sqrt {a x -b}}\right )}{128}-\frac {b \left (\frac {\left (2 \sqrt {a x -b}+1\right ) \sqrt {a x +\sqrt {a x -b}}}{4}+\frac {\left (4 b -1\right ) \ln \left (\frac {1}{2}+\sqrt {a x -b}+\sqrt {a x +\sqrt {a x -b}}\right )}{8}\right )}{4}\right )}{a^{3}}\)	\(565\)
default	\(\frac {2 b^{2} \left (\frac {\left (2 \sqrt {a x -b}+1\right ) \sqrt {a x +\sqrt {a x -b}}}{4}+\frac {\left (4 b -1\right ) \ln \left (\frac {1}{2}+\sqrt {a x -b}+\sqrt {a x +\sqrt {a x -b}}\right )}{8}\right )+\frac {\left (a x -b \right )^{\frac {3}{2}} \left (a x +\sqrt {a x -b}\right )^{\frac {3}{2}}}{3}-\frac {3 \left (a x -b \right ) \left (a x +\sqrt {a x -b}\right )^{\frac {3}{2}}}{10}+\frac {21 \sqrt {a x -b}\, \left (a x +\sqrt {a x -b}\right )^{\frac {3}{2}}}{80}-\frac {7 \left (a x +\sqrt {a x -b}\right )^{\frac {3}{2}}}{32}+\frac {21 \left (2 \sqrt {a x -b}+1\right ) \sqrt {a x +\sqrt {a x -b}}}{256}+\frac {21 \left (4 b -1\right ) \ln \left (\frac {1}{2}+\sqrt {a x -b}+\sqrt {a x +\sqrt {a x -b}}\right )}{512}-\frac {21 b \left (\frac {\left (2 \sqrt {a x -b}+1\right ) \sqrt {a x +\sqrt {a x -b}}}{4}+\frac {\left (4 b -1\right ) \ln \left (\frac {1}{2}+\sqrt {a x -b}+\sqrt {a x +\sqrt {a x -b}}\right )}{8}\right )}{80}+\frac {3 b \left (\frac {\left (a x +\sqrt {a x -b}\right )^{\frac {3}{2}}}{3}-\frac {\left (2 \sqrt {a x -b}+1\right ) \sqrt {a x +\sqrt {a x -b}}}{8}-\frac {\left (4 b -1\right ) \ln \left (\frac {1}{2}+\sqrt {a x -b}+\sqrt {a x +\sqrt {a x -b}}\right )}{16}\right )}{5}+3 b \left (\frac {\sqrt {a x -b}\, \left (a x +\sqrt {a x -b}\right )^{\frac {3}{2}}}{4}-\frac {5 \left (a x +\sqrt {a x -b}\right )^{\frac {3}{2}}}{24}+\frac {5 \left (2 \sqrt {a x -b}+1\right ) \sqrt {a x +\sqrt {a x -b}}}{64}+\frac {5 \left (4 b -1\right ) \ln \left (\frac {1}{2}+\sqrt {a x -b}+\sqrt {a x +\sqrt {a x -b}}\right )}{128}-\frac {b \left (\frac {\left (2 \sqrt {a x -b}+1\right ) \sqrt {a x +\sqrt {a x -b}}}{4}+\frac {\left (4 b -1\right ) \ln \left (\frac {1}{2}+\sqrt {a x -b}+\sqrt {a x +\sqrt {a x -b}}\right )}{8}\right )}{4}\right )}{a^{3}}\)	\(565\)

[In]

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

[Out]

2/a^3*(b^2*(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)))+1/6*(a*x-b)^(3/2)*(a*x+(a*x-b)^(1/2))^(3/2)-3/20*(a*x-b)*(a*x+(a*x-b)^(1/2))^(3/2)+21/160*(a*x-b)
^(1/2)*(a*x+(a*x-b)^(1/2))^(3/2)-7/64*(a*x+(a*x-b)^(1/2))^(3/2)+21/512*(2*(a*x-b)^(1/2)+1)*(a*x+(a*x-b)^(1/2))
^(1/2)+21/1024*(4*b-1)*ln(1/2+(a*x-b)^(1/2)+(a*x+(a*x-b)^(1/2))^(1/2))-21/160*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)))+3/10*b*(1/3*(a*x+(a*x-b)^(1/
2))^(3/2)-1/8*(2*(a*x-b)^(1/2)+1)*(a*x+(a*x-b)^(1/2))^(1/2)-1/16*(4*b-1)*ln(1/2+(a*x-b)^(1/2)+(a*x+(a*x-b)^(1/
2))^(1/2)))+3/2*b*(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))-1/4*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)))))

Fricas [F(-1)]

Timed out. \[ \int \frac {x^2 \sqrt {a x+\sqrt {-b+a x}}}{\sqrt {-b+a x}} \, dx=\text {Timed out} \]

[In]

integrate(x^2*(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 = 0.58 (sec) , antiderivative size = 299, normalized size of antiderivative = 1.69 \[ \int \frac {x^2 \sqrt {a x+\sqrt {-b+a x}}}{\sqrt {-b+a x}} \, dx=\begin {cases} \frac {2 \left (\sqrt {a x + \sqrt {a x - b}} \left (- \frac {5 b^{2}}{4} - 2 b \left (\frac {b}{80} + \frac {7}{320}\right ) + \frac {9 b \left (\frac {13 b}{24} - \frac {3}{160}\right )}{4} + \frac {3 b}{128} + \left (\frac {b}{80} + \frac {7}{320}\right ) \left (a x - b\right ) + \left (\frac {13 b}{24} - \frac {3}{160}\right ) \left (a x - b\right )^{\frac {3}{2}} + \frac {\left (a x - b\right )^{\frac {5}{2}}}{6} + \sqrt {a x - b} \left (\frac {3 b^{2}}{2} - \frac {3 b \left (\frac {13 b}{24} - \frac {3}{160}\right )}{2} - \frac {b}{64} - \frac {7}{256}\right ) + \frac {\left (a x - b\right )^{2}}{60} + \frac {21}{512}\right ) + \left (b^{3} + \frac {5 b^{2}}{8} + b \left (\frac {b}{80} + \frac {7}{320}\right ) - \frac {9 b \left (\frac {13 b}{24} - \frac {3}{160}\right )}{8} - b \left (\frac {3 b^{2}}{2} - \frac {3 b \left (\frac {13 b}{24} - \frac {3}{160}\right )}{2} - \frac {b}{64} - \frac {7}{256}\right ) - \frac {3 b}{256} - \frac {21}{1024}\right ) \left (\begin {cases} \log {\left (2 \sqrt {a x - b} + 2 \sqrt {a x + \sqrt {a x - b}} + 1 \right )} & \text {for}\: b \neq \frac {1}{4} \\\frac {\left (\sqrt {a x - b} + \frac {1}{2}\right ) \log {\left (\sqrt {a x - b} + \frac {1}{2} \right )}}{\sqrt {\left (\sqrt {a x - b} + \frac {1}{2}\right )^{2}}} & \text {otherwise} \end {cases}\right )\right )}{a^{3}} & \text {for}\: a \neq 0 \\\frac {x^{3}}{3 \sqrt [4]{- b}} & \text {otherwise} \end {cases} \]

[In]

integrate(x**2*(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))*(-5*b**2/4 - 2*b*(b/80 + 7/320) + 9*b*(13*b/24 - 3/160)/4 + 3*b/128 +
(b/80 + 7/320)*(a*x - b) + (13*b/24 - 3/160)*(a*x - b)**(3/2) + (a*x - b)**(5/2)/6 + sqrt(a*x - b)*(3*b**2/2 -
3*b*(13*b/24 - 3/160)/2 - b/64 - 7/256) + (a*x - b)**2/60 + 21/512) + (b**3 + 5*b**2/8 + b*(b/80 + 7/320) - 9
*b*(13*b/24 - 3/160)/8 - b*(3*b**2/2 - 3*b*(13*b/24 - 3/160)/2 - b/64 - 7/256) - 3*b/256 - 21/1024)*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**3, Ne(a, 0)), (x**3/(3*(-b)**(1/4)), True))

Maxima [F]

\[ \int \frac {x^2 \sqrt {a x+\sqrt {-b+a x}}}{\sqrt {-b+a x}} \, dx=\int { \frac {\sqrt {a x + \sqrt {a x - b}} x^{2}}{\sqrt {a x - b}} \,d x } \]

[In]

integrate(x^2*(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))*x^2/sqrt(a*x - b), x)

Giac [A] (verification not implemented)

none

Time = 0.65 (sec) , antiderivative size = 224, normalized size of antiderivative = 1.27 \[ \int \frac {x^2 \sqrt {a x+\sqrt {-b+a x}}}{\sqrt {-b+a x}} \, dx=\frac {2 \, \sqrt {a x + \sqrt {a x - b}} {\left (2 \, \sqrt {a x - b} {\left (4 \, \sqrt {a x - b} {\left (2 \, \sqrt {a x - b} {\left (8 \, \sqrt {a x - b} {\left (\frac {10 \, \sqrt {a x - b}}{a^{2}} + \frac {1}{a^{2}}\right )} + \frac {260 \, a^{12} b - 9 \, a^{12}}{a^{14}}\right )} + \frac {3 \, {\left (4 \, a^{12} b + 7 \, a^{12}\right )}}{a^{14}}\right )} + \frac {3 \, {\left (880 \, a^{12} b^{2} + 16 \, a^{12} b - 35 \, a^{12}\right )}}{a^{14}}\right )} - \frac {3 \, {\left (144 \, a^{12} b^{2} + 160 \, a^{12} b - 105 \, a^{12}\right )}}{a^{14}}\right )} - \frac {15 \, {\left (320 \, b^{3} + 16 \, b^{2} + 60 \, b - 21\right )} \log \left ({\left | -2 \, \sqrt {a x - b} + 2 \, \sqrt {a x + \sqrt {a x - b}} - 1 \right |}\right )}{a^{2}}}{7680 \, a} \]

[In]

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

[Out]

1/7680*(2*sqrt(a*x + sqrt(a*x - b))*(2*sqrt(a*x - b)*(4*sqrt(a*x - b)*(2*sqrt(a*x - b)*(8*sqrt(a*x - b)*(10*sq
rt(a*x - b)/a^2 + 1/a^2) + (260*a^12*b - 9*a^12)/a^14) + 3*(4*a^12*b + 7*a^12)/a^14) + 3*(880*a^12*b^2 + 16*a^
12*b - 35*a^12)/a^14) - 3*(144*a^12*b^2 + 160*a^12*b - 105*a^12)/a^14) - 15*(320*b^3 + 16*b^2 + 60*b - 21)*log
(abs(-2*sqrt(a*x - b) + 2*sqrt(a*x + sqrt(a*x - b)) - 1))/a^2)/a

Mupad [F(-1)]

Timed out. \[ \int \frac {x^2 \sqrt {a x+\sqrt {-b+a x}}}{\sqrt {-b+a x}} \, dx=\int \frac {x^2\,\sqrt {a\,x+\sqrt {a\,x-b}}}{\sqrt {a\,x-b}} \,d x \]

[In]

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

[Out]

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