3.3054 \(\int \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}} x \, dx\)

Optimal. Leaf size=209 \[ -\frac{x \left (4 a c-5 b^2 d\right ) \left (2 a+b \sqrt{\frac{d}{x}}\right ) \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}{32 a^3}-\frac{\left (4 a c-5 b^2 d\right ) \left (4 a c-b^2 d\right ) \tanh ^{-1}\left (\frac{2 a+b \sqrt{\frac{d}{x}}}{2 \sqrt{a} \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}\right )}{64 a^{7/2}}-\frac{5 b d^2 \left (a+b \sqrt{\frac{d}{x}}+\frac{c}{x}\right )^{3/2}}{12 a^2 \left (\frac{d}{x}\right )^{3/2}}+\frac{x^2 \left (a+b \sqrt{\frac{d}{x}}+\frac{c}{x}\right )^{3/2}}{2 a} \]

[Out]

(-5*b*d^2*(a + b*Sqrt[d/x] + c/x)^(3/2))/(12*a^2*(d/x)^(3/2)) - ((4*a*c - 5*b^2*d)*(2*a + b*Sqrt[d/x])*Sqrt[a
+ b*Sqrt[d/x] + c/x]*x)/(32*a^3) + ((a + b*Sqrt[d/x] + c/x)^(3/2)*x^2)/(2*a) - ((4*a*c - 5*b^2*d)*(4*a*c - b^2
*d)*ArcTanh[(2*a + b*Sqrt[d/x])/(2*Sqrt[a]*Sqrt[a + b*Sqrt[d/x] + c/x])])/(64*a^(7/2))

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Rubi [A]  time = 0.284214, antiderivative size = 209, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.292, Rules used = {1970, 1357, 744, 806, 720, 724, 206} \[ -\frac{x \left (4 a c-5 b^2 d\right ) \left (2 a+b \sqrt{\frac{d}{x}}\right ) \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}{32 a^3}-\frac{\left (4 a c-5 b^2 d\right ) \left (4 a c-b^2 d\right ) \tanh ^{-1}\left (\frac{2 a+b \sqrt{\frac{d}{x}}}{2 \sqrt{a} \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}\right )}{64 a^{7/2}}-\frac{5 b d^2 \left (a+b \sqrt{\frac{d}{x}}+\frac{c}{x}\right )^{3/2}}{12 a^2 \left (\frac{d}{x}\right )^{3/2}}+\frac{x^2 \left (a+b \sqrt{\frac{d}{x}}+\frac{c}{x}\right )^{3/2}}{2 a} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a + b*Sqrt[d/x] + c/x]*x,x]

[Out]

(-5*b*d^2*(a + b*Sqrt[d/x] + c/x)^(3/2))/(12*a^2*(d/x)^(3/2)) - ((4*a*c - 5*b^2*d)*(2*a + b*Sqrt[d/x])*Sqrt[a
+ b*Sqrt[d/x] + c/x]*x)/(32*a^3) + ((a + b*Sqrt[d/x] + c/x)^(3/2)*x^2)/(2*a) - ((4*a*c - 5*b^2*d)*(4*a*c - b^2
*d)*ArcTanh[(2*a + b*Sqrt[d/x])/(2*Sqrt[a]*Sqrt[a + b*Sqrt[d/x] + c/x])])/(64*a^(7/2))

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 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 \, dx &=-\left (d^2 \operatorname{Subst}\left (\int \frac{\sqrt{a+b \sqrt{x}+\frac{c x}{d}}}{x^3} \, dx,x,\frac{d}{x}\right )\right )\\ &=-\left (\left (2 d^2\right ) \operatorname{Subst}\left (\int \frac{\sqrt{a+b x+\frac{c x^2}{d}}}{x^5} \, dx,x,\sqrt{\frac{d}{x}}\right )\right )\\ &=\frac{\left (a+b \sqrt{\frac{d}{x}}+\frac{c}{x}\right )^{3/2} x^2}{2 a}+\frac{d^2 \operatorname{Subst}\left (\int \frac{\left (\frac{5 b}{2}+\frac{c x}{d}\right ) \sqrt{a+b x+\frac{c x^2}{d}}}{x^4} \, dx,x,\sqrt{\frac{d}{x}}\right )}{2 a}\\ &=-\frac{5 b d^2 \left (a+b \sqrt{\frac{d}{x}}+\frac{c}{x}\right )^{3/2}}{12 a^2 \left (\frac{d}{x}\right )^{3/2}}+\frac{\left (a+b \sqrt{\frac{d}{x}}+\frac{c}{x}\right )^{3/2} x^2}{2 a}+\frac{\left (d \left (4 a c-5 b^2 d\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 )}{8 a^2}\\ &=-\frac{5 b d^2 \left (a+b \sqrt{\frac{d}{x}}+\frac{c}{x}\right )^{3/2}}{12 a^2 \left (\frac{d}{x}\right )^{3/2}}-\frac{\left (4 a c-5 b^2 d\right ) \left (2 a+b \sqrt{\frac{d}{x}}\right ) \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}} x}{32 a^3}+\frac{\left (a+b \sqrt{\frac{d}{x}}+\frac{c}{x}\right )^{3/2} x^2}{2 a}+\frac{\left (\left (4 a c-5 b^2 d\right ) \left (4 a c-b^2 d\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 )}{64 a^3}\\ &=-\frac{5 b d^2 \left (a+b \sqrt{\frac{d}{x}}+\frac{c}{x}\right )^{3/2}}{12 a^2 \left (\frac{d}{x}\right )^{3/2}}-\frac{\left (4 a c-5 b^2 d\right ) \left (2 a+b \sqrt{\frac{d}{x}}\right ) \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}} x}{32 a^3}+\frac{\left (a+b \sqrt{\frac{d}{x}}+\frac{c}{x}\right )^{3/2} x^2}{2 a}-\frac{\left (\left (4 a c-5 b^2 d\right ) \left (4 a c-b^2 d\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 )}{32 a^3}\\ &=-\frac{5 b d^2 \left (a+b \sqrt{\frac{d}{x}}+\frac{c}{x}\right )^{3/2}}{12 a^2 \left (\frac{d}{x}\right )^{3/2}}-\frac{\left (4 a c-5 b^2 d\right ) \left (2 a+b \sqrt{\frac{d}{x}}\right ) \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}} x}{32 a^3}+\frac{\left (a+b \sqrt{\frac{d}{x}}+\frac{c}{x}\right )^{3/2} x^2}{2 a}-\frac{\left (4 a c-5 b^2 d\right ) \left (4 a c-b^2 d\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^{7/2}}\\ \end{align*}

Mathematica [F]  time = 0.241565, size = 0, normalized size = 0. \[ \int \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}} x \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [B]  time = 0.126, size = 398, normalized size = 1.9 \begin{align*} -{\frac{1}{192}\sqrt{{\frac{1}{x} \left ( b\sqrt{{\frac{d}{x}}}x+ax+c \right ) }}\sqrt{x} \left ( -30\,{a}^{3/2}\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c} \left ({\frac{d}{x}} \right ) ^{3/2}{x}^{3/2}{b}^{3}+80\,{a}^{5/2} \left ( b\sqrt{{\frac{d}{x}}}x+ax+c \right ) ^{3/2}\sqrt{{\frac{d}{x}}}\sqrt{x}b-96\,\sqrt{x} \left ( b\sqrt{{\frac{d}{x}}}x+ax+c \right ) ^{3/2}{a}^{7/2}+24\,{a}^{5/2}\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c}\sqrt{{\frac{d}{x}}}\sqrt{x}bc+48\,{a}^{7/2}\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c}\sqrt{x}c-60\,{a}^{5/2}\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c}d\sqrt{x}{b}^{2}+15\,\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}-72\,\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+48\,\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{9}{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)*(-30*a^(3/2)*(b*(d/x)^(1/2)*x+a*x+c)^(1/2)*(d/x)^(3/2)*x^(3/2
)*b^3+80*a^(5/2)*(b*(d/x)^(1/2)*x+a*x+c)^(3/2)*(d/x)^(1/2)*x^(1/2)*b-96*x^(1/2)*(b*(d/x)^(1/2)*x+a*x+c)^(3/2)*
a^(7/2)+24*a^(5/2)*(b*(d/x)^(1/2)*x+a*x+c)^(1/2)*(d/x)^(1/2)*x^(1/2)*b*c+48*a^(7/2)*(b*(d/x)^(1/2)*x+a*x+c)^(1
/2)*x^(1/2)*c-60*a^(5/2)*(b*(d/x)^(1/2)*x+a*x+c)^(1/2)*d*x^(1/2)*b^2+15*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^4-72*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+48*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^(9/2)

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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\,{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(sqrt(b*sqrt(d/x) + a + c/x)*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 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 [A]  time = 1.37033, size = 360, normalized size = 1.72 \begin{align*} \frac{1}{192} \,{\left (2 \, \sqrt{b \sqrt{d} \sqrt{x} + a x + c}{\left (2 \,{\left (4 \, \sqrt{x}{\left (\frac{b \sqrt{d}}{a} + 6 \, \sqrt{x}\right )} - \frac{5 \, a b^{2} d - 12 \, a^{2} c}{a^{3}}\right )} \sqrt{x} + \frac{15 \, b^{3} d^{\frac{3}{2}} - 52 \, a b c \sqrt{d}}{a^{3}}\right )} + \frac{3 \,{\left (5 \, b^{4} d^{2} - 24 \, a b^{2} c d + 16 \, a^{2} c^{2}\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{7}{2}}}\right )} \mathrm{sgn}\left (x\right ) - \frac{{\left (15 \, b^{4} d^{2} \log \left ({\left | -b \sqrt{d} + 2 \, \sqrt{a} \sqrt{c} \right |}\right ) - 72 \, a b^{2} c d \log \left ({\left | -b \sqrt{d} + 2 \, \sqrt{a} \sqrt{c} \right |}\right ) + 30 \, \sqrt{a} b^{3} \sqrt{c} d^{\frac{3}{2}} + 48 \, a^{2} c^{2} \log \left ({\left | -b \sqrt{d} + 2 \, \sqrt{a} \sqrt{c} \right |}\right ) - 104 \, a^{\frac{3}{2}} b c^{\frac{3}{2}} \sqrt{d}\right )} \mathrm{sgn}\left (x\right )}{192 \, a^{\frac{7}{2}}} \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]

1/192*(2*sqrt(b*sqrt(d)*sqrt(x) + a*x + c)*(2*(4*sqrt(x)*(b*sqrt(d)/a + 6*sqrt(x)) - (5*a*b^2*d - 12*a^2*c)/a^
3)*sqrt(x) + (15*b^3*d^(3/2) - 52*a*b*c*sqrt(d))/a^3) + 3*(5*b^4*d^2 - 24*a*b^2*c*d + 16*a^2*c^2)*log(abs(-b*s
qrt(d) - 2*sqrt(a)*(sqrt(a)*sqrt(x) - sqrt(b*sqrt(d)*sqrt(x) + a*x + c))))/a^(7/2))*sgn(x) - 1/192*(15*b^4*d^2
*log(abs(-b*sqrt(d) + 2*sqrt(a)*sqrt(c))) - 72*a*b^2*c*d*log(abs(-b*sqrt(d) + 2*sqrt(a)*sqrt(c))) + 30*sqrt(a)
*b^3*sqrt(c)*d^(3/2) + 48*a^2*c^2*log(abs(-b*sqrt(d) + 2*sqrt(a)*sqrt(c))) - 104*a^(3/2)*b*c^(3/2)*sqrt(d))*sg
n(x)/a^(7/2)