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

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

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Rubi [A]  time = 0.90, antiderivative size = 255, normalized size of antiderivative = 1.44, number of steps used = 8, number of rules used = 5, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.147, Rules used = {1661, 640, 612, 621, 206} \begin {gather*} \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 {(1-4 b) \left (80 b^2+24 b+21\right ) \tanh ^{-1}\left (\frac {2 \sqrt {a x-b}+1}{2 \sqrt {\sqrt {a x-b}+a x}}\right )}{512 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} \end {gather*}

Antiderivative was successfully verified.

[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 206

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

Rule 612

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 621

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 640

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 1661

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*} \int \frac {x^2 \sqrt {a x+\sqrt {-b+a x}}}{\sqrt {-b+a x}} \, dx &=\frac {2 \operatorname {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 {\operatorname {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 {\operatorname {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 {\operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \tanh ^{-1}\left (\frac {1+2 \sqrt {-b+a x}}{2 \sqrt {a x+\sqrt {-b+a x}}}\right )}{512 a^3}\\ \end {align*}

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Mathematica [A]  time = 0.37, size = 199, normalized size = 1.12 \begin {gather*} \frac {2 \sqrt {\sqrt {a x-b}+a x} \left (128 a^2 x^2 \left (10 \sqrt {a x-b}+1\right )+400 b^2 \left (6 \sqrt {a x-b}-1\right )+8 b \left (20 a x \left (10 \sqrt {a x-b}-1\right )+30 \sqrt {a x-b}-81\right )-24 a x \left (6 \sqrt {a x-b}-7\right )-210 \sqrt {a x-b}+315\right )+15 \left (320 b^3+16 b^2+60 b-21\right ) \tanh ^{-1}\left (\frac {2 \sqrt {a x-b}+1}{2 \sqrt {\sqrt {a x-b}+a x}}\right )}{7680 a^3} \end {gather*}

Antiderivative was successfully verified.

[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)*ArcTanh[(1 + 2*Sqrt[-b + a*x])/(2*Sqrt[a*x + Sqrt[-b + a*x]])])/(
7680*a^3)

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IntegrateAlgebraic [A]  time = 0.53, size = 209, normalized size = 1.18 \begin {gather*} \frac {\sqrt {a x+\sqrt {-b+a x}} \left (315-480 b-432 b^2-210 \sqrt {-b+a x}+96 b \sqrt {-b+a x}+5280 b^2 \sqrt {-b+a x}+168 (-b+a x)+96 b (-b+a x)-144 (-b+a x)^{3/2}+4160 b (-b+a x)^{3/2}+128 (-b+a x)^2+1280 (-b+a x)^{5/2}\right )}{3840 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} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[a*x + Sqrt[-b + a*x]]*(315 - 480*b - 432*b^2 - 210*Sqrt[-b + a*x] + 96*b*Sqrt[-b + a*x] + 5280*b^2*Sqrt[
-b + a*x] + 168*(-b + a*x) + 96*b*(-b + a*x) - 144*(-b + a*x)^(3/2) + 4160*b*(-b + a*x)^(3/2) + 128*(-b + a*x)
^2 + 1280*(-b + a*x)^(5/2)))/(3840*a^3) + ((21 - 60*b - 16*b^2 - 320*b^3)*Log[-1 - 2*Sqrt[-b + a*x] + 2*Sqrt[a
*x + Sqrt[-b + a*x]]])/(512*a^3)

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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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giac [A]  time = 2.09, size = 224, normalized size = 1.27 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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maple [B]  time = 0.12, size = 565, normalized size = 3.19

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

Verification of antiderivative is not currently implemented for this CAS.

[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*(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))))+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))))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {a x + \sqrt {a x - b}} x^{2}}{\sqrt {a x - b}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^2\,\sqrt {a\,x+\sqrt {a\,x-b}}}{\sqrt {a\,x-b}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2} \sqrt {a x + \sqrt {a x - b}}}{\sqrt {a x - b}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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