3.405 \(\int \frac {1}{x^2 (\sqrt {a+b x}+\sqrt {c+b x})} \, dx\)

Optimal. Leaf size=103 \[ -\frac {\sqrt {a+b x}}{x (a-c)}+\frac {\sqrt {b x+c}}{x (a-c)}-\frac {b \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{\sqrt {a} (a-c)}+\frac {b \tanh ^{-1}\left (\frac {\sqrt {b x+c}}{\sqrt {c}}\right )}{\sqrt {c} (a-c)} \]

[Out]

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

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Rubi [A]  time = 0.10, antiderivative size = 103, 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, 47, 63, 208} \[ -\frac {\sqrt {a+b x}}{x (a-c)}+\frac {\sqrt {b x+c}}{x (a-c)}-\frac {b \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{\sqrt {a} (a-c)}+\frac {b \tanh ^{-1}\left (\frac {\sqrt {b x+c}}{\sqrt {c}}\right )}{\sqrt {c} (a-c)} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^2*(Sqrt[a + b*x] + Sqrt[c + b*x])),x]

[Out]

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

Rule 47

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + 1)), x] - Dist[(d*n)/(b*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d},
x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && LtQ[m, -1] &&  !(IntegerQ[n] &&  !IntegerQ[m]) &&  !(ILeQ[m + n + 2, 0
] && (FractionQ[m] || GeQ[2*n + m + 1, 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^2 \left (\sqrt {a+b x}+\sqrt {c+b x}\right )} \, dx &=-\frac {b \int \frac {\sqrt {a+b x}}{x^2} \, dx}{-a b+b c}+\frac {b \int \frac {\sqrt {c+b x}}{x^2} \, dx}{-a b+b c}\\ &=-\frac {\sqrt {a+b x}}{(a-c) x}+\frac {\sqrt {c+b x}}{(a-c) x}+\frac {b \int \frac {1}{x \sqrt {a+b x}} \, dx}{2 (a-c)}-\frac {b \int \frac {1}{x \sqrt {c+b x}} \, dx}{2 (a-c)}\\ &=-\frac {\sqrt {a+b x}}{(a-c) x}+\frac {\sqrt {c+b x}}{(a-c) x}+\frac {\operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x}\right )}{a-c}-\frac {\operatorname {Subst}\left (\int \frac {1}{-\frac {c}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {c+b x}\right )}{a-c}\\ &=-\frac {\sqrt {a+b x}}{(a-c) x}+\frac {\sqrt {c+b x}}{(a-c) x}-\frac {b \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{\sqrt {a} (a-c)}+\frac {b \tanh ^{-1}\left (\frac {\sqrt {c+b x}}{\sqrt {c}}\right )}{(a-c) \sqrt {c}}\\ \end {align*}

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Mathematica [A]  time = 0.27, size = 99, normalized size = 0.96 \[ \frac {\frac {b x \sqrt {\frac {b x}{c}+1} \tanh ^{-1}\left (\sqrt {\frac {b x}{c}+1}\right )+b x+c}{\sqrt {b x+c}}-\frac {b x \sqrt {\frac {b x}{a}+1} \tanh ^{-1}\left (\sqrt {\frac {b x}{a}+1}\right )+a+b x}{\sqrt {a+b x}}}{x (a-c)} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^2*(Sqrt[a + b*x] + Sqrt[c + b*x])),x]

[Out]

(-((a + b*x + b*x*Sqrt[1 + (b*x)/a]*ArcTanh[Sqrt[1 + (b*x)/a]])/Sqrt[a + b*x]) + (c + b*x + b*x*Sqrt[1 + (b*x)
/c]*ArcTanh[Sqrt[1 + (b*x)/c]])/Sqrt[c + b*x])/((a - c)*x)

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fricas [A]  time = 0.51, size = 399, normalized size = 3.87 \[ \left [-\frac {\sqrt {a} b c x \log \left (\frac {b x + 2 \, \sqrt {b x + a} \sqrt {a} + 2 \, a}{x}\right ) + a b \sqrt {c} x \log \left (\frac {b x - 2 \, \sqrt {b x + c} \sqrt {c} + 2 \, c}{x}\right ) + 2 \, \sqrt {b x + a} a c - 2 \, \sqrt {b x + c} a c}{2 \, {\left (a^{2} c - a c^{2}\right )} x}, -\frac {2 \, a b \sqrt {-c} x \arctan \left (\frac {\sqrt {b x + c} \sqrt {-c}}{c}\right ) + \sqrt {a} b c x \log \left (\frac {b x + 2 \, \sqrt {b x + a} \sqrt {a} + 2 \, a}{x}\right ) + 2 \, \sqrt {b x + a} a c - 2 \, \sqrt {b x + c} a c}{2 \, {\left (a^{2} c - a c^{2}\right )} x}, \frac {2 \, \sqrt {-a} b c x \arctan \left (\frac {\sqrt {b x + a} \sqrt {-a}}{a}\right ) - a b \sqrt {c} x \log \left (\frac {b x - 2 \, \sqrt {b x + c} \sqrt {c} + 2 \, c}{x}\right ) - 2 \, \sqrt {b x + a} a c + 2 \, \sqrt {b x + c} a c}{2 \, {\left (a^{2} c - a c^{2}\right )} x}, \frac {\sqrt {-a} b c x \arctan \left (\frac {\sqrt {b x + a} \sqrt {-a}}{a}\right ) - a b \sqrt {-c} x \arctan \left (\frac {\sqrt {b x + c} \sqrt {-c}}{c}\right ) - \sqrt {b x + a} a c + \sqrt {b x + c} a c}{{\left (a^{2} c - a c^{2}\right )} x}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/2*(sqrt(a)*b*c*x*log((b*x + 2*sqrt(b*x + a)*sqrt(a) + 2*a)/x) + a*b*sqrt(c)*x*log((b*x - 2*sqrt(b*x + c)*s
qrt(c) + 2*c)/x) + 2*sqrt(b*x + a)*a*c - 2*sqrt(b*x + c)*a*c)/((a^2*c - a*c^2)*x), -1/2*(2*a*b*sqrt(-c)*x*arct
an(sqrt(b*x + c)*sqrt(-c)/c) + sqrt(a)*b*c*x*log((b*x + 2*sqrt(b*x + a)*sqrt(a) + 2*a)/x) + 2*sqrt(b*x + a)*a*
c - 2*sqrt(b*x + c)*a*c)/((a^2*c - a*c^2)*x), 1/2*(2*sqrt(-a)*b*c*x*arctan(sqrt(b*x + a)*sqrt(-a)/a) - a*b*sqr
t(c)*x*log((b*x - 2*sqrt(b*x + c)*sqrt(c) + 2*c)/x) - 2*sqrt(b*x + a)*a*c + 2*sqrt(b*x + c)*a*c)/((a^2*c - a*c
^2)*x), (sqrt(-a)*b*c*x*arctan(sqrt(b*x + a)*sqrt(-a)/a) - a*b*sqrt(-c)*x*arctan(sqrt(b*x + c)*sqrt(-c)/c) - s
qrt(b*x + a)*a*c + sqrt(b*x + c)*a*c)/((a^2*c - a*c^2)*x)]

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giac [B]  time = 10.21, size = 1190, normalized size = 11.55 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b*arctan(sqrt(b*x + a)/sqrt(-a))/(sqrt(-a)*(a - c)) + (2*(a*c^2 + sqrt(a*c)*c^2)*(a - c)^2*b*sgn(2*a - 2*c) +
2*(a*c^2 + sqrt(a*c)*a*c)*(a - c)^2*b + (a^2*c^2 - 2*a*c^3 + c^4 + (a^2*c - 2*a*c^2 + c^3)*sqrt(a*c))*b*abs(a
- c)*sgn(2*a - 2*c) + (a^3*c - 2*a^2*c^2 + a*c^3 + (a^2*c - 2*a*c^2 + c^3)*sqrt(a*c))*b*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))*b*sgn(2*a - 2*c) - (a^4*c - a^3*c^2 - a
^2*c^3 + a*c^4 + (a^4 - a^3*c - a^2*c^2 + a*c^3)*sqrt(a*c))*b)*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*c - a^4*sq
rt(-c)*c - 4*sqrt(-a)*a^3*c^2 + 4*a^3*sqrt(-c)*c^2 + 6*sqrt(-a)*a^2*c^3 - 6*a^2*sqrt(-c)*c^3 - 4*sqrt(-a)*a*c^
4 + 4*a*sqrt(-c)*c^4 + sqrt(-a)*c^5 - sqrt(-c)*c^5)*abs(a - c)) - (2*(a*c^2 + sqrt(a*c)*c^2)*(a - c)^2*b*sgn(2
*a - 2*c) - 2*(a*c^2 + sqrt(a*c)*a*c)*(a - c)^2*b + (a^2*c^2 - 2*a*c^3 + c^4 - (a^2*c - 2*a*c^2 + c^3)*sqrt(a*
c))*b*abs(a - c)*sgn(2*a - 2*c) - (a^3*c - 2*a^2*c^2 + a*c^3 + (a^2*c - 2*a*c^2 + c^3)*sqrt(a*c))*b*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))*b*sgn(2*a - 2*c) + (a^4*c -
a^3*c^2 - a^2*c^3 + a*c^4 - (a^4 - a^3*c - a^2*c^2 + a*c^3)*sqrt(a*c))*b)*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*c - a^4*sqrt(-c)*c - 4*sqrt(-a)*a^3*c^2 + 4*a^3*sqrt(-c)*c^2 + 6*sqrt(-a)*a^2*c^3 - 6*a^2*sqrt(-c)*c^3 - 4*s
qrt(-a)*a*c^4 + 4*a*sqrt(-c)*c^4 + sqrt(-a)*c^5 - sqrt(-c)*c^5)*abs(a - c)) - 2*(b*(sqrt(b*x + a) - sqrt(b*x +
c))^3 - a*b*(sqrt(b*x + a) - sqrt(b*x + c)) + b*c*(sqrt(b*x + a) - sqrt(b*x + c)))/(((sqrt(b*x + a) - sqrt(b*
x + c))^4 - 2*a*(sqrt(b*x + a) - sqrt(b*x + c))^2 - 2*c*(sqrt(b*x + a) - sqrt(b*x + c))^2 + a^2 - 2*a*c + c^2)
*(a - c)) - sqrt(b*x + a)/((a - c)*x)

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maple [A]  time = 0.02, size = 88, normalized size = 0.85 \[ \frac {2 \left (-\frac {\arctanh \left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{2 \sqrt {a}}-\frac {\sqrt {b x +a}}{2 b x}\right ) b}{a -c}-\frac {2 \left (-\frac {\arctanh \left (\frac {\sqrt {b x +c}}{\sqrt {c}}\right )}{2 \sqrt {c}}-\frac {\sqrt {b x +c}}{2 b x}\right ) b}{a -c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/(a-c)*b*(-1/2*(b*x+a)^(1/2)/b/x-1/2*arctanh((b*x+a)^(1/2)/a^(1/2))/a^(1/2))-2/(a-c)*b*(-1/2*(b*x+c)^(1/2)/b/
x-1/2/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 \[ \int \frac {1}{x^{2} {\left (\sqrt {b x + a} + \sqrt {b x + c}\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 18.88, size = 2642, normalized size = 25.65 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b*atan(((b*(a*c^(1/2) + a^(1/2)*c)*((a^(1/2)*c^(3/2) - 2*a*c + a^(3/2)*c^(1/2))*(2*a*c + a^(1/2)*c^(3/2) + a^
(3/2)*c^(1/2)))^(1/2)*((a^3*b*c^(7/2) - a^(7/2)*b*c^3 - a^2*b*c^(9/2) + a^(9/2)*b*c^2)/(a^3*c^5 - 2*a^4*c^4 +
a^5*c^3) + (((a + b*x)^(1/2) - a^(1/2))*(2*a^(3/2)*b*c^5 - 2*a^5*b*c^(3/2) + 2*a^4*b*c^(5/2) - 2*a^(5/2)*b*c^4
))/(2*((c + b*x)^(1/2) - c^(1/2))*(a^3*c^5 - 2*a^4*c^4 + a^5*c^3)) - (b*(a*c^(1/2) + a^(1/2)*c)*((a^(5/2)*c^(1
1/2) - a^(7/2)*c^(9/2) - a^(9/2)*c^(7/2) + a^(11/2)*c^(5/2))/(a^3*c^5 - 2*a^4*c^4 + a^5*c^3) - (((a + b*x)^(1/
2) - a^(1/2))*(4*a^2*c^6 - 12*a^3*c^5 + 16*a^4*c^4 - 12*a^5*c^3 + 4*a^6*c^2))/(2*((c + b*x)^(1/2) - c^(1/2))*(
a^3*c^5 - 2*a^4*c^4 + a^5*c^3)))*((a^(1/2)*c^(3/2) - 2*a*c + a^(3/2)*c^(1/2))*(2*a*c + a^(1/2)*c^(3/2) + a^(3/
2)*c^(1/2)))^(1/2))/(2*(2*a^2*c^3 - 2*a^3*c^2 + a^(3/2)*c^(7/2) - a^(7/2)*c^(3/2))))*1i)/(2*(2*a^2*c^3 - 2*a^3
*c^2 + a^(3/2)*c^(7/2) - a^(7/2)*c^(3/2))) + (b*(a*c^(1/2) + a^(1/2)*c)*((a^(1/2)*c^(3/2) - 2*a*c + a^(3/2)*c^
(1/2))*(2*a*c + a^(1/2)*c^(3/2) + a^(3/2)*c^(1/2)))^(1/2)*((a^3*b*c^(7/2) - a^(7/2)*b*c^3 - a^2*b*c^(9/2) + a^
(9/2)*b*c^2)/(a^3*c^5 - 2*a^4*c^4 + a^5*c^3) + (((a + b*x)^(1/2) - a^(1/2))*(2*a^(3/2)*b*c^5 - 2*a^5*b*c^(3/2)
+ 2*a^4*b*c^(5/2) - 2*a^(5/2)*b*c^4))/(2*((c + b*x)^(1/2) - c^(1/2))*(a^3*c^5 - 2*a^4*c^4 + a^5*c^3)) + (b*(a
*c^(1/2) + a^(1/2)*c)*((a^(5/2)*c^(11/2) - a^(7/2)*c^(9/2) - a^(9/2)*c^(7/2) + a^(11/2)*c^(5/2))/(a^3*c^5 - 2*
a^4*c^4 + a^5*c^3) - (((a + b*x)^(1/2) - a^(1/2))*(4*a^2*c^6 - 12*a^3*c^5 + 16*a^4*c^4 - 12*a^5*c^3 + 4*a^6*c^
2))/(2*((c + b*x)^(1/2) - c^(1/2))*(a^3*c^5 - 2*a^4*c^4 + a^5*c^3)))*((a^(1/2)*c^(3/2) - 2*a*c + a^(3/2)*c^(1/
2))*(2*a*c + a^(1/2)*c^(3/2) + a^(3/2)*c^(1/2)))^(1/2))/(2*(2*a^2*c^3 - 2*a^3*c^2 + a^(3/2)*c^(7/2) - a^(7/2)*
c^(3/2))))*1i)/(2*(2*a^2*c^3 - 2*a^3*c^2 + a^(3/2)*c^(7/2) - a^(7/2)*c^(3/2))))/(((a^(3/2)*b^2*c^(7/2))/2 - a^
(5/2)*b^2*c^(5/2) + (a^(7/2)*b^2*c^(3/2))/2)/(a^3*c^5 - 2*a^4*c^4 + a^5*c^3) - (((a + b*x)^(1/2) - a^(1/2))*(a
^(3/2)*b^2*c^(7/2) - 2*a^(5/2)*b^2*c^(5/2) + a^(7/2)*b^2*c^(3/2)))/(((c + b*x)^(1/2) - c^(1/2))*(a^3*c^5 - 2*a
^4*c^4 + a^5*c^3)) - (b*(a*c^(1/2) + a^(1/2)*c)*((a^(1/2)*c^(3/2) - 2*a*c + a^(3/2)*c^(1/2))*(2*a*c + a^(1/2)*
c^(3/2) + a^(3/2)*c^(1/2)))^(1/2)*((a^3*b*c^(7/2) - a^(7/2)*b*c^3 - a^2*b*c^(9/2) + a^(9/2)*b*c^2)/(a^3*c^5 -
2*a^4*c^4 + a^5*c^3) + (((a + b*x)^(1/2) - a^(1/2))*(2*a^(3/2)*b*c^5 - 2*a^5*b*c^(3/2) + 2*a^4*b*c^(5/2) - 2*a
^(5/2)*b*c^4))/(2*((c + b*x)^(1/2) - c^(1/2))*(a^3*c^5 - 2*a^4*c^4 + a^5*c^3)) - (b*(a*c^(1/2) + a^(1/2)*c)*((
a^(5/2)*c^(11/2) - a^(7/2)*c^(9/2) - a^(9/2)*c^(7/2) + a^(11/2)*c^(5/2))/(a^3*c^5 - 2*a^4*c^4 + a^5*c^3) - (((
a + b*x)^(1/2) - a^(1/2))*(4*a^2*c^6 - 12*a^3*c^5 + 16*a^4*c^4 - 12*a^5*c^3 + 4*a^6*c^2))/(2*((c + b*x)^(1/2)
- c^(1/2))*(a^3*c^5 - 2*a^4*c^4 + a^5*c^3)))*((a^(1/2)*c^(3/2) - 2*a*c + a^(3/2)*c^(1/2))*(2*a*c + a^(1/2)*c^(
3/2) + a^(3/2)*c^(1/2)))^(1/2))/(2*(2*a^2*c^3 - 2*a^3*c^2 + a^(3/2)*c^(7/2) - a^(7/2)*c^(3/2)))))/(2*(2*a^2*c^
3 - 2*a^3*c^2 + a^(3/2)*c^(7/2) - a^(7/2)*c^(3/2))) + (b*(a*c^(1/2) + a^(1/2)*c)*((a^(1/2)*c^(3/2) - 2*a*c + a
^(3/2)*c^(1/2))*(2*a*c + a^(1/2)*c^(3/2) + a^(3/2)*c^(1/2)))^(1/2)*((a^3*b*c^(7/2) - a^(7/2)*b*c^3 - a^2*b*c^(
9/2) + a^(9/2)*b*c^2)/(a^3*c^5 - 2*a^4*c^4 + a^5*c^3) + (((a + b*x)^(1/2) - a^(1/2))*(2*a^(3/2)*b*c^5 - 2*a^5*
b*c^(3/2) + 2*a^4*b*c^(5/2) - 2*a^(5/2)*b*c^4))/(2*((c + b*x)^(1/2) - c^(1/2))*(a^3*c^5 - 2*a^4*c^4 + a^5*c^3)
) + (b*(a*c^(1/2) + a^(1/2)*c)*((a^(5/2)*c^(11/2) - a^(7/2)*c^(9/2) - a^(9/2)*c^(7/2) + a^(11/2)*c^(5/2))/(a^3
*c^5 - 2*a^4*c^4 + a^5*c^3) - (((a + b*x)^(1/2) - a^(1/2))*(4*a^2*c^6 - 12*a^3*c^5 + 16*a^4*c^4 - 12*a^5*c^3 +
4*a^6*c^2))/(2*((c + b*x)^(1/2) - c^(1/2))*(a^3*c^5 - 2*a^4*c^4 + a^5*c^3)))*((a^(1/2)*c^(3/2) - 2*a*c + a^(3
/2)*c^(1/2))*(2*a*c + a^(1/2)*c^(3/2) + a^(3/2)*c^(1/2)))^(1/2))/(2*(2*a^2*c^3 - 2*a^3*c^2 + a^(3/2)*c^(7/2) -
a^(7/2)*c^(3/2)))))/(2*(2*a^2*c^3 - 2*a^3*c^2 + a^(3/2)*c^(7/2) - a^(7/2)*c^(3/2)))))*(a*c^(1/2) + a^(1/2)*c)
*((a^(1/2)*c^(3/2) - 2*a*c + a^(3/2)*c^(1/2))*(2*a*c + a^(1/2)*c^(3/2) + a^(3/2)*c^(1/2)))^(1/2)*1i)/(2*a^2*c^
3 - 2*a^3*c^2 + a^(3/2)*c^(7/2) - a^(7/2)*c^(3/2)) - ((a^(1/2)*b)/(4*(a*c - a^2)) - (b*c^(1/2))/(4*(a*c - c^2)
) - (((a^(1/2)*((a^2*b)/4 - (b*c^2)/4 + (a*b*c)/4))/(a^3*c - a^2*c^2) - (c^(1/2)*((b*c^2)/4 - (a^2*b)/4 + (a*b
*c)/4))/(a*c^3 - a^2*c^2))*((a + b*x)^(1/2) - a^(1/2))^2)/((c + b*x)^(1/2) - c^(1/2))^2 + (((a^(1/2)*((a*b)/4
- (3*b*c)/4))/(a*c^2 - a^2*c) - (c^(1/2)*((3*a*b)/4 - (b*c)/4))/(a*c^2 - a^2*c))*((a + b*x)^(1/2) - a^(1/2)))/
((c + b*x)^(1/2) - c^(1/2)))/(((a + b*x)^(1/2) - a^(1/2))/((c + b*x)^(1/2) - c^(1/2)) + ((a + b*x)^(1/2) - a^(
1/2))^3/((c + b*x)^(1/2) - c^(1/2))^3 - ((a + c)*((a + b*x)^(1/2) - a^(1/2))^2)/(a^(1/2)*c^(1/2)*((c + b*x)^(1
/2) - c^(1/2))^2)) - log(((a + b*x)^(1/2) - a^(1/2))/((c + b*x)^(1/2) - c^(1/2)))*(b/(2*a^(1/2)*c) - (b*(a^(1/
2) + c^(1/2)))/(2*c*(a - c))) - (b*((a + b*x)^(1/2) - a^(1/2)))/(4*a^(1/2)*c^(1/2)*(a^(1/2) - c^(1/2))*((c + b
*x)^(1/2) - c^(1/2)))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x^{2} \left (\sqrt {a + b x} + \sqrt {b x + c}\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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