3.2992 \(\int \frac{1}{\sqrt{a+b \sqrt{\frac{c}{x}}} x^3} \, dx\)

Optimal. Leaf size=112 \[ -\frac{4 a^2 \left (a+b \sqrt{\frac{c}{x}}\right )^{3/2}}{b^4 c^2}+\frac{4 a^3 \sqrt{a+b \sqrt{\frac{c}{x}}}}{b^4 c^2}-\frac{4 \left (a+b \sqrt{\frac{c}{x}}\right )^{7/2}}{7 b^4 c^2}+\frac{12 a \left (a+b \sqrt{\frac{c}{x}}\right )^{5/2}}{5 b^4 c^2} \]

[Out]

(4*a^3*Sqrt[a + b*Sqrt[c/x]])/(b^4*c^2) - (4*a^2*(a + b*Sqrt[c/x])^(3/2))/(b^4*c^2) + (12*a*(a + b*Sqrt[c/x])^
(5/2))/(5*b^4*c^2) - (4*(a + b*Sqrt[c/x])^(7/2))/(7*b^4*c^2)

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Rubi [A]  time = 0.0705248, antiderivative size = 112, 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} \[ -\frac{4 a^2 \left (a+b \sqrt{\frac{c}{x}}\right )^{3/2}}{b^4 c^2}+\frac{4 a^3 \sqrt{a+b \sqrt{\frac{c}{x}}}}{b^4 c^2}-\frac{4 \left (a+b \sqrt{\frac{c}{x}}\right )^{7/2}}{7 b^4 c^2}+\frac{12 a \left (a+b \sqrt{\frac{c}{x}}\right )^{5/2}}{5 b^4 c^2} \]

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[a + b*Sqrt[c/x]]*x^3),x]

[Out]

(4*a^3*Sqrt[a + b*Sqrt[c/x]])/(b^4*c^2) - (4*a^2*(a + b*Sqrt[c/x])^(3/2))/(b^4*c^2) + (12*a*(a + b*Sqrt[c/x])^
(5/2))/(5*b^4*c^2) - (4*(a + b*Sqrt[c/x])^(7/2))/(7*b^4*c^2)

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{1}{\sqrt{a+b \sqrt{\frac{c}{x}}} x^3} \, dx &=\operatorname{Subst}\left (\int \frac{1}{\sqrt{a+\frac{b \sqrt{c}}{\sqrt{x}}} x^3} \, dx,\sqrt{x},\frac{\sqrt{\frac{c}{x}} x}{\sqrt{c}}\right )\\ &=-\operatorname{Subst}\left (2 \operatorname{Subst}\left (\int \frac{x^3}{\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^3}{b^3 c^{3/2} \sqrt{a+b \sqrt{c} x}}+\frac{3 a^2 \sqrt{a+b \sqrt{c} x}}{b^3 c^{3/2}}-\frac{3 a \left (a+b \sqrt{c} x\right )^{3/2}}{b^3 c^{3/2}}+\frac{\left (a+b \sqrt{c} x\right )^{5/2}}{b^3 c^{3/2}}\right ) \, dx,x,\frac{1}{\sqrt{x}}\right ),\sqrt{x},\frac{\sqrt{\frac{c}{x}} x}{\sqrt{c}}\right )\\ &=\frac{4 a^3 \sqrt{a+b \sqrt{\frac{c}{x}}}}{b^4 c^2}-\frac{4 a^2 \left (a+b \sqrt{\frac{c}{x}}\right )^{3/2}}{b^4 c^2}+\frac{12 a \left (a+b \sqrt{\frac{c}{x}}\right )^{5/2}}{5 b^4 c^2}-\frac{4 \left (a+b \sqrt{\frac{c}{x}}\right )^{7/2}}{7 b^4 c^2}\\ \end{align*}

Mathematica [A]  time = 0.0700009, size = 75, normalized size = 0.67 \[ \frac{4 \sqrt{a+b \sqrt{\frac{c}{x}}} \left (-8 a^2 b x \sqrt{\frac{c}{x}}+16 a^3 x+6 a b^2 c-5 b^3 c \sqrt{\frac{c}{x}}\right )}{35 b^4 c^2 x} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[a + b*Sqrt[c/x]]*x^3),x]

[Out]

(4*Sqrt[a + b*Sqrt[c/x]]*(6*a*b^2*c - 5*b^3*c*Sqrt[c/x] + 16*a^3*x - 8*a^2*b*Sqrt[c/x]*x))/(35*b^4*c^2*x)

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Maple [C]  time = 0.049, size = 336, normalized size = 3. \begin{align*} -{\frac{1}{35\,{b}^{5}}\sqrt{a+b\sqrt{{\frac{c}{x}}}} \left ( 70\,\sqrt{ax+b\sqrt{{\frac{c}{x}}}x}{a}^{9/2}{x}^{5/2}+70\,\sqrt{x \left ( a+b\sqrt{{\frac{c}{x}}} \right ) }{a}^{9/2}{x}^{5/2}+20\, \left ( ax+b\sqrt{{\frac{c}{x}}}x \right ) ^{3/2} \left ({\frac{c}{x}} \right ) ^{3/2}\sqrt{a}{x}^{3/2}{b}^{3}+76\, \left ( ax+b\sqrt{{\frac{c}{x}}}x \right ) ^{3/2}\sqrt{{\frac{c}{x}}}{a}^{5/2}{x}^{3/2}b-140\, \left ( ax+b\sqrt{{\frac{c}{x}}}x \right ) ^{3/2}{a}^{7/2}{x}^{3/2}+35\,\ln \left ( 1/2\,{\frac{1}{\sqrt{a}} \left ( b\sqrt{{\frac{c}{x}}}\sqrt{x}+2\,\sqrt{ax+b\sqrt{{\frac{c}{x}}}x}\sqrt{a}+2\,a\sqrt{x} \right ) } \right ) \sqrt{{\frac{c}{x}}}{x}^{3}{a}^{4}b-35\,\ln \left ( 1/2\,{\frac{1}{\sqrt{a}} \left ( b\sqrt{{\frac{c}{x}}}\sqrt{x}+2\,\sqrt{x \left ( a+b\sqrt{{\frac{c}{x}}} \right ) }\sqrt{a}+2\,a\sqrt{x} \right ) } \right ) \sqrt{{\frac{c}{x}}}{x}^{3}{a}^{4}b-44\, \left ( ax+b\sqrt{{\frac{c}{x}}}x \right ) ^{3/2}{a}^{3/2}\sqrt{x}{b}^{2}c \right ){x}^{-{\frac{9}{2}}}{\frac{1}{\sqrt{x \left ( a+b\sqrt{{\frac{c}{x}}} \right ) }}} \left ({\frac{c}{x}} \right ) ^{-{\frac{5}{2}}}{\frac{1}{\sqrt{a}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^3/(a+b*(c/x)^(1/2))^(1/2),x)

[Out]

-1/35*(a+b*(c/x)^(1/2))^(1/2)*(70*(a*x+b*(c/x)^(1/2)*x)^(1/2)*a^(9/2)*x^(5/2)+70*(x*(a+b*(c/x)^(1/2)))^(1/2)*a
^(9/2)*x^(5/2)+20*(a*x+b*(c/x)^(1/2)*x)^(3/2)*(c/x)^(3/2)*a^(1/2)*x^(3/2)*b^3+76*(a*x+b*(c/x)^(1/2)*x)^(3/2)*(
c/x)^(1/2)*a^(5/2)*x^(3/2)*b-140*(a*x+b*(c/x)^(1/2)*x)^(3/2)*a^(7/2)*x^(3/2)+35*ln(1/2*(b*(c/x)^(1/2)*x^(1/2)+
2*(a*x+b*(c/x)^(1/2)*x)^(1/2)*a^(1/2)+2*a*x^(1/2))/a^(1/2))*(c/x)^(1/2)*x^3*a^4*b-35*ln(1/2*(b*(c/x)^(1/2)*x^(
1/2)+2*(x*(a+b*(c/x)^(1/2)))^(1/2)*a^(1/2)+2*a*x^(1/2))/a^(1/2))*(c/x)^(1/2)*x^3*a^4*b-44*(a*x+b*(c/x)^(1/2)*x
)^(3/2)*a^(3/2)*x^(1/2)*b^2*c)/x^(9/2)/(x*(a+b*(c/x)^(1/2)))^(1/2)/b^5/(c/x)^(5/2)/a^(1/2)

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Maxima [A]  time = 0.92672, size = 115, normalized size = 1.03 \begin{align*} -\frac{4 \,{\left (\frac{5 \,{\left (b \sqrt{\frac{c}{x}} + a\right )}^{\frac{7}{2}}}{b^{4}} - \frac{21 \,{\left (b \sqrt{\frac{c}{x}} + a\right )}^{\frac{5}{2}} a}{b^{4}} + \frac{35 \,{\left (b \sqrt{\frac{c}{x}} + a\right )}^{\frac{3}{2}} a^{2}}{b^{4}} - \frac{35 \, \sqrt{b \sqrt{\frac{c}{x}} + a} a^{3}}{b^{4}}\right )}}{35 \, c^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/(a+b*(c/x)^(1/2))^(1/2),x, algorithm="maxima")

[Out]

-4/35*(5*(b*sqrt(c/x) + a)^(7/2)/b^4 - 21*(b*sqrt(c/x) + a)^(5/2)*a/b^4 + 35*(b*sqrt(c/x) + a)^(3/2)*a^2/b^4 -
35*sqrt(b*sqrt(c/x) + a)*a^3/b^4)/c^2

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Fricas [A]  time = 1.45383, size = 131, normalized size = 1.17 \begin{align*} \frac{4 \,{\left (6 \, a b^{2} c + 16 \, a^{3} x -{\left (5 \, b^{3} c + 8 \, a^{2} b x\right )} \sqrt{\frac{c}{x}}\right )} \sqrt{b \sqrt{\frac{c}{x}} + a}}{35 \, b^{4} c^{2} x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/(a+b*(c/x)^(1/2))^(1/2),x, algorithm="fricas")

[Out]

4/35*(6*a*b^2*c + 16*a^3*x - (5*b^3*c + 8*a^2*b*x)*sqrt(c/x))*sqrt(b*sqrt(c/x) + a)/(b^4*c^2*x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x^{3} \sqrt{a + b \sqrt{\frac{c}{x}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**3/(a+b*(c/x)**(1/2))**(1/2),x)

[Out]

Integral(1/(x**3*sqrt(a + b*sqrt(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{c}{x}} + a} x^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/(a+b*(c/x)^(1/2))^(1/2),x, algorithm="giac")

[Out]

integrate(1/(sqrt(b*sqrt(c/x) + a)*x^3), x)