3.3059 \(\int \frac{\sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}{x^4} \, dx\)

Optimal. Leaf size=371 \[ -\frac{\left (1024 a^2 c^2+18 b c \sqrt{\frac{d}{x}} \left (148 a c-77 b^2 d\right )-3276 a b^2 c d+1155 b^4 d^2\right ) \left (a+b \sqrt{\frac{d}{x}}+\frac{c}{x}\right )^{3/2}}{6720 c^5}+\frac{b \left (80 a^2 c^2-120 a b^2 c d+33 b^4 d^2\right ) \left (b d+2 c \sqrt{\frac{d}{x}}\right ) \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}{512 c^6}+\frac{b \sqrt{d} \left (4 a c-b^2 d\right ) \left (80 a^2 c^2-120 a b^2 c d+33 b^4 d^2\right ) \tanh ^{-1}\left (\frac{b d+2 c \sqrt{\frac{d}{x}}}{2 \sqrt{c} \sqrt{d} \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}\right )}{1024 c^{13/2}}+\frac{\left (32 a c-33 b^2 d\right ) \left (a+b \sqrt{\frac{d}{x}}+\frac{c}{x}\right )^{3/2}}{140 c^3 x}+\frac{11 b \left (\frac{d}{x}\right )^{3/2} \left (a+b \sqrt{\frac{d}{x}}+\frac{c}{x}\right )^{3/2}}{42 c^2 d}-\frac{2 \left (a+b \sqrt{\frac{d}{x}}+\frac{c}{x}\right )^{3/2}}{7 c x^2} \]

[Out]

(b*(80*a^2*c^2 - 120*a*b^2*c*d + 33*b^4*d^2)*(b*d + 2*c*Sqrt[d/x])*Sqrt[a + b*Sqrt[d/x] + c/x])/(512*c^6) - ((
1024*a^2*c^2 - 3276*a*b^2*c*d + 1155*b^4*d^2 + 18*b*c*(148*a*c - 77*b^2*d)*Sqrt[d/x])*(a + b*Sqrt[d/x] + c/x)^
(3/2))/(6720*c^5) + (11*b*(a + b*Sqrt[d/x] + c/x)^(3/2)*(d/x)^(3/2))/(42*c^2*d) - (2*(a + b*Sqrt[d/x] + c/x)^(
3/2))/(7*c*x^2) + ((32*a*c - 33*b^2*d)*(a + b*Sqrt[d/x] + c/x)^(3/2))/(140*c^3*x) + (b*Sqrt[d]*(4*a*c - b^2*d)
*(80*a^2*c^2 - 120*a*b^2*c*d + 33*b^4*d^2)*ArcTanh[(b*d + 2*c*Sqrt[d/x])/(2*Sqrt[c]*Sqrt[d]*Sqrt[a + b*Sqrt[d/
x] + c/x])])/(1024*c^(13/2))

________________________________________________________________________________________

Rubi [A]  time = 0.664856, antiderivative size = 371, normalized size of antiderivative = 1., 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, 742, 832, 779, 612, 621, 206} \[ -\frac{\left (1024 a^2 c^2+18 b c \sqrt{\frac{d}{x}} \left (148 a c-77 b^2 d\right )-3276 a b^2 c d+1155 b^4 d^2\right ) \left (a+b \sqrt{\frac{d}{x}}+\frac{c}{x}\right )^{3/2}}{6720 c^5}+\frac{b \left (80 a^2 c^2-120 a b^2 c d+33 b^4 d^2\right ) \left (b d+2 c \sqrt{\frac{d}{x}}\right ) \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}{512 c^6}+\frac{b \sqrt{d} \left (4 a c-b^2 d\right ) \left (80 a^2 c^2-120 a b^2 c d+33 b^4 d^2\right ) \tanh ^{-1}\left (\frac{b d+2 c \sqrt{\frac{d}{x}}}{2 \sqrt{c} \sqrt{d} \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}\right )}{1024 c^{13/2}}+\frac{\left (32 a c-33 b^2 d\right ) \left (a+b \sqrt{\frac{d}{x}}+\frac{c}{x}\right )^{3/2}}{140 c^3 x}+\frac{11 b \left (\frac{d}{x}\right )^{3/2} \left (a+b \sqrt{\frac{d}{x}}+\frac{c}{x}\right )^{3/2}}{42 c^2 d}-\frac{2 \left (a+b \sqrt{\frac{d}{x}}+\frac{c}{x}\right )^{3/2}}{7 c x^2} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a + b*Sqrt[d/x] + c/x]/x^4,x]

[Out]

(b*(80*a^2*c^2 - 120*a*b^2*c*d + 33*b^4*d^2)*(b*d + 2*c*Sqrt[d/x])*Sqrt[a + b*Sqrt[d/x] + c/x])/(512*c^6) - ((
1024*a^2*c^2 - 3276*a*b^2*c*d + 1155*b^4*d^2 + 18*b*c*(148*a*c - 77*b^2*d)*Sqrt[d/x])*(a + b*Sqrt[d/x] + c/x)^
(3/2))/(6720*c^5) + (11*b*(a + b*Sqrt[d/x] + c/x)^(3/2)*(d/x)^(3/2))/(42*c^2*d) - (2*(a + b*Sqrt[d/x] + c/x)^(
3/2))/(7*c*x^2) + ((32*a*c - 33*b^2*d)*(a + b*Sqrt[d/x] + c/x)^(3/2))/(140*c^3*x) + (b*Sqrt[d]*(4*a*c - b^2*d)
*(80*a^2*c^2 - 120*a*b^2*c*d + 33*b^4*d^2)*ArcTanh[(b*d + 2*c*Sqrt[d/x])/(2*Sqrt[c]*Sqrt[d]*Sqrt[a + b*Sqrt[d/
x] + c/x])])/(1024*c^(13/2))

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 832

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[(g*(d + e*x)^m*(a + b*x + c*x^2)^(p + 1))/(c*(m + 2*p + 2)), x] + Dist[1/(c*(m + 2*p + 2)), Int[(d + e*x)^(m
- 1)*(a + b*x + c*x^2)^p*Simp[m*(c*d*f - a*e*g) + d*(2*c*f - b*g)*(p + 1) + (m*(c*e*f + c*d*g - b*e*g) + e*(p
+ 1)*(2*c*f - b*g))*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] && GtQ[m, 0] && NeQ[m + 2*p + 2, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
&&  !(IGtQ[m, 0] && EqQ[f, 0])

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^4} \, dx &=-\frac{\operatorname{Subst}\left (\int x^2 \sqrt{a+b \sqrt{x}+\frac{c x}{d}} \, dx,x,\frac{d}{x}\right )}{d^3}\\ &=-\frac{2 \operatorname{Subst}\left (\int x^5 \sqrt{a+b x+\frac{c x^2}{d}} \, dx,x,\sqrt{\frac{d}{x}}\right )}{d^3}\\ &=-\frac{2 \left (a+b \sqrt{\frac{d}{x}}+\frac{c}{x}\right )^{3/2}}{7 c x^2}-\frac{2 \operatorname{Subst}\left (\int x^3 \left (-4 a-\frac{11 b x}{2}\right ) \sqrt{a+b x+\frac{c x^2}{d}} \, dx,x,\sqrt{\frac{d}{x}}\right )}{7 c d^2}\\ &=\frac{11 b \left (a+b \sqrt{\frac{d}{x}}+\frac{c}{x}\right )^{3/2} \left (\frac{d}{x}\right )^{3/2}}{42 c^2 d}-\frac{2 \left (a+b \sqrt{\frac{d}{x}}+\frac{c}{x}\right )^{3/2}}{7 c x^2}-\frac{\operatorname{Subst}\left (\int x^2 \left (\frac{33 a b}{2}-\frac{3 \left (32 a c-33 b^2 d\right ) x}{4 d}\right ) \sqrt{a+b x+\frac{c x^2}{d}} \, dx,x,\sqrt{\frac{d}{x}}\right )}{21 c^2 d}\\ &=\frac{11 b \left (a+b \sqrt{\frac{d}{x}}+\frac{c}{x}\right )^{3/2} \left (\frac{d}{x}\right )^{3/2}}{42 c^2 d}-\frac{2 \left (a+b \sqrt{\frac{d}{x}}+\frac{c}{x}\right )^{3/2}}{7 c x^2}+\frac{\left (32 a c-33 b^2 d\right ) \left (a+b \sqrt{\frac{d}{x}}+\frac{c}{x}\right )^{3/2}}{140 c^3 x}-\frac{\operatorname{Subst}\left (\int x \left (-\frac{3}{2} a \left (33 b^2-\frac{32 a c}{d}\right )+\frac{9 b \left (148 a c-77 b^2 d\right ) x}{8 d}\right ) \sqrt{a+b x+\frac{c x^2}{d}} \, dx,x,\sqrt{\frac{d}{x}}\right )}{105 c^3}\\ &=-\frac{\left (1024 a^2 c^2-3276 a b^2 c d+1155 b^4 d^2+18 b c \left (148 a c-77 b^2 d\right ) \sqrt{\frac{d}{x}}\right ) \left (a+b \sqrt{\frac{d}{x}}+\frac{c}{x}\right )^{3/2}}{6720 c^5}+\frac{11 b \left (a+b \sqrt{\frac{d}{x}}+\frac{c}{x}\right )^{3/2} \left (\frac{d}{x}\right )^{3/2}}{42 c^2 d}-\frac{2 \left (a+b \sqrt{\frac{d}{x}}+\frac{c}{x}\right )^{3/2}}{7 c x^2}+\frac{\left (32 a c-33 b^2 d\right ) \left (a+b \sqrt{\frac{d}{x}}+\frac{c}{x}\right )^{3/2}}{140 c^3 x}+\frac{\left (b \left (80 a^2 c^2-120 a b^2 c d+33 b^4 d^2\right )\right ) \operatorname{Subst}\left (\int \sqrt{a+b x+\frac{c x^2}{d}} \, dx,x,\sqrt{\frac{d}{x}}\right )}{128 c^5}\\ &=\frac{b \left (80 a^2 c^2-120 a b^2 c d+33 b^4 d^2\right ) \left (b d+2 c \sqrt{\frac{d}{x}}\right ) \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}{512 c^6}-\frac{\left (1024 a^2 c^2-3276 a b^2 c d+1155 b^4 d^2+18 b c \left (148 a c-77 b^2 d\right ) \sqrt{\frac{d}{x}}\right ) \left (a+b \sqrt{\frac{d}{x}}+\frac{c}{x}\right )^{3/2}}{6720 c^5}+\frac{11 b \left (a+b \sqrt{\frac{d}{x}}+\frac{c}{x}\right )^{3/2} \left (\frac{d}{x}\right )^{3/2}}{42 c^2 d}-\frac{2 \left (a+b \sqrt{\frac{d}{x}}+\frac{c}{x}\right )^{3/2}}{7 c x^2}+\frac{\left (32 a c-33 b^2 d\right ) \left (a+b \sqrt{\frac{d}{x}}+\frac{c}{x}\right )^{3/2}}{140 c^3 x}+\frac{\left (b \left (4 a c-b^2 d\right ) \left (80 a^2 c^2-120 a b^2 c d+33 b^4 d^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x+\frac{c x^2}{d}}} \, dx,x,\sqrt{\frac{d}{x}}\right )}{1024 c^6}\\ &=\frac{b \left (80 a^2 c^2-120 a b^2 c d+33 b^4 d^2\right ) \left (b d+2 c \sqrt{\frac{d}{x}}\right ) \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}{512 c^6}-\frac{\left (1024 a^2 c^2-3276 a b^2 c d+1155 b^4 d^2+18 b c \left (148 a c-77 b^2 d\right ) \sqrt{\frac{d}{x}}\right ) \left (a+b \sqrt{\frac{d}{x}}+\frac{c}{x}\right )^{3/2}}{6720 c^5}+\frac{11 b \left (a+b \sqrt{\frac{d}{x}}+\frac{c}{x}\right )^{3/2} \left (\frac{d}{x}\right )^{3/2}}{42 c^2 d}-\frac{2 \left (a+b \sqrt{\frac{d}{x}}+\frac{c}{x}\right )^{3/2}}{7 c x^2}+\frac{\left (32 a c-33 b^2 d\right ) \left (a+b \sqrt{\frac{d}{x}}+\frac{c}{x}\right )^{3/2}}{140 c^3 x}+\frac{\left (b \left (4 a c-b^2 d\right ) \left (80 a^2 c^2-120 a b^2 c d+33 b^4 d^2\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 )}{512 c^6}\\ &=\frac{b \left (80 a^2 c^2-120 a b^2 c d+33 b^4 d^2\right ) \left (b d+2 c \sqrt{\frac{d}{x}}\right ) \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}{512 c^6}-\frac{\left (1024 a^2 c^2-3276 a b^2 c d+1155 b^4 d^2+18 b c \left (148 a c-77 b^2 d\right ) \sqrt{\frac{d}{x}}\right ) \left (a+b \sqrt{\frac{d}{x}}+\frac{c}{x}\right )^{3/2}}{6720 c^5}+\frac{11 b \left (a+b \sqrt{\frac{d}{x}}+\frac{c}{x}\right )^{3/2} \left (\frac{d}{x}\right )^{3/2}}{42 c^2 d}-\frac{2 \left (a+b \sqrt{\frac{d}{x}}+\frac{c}{x}\right )^{3/2}}{7 c x^2}+\frac{\left (32 a c-33 b^2 d\right ) \left (a+b \sqrt{\frac{d}{x}}+\frac{c}{x}\right )^{3/2}}{140 c^3 x}+\frac{b \sqrt{d} \left (4 a c-b^2 d\right ) \left (80 a^2 c^2-120 a b^2 c d+33 b^4 d^2\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 )}{1024 c^{13/2}}\\ \end{align*}

Mathematica [F]  time = 0.128154, size = 0, normalized size = 0. \[ \int \frac{\sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}{x^4} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[Sqrt[a + b*Sqrt[d/x] + c/x]/x^4,x]

[Out]

Integrate[Sqrt[a + b*Sqrt[d/x] + c/x]/x^4, x]

________________________________________________________________________________________

Maple [B]  time = 0.136, size = 979, normalized size = 2.6 \begin{align*} \text{result too large to display} \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^4,x)

[Out]

1/107520*((b*(d/x)^(1/2)*x+a*x+c)/x)^(1/2)*(-3465*c^(1/2)*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))*(d/x)^(7/2)*x^7*b^7+22176*(b*(d/x)^(1/2)*x+a*x+c)^(3/2)*(d/x)^(3/2)*x^3*b^3*c^3+28160*(b
*(d/x)^(1/2)*x+a*x+c)^(3/2)*(d/x)^(1/2)*x*b*c^5+13860*(b*(d/x)^(1/2)*x+a*x+c)^(3/2)*(d/x)^(5/2)*x^5*b^5*c+6930
*a*(b*(d/x)^(1/2)*x+a*x+c)^(1/2)*d^3*x^4*b^6-18480*(b*(d/x)^(1/2)*x+a*x+c)^(3/2)*d^2*x^2*b^4*c^2-25344*(b*(d/x
)^(1/2)*x+a*x+c)^(3/2)*d*x*b^2*c^4-16384*a^2*(b*(d/x)^(1/2)*x+a*x+c)^(3/2)*x^2*c^4+24576*a*(b*(d/x)^(1/2)*x+a*
x+c)^(3/2)*x*c^5-6930*(b*(d/x)^(1/2)*x+a*x+c)^(3/2)*d^3*x^3*b^6+6930*(b*(d/x)^(1/2)*x+a*x+c)^(1/2)*(d/x)^(7/2)
*x^7*b^7-30720*(b*(d/x)^(1/2)*x+a*x+c)^(3/2)*c^6-16800*a^2*(b*(d/x)^(1/2)*x+a*x+c)^(3/2)*d*x^3*b^2*c^2+16800*a
^3*(b*(d/x)^(1/2)*x+a*x+c)^(1/2)*d*x^4*b^2*c^2+25200*a*(b*(d/x)^(1/2)*x+a*x+c)^(3/2)*d^2*x^3*b^4*c+52416*a*(b*
(d/x)^(1/2)*x+a*x+c)^(3/2)*d*x^2*b^2*c^3-25200*a^2*(b*(d/x)^(1/2)*x+a*x+c)^(1/2)*d^2*x^4*b^4*c+67200*a^2*(b*(d
/x)^(1/2)*x+a*x+c)^(1/2)*(d/x)^(3/2)*x^5*b^3*c^2+33600*c^(7/2)*a^3*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))*(d/x)^(1/2)*x^4*b-50400*a*(b*(d/x)^(1/2)*x+a*x+c)^(3/2)*(d/x)^(3/2)*x^4*b^3*c^2
-33600*a^3*(b*(d/x)^(1/2)*x+a*x+c)^(1/2)*(d/x)^(1/2)*x^4*b*c^3+33600*a^2*(b*(d/x)^(1/2)*x+a*x+c)^(3/2)*(d/x)^(
1/2)*x^3*b*c^3-39060*a*(b*(d/x)^(1/2)*x+a*x+c)^(1/2)*(d/x)^(5/2)*x^6*b^5*c+26460*c^(3/2)*a*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))*(d/x)^(5/2)*x^6*b^5-42624*a*(b*(d/x)^(1/2)*x+a*x+c)^(3/
2)*(d/x)^(1/2)*x^2*b*c^4-58800*c^(5/2)*a^2*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))*(d/x)^(3/2)*x^5*b^3)/x^3/(b*(d/x)^(1/2)*x+a*x+c)^(1/2)/c^7

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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^{4}}\,{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^4,x, algorithm="maxima")

[Out]

integrate(sqrt(b*sqrt(d/x) + a + c/x)/x^4, 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^4,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^{4}}\, 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**4,x)

[Out]

Integral(sqrt(a + b*sqrt(d/x) + c/x)/x**4, 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^4,x, algorithm="giac")

[Out]

Timed out