3.24.94 \(\int \frac {x}{\sqrt {a+b \sqrt {\frac {c}{x}}}} \, dx\)

Optimal. Leaf size=172 \[ \frac {35 b^4 c^2 \tanh ^{-1}\left (\frac {\sqrt {a+b \sqrt {\frac {c}{x}}}}{\sqrt {a}}\right )}{32 a^{9/2}}-\frac {35 b^3 c^2 \sqrt {a+b \sqrt {\frac {c}{x}}}}{32 a^4 \sqrt {\frac {c}{x}}}+\frac {35 b^2 c x \sqrt {a+b \sqrt {\frac {c}{x}}}}{48 a^3}-\frac {7 b c^2 \sqrt {a+b \sqrt {\frac {c}{x}}}}{12 a^2 \left (\frac {c}{x}\right )^{3/2}}+\frac {x^2 \sqrt {a+b \sqrt {\frac {c}{x}}}}{2 a} \]

________________________________________________________________________________________

Rubi [A]  time = 0.12, antiderivative size = 175, normalized size of antiderivative = 1.02, number of steps used = 8, number of rules used = 5, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {369, 266, 51, 63, 208} \begin {gather*} -\frac {35 b^3 c^2 \sqrt {a+b \sqrt {\frac {c}{x}}}}{32 a^4 \sqrt {\frac {c}{x}}}+\frac {35 b^4 c^2 \tanh ^{-1}\left (\frac {\sqrt {a+b \sqrt {\frac {c}{x}}}}{\sqrt {a}}\right )}{32 a^{9/2}}+\frac {35 b^2 c x \sqrt {a+b \sqrt {\frac {c}{x}}}}{48 a^3}-\frac {7 b x^3 \left (\frac {c}{x}\right )^{3/2} \sqrt {a+b \sqrt {\frac {c}{x}}}}{12 a^2 c}+\frac {x^2 \sqrt {a+b \sqrt {\frac {c}{x}}}}{2 a} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x/Sqrt[a + b*Sqrt[c/x]],x]

[Out]

(-35*b^3*c^2*Sqrt[a + b*Sqrt[c/x]])/(32*a^4*Sqrt[c/x]) + (35*b^2*c*Sqrt[a + b*Sqrt[c/x]]*x)/(48*a^3) + (Sqrt[a
+ b*Sqrt[c/x]]*x^2)/(2*a) - (7*b*Sqrt[a + b*Sqrt[c/x]]*(c/x)^(3/2)*x^3)/(12*a^2*c) + (35*b^4*c^2*ArcTanh[Sqrt
[a + b*Sqrt[c/x]]/Sqrt[a]])/(32*a^(9/2))

Rule 51

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*(m + n + 2))/((b*c - a*d)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

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 {x}{\sqrt {a+b \sqrt {\frac {c}{x}}}} \, dx &=\operatorname {Subst}\left (\int \frac {x}{\sqrt {a+\frac {b \sqrt {c}}{\sqrt {x}}}} \, dx,\sqrt {x},\frac {\sqrt {\frac {c}{x}} x}{\sqrt {c}}\right )\\ &=-\operatorname {Subst}\left (2 \operatorname {Subst}\left (\int \frac {1}{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 )\\ &=\frac {\sqrt {a+b \sqrt {\frac {c}{x}}} x^2}{2 a}+\operatorname {Subst}\left (\frac {\left (7 b \sqrt {c}\right ) \operatorname {Subst}\left (\int \frac {1}{x^4 \sqrt {a+b \sqrt {c} x}} \, dx,x,\frac {1}{\sqrt {x}}\right )}{4 a},\sqrt {x},\frac {\sqrt {\frac {c}{x}} x}{\sqrt {c}}\right )\\ &=\frac {\sqrt {a+b \sqrt {\frac {c}{x}}} x^2}{2 a}-\frac {7 b \sqrt {a+b \sqrt {\frac {c}{x}}} \left (\frac {c}{x}\right )^{3/2} x^3}{12 a^2 c}-\operatorname {Subst}\left (\frac {\left (35 b^2 c\right ) \operatorname {Subst}\left (\int \frac {1}{x^3 \sqrt {a+b \sqrt {c} x}} \, dx,x,\frac {1}{\sqrt {x}}\right )}{24 a^2},\sqrt {x},\frac {\sqrt {\frac {c}{x}} x}{\sqrt {c}}\right )\\ &=\frac {35 b^2 c \sqrt {a+b \sqrt {\frac {c}{x}}} x}{48 a^3}+\frac {\sqrt {a+b \sqrt {\frac {c}{x}}} x^2}{2 a}-\frac {7 b \sqrt {a+b \sqrt {\frac {c}{x}}} \left (\frac {c}{x}\right )^{3/2} x^3}{12 a^2 c}+\operatorname {Subst}\left (\frac {\left (35 b^3 c^{3/2}\right ) \operatorname {Subst}\left (\int \frac {1}{x^2 \sqrt {a+b \sqrt {c} x}} \, dx,x,\frac {1}{\sqrt {x}}\right )}{32 a^3},\sqrt {x},\frac {\sqrt {\frac {c}{x}} x}{\sqrt {c}}\right )\\ &=-\frac {35 b^3 c^2 \sqrt {a+b \sqrt {\frac {c}{x}}}}{32 a^4 \sqrt {\frac {c}{x}}}+\frac {35 b^2 c \sqrt {a+b \sqrt {\frac {c}{x}}} x}{48 a^3}+\frac {\sqrt {a+b \sqrt {\frac {c}{x}}} x^2}{2 a}-\frac {7 b \sqrt {a+b \sqrt {\frac {c}{x}}} \left (\frac {c}{x}\right )^{3/2} x^3}{12 a^2 c}-\operatorname {Subst}\left (\frac {\left (35 b^4 c^2\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b \sqrt {c} x}} \, dx,x,\frac {1}{\sqrt {x}}\right )}{64 a^4},\sqrt {x},\frac {\sqrt {\frac {c}{x}} x}{\sqrt {c}}\right )\\ &=-\frac {35 b^3 c^2 \sqrt {a+b \sqrt {\frac {c}{x}}}}{32 a^4 \sqrt {\frac {c}{x}}}+\frac {35 b^2 c \sqrt {a+b \sqrt {\frac {c}{x}}} x}{48 a^3}+\frac {\sqrt {a+b \sqrt {\frac {c}{x}}} x^2}{2 a}-\frac {7 b \sqrt {a+b \sqrt {\frac {c}{x}}} \left (\frac {c}{x}\right )^{3/2} x^3}{12 a^2 c}-\operatorname {Subst}\left (\frac {\left (35 b^3 c^{3/2}\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b \sqrt {c}}+\frac {x^2}{b \sqrt {c}}} \, dx,x,\sqrt {a+\frac {b \sqrt {c}}{\sqrt {x}}}\right )}{32 a^4},\sqrt {x},\frac {\sqrt {\frac {c}{x}} x}{\sqrt {c}}\right )\\ &=-\frac {35 b^3 c^2 \sqrt {a+b \sqrt {\frac {c}{x}}}}{32 a^4 \sqrt {\frac {c}{x}}}+\frac {35 b^2 c \sqrt {a+b \sqrt {\frac {c}{x}}} x}{48 a^3}+\frac {\sqrt {a+b \sqrt {\frac {c}{x}}} x^2}{2 a}-\frac {7 b \sqrt {a+b \sqrt {\frac {c}{x}}} \left (\frac {c}{x}\right )^{3/2} x^3}{12 a^2 c}+\frac {35 b^4 c^2 \tanh ^{-1}\left (\frac {\sqrt {a+b \sqrt {\frac {c}{x}}}}{\sqrt {a}}\right )}{32 a^{9/2}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.16, size = 126, normalized size = 0.73 \begin {gather*} \frac {35 b^4 c^2 \tanh ^{-1}\left (\frac {\sqrt {a}}{\sqrt {a+b \sqrt {\frac {c}{x}}}}\right )}{32 a^{9/2}}+\frac {48 a^4 x^2-8 a^3 b x^2 \sqrt {\frac {c}{x}}+14 a^2 b^2 c x-35 a b^3 c x \sqrt {\frac {c}{x}}-105 b^4 c^2}{96 a^4 \sqrt {a+b \sqrt {\frac {c}{x}}}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x/Sqrt[a + b*Sqrt[c/x]],x]

[Out]

(-105*b^4*c^2 + 14*a^2*b^2*c*x - 35*a*b^3*c*Sqrt[c/x]*x + 48*a^4*x^2 - 8*a^3*b*Sqrt[c/x]*x^2)/(96*a^4*Sqrt[a +
b*Sqrt[c/x]]) + (35*b^4*c^2*ArcTanh[Sqrt[a]/Sqrt[a + b*Sqrt[c/x]]])/(32*a^(9/2))

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.11, size = 112, normalized size = 0.65 \begin {gather*} \frac {35 b^4 c^2 \tanh ^{-1}\left (\frac {\sqrt {a+b \sqrt {\frac {c}{x}}}}{\sqrt {a}}\right )}{32 a^{9/2}}+\frac {x^2 \sqrt {a+b \sqrt {\frac {c}{x}}} \left (48 a^3-56 a^2 b \sqrt {\frac {c}{x}}+\frac {70 a b^2 c}{x}-105 b^3 \left (\frac {c}{x}\right )^{3/2}\right )}{96 a^4} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[x/Sqrt[a + b*Sqrt[c/x]],x]

[Out]

(Sqrt[a + b*Sqrt[c/x]]*(48*a^3 - 56*a^2*b*Sqrt[c/x] - 105*b^3*(c/x)^(3/2) + (70*a*b^2*c)/x)*x^2)/(96*a^4) + (3
5*b^4*c^2*ArcTanh[Sqrt[a + b*Sqrt[c/x]]/Sqrt[a]])/(32*a^(9/2))

________________________________________________________________________________________

fricas [A]  time = 1.48, size = 224, normalized size = 1.30 \begin {gather*} \left [\frac {105 \, \sqrt {a} b^{4} c^{2} \log \left (2 \, \sqrt {b \sqrt {\frac {c}{x}} + a} \sqrt {a} x \sqrt {\frac {c}{x}} + 2 \, a x \sqrt {\frac {c}{x}} + b c\right ) + 2 \, {\left (70 \, a^{2} b^{2} c x + 48 \, a^{4} x^{2} - 7 \, {\left (15 \, a b^{3} c x + 8 \, a^{3} b x^{2}\right )} \sqrt {\frac {c}{x}}\right )} \sqrt {b \sqrt {\frac {c}{x}} + a}}{192 \, a^{5}}, -\frac {105 \, \sqrt {-a} b^{4} c^{2} \arctan \left (\frac {\sqrt {b \sqrt {\frac {c}{x}} + a} \sqrt {-a}}{a}\right ) - {\left (70 \, a^{2} b^{2} c x + 48 \, a^{4} x^{2} - 7 \, {\left (15 \, a b^{3} c x + 8 \, a^{3} b x^{2}\right )} \sqrt {\frac {c}{x}}\right )} \sqrt {b \sqrt {\frac {c}{x}} + a}}{96 \, a^{5}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/192*(105*sqrt(a)*b^4*c^2*log(2*sqrt(b*sqrt(c/x) + a)*sqrt(a)*x*sqrt(c/x) + 2*a*x*sqrt(c/x) + b*c) + 2*(70*a
^2*b^2*c*x + 48*a^4*x^2 - 7*(15*a*b^3*c*x + 8*a^3*b*x^2)*sqrt(c/x))*sqrt(b*sqrt(c/x) + a))/a^5, -1/96*(105*sqr
t(-a)*b^4*c^2*arctan(sqrt(b*sqrt(c/x) + a)*sqrt(-a)/a) - (70*a^2*b^2*c*x + 48*a^4*x^2 - 7*(15*a*b^3*c*x + 8*a^
3*b*x^2)*sqrt(c/x))*sqrt(b*sqrt(c/x) + a))/a^5]

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x}{\sqrt {b \sqrt {\frac {c}{x}} + a}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x/sqrt(b*sqrt(c/x) + a), x)

________________________________________________________________________________________

maple [B]  time = 0.05, size = 298, normalized size = 1.73 \begin {gather*} -\frac {\sqrt {a +\sqrt {\frac {c}{x}}\, b}\, \left (-192 a \,b^{4} c^{2} \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 )+87 a \,b^{4} c^{2} \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 )-348 \sqrt {a x +\sqrt {\frac {c}{x}}\, b x}\, a^{\frac {5}{2}} b^{2} c \sqrt {x}+384 \sqrt {\left (a +\sqrt {\frac {c}{x}}\, b \right ) x}\, \left (\frac {c}{x}\right )^{\frac {3}{2}} a^{\frac {3}{2}} b^{3} x^{\frac {3}{2}}-174 \sqrt {a x +\sqrt {\frac {c}{x}}\, b x}\, \left (\frac {c}{x}\right )^{\frac {3}{2}} a^{\frac {3}{2}} b^{3} x^{\frac {3}{2}}-96 \left (a x +\sqrt {\frac {c}{x}}\, b x \right )^{\frac {3}{2}} a^{\frac {7}{2}} \sqrt {x}+208 \left (a x +\sqrt {\frac {c}{x}}\, b x \right )^{\frac {3}{2}} \sqrt {\frac {c}{x}}\, a^{\frac {5}{2}} b \sqrt {x}\right ) \sqrt {x}}{192 \sqrt {\left (a +\sqrt {\frac {c}{x}}\, b \right ) x}\, a^{\frac {11}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/192*(a+(c/x)^(1/2)*b)^(1/2)*x^(1/2)*(384*((a+(c/x)^(1/2)*b)*x)^(1/2)*a^(3/2)*(c/x)^(3/2)*x^(3/2)*b^3-174*(a
*x+(c/x)^(1/2)*b*x)^(1/2)*(c/x)^(3/2)*a^(3/2)*b^3*x^(3/2)+208*(a*x+(c/x)^(1/2)*b*x)^(3/2)*(c/x)^(1/2)*a^(5/2)*
b*x^(1/2)-96*(a*x+(c/x)^(1/2)*b*x)^(3/2)*a^(7/2)*x^(1/2)-348*(a*x+(c/x)^(1/2)*b*x)^(1/2)*a^(5/2)*b^2*c*x^(1/2)
-192*ln(1/2*((c/x)^(1/2)*b*x^(1/2)+2*((a+(c/x)^(1/2)*b)*x)^(1/2)*a^(1/2)+2*a*x^(1/2))/a^(1/2))*c^2*a*b^4+87*a*
b^4*c^2*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)))/((a+(c/x)^(
1/2)*b)*x)^(1/2)/a^(11/2)

________________________________________________________________________________________

maxima [A]  time = 1.34, size = 211, normalized size = 1.23 \begin {gather*} -\frac {1}{192} \, c^{2} {\left (\frac {105 \, b^{4} \log \left (\frac {\sqrt {b \sqrt {\frac {c}{x}} + a} - \sqrt {a}}{\sqrt {b \sqrt {\frac {c}{x}} + a} + \sqrt {a}}\right )}{a^{\frac {9}{2}}} + \frac {2 \, {\left (105 \, {\left (b \sqrt {\frac {c}{x}} + a\right )}^{\frac {7}{2}} b^{4} - 385 \, {\left (b \sqrt {\frac {c}{x}} + a\right )}^{\frac {5}{2}} a b^{4} + 511 \, {\left (b \sqrt {\frac {c}{x}} + a\right )}^{\frac {3}{2}} a^{2} b^{4} - 279 \, \sqrt {b \sqrt {\frac {c}{x}} + a} a^{3} b^{4}\right )}}{{\left (b \sqrt {\frac {c}{x}} + a\right )}^{4} a^{4} - 4 \, {\left (b \sqrt {\frac {c}{x}} + a\right )}^{3} a^{5} + 6 \, {\left (b \sqrt {\frac {c}{x}} + a\right )}^{2} a^{6} - 4 \, {\left (b \sqrt {\frac {c}{x}} + a\right )} a^{7} + a^{8}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/192*c^2*(105*b^4*log((sqrt(b*sqrt(c/x) + a) - sqrt(a))/(sqrt(b*sqrt(c/x) + a) + sqrt(a)))/a^(9/2) + 2*(105*
(b*sqrt(c/x) + a)^(7/2)*b^4 - 385*(b*sqrt(c/x) + a)^(5/2)*a*b^4 + 511*(b*sqrt(c/x) + a)^(3/2)*a^2*b^4 - 279*sq
rt(b*sqrt(c/x) + a)*a^3*b^4)/((b*sqrt(c/x) + a)^4*a^4 - 4*(b*sqrt(c/x) + a)^3*a^5 + 6*(b*sqrt(c/x) + a)^2*a^6
- 4*(b*sqrt(c/x) + a)*a^7 + a^8))

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(a + b*(c/x)^(1/2))^(1/2),x)

[Out]

int(x/(a + b*(c/x)^(1/2))^(1/2), x)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x}{\sqrt {a + b \sqrt {\frac {c}{x}}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x/sqrt(a + b*sqrt(c/x)), x)

________________________________________________________________________________________