3.3067 \(\int \frac {1}{\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} x^4} \, dx\)

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

[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 = 0.54, 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 = {1970, 1357, 742, 832, 779, 621, 206} \[ -\frac {\left (1024 a^2 c^2+14 b c \sqrt {\frac {d}{x}} \left (92 a c-45 b^2 d\right )-2940 a b^2 c d+945 b^4 d^2\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{960 c^5}+\frac {b \sqrt {d} \left (240 a^2 c^2-280 a b^2 c d+63 b^4 d^2\right ) \tanh ^{-1}\left (\frac {b d+2 c \sqrt {\frac {d}{x}}}{2 \sqrt {c} \sqrt {d} \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}\right )}{128 c^{11/2}}+\frac {\left (64 a c-63 b^2 d\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{120 c^3 x}+\frac {9 b \left (\frac {d}{x}\right )^{3/2} \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{20 c^2 d}-\frac {2 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{5 c x^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((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])/(960*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 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])

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 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 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 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 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]

Rubi steps

\begin {align*} \int \frac {1}{\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} x^4} \, dx &=-\frac {\operatorname {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 \operatorname {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 \operatorname {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 {\operatorname {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 {\operatorname {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 ) \operatorname {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 ) \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^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 [F]  time = 0.21, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} x^4} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

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]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:sym2
poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Valuesym2poly/r2sym(const ge
n & e,const index_m & i,const vecteur & l) Error: Bad Argument ValueWarning, integration of abs or sign assume
s constant sign by intervals (correct if the argument is real):Check [abs(t_nostep)]Warning, integration of ab
s or sign assumes constant sign by intervals (correct if the argument is real):Check [abs(t_nostep)]Evaluation
time: 0.5Done

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maple [A]  time = 0.16, size = 487, normalized size = 1.69 \[ \frac {\sqrt {\frac {a x +\sqrt {\frac {d}{x}}\, b x +c}{x}}\, \left (945 \left (\frac {d}{x}\right )^{\frac {5}{2}} b^{5} c \,x^{5} \ln \left (\frac {\sqrt {\frac {d}{x}}\, b x +2 c +2 \sqrt {a x +\sqrt {\frac {d}{x}}\, b x +c}\, \sqrt {c}}{\sqrt {x}}\right )-4200 \left (\frac {d}{x}\right )^{\frac {3}{2}} a \,b^{3} c^{2} x^{4} \ln \left (\frac {\sqrt {\frac {d}{x}}\, b x +2 c +2 \sqrt {a x +\sqrt {\frac {d}{x}}\, b x +c}\, \sqrt {c}}{\sqrt {x}}\right )+3600 \sqrt {\frac {d}{x}}\, a^{2} b \,c^{3} x^{3} \ln \left (\frac {\sqrt {\frac {d}{x}}\, b x +2 c +2 \sqrt {a x +\sqrt {\frac {d}{x}}\, b x +c}\, \sqrt {c}}{\sqrt {x}}\right )-1890 \sqrt {a x +\sqrt {\frac {d}{x}}\, b x +c}\, b^{4} c^{\frac {3}{2}} d^{2} x^{2}+5880 \sqrt {a x +\sqrt {\frac {d}{x}}\, b x +c}\, a \,b^{2} c^{\frac {5}{2}} d \,x^{2}+1260 \sqrt {a x +\sqrt {\frac {d}{x}}\, b x +c}\, \left (\frac {d}{x}\right )^{\frac {3}{2}} b^{3} c^{\frac {5}{2}} x^{3}-2048 \sqrt {a x +\sqrt {\frac {d}{x}}\, b x +c}\, a^{2} c^{\frac {7}{2}} x^{2}-2576 \sqrt {a x +\sqrt {\frac {d}{x}}\, b x +c}\, \sqrt {\frac {d}{x}}\, a b \,c^{\frac {7}{2}} x^{2}-1008 \sqrt {a x +\sqrt {\frac {d}{x}}\, b x +c}\, b^{2} c^{\frac {7}{2}} d x +1024 \sqrt {a x +\sqrt {\frac {d}{x}}\, b x +c}\, a \,c^{\frac {9}{2}} x +864 \sqrt {a x +\sqrt {\frac {d}{x}}\, b x +c}\, \sqrt {\frac {d}{x}}\, b \,c^{\frac {9}{2}} x -768 \sqrt {a x +\sqrt {\frac {d}{x}}\, b x +c}\, c^{\frac {11}{2}}\right )}{1920 \sqrt {a x +\sqrt {\frac {d}{x}}\, b x +c}\, c^{\frac {13}{2}} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {b \sqrt {\frac {d}{x}} + a + \frac {c}{x}} x^{4}}\,{d x} \]

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

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

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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