3.3062 \(\int \frac {x}{\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}} \, dx\)

Optimal. Leaf size=248 \[ \frac {5 b d \left (44 a c-21 b^2 d\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{96 a^4 \sqrt {\frac {d}{x}}}-\frac {x \left (36 a c-35 b^2 d\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{48 a^3}-\frac {7 b d^2 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{12 a^2 \left (\frac {d}{x}\right )^{3/2}}+\frac {\left (48 a^2 c^2-120 a b^2 c d+35 b^4 d^2\right ) \tanh ^{-1}\left (\frac {2 a+b \sqrt {\frac {d}{x}}}{2 \sqrt {a} \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}\right )}{64 a^{9/2}}+\frac {x^2 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{2 a} \]

[Out]

1/64*(35*b^4*d^2-120*a*b^2*c*d+48*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^(9/2)-7/12*b*d^2*(a+c/x+b*(d/x)^(1/2))^(1/2)/a^2/(d/x)^(3/2)-1/48*(-35*b^2*d+36*a*c)*x*(a+c/x+b*(d/x)^(1/2
))^(1/2)/a^3+1/2*x^2*(a+c/x+b*(d/x)^(1/2))^(1/2)/a+5/96*b*d*(-21*b^2*d+44*a*c)*(a+c/x+b*(d/x)^(1/2))^(1/2)/a^4
/(d/x)^(1/2)

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Rubi [A]  time = 0.43, antiderivative size = 248, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {1970, 1357, 744, 834, 806, 724, 206} \[ \frac {\left (48 a^2 c^2-120 a b^2 c d+35 b^4 d^2\right ) \tanh ^{-1}\left (\frac {2 a+b \sqrt {\frac {d}{x}}}{2 \sqrt {a} \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}\right )}{64 a^{9/2}}+\frac {5 b d \left (44 a c-21 b^2 d\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{96 a^4 \sqrt {\frac {d}{x}}}-\frac {x \left (36 a c-35 b^2 d\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{48 a^3}-\frac {7 b d^2 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{12 a^2 \left (\frac {d}{x}\right )^{3/2}}+\frac {x^2 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{2 a} \]

Antiderivative was successfully verified.

[In]

Int[x/Sqrt[a + b*Sqrt[d/x] + c/x],x]

[Out]

(-7*b*d^2*Sqrt[a + b*Sqrt[d/x] + c/x])/(12*a^2*(d/x)^(3/2)) + (5*b*d*(44*a*c - 21*b^2*d)*Sqrt[a + b*Sqrt[d/x]
+ c/x])/(96*a^4*Sqrt[d/x]) - ((36*a*c - 35*b^2*d)*Sqrt[a + b*Sqrt[d/x] + c/x]*x)/(48*a^3) + (Sqrt[a + b*Sqrt[d
/x] + c/x]*x^2)/(2*a) + ((48*a^2*c^2 - 120*a*b^2*c*d + 35*b^4*d^2)*ArcTanh[(2*a + b*Sqrt[d/x])/(2*Sqrt[a]*Sqrt
[a + b*Sqrt[d/x] + c/x])])/(64*a^(9/2))

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 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 \frac {x}{\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}} \, dx &=-\left (d^2 \operatorname {Subst}\left (\int \frac {1}{x^3 \sqrt {a+b \sqrt {x}+\frac {c x}{d}}} \, dx,x,\frac {d}{x}\right )\right )\\ &=-\left (\left (2 d^2\right ) \operatorname {Subst}\left (\int \frac {1}{x^5 \sqrt {a+b x+\frac {c x^2}{d}}} \, dx,x,\sqrt {\frac {d}{x}}\right )\right )\\ &=\frac {\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} x^2}{2 a}+\frac {d^2 \operatorname {Subst}\left (\int \frac {\frac {7 b}{2}+\frac {3 c x}{d}}{x^4 \sqrt {a+b x+\frac {c x^2}{d}}} \, dx,x,\sqrt {\frac {d}{x}}\right )}{2 a}\\ &=-\frac {7 b d^2 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{12 a^2 \left (\frac {d}{x}\right )^{3/2}}+\frac {\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} x^2}{2 a}-\frac {d^2 \operatorname {Subst}\left (\int \frac {\frac {1}{4} \left (35 b^2-\frac {36 a c}{d}\right )+\frac {7 b c x}{d}}{x^3 \sqrt {a+b x+\frac {c x^2}{d}}} \, dx,x,\sqrt {\frac {d}{x}}\right )}{6 a^2}\\ &=-\frac {7 b d^2 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{12 a^2 \left (\frac {d}{x}\right )^{3/2}}-\frac {\left (36 a c-35 b^2 d\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} x}{48 a^3}+\frac {\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} x^2}{2 a}+\frac {d^2 \operatorname {Subst}\left (\int \frac {-\frac {5 b \left (44 a c-21 b^2 d\right )}{8 d}-\frac {c \left (36 a c-35 b^2 d\right ) x}{4 d^2}}{x^2 \sqrt {a+b x+\frac {c x^2}{d}}} \, dx,x,\sqrt {\frac {d}{x}}\right )}{12 a^3}\\ &=-\frac {7 b d^2 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{12 a^2 \left (\frac {d}{x}\right )^{3/2}}+\frac {5 b d \left (44 a c-21 b^2 d\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{96 a^4 \sqrt {\frac {d}{x}}}-\frac {\left (36 a c-35 b^2 d\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} x}{48 a^3}+\frac {\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} x^2}{2 a}-\frac {\left (48 a^2 c^2-120 a b^2 c d+35 b^4 d^2\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x+\frac {c x^2}{d}}} \, dx,x,\sqrt {\frac {d}{x}}\right )}{64 a^4}\\ &=-\frac {7 b d^2 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{12 a^2 \left (\frac {d}{x}\right )^{3/2}}+\frac {5 b d \left (44 a c-21 b^2 d\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{96 a^4 \sqrt {\frac {d}{x}}}-\frac {\left (36 a c-35 b^2 d\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} x}{48 a^3}+\frac {\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} x^2}{2 a}+\frac {\left (48 a^2 c^2-120 a b^2 c d+35 b^4 d^2\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 )}{32 a^4}\\ &=-\frac {7 b d^2 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{12 a^2 \left (\frac {d}{x}\right )^{3/2}}+\frac {5 b d \left (44 a c-21 b^2 d\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{96 a^4 \sqrt {\frac {d}{x}}}-\frac {\left (36 a c-35 b^2 d\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} x}{48 a^3}+\frac {\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} x^2}{2 a}+\frac {\left (48 a^2 c^2-120 a b^2 c d+35 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 )}{64 a^{9/2}}\\ \end {align*}

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Mathematica [F]  time = 0.33, size = 0, normalized size = 0.00 \[ \int \frac {x}{\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

Integrate[x/Sqrt[a + b*Sqrt[d/x] + c/x], 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/(a+c/x+b*(d/x)^(1/2))^(1/2),x, algorithm="fricas")

[Out]

Timed out

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:sym2
poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Valuesym2poly/r2sym(const ge
n & e,const index_m & i,const vecteur & l) Error: Bad Argument ValueWarning, integration of abs or sign assume
s constant sign by intervals (correct if the argument is real):Check [abs(t_nostep)]Warning, integration of ab
s or sign assumes constant sign by intervals (correct if the argument is real):Check [abs(t_nostep)]Evaluation
time: 0.47Done

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maple [A]  time = 0.14, size = 398, normalized size = 1.60 \[ \frac {\sqrt {\frac {a x +\sqrt {\frac {d}{x}}\, b x +c}{x}}\, \left (105 a \,b^{4} d^{2} \ln \left (\frac {2 a \sqrt {x}+\sqrt {\frac {d}{x}}\, b \sqrt {x}+2 \sqrt {a x +\sqrt {\frac {d}{x}}\, b x +c}\, \sqrt {a}}{2 \sqrt {a}}\right )-360 a^{2} b^{2} c d \ln \left (\frac {2 a \sqrt {x}+\sqrt {\frac {d}{x}}\, b \sqrt {x}+2 \sqrt {a x +\sqrt {\frac {d}{x}}\, b x +c}\, \sqrt {a}}{2 \sqrt {a}}\right )+96 \sqrt {a x +\sqrt {\frac {d}{x}}\, b x +c}\, a^{\frac {9}{2}} x^{\frac {3}{2}}-112 \sqrt {a x +\sqrt {\frac {d}{x}}\, b x +c}\, \sqrt {\frac {d}{x}}\, a^{\frac {7}{2}} b \,x^{\frac {3}{2}}+144 a^{3} c^{2} \ln \left (\frac {2 a \sqrt {x}+\sqrt {\frac {d}{x}}\, b \sqrt {x}+2 \sqrt {a x +\sqrt {\frac {d}{x}}\, b x +c}\, \sqrt {a}}{2 \sqrt {a}}\right )+140 \sqrt {a x +\sqrt {\frac {d}{x}}\, b x +c}\, a^{\frac {5}{2}} b^{2} d \sqrt {x}-210 \sqrt {a x +\sqrt {\frac {d}{x}}\, b x +c}\, \left (\frac {d}{x}\right )^{\frac {3}{2}} a^{\frac {3}{2}} b^{3} x^{\frac {3}{2}}-144 \sqrt {a x +\sqrt {\frac {d}{x}}\, b x +c}\, a^{\frac {7}{2}} c \sqrt {x}+440 \sqrt {a x +\sqrt {\frac {d}{x}}\, b x +c}\, \sqrt {\frac {d}{x}}\, a^{\frac {5}{2}} b c \sqrt {x}\right ) \sqrt {x}}{192 \sqrt {a x +\sqrt {\frac {d}{x}}\, b x +c}\, a^{\frac {11}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/192*((a*x+(d/x)^(1/2)*b*x+c)/x)^(1/2)*x^(1/2)*(96*x^(3/2)*(a*x+(d/x)^(1/2)*b*x+c)^(1/2)*a^(9/2)-112*a^(7/2)*
(a*x+(d/x)^(1/2)*b*x+c)^(1/2)*(d/x)^(1/2)*x^(3/2)*b-210*(a*x+(d/x)^(1/2)*b*x+c)^(1/2)*(d/x)^(3/2)*a^(3/2)*b^3*
x^(3/2)+140*(a*x+(d/x)^(1/2)*b*x+c)^(1/2)*a^(5/2)*b^2*d*x^(1/2)+105*a*b^4*d^2*ln(1/2*(2*a*x^(1/2)+(d/x)^(1/2)*
b*x^(1/2)+2*(a*x+(d/x)^(1/2)*b*x+c)^(1/2)*a^(1/2))/a^(1/2))-144*(a*x+(d/x)^(1/2)*b*x+c)^(1/2)*a^(7/2)*c*x^(1/2
)+440*(a*x+(d/x)^(1/2)*b*x+c)^(1/2)*(d/x)^(1/2)*a^(5/2)*b*c*x^(1/2)-360*a^2*b^2*c*d*ln(1/2*(2*a*x^(1/2)+(d/x)^
(1/2)*b*x^(1/2)+2*(a*x+(d/x)^(1/2)*b*x+c)^(1/2)*a^(1/2))/a^(1/2))+144*a^3*c^2*ln(1/2*(2*a*x^(1/2)+(d/x)^(1/2)*
b*x^(1/2)+2*(a*x+(d/x)^(1/2)*b*x+c)^(1/2)*a^(1/2))/a^(1/2)))/(a*x+(d/x)^(1/2)*b*x+c)^(1/2)/a^(11/2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(x/(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 \[ \int \frac {x}{\sqrt {a + b \sqrt {\frac {d}{x}} + \frac {c}{x}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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