3.3058 \(\int \frac{\sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}{x^3} \, dx\)

Optimal. Leaf size=233 \[ \frac{\left (32 a c-35 b^2 d+42 b c \sqrt{\frac{d}{x}}\right ) \left (a+b \sqrt{\frac{d}{x}}+\frac{c}{x}\right )^{3/2}}{120 c^3}-\frac{b \left (12 a c-7 b^2 d\right ) \left (b d+2 c \sqrt{\frac{d}{x}}\right ) \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}{64 c^4}-\frac{b \sqrt{d} \left (12 a c-7 b^2 d\right ) \left (4 a c-b^2 d\right ) \tanh ^{-1}\left (\frac{b d+2 c \sqrt{\frac{d}{x}}}{2 \sqrt{c} \sqrt{d} \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}\right )}{128 c^{9/2}}-\frac{2 \left (a+b \sqrt{\frac{d}{x}}+\frac{c}{x}\right )^{3/2}}{5 c x} \]

[Out]

-(b*(12*a*c - 7*b^2*d)*(b*d + 2*c*Sqrt[d/x])*Sqrt[a + b*Sqrt[d/x] + c/x])/(64*c^4) + ((32*a*c - 35*b^2*d + 42*
b*c*Sqrt[d/x])*(a + b*Sqrt[d/x] + c/x)^(3/2))/(120*c^3) - (2*(a + b*Sqrt[d/x] + c/x)^(3/2))/(5*c*x) - (b*Sqrt[
d]*(12*a*c - 7*b^2*d)*(4*a*c - b^2*d)*ArcTanh[(b*d + 2*c*Sqrt[d/x])/(2*Sqrt[c]*Sqrt[d]*Sqrt[a + b*Sqrt[d/x] +
c/x])])/(128*c^(9/2))

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Rubi [A]  time = 0.307696, antiderivative size = 233, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.269, Rules used = {1970, 1357, 742, 779, 612, 621, 206} \[ \frac{\left (32 a c-35 b^2 d+42 b c \sqrt{\frac{d}{x}}\right ) \left (a+b \sqrt{\frac{d}{x}}+\frac{c}{x}\right )^{3/2}}{120 c^3}-\frac{b \left (12 a c-7 b^2 d\right ) \left (b d+2 c \sqrt{\frac{d}{x}}\right ) \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}{64 c^4}-\frac{b \sqrt{d} \left (12 a c-7 b^2 d\right ) \left (4 a c-b^2 d\right ) \tanh ^{-1}\left (\frac{b d+2 c \sqrt{\frac{d}{x}}}{2 \sqrt{c} \sqrt{d} \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}\right )}{128 c^{9/2}}-\frac{2 \left (a+b \sqrt{\frac{d}{x}}+\frac{c}{x}\right )^{3/2}}{5 c x} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a + b*Sqrt[d/x] + c/x]/x^3,x]

[Out]

-(b*(12*a*c - 7*b^2*d)*(b*d + 2*c*Sqrt[d/x])*Sqrt[a + b*Sqrt[d/x] + c/x])/(64*c^4) + ((32*a*c - 35*b^2*d + 42*
b*c*Sqrt[d/x])*(a + b*Sqrt[d/x] + c/x)^(3/2))/(120*c^3) - (2*(a + b*Sqrt[d/x] + c/x)^(3/2))/(5*c*x) - (b*Sqrt[
d]*(12*a*c - 7*b^2*d)*(4*a*c - b^2*d)*ArcTanh[(b*d + 2*c*Sqrt[d/x])/(2*Sqrt[c]*Sqrt[d]*Sqrt[a + b*Sqrt[d/x] +
c/x])])/(128*c^(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 742

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))/(c*(m + 2*p + 1)), x] + Dist[1/(c*(m + 2*p + 1)), Int[(d + e*x)^(m - 2)*Simp[c*d^2
*(m + 2*p + 1) - e*(a*e*(m - 1) + b*d*(p + 1)) + e*(2*c*d - b*e)*(m + p)*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]
&& If[RationalQ[m], GtQ[m, 1], SumSimplerQ[m, -2]] && NeQ[m + 2*p + 1, 0] && IntQuadraticQ[a, b, c, d, e, m,
p, x]

Rule 779

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[((b
*e*g*(p + 2) - c*(e*f + d*g)*(2*p + 3) - 2*c*e*g*(p + 1)*x)*(a + b*x + c*x^2)^(p + 1))/(2*c^2*(p + 1)*(2*p + 3
)), x] + Dist[(b^2*e*g*(p + 2) - 2*a*c*e*g + c*(2*c*d*f - b*(e*f + d*g))*(2*p + 3))/(2*c^2*(2*p + 3)), Int[(a
+ b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && NeQ[b^2 - 4*a*c, 0] &&  !LeQ[p, -1]

Rule 612

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((b + 2*c*x)*(a + b*x + c*x^2)^p)/(2*c*(2*p +
1)), x] - Dist[(p*(b^2 - 4*a*c))/(2*c*(2*p + 1)), Int[(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x]
&& NeQ[b^2 - 4*a*c, 0] && GtQ[p, 0] && IntegerQ[4*p]

Rule 621

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[1/(4*c - x^2), x], x, (b + 2*c*x)
/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 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{\sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}{x^3} \, dx &=-\frac{\operatorname{Subst}\left (\int x \sqrt{a+b \sqrt{x}+\frac{c x}{d}} \, dx,x,\frac{d}{x}\right )}{d^2}\\ &=-\frac{2 \operatorname{Subst}\left (\int x^3 \sqrt{a+b x+\frac{c x^2}{d}} \, dx,x,\sqrt{\frac{d}{x}}\right )}{d^2}\\ &=-\frac{2 \left (a+b \sqrt{\frac{d}{x}}+\frac{c}{x}\right )^{3/2}}{5 c x}-\frac{2 \operatorname{Subst}\left (\int x \left (-2 a-\frac{7 b x}{2}\right ) \sqrt{a+b x+\frac{c x^2}{d}} \, dx,x,\sqrt{\frac{d}{x}}\right )}{5 c d}\\ &=\frac{\left (32 a c-7 b \left (5 b d-6 c \sqrt{\frac{d}{x}}\right )\right ) \left (a+b \sqrt{\frac{d}{x}}+\frac{c}{x}\right )^{3/2}}{120 c^3}-\frac{2 \left (a+b \sqrt{\frac{d}{x}}+\frac{c}{x}\right )^{3/2}}{5 c x}-\frac{\left (b \left (12 a c-7 b^2 d\right )\right ) \operatorname{Subst}\left (\int \sqrt{a+b x+\frac{c x^2}{d}} \, dx,x,\sqrt{\frac{d}{x}}\right )}{16 c^3}\\ &=-\frac{b \left (12 a c-7 b^2 d\right ) \left (b d+2 c \sqrt{\frac{d}{x}}\right ) \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}{64 c^4}+\frac{\left (32 a c-7 b \left (5 b d-6 c \sqrt{\frac{d}{x}}\right )\right ) \left (a+b \sqrt{\frac{d}{x}}+\frac{c}{x}\right )^{3/2}}{120 c^3}-\frac{2 \left (a+b \sqrt{\frac{d}{x}}+\frac{c}{x}\right )^{3/2}}{5 c x}-\frac{\left (b \left (12 a c-7 b^2 d\right ) \left (4 a c-b^2 d\right )\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x+\frac{c x^2}{d}}} \, dx,x,\sqrt{\frac{d}{x}}\right )}{128 c^4}\\ &=-\frac{b \left (12 a c-7 b^2 d\right ) \left (b d+2 c \sqrt{\frac{d}{x}}\right ) \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}{64 c^4}+\frac{\left (32 a c-7 b \left (5 b d-6 c \sqrt{\frac{d}{x}}\right )\right ) \left (a+b \sqrt{\frac{d}{x}}+\frac{c}{x}\right )^{3/2}}{120 c^3}-\frac{2 \left (a+b \sqrt{\frac{d}{x}}+\frac{c}{x}\right )^{3/2}}{5 c x}-\frac{\left (b \left (12 a c-7 b^2 d\right ) \left (4 a c-b^2 d\right )\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{4 c}{d}-x^2} \, dx,x,\frac{b+\frac{2 c \sqrt{\frac{d}{x}}}{d}}{\sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}\right )}{64 c^4}\\ &=-\frac{b \left (12 a c-7 b^2 d\right ) \left (b d+2 c \sqrt{\frac{d}{x}}\right ) \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}{64 c^4}+\frac{\left (32 a c-7 b \left (5 b d-6 c \sqrt{\frac{d}{x}}\right )\right ) \left (a+b \sqrt{\frac{d}{x}}+\frac{c}{x}\right )^{3/2}}{120 c^3}-\frac{2 \left (a+b \sqrt{\frac{d}{x}}+\frac{c}{x}\right )^{3/2}}{5 c x}-\frac{b \sqrt{d} \left (12 a c-7 b^2 d\right ) \left (4 a c-b^2 d\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \left (b+\frac{2 c \sqrt{\frac{d}{x}}}{d}\right )}{2 \sqrt{c} \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}\right )}{128 c^{9/2}}\\ \end{align*}

Mathematica [F]  time = 0.119584, size = 0, normalized size = 0. \[ \int \frac{\sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}{x^3} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[Sqrt[a + b*Sqrt[d/x] + c/x]/x^3,x]

[Out]

Integrate[Sqrt[a + b*Sqrt[d/x] + c/x]/x^3, x]

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Maple [B]  time = 0.136, size = 615, normalized size = 2.6 \begin{align*} -{\frac{1}{1920\,{x}^{2}{c}^{5}}\sqrt{{\frac{1}{x} \left ( b\sqrt{{\frac{d}{x}}}x+ax+c \right ) }} \left ( 105\,\ln \left ({\frac{1}{\sqrt{x}} \left ( 2\,c+b\sqrt{{\frac{d}{x}}}x+2\,\sqrt{c}\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c} \right ) } \right ) \sqrt{c} \left ({\frac{d}{x}} \right ) ^{5/2}{x}^{5}{b}^{5}-210\,\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c} \left ({\frac{d}{x}} \right ) ^{5/2}{x}^{5}{b}^{5}-600\,\ln \left ({\frac{1}{\sqrt{x}} \left ( 2\,c+b\sqrt{{\frac{d}{x}}}x+2\,\sqrt{c}\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c} \right ) } \right ){c}^{3/2}a \left ({\frac{d}{x}} \right ) ^{3/2}{x}^{4}{b}^{3}-210\,a\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c}{d}^{2}{x}^{3}{b}^{4}+720\,\ln \left ({\frac{1}{\sqrt{x}} \left ( 2\,c+b\sqrt{{\frac{d}{x}}}x+2\,\sqrt{c}\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c} \right ) } \right ){c}^{5/2}{a}^{2}\sqrt{{\frac{d}{x}}}{x}^{3}b+780\,a\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c} \left ({\frac{d}{x}} \right ) ^{3/2}{x}^{4}{b}^{3}c+360\,{a}^{2}\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c}d{x}^{3}{b}^{2}c+210\, \left ( b\sqrt{{\frac{d}{x}}}x+ax+c \right ) ^{3/2}{d}^{2}{x}^{2}{b}^{4}-420\, \left ( b\sqrt{{\frac{d}{x}}}x+ax+c \right ) ^{3/2} \left ({\frac{d}{x}} \right ) ^{3/2}{x}^{3}{b}^{3}c-720\,{a}^{2}\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c}\sqrt{{\frac{d}{x}}}{x}^{3}b{c}^{2}-360\,a \left ( b\sqrt{{\frac{d}{x}}}x+ax+c \right ) ^{3/2}d{x}^{2}{b}^{2}c+720\,a \left ( b\sqrt{{\frac{d}{x}}}x+ax+c \right ) ^{3/2}\sqrt{{\frac{d}{x}}}{x}^{2}b{c}^{2}+560\, \left ( b\sqrt{{\frac{d}{x}}}x+ax+c \right ) ^{3/2}dx{b}^{2}{c}^{2}-512\,a \left ( b\sqrt{{\frac{d}{x}}}x+ax+c \right ) ^{3/2}x{c}^{3}-672\, \left ( b\sqrt{{\frac{d}{x}}}x+ax+c \right ) ^{3/2}\sqrt{{\frac{d}{x}}}xb{c}^{3}+768\, \left ( b\sqrt{{\frac{d}{x}}}x+ax+c \right ) ^{3/2}{c}^{4} \right ){\frac{1}{\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+c/x+b*(d/x)^(1/2))^(1/2)/x^3,x)

[Out]

-1/1920*((b*(d/x)^(1/2)*x+a*x+c)/x)^(1/2)*(105*ln((2*c+b*(d/x)^(1/2)*x+2*c^(1/2)*(b*(d/x)^(1/2)*x+a*x+c)^(1/2)
)/x^(1/2))*c^(1/2)*(d/x)^(5/2)*x^5*b^5-210*(b*(d/x)^(1/2)*x+a*x+c)^(1/2)*(d/x)^(5/2)*x^5*b^5-600*ln((2*c+b*(d/
x)^(1/2)*x+2*c^(1/2)*(b*(d/x)^(1/2)*x+a*x+c)^(1/2))/x^(1/2))*c^(3/2)*a*(d/x)^(3/2)*x^4*b^3-210*a*(b*(d/x)^(1/2
)*x+a*x+c)^(1/2)*d^2*x^3*b^4+720*ln((2*c+b*(d/x)^(1/2)*x+2*c^(1/2)*(b*(d/x)^(1/2)*x+a*x+c)^(1/2))/x^(1/2))*c^(
5/2)*a^2*(d/x)^(1/2)*x^3*b+780*a*(b*(d/x)^(1/2)*x+a*x+c)^(1/2)*(d/x)^(3/2)*x^4*b^3*c+360*a^2*(b*(d/x)^(1/2)*x+
a*x+c)^(1/2)*d*x^3*b^2*c+210*(b*(d/x)^(1/2)*x+a*x+c)^(3/2)*d^2*x^2*b^4-420*(b*(d/x)^(1/2)*x+a*x+c)^(3/2)*(d/x)
^(3/2)*x^3*b^3*c-720*a^2*(b*(d/x)^(1/2)*x+a*x+c)^(1/2)*(d/x)^(1/2)*x^3*b*c^2-360*a*(b*(d/x)^(1/2)*x+a*x+c)^(3/
2)*d*x^2*b^2*c+720*a*(b*(d/x)^(1/2)*x+a*x+c)^(3/2)*(d/x)^(1/2)*x^2*b*c^2+560*(b*(d/x)^(1/2)*x+a*x+c)^(3/2)*d*x
*b^2*c^2-512*a*(b*(d/x)^(1/2)*x+a*x+c)^(3/2)*x*c^3-672*(b*(d/x)^(1/2)*x+a*x+c)^(3/2)*(d/x)^(1/2)*x*b*c^3+768*(
b*(d/x)^(1/2)*x+a*x+c)^(3/2)*c^4)/x^2/(b*(d/x)^(1/2)*x+a*x+c)^(1/2)/c^5

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{b \sqrt{\frac{d}{x}} + a + \frac{c}{x}}}{x^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+c/x+b*(d/x)^(1/2))^(1/2)/x^3,x, algorithm="maxima")

[Out]

integrate(sqrt(b*sqrt(d/x) + a + c/x)/x^3, 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((a+c/x+b*(d/x)^(1/2))^(1/2)/x^3,x, algorithm="fricas")

[Out]

Timed out

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{a + b \sqrt{\frac{d}{x}} + \frac{c}{x}}}{x^{3}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+c/x+b*(d/x)**(1/2))**(1/2)/x**3,x)

[Out]

Integral(sqrt(a + b*sqrt(d/x) + c/x)/x**3, x)

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Giac [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((a+c/x+b*(d/x)^(1/2))^(1/2)/x^3,x, algorithm="giac")

[Out]

Timed out