3.31.53 \(\int \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} x^2 \, dx\) [3053]

3.31.53.1 Optimal result

Integrand size = 26, antiderivative size = 333 \[ \int \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} x^2 \, dx=-\frac {3 b d^3 \left (a+b \sqrt {\frac {d}{x}}+\frac {c}{x}\right )^{3/2}}{10 a^2 \left (\frac {d}{x}\right )^{5/2}}+\frac {7 b d^2 \left (28 a c-15 b^2 d\right ) \left (a+b \sqrt {\frac {d}{x}}+\frac {c}{x}\right )^{3/2}}{480 a^4 \left (\frac {d}{x}\right )^{3/2}}+\frac {\left (16 a^2 c^2-56 a b^2 c d+21 b^4 d^2\right ) \left (2 a+b \sqrt {\frac {d}{x}}\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} x}{256 a^5}-\frac {\left (20 a c-21 b^2 d\right ) \left (a+b \sqrt {\frac {d}{x}}+\frac {c}{x}\right )^{3/2} x^2}{80 a^3}+\frac {\left (a+b \sqrt {\frac {d}{x}}+\frac {c}{x}\right )^{3/2} x^3}{3 a}+\frac {\left (4 a c-b^2 d\right ) \left (16 a^2 c^2-56 a b^2 c d+21 b^4 d^2\right ) \text {arctanh}\left (\frac {2 a+b \sqrt {\frac {d}{x}}}{2 \sqrt {a} \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}\right )}{512 a^{11/2}} \]

output

1/512*(-b^2*d+4*a*c)*(21*b^4*d^2-56*a*b^2*c*d+16*a^2*c^2)*arctanh(1/2*(2*a 
+b*(d/x)^(1/2))/a^(1/2)/(a+c/x+b*(d/x)^(1/2))^(1/2))/a^(11/2)-3/10*b*d^3*( 
a+c/x+b*(d/x)^(1/2))^(3/2)/a^2/(d/x)^(5/2)+7/480*b*d^2*(-15*b^2*d+28*a*c)* 
(a+c/x+b*(d/x)^(1/2))^(3/2)/a^4/(d/x)^(3/2)-1/80*(-21*b^2*d+20*a*c)*x^2*(a 
+c/x+b*(d/x)^(1/2))^(3/2)/a^3+1/3*x^3*(a+c/x+b*(d/x)^(1/2))^(3/2)/a+1/256* 
(21*b^4*d^2-56*a*b^2*c*d+16*a^2*c^2)*x*(2*a+b*(d/x)^(1/2))*(a+c/x+b*(d/x)^ 
(1/2))^(1/2)/a^5
 

3.31.53.2 Mathematica [A] (verified)

Time = 2.68 (sec) , antiderivative size = 312, normalized size of antiderivative = 0.94 \[ \int \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} x^2 \, dx=\frac {\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} \left (\sqrt {a} x \left (-210 a b^3 d \left (b d+8 c \sqrt {\frac {d}{x}}\right )+315 b^5 d \left (\frac {d}{x}\right )^{3/2} x+1280 a^5 x^2+64 a^4 x \left (5 c+2 b \sqrt {\frac {d}{x}} x\right )-16 a^3 \left (30 c^2+9 b^2 d x+34 b c \sqrt {\frac {d}{x}} x\right )+8 a^2 b \left (112 b c d+226 c^2 \sqrt {\frac {d}{x}}+21 b^2 d \sqrt {\frac {d}{x}} x\right )\right )+\frac {15 \sqrt {d} \left (-64 a^3 c^3+240 a^2 b^2 c^2 d-140 a b^4 c d^2+21 b^6 d^3\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {\frac {d}{x}}-\sqrt {\frac {d \left (c+a x+b \sqrt {\frac {d}{x}} x\right )}{x}}}{\sqrt {a} \sqrt {d}}\right )}{\sqrt {\frac {d \left (c+\left (a+b \sqrt {\frac {d}{x}}\right ) x\right )}{x}}}\right )}{3840 a^{11/2}} \]

input

Integrate[Sqrt[a + b*Sqrt[d/x] + c/x]*x^2,x]
 

output

(Sqrt[a + b*Sqrt[d/x] + c/x]*(Sqrt[a]*x*(-210*a*b^3*d*(b*d + 8*c*Sqrt[d/x] 
) + 315*b^5*d*(d/x)^(3/2)*x + 1280*a^5*x^2 + 64*a^4*x*(5*c + 2*b*Sqrt[d/x] 
*x) - 16*a^3*(30*c^2 + 9*b^2*d*x + 34*b*c*Sqrt[d/x]*x) + 8*a^2*b*(112*b*c* 
d + 226*c^2*Sqrt[d/x] + 21*b^2*d*Sqrt[d/x]*x)) + (15*Sqrt[d]*(-64*a^3*c^3 
+ 240*a^2*b^2*c^2*d - 140*a*b^4*c*d^2 + 21*b^6*d^3)*ArcTanh[(Sqrt[c]*Sqrt[ 
d/x] - Sqrt[(d*(c + a*x + b*Sqrt[d/x]*x))/x])/(Sqrt[a]*Sqrt[d])])/Sqrt[(d* 
(c + (a + b*Sqrt[d/x])*x))/x]))/(3840*a^(11/2))
 

3.31.53.3 Rubi [A] (warning: unable to verify)

Time = 0.63 (sec) , antiderivative size = 356, normalized size of antiderivative = 1.07, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {2066, 1693, 1167, 27, 1237, 27, 1237, 27, 1228, 1152, 1154, 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.

\(\displaystyle \int x^2 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} \, dx\)

\(\Big \downarrow \) 2066

\(\displaystyle -d^3 \int \frac {\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} x^4}{d^4}d\frac {d}{x}\)

\(\Big \downarrow \) 1693

\(\displaystyle -2 d^3 \int \frac {\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}} x^7}{d^7}d\sqrt {\frac {d}{x}}\)

\(\Big \downarrow \) 1167

\(\displaystyle -2 d^3 \left (-\frac {\int \frac {3 \left (2 \sqrt {\frac {d}{x}} c+3 b d\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}} x^6}{2 d^7}d\sqrt {\frac {d}{x}}}{6 a}-\frac {x^6 \left (a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}\right )^{3/2}}{6 a d^6}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle -2 d^3 \left (-\frac {\int \frac {\left (2 \sqrt {\frac {d}{x}} c+3 b d\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}} x^6}{d^6}d\sqrt {\frac {d}{x}}}{4 a d}-\frac {x^6 \left (a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}\right )^{3/2}}{6 a d^6}\right )\)

\(\Big \downarrow \) 1237

\(\displaystyle -2 d^3 \left (-\frac {-\frac {\int -\frac {\left (-21 d b^2-12 c \sqrt {\frac {d}{x}} b+20 a c\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}} x^5}{2 d^5}d\sqrt {\frac {d}{x}}}{5 a}-\frac {3 b x^5 \left (a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}\right )^{3/2}}{5 a d^4}}{4 a d}-\frac {x^6 \left (a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}\right )^{3/2}}{6 a d^6}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle -2 d^3 \left (-\frac {\frac {\int \frac {\left (-21 d b^2-12 c \sqrt {\frac {d}{x}} b+20 a c\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}} x^5}{d^5}d\sqrt {\frac {d}{x}}}{10 a}-\frac {3 b x^5 \left (a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}\right )^{3/2}}{5 a d^4}}{4 a d}-\frac {x^6 \left (a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}\right )^{3/2}}{6 a d^6}\right )\)

\(\Big \downarrow \) 1237

\(\displaystyle -2 d^3 \left (-\frac {\frac {-\frac {\int \frac {\left (2 c \sqrt {\frac {d}{x}} \left (20 a c-21 b^2 d\right )+7 b d \left (28 a c-15 b^2 d\right )\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}} x^4}{2 d^5}d\sqrt {\frac {d}{x}}}{4 a}-\frac {x^4 \left (20 a c-21 b^2 d\right ) \left (a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}\right )^{3/2}}{4 a d^4}}{10 a}-\frac {3 b x^5 \left (a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}\right )^{3/2}}{5 a d^4}}{4 a d}-\frac {x^6 \left (a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}\right )^{3/2}}{6 a d^6}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle -2 d^3 \left (-\frac {\frac {-\frac {\int \frac {\left (2 c \sqrt {\frac {d}{x}} \left (20 a c-21 b^2 d\right )+7 b d \left (28 a c-15 b^2 d\right )\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}} x^4}{d^4}d\sqrt {\frac {d}{x}}}{8 a d}-\frac {x^4 \left (20 a c-21 b^2 d\right ) \left (a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}\right )^{3/2}}{4 a d^4}}{10 a}-\frac {3 b x^5 \left (a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}\right )^{3/2}}{5 a d^4}}{4 a d}-\frac {x^6 \left (a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}\right )^{3/2}}{6 a d^6}\right )\)

\(\Big \downarrow \) 1228

\(\displaystyle -2 d^3 \left (-\frac {\frac {-\frac {\frac {5 \left (16 a^2 c^2-56 a b^2 c d+21 b^4 d^2\right ) \int \frac {\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}} x^3}{d^3}d\sqrt {\frac {d}{x}}}{2 a}-\frac {7 b x^3 \left (28 a c-15 b^2 d\right ) \left (a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}\right )^{3/2}}{3 a d^2}}{8 a d}-\frac {x^4 \left (20 a c-21 b^2 d\right ) \left (a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}\right )^{3/2}}{4 a d^4}}{10 a}-\frac {3 b x^5 \left (a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}\right )^{3/2}}{5 a d^4}}{4 a d}-\frac {x^6 \left (a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}\right )^{3/2}}{6 a d^6}\right )\)

\(\Big \downarrow \) 1152

\(\displaystyle -2 d^3 \left (-\frac {\frac {-\frac {\frac {5 \left (16 a^2 c^2-56 a b^2 c d+21 b^4 d^2\right ) \left (-\frac {\left (b^2-\frac {4 a c}{d}\right ) \int \frac {x}{d \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}d\sqrt {\frac {d}{x}}}{8 a}-\frac {x^2 \left (2 a+b \sqrt {\frac {d}{x}}\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}{4 a d^2}\right )}{2 a}-\frac {7 b x^3 \left (28 a c-15 b^2 d\right ) \left (a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}\right )^{3/2}}{3 a d^2}}{8 a d}-\frac {x^4 \left (20 a c-21 b^2 d\right ) \left (a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}\right )^{3/2}}{4 a d^4}}{10 a}-\frac {3 b x^5 \left (a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}\right )^{3/2}}{5 a d^4}}{4 a d}-\frac {x^6 \left (a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}\right )^{3/2}}{6 a d^6}\right )\)

\(\Big \downarrow \) 1154

\(\displaystyle -2 d^3 \left (-\frac {\frac {-\frac {\frac {5 \left (16 a^2 c^2-56 a b^2 c d+21 b^4 d^2\right ) \left (\frac {\left (b^2-\frac {4 a c}{d}\right ) \int \frac {1}{4 a-\frac {d^2}{x^2}}d\frac {2 a+b \sqrt {\frac {d}{x}}}{\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}}{4 a}-\frac {x^2 \left (2 a+b \sqrt {\frac {d}{x}}\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}{4 a d^2}\right )}{2 a}-\frac {7 b x^3 \left (28 a c-15 b^2 d\right ) \left (a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}\right )^{3/2}}{3 a d^2}}{8 a d}-\frac {x^4 \left (20 a c-21 b^2 d\right ) \left (a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}\right )^{3/2}}{4 a d^4}}{10 a}-\frac {3 b x^5 \left (a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}\right )^{3/2}}{5 a d^4}}{4 a d}-\frac {x^6 \left (a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}\right )^{3/2}}{6 a d^6}\right )\)

\(\Big \downarrow \) 219

\(\displaystyle -2 d^3 \left (-\frac {\frac {-\frac {\frac {5 \left (16 a^2 c^2-56 a b^2 c d+21 b^4 d^2\right ) \left (\frac {\left (b^2-\frac {4 a c}{d}\right ) \text {arctanh}\left (\frac {2 a+b \sqrt {\frac {d}{x}}}{2 \sqrt {a} \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}\right )}{8 a^{3/2}}-\frac {x^2 \left (2 a+b \sqrt {\frac {d}{x}}\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}}}{4 a d^2}\right )}{2 a}-\frac {7 b x^3 \left (28 a c-15 b^2 d\right ) \left (a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}\right )^{3/2}}{3 a d^2}}{8 a d}-\frac {x^4 \left (20 a c-21 b^2 d\right ) \left (a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}\right )^{3/2}}{4 a d^4}}{10 a}-\frac {3 b x^5 \left (a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}\right )^{3/2}}{5 a d^4}}{4 a d}-\frac {x^6 \left (a+b \sqrt {\frac {d}{x}}+\frac {c d}{x^2}\right )^{3/2}}{6 a d^6}\right )\)

input

Int[Sqrt[a + b*Sqrt[d/x] + c/x]*x^2,x]
 

output

-2*d^3*(-1/6*((a + b*Sqrt[d/x] + (c*d)/x^2)^(3/2)*x^6)/(a*d^6) - ((-3*b*(a 
 + b*Sqrt[d/x] + (c*d)/x^2)^(3/2)*x^5)/(5*a*d^4) + (-1/4*((20*a*c - 21*b^2 
*d)*(a + b*Sqrt[d/x] + (c*d)/x^2)^(3/2)*x^4)/(a*d^4) - ((-7*b*(28*a*c - 15 
*b^2*d)*(a + b*Sqrt[d/x] + (c*d)/x^2)^(3/2)*x^3)/(3*a*d^2) + (5*(16*a^2*c^ 
2 - 56*a*b^2*c*d + 21*b^4*d^2)*(-1/4*((2*a + b*Sqrt[d/x])*Sqrt[a + b*Sqrt[ 
d/x] + (c*d)/x^2]*x^2)/(a*d^2) + ((b^2 - (4*a*c)/d)*ArcTanh[(2*a + b*Sqrt[ 
d/x])/(2*Sqrt[a]*Sqrt[a + b*Sqrt[d/x] + (c*d)/x^2])])/(8*a^(3/2))))/(2*a)) 
/(8*a*d))/(10*a))/(4*a*d))
 

3.31.53.3.1 Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 219

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])
 

rule 1152

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S 
ymbol] :> Simp[(-(d + e*x)^(m + 1))*(d*b - 2*a*e + (2*c*d - b*e)*x)*((a + b 
*x + c*x^2)^p/(2*(m + 1)*(c*d^2 - b*d*e + a*e^2))), x] + Simp[p*((b^2 - 4*a 
*c)/(2*(m + 1)*(c*d^2 - b*d*e + a*e^2)))   Int[(d + e*x)^(m + 2)*(a + b*x + 
 c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[m + 2*p + 2, 0] 
 && GtQ[p, 0]
 

rule 1154

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Sym 
bol] :> Simp[-2   Subst[Int[1/(4*c*d^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, ( 
2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c 
, d, e}, x]
 

rule 1167

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S 
ymbol] :> Simp[e*(d + e*x)^(m + 1)*((a + b*x + c*x^2)^(p + 1)/((m + 1)*(c*d 
^2 - b*d*e + a*e^2))), x] + Simp[1/((m + 1)*(c*d^2 - b*d*e + a*e^2))   Int[ 
(d + e*x)^(m + 1)*Simp[c*d*(m + 1) - b*e*(m + p + 2) - c*e*(m + 2*p + 3)*x, 
 x]*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[m 
, -1] && ((LtQ[m, -1] && IntQuadraticQ[a, b, c, d, e, m, p, x]) || (SumSimp 
lerQ[m, 1] && IntegerQ[p]) || ILtQ[Simplify[m + 2*p + 3], 0])
 

rule 1228

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(-(e*f - d*g))*(d + e*x)^(m + 1)*((a + 
 b*x + c*x^2)^(p + 1)/(2*(p + 1)*(c*d^2 - b*d*e + a*e^2))), x] - Simp[(b*(e 
*f + d*g) - 2*(c*d*f + a*e*g))/(2*(c*d^2 - b*d*e + a*e^2))   Int[(d + e*x)^ 
(m + 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x 
] && EqQ[Simplify[m + 2*p + 3], 0]
 

rule 1237

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(e*f - d*g)*(d + e*x)^(m + 1)*((a + b* 
x + c*x^2)^(p + 1)/((m + 1)*(c*d^2 - b*d*e + a*e^2))), x] + Simp[1/((m + 1) 
*(c*d^2 - b*d*e + a*e^2))   Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p*Simp[ 
(c*d*f - f*b*e + a*e*g)*(m + 1) + b*(d*g - e*f)*(p + 1) - c*(e*f - d*g)*(m 
+ 2*p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && LtQ[m, -1 
] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
 

rule 1693

Int[(x_)^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol 
] :> Simp[1/n   Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*x + c*x^2)^p, 
x], x, x^n], x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[n2, 2*n] && IntegerQ 
[Simplify[(m + 1)/n]]
 

rule 2066

Int[(x_)^(m_.)*((a_) + (b_.)*((d_.)/(x_))^(n_) + (c_.)*(x_)^(n2_.))^(p_), x 
_Symbol] :> Simp[-d^(m + 1)   Subst[Int[(a + b*x^n + (c/d^(2*n))*x^(2*n))^p 
/x^(m + 2), x], x, d/x], x] /; FreeQ[{a, b, c, d, n, p}, x] && EqQ[n2, -2*n 
] && IntegerQ[2*n] && IntegerQ[m]
 

3.31.53.4 Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(654\) vs. \(2(283)=566\).

Time = 0.30 (sec) , antiderivative size = 655, normalized size of antiderivative = 1.97

method	result	size
default	\(\frac {\sqrt {\frac {b \sqrt {\frac {d}{x}}\, x +a x +c}{x}}\, \sqrt {x}\, \left (630 \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}\, a^{\frac {3}{2}} \left (\frac {d}{x}\right )^{\frac {5}{2}} x^{\frac {5}{2}} b^{5}+2560 x^{\frac {3}{2}} \left (b \sqrt {\frac {d}{x}}\, x +a x +c \right )^{\frac {3}{2}} a^{\frac {11}{2}}-2304 \left (b \sqrt {\frac {d}{x}}\, x +a x +c \right )^{\frac {3}{2}} a^{\frac {9}{2}} \sqrt {\frac {d}{x}}\, x^{\frac {3}{2}} b -1680 \left (b \sqrt {\frac {d}{x}}\, x +a x +c \right )^{\frac {3}{2}} a^{\frac {5}{2}} \left (\frac {d}{x}\right )^{\frac {3}{2}} x^{\frac {3}{2}} b^{3}+1260 \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}\, a^{\frac {5}{2}} d^{2} \sqrt {x}\, b^{4}-315 d^{3} \ln \left (\frac {\sqrt {\frac {d}{x}}\, \sqrt {x}\, b +2 \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}\, \sqrt {a}+2 a \sqrt {x}}{2 \sqrt {a}}\right ) a \,b^{6}+2016 \left (b \sqrt {\frac {d}{x}}\, x +a x +c \right )^{\frac {3}{2}} a^{\frac {7}{2}} d \sqrt {x}\, b^{2}-1680 \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}\, a^{\frac {5}{2}} \left (\frac {d}{x}\right )^{\frac {3}{2}} x^{\frac {3}{2}} b^{3} c -3360 \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}\, a^{\frac {7}{2}} d \sqrt {x}\, b^{2} c -1920 \left (b \sqrt {\frac {d}{x}}\, x +a x +c \right )^{\frac {3}{2}} a^{\frac {9}{2}} c \sqrt {x}+3136 \left (b \sqrt {\frac {d}{x}}\, x +a x +c \right )^{\frac {3}{2}} a^{\frac {7}{2}} \sqrt {\frac {d}{x}}\, \sqrt {x}\, b c +2100 d^{2} \ln \left (\frac {\sqrt {\frac {d}{x}}\, \sqrt {x}\, b +2 \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}\, \sqrt {a}+2 a \sqrt {x}}{2 \sqrt {a}}\right ) a^{2} b^{4} c +960 \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}\, a^{\frac {9}{2}} c^{2} \sqrt {x}+480 \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}\, a^{\frac {7}{2}} \sqrt {\frac {d}{x}}\, \sqrt {x}\, b \,c^{2}-3600 d \ln \left (\frac {\sqrt {\frac {d}{x}}\, \sqrt {x}\, b +2 \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}\, \sqrt {a}+2 a \sqrt {x}}{2 \sqrt {a}}\right ) a^{3} b^{2} c^{2}+960 \ln \left (\frac {\sqrt {\frac {d}{x}}\, \sqrt {x}\, b +2 \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}\, \sqrt {a}+2 a \sqrt {x}}{2 \sqrt {a}}\right ) a^{4} c^{3}\right )}{7680 \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}\, a^{\frac {13}{2}}}\)	\(655\)

input

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

output

1/7680*((b*(d/x)^(1/2)*x+a*x+c)/x)^(1/2)*x^(1/2)*(630*(b*(d/x)^(1/2)*x+a*x 
+c)^(1/2)*a^(3/2)*(d/x)^(5/2)*x^(5/2)*b^5+2560*x^(3/2)*(b*(d/x)^(1/2)*x+a* 
x+c)^(3/2)*a^(11/2)-2304*(b*(d/x)^(1/2)*x+a*x+c)^(3/2)*a^(9/2)*(d/x)^(1/2) 
*x^(3/2)*b-1680*(b*(d/x)^(1/2)*x+a*x+c)^(3/2)*a^(5/2)*(d/x)^(3/2)*x^(3/2)* 
b^3+1260*(b*(d/x)^(1/2)*x+a*x+c)^(1/2)*a^(5/2)*d^2*x^(1/2)*b^4-315*d^3*ln( 
1/2*((d/x)^(1/2)*x^(1/2)*b+2*(b*(d/x)^(1/2)*x+a*x+c)^(1/2)*a^(1/2)+2*a*x^( 
1/2))/a^(1/2))*a*b^6+2016*(b*(d/x)^(1/2)*x+a*x+c)^(3/2)*a^(7/2)*d*x^(1/2)* 
b^2-1680*(b*(d/x)^(1/2)*x+a*x+c)^(1/2)*a^(5/2)*(d/x)^(3/2)*x^(3/2)*b^3*c-3 
360*(b*(d/x)^(1/2)*x+a*x+c)^(1/2)*a^(7/2)*d*x^(1/2)*b^2*c-1920*(b*(d/x)^(1 
/2)*x+a*x+c)^(3/2)*a^(9/2)*c*x^(1/2)+3136*(b*(d/x)^(1/2)*x+a*x+c)^(3/2)*a^ 
(7/2)*(d/x)^(1/2)*x^(1/2)*b*c+2100*d^2*ln(1/2*((d/x)^(1/2)*x^(1/2)*b+2*(b* 
(d/x)^(1/2)*x+a*x+c)^(1/2)*a^(1/2)+2*a*x^(1/2))/a^(1/2))*a^2*b^4*c+960*(b* 
(d/x)^(1/2)*x+a*x+c)^(1/2)*a^(9/2)*c^2*x^(1/2)+480*(b*(d/x)^(1/2)*x+a*x+c) 
^(1/2)*a^(7/2)*(d/x)^(1/2)*x^(1/2)*b*c^2-3600*d*ln(1/2*((d/x)^(1/2)*x^(1/2 
)*b+2*(b*(d/x)^(1/2)*x+a*x+c)^(1/2)*a^(1/2)+2*a*x^(1/2))/a^(1/2))*a^3*b^2* 
c^2+960*ln(1/2*((d/x)^(1/2)*x^(1/2)*b+2*(b*(d/x)^(1/2)*x+a*x+c)^(1/2)*a^(1 
/2)+2*a*x^(1/2))/a^(1/2))*a^4*c^3)/(b*(d/x)^(1/2)*x+a*x+c)^(1/2)/a^(13/2)
 

3.31.53.5 Fricas [F(-1)]

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

input

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

output

Timed out
 

3.31.53.6 Sympy [F]

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

input

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

output

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

3.31.53.7 Maxima [F]

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

input

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

output

integrate(sqrt(b*sqrt(d/x) + a + c/x)*x^2, x)
 

3.31.53.8 Giac [F]

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

input

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

output

sage0*x
 

3.31.53.9 Mupad [F(-1)]

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

input

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

output

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