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

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

[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))

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Rubi [A]  time = 0.539124, antiderivative size = 289, normalized size of antiderivative = 1., 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 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 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{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*}

Mathematica [F]  time = 0.197875, size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}} x^4} \, dx \]

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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Maple [A]  time = 0.143, size = 487, normalized size = 1.7 \begin{align*}{\frac{1}{1920\,{x}^{2}}\sqrt{{\frac{1}{x} \left ( b\sqrt{{\frac{d}{x}}}x+ax+c \right ) }} \left ( 945\,\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 ) \left ({\frac{d}{x}} \right ) ^{5/2}{x}^{5}{b}^{5}c+1260\,{c}^{5/2}\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c} \left ({\frac{d}{x}} \right ) ^{3/2}{x}^{3}{b}^{3}-4200\,\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 ) \left ({\frac{d}{x}} \right ) ^{3/2}{x}^{4}a{b}^{3}{c}^{2}+864\,{c}^{9/2}\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c}\sqrt{{\frac{d}{x}}}xb-2576\,{c}^{7/2}\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c}\sqrt{{\frac{d}{x}}}{x}^{2}ab-768\,\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c}{c}^{11/2}+1024\,{c}^{9/2}\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c}xa-2048\,{c}^{7/2}\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c}{x}^{2}{a}^{2}-1008\,{c}^{7/2}\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c}dx{b}^{2}+5880\,{c}^{5/2}\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c}d{x}^{2}a{b}^{2}-1890\,{c}^{3/2}\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c}{d}^{2}{x}^{2}{b}^{4}+3600\,\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{{\frac{d}{x}}}{x}^{3}{a}^{2}b{c}^{3} \right ){\frac{1}{\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c}}}{c}^{-{\frac{13}{2}}}} \end{align*}

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]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\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(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)]  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(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., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x^{4} \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(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]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\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(1/x^4/(a+c/x+b*(d/x)^(1/2))^(1/2),x, algorithm="giac")

[Out]

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