3.650 \(\int \frac {1}{x \sqrt {a+b \sqrt {c+d x}}} \, dx\)

Optimal. Leaf size=97 \[ -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+b \sqrt {c+d x}}}{\sqrt {a-b \sqrt {c}}}\right )}{\sqrt {a-b \sqrt {c}}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+b \sqrt {c+d x}}}{\sqrt {a+b \sqrt {c}}}\right )}{\sqrt {a+b \sqrt {c}}} \]

[Out]

-2*arctanh((a+b*(d*x+c)^(1/2))^(1/2)/(a-b*c^(1/2))^(1/2))/(a-b*c^(1/2))^(1/2)-2*arctanh((a+b*(d*x+c)^(1/2))^(1
/2)/(a+b*c^(1/2))^(1/2))/(a+b*c^(1/2))^(1/2)

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Rubi [A]  time = 0.09, antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {371, 1398, 827, 1166, 207} \[ -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+b \sqrt {c+d x}}}{\sqrt {a-b \sqrt {c}}}\right )}{\sqrt {a-b \sqrt {c}}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+b \sqrt {c+d x}}}{\sqrt {a+b \sqrt {c}}}\right )}{\sqrt {a+b \sqrt {c}}} \]

Antiderivative was successfully verified.

[In]

Int[1/(x*Sqrt[a + b*Sqrt[c + d*x]]),x]

[Out]

(-2*ArcTanh[Sqrt[a + b*Sqrt[c + d*x]]/Sqrt[a - b*Sqrt[c]]])/Sqrt[a - b*Sqrt[c]] - (2*ArcTanh[Sqrt[a + b*Sqrt[c
+ d*x]]/Sqrt[a + b*Sqrt[c]]])/Sqrt[a + b*Sqrt[c]]

Rule 207

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

Rule 371

Int[((a_) + (b_.)*(v_)^(n_))^(p_.)*(x_)^(m_.), x_Symbol] :> With[{c = Coefficient[v, x, 0], d = Coefficient[v,
x, 1]}, Dist[1/d^(m + 1), Subst[Int[SimplifyIntegrand[(x - c)^m*(a + b*x^n)^p, x], x], x, v], x] /; NeQ[c, 0]
] /; FreeQ[{a, b, n, p}, x] && LinearQ[v, x] && IntegerQ[m]

Rule 827

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

Rule 1166

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

Rule 1398

Int[((a_) + (c_.)*(x_)^(n2_.))^(p_.)*((d_) + (e_.)*(x_)^(n_))^(q_.), x_Symbol] :> With[{g = Denominator[n]}, D
ist[g, Subst[Int[x^(g - 1)*(d + e*x^(g*n))^q*(a + c*x^(2*g*n))^p, x], x, x^(1/g)], x]] /; FreeQ[{a, c, d, e, p
, q}, x] && EqQ[n2, 2*n] && FractionQ[n]

Rubi steps

\begin {align*} \int \frac {1}{x \sqrt {a+b \sqrt {c+d x}}} \, dx &=\operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b \sqrt {x}} (-c+x)} \, dx,x,c+d x\right )\\ &=2 \operatorname {Subst}\left (\int \frac {x}{\sqrt {a+b x} \left (-c+x^2\right )} \, dx,x,\sqrt {c+d x}\right )\\ &=4 \operatorname {Subst}\left (\int \frac {-a+x^2}{a^2-b^2 c-2 a x^2+x^4} \, dx,x,\sqrt {a+b \sqrt {c+d x}}\right )\\ &=2 \operatorname {Subst}\left (\int \frac {1}{-a-b \sqrt {c}+x^2} \, dx,x,\sqrt {a+b \sqrt {c+d x}}\right )+2 \operatorname {Subst}\left (\int \frac {1}{-a+b \sqrt {c}+x^2} \, dx,x,\sqrt {a+b \sqrt {c+d x}}\right )\\ &=-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+b \sqrt {c+d x}}}{\sqrt {a-b \sqrt {c}}}\right )}{\sqrt {a-b \sqrt {c}}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+b \sqrt {c+d x}}}{\sqrt {a+b \sqrt {c}}}\right )}{\sqrt {a+b \sqrt {c}}}\\ \end {align*}

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Mathematica [A]  time = 0.10, size = 97, normalized size = 1.00 \[ -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+b \sqrt {c+d x}}}{\sqrt {a-b \sqrt {c}}}\right )}{\sqrt {a-b \sqrt {c}}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+b \sqrt {c+d x}}}{\sqrt {a+b \sqrt {c}}}\right )}{\sqrt {a+b \sqrt {c}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*ArcTanh[Sqrt[a + b*Sqrt[c + d*x]]/Sqrt[a - b*Sqrt[c]]])/Sqrt[a - b*Sqrt[c]] - (2*ArcTanh[Sqrt[a + b*Sqrt[c
+ d*x]]/Sqrt[a + b*Sqrt[c]]])/Sqrt[a + b*Sqrt[c]]

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fricas [B]  time = 0.56, size = 743, normalized size = 7.66 \[ \sqrt {-\frac {{\left (b^{2} c - a^{2}\right )} \sqrt {\frac {b^{2} c}{b^{4} c^{2} - 2 \, a^{2} b^{2} c + a^{4}}} + a}{b^{2} c - a^{2}}} \log \left (4 \, {\left ({\left (b^{2} c - a^{2}\right )} \sqrt {\frac {b^{2} c}{b^{4} c^{2} - 2 \, a^{2} b^{2} c + a^{4}}} - a\right )} \sqrt {-\frac {{\left (b^{2} c - a^{2}\right )} \sqrt {\frac {b^{2} c}{b^{4} c^{2} - 2 \, a^{2} b^{2} c + a^{4}}} + a}{b^{2} c - a^{2}}} + 4 \, \sqrt {\sqrt {d x + c} b + a}\right ) - \sqrt {-\frac {{\left (b^{2} c - a^{2}\right )} \sqrt {\frac {b^{2} c}{b^{4} c^{2} - 2 \, a^{2} b^{2} c + a^{4}}} + a}{b^{2} c - a^{2}}} \log \left (-4 \, {\left ({\left (b^{2} c - a^{2}\right )} \sqrt {\frac {b^{2} c}{b^{4} c^{2} - 2 \, a^{2} b^{2} c + a^{4}}} - a\right )} \sqrt {-\frac {{\left (b^{2} c - a^{2}\right )} \sqrt {\frac {b^{2} c}{b^{4} c^{2} - 2 \, a^{2} b^{2} c + a^{4}}} + a}{b^{2} c - a^{2}}} + 4 \, \sqrt {\sqrt {d x + c} b + a}\right ) - \sqrt {\frac {{\left (b^{2} c - a^{2}\right )} \sqrt {\frac {b^{2} c}{b^{4} c^{2} - 2 \, a^{2} b^{2} c + a^{4}}} - a}{b^{2} c - a^{2}}} \log \left (4 \, {\left ({\left (b^{2} c - a^{2}\right )} \sqrt {\frac {b^{2} c}{b^{4} c^{2} - 2 \, a^{2} b^{2} c + a^{4}}} + a\right )} \sqrt {\frac {{\left (b^{2} c - a^{2}\right )} \sqrt {\frac {b^{2} c}{b^{4} c^{2} - 2 \, a^{2} b^{2} c + a^{4}}} - a}{b^{2} c - a^{2}}} + 4 \, \sqrt {\sqrt {d x + c} b + a}\right ) + \sqrt {\frac {{\left (b^{2} c - a^{2}\right )} \sqrt {\frac {b^{2} c}{b^{4} c^{2} - 2 \, a^{2} b^{2} c + a^{4}}} - a}{b^{2} c - a^{2}}} \log \left (-4 \, {\left ({\left (b^{2} c - a^{2}\right )} \sqrt {\frac {b^{2} c}{b^{4} c^{2} - 2 \, a^{2} b^{2} c + a^{4}}} + a\right )} \sqrt {\frac {{\left (b^{2} c - a^{2}\right )} \sqrt {\frac {b^{2} c}{b^{4} c^{2} - 2 \, a^{2} b^{2} c + a^{4}}} - a}{b^{2} c - a^{2}}} + 4 \, \sqrt {\sqrt {d x + c} b + a}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(-((b^2*c - a^2)*sqrt(b^2*c/(b^4*c^2 - 2*a^2*b^2*c + a^4)) + a)/(b^2*c - a^2))*log(4*((b^2*c - a^2)*sqrt(b
^2*c/(b^4*c^2 - 2*a^2*b^2*c + a^4)) - a)*sqrt(-((b^2*c - a^2)*sqrt(b^2*c/(b^4*c^2 - 2*a^2*b^2*c + a^4)) + a)/(
b^2*c - a^2)) + 4*sqrt(sqrt(d*x + c)*b + a)) - sqrt(-((b^2*c - a^2)*sqrt(b^2*c/(b^4*c^2 - 2*a^2*b^2*c + a^4))
+ a)/(b^2*c - a^2))*log(-4*((b^2*c - a^2)*sqrt(b^2*c/(b^4*c^2 - 2*a^2*b^2*c + a^4)) - a)*sqrt(-((b^2*c - a^2)*
sqrt(b^2*c/(b^4*c^2 - 2*a^2*b^2*c + a^4)) + a)/(b^2*c - a^2)) + 4*sqrt(sqrt(d*x + c)*b + a)) - sqrt(((b^2*c -
a^2)*sqrt(b^2*c/(b^4*c^2 - 2*a^2*b^2*c + a^4)) - a)/(b^2*c - a^2))*log(4*((b^2*c - a^2)*sqrt(b^2*c/(b^4*c^2 -
2*a^2*b^2*c + a^4)) + a)*sqrt(((b^2*c - a^2)*sqrt(b^2*c/(b^4*c^2 - 2*a^2*b^2*c + a^4)) - a)/(b^2*c - a^2)) + 4
*sqrt(sqrt(d*x + c)*b + a)) + sqrt(((b^2*c - a^2)*sqrt(b^2*c/(b^4*c^2 - 2*a^2*b^2*c + a^4)) - a)/(b^2*c - a^2)
)*log(-4*((b^2*c - a^2)*sqrt(b^2*c/(b^4*c^2 - 2*a^2*b^2*c + a^4)) + a)*sqrt(((b^2*c - a^2)*sqrt(b^2*c/(b^4*c^2
- 2*a^2*b^2*c + a^4)) - a)/(b^2*c - a^2)) + 4*sqrt(sqrt(d*x + c)*b + a))

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giac [A]  time = 0.50, size = 140, normalized size = 1.44 \[ \frac {2 \, {\left (\frac {{\left (b^{2} \sqrt {c} {\left | b \right |} + a b^{2}\right )} \arctan \left (\frac {\sqrt {\sqrt {d x + c} b + a}}{\sqrt {-a + \sqrt {b^{2} c}}}\right )}{{\left (b \sqrt {c} + a\right )} \sqrt {b \sqrt {c} - a}} + \frac {{\left (b^{2} \sqrt {c} {\left | b \right |} - a b^{2}\right )} \arctan \left (\frac {\sqrt {\sqrt {d x + c} b + a}}{\sqrt {-a - \sqrt {b^{2} c}}}\right )}{{\left (b \sqrt {c} - a\right )} \sqrt {-b \sqrt {c} - a}}\right )}}{b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*((b^2*sqrt(c)*abs(b) + a*b^2)*arctan(sqrt(sqrt(d*x + c)*b + a)/sqrt(-a + sqrt(b^2*c)))/((b*sqrt(c) + a)*sqrt
(b*sqrt(c) - a)) + (b^2*sqrt(c)*abs(b) - a*b^2)*arctan(sqrt(sqrt(d*x + c)*b + a)/sqrt(-a - sqrt(b^2*c)))/((b*s
qrt(c) - a)*sqrt(-b*sqrt(c) - a)))/b^2

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maple [A]  time = 0.02, size = 92, normalized size = 0.95 \[ \frac {2 \arctan \left (\frac {\sqrt {a +\sqrt {d x +c}\, b}}{\sqrt {-a -\sqrt {b^{2} c}}}\right )}{\sqrt {-a -\sqrt {b^{2} c}}}+\frac {2 \arctan \left (\frac {\sqrt {a +\sqrt {d x +c}\, b}}{\sqrt {-a +\sqrt {b^{2} c}}}\right )}{\sqrt {-a +\sqrt {b^{2} c}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/(-a-(b^2*c)^(1/2))^(1/2)*arctan((a+(d*x+c)^(1/2)*b)^(1/2)/(-a-(b^2*c)^(1/2))^(1/2))+2/(-a+(b^2*c)^(1/2))^(1/
2)*arctan((a+(d*x+c)^(1/2)*b)^(1/2)/(-a+(b^2*c)^(1/2))^(1/2))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {\sqrt {d x + c} b + a} x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/(x*sqrt(a + b*sqrt(c + d*x))), x)

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