3.2988 \(\int \frac{x}{\sqrt{a+b \sqrt{\frac{c}{x}}}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.121423, 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} \[ -\frac{35 b^3 c^2 \sqrt{a+b \sqrt{\frac{c}{x}}}}{32 a^4 \sqrt{\frac{c}{x}}}+\frac{35 b^4 c^2 \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{\frac{c}{x}}}}{\sqrt{a}}\right )}{32 a^{9/2}}+\frac{35 b^2 c x \sqrt{a+b \sqrt{\frac{c}{x}}}}{48 a^3}-\frac{7 b x^3 \left (\frac{c}{x}\right )^{3/2} \sqrt{a+b \sqrt{\frac{c}{x}}}}{12 a^2 c}+\frac{x^2 \sqrt{a+b \sqrt{\frac{c}{x}}}}{2 a} \]

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 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 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]

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 = 0.14556, size = 126, normalized size = 0.73 \[ \frac{14 a^2 b^2 c x-8 a^3 b x^2 \sqrt{\frac{c}{x}}+48 a^4 x^2-35 a b^3 c x \sqrt{\frac{c}{x}}-105 b^4 c^2}{96 a^4 \sqrt{a+b \sqrt{\frac{c}{x}}}}+\frac{35 b^4 c^2 \tanh ^{-1}\left (\frac{\sqrt{a}}{\sqrt{a+b \sqrt{\frac{c}{x}}}}\right )}{32 a^{9/2}} \]

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))

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Maple [B]  time = 0.041, size = 298, normalized size = 1.7 \begin{align*} -{\frac{1}{192}\sqrt{a+b\sqrt{{\frac{c}{x}}}}\sqrt{x} \left ( 384\,{a}^{3/2}\sqrt{x \left ( a+b\sqrt{{\frac{c}{x}}} \right ) } \left ({\frac{c}{x}} \right ) ^{3/2}{x}^{3/2}{b}^{3}-174\,{a}^{3/2}\sqrt{ax+b\sqrt{{\frac{c}{x}}}x} \left ({\frac{c}{x}} \right ) ^{3/2}{x}^{3/2}{b}^{3}-348\,{a}^{5/2}\sqrt{ax+b\sqrt{{\frac{c}{x}}}x}c\sqrt{x}{b}^{2}-192\,\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 ) a{c}^{2}{b}^{4}+87\,\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 ){c}^{2}a{b}^{4}-96\,\sqrt{x} \left ( ax+b\sqrt{{\frac{c}{x}}}x \right ) ^{3/2}{a}^{7/2}+208\,{a}^{5/2} \left ( ax+b\sqrt{{\frac{c}{x}}}x \right ) ^{3/2}\sqrt{{\frac{c}{x}}}\sqrt{x}b \right ){\frac{1}{\sqrt{x \left ( a+b\sqrt{{\frac{c}{x}}} \right ) }}}{a}^{-{\frac{11}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

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]

Exception raised: ValueError

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Fricas [A]  time = 1.49442, size = 514, normalized size = 2.99 \begin{align*} \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{align*}

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]

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

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)

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

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)