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

Optimal. Leaf size=138 \[ -\frac {2 a}{x (b-c)^2}+\frac {2 \sqrt {a+b x} \sqrt {a+c x}}{x (b-c)^2}+\frac {2 (b+c) \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a+c x}}\right )}{(b-c)^2}-\frac {4 \sqrt {b} \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {b} \sqrt {a+c x}}\right )}{(b-c)^2}+\frac {(b+c) \log (x)}{(b-c)^2} \]

[Out]

-2*a/(b-c)^2/x+2*(b+c)*arctanh((b*x+a)^(1/2)/(c*x+a)^(1/2))/(b-c)^2+(b+c)*ln(x)/(b-c)^2-4*arctanh(c^(1/2)*(b*x
+a)^(1/2)/b^(1/2)/(c*x+a)^(1/2))*b^(1/2)*c^(1/2)/(b-c)^2+2*(b*x+a)^(1/2)*(c*x+a)^(1/2)/(b-c)^2/x

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Rubi [A]  time = 0.11, antiderivative size = 138, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {6690, 97, 157, 63, 217, 206, 93, 208} \[ -\frac {2 a}{x (b-c)^2}+\frac {2 \sqrt {a+b x} \sqrt {a+c x}}{x (b-c)^2}+\frac {2 (b+c) \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a+c x}}\right )}{(b-c)^2}-\frac {4 \sqrt {b} \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {b} \sqrt {a+c x}}\right )}{(b-c)^2}+\frac {(b+c) \log (x)}{(b-c)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*a)/((b - c)^2*x) + (2*Sqrt[a + b*x]*Sqrt[a + c*x])/((b - c)^2*x) + (2*(b + c)*ArcTanh[Sqrt[a + b*x]/Sqrt[a
+ c*x]])/(b - c)^2 - (4*Sqrt[b]*Sqrt[c]*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[a + c*x])])/(b - c)^2 +
((b + c)*Log[x])/(b - c)^2

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 93

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[q, Subst[Int[x^(q*(m + 1) - 1)/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^
(1/q)], x]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*x]

Rule 97

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

Rule 157

Int[(((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/((a_.) + (b_.)*(x_)), x_Symbol]
:> Dist[h/b, Int[(c + d*x)^n*(e + f*x)^p, x], x] + Dist[(b*g - a*h)/b, Int[((c + d*x)^n*(e + f*x)^p)/(a + b*x
), x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x]

Rule 206

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

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 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 6690

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

Rubi steps

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

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Mathematica [A]  time = 0.71, size = 178, normalized size = 1.29 \[ \frac {2 c \sqrt {a+b x}+\frac {2 a \left (\sqrt {a+b x}-\sqrt {a+c x}\right )}{x}+(b+c) \log (x) \sqrt {a+c x}-\frac {4 b \sqrt {c} (a+c x) \sinh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a (b-c)}}\right )}{\sqrt {a (b-c)} \sqrt {\frac {b (a+c x)}{a (b-c)}}}+2 (b+c) \sqrt {a+c x} \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a+c x}}\right )}{(b-c)^2 \sqrt {a+c x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*c*Sqrt[a + b*x] + (2*a*(Sqrt[a + b*x] - Sqrt[a + c*x]))/x - (4*b*Sqrt[c]*(a + c*x)*ArcSinh[(Sqrt[c]*Sqrt[a
+ b*x])/Sqrt[a*(b - c)]])/(Sqrt[a*(b - c)]*Sqrt[(b*(a + c*x))/(a*(b - c))]) + 2*(b + c)*Sqrt[a + c*x]*ArcTanh[
Sqrt[a + b*x]/Sqrt[a + c*x]] + (b + c)*Sqrt[a + c*x]*Log[x])/((b - c)^2*Sqrt[a + c*x])

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fricas [A]  time = 0.64, size = 317, normalized size = 2.30 \[ \left [\frac {2 \, {\left (b + c\right )} x \log \relax (x) - 2 \, {\left (b + c\right )} x \log \left (-\frac {{\left (b + c\right )} x - 2 \, \sqrt {b x + a} \sqrt {c x + a} + 2 \, a}{x}\right ) + 4 \, \sqrt {b c} x \log \left (a b^{2} + 2 \, a b c + a c^{2} + 2 \, {\left (2 \, b c - \sqrt {b c} {\left (b + c\right )}\right )} \sqrt {b x + a} \sqrt {c x + a} + 2 \, {\left (b^{2} c + b c^{2}\right )} x - 2 \, {\left (2 \, b c x + a b + a c\right )} \sqrt {b c}\right ) + {\left (b + c\right )} x + 4 \, \sqrt {b x + a} \sqrt {c x + a} - 4 \, a}{2 \, {\left (b^{2} - 2 \, b c + c^{2}\right )} x}, \frac {2 \, {\left (b + c\right )} x \log \relax (x) - 2 \, {\left (b + c\right )} x \log \left (-\frac {{\left (b + c\right )} x - 2 \, \sqrt {b x + a} \sqrt {c x + a} + 2 \, a}{x}\right ) + 8 \, \sqrt {-b c} x \arctan \left (\frac {\sqrt {-b c} \sqrt {b x + a} \sqrt {c x + a} - \sqrt {-b c} a}{b c x}\right ) + {\left (b + c\right )} x + 4 \, \sqrt {b x + a} \sqrt {c x + a} - 4 \, a}{2 \, {\left (b^{2} - 2 \, b c + c^{2}\right )} x}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/2*(2*(b + c)*x*log(x) - 2*(b + c)*x*log(-((b + c)*x - 2*sqrt(b*x + a)*sqrt(c*x + a) + 2*a)/x) + 4*sqrt(b*c)
*x*log(a*b^2 + 2*a*b*c + a*c^2 + 2*(2*b*c - sqrt(b*c)*(b + c))*sqrt(b*x + a)*sqrt(c*x + a) + 2*(b^2*c + b*c^2)
*x - 2*(2*b*c*x + a*b + a*c)*sqrt(b*c)) + (b + c)*x + 4*sqrt(b*x + a)*sqrt(c*x + a) - 4*a)/((b^2 - 2*b*c + c^2
)*x), 1/2*(2*(b + c)*x*log(x) - 2*(b + c)*x*log(-((b + c)*x - 2*sqrt(b*x + a)*sqrt(c*x + a) + 2*a)/x) + 8*sqrt
(-b*c)*x*arctan((sqrt(-b*c)*sqrt(b*x + a)*sqrt(c*x + a) - sqrt(-b*c)*a)/(b*c*x)) + (b + c)*x + 4*sqrt(b*x + a)
*sqrt(c*x + a) - 4*a)/((b^2 - 2*b*c + c^2)*x)]

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giac [B]  time = 4.50, size = 438, normalized size = 3.17 \[ \frac {2 \, \sqrt {b c} {\left | b \right |} \log \left ({\left (\sqrt {b c} \sqrt {b x + a} - \sqrt {a b^{2} + {\left (b x + a\right )} b c - a b c}\right )}^{2}\right )}{b^{3} - 2 \, b^{2} c + b c^{2}} - \frac {2 \, \sqrt {b c} {\left (b + c\right )} {\left | b \right |} \arctan \left (\frac {a b^{2} + a b c - {\left (\sqrt {b c} \sqrt {b x + a} - \sqrt {a b^{2} + {\left (b x + a\right )} b c - a b c}\right )}^{2}}{2 \, \sqrt {-b c} a b}\right )}{{\left (b^{2} - 2 \, b c + c^{2}\right )} \sqrt {-b c} b} + \frac {{\left (b + c\right )} \log \left ({\left | b x \right |}\right )}{b^{2} - 2 \, b c + c^{2}} - \frac {4 \, {\left (\sqrt {b c} {\left (\sqrt {b c} \sqrt {b x + a} - \sqrt {a b^{2} + {\left (b x + a\right )} b c - a b c}\right )}^{2} a {\left (b + c\right )} {\left | b \right |} - {\left (b^{3} - 2 \, b^{2} c + b c^{2}\right )} \sqrt {b c} a^{2} {\left | b \right |}\right )}}{{\left ({\left (\sqrt {b c} \sqrt {b x + a} - \sqrt {a b^{2} + {\left (b x + a\right )} b c - a b c}\right )}^{4} - 2 \, {\left (b^{2} + b c\right )} {\left (\sqrt {b c} \sqrt {b x + a} - \sqrt {a b^{2} + {\left (b x + a\right )} b c - a b c}\right )}^{2} a + {\left (b^{4} - 2 \, b^{3} c + b^{2} c^{2}\right )} a^{2}\right )} {\left (b^{2} - 2 \, b c + c^{2}\right )}} - \frac {{\left (b x + a\right )} b + a b + {\left (b x + a\right )} c - a c}{{\left (b^{2} - 2 \, b c + c^{2}\right )} b x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(b*c)*abs(b)*log((sqrt(b*c)*sqrt(b*x + a) - sqrt(a*b^2 + (b*x + a)*b*c - a*b*c))^2)/(b^3 - 2*b^2*c + b*c
^2) - 2*sqrt(b*c)*(b + c)*abs(b)*arctan(1/2*(a*b^2 + a*b*c - (sqrt(b*c)*sqrt(b*x + a) - sqrt(a*b^2 + (b*x + a)
*b*c - a*b*c))^2)/(sqrt(-b*c)*a*b))/((b^2 - 2*b*c + c^2)*sqrt(-b*c)*b) + (b + c)*log(abs(b*x))/(b^2 - 2*b*c +
c^2) - 4*(sqrt(b*c)*(sqrt(b*c)*sqrt(b*x + a) - sqrt(a*b^2 + (b*x + a)*b*c - a*b*c))^2*a*(b + c)*abs(b) - (b^3
- 2*b^2*c + b*c^2)*sqrt(b*c)*a^2*abs(b))/(((sqrt(b*c)*sqrt(b*x + a) - sqrt(a*b^2 + (b*x + a)*b*c - a*b*c))^4 -
2*(b^2 + b*c)*(sqrt(b*c)*sqrt(b*x + a) - sqrt(a*b^2 + (b*x + a)*b*c - a*b*c))^2*a + (b^4 - 2*b^3*c + b^2*c^2)
*a^2)*(b^2 - 2*b*c + c^2)) - ((b*x + a)*b + a*b + (b*x + a)*c - a*c)/((b^2 - 2*b*c + c^2)*b*x)

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maple [C]  time = 0.02, size = 272, normalized size = 1.97 \[ \frac {b \ln \relax (x )}{\left (b -c \right )^{2}}+\frac {c \ln \relax (x )}{\left (b -c \right )^{2}}-\frac {2 a}{\left (b -c \right )^{2} x}-\frac {\sqrt {b x +a}\, \sqrt {c x +a}\, \left (2 b c x \,\mathrm {csgn}\relax (a ) \ln \left (\frac {2 b c x +a b +a c +2 \sqrt {b c \,x^{2}+a b x +a c x +a^{2}}\, \sqrt {b c}}{2 \sqrt {b c}}\right )-\sqrt {b c}\, b x \ln \left (\frac {\left (b x +c x +2 a +2 \sqrt {b c \,x^{2}+a b x +a c x +a^{2}}\, \mathrm {csgn}\relax (a )\right ) a}{x}\right )-\sqrt {b c}\, c x \ln \left (\frac {\left (b x +c x +2 a +2 \sqrt {b c \,x^{2}+a b x +a c x +a^{2}}\, \mathrm {csgn}\relax (a )\right ) a}{x}\right )-2 \sqrt {b c \,x^{2}+a b x +a c x +a^{2}}\, \sqrt {b c}\, \mathrm {csgn}\relax (a )\right ) \mathrm {csgn}\relax (a )}{\left (b -c \right )^{2} \sqrt {b c \,x^{2}+a b x +a c x +a^{2}}\, \sqrt {b c}\, x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 17.44, size = 4285, normalized size = 31.05 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(atan((((b*c)^(1/2)*((4*(b*c)^(1/2)*((4*(b^4*c^12 + 16*b^5*c^11 - 42*b^6*c^10 + 25*b^7*c^9 + 25*b^8*c^8 - 42*b
^9*c^7 + 16*b^10*c^6 + b^11*c^5))/(b^4 - 4*b^3*c - 4*b*c^3 + c^4 + 6*b^2*c^2) + (4*(b*c)^(1/2)*((4*(4*b^5*c^12
- 36*b^7*c^10 + 64*b^8*c^9 - 36*b^9*c^8 + 4*b^11*c^6))/(b^4 - 4*b^3*c - 4*b*c^3 + c^4 + 6*b^2*c^2) + (4*(b*c)
^(1/2)*((4*(4*b^5*c^13 - b^4*c^14 - 5*b^6*c^12 + b^7*c^11 + b^8*c^10 + b^9*c^9 + b^10*c^8 - 5*b^11*c^7 + 4*b^1
2*c^6 - b^13*c^5))/(b^4 - 4*b^3*c - 4*b*c^3 + c^4 + 6*b^2*c^2) + (2*((a + b*x)^(1/2) - a^(1/2))*(4*b^3*c^15 -
31*b^4*c^14 + 120*b^5*c^13 - 300*b^6*c^12 + 516*b^7*c^11 - 618*b^8*c^10 + 516*b^9*c^9 - 300*b^10*c^8 + 120*b^1
1*c^7 - 31*b^12*c^6 + 4*b^13*c^5))/(((a + c*x)^(1/2) - a^(1/2))*(b^4 - 4*b^3*c - 4*b*c^3 + c^4 + 6*b^2*c^2))))
/(b - c)^2 - (2*((a + b*x)^(1/2) - a^(1/2))*(4*b^3*c^14 - 27*b^4*c^13 + 99*b^5*c^12 - 175*b^6*c^11 + 99*b^7*c^
10 + 99*b^8*c^9 - 175*b^9*c^8 + 99*b^10*c^7 - 27*b^11*c^6 + 4*b^12*c^5))/(((a + c*x)^(1/2) - a^(1/2))*(b^4 - 4
*b^3*c - 4*b*c^3 + c^4 + 6*b^2*c^2))))/(b - c)^2 - (2*((a + b*x)^(1/2) - a^(1/2))*(73*b^4*c^12 - 278*b^5*c^11
+ 503*b^6*c^10 - 596*b^7*c^9 + 503*b^8*c^8 - 278*b^9*c^7 + 73*b^10*c^6))/(((a + c*x)^(1/2) - a^(1/2))*(b^4 - 4
*b^3*c - 4*b*c^3 + c^4 + 6*b^2*c^2))))/(b - c)^2 - (4*(4*b^5*c^10 + 24*b^6*c^9 + 40*b^7*c^8 + 24*b^8*c^7 + 4*b
^9*c^6))/(b^4 - 4*b^3*c - 4*b*c^3 + c^4 + 6*b^2*c^2) + (2*((a + b*x)^(1/2) - a^(1/2))*(65*b^4*c^11 - 167*b^5*c
^10 + 198*b^6*c^9 + 198*b^7*c^8 - 167*b^8*c^7 + 65*b^9*c^6))/(((a + c*x)^(1/2) - a^(1/2))*(b^4 - 4*b^3*c - 4*b
*c^3 + c^4 + 6*b^2*c^2)))*4i)/(b - c)^2 - ((b*c)^(1/2)*((4*(4*b^5*c^10 + 24*b^6*c^9 + 40*b^7*c^8 + 24*b^8*c^7
+ 4*b^9*c^6))/(b^4 - 4*b^3*c - 4*b*c^3 + c^4 + 6*b^2*c^2) + (4*(b*c)^(1/2)*((4*(b^4*c^12 + 16*b^5*c^11 - 42*b^
6*c^10 + 25*b^7*c^9 + 25*b^8*c^8 - 42*b^9*c^7 + 16*b^10*c^6 + b^11*c^5))/(b^4 - 4*b^3*c - 4*b*c^3 + c^4 + 6*b^
2*c^2) + (4*(b*c)^(1/2)*((4*(b*c)^(1/2)*((4*(4*b^5*c^13 - b^4*c^14 - 5*b^6*c^12 + b^7*c^11 + b^8*c^10 + b^9*c^
9 + b^10*c^8 - 5*b^11*c^7 + 4*b^12*c^6 - b^13*c^5))/(b^4 - 4*b^3*c - 4*b*c^3 + c^4 + 6*b^2*c^2) + (2*((a + b*x
)^(1/2) - a^(1/2))*(4*b^3*c^15 - 31*b^4*c^14 + 120*b^5*c^13 - 300*b^6*c^12 + 516*b^7*c^11 - 618*b^8*c^10 + 516
*b^9*c^9 - 300*b^10*c^8 + 120*b^11*c^7 - 31*b^12*c^6 + 4*b^13*c^5))/(((a + c*x)^(1/2) - a^(1/2))*(b^4 - 4*b^3*
c - 4*b*c^3 + c^4 + 6*b^2*c^2))))/(b - c)^2 - (4*(4*b^5*c^12 - 36*b^7*c^10 + 64*b^8*c^9 - 36*b^9*c^8 + 4*b^11*
c^6))/(b^4 - 4*b^3*c - 4*b*c^3 + c^4 + 6*b^2*c^2) + (2*((a + b*x)^(1/2) - a^(1/2))*(4*b^3*c^14 - 27*b^4*c^13 +
99*b^5*c^12 - 175*b^6*c^11 + 99*b^7*c^10 + 99*b^8*c^9 - 175*b^9*c^8 + 99*b^10*c^7 - 27*b^11*c^6 + 4*b^12*c^5)
)/(((a + c*x)^(1/2) - a^(1/2))*(b^4 - 4*b^3*c - 4*b*c^3 + c^4 + 6*b^2*c^2))))/(b - c)^2 - (2*((a + b*x)^(1/2)
- a^(1/2))*(73*b^4*c^12 - 278*b^5*c^11 + 503*b^6*c^10 - 596*b^7*c^9 + 503*b^8*c^8 - 278*b^9*c^7 + 73*b^10*c^6)
)/(((a + c*x)^(1/2) - a^(1/2))*(b^4 - 4*b^3*c - 4*b*c^3 + c^4 + 6*b^2*c^2))))/(b - c)^2 - (2*((a + b*x)^(1/2)
- a^(1/2))*(65*b^4*c^11 - 167*b^5*c^10 + 198*b^6*c^9 + 198*b^7*c^8 - 167*b^8*c^7 + 65*b^9*c^6))/(((a + c*x)^(1
/2) - a^(1/2))*(b^4 - 4*b^3*c - 4*b*c^3 + c^4 + 6*b^2*c^2)))*4i)/(b - c)^2)/((4*(b*c)^(1/2)*((4*(b*c)^(1/2)*((
4*(b^4*c^12 + 16*b^5*c^11 - 42*b^6*c^10 + 25*b^7*c^9 + 25*b^8*c^8 - 42*b^9*c^7 + 16*b^10*c^6 + b^11*c^5))/(b^4
- 4*b^3*c - 4*b*c^3 + c^4 + 6*b^2*c^2) + (4*(b*c)^(1/2)*((4*(4*b^5*c^12 - 36*b^7*c^10 + 64*b^8*c^9 - 36*b^9*c
^8 + 4*b^11*c^6))/(b^4 - 4*b^3*c - 4*b*c^3 + c^4 + 6*b^2*c^2) + (4*(b*c)^(1/2)*((4*(4*b^5*c^13 - b^4*c^14 - 5*
b^6*c^12 + b^7*c^11 + b^8*c^10 + b^9*c^9 + b^10*c^8 - 5*b^11*c^7 + 4*b^12*c^6 - b^13*c^5))/(b^4 - 4*b^3*c - 4*
b*c^3 + c^4 + 6*b^2*c^2) + (2*((a + b*x)^(1/2) - a^(1/2))*(4*b^3*c^15 - 31*b^4*c^14 + 120*b^5*c^13 - 300*b^6*c
^12 + 516*b^7*c^11 - 618*b^8*c^10 + 516*b^9*c^9 - 300*b^10*c^8 + 120*b^11*c^7 - 31*b^12*c^6 + 4*b^13*c^5))/(((
a + c*x)^(1/2) - a^(1/2))*(b^4 - 4*b^3*c - 4*b*c^3 + c^4 + 6*b^2*c^2))))/(b - c)^2 - (2*((a + b*x)^(1/2) - a^(
1/2))*(4*b^3*c^14 - 27*b^4*c^13 + 99*b^5*c^12 - 175*b^6*c^11 + 99*b^7*c^10 + 99*b^8*c^9 - 175*b^9*c^8 + 99*b^1
0*c^7 - 27*b^11*c^6 + 4*b^12*c^5))/(((a + c*x)^(1/2) - a^(1/2))*(b^4 - 4*b^3*c - 4*b*c^3 + c^4 + 6*b^2*c^2))))
/(b - c)^2 - (2*((a + b*x)^(1/2) - a^(1/2))*(73*b^4*c^12 - 278*b^5*c^11 + 503*b^6*c^10 - 596*b^7*c^9 + 503*b^8
*c^8 - 278*b^9*c^7 + 73*b^10*c^6))/(((a + c*x)^(1/2) - a^(1/2))*(b^4 - 4*b^3*c - 4*b*c^3 + c^4 + 6*b^2*c^2))))
/(b - c)^2 - (4*(4*b^5*c^10 + 24*b^6*c^9 + 40*b^7*c^8 + 24*b^8*c^7 + 4*b^9*c^6))/(b^4 - 4*b^3*c - 4*b*c^3 + c^
4 + 6*b^2*c^2) + (2*((a + b*x)^(1/2) - a^(1/2))*(65*b^4*c^11 - 167*b^5*c^10 + 198*b^6*c^9 + 198*b^7*c^8 - 167*
b^8*c^7 + 65*b^9*c^6))/(((a + c*x)^(1/2) - a^(1/2))*(b^4 - 4*b^3*c - 4*b*c^3 + c^4 + 6*b^2*c^2))))/(b - c)^2 -
(8*(14*b^5*c^9 + 42*b^6*c^8 + 42*b^7*c^7 + 14*b^8*c^6))/(b^4 - 4*b^3*c - 4*b*c^3 + c^4 + 6*b^2*c^2) + (4*(b*c
)^(1/2)*((4*(4*b^5*c^10 + 24*b^6*c^9 + 40*b^7*c^8 + 24*b^8*c^7 + 4*b^9*c^6))/(b^4 - 4*b^3*c - 4*b*c^3 + c^4 +
6*b^2*c^2) + (4*(b*c)^(1/2)*((4*(b^4*c^12 + 16*b^5*c^11 - 42*b^6*c^10 + 25*b^7*c^9 + 25*b^8*c^8 - 42*b^9*c^7 +
16*b^10*c^6 + b^11*c^5))/(b^4 - 4*b^3*c - 4*b*c^3 + c^4 + 6*b^2*c^2) + (4*(b*c)^(1/2)*((4*(b*c)^(1/2)*((4*(4*
b^5*c^13 - b^4*c^14 - 5*b^6*c^12 + b^7*c^11 + b^8*c^10 + b^9*c^9 + b^10*c^8 - 5*b^11*c^7 + 4*b^12*c^6 - b^13*c
^5))/(b^4 - 4*b^3*c - 4*b*c^3 + c^4 + 6*b^2*c^2) + (2*((a + b*x)^(1/2) - a^(1/2))*(4*b^3*c^15 - 31*b^4*c^14 +
120*b^5*c^13 - 300*b^6*c^12 + 516*b^7*c^11 - 618*b^8*c^10 + 516*b^9*c^9 - 300*b^10*c^8 + 120*b^11*c^7 - 31*b^1
2*c^6 + 4*b^13*c^5))/(((a + c*x)^(1/2) - a^(1/2))*(b^4 - 4*b^3*c - 4*b*c^3 + c^4 + 6*b^2*c^2))))/(b - c)^2 - (
4*(4*b^5*c^12 - 36*b^7*c^10 + 64*b^8*c^9 - 36*b^9*c^8 + 4*b^11*c^6))/(b^4 - 4*b^3*c - 4*b*c^3 + c^4 + 6*b^2*c^
2) + (2*((a + b*x)^(1/2) - a^(1/2))*(4*b^3*c^14 - 27*b^4*c^13 + 99*b^5*c^12 - 175*b^6*c^11 + 99*b^7*c^10 + 99*
b^8*c^9 - 175*b^9*c^8 + 99*b^10*c^7 - 27*b^11*c^6 + 4*b^12*c^5))/(((a + c*x)^(1/2) - a^(1/2))*(b^4 - 4*b^3*c -
4*b*c^3 + c^4 + 6*b^2*c^2))))/(b - c)^2 - (2*((a + b*x)^(1/2) - a^(1/2))*(73*b^4*c^12 - 278*b^5*c^11 + 503*b^
6*c^10 - 596*b^7*c^9 + 503*b^8*c^8 - 278*b^9*c^7 + 73*b^10*c^6))/(((a + c*x)^(1/2) - a^(1/2))*(b^4 - 4*b^3*c -
4*b*c^3 + c^4 + 6*b^2*c^2))))/(b - c)^2 - (2*((a + b*x)^(1/2) - a^(1/2))*(65*b^4*c^11 - 167*b^5*c^10 + 198*b^
6*c^9 + 198*b^7*c^8 - 167*b^8*c^7 + 65*b^9*c^6))/(((a + c*x)^(1/2) - a^(1/2))*(b^4 - 4*b^3*c - 4*b*c^3 + c^4 +
6*b^2*c^2))))/(b - c)^2 + (4*((a + b*x)^(1/2) - a^(1/2))*(112*b^5*c^9 + 224*b^6*c^8 + 112*b^7*c^7))/(((a + c*
x)^(1/2) - a^(1/2))*(b^4 - 4*b^3*c - 4*b*c^3 + c^4 + 6*b^2*c^2))))*(b*c)^(1/2)*8i)/(b - c)^2 - (((b*c + b^2)*(
(a + b*x)^(1/2) - a^(1/2)))/((a + c*x)^(1/2) - a^(1/2)) - b^2 + (((a + b*x)^(1/2) - a^(1/2))^2*(b^2 - 3*b*c +
c^2))/((a + c*x)^(1/2) - a^(1/2))^2)/((((a + b*x)^(1/2) - a^(1/2))^3*(2*b^2*c - 4*b*c^2 + 2*c^3))/((a + c*x)^(
1/2) - a^(1/2))^3 + (((a + b*x)^(1/2) - a^(1/2))^2*(2*b*c^2 + 2*b^2*c - 2*b^3 - 2*c^3))/((a + c*x)^(1/2) - a^(
1/2))^2 + (((a + b*x)^(1/2) - a^(1/2))*(2*b*c^2 - 4*b^2*c + 2*b^3))/((a + c*x)^(1/2) - a^(1/2))) + (log(((a +
b*x)^(1/2) - a^(1/2))/((a + c*x)^(1/2) - a^(1/2)))*(b + c))/(b - c)^2 + (log(x)*(b + c))/(b^2 - 2*b*c + c^2) -
(log((((a + b*x)^(1/2) - (a + c*x)^(1/2))*(b - (c*((a + b*x)^(1/2) - a^(1/2)))/((a + c*x)^(1/2) - a^(1/2))))/
((a + c*x)^(1/2) - a^(1/2)))*(b + c))/(b^2 - 2*b*c + c^2) - (2*a)/(x*(b^2 - 2*b*c + c^2)) + (c*((a + b*x)^(1/2
) - a^(1/2)))/(2*(b - c)^2*((a + c*x)^(1/2) - a^(1/2)))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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