3.31.67 \(\int \frac {1}{\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} x^4} \, dx\) [3067]

Optimal. Leaf size=289 \[ -\frac {\left (1024 a^2 c^2-2940 a b^2 c d+945 b^4 d^2+14 b c \left (92 a c-45 b^2 d\right ) \sqrt {\frac {d}{x}}\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{960 c^5}+\frac {9 b \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} \left (\frac {d}{x}\right )^{3/2}}{20 c^2 d}-\frac {2 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{5 c x^2}+\frac {\left (64 a c-63 b^2 d\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{120 c^3 x}+\frac {b \sqrt {d} \left (240 a^2 c^2-280 a b^2 c d+63 b^4 d^2\right ) \tanh ^{-1}\left (\frac {b d+2 c \sqrt {\frac {d}{x}}}{2 \sqrt {c} \sqrt {d} \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}\right )}{128 c^{11/2}} \]

[Out]

1/128*b*(63*b^4*d^2-280*a*b^2*c*d+240*a^2*c^2)*arctanh(1/2*(b*d+2*c*(d/x)^(1/2))/c^(1/2)/d^(1/2)/(a+c/x+b*(d/x
)^(1/2))^(1/2))*d^(1/2)/c^(11/2)+9/20*b*(d/x)^(3/2)*(a+c/x+b*(d/x)^(1/2))^(1/2)/c^2/d-2/5*(a+c/x+b*(d/x)^(1/2)
)^(1/2)/c/x^2+1/120*(-63*b^2*d+64*a*c)*(a+c/x+b*(d/x)^(1/2))^(1/2)/c^3/x-1/960*(1024*a^2*c^2-2940*a*b^2*c*d+94
5*b^4*d^2+14*b*c*(-45*b^2*d+92*a*c)*(d/x)^(1/2))*(a+c/x+b*(d/x)^(1/2))^(1/2)/c^5

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Rubi [A]
time = 0.37, antiderivative size = 289, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {1994, 1371, 756, 846, 793, 635, 212} \begin {gather*} \frac {b \sqrt {d} \left (240 a^2 c^2-280 a b^2 c d+63 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 )}{128 c^{11/2}}-\frac {\left (1024 a^2 c^2+14 b c \sqrt {\frac {d}{x}} \left (92 a c-45 b^2 d\right )-2940 a b^2 c d+945 b^4 d^2\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{960 c^5}+\frac {\left (64 a c-63 b^2 d\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{120 c^3 x}+\frac {9 b \left (\frac {d}{x}\right )^{3/2} \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{20 c^2 d}-\frac {2 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{5 c x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[a + b*Sqrt[d/x] + c/x]*x^4),x]

[Out]

-1/960*((1024*a^2*c^2 - 2940*a*b^2*c*d + 945*b^4*d^2 + 14*b*c*(92*a*c - 45*b^2*d)*Sqrt[d/x])*Sqrt[a + b*Sqrt[d
/x] + c/x])/c^5 + (9*b*Sqrt[a + b*Sqrt[d/x] + c/x]*(d/x)^(3/2))/(20*c^2*d) - (2*Sqrt[a + b*Sqrt[d/x] + c/x])/(
5*c*x^2) + ((64*a*c - 63*b^2*d)*Sqrt[a + b*Sqrt[d/x] + c/x])/(120*c^3*x) + (b*Sqrt[d]*(240*a^2*c^2 - 280*a*b^2
*c*d + 63*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])])/(128*c^(11/
2))

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 635

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 756

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 793

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 846

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 1371

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 1994

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/d^(2*n))*x^(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 {1}{\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} x^4} \, dx &=-\frac {\text {Subst}\left (\int \frac {x^2}{\sqrt {a+b \sqrt {x}+\frac {c x}{d}}} \, dx,x,\frac {d}{x}\right )}{d^3}\\ &=-\frac {2 \text {Subst}\left (\int \frac {x^5}{\sqrt {a+b x+\frac {c x^2}{d}}} \, dx,x,\sqrt {\frac {d}{x}}\right )}{d^3}\\ &=-\frac {2 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{5 c x^2}-\frac {2 \text {Subst}\left (\int \frac {x^3 \left (-4 a-\frac {9 b x}{2}\right )}{\sqrt {a+b x+\frac {c x^2}{d}}} \, dx,x,\sqrt {\frac {d}{x}}\right )}{5 c d^2}\\ &=\frac {9 b \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} \left (\frac {d}{x}\right )^{3/2}}{20 c^2 d}-\frac {2 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{5 c x^2}-\frac {\text {Subst}\left (\int \frac {x^2 \left (\frac {27 a b}{2}-\frac {\left (64 a c-63 b^2 d\right ) x}{4 d}\right )}{\sqrt {a+b x+\frac {c x^2}{d}}} \, dx,x,\sqrt {\frac {d}{x}}\right )}{10 c^2 d}\\ &=\frac {9 b \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} \left (\frac {d}{x}\right )^{3/2}}{20 c^2 d}-\frac {2 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{5 c x^2}+\frac {\left (64 a c-63 b^2 d\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{120 c^3 x}-\frac {\text {Subst}\left (\int \frac {x \left (-\frac {1}{2} a \left (63 b^2-\frac {64 a c}{d}\right )+\frac {7 b \left (92 a c-45 b^2 d\right ) x}{8 d}\right )}{\sqrt {a+b x+\frac {c x^2}{d}}} \, dx,x,\sqrt {\frac {d}{x}}\right )}{30 c^3}\\ &=-\frac {\left (1024 a^2 c^2-2940 a b^2 c d+945 b^4 d^2+14 b c \left (92 a c-45 b^2 d\right ) \sqrt {\frac {d}{x}}\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{960 c^5}+\frac {9 b \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} \left (\frac {d}{x}\right )^{3/2}}{20 c^2 d}-\frac {2 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{5 c x^2}+\frac {\left (64 a c-63 b^2 d\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{120 c^3 x}+\frac {\left (b \left (240 a^2 c^2-280 a b^2 c d+63 b^4 d^2\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x+\frac {c x^2}{d}}} \, dx,x,\sqrt {\frac {d}{x}}\right )}{128 c^5}\\ &=-\frac {\left (1024 a^2 c^2-2940 a b^2 c d+945 b^4 d^2+14 b c \left (92 a c-45 b^2 d\right ) \sqrt {\frac {d}{x}}\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{960 c^5}+\frac {9 b \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} \left (\frac {d}{x}\right )^{3/2}}{20 c^2 d}-\frac {2 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{5 c x^2}+\frac {\left (64 a c-63 b^2 d\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{120 c^3 x}+\frac {\left (b \left (240 a^2 c^2-280 a b^2 c d+63 b^4 d^2\right )\right ) \text {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^5}\\ &=-\frac {\left (1024 a^2 c^2-2940 a b^2 c d+945 b^4 d^2+14 b c \left (92 a c-45 b^2 d\right ) \sqrt {\frac {d}{x}}\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{960 c^5}+\frac {9 b \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} \left (\frac {d}{x}\right )^{3/2}}{20 c^2 d}-\frac {2 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{5 c x^2}+\frac {\left (64 a c-63 b^2 d\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{120 c^3 x}+\frac {b \sqrt {d} \left (240 a^2 c^2-280 a b^2 c d+63 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 )}{128 c^{11/2}}\\ \end {align*}

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Mathematica [A]
time = 1.01, size = 324, normalized size = 1.12 \begin {gather*} \frac {-2 \sqrt {c} \left (384 c^5-16 c^4 \left (8 a+3 b \sqrt {\frac {d}{x}}\right ) x+945 b^4 d^2 \left (a+b \sqrt {\frac {d}{x}}\right ) x^3-105 b^2 c d x^2 \left (-3 b^2 d+28 a^2 x+34 a b \sqrt {\frac {d}{x}} x\right )+8 c^3 x \left (9 b^2 d+64 a^2 x+43 a b \sqrt {\frac {d}{x}} x\right )+2 c^2 x^2 \left (-574 a b^2 d-63 b^3 d \sqrt {\frac {d}{x}}+512 a^3 x+1156 a^2 b \sqrt {\frac {d}{x}} x\right )\right )-15 b \left (240 a^2 c^2-280 a b^2 c d+63 b^4 d^2\right ) x^3 \sqrt {\frac {d \left (c+\left (a+b \sqrt {\frac {d}{x}}\right ) x\right )}{x}} \log \left (b d+2 c \sqrt {\frac {d}{x}}-2 \sqrt {c} \sqrt {\frac {d \left (c+a x+b \sqrt {\frac {d}{x}} x\right )}{x}}\right )}{1920 c^{11/2} \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[a + b*Sqrt[d/x] + c/x]*x^4),x]

[Out]

(-2*Sqrt[c]*(384*c^5 - 16*c^4*(8*a + 3*b*Sqrt[d/x])*x + 945*b^4*d^2*(a + b*Sqrt[d/x])*x^3 - 105*b^2*c*d*x^2*(-
3*b^2*d + 28*a^2*x + 34*a*b*Sqrt[d/x]*x) + 8*c^3*x*(9*b^2*d + 64*a^2*x + 43*a*b*Sqrt[d/x]*x) + 2*c^2*x^2*(-574
*a*b^2*d - 63*b^3*d*Sqrt[d/x] + 512*a^3*x + 1156*a^2*b*Sqrt[d/x]*x)) - 15*b*(240*a^2*c^2 - 280*a*b^2*c*d + 63*
b^4*d^2)*x^3*Sqrt[(d*(c + (a + b*Sqrt[d/x])*x))/x]*Log[b*d + 2*c*Sqrt[d/x] - 2*Sqrt[c]*Sqrt[(d*(c + a*x + b*Sq
rt[d/x]*x))/x]])/(1920*c^(11/2)*Sqrt[a + b*Sqrt[d/x] + c/x]*x^3)

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Maple [A]
time = 0.07, size = 487, normalized size = 1.69

method	result	size
default	\(\frac {\sqrt {\frac {b \sqrt {\frac {d}{x}}\, x +a x +c}{x}}\, \left (945 \ln \left (\frac {2 c +b \sqrt {\frac {d}{x}}\, x +2 \sqrt {c}\, \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}}{\sqrt {x}}\right ) \left (\frac {d}{x}\right )^{\frac {5}{2}} x^{5} b^{5} c -1890 d^{2} c^{\frac {3}{2}} \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}\, x^{2} b^{4}-4200 \ln \left (\frac {2 c +b \sqrt {\frac {d}{x}}\, x +2 \sqrt {c}\, \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}}{\sqrt {x}}\right ) \left (\frac {d}{x}\right )^{\frac {3}{2}} x^{4} a \,b^{3} c^{2}+1260 c^{\frac {5}{2}} \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}\, \left (\frac {d}{x}\right )^{\frac {3}{2}} x^{3} b^{3}+5880 d \,c^{\frac {5}{2}} \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}\, x^{2} a \,b^{2}-1008 d \,c^{\frac {7}{2}} \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}\, x \,b^{2}+3600 \ln \left (\frac {2 c +b \sqrt {\frac {d}{x}}\, x +2 \sqrt {c}\, \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}}{\sqrt {x}}\right ) \sqrt {\frac {d}{x}}\, x^{3} a^{2} b \,c^{3}-2576 c^{\frac {7}{2}} \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}\, \sqrt {\frac {d}{x}}\, x^{2} a b -2048 c^{\frac {7}{2}} \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}\, a^{2} x^{2}+864 c^{\frac {9}{2}} \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}\, \sqrt {\frac {d}{x}}\, x b +1024 c^{\frac {9}{2}} \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}\, a x -768 \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}\, c^{\frac {11}{2}}\right )}{1920 x^{2} \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}\, c^{\frac {13}{2}}}\)	\(487\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^4/(a+c/x+b*(d/x)^(1/2))^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/1920*((b*(d/x)^(1/2)*x+a*x+c)/x)^(1/2)/x^2*(945*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^5*b^5*c-1890*d^2*c^(3/2)*(b*(d/x)^(1/2)*x+a*x+c)^(1/2)*x^2*b^4-4200*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^4*a*b^3*c^2+1260*c^(5/2)*(b*(d/x)^(1
/2)*x+a*x+c)^(1/2)*(d/x)^(3/2)*x^3*b^3+5880*d*c^(5/2)*(b*(d/x)^(1/2)*x+a*x+c)^(1/2)*x^2*a*b^2-1008*d*c^(7/2)*(
b*(d/x)^(1/2)*x+a*x+c)^(1/2)*x*b^2+3600*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^3*a^2*b*c^3-2576*c^(7/2)*(b*(d/x)^(1/2)*x+a*x+c)^(1/2)*(d/x)^(1/2)*x^2*a*b-2048*c^(7/2)*(b*(
d/x)^(1/2)*x+a*x+c)^(1/2)*a^2*x^2+864*c^(9/2)*(b*(d/x)^(1/2)*x+a*x+c)^(1/2)*(d/x)^(1/2)*x*b+1024*c^(9/2)*(b*(d
/x)^(1/2)*x+a*x+c)^(1/2)*a*x-768*(b*(d/x)^(1/2)*x+a*x+c)^(1/2)*c^(11/2))/(b*(d/x)^(1/2)*x+a*x+c)^(1/2)/c^(13/2
)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(a+c/x+b*(d/x)^(1/2))^(1/2),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(b*sqrt(d/x) + a + c/x)*x^4), x)

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Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(a+c/x+b*(d/x)^(1/2))^(1/2),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^{4} \sqrt {a + b \sqrt {\frac {d}{x}} + \frac {c}{x}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**4/(a+c/x+b*(d/x)**(1/2))**(1/2),x)

[Out]

Integral(1/(x**4*sqrt(a + b*sqrt(d/x) + c/x)), x)

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Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(a+c/x+b*(d/x)^(1/2))^(1/2),x, algorithm="giac")

[Out]

Timed out

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {1}{x^4\,\sqrt {a+\frac {c}{x}+b\,\sqrt {\frac {d}{x}}}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^4*(a + c/x + b*(d/x)^(1/2))^(1/2)),x)

[Out]

int(1/(x^4*(a + c/x + b*(d/x)^(1/2))^(1/2)), x)

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