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

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

[Out]

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

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Rubi [A]  time = 0.10, antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {2104, 50, 63, 208} \[ \frac {2 \sqrt {a+b x}}{a-c}-\frac {2 \sqrt {b x+c}}{a-c}-\frac {2 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{a-c}+\frac {2 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {b x+c}}{\sqrt {c}}\right )}{a-c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && 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 2104

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

Rubi steps

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

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Mathematica [A]  time = 0.07, size = 75, normalized size = 0.77 \[ \frac {2 \left (\sqrt {a+b x}-\sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )-\sqrt {b x+c}+\sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {b x+c}}{\sqrt {c}}\right )\right )}{a-c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.52, size = 318, normalized size = 3.28 \[ \left [-\frac {\sqrt {a} \log \left (\frac {b x + 2 \, \sqrt {b x + a} \sqrt {a} + 2 \, a}{x}\right ) + \sqrt {c} \log \left (\frac {b x - 2 \, \sqrt {b x + c} \sqrt {c} + 2 \, c}{x}\right ) - 2 \, \sqrt {b x + a} + 2 \, \sqrt {b x + c}}{a - c}, -\frac {2 \, \sqrt {-c} \arctan \left (\frac {\sqrt {b x + c} \sqrt {-c}}{c}\right ) + \sqrt {a} \log \left (\frac {b x + 2 \, \sqrt {b x + a} \sqrt {a} + 2 \, a}{x}\right ) - 2 \, \sqrt {b x + a} + 2 \, \sqrt {b x + c}}{a - c}, \frac {2 \, \sqrt {-a} \arctan \left (\frac {\sqrt {b x + a} \sqrt {-a}}{a}\right ) - \sqrt {c} \log \left (\frac {b x - 2 \, \sqrt {b x + c} \sqrt {c} + 2 \, c}{x}\right ) + 2 \, \sqrt {b x + a} - 2 \, \sqrt {b x + c}}{a - c}, \frac {2 \, {\left (\sqrt {-a} \arctan \left (\frac {\sqrt {b x + a} \sqrt {-a}}{a}\right ) - \sqrt {-c} \arctan \left (\frac {\sqrt {b x + c} \sqrt {-c}}{c}\right ) + \sqrt {b x + a} - \sqrt {b x + c}\right )}}{a - c}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-(sqrt(a)*log((b*x + 2*sqrt(b*x + a)*sqrt(a) + 2*a)/x) + sqrt(c)*log((b*x - 2*sqrt(b*x + c)*sqrt(c) + 2*c)/x)
- 2*sqrt(b*x + a) + 2*sqrt(b*x + c))/(a - c), -(2*sqrt(-c)*arctan(sqrt(b*x + c)*sqrt(-c)/c) + sqrt(a)*log((b*
x + 2*sqrt(b*x + a)*sqrt(a) + 2*a)/x) - 2*sqrt(b*x + a) + 2*sqrt(b*x + c))/(a - c), (2*sqrt(-a)*arctan(sqrt(b*
x + a)*sqrt(-a)/a) - sqrt(c)*log((b*x - 2*sqrt(b*x + c)*sqrt(c) + 2*c)/x) + 2*sqrt(b*x + a) - 2*sqrt(b*x + c))
/(a - c), 2*(sqrt(-a)*arctan(sqrt(b*x + a)*sqrt(-a)/a) - sqrt(-c)*arctan(sqrt(b*x + c)*sqrt(-c)/c) + sqrt(b*x
+ a) - sqrt(b*x + c))/(a - c)]

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giac [B]  time = 0.86, size = 1016, normalized size = 10.47 \[ \frac {2 \, a \arctan \left (\frac {\sqrt {b x + a}}{\sqrt {-a}}\right )}{\sqrt {-a} {\left (a - c\right )}} - \frac {2 \, {\left (a^{4} c - a^{3} c^{2} - a^{2} c^{3} + a c^{4} + 2 \, {\left (a c^{2} + \sqrt {a c} c^{2}\right )} {\left (a - c\right )}^{2} \mathrm {sgn}\left (-a + c\right ) - 2 \, {\left (a c^{2} + \sqrt {a c} a c\right )} {\left (a - c\right )}^{2} + {\left (a^{2} c^{2} - 2 \, a c^{3} + c^{4} - {\left (a^{2} c - 2 \, a c^{2} + c^{3}\right )} \sqrt {a c}\right )} {\left | -a + c \right |} \mathrm {sgn}\left (-a + c\right ) - {\left (a^{3} c - 2 \, a^{2} c^{2} + a c^{3} + {\left (a^{2} c - 2 \, a c^{2} + c^{3}\right )} \sqrt {a c}\right )} {\left | -a + c \right |} - {\left (a^{4} c - a^{3} c^{2} - a^{2} c^{3} + a c^{4} + {\left (a^{3} c - a^{2} c^{2} - a c^{3} + c^{4}\right )} \sqrt {a c}\right )} \mathrm {sgn}\left (-a + c\right ) + {\left (a^{4} - a^{3} c - a^{2} c^{2} + a c^{3}\right )} \sqrt {a c}\right )} \arctan \left (-\frac {\sqrt {b x + a} - \sqrt {b x + c}}{\sqrt {-\frac {a^{2} - c^{2} + \sqrt {{\left (a^{2} - c^{2}\right )}^{2} - {\left (a^{3} - 3 \, a^{2} c + 3 \, a c^{2} - c^{3}\right )} {\left (a - c\right )}}}{a - c}}}\right )}{{\left (\sqrt {-a} a^{4} - a^{4} \sqrt {-c} - 4 \, \sqrt {-a} a^{3} c + 4 \, a^{3} \sqrt {-c} c + 6 \, \sqrt {-a} a^{2} c^{2} - 6 \, a^{2} \sqrt {-c} c^{2} - 4 \, \sqrt {-a} a c^{3} + 4 \, a \sqrt {-c} c^{3} + \sqrt {-a} c^{4} - \sqrt {-c} c^{4}\right )} {\left | -a + c \right |}} + \frac {2 \, {\left (a^{4} c - a^{3} c^{2} - a^{2} c^{3} + a c^{4} - 2 \, {\left (a c^{2} + \sqrt {a c} c^{2}\right )} {\left (a - c\right )}^{2} \mathrm {sgn}\left (-a + c\right ) - 2 \, {\left (a c^{2} - \sqrt {a c} a c\right )} {\left (a - c\right )}^{2} + {\left (a^{2} c^{2} - 2 \, a c^{3} + c^{4} - {\left (a^{2} c - 2 \, a c^{2} + c^{3}\right )} \sqrt {a c}\right )} {\left | -a + c \right |} \mathrm {sgn}\left (-a + c\right ) + {\left (a^{3} c - 2 \, a^{2} c^{2} + a c^{3} + {\left (a^{2} c - 2 \, a c^{2} + c^{3}\right )} \sqrt {a c}\right )} {\left | -a + c \right |} + {\left (a^{4} c - a^{3} c^{2} - a^{2} c^{3} + a c^{4} - {\left (a^{3} c - a^{2} c^{2} - a c^{3} + c^{4}\right )} \sqrt {a c}\right )} \mathrm {sgn}\left (-a + c\right ) + {\left (a^{4} - a^{3} c - a^{2} c^{2} + a c^{3}\right )} \sqrt {a c}\right )} \arctan \left (-\frac {\sqrt {b x + a} - \sqrt {b x + c}}{\sqrt {-\frac {a^{2} - c^{2} - \sqrt {{\left (a^{2} - c^{2}\right )}^{2} - {\left (a^{3} - 3 \, a^{2} c + 3 \, a c^{2} - c^{3}\right )} {\left (a - c\right )}}}{a - c}}}\right )}{{\left (\sqrt {-a} a^{4} - a^{4} \sqrt {-c} - 4 \, \sqrt {-a} a^{3} c + 4 \, a^{3} \sqrt {-c} c + 6 \, \sqrt {-a} a^{2} c^{2} - 6 \, a^{2} \sqrt {-c} c^{2} - 4 \, \sqrt {-a} a c^{3} + 4 \, a \sqrt {-c} c^{3} + \sqrt {-a} c^{4} - \sqrt {-c} c^{4}\right )} {\left | -a + c \right |}} + \frac {2 \, \sqrt {b x + a}}{a - c} - \frac {2 \, \sqrt {b x + c}}{a - c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*a*arctan(sqrt(b*x + a)/sqrt(-a))/(sqrt(-a)*(a - c)) - 2*(a^4*c - a^3*c^2 - a^2*c^3 + a*c^4 + 2*(a*c^2 + sqrt
(a*c)*c^2)*(a - c)^2*sgn(-a + c) - 2*(a*c^2 + sqrt(a*c)*a*c)*(a - c)^2 + (a^2*c^2 - 2*a*c^3 + c^4 - (a^2*c - 2
*a*c^2 + c^3)*sqrt(a*c))*abs(-a + c)*sgn(-a + c) - (a^3*c - 2*a^2*c^2 + a*c^3 + (a^2*c - 2*a*c^2 + c^3)*sqrt(a
*c))*abs(-a + c) - (a^4*c - a^3*c^2 - a^2*c^3 + a*c^4 + (a^3*c - a^2*c^2 - a*c^3 + c^4)*sqrt(a*c))*sgn(-a + c)
+ (a^4 - a^3*c - a^2*c^2 + a*c^3)*sqrt(a*c))*arctan(-(sqrt(b*x + a) - sqrt(b*x + c))/sqrt(-(a^2 - c^2 + sqrt(
(a^2 - c^2)^2 - (a^3 - 3*a^2*c + 3*a*c^2 - c^3)*(a - c)))/(a - c)))/((sqrt(-a)*a^4 - a^4*sqrt(-c) - 4*sqrt(-a)
*a^3*c + 4*a^3*sqrt(-c)*c + 6*sqrt(-a)*a^2*c^2 - 6*a^2*sqrt(-c)*c^2 - 4*sqrt(-a)*a*c^3 + 4*a*sqrt(-c)*c^3 + sq
rt(-a)*c^4 - sqrt(-c)*c^4)*abs(-a + c)) + 2*(a^4*c - a^3*c^2 - a^2*c^3 + a*c^4 - 2*(a*c^2 + sqrt(a*c)*c^2)*(a
- c)^2*sgn(-a + c) - 2*(a*c^2 - sqrt(a*c)*a*c)*(a - c)^2 + (a^2*c^2 - 2*a*c^3 + c^4 - (a^2*c - 2*a*c^2 + c^3)*
sqrt(a*c))*abs(-a + c)*sgn(-a + c) + (a^3*c - 2*a^2*c^2 + a*c^3 + (a^2*c - 2*a*c^2 + c^3)*sqrt(a*c))*abs(-a +
c) + (a^4*c - a^3*c^2 - a^2*c^3 + a*c^4 - (a^3*c - a^2*c^2 - a*c^3 + c^4)*sqrt(a*c))*sgn(-a + c) + (a^4 - a^3*
c - a^2*c^2 + a*c^3)*sqrt(a*c))*arctan(-(sqrt(b*x + a) - sqrt(b*x + c))/sqrt(-(a^2 - c^2 - sqrt((a^2 - c^2)^2
- (a^3 - 3*a^2*c + 3*a*c^2 - c^3)*(a - c)))/(a - c)))/((sqrt(-a)*a^4 - a^4*sqrt(-c) - 4*sqrt(-a)*a^3*c + 4*a^3
*sqrt(-c)*c + 6*sqrt(-a)*a^2*c^2 - 6*a^2*sqrt(-c)*c^2 - 4*sqrt(-a)*a*c^3 + 4*a*sqrt(-c)*c^3 + sqrt(-a)*c^4 - s
qrt(-c)*c^4)*abs(-a + c)) + 2*sqrt(b*x + a)/(a - c) - 2*sqrt(b*x + c)/(a - c)

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maple [A]  time = 0.01, size = 73, normalized size = 0.75 \[ \frac {-2 \sqrt {a}\, \arctanh \left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )+2 \sqrt {b x +a}}{a -c}-\frac {-2 \sqrt {c}\, \arctanh \left (\frac {\sqrt {b x +c}}{\sqrt {c}}\right )+2 \sqrt {b x +c}}{a -c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 18.08, size = 2983, normalized size = 30.75 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(atan((a^2*c^(5/2)*(a*c^3 + a^3*c - 2*a^2*c^2)^(1/2)*2i - a^3*c^(3/2)*(a*c^3 + a^3*c - 2*a^2*c^2)^(1/2)*2i - a
^(7/2)*c*(a*c^3 + a^3*c - 2*a^2*c^2)^(1/2)*2i + a^(5/2)*c^2*(a*c^3 + a^3*c - 2*a^2*c^2)^(1/2)*2i + a*c^3*(a +
b*x)^(1/2)*(a*c^3 + a^3*c - 2*a^2*c^2)^(1/2)*2i + a^3*c*(a + b*x)^(1/2)*(a*c^3 + a^3*c - 2*a^2*c^2)^(1/2)*2i +
a^(3/2)*c^(5/2)*(a + b*x)^(1/2)*(a*c^3 + a^3*c - 2*a^2*c^2)^(1/2)*2i + a^(5/2)*c^(3/2)*(a + b*x)^(1/2)*(a*c^3
+ a^3*c - 2*a^2*c^2)^(1/2)*2i - a^2*c^2*(c + b*x)^(1/2)*(a*c^3 + a^3*c - 2*a^2*c^2)^(1/2)*4i - a^(3/2)*c^(5/2
)*(c + b*x)^(1/2)*(a*c^3 + a^3*c - 2*a^2*c^2)^(1/2)*2i - a^(5/2)*c^(3/2)*(c + b*x)^(1/2)*(a*c^3 + a^3*c - 2*a^
2*c^2)^(1/2)*2i)/(2*a^5*c^(3/2) - 4*a^4*c^(5/2) + 2*a^(5/2)*c^4 + 2*a^3*c^(7/2) - 4*a^(7/2)*c^3 + 2*a^(9/2)*c^
2 - 2*a^2*c^4*(a + b*x)^(1/2) + 4*a^3*c^3*(a + b*x)^(1/2) - 2*a^4*c^2*(a + b*x)^(1/2) - 2*a^(3/2)*c^(9/2)*(a +
b*x)^(1/2) + 2*a^(5/2)*c^(7/2)*(a + b*x)^(1/2) + 2*a^(7/2)*c^(5/2)*(a + b*x)^(1/2) - 2*a^(9/2)*c^(3/2)*(a + b
*x)^(1/2) + 2*a^2*c^4*(c + b*x)^(1/2) - 4*a^3*c^3*(c + b*x)^(1/2) + 2*a^4*c^2*(c + b*x)^(1/2)))*(a + b*x)^(1/2
)*(a*c^3 + a^3*c - 2*a^2*c^2)^(1/2)*2i - 4*a^(3/2)*c - 8*a*c^(3/2) + atan((a^2*c^(5/2)*(a*c^3 + a^3*c - 2*a^2*
c^2)^(1/2)*2i - a^3*c^(3/2)*(a*c^3 + a^3*c - 2*a^2*c^2)^(1/2)*2i - a^(7/2)*c*(a*c^3 + a^3*c - 2*a^2*c^2)^(1/2)
*2i + a^(5/2)*c^2*(a*c^3 + a^3*c - 2*a^2*c^2)^(1/2)*2i + a*c^3*(a + b*x)^(1/2)*(a*c^3 + a^3*c - 2*a^2*c^2)^(1/
2)*2i + a^3*c*(a + b*x)^(1/2)*(a*c^3 + a^3*c - 2*a^2*c^2)^(1/2)*2i + a^(3/2)*c^(5/2)*(a + b*x)^(1/2)*(a*c^3 +
a^3*c - 2*a^2*c^2)^(1/2)*2i + a^(5/2)*c^(3/2)*(a + b*x)^(1/2)*(a*c^3 + a^3*c - 2*a^2*c^2)^(1/2)*2i - a^2*c^2*(
c + b*x)^(1/2)*(a*c^3 + a^3*c - 2*a^2*c^2)^(1/2)*4i - a^(3/2)*c^(5/2)*(c + b*x)^(1/2)*(a*c^3 + a^3*c - 2*a^2*c
^2)^(1/2)*2i - a^(5/2)*c^(3/2)*(c + b*x)^(1/2)*(a*c^3 + a^3*c - 2*a^2*c^2)^(1/2)*2i)/(2*a^5*c^(3/2) - 4*a^4*c^
(5/2) + 2*a^(5/2)*c^4 + 2*a^3*c^(7/2) - 4*a^(7/2)*c^3 + 2*a^(9/2)*c^2 - 2*a^2*c^4*(a + b*x)^(1/2) + 4*a^3*c^3*
(a + b*x)^(1/2) - 2*a^4*c^2*(a + b*x)^(1/2) - 2*a^(3/2)*c^(9/2)*(a + b*x)^(1/2) + 2*a^(5/2)*c^(7/2)*(a + b*x)^
(1/2) + 2*a^(7/2)*c^(5/2)*(a + b*x)^(1/2) - 2*a^(9/2)*c^(3/2)*(a + b*x)^(1/2) + 2*a^2*c^4*(c + b*x)^(1/2) - 4*
a^3*c^3*(c + b*x)^(1/2) + 2*a^4*c^2*(c + b*x)^(1/2)))*(c + b*x)^(1/2)*(a*c^3 + a^3*c - 2*a^2*c^2)^(1/2)*2i + 4
*a^(3/2)*c^(1/2)*(c + b*x)^(1/2) - 3*a*c^(3/2)*log(((a + b*x)^(1/2) - a^(1/2))/((c + b*x)^(1/2) - c^(1/2))) -
3*a^(3/2)*c*log(((a + b*x)^(1/2) - a^(1/2))/((c + b*x)^(1/2) - c^(1/2))) + 8*a*c*(c + b*x)^(1/2) - c^(1/2)*ata
n((a^2*c^(5/2)*(a*c^3 + a^3*c - 2*a^2*c^2)^(1/2)*2i - a^3*c^(3/2)*(a*c^3 + a^3*c - 2*a^2*c^2)^(1/2)*2i - a^(7/
2)*c*(a*c^3 + a^3*c - 2*a^2*c^2)^(1/2)*2i + a^(5/2)*c^2*(a*c^3 + a^3*c - 2*a^2*c^2)^(1/2)*2i + a*c^3*(a + b*x)
^(1/2)*(a*c^3 + a^3*c - 2*a^2*c^2)^(1/2)*2i + a^3*c*(a + b*x)^(1/2)*(a*c^3 + a^3*c - 2*a^2*c^2)^(1/2)*2i + a^(
3/2)*c^(5/2)*(a + b*x)^(1/2)*(a*c^3 + a^3*c - 2*a^2*c^2)^(1/2)*2i + a^(5/2)*c^(3/2)*(a + b*x)^(1/2)*(a*c^3 + a
^3*c - 2*a^2*c^2)^(1/2)*2i - a^2*c^2*(c + b*x)^(1/2)*(a*c^3 + a^3*c - 2*a^2*c^2)^(1/2)*4i - a^(3/2)*c^(5/2)*(c
+ b*x)^(1/2)*(a*c^3 + a^3*c - 2*a^2*c^2)^(1/2)*2i - a^(5/2)*c^(3/2)*(c + b*x)^(1/2)*(a*c^3 + a^3*c - 2*a^2*c^
2)^(1/2)*2i)/(2*a^5*c^(3/2) - 4*a^4*c^(5/2) + 2*a^(5/2)*c^4 + 2*a^3*c^(7/2) - 4*a^(7/2)*c^3 + 2*a^(9/2)*c^2 -
2*a^2*c^4*(a + b*x)^(1/2) + 4*a^3*c^3*(a + b*x)^(1/2) - 2*a^4*c^2*(a + b*x)^(1/2) - 2*a^(3/2)*c^(9/2)*(a + b*x
)^(1/2) + 2*a^(5/2)*c^(7/2)*(a + b*x)^(1/2) + 2*a^(7/2)*c^(5/2)*(a + b*x)^(1/2) - 2*a^(9/2)*c^(3/2)*(a + b*x)^
(1/2) + 2*a^2*c^4*(c + b*x)^(1/2) - 4*a^3*c^3*(c + b*x)^(1/2) + 2*a^4*c^2*(c + b*x)^(1/2)))*(a*c^3 + a^3*c - 2
*a^2*c^2)^(1/2)*2i - a^2*c^(1/2)*log(((a + b*x)^(1/2) - a^(1/2))/((c + b*x)^(1/2) - c^(1/2))) + a^(3/2)*c^(1/2
)*log(((a + b*x)^(1/2) - a^(1/2))/((c + b*x)^(1/2) - c^(1/2)))*(a + b*x)^(1/2) + a^(3/2)*c^(1/2)*log(((a + b*x
)^(1/2) - a^(1/2))/((c + b*x)^(1/2) - c^(1/2)))*(c + b*x)^(1/2) + 2*a*c*log(((a + b*x)^(1/2) - a^(1/2))/((c +
b*x)^(1/2) - c^(1/2)))*(a + b*x)^(1/2) + 2*a*c*log(((a + b*x)^(1/2) - a^(1/2))/((c + b*x)^(1/2) - c^(1/2)))*(c
+ b*x)^(1/2))/(a^(1/2)*c^(1/2)*(a^(1/2) - c^(1/2))*(a^(1/2) + c^(1/2))^2*((a + b*x)^(1/2) + (c + b*x)^(1/2) -
a^(1/2) - c^(1/2))) - (c^2*log(((a + b*x)^(1/2) - a^(1/2))/((c + b*x)^(1/2) - c^(1/2))) - 4*c^(3/2)*(c + b*x)
^(1/2) + 4*c^2 + atan((a^2*c^(5/2)*(a*c^3 + a^3*c - 2*a^2*c^2)^(1/2)*2i - a^3*c^(3/2)*(a*c^3 + a^3*c - 2*a^2*c
^2)^(1/2)*2i - a^(7/2)*c*(a*c^3 + a^3*c - 2*a^2*c^2)^(1/2)*2i + a^(5/2)*c^2*(a*c^3 + a^3*c - 2*a^2*c^2)^(1/2)*
2i + a*c^3*(a + b*x)^(1/2)*(a*c^3 + a^3*c - 2*a^2*c^2)^(1/2)*2i + a^3*c*(a + b*x)^(1/2)*(a*c^3 + a^3*c - 2*a^2
*c^2)^(1/2)*2i + a^(3/2)*c^(5/2)*(a + b*x)^(1/2)*(a*c^3 + a^3*c - 2*a^2*c^2)^(1/2)*2i + a^(5/2)*c^(3/2)*(a + b
*x)^(1/2)*(a*c^3 + a^3*c - 2*a^2*c^2)^(1/2)*2i - a^2*c^2*(c + b*x)^(1/2)*(a*c^3 + a^3*c - 2*a^2*c^2)^(1/2)*4i
- a^(3/2)*c^(5/2)*(c + b*x)^(1/2)*(a*c^3 + a^3*c - 2*a^2*c^2)^(1/2)*2i - a^(5/2)*c^(3/2)*(c + b*x)^(1/2)*(a*c^
3 + a^3*c - 2*a^2*c^2)^(1/2)*2i)/(2*a^5*c^(3/2) - 4*a^4*c^(5/2) + 2*a^(5/2)*c^4 + 2*a^3*c^(7/2) - 4*a^(7/2)*c^
3 + 2*a^(9/2)*c^2 - 2*a^2*c^4*(a + b*x)^(1/2) + 4*a^3*c^3*(a + b*x)^(1/2) - 2*a^4*c^2*(a + b*x)^(1/2) - 2*a^(3
/2)*c^(9/2)*(a + b*x)^(1/2) + 2*a^(5/2)*c^(7/2)*(a + b*x)^(1/2) + 2*a^(7/2)*c^(5/2)*(a + b*x)^(1/2) - 2*a^(9/2
)*c^(3/2)*(a + b*x)^(1/2) + 2*a^2*c^4*(c + b*x)^(1/2) - 4*a^3*c^3*(c + b*x)^(1/2) + 2*a^4*c^2*(c + b*x)^(1/2))
)*(a*c^3 + a^3*c - 2*a^2*c^2)^(1/2)*2i - c^(3/2)*log(((a + b*x)^(1/2) - a^(1/2))/((c + b*x)^(1/2) - c^(1/2)))*
(a + b*x)^(1/2) - c^(3/2)*log(((a + b*x)^(1/2) - a^(1/2))/((c + b*x)^(1/2) - c^(1/2)))*(c + b*x)^(1/2))/(c^(1/
2)*(a^(1/2) - c^(1/2))*(a^(1/2) + c^(1/2))^2*((a + b*x)^(1/2) + (c + b*x)^(1/2) - a^(1/2) - c^(1/2)))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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