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

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

[Out]

-4*a^2*(a+b*(c/x)^(1/2))^(3/2)/b^4/c^2+12/5*a*(a+b*(c/x)^(1/2))^(5/2)/b^4/c^2-4/7*(a+b*(c/x)^(1/2))^(7/2)/b^4/
c^2+4*a^3*(a+b*(c/x)^(1/2))^(1/2)/b^4/c^2

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Rubi [A]  time = 0.07, antiderivative size = 112, normalized size of antiderivative = 1.00, 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 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])

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 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]

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*}

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Mathematica [A]  time = 0.08, size = 75, normalized size = 0.67 \[ \frac {4 \sqrt {a+b \sqrt {\frac {c}{x}}} \left (16 a^3 x-8 a^2 b x \sqrt {\frac {c}{x}}+6 a b^2 c-5 b^3 c \sqrt {\frac {c}{x}}\right )}{35 b^4 c^2 x} \]

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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fricas [A]  time = 1.08, size = 61, normalized size = 0.54 \[ \frac {4 \, {\left (6 \, a b^{2} c + 16 \, a^{3} x - {\left (5 \, b^{3} c + 8 \, a^{2} b x\right )} \sqrt {\frac {c}{x}}\right )} \sqrt {b \sqrt {\frac {c}{x}} + a}}{35 \, b^{4} c^{2} x} \]

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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giac [A]  time = 0.25, size = 152, normalized size = 1.36 \[ -\frac {4 \, {\left (5 \, {\left (b \sqrt {\frac {c}{x}} + a\right )}^{\frac {7}{2}} \mathrm {sgn}\left ({\left (b \sqrt {\frac {c}{x}} + a\right )} b - a b\right ) - 21 \, {\left (b \sqrt {\frac {c}{x}} + a\right )}^{\frac {5}{2}} a \mathrm {sgn}\left ({\left (b \sqrt {\frac {c}{x}} + a\right )} b - a b\right ) + 35 \, {\left (b \sqrt {\frac {c}{x}} + a\right )}^{\frac {3}{2}} a^{2} \mathrm {sgn}\left ({\left (b \sqrt {\frac {c}{x}} + a\right )} b - a b\right ) - 35 \, \sqrt {b \sqrt {\frac {c}{x}} + a} a^{3} \mathrm {sgn}\left ({\left (b \sqrt {\frac {c}{x}} + a\right )} b - a b\right )\right )}}{35 \, b^{4} c^{2}} \]

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]

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

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maple [C]  time = 0.05, size = 336, normalized size = 3.00 \[ -\frac {\sqrt {a +\sqrt {\frac {c}{x}}\, b}\, \left (-35 \sqrt {\frac {c}{x}}\, a^{4} b \,x^{3} \ln \left (\frac {2 a \sqrt {x}+\sqrt {\frac {c}{x}}\, b \sqrt {x}+2 \sqrt {\left (a +\sqrt {\frac {c}{x}}\, b \right ) x}\, \sqrt {a}}{2 \sqrt {a}}\right )+35 \sqrt {\frac {c}{x}}\, a^{4} b \,x^{3} \ln \left (\frac {2 a \sqrt {x}+\sqrt {\frac {c}{x}}\, b \sqrt {x}+2 \sqrt {a x +\sqrt {\frac {c}{x}}\, b x}\, \sqrt {a}}{2 \sqrt {a}}\right )+70 \sqrt {a x +\sqrt {\frac {c}{x}}\, b x}\, a^{\frac {9}{2}} x^{\frac {5}{2}}+70 \sqrt {\left (a +\sqrt {\frac {c}{x}}\, b \right ) x}\, a^{\frac {9}{2}} x^{\frac {5}{2}}-140 \left (a x +\sqrt {\frac {c}{x}}\, b x \right )^{\frac {3}{2}} a^{\frac {7}{2}} x^{\frac {3}{2}}+76 \left (a x +\sqrt {\frac {c}{x}}\, b x \right )^{\frac {3}{2}} \sqrt {\frac {c}{x}}\, a^{\frac {5}{2}} b \,x^{\frac {3}{2}}-44 \left (a x +\sqrt {\frac {c}{x}}\, b x \right )^{\frac {3}{2}} a^{\frac {3}{2}} b^{2} c \sqrt {x}+20 \left (a x +\sqrt {\frac {c}{x}}\, b x \right )^{\frac {3}{2}} \left (\frac {c}{x}\right )^{\frac {3}{2}} \sqrt {a}\, b^{3} x^{\frac {3}{2}}\right )}{35 \sqrt {\left (a +\sqrt {\frac {c}{x}}\, b \right ) x}\, \left (\frac {c}{x}\right )^{\frac {5}{2}} \sqrt {a}\, b^{5} x^{\frac {9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.53, size = 85, normalized size = 0.76 \[ -\frac {4 \, {\left (\frac {5 \, {\left (b \sqrt {\frac {c}{x}} + a\right )}^{\frac {7}{2}}}{b^{4}} - \frac {21 \, {\left (b \sqrt {\frac {c}{x}} + a\right )}^{\frac {5}{2}} a}{b^{4}} + \frac {35 \, {\left (b \sqrt {\frac {c}{x}} + a\right )}^{\frac {3}{2}} a^{2}}{b^{4}} - \frac {35 \, \sqrt {b \sqrt {\frac {c}{x}} + a} a^{3}}{b^{4}}\right )}}{35 \, c^{2}} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{x^3\,\sqrt {a+b\,\sqrt {\frac {c}{x}}}} \,d x \]

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]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x^{3} \sqrt {a + b \sqrt {\frac {c}{x}}}}\, dx \]

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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