3.2987 \(\int \frac{\sqrt{a+b \sqrt{\frac{c}{x}}}}{x^4} \, dx\)

Optimal. Leaf size=174 \[ -\frac{40 a^2 \left (a+b \sqrt{\frac{c}{x}}\right )^{9/2}}{9 b^6 c^3}+\frac{40 a^3 \left (a+b \sqrt{\frac{c}{x}}\right )^{7/2}}{7 b^6 c^3}-\frac{4 a^4 \left (a+b \sqrt{\frac{c}{x}}\right )^{5/2}}{b^6 c^3}+\frac{4 a^5 \left (a+b \sqrt{\frac{c}{x}}\right )^{3/2}}{3 b^6 c^3}-\frac{4 \left (a+b \sqrt{\frac{c}{x}}\right )^{13/2}}{13 b^6 c^3}+\frac{20 a \left (a+b \sqrt{\frac{c}{x}}\right )^{11/2}}{11 b^6 c^3} \]

[Out]

(4*a^5*(a + b*Sqrt[c/x])^(3/2))/(3*b^6*c^3) - (4*a^4*(a + b*Sqrt[c/x])^(5/2))/(b^6*c^3) + (40*a^3*(a + b*Sqrt[
c/x])^(7/2))/(7*b^6*c^3) - (40*a^2*(a + b*Sqrt[c/x])^(9/2))/(9*b^6*c^3) + (20*a*(a + b*Sqrt[c/x])^(11/2))/(11*
b^6*c^3) - (4*(a + b*Sqrt[c/x])^(13/2))/(13*b^6*c^3)

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Rubi [A]  time = 0.108447, antiderivative size = 174, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {369, 266, 43} \[ -\frac{40 a^2 \left (a+b \sqrt{\frac{c}{x}}\right )^{9/2}}{9 b^6 c^3}+\frac{40 a^3 \left (a+b \sqrt{\frac{c}{x}}\right )^{7/2}}{7 b^6 c^3}-\frac{4 a^4 \left (a+b \sqrt{\frac{c}{x}}\right )^{5/2}}{b^6 c^3}+\frac{4 a^5 \left (a+b \sqrt{\frac{c}{x}}\right )^{3/2}}{3 b^6 c^3}-\frac{4 \left (a+b \sqrt{\frac{c}{x}}\right )^{13/2}}{13 b^6 c^3}+\frac{20 a \left (a+b \sqrt{\frac{c}{x}}\right )^{11/2}}{11 b^6 c^3} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a + b*Sqrt[c/x]]/x^4,x]

[Out]

(4*a^5*(a + b*Sqrt[c/x])^(3/2))/(3*b^6*c^3) - (4*a^4*(a + b*Sqrt[c/x])^(5/2))/(b^6*c^3) + (40*a^3*(a + b*Sqrt[
c/x])^(7/2))/(7*b^6*c^3) - (40*a^2*(a + b*Sqrt[c/x])^(9/2))/(9*b^6*c^3) + (20*a*(a + b*Sqrt[c/x])^(11/2))/(11*
b^6*c^3) - (4*(a + b*Sqrt[c/x])^(13/2))/(13*b^6*c^3)

Rule 369

Int[((d_.)*(x_))^(m_.)*((a_) + (b_.)*((c_.)*(x_)^(q_))^(n_))^(p_.), x_Symbol] :> With[{k = Denominator[n]}, Su
bst[Int[(d*x)^m*(a + b*c^n*x^(n*q))^p, x], x^(1/k), (c*x^q)^(1/k)/(c^(1/k)*(x^(1/k))^(q - 1))]] /; FreeQ[{a, b
, c, d, m, p, q}, x] && FractionQ[n]

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{\sqrt{a+b \sqrt{\frac{c}{x}}}}{x^4} \, dx &=\operatorname{Subst}\left (\int \frac{\sqrt{a+\frac{b \sqrt{c}}{\sqrt{x}}}}{x^4} \, dx,\sqrt{x},\frac{\sqrt{\frac{c}{x}} x}{\sqrt{c}}\right )\\ &=-\operatorname{Subst}\left (2 \operatorname{Subst}\left (\int x^5 \sqrt{a+b \sqrt{c} x} \, dx,x,\frac{1}{\sqrt{x}}\right ),\sqrt{x},\frac{\sqrt{\frac{c}{x}} x}{\sqrt{c}}\right )\\ &=-\operatorname{Subst}\left (2 \operatorname{Subst}\left (\int \left (-\frac{a^5 \sqrt{a+b \sqrt{c} x}}{b^5 c^{5/2}}+\frac{5 a^4 \left (a+b \sqrt{c} x\right )^{3/2}}{b^5 c^{5/2}}-\frac{10 a^3 \left (a+b \sqrt{c} x\right )^{5/2}}{b^5 c^{5/2}}+\frac{10 a^2 \left (a+b \sqrt{c} x\right )^{7/2}}{b^5 c^{5/2}}-\frac{5 a \left (a+b \sqrt{c} x\right )^{9/2}}{b^5 c^{5/2}}+\frac{\left (a+b \sqrt{c} x\right )^{11/2}}{b^5 c^{5/2}}\right ) \, dx,x,\frac{1}{\sqrt{x}}\right ),\sqrt{x},\frac{\sqrt{\frac{c}{x}} x}{\sqrt{c}}\right )\\ &=\frac{4 a^5 \left (a+b \sqrt{\frac{c}{x}}\right )^{3/2}}{3 b^6 c^3}-\frac{4 a^4 \left (a+b \sqrt{\frac{c}{x}}\right )^{5/2}}{b^6 c^3}+\frac{40 a^3 \left (a+b \sqrt{\frac{c}{x}}\right )^{7/2}}{7 b^6 c^3}-\frac{40 a^2 \left (a+b \sqrt{\frac{c}{x}}\right )^{9/2}}{9 b^6 c^3}+\frac{20 a \left (a+b \sqrt{\frac{c}{x}}\right )^{11/2}}{11 b^6 c^3}-\frac{4 \left (a+b \sqrt{\frac{c}{x}}\right )^{13/2}}{13 b^6 c^3}\\ \end{align*}

Mathematica [A]  time = 0.0607969, size = 111, normalized size = 0.64 \[ \frac{4 \left (a+b \sqrt{\frac{c}{x}}\right )^{3/2} \left (480 a^3 b^2 c x-560 a^2 b^3 c x \sqrt{\frac{c}{x}}-384 a^4 b x^2 \sqrt{\frac{c}{x}}+256 a^5 x^2+630 a b^4 c^2-693 b^5 c x \left (\frac{c}{x}\right )^{3/2}\right )}{9009 b^6 c^3 x^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(4*(a + b*Sqrt[c/x])^(3/2)*(630*a*b^4*c^2 + 480*a^3*b^2*c*x - 560*a^2*b^3*c*Sqrt[c/x]*x - 693*b^5*c*(c/x)^(3/2
)*x + 256*a^5*x^2 - 384*a^4*b*Sqrt[c/x]*x^2))/(9009*b^6*c^3*x^2)

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Maple [A]  time = 0.021, size = 133, normalized size = 0.8 \begin{align*} -{\frac{4}{9009\,{c}^{3}{x}^{3}{b}^{6}}\sqrt{a+b\sqrt{{\frac{c}{x}}}} \left ( ax+b\sqrt{{\frac{c}{x}}}x \right ) ^{{\frac{3}{2}}} \left ( 693\, \left ({\frac{c}{x}} \right ) ^{5/2}{x}^{2}{b}^{5}+560\, \left ({\frac{c}{x}} \right ) ^{3/2}{x}^{2}{a}^{2}{b}^{3}+384\,\sqrt{{\frac{c}{x}}}{x}^{2}{a}^{4}b-630\,{c}^{2}a{b}^{4}-480\,cx{a}^{3}{b}^{2}-256\,{a}^{5}{x}^{2} \right ){\frac{1}{\sqrt{x \left ( a+b\sqrt{{\frac{c}{x}}} \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4/9009*(a+b*(c/x)^(1/2))^(1/2)*(a*x+b*(c/x)^(1/2)*x)^(3/2)/c^3/x^3*(693*(c/x)^(5/2)*x^2*b^5+560*(c/x)^(3/2)*x
^2*a^2*b^3+384*(c/x)^(1/2)*x^2*a^4*b-630*c^2*a*b^4-480*c*x*a^3*b^2-256*a^5*x^2)/(x*(a+b*(c/x)^(1/2)))^(1/2)/b^
6

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Maxima [A]  time = 0.953233, size = 171, normalized size = 0.98 \begin{align*} -\frac{4 \,{\left (\frac{693 \,{\left (b \sqrt{\frac{c}{x}} + a\right )}^{\frac{13}{2}}}{b^{6}} - \frac{4095 \,{\left (b \sqrt{\frac{c}{x}} + a\right )}^{\frac{11}{2}} a}{b^{6}} + \frac{10010 \,{\left (b \sqrt{\frac{c}{x}} + a\right )}^{\frac{9}{2}} a^{2}}{b^{6}} - \frac{12870 \,{\left (b \sqrt{\frac{c}{x}} + a\right )}^{\frac{7}{2}} a^{3}}{b^{6}} + \frac{9009 \,{\left (b \sqrt{\frac{c}{x}} + a\right )}^{\frac{5}{2}} a^{4}}{b^{6}} - \frac{3003 \,{\left (b \sqrt{\frac{c}{x}} + a\right )}^{\frac{3}{2}} a^{5}}{b^{6}}\right )}}{9009 \, c^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4/9009*(693*(b*sqrt(c/x) + a)^(13/2)/b^6 - 4095*(b*sqrt(c/x) + a)^(11/2)*a/b^6 + 10010*(b*sqrt(c/x) + a)^(9/2
)*a^2/b^6 - 12870*(b*sqrt(c/x) + a)^(7/2)*a^3/b^6 + 9009*(b*sqrt(c/x) + a)^(5/2)*a^4/b^6 - 3003*(b*sqrt(c/x) +
a)^(3/2)*a^5/b^6)/c^3

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Fricas [A]  time = 1.50848, size = 236, normalized size = 1.36 \begin{align*} -\frac{4 \,{\left (693 \, b^{6} c^{3} - 70 \, a^{2} b^{4} c^{2} x - 96 \, a^{4} b^{2} c x^{2} - 256 \, a^{6} x^{3} +{\left (63 \, a b^{5} c^{2} x + 80 \, a^{3} b^{3} c x^{2} + 128 \, a^{5} b x^{3}\right )} \sqrt{\frac{c}{x}}\right )} \sqrt{b \sqrt{\frac{c}{x}} + a}}{9009 \, b^{6} c^{3} x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4/9009*(693*b^6*c^3 - 70*a^2*b^4*c^2*x - 96*a^4*b^2*c*x^2 - 256*a^6*x^3 + (63*a*b^5*c^2*x + 80*a^3*b^3*c*x^2
+ 128*a^5*b*x^3)*sqrt(c/x))*sqrt(b*sqrt(c/x) + a)/(b^6*c^3*x^3)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out