3.31.62 \(\int \frac {x}{\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}} \, dx\) [3062]

Optimal. Leaf size=248 \[ -\frac {7 b d^2 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{12 a^2 \left (\frac {d}{x}\right )^{3/2}}+\frac {5 b d \left (44 a c-21 b^2 d\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{96 a^4 \sqrt {\frac {d}{x}}}-\frac {\left (36 a c-35 b^2 d\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} x}{48 a^3}+\frac {\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} x^2}{2 a}+\frac {\left (48 a^2 c^2-120 a b^2 c d+35 b^4 d^2\right ) \tanh ^{-1}\left (\frac {2 a+b \sqrt {\frac {d}{x}}}{2 \sqrt {a} \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}\right )}{64 a^{9/2}} \]

[Out]

1/64*(35*b^4*d^2-120*a*b^2*c*d+48*a^2*c^2)*arctanh(1/2*(2*a+b*(d/x)^(1/2))/a^(1/2)/(a+c/x+b*(d/x)^(1/2))^(1/2)
)/a^(9/2)-7/12*b*d^2*(a+c/x+b*(d/x)^(1/2))^(1/2)/a^2/(d/x)^(3/2)-1/48*(-35*b^2*d+36*a*c)*x*(a+c/x+b*(d/x)^(1/2
))^(1/2)/a^3+1/2*x^2*(a+c/x+b*(d/x)^(1/2))^(1/2)/a+5/96*b*d*(-21*b^2*d+44*a*c)*(a+c/x+b*(d/x)^(1/2))^(1/2)/a^4
/(d/x)^(1/2)

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Rubi [A]
time = 0.31, antiderivative size = 248, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {1994, 1371, 758, 848, 820, 738, 212} \begin {gather*} \frac {5 b d \left (44 a c-21 b^2 d\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{96 a^4 \sqrt {\frac {d}{x}}}-\frac {x \left (36 a c-35 b^2 d\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{48 a^3}-\frac {7 b d^2 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{12 a^2 \left (\frac {d}{x}\right )^{3/2}}+\frac {\left (48 a^2 c^2-120 a b^2 c d+35 b^4 d^2\right ) \tanh ^{-1}\left (\frac {2 a+b \sqrt {\frac {d}{x}}}{2 \sqrt {a} \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}\right )}{64 a^{9/2}}+\frac {x^2 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{2 a} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x/Sqrt[a + b*Sqrt[d/x] + c/x],x]

[Out]

(-7*b*d^2*Sqrt[a + b*Sqrt[d/x] + c/x])/(12*a^2*(d/x)^(3/2)) + (5*b*d*(44*a*c - 21*b^2*d)*Sqrt[a + b*Sqrt[d/x]
+ c/x])/(96*a^4*Sqrt[d/x]) - ((36*a*c - 35*b^2*d)*Sqrt[a + b*Sqrt[d/x] + c/x]*x)/(48*a^3) + (Sqrt[a + b*Sqrt[d
/x] + c/x]*x^2)/(2*a) + ((48*a^2*c^2 - 120*a*b^2*c*d + 35*b^4*d^2)*ArcTanh[(2*a + b*Sqrt[d/x])/(2*Sqrt[a]*Sqrt
[a + b*Sqrt[d/x] + c/x])])/(64*a^(9/2))

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 738

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[-2, Subst[Int[1/(4*c*d
^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, (2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a,
b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[2*c*d - b*e, 0]

Rule 758

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[e*(d + e*x)^(m + 1)*
((a + b*x + c*x^2)^(p + 1)/((m + 1)*(c*d^2 - b*d*e + a*e^2))), x] + Dist[1/((m + 1)*(c*d^2 - b*d*e + a*e^2)),
Int[(d + e*x)^(m + 1)*Simp[c*d*(m + 1) - b*e*(m + p + 2) - c*e*(m + 2*p + 3)*x, x]*(a + b*x + c*x^2)^p, x], x]
/; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e
, 0] && NeQ[m, -1] && ((LtQ[m, -1] && IntQuadraticQ[a, b, c, d, e, m, p, x]) || (SumSimplerQ[m, 1] && IntegerQ
[p]) || ILtQ[Simplify[m + 2*p + 3], 0])

Rule 820

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[(-(e*f - d*g))*(d + e*x)^(m + 1)*((a + b*x + c*x^2)^(p + 1)/(2*(p + 1)*(c*d^2 - b*d*e + a*e^2))), x] - Dist[
(b*(e*f + d*g) - 2*(c*d*f + a*e*g))/(2*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x]
, x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && EqQ[S
implify[m + 2*p + 3], 0]

Rule 848

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[(e*f - d*g)*(d + e*x)^(m + 1)*((a + b*x + c*x^2)^(p + 1)/((m + 1)*(c*d^2 - b*d*e + a*e^2))), x] + Dist[1/((m
+ 1)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p*Simp[(c*d*f - f*b*e + a*e*g)*(m + 1)
+ b*(d*g - e*f)*(p + 1) - c*(e*f - d*g)*(m + 2*p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] &&
NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[m, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ
[2*m, 2*p])

Rule 1371

Int[(x_)^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplif
y[(m + 1)/n] - 1)*(a + b*x + c*x^2)^p, x], x, x^n], x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[n2, 2*n] && NeQ[
b^2 - 4*a*c, 0] && IntegerQ[Simplify[(m + 1)/n]]

Rule 1994

Int[(x_)^(m_.)*((a_) + (b_.)*((d_.)/(x_))^(n_) + (c_.)*(x_)^(n2_.))^(p_), x_Symbol] :> Dist[-d^(m + 1), Subst[
Int[(a + b*x^n + (c/d^(2*n))*x^(2*n))^p/x^(m + 2), x], x, d/x], x] /; FreeQ[{a, b, c, d, n, p}, x] && EqQ[n2,
-2*n] && IntegerQ[2*n] && IntegerQ[m]

Rubi steps

\begin {align*} \int \frac {x}{\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}} \, dx &=-\left (d^2 \text {Subst}\left (\int \frac {1}{x^3 \sqrt {a+b \sqrt {x}+\frac {c x}{d}}} \, dx,x,\frac {d}{x}\right )\right )\\ &=-\left (\left (2 d^2\right ) \text {Subst}\left (\int \frac {1}{x^5 \sqrt {a+b x+\frac {c x^2}{d}}} \, dx,x,\sqrt {\frac {d}{x}}\right )\right )\\ &=\frac {\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} x^2}{2 a}+\frac {d^2 \text {Subst}\left (\int \frac {\frac {7 b}{2}+\frac {3 c x}{d}}{x^4 \sqrt {a+b x+\frac {c x^2}{d}}} \, dx,x,\sqrt {\frac {d}{x}}\right )}{2 a}\\ &=-\frac {7 b d^2 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{12 a^2 \left (\frac {d}{x}\right )^{3/2}}+\frac {\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} x^2}{2 a}-\frac {d^2 \text {Subst}\left (\int \frac {\frac {1}{4} \left (35 b^2-\frac {36 a c}{d}\right )+\frac {7 b c x}{d}}{x^3 \sqrt {a+b x+\frac {c x^2}{d}}} \, dx,x,\sqrt {\frac {d}{x}}\right )}{6 a^2}\\ &=-\frac {7 b d^2 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{12 a^2 \left (\frac {d}{x}\right )^{3/2}}-\frac {\left (36 a c-35 b^2 d\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} x}{48 a^3}+\frac {\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} x^2}{2 a}+\frac {d^2 \text {Subst}\left (\int \frac {-\frac {5 b \left (44 a c-21 b^2 d\right )}{8 d}-\frac {c \left (36 a c-35 b^2 d\right ) x}{4 d^2}}{x^2 \sqrt {a+b x+\frac {c x^2}{d}}} \, dx,x,\sqrt {\frac {d}{x}}\right )}{12 a^3}\\ &=-\frac {7 b d^2 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{12 a^2 \left (\frac {d}{x}\right )^{3/2}}+\frac {5 b d \left (44 a c-21 b^2 d\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{96 a^4 \sqrt {\frac {d}{x}}}-\frac {\left (36 a c-35 b^2 d\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} x}{48 a^3}+\frac {\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} x^2}{2 a}-\frac {\left (48 a^2 c^2-120 a b^2 c d+35 b^4 d^2\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x+\frac {c x^2}{d}}} \, dx,x,\sqrt {\frac {d}{x}}\right )}{64 a^4}\\ &=-\frac {7 b d^2 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{12 a^2 \left (\frac {d}{x}\right )^{3/2}}+\frac {5 b d \left (44 a c-21 b^2 d\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{96 a^4 \sqrt {\frac {d}{x}}}-\frac {\left (36 a c-35 b^2 d\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} x}{48 a^3}+\frac {\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} x^2}{2 a}+\frac {\left (48 a^2 c^2-120 a b^2 c d+35 b^4 d^2\right ) \text {Subst}\left (\int \frac {1}{4 a-x^2} \, dx,x,\frac {2 a+b \sqrt {\frac {d}{x}}}{\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}\right )}{32 a^4}\\ &=-\frac {7 b d^2 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{12 a^2 \left (\frac {d}{x}\right )^{3/2}}+\frac {5 b d \left (44 a c-21 b^2 d\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{96 a^4 \sqrt {\frac {d}{x}}}-\frac {\left (36 a c-35 b^2 d\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} x}{48 a^3}+\frac {\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} x^2}{2 a}+\frac {\left (48 a^2 c^2-120 a b^2 c d+35 b^4 d^2\right ) \tanh ^{-1}\left (\frac {2 a+b \sqrt {\frac {d}{x}}}{2 \sqrt {a} \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}\right )}{64 a^{9/2}}\\ \end {align*}

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Mathematica [A]
time = 1.11, size = 279, normalized size = 1.12 \begin {gather*} \frac {\sqrt {a} d \left (-105 b^3 d \left (b d+c \sqrt {\frac {d}{x}}\right )+48 a^4 x^2-8 a^3 x \left (3 c+b \sqrt {\frac {d}{x}} x\right )+a^2 \left (-72 c^2+14 b^2 d x+92 b c \sqrt {\frac {d}{x}} x\right )-5 a b \left (-58 b c d-44 c^2 \sqrt {\frac {d}{x}}+7 b^2 d \sqrt {\frac {d}{x}} x\right )\right )-3 \sqrt {d} \left (48 a^2 c^2-120 a b^2 c d+35 b^4 d^2\right ) \sqrt {\frac {d \left (c+\left (a+b \sqrt {\frac {d}{x}}\right ) x\right )}{x}} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {\frac {d}{x}}-\sqrt {\frac {d \left (c+a x+b \sqrt {\frac {d}{x}} x\right )}{x}}}{\sqrt {a} \sqrt {d}}\right )}{96 a^{9/2} d \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[a]*d*(-105*b^3*d*(b*d + c*Sqrt[d/x]) + 48*a^4*x^2 - 8*a^3*x*(3*c + b*Sqrt[d/x]*x) + a^2*(-72*c^2 + 14*b^
2*d*x + 92*b*c*Sqrt[d/x]*x) - 5*a*b*(-58*b*c*d - 44*c^2*Sqrt[d/x] + 7*b^2*d*Sqrt[d/x]*x)) - 3*Sqrt[d]*(48*a^2*
c^2 - 120*a*b^2*c*d + 35*b^4*d^2)*Sqrt[(d*(c + (a + b*Sqrt[d/x])*x))/x]*ArcTanh[(Sqrt[c]*Sqrt[d/x] - Sqrt[(d*(
c + a*x + b*Sqrt[d/x]*x))/x])/(Sqrt[a]*Sqrt[d])])/(96*a^(9/2)*d*Sqrt[a + b*Sqrt[d/x] + c/x])

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Maple [A]
time = 0.07, size = 398, normalized size = 1.60

method	result	size
default	\(-\frac {\sqrt {\frac {b \sqrt {\frac {d}{x}}\, x +a x +c}{x}}\, \sqrt {x}\, \left (-105 d^{2} \ln \left (\frac {b \sqrt {\frac {d}{x}}\, \sqrt {x}+2 \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}\, \sqrt {a}+2 a \sqrt {x}}{2 \sqrt {a}}\right ) a \,b^{4}+210 \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}\, a^{\frac {3}{2}} \left (\frac {d}{x}\right )^{\frac {3}{2}} x^{\frac {3}{2}} b^{3}-140 \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}\, d \,a^{\frac {5}{2}} \sqrt {x}\, b^{2}+112 \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}\, a^{\frac {7}{2}} \sqrt {\frac {d}{x}}\, x^{\frac {3}{2}} b -96 x^{\frac {3}{2}} \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}\, a^{\frac {9}{2}}+360 d \ln \left (\frac {b \sqrt {\frac {d}{x}}\, \sqrt {x}+2 \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}\, \sqrt {a}+2 a \sqrt {x}}{2 \sqrt {a}}\right ) a^{2} b^{2} c -440 \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}\, a^{\frac {5}{2}} \sqrt {\frac {d}{x}}\, \sqrt {x}\, b c +144 \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}\, a^{\frac {7}{2}} c \sqrt {x}-144 \ln \left (\frac {b \sqrt {\frac {d}{x}}\, \sqrt {x}+2 \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}\, \sqrt {a}+2 a \sqrt {x}}{2 \sqrt {a}}\right ) a^{3} c^{2}\right )}{192 \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}\, a^{\frac {11}{2}}}\)	\(398\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(a+c/x+b*(d/x)^(1/2))^(1/2),x,method=_RETURNVERBOSE)

[Out]

-1/192*((b*(d/x)^(1/2)*x+a*x+c)/x)^(1/2)*x^(1/2)*(-105*d^2*ln(1/2*(b*(d/x)^(1/2)*x^(1/2)+2*(b*(d/x)^(1/2)*x+a*
x+c)^(1/2)*a^(1/2)+2*a*x^(1/2))/a^(1/2))*a*b^4+210*(b*(d/x)^(1/2)*x+a*x+c)^(1/2)*a^(3/2)*(d/x)^(3/2)*x^(3/2)*b
^3-140*(b*(d/x)^(1/2)*x+a*x+c)^(1/2)*d*a^(5/2)*x^(1/2)*b^2+112*(b*(d/x)^(1/2)*x+a*x+c)^(1/2)*a^(7/2)*(d/x)^(1/
2)*x^(3/2)*b-96*x^(3/2)*(b*(d/x)^(1/2)*x+a*x+c)^(1/2)*a^(9/2)+360*d*ln(1/2*(b*(d/x)^(1/2)*x^(1/2)+2*(b*(d/x)^(
1/2)*x+a*x+c)^(1/2)*a^(1/2)+2*a*x^(1/2))/a^(1/2))*a^2*b^2*c-440*(b*(d/x)^(1/2)*x+a*x+c)^(1/2)*a^(5/2)*(d/x)^(1
/2)*x^(1/2)*b*c+144*(b*(d/x)^(1/2)*x+a*x+c)^(1/2)*a^(7/2)*c*x^(1/2)-144*ln(1/2*(b*(d/x)^(1/2)*x^(1/2)+2*(b*(d/
x)^(1/2)*x+a*x+c)^(1/2)*a^(1/2)+2*a*x^(1/2))/a^(1/2))*a^3*c^2)/(b*(d/x)^(1/2)*x+a*x+c)^(1/2)/a^(11/2)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x/sqrt(b*sqrt(d/x) + a + c/x), x)

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Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x/sqrt(a + b*sqrt(d/x) + c/x), x)

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Giac [A]
time = 1.93, size = 363, normalized size = 1.46 \begin {gather*} -\frac {2 \, \sqrt {a d^{2} x + \sqrt {d x} b d^{2} + c d^{2}} {\left (2 \, \sqrt {d x} {\left (4 \, \sqrt {d x} {\left (\frac {7 \, b}{a^{2}} - \frac {6 \, \sqrt {d x}}{a d}\right )} - \frac {35 \, a b^{2} d^{2} - 36 \, a^{2} c d}{a^{4} d}\right )} + \frac {5 \, {\left (21 \, b^{3} d^{3} - 44 \, a b c d^{2}\right )}}{a^{4} d}\right )} + \frac {3 \, {\left (35 \, b^{4} d^{4} - 120 \, a b^{2} c d^{3} + 48 \, a^{2} c^{2} d^{2}\right )} \log \left ({\left | -b d^{2} - 2 \, \sqrt {a d} {\left (\sqrt {a d} \sqrt {d x} - \sqrt {a d^{2} x + \sqrt {d x} b d^{2} + c d^{2}}\right )} \right |}\right )}{\sqrt {a d} a^{4}} - \frac {105 \, b^{4} d^{4} \log \left ({\left | -b d^{2} + 2 \, \sqrt {c d^{2}} \sqrt {a d} \right |}\right ) - 360 \, a b^{2} c d^{3} \log \left ({\left | -b d^{2} + 2 \, \sqrt {c d^{2}} \sqrt {a d} \right |}\right ) + 144 \, a^{2} c^{2} d^{2} \log \left ({\left | -b d^{2} + 2 \, \sqrt {c d^{2}} \sqrt {a d} \right |}\right ) + 210 \, \sqrt {c d^{2}} \sqrt {a d} b^{3} d^{2} - 440 \, \sqrt {c d^{2}} \sqrt {a d} a b c d}{\sqrt {a d} a^{4}}}{192 \, d^{\frac {3}{2}} \mathrm {sgn}\left (x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/192*(2*sqrt(a*d^2*x + sqrt(d*x)*b*d^2 + c*d^2)*(2*sqrt(d*x)*(4*sqrt(d*x)*(7*b/a^2 - 6*sqrt(d*x)/(a*d)) - (3
5*a*b^2*d^2 - 36*a^2*c*d)/(a^4*d)) + 5*(21*b^3*d^3 - 44*a*b*c*d^2)/(a^4*d)) + 3*(35*b^4*d^4 - 120*a*b^2*c*d^3
+ 48*a^2*c^2*d^2)*log(abs(-b*d^2 - 2*sqrt(a*d)*(sqrt(a*d)*sqrt(d*x) - sqrt(a*d^2*x + sqrt(d*x)*b*d^2 + c*d^2))
))/(sqrt(a*d)*a^4) - (105*b^4*d^4*log(abs(-b*d^2 + 2*sqrt(c*d^2)*sqrt(a*d))) - 360*a*b^2*c*d^3*log(abs(-b*d^2
+ 2*sqrt(c*d^2)*sqrt(a*d))) + 144*a^2*c^2*d^2*log(abs(-b*d^2 + 2*sqrt(c*d^2)*sqrt(a*d))) + 210*sqrt(c*d^2)*sqr
t(a*d)*b^3*d^2 - 440*sqrt(c*d^2)*sqrt(a*d)*a*b*c*d)/(sqrt(a*d)*a^4))/(d^(3/2)*sgn(x))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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