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

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

[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))

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Rubi [A]  time = 0.430493, antiderivative size = 248, normalized size of antiderivative = 1., 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 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 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 \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*}

Mathematica [F]  time = 0.224756, size = 0, normalized size = 0. \[ \int \frac{x}{\sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}} \, dx \]

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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Maple [A]  time = 0.139, size = 398, normalized size = 1.6 \begin{align*} -{\frac{1}{192}\sqrt{{\frac{1}{x} \left ( b\sqrt{{\frac{d}{x}}}x+ax+c \right ) }}\sqrt{x} \left ( 210\,{a}^{3/2}\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c} \left ({\frac{d}{x}} \right ) ^{3/2}{x}^{3/2}{b}^{3}+112\,{a}^{7/2}\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c}\sqrt{{\frac{d}{x}}}{x}^{3/2}b-96\,{x}^{3/2}\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c}{a}^{9/2}-440\,{a}^{5/2}\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c}\sqrt{{\frac{d}{x}}}\sqrt{x}bc+144\,{a}^{7/2}\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c}\sqrt{x}c-140\,{a}^{5/2}\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c}d\sqrt{x}{b}^{2}-105\,\ln \left ( 1/2\,{\frac{1}{\sqrt{a}} \left ( b\sqrt{{\frac{d}{x}}}\sqrt{x}+2\,\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c}\sqrt{a}+2\,a\sqrt{x} \right ) } \right ){d}^{2}a{b}^{4}+360\,\ln \left ( 1/2\,{\frac{1}{\sqrt{a}} \left ( b\sqrt{{\frac{d}{x}}}\sqrt{x}+2\,\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c}\sqrt{a}+2\,a\sqrt{x} \right ) } \right ) d{a}^{2}{b}^{2}c-144\,\ln \left ( 1/2\,{\frac{1}{\sqrt{a}} \left ( b\sqrt{{\frac{d}{x}}}\sqrt{x}+2\,\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c}\sqrt{a}+2\,a\sqrt{x} \right ) } \right ){a}^{3}{c}^{2} \right ){\frac{1}{\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c}}}{a}^{-{\frac{11}{2}}}} \end{align*}

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]

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

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

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

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]

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