3.3053 \(\int \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}} x^2 \, dx\)

Optimal. Leaf size=333 \[ \frac{x \left (16 a^2 c^2-56 a b^2 c d+21 b^4 d^2\right ) \left (2 a+b \sqrt{\frac{d}{x}}\right ) \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}{256 a^5}+\frac{\left (4 a c-b^2 d\right ) \left (16 a^2 c^2-56 a b^2 c d+21 b^4 d^2\right ) \tanh ^{-1}\left (\frac{2 a+b \sqrt{\frac{d}{x}}}{2 \sqrt{a} \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}\right )}{512 a^{11/2}}+\frac{7 b d^2 \left (28 a c-15 b^2 d\right ) \left (a+b \sqrt{\frac{d}{x}}+\frac{c}{x}\right )^{3/2}}{480 a^4 \left (\frac{d}{x}\right )^{3/2}}-\frac{x^2 \left (20 a c-21 b^2 d\right ) \left (a+b \sqrt{\frac{d}{x}}+\frac{c}{x}\right )^{3/2}}{80 a^3}-\frac{3 b d^3 \left (a+b \sqrt{\frac{d}{x}}+\frac{c}{x}\right )^{3/2}}{10 a^2 \left (\frac{d}{x}\right )^{5/2}}+\frac{x^3 \left (a+b \sqrt{\frac{d}{x}}+\frac{c}{x}\right )^{3/2}}{3 a} \]

[Out]

(-3*b*d^3*(a + b*Sqrt[d/x] + c/x)^(3/2))/(10*a^2*(d/x)^(5/2)) + (7*b*d^2*(28*a*c - 15*b^2*d)*(a + b*Sqrt[d/x]
+ c/x)^(3/2))/(480*a^4*(d/x)^(3/2)) + ((16*a^2*c^2 - 56*a*b^2*c*d + 21*b^4*d^2)*(2*a + b*Sqrt[d/x])*Sqrt[a + b
*Sqrt[d/x] + c/x]*x)/(256*a^5) - ((20*a*c - 21*b^2*d)*(a + b*Sqrt[d/x] + c/x)^(3/2)*x^2)/(80*a^3) + ((a + b*Sq
rt[d/x] + c/x)^(3/2)*x^3)/(3*a) + ((4*a*c - b^2*d)*(16*a^2*c^2 - 56*a*b^2*c*d + 21*b^4*d^2)*ArcTanh[(2*a + b*S
qrt[d/x])/(2*Sqrt[a]*Sqrt[a + b*Sqrt[d/x] + c/x])])/(512*a^(11/2))

________________________________________________________________________________________

Rubi [A]  time = 0.59977, antiderivative size = 333, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308, Rules used = {1970, 1357, 744, 834, 806, 720, 724, 206} \[ \frac{x \left (16 a^2 c^2-56 a b^2 c d+21 b^4 d^2\right ) \left (2 a+b \sqrt{\frac{d}{x}}\right ) \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}{256 a^5}+\frac{\left (4 a c-b^2 d\right ) \left (16 a^2 c^2-56 a b^2 c d+21 b^4 d^2\right ) \tanh ^{-1}\left (\frac{2 a+b \sqrt{\frac{d}{x}}}{2 \sqrt{a} \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}\right )}{512 a^{11/2}}+\frac{7 b d^2 \left (28 a c-15 b^2 d\right ) \left (a+b \sqrt{\frac{d}{x}}+\frac{c}{x}\right )^{3/2}}{480 a^4 \left (\frac{d}{x}\right )^{3/2}}-\frac{x^2 \left (20 a c-21 b^2 d\right ) \left (a+b \sqrt{\frac{d}{x}}+\frac{c}{x}\right )^{3/2}}{80 a^3}-\frac{3 b d^3 \left (a+b \sqrt{\frac{d}{x}}+\frac{c}{x}\right )^{3/2}}{10 a^2 \left (\frac{d}{x}\right )^{5/2}}+\frac{x^3 \left (a+b \sqrt{\frac{d}{x}}+\frac{c}{x}\right )^{3/2}}{3 a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*b*d^3*(a + b*Sqrt[d/x] + c/x)^(3/2))/(10*a^2*(d/x)^(5/2)) + (7*b*d^2*(28*a*c - 15*b^2*d)*(a + b*Sqrt[d/x]
+ c/x)^(3/2))/(480*a^4*(d/x)^(3/2)) + ((16*a^2*c^2 - 56*a*b^2*c*d + 21*b^4*d^2)*(2*a + b*Sqrt[d/x])*Sqrt[a + b
*Sqrt[d/x] + c/x]*x)/(256*a^5) - ((20*a*c - 21*b^2*d)*(a + b*Sqrt[d/x] + c/x)^(3/2)*x^2)/(80*a^3) + ((a + b*Sq
rt[d/x] + c/x)^(3/2)*x^3)/(3*a) + ((4*a*c - b^2*d)*(16*a^2*c^2 - 56*a*b^2*c*d + 21*b^4*d^2)*ArcTanh[(2*a + b*S
qrt[d/x])/(2*Sqrt[a]*Sqrt[a + b*Sqrt[d/x] + c/x])])/(512*a^(11/2))

Rule 1970

Int[(x_)^(m_.)*((a_) + (b_.)*((d_.)/(x_))^(n_) + (c_.)*(x_)^(n2_.))^(p_.), x_Symbol] :> -Dist[d^(m + 1), Subst
[Int[(a + b*x^n + (c*x^(2*n))/d^(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]

Rule 1357

Int[(x_)^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplif
y[(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] && NeQ[
b^2 - 4*a*c, 0] && IntegerQ[Simplify[(m + 1)/n]]

Rule 744

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> 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] + Dist[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[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e
, 0] && NeQ[m, -1] && ((LtQ[m, -1] && IntQuadraticQ[a, b, c, d, e, m, p, x]) || (SumSimplerQ[m, 1] && IntegerQ
[p]) || ILtQ[Simplify[m + 2*p + 3], 0])

Rule 834

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[((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] + Dist[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] &&
NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[m, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ
[2*m, 2*p])

Rule 806

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> -Si
mp[((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] - Dist[(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] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && EqQ[Sim
plify[m + 2*p + 3], 0]

Rule 720

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -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] + Dist[(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] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && EqQ[m +
2*p + 2, 0] && GtQ[p, 0]

Rule 724

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[-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] && NeQ[b^2 - 4*a*c, 0] && NeQ[2*c*d - b*e, 0]

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])

Rubi steps

\begin{align*} \int \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}} x^2 \, dx &=-\left (d^3 \operatorname{Subst}\left (\int \frac{\sqrt{a+b \sqrt{x}+\frac{c x}{d}}}{x^4} \, dx,x,\frac{d}{x}\right )\right )\\ &=-\left (\left (2 d^3\right ) \operatorname{Subst}\left (\int \frac{\sqrt{a+b x+\frac{c x^2}{d}}}{x^7} \, dx,x,\sqrt{\frac{d}{x}}\right )\right )\\ &=\frac{\left (a+b \sqrt{\frac{d}{x}}+\frac{c}{x}\right )^{3/2} x^3}{3 a}+\frac{d^3 \operatorname{Subst}\left (\int \frac{\left (\frac{9 b}{2}+\frac{3 c x}{d}\right ) \sqrt{a+b x+\frac{c x^2}{d}}}{x^6} \, dx,x,\sqrt{\frac{d}{x}}\right )}{3 a}\\ &=-\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{\left (a+b \sqrt{\frac{d}{x}}+\frac{c}{x}\right )^{3/2} x^3}{3 a}-\frac{d^3 \operatorname{Subst}\left (\int \frac{\left (\frac{3}{4} \left (21 b^2-\frac{20 a c}{d}\right )+\frac{9 b c x}{d}\right ) \sqrt{a+b x+\frac{c x^2}{d}}}{x^5} \, dx,x,\sqrt{\frac{d}{x}}\right )}{15 a^2}\\ &=-\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{\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{d^3 \operatorname{Subst}\left (\int \frac{\left (-\frac{21 b \left (28 a c-15 b^2 d\right )}{8 d}-\frac{3 c \left (20 a c-21 b^2 d\right ) x}{4 d^2}\right ) \sqrt{a+b x+\frac{c x^2}{d}}}{x^4} \, dx,x,\sqrt{\frac{d}{x}}\right )}{60 a^3}\\ &=-\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 (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 (d \left (16 a^2 c^2-56 a b^2 c d+21 b^4 d^2\right )\right ) \operatorname{Subst}\left (\int \frac{\sqrt{a+b x+\frac{c x^2}{d}}}{x^3} \, dx,x,\sqrt{\frac{d}{x}}\right )}{64 a^4}\\ &=-\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 (\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 )\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x+\frac{c x^2}{d}}} \, dx,x,\sqrt{\frac{d}{x}}\right )}{512 a^5}\\ &=-\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 (\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 )\right ) \operatorname{Subst}\left (\int \frac{1}{4 a-x^2} \, dx,x,\frac{2 a+b \sqrt{\frac{d}{x}}}{\sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}\right )}{256 a^5}\\ &=-\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 ) \tanh ^{-1}\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}}\\ \end{align*}

Mathematica [F]  time = 0.153394, size = 0, normalized size = 0. \[ \int \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}} x^2 \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [B]  time = 0.178, size = 655, normalized size = 2. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/7680*((b*(d/x)^(1/2)*x+a*x+c)/x)^(1/2)*x^(1/2)*(-630*a^(3/2)*(b*(d/x)^(1/2)*x+a*x+c)^(1/2)*(d/x)^(5/2)*x^(5
/2)*b^5+1680*a^(5/2)*(b*(d/x)^(1/2)*x+a*x+c)^(3/2)*(d/x)^(3/2)*x^(3/2)*b^3+2304*a^(9/2)*(b*(d/x)^(1/2)*x+a*x+c
)^(3/2)*(d/x)^(1/2)*x^(3/2)*b-2560*x^(3/2)*(b*(d/x)^(1/2)*x+a*x+c)^(3/2)*a^(11/2)+1680*a^(5/2)*(b*(d/x)^(1/2)*
x+a*x+c)^(1/2)*(d/x)^(3/2)*x^(3/2)*b^3*c-3136*a^(7/2)*(b*(d/x)^(1/2)*x+a*x+c)^(3/2)*(d/x)^(1/2)*x^(1/2)*b*c+19
20*a^(9/2)*(b*(d/x)^(1/2)*x+a*x+c)^(3/2)*x^(1/2)*c-2016*a^(7/2)*(b*(d/x)^(1/2)*x+a*x+c)^(3/2)*d*x^(1/2)*b^2-48
0*a^(7/2)*(b*(d/x)^(1/2)*x+a*x+c)^(1/2)*(d/x)^(1/2)*x^(1/2)*b*c^2-960*a^(9/2)*(b*(d/x)^(1/2)*x+a*x+c)^(1/2)*x^
(1/2)*c^2+3360*a^(7/2)*(b*(d/x)^(1/2)*x+a*x+c)^(1/2)*d*x^(1/2)*b^2*c-1260*a^(5/2)*(b*(d/x)^(1/2)*x+a*x+c)^(1/2
)*d^2*x^(1/2)*b^4+315*ln(1/2*(b*(d/x)^(1/2)*x^(1/2)+2*(b*(d/x)^(1/2)*x+a*x+c)^(1/2)*a^(1/2)+2*a*x^(1/2))/a^(1/
2))*d^3*a*b^6-2100*ln(1/2*(b*(d/x)^(1/2)*x^(1/2)+2*(b*(d/x)^(1/2)*x+a*x+c)^(1/2)*a^(1/2)+2*a*x^(1/2))/a^(1/2))
*d^2*a^2*b^4*c+3600*ln(1/2*(b*(d/x)^(1/2)*x^(1/2)+2*(b*(d/x)^(1/2)*x+a*x+c)^(1/2)*a^(1/2)+2*a*x^(1/2))/a^(1/2)
)*d*a^3*b^2*c^2-960*ln(1/2*(b*(d/x)^(1/2)*x^(1/2)+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)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{b \sqrt{\frac{d}{x}} + a + \frac{c}{x}} x^{2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{2} \sqrt{a + b \sqrt{\frac{d}{x}} + \frac{c}{x}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.39277, size = 558, normalized size = 1.68 \begin{align*} \frac{1}{7680} \,{\left (2 \, \sqrt{b \sqrt{d} \sqrt{x} + a x + c}{\left (2 \,{\left (4 \,{\left (2 \,{\left (8 \, \sqrt{x}{\left (\frac{b \sqrt{d}}{a} + 10 \, \sqrt{x}\right )} - \frac{9 \, a^{3} b^{2} d - 20 \, a^{4} c}{a^{5}}\right )} \sqrt{x} + \frac{21 \, a^{2} b^{3} d^{\frac{3}{2}} - 68 \, a^{3} b c \sqrt{d}}{a^{5}}\right )} \sqrt{x} - \frac{105 \, a b^{4} d^{2} - 448 \, a^{2} b^{2} c d + 240 \, a^{3} c^{2}}{a^{5}}\right )} \sqrt{x} + \frac{315 \, b^{5} d^{\frac{5}{2}} - 1680 \, a b^{3} c d^{\frac{3}{2}} + 1808 \, a^{2} b c^{2} \sqrt{d}}{a^{5}}\right )} + \frac{15 \,{\left (21 \, b^{6} d^{3} - 140 \, a b^{4} c d^{2} + 240 \, a^{2} b^{2} c^{2} d - 64 \, a^{3} c^{3}\right )} \log \left ({\left | -b \sqrt{d} - 2 \, \sqrt{a}{\left (\sqrt{a} \sqrt{x} - \sqrt{b \sqrt{d} \sqrt{x} + a x + c}\right )} \right |}\right )}{a^{\frac{11}{2}}}\right )} \mathrm{sgn}\left (x\right ) - \frac{{\left (315 \, b^{6} d^{3} \log \left ({\left | -b \sqrt{d} + 2 \, \sqrt{a} \sqrt{c} \right |}\right ) - 2100 \, a b^{4} c d^{2} \log \left ({\left | -b \sqrt{d} + 2 \, \sqrt{a} \sqrt{c} \right |}\right ) + 630 \, \sqrt{a} b^{5} \sqrt{c} d^{\frac{5}{2}} + 3600 \, a^{2} b^{2} c^{2} d \log \left ({\left | -b \sqrt{d} + 2 \, \sqrt{a} \sqrt{c} \right |}\right ) - 3360 \, a^{\frac{3}{2}} b^{3} c^{\frac{3}{2}} d^{\frac{3}{2}} - 960 \, a^{3} c^{3} \log \left ({\left | -b \sqrt{d} + 2 \, \sqrt{a} \sqrt{c} \right |}\right ) + 3616 \, a^{\frac{5}{2}} b c^{\frac{5}{2}} \sqrt{d}\right )} \mathrm{sgn}\left (x\right )}{7680 \, a^{\frac{11}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7680*(2*sqrt(b*sqrt(d)*sqrt(x) + a*x + c)*(2*(4*(2*(8*sqrt(x)*(b*sqrt(d)/a + 10*sqrt(x)) - (9*a^3*b^2*d - 20
*a^4*c)/a^5)*sqrt(x) + (21*a^2*b^3*d^(3/2) - 68*a^3*b*c*sqrt(d))/a^5)*sqrt(x) - (105*a*b^4*d^2 - 448*a^2*b^2*c
*d + 240*a^3*c^2)/a^5)*sqrt(x) + (315*b^5*d^(5/2) - 1680*a*b^3*c*d^(3/2) + 1808*a^2*b*c^2*sqrt(d))/a^5) + 15*(
21*b^6*d^3 - 140*a*b^4*c*d^2 + 240*a^2*b^2*c^2*d - 64*a^3*c^3)*log(abs(-b*sqrt(d) - 2*sqrt(a)*(sqrt(a)*sqrt(x)
- sqrt(b*sqrt(d)*sqrt(x) + a*x + c))))/a^(11/2))*sgn(x) - 1/7680*(315*b^6*d^3*log(abs(-b*sqrt(d) + 2*sqrt(a)*
sqrt(c))) - 2100*a*b^4*c*d^2*log(abs(-b*sqrt(d) + 2*sqrt(a)*sqrt(c))) + 630*sqrt(a)*b^5*sqrt(c)*d^(5/2) + 3600
*a^2*b^2*c^2*d*log(abs(-b*sqrt(d) + 2*sqrt(a)*sqrt(c))) - 3360*a^(3/2)*b^3*c^(3/2)*d^(3/2) - 960*a^3*c^3*log(a
bs(-b*sqrt(d) + 2*sqrt(a)*sqrt(c))) + 3616*a^(5/2)*b*c^(5/2)*sqrt(d))*sgn(x)/a^(11/2)