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

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

[Out]

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

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Rubi [A]  time = 0.60, antiderivative size = 333, normalized size of antiderivative = 1.00, 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 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 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 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 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 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 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 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]

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*}

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Mathematica [F]  time = 0.21, size = 0, normalized size = 0.00 \[ \int \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} x^2 \, dx \]

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

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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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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]

Timed out

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maple [B]  time = 0.19, size = 655, normalized size = 1.97 \[ \frac {\sqrt {\frac {a x +\sqrt {\frac {d}{x}}\, b x +c}{x}}\, \left (-315 a \,b^{6} d^{3} \ln \left (\frac {2 a \sqrt {x}+\sqrt {\frac {d}{x}}\, b \sqrt {x}+2 \sqrt {a x +\sqrt {\frac {d}{x}}\, b x +c}\, \sqrt {a}}{2 \sqrt {a}}\right )+2100 a^{2} b^{4} c \,d^{2} \ln \left (\frac {2 a \sqrt {x}+\sqrt {\frac {d}{x}}\, b \sqrt {x}+2 \sqrt {a x +\sqrt {\frac {d}{x}}\, b x +c}\, \sqrt {a}}{2 \sqrt {a}}\right )-3600 a^{3} b^{2} c^{2} d \ln \left (\frac {2 a \sqrt {x}+\sqrt {\frac {d}{x}}\, b \sqrt {x}+2 \sqrt {a x +\sqrt {\frac {d}{x}}\, b x +c}\, \sqrt {a}}{2 \sqrt {a}}\right )+1260 \sqrt {a x +\sqrt {\frac {d}{x}}\, b x +c}\, a^{\frac {5}{2}} b^{4} d^{2} \sqrt {x}+630 \sqrt {a x +\sqrt {\frac {d}{x}}\, b x +c}\, \left (\frac {d}{x}\right )^{\frac {5}{2}} a^{\frac {3}{2}} b^{5} x^{\frac {5}{2}}+960 a^{4} c^{3} \ln \left (\frac {2 a \sqrt {x}+\sqrt {\frac {d}{x}}\, b \sqrt {x}+2 \sqrt {a x +\sqrt {\frac {d}{x}}\, b x +c}\, \sqrt {a}}{2 \sqrt {a}}\right )-3360 \sqrt {a x +\sqrt {\frac {d}{x}}\, b x +c}\, a^{\frac {7}{2}} b^{2} c d \sqrt {x}-1680 \sqrt {a x +\sqrt {\frac {d}{x}}\, b x +c}\, \left (\frac {d}{x}\right )^{\frac {3}{2}} a^{\frac {5}{2}} b^{3} c \,x^{\frac {3}{2}}+2560 \left (a x +\sqrt {\frac {d}{x}}\, b x +c \right )^{\frac {3}{2}} a^{\frac {11}{2}} x^{\frac {3}{2}}-2304 \left (a x +\sqrt {\frac {d}{x}}\, b x +c \right )^{\frac {3}{2}} \sqrt {\frac {d}{x}}\, a^{\frac {9}{2}} b \,x^{\frac {3}{2}}+960 \sqrt {a x +\sqrt {\frac {d}{x}}\, b x +c}\, a^{\frac {9}{2}} c^{2} \sqrt {x}+2016 \left (a x +\sqrt {\frac {d}{x}}\, b x +c \right )^{\frac {3}{2}} a^{\frac {7}{2}} b^{2} d \sqrt {x}+480 \sqrt {a x +\sqrt {\frac {d}{x}}\, b x +c}\, \sqrt {\frac {d}{x}}\, a^{\frac {7}{2}} b \,c^{2} \sqrt {x}-1680 \left (a x +\sqrt {\frac {d}{x}}\, b x +c \right )^{\frac {3}{2}} \left (\frac {d}{x}\right )^{\frac {3}{2}} a^{\frac {5}{2}} b^{3} x^{\frac {3}{2}}-1920 \left (a x +\sqrt {\frac {d}{x}}\, b x +c \right )^{\frac {3}{2}} a^{\frac {9}{2}} c \sqrt {x}+3136 \left (a x +\sqrt {\frac {d}{x}}\, b x +c \right )^{\frac {3}{2}} \sqrt {\frac {d}{x}}\, a^{\frac {7}{2}} b c \sqrt {x}\right ) \sqrt {x}}{7680 \sqrt {a x +\sqrt {\frac {d}{x}}\, b x +c}\, a^{\frac {13}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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^5-1680*(b*(d/x)^(1/2)*x+a*x+c)^(3/2)*a^(5/2)*(d/x)^(3/2)*x^(3/2)*b^3-2304*(b*(d/x)^(1/2)*x+a*x+c)^(3/2)*a^
(9/2)*(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*(b*(d/x)^(1/2)*x+a*x+c)^(
1/2)*a^(5/2)*(d/x)^(3/2)*x^(3/2)*b^3*c+3136*(b*(d/x)^(1/2)*x+a*x+c)^(3/2)*a^(7/2)*(d/x)^(1/2)*x^(1/2)*b*c-1920
*(b*(d/x)^(1/2)*x+a*x+c)^(3/2)*a^(9/2)*x^(1/2)*c+2016*(b*(d/x)^(1/2)*x+a*x+c)^(3/2)*d*a^(7/2)*x^(1/2)*b^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+960*(b*(d/x)^(1/2)*x+a*x+c)^(1/2)*a^(9/2)*x^(1
/2)*c^2-3360*(b*(d/x)^(1/2)*x+a*x+c)^(1/2)*d*a^(7/2)*x^(1/2)*b^2*c+1260*(b*(d/x)^(1/2)*x+a*x+c)^(1/2)*d^2*a^(5
/2)*x^(1/2)*b^4-315*d^3*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*b^6+2100*d^2*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^2*b^4*c-3600*d*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^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 = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {b \sqrt {\frac {d}{x}} + a + \frac {c}{x}} x^{2}\,{d x} \]

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

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]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{2} \sqrt {a + b \sqrt {\frac {d}{x}} + \frac {c}{x}}\, dx \]

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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