3.5.37 \(\int \frac {1}{x^2 (\sqrt {a+b x}+\sqrt {a+c x})^2} \, dx\) [437]

Optimal. Leaf size=174 \[ -\frac {2 a}{3 (b-c)^2 x^3}-\frac {b+c}{2 (b-c)^2 x^2}-\frac {(b+c) \sqrt {a+b x} \sqrt {a+c x}}{4 a^2 (b-c) x}-\frac {(b+c) \sqrt {a+b x} (a+c x)^{3/2}}{2 a^2 (b-c)^2 x^2}+\frac {2 (a+b x)^{3/2} (a+c x)^{3/2}}{3 a^2 (b-c)^2 x^3}+\frac {(b+c) \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a+c x}}\right )}{4 a^2} \]

[Out]

-2/3*a/(b-c)^2/x^3+1/2*(-b-c)/(b-c)^2/x^2+2/3*(b*x+a)^(3/2)*(c*x+a)^(3/2)/a^2/(b-c)^2/x^3+1/4*(b+c)*arctanh((b
*x+a)^(1/2)/(c*x+a)^(1/2))/a^2-1/2*(b+c)*(c*x+a)^(3/2)*(b*x+a)^(1/2)/a^2/(b-c)^2/x^2-1/4*(b+c)*(b*x+a)^(1/2)*(
c*x+a)^(1/2)/a^2/(b-c)/x

________________________________________________________________________________________

Rubi [A]
time = 0.16, antiderivative size = 174, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {6822, 98, 96, 95, 214} \begin {gather*} \frac {2 (a+b x)^{3/2} (a+c x)^{3/2}}{3 a^2 x^3 (b-c)^2}-\frac {(b+c) \sqrt {a+b x} (a+c x)^{3/2}}{2 a^2 x^2 (b-c)^2}-\frac {(b+c) \sqrt {a+b x} \sqrt {a+c x}}{4 a^2 x (b-c)}+\frac {(b+c) \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a+c x}}\right )}{4 a^2}-\frac {2 a}{3 x^3 (b-c)^2}-\frac {b+c}{2 x^2 (b-c)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*a)/(3*(b - c)^2*x^3) - (b + c)/(2*(b - c)^2*x^2) - ((b + c)*Sqrt[a + b*x]*Sqrt[a + c*x])/(4*a^2*(b - c)*x)
- ((b + c)*Sqrt[a + b*x]*(a + c*x)^(3/2))/(2*a^2*(b - c)^2*x^2) + (2*(a + b*x)^(3/2)*(a + c*x)^(3/2))/(3*a^2*
(b - c)^2*x^3) + ((b + c)*ArcTanh[Sqrt[a + b*x]/Sqrt[a + c*x]])/(4*a^2)

Rule 95

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 96

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 + 1)/((m + 1)*(b*e - a*f))), x] - Dist[n*((d*e - c*f)/((m + 1)*(b*e - a*f
))), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[
m + n + p + 2, 0] && GtQ[n, 0] && (SumSimplerQ[m, 1] ||  !SumSimplerQ[p, 1]) && NeQ[m, -1]

Rule 98

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[b*(a +
b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Dist[(a*d*f*(m + 1)
+ b*c*f*(n + 1) + b*d*e*(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*
x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[Simplify[m + n + p + 3], 0] && (LtQ[m, -1] || Sum
SimplerQ[m, 1])

Rule 214

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

Rule 6822

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}{x^2 \left (\sqrt {a+b x}+\sqrt {a+c x}\right )^2} \, dx &=\frac {\int \left (\frac {2 a}{x^4}+\frac {b \left (1+\frac {c}{b}\right )}{x^3}-\frac {2 \sqrt {a+b x} \sqrt {a+c x}}{x^4}\right ) \, dx}{(b-c)^2}\\ &=-\frac {2 a}{3 (b-c)^2 x^3}-\frac {b+c}{2 (b-c)^2 x^2}-\frac {2 \int \frac {\sqrt {a+b x} \sqrt {a+c x}}{x^4} \, dx}{(b-c)^2}\\ &=-\frac {2 a}{3 (b-c)^2 x^3}-\frac {b+c}{2 (b-c)^2 x^2}+\frac {2 (a+b x)^{3/2} (a+c x)^{3/2}}{3 a^2 (b-c)^2 x^3}+\frac {(b+c) \int \frac {\sqrt {a+b x} \sqrt {a+c x}}{x^3} \, dx}{a (b-c)^2}\\ &=-\frac {2 a}{3 (b-c)^2 x^3}-\frac {b+c}{2 (b-c)^2 x^2}-\frac {(b+c) \sqrt {a+b x} (a+c x)^{3/2}}{2 a^2 (b-c)^2 x^2}+\frac {2 (a+b x)^{3/2} (a+c x)^{3/2}}{3 a^2 (b-c)^2 x^3}+\frac {(b+c) \int \frac {\sqrt {a+c x}}{x^2 \sqrt {a+b x}} \, dx}{4 a (b-c)}\\ &=-\frac {2 a}{3 (b-c)^2 x^3}-\frac {b+c}{2 (b-c)^2 x^2}-\frac {(b+c) \sqrt {a+b x} \sqrt {a+c x}}{4 a^2 (b-c) x}-\frac {(b+c) \sqrt {a+b x} (a+c x)^{3/2}}{2 a^2 (b-c)^2 x^2}+\frac {2 (a+b x)^{3/2} (a+c x)^{3/2}}{3 a^2 (b-c)^2 x^3}-\frac {(b+c) \int \frac {1}{x \sqrt {a+b x} \sqrt {a+c x}} \, dx}{8 a}\\ &=-\frac {2 a}{3 (b-c)^2 x^3}-\frac {b+c}{2 (b-c)^2 x^2}-\frac {(b+c) \sqrt {a+b x} \sqrt {a+c x}}{4 a^2 (b-c) x}-\frac {(b+c) \sqrt {a+b x} (a+c x)^{3/2}}{2 a^2 (b-c)^2 x^2}+\frac {2 (a+b x)^{3/2} (a+c x)^{3/2}}{3 a^2 (b-c)^2 x^3}-\frac {(b+c) \text {Subst}\left (\int \frac {1}{-a+a x^2} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {a+c x}}\right )}{4 a}\\ &=-\frac {2 a}{3 (b-c)^2 x^3}-\frac {b+c}{2 (b-c)^2 x^2}-\frac {(b+c) \sqrt {a+b x} \sqrt {a+c x}}{4 a^2 (b-c) x}-\frac {(b+c) \sqrt {a+b x} (a+c x)^{3/2}}{2 a^2 (b-c)^2 x^2}+\frac {2 (a+b x)^{3/2} (a+c x)^{3/2}}{3 a^2 (b-c)^2 x^3}+\frac {(b+c) \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a+c x}}\right )}{4 a^2}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 10.27, size = 164, normalized size = 0.94 \begin {gather*} \frac {\frac {2 \left (-8 a^3+2 a (b+c) x \sqrt {a+b x} \sqrt {a+c x}+\left (-3 b^2+2 b c-3 c^2\right ) x^2 \sqrt {a+b x} \sqrt {a+c x}+a^2 \left (-6 b x-6 c x+8 \sqrt {a+b x} \sqrt {a+c x}\right )\right )}{(b-c)^2 x^3}-3 (b+c) \log (x)+3 (b+c) \log \left (2 a+b x+c x+2 \sqrt {a+b x} \sqrt {a+c x}\right )}{24 a^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((2*(-8*a^3 + 2*a*(b + c)*x*Sqrt[a + b*x]*Sqrt[a + c*x] + (-3*b^2 + 2*b*c - 3*c^2)*x^2*Sqrt[a + b*x]*Sqrt[a +
c*x] + a^2*(-6*b*x - 6*c*x + 8*Sqrt[a + b*x]*Sqrt[a + c*x])))/((b - c)^2*x^3) - 3*(b + c)*Log[x] + 3*(b + c)*L
og[2*a + b*x + c*x + 2*Sqrt[a + b*x]*Sqrt[a + c*x]])/(24*a^2)

________________________________________________________________________________________

Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.02, size = 457, normalized size = 2.63

method	result	size
default	\(-\frac {b}{2 x^{2} \left (b -c \right )^{2}}-\frac {c}{2 x^{2} \left (b -c \right )^{2}}-\frac {2 a}{3 \left (b -c \right )^{2} x^{3}}-\frac {\sqrt {b x +a}\, \sqrt {c x +a}\, \left (-3 \ln \left (\frac {a \left (2 \sqrt {b c \,x^{2}+a b x +a c x +a^{2}}\, \mathrm {csgn}\left (a \right )+b x +c x +2 a \right )}{x}\right ) x^{3} b^{3}+3 \ln \left (\frac {a \left (2 \sqrt {b c \,x^{2}+a b x +a c x +a^{2}}\, \mathrm {csgn}\left (a \right )+b x +c x +2 a \right )}{x}\right ) x^{3} b^{2} c +3 \ln \left (\frac {a \left (2 \sqrt {b c \,x^{2}+a b x +a c x +a^{2}}\, \mathrm {csgn}\left (a \right )+b x +c x +2 a \right )}{x}\right ) x^{3} b \,c^{2}-3 \ln \left (\frac {a \left (2 \sqrt {b c \,x^{2}+a b x +a c x +a^{2}}\, \mathrm {csgn}\left (a \right )+b x +c x +2 a \right )}{x}\right ) x^{3} c^{3}+6 \,\mathrm {csgn}\left (a \right ) \sqrt {b c \,x^{2}+a b x +a c x +a^{2}}\, x^{2} b^{2}-4 \,\mathrm {csgn}\left (a \right ) \sqrt {b c \,x^{2}+a b x +a c x +a^{2}}\, x^{2} b c +6 \,\mathrm {csgn}\left (a \right ) \sqrt {b c \,x^{2}+a b x +a c x +a^{2}}\, x^{2} c^{2}-4 \,\mathrm {csgn}\left (a \right ) a \sqrt {b c \,x^{2}+a b x +a c x +a^{2}}\, x b -4 \,\mathrm {csgn}\left (a \right ) a \sqrt {b c \,x^{2}+a b x +a c x +a^{2}}\, x c -16 \sqrt {b c \,x^{2}+a b x +a c x +a^{2}}\, a^{2} \mathrm {csgn}\left (a \right )\right ) \mathrm {csgn}\left (a \right )}{24 \left (b -c \right )^{2} a^{2} \sqrt {b c \,x^{2}+a b x +a c x +a^{2}}\, x^{3}}\)	\(457\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2/x^2/(b-c)^2*b-1/2/x^2/(b-c)^2*c-2/3*a/(b-c)^2/x^3-1/24/(b-c)^2*(b*x+a)^(1/2)*(c*x+a)^(1/2)/a^2*(-3*ln(a*(
2*(b*c*x^2+a*b*x+a*c*x+a^2)^(1/2)*csgn(a)+b*x+c*x+2*a)/x)*x^3*b^3+3*ln(a*(2*(b*c*x^2+a*b*x+a*c*x+a^2)^(1/2)*cs
gn(a)+b*x+c*x+2*a)/x)*x^3*b^2*c+3*ln(a*(2*(b*c*x^2+a*b*x+a*c*x+a^2)^(1/2)*csgn(a)+b*x+c*x+2*a)/x)*x^3*b*c^2-3*
ln(a*(2*(b*c*x^2+a*b*x+a*c*x+a^2)^(1/2)*csgn(a)+b*x+c*x+2*a)/x)*x^3*c^3+6*csgn(a)*(b*c*x^2+a*b*x+a*c*x+a^2)^(1
/2)*x^2*b^2-4*csgn(a)*(b*c*x^2+a*b*x+a*c*x+a^2)^(1/2)*x^2*b*c+6*csgn(a)*(b*c*x^2+a*b*x+a*c*x+a^2)^(1/2)*x^2*c^
2-4*csgn(a)*a*(b*c*x^2+a*b*x+a*c*x+a^2)^(1/2)*x*b-4*csgn(a)*a*(b*c*x^2+a*b*x+a*c*x+a^2)^(1/2)*x*c-16*(b*c*x^2+
a*b*x+a*c*x+a^2)^(1/2)*a^2*csgn(a))*csgn(a)/(b*c*x^2+a*b*x+a*c*x+a^2)^(1/2)/x^3

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

Fricas [A]
time = 0.33, size = 182, normalized size = 1.05 \begin {gather*} -\frac {12 \, {\left (b^{3} - b^{2} c - b c^{2} + c^{3}\right )} x^{3} \log \left (-\frac {{\left (b + c\right )} x - 2 \, \sqrt {b x + a} \sqrt {c x + a} + 2 \, a}{x}\right ) + {\left (5 \, b^{3} + 3 \, b^{2} c + 3 \, b c^{2} + 5 \, c^{3}\right )} x^{3} + 64 \, a^{3} + 8 \, {\left ({\left (3 \, b^{2} - 2 \, b c + 3 \, c^{2}\right )} x^{2} - 8 \, a^{2} - 2 \, {\left (a b + a c\right )} x\right )} \sqrt {b x + a} \sqrt {c x + a} + 48 \, {\left (a^{2} b + a^{2} c\right )} x}{96 \, {\left (a^{2} b^{2} - 2 \, a^{2} b c + a^{2} c^{2}\right )} x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/96*(12*(b^3 - b^2*c - b*c^2 + c^3)*x^3*log(-((b + c)*x - 2*sqrt(b*x + a)*sqrt(c*x + a) + 2*a)/x) + (5*b^3 +
3*b^2*c + 3*b*c^2 + 5*c^3)*x^3 + 64*a^3 + 8*((3*b^2 - 2*b*c + 3*c^2)*x^2 - 8*a^2 - 2*(a*b + a*c)*x)*sqrt(b*x
+ a)*sqrt(c*x + a) + 48*(a^2*b + a^2*c)*x)/((a^2*b^2 - 2*a^2*b*c + a^2*c^2)*x^3)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 802 vs. \(2 (146) = 292\).
time = 6.36, size = 802, normalized size = 4.61 \begin {gather*} -\frac {\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 )}{4 \, \sqrt {-b c} a^{2} b} + \frac {3 \, {\left (b^{3} - b^{2} c - b c^{2} + c^{3}\right )} \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 )}^{10} {\left | b \right |} - 3 \, {\left (5 \, b^{5} + 22 \, b^{3} c^{2} + 5 \, b c^{4}\right )} \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 )}^{8} a {\left | b \right |} + 2 \, {\left (15 \, b^{7} - b^{6} c + 18 \, b^{5} c^{2} + 18 \, b^{4} c^{3} - b^{3} c^{4} + 15 \, b^{2} c^{5}\right )} \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 )}^{6} a^{2} {\left | b \right |} - 6 \, {\left (5 \, b^{9} - 6 \, b^{8} c - 5 \, b^{7} c^{2} + 12 \, b^{6} c^{3} - 5 \, b^{5} c^{4} - 6 \, b^{4} c^{5} + 5 \, b^{3} c^{6}\right )} \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 )}^{4} a^{3} {\left | b \right |} + 3 \, {\left (5 \, b^{11} - 17 \, b^{10} c + 21 \, b^{9} c^{2} - 9 \, b^{8} c^{3} - 9 \, b^{7} c^{4} + 21 \, b^{6} c^{5} - 17 \, b^{5} c^{6} + 5 \, b^{4} c^{7}\right )} \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^{4} {\left | b \right |} - {\left (3 \, b^{13} - 20 \, b^{12} c + 60 \, b^{11} c^{2} - 108 \, b^{10} c^{3} + 130 \, b^{9} c^{4} - 108 \, b^{8} c^{5} + 60 \, b^{7} c^{6} - 20 \, b^{6} c^{7} + 3 \, b^{5} c^{8}\right )} \sqrt {b c} a^{5} {\left | b \right |}}{6 \, {\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 )}^{3} {\left (b^{2} - 2 \, b c + c^{2}\right )} a} - \frac {3 \, {\left (b x + a\right )} b^{3} + a b^{3} + 3 \, {\left (b x + a\right )} b^{2} c - 3 \, a b^{2} c}{6 \, {\left (b^{2} - 2 \, b c + c^{2}\right )} b^{3} x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*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))/(sqrt(-b*c)*a^2*b) + 1/6*(3*(b^3 - b^2*c - b*c^2 + c^3)*sqrt(b*c)*(sqrt(b*c)*
sqrt(b*x + a) - sqrt(a*b^2 + (b*x + a)*b*c - a*b*c))^10*abs(b) - 3*(5*b^5 + 22*b^3*c^2 + 5*b*c^4)*sqrt(b*c)*(s
qrt(b*c)*sqrt(b*x + a) - sqrt(a*b^2 + (b*x + a)*b*c - a*b*c))^8*a*abs(b) + 2*(15*b^7 - b^6*c + 18*b^5*c^2 + 18
*b^4*c^3 - b^3*c^4 + 15*b^2*c^5)*sqrt(b*c)*(sqrt(b*c)*sqrt(b*x + a) - sqrt(a*b^2 + (b*x + a)*b*c - a*b*c))^6*a
^2*abs(b) - 6*(5*b^9 - 6*b^8*c - 5*b^7*c^2 + 12*b^6*c^3 - 5*b^5*c^4 - 6*b^4*c^5 + 5*b^3*c^6)*sqrt(b*c)*(sqrt(b
*c)*sqrt(b*x + a) - sqrt(a*b^2 + (b*x + a)*b*c - a*b*c))^4*a^3*abs(b) + 3*(5*b^11 - 17*b^10*c + 21*b^9*c^2 - 9
*b^8*c^3 - 9*b^7*c^4 + 21*b^6*c^5 - 17*b^5*c^6 + 5*b^4*c^7)*sqrt(b*c)*(sqrt(b*c)*sqrt(b*x + a) - sqrt(a*b^2 +
(b*x + a)*b*c - a*b*c))^2*a^4*abs(b) - (3*b^13 - 20*b^12*c + 60*b^11*c^2 - 108*b^10*c^3 + 130*b^9*c^4 - 108*b^
8*c^5 + 60*b^7*c^6 - 20*b^6*c^7 + 3*b^5*c^8)*sqrt(b*c)*a^5*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)^3*(b^2 - 2*b*c + c^2)*a) - 1/6*(3*(b*x + a)*b^3 + a*b^3 + 3*(b*x + a)*b^2*c -
3*a*b^2*c)/((b^2 - 2*b*c + c^2)*b^3*x^3)

________________________________________________________________________________________

Mupad [B]
time = 18.74, size = 1290, normalized size = 7.41 \begin {gather*} \frac {\ln \left (\frac {\sqrt {a+b\,x}-\sqrt {a}}{\sqrt {a+c\,x}-\sqrt {a}}\right )\,\left (b+c\right )}{8\,a^2}-\frac {\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^7\,\left (3\,b^5\,c-15\,b^4\,c^2+3\,b^3\,c^3+3\,b^2\,c^4-15\,b\,c^5+3\,c^6\right )}{{\left (\sqrt {a+c\,x}-\sqrt {a}\right )}^7}-\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^5\,\left (-b^6+26\,b^5\,c+4\,b^4\,c^2+4\,b^3\,c^3+26\,b^2\,c^4-b\,c^5\right )}{{\left (\sqrt {a+c\,x}-\sqrt {a}\right )}^5}-\frac {b^6}{3}+\frac {\left (b^6+c\,b^5\right )\,\left (\sqrt {a+b\,x}-\sqrt {a}\right )}{\sqrt {a+c\,x}-\sqrt {a}}-\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^8\,\left (b^4\,c^2-6\,b^3\,c^3+7\,b^2\,c^4-6\,b\,c^5+c^6\right )}{{\left (\sqrt {a+c\,x}-\sqrt {a}\right )}^8}+\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^6\,\left (-\frac {5\,b^6}{3}+6\,b^5\,c+30\,b^4\,c^2-24\,b^3\,c^3+30\,b^2\,c^4+6\,b\,c^5-\frac {5\,c^6}{3}\right )}{{\left (\sqrt {a+c\,x}-\sqrt {a}\right )}^6}-\frac {\left (\frac {17\,b^6}{3}+\frac {17\,b^3\,c^3}{3}\right )\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^3}{{\left (\sqrt {a+c\,x}-\sqrt {a}\right )}^3}+\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^2\,\left (b^6-4\,b^5\,c+b^4\,c^2\right )}{{\left (\sqrt {a+c\,x}-\sqrt {a}\right )}^2}+\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^4\,\left (5\,b^6+18\,b^5\,c-6\,b^4\,c^2+18\,b^3\,c^3+5\,b^2\,c^4\right )}{{\left (\sqrt {a+c\,x}-\sqrt {a}\right )}^4}}{\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^5\,\left (96\,a^2\,b^5+96\,a^2\,b^4\,c-384\,a^2\,b^3\,c^2+96\,a^2\,b^2\,c^3+96\,a^2\,b\,c^4\right )}{{\left (\sqrt {a+c\,x}-\sqrt {a}\right )}^5}-\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^8\,\left (96\,a^2\,b^3\,c^2-96\,a^2\,b^2\,c^3-96\,a^2\,b\,c^4+96\,a^2\,c^5\right )}{{\left (\sqrt {a+c\,x}-\sqrt {a}\right )}^8}-\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^6\,\left (32\,a^2\,b^5+224\,a^2\,b^4\,c-256\,a^2\,b^3\,c^2-256\,a^2\,b^2\,c^3+224\,a^2\,b\,c^4+32\,a^2\,c^5\right )}{{\left (\sqrt {a+c\,x}-\sqrt {a}\right )}^6}-\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^4\,\left (96\,a^2\,b^5-96\,a^2\,b^4\,c-96\,a^2\,b^3\,c^2+96\,a^2\,b^2\,c^3\right )}{{\left (\sqrt {a+c\,x}-\sqrt {a}\right )}^4}+\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^7\,\left (96\,a^2\,b^4\,c+96\,a^2\,b^3\,c^2-384\,a^2\,b^2\,c^3+96\,a^2\,b\,c^4+96\,a^2\,c^5\right )}{{\left (\sqrt {a+c\,x}-\sqrt {a}\right )}^7}+\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^3\,\left (32\,a^2\,b^5-64\,a^2\,b^4\,c+32\,a^2\,b^3\,c^2\right )}{{\left (\sqrt {a+c\,x}-\sqrt {a}\right )}^3}+\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^9\,\left (32\,a^2\,b^2\,c^3-64\,a^2\,b\,c^4+32\,a^2\,c^5\right )}{{\left (\sqrt {a+c\,x}-\sqrt {a}\right )}^9}}-\frac {\left (\frac {c\,\left (3\,b^2+8\,b\,c+3\,c^2\right )}{16\,a^2\,{\left (b-c\right )}^2}-\frac {c\,\left (4\,b^2+17\,b\,c+4\,c^2\right )}{32\,a^2\,{\left (b-c\right )}^2}\right )\,\left (\sqrt {a+b\,x}-\sqrt {a}\right )}{\sqrt {a+c\,x}-\sqrt {a}}-\frac {\ln \left (\frac {\left (\sqrt {a+b\,x}-\sqrt {a+c\,x}\right )\,\left (b-\frac {c\,\left (\sqrt {a+b\,x}-\sqrt {a}\right )}{\sqrt {a+c\,x}-\sqrt {a}}\right )}{\sqrt {a+c\,x}-\sqrt {a}}\right )\,\left (b+c\right )}{8\,a^2}-\frac {\frac {2\,a}{3}+x\,\left (\frac {b}{2}+\frac {c}{2}\right )}{x^3\,\left (b^2-2\,b\,c+c^2\right )}+\frac {c^3\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^3}{96\,a^2\,{\left (b-c\right )}^2\,{\left (\sqrt {a+c\,x}-\sqrt {a}\right )}^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(log(((a + b*x)^(1/2) - a^(1/2))/((a + c*x)^(1/2) - a^(1/2)))*(b + c))/(8*a^2) - ((((a + b*x)^(1/2) - a^(1/2))
^7*(3*b^5*c - 15*b*c^5 + 3*c^6 + 3*b^2*c^4 + 3*b^3*c^3 - 15*b^4*c^2))/((a + c*x)^(1/2) - a^(1/2))^7 - (((a + b
*x)^(1/2) - a^(1/2))^5*(26*b^5*c - b*c^5 - b^6 + 26*b^2*c^4 + 4*b^3*c^3 + 4*b^4*c^2))/((a + c*x)^(1/2) - a^(1/
2))^5 - b^6/3 + ((b^5*c + b^6)*((a + b*x)^(1/2) - a^(1/2)))/((a + c*x)^(1/2) - a^(1/2)) - (((a + b*x)^(1/2) -
a^(1/2))^8*(c^6 - 6*b*c^5 + 7*b^2*c^4 - 6*b^3*c^3 + b^4*c^2))/((a + c*x)^(1/2) - a^(1/2))^8 + (((a + b*x)^(1/2
) - a^(1/2))^6*(6*b*c^5 + 6*b^5*c - (5*b^6)/3 - (5*c^6)/3 + 30*b^2*c^4 - 24*b^3*c^3 + 30*b^4*c^2))/((a + c*x)^
(1/2) - a^(1/2))^6 - (((17*b^6)/3 + (17*b^3*c^3)/3)*((a + b*x)^(1/2) - a^(1/2))^3)/((a + c*x)^(1/2) - a^(1/2))
^3 + (((a + b*x)^(1/2) - a^(1/2))^2*(b^6 - 4*b^5*c + b^4*c^2))/((a + c*x)^(1/2) - a^(1/2))^2 + (((a + b*x)^(1/
2) - a^(1/2))^4*(18*b^5*c + 5*b^6 + 5*b^2*c^4 + 18*b^3*c^3 - 6*b^4*c^2))/((a + c*x)^(1/2) - a^(1/2))^4)/((((a
+ b*x)^(1/2) - a^(1/2))^5*(96*a^2*b^5 + 96*a^2*b*c^4 + 96*a^2*b^4*c + 96*a^2*b^2*c^3 - 384*a^2*b^3*c^2))/((a +
c*x)^(1/2) - a^(1/2))^5 - (((a + b*x)^(1/2) - a^(1/2))^8*(96*a^2*c^5 - 96*a^2*b*c^4 - 96*a^2*b^2*c^3 + 96*a^2
*b^3*c^2))/((a + c*x)^(1/2) - a^(1/2))^8 - (((a + b*x)^(1/2) - a^(1/2))^6*(32*a^2*b^5 + 32*a^2*c^5 + 224*a^2*b
*c^4 + 224*a^2*b^4*c - 256*a^2*b^2*c^3 - 256*a^2*b^3*c^2))/((a + c*x)^(1/2) - a^(1/2))^6 - (((a + b*x)^(1/2) -
a^(1/2))^4*(96*a^2*b^5 - 96*a^2*b^4*c + 96*a^2*b^2*c^3 - 96*a^2*b^3*c^2))/((a + c*x)^(1/2) - a^(1/2))^4 + (((
a + b*x)^(1/2) - a^(1/2))^7*(96*a^2*c^5 + 96*a^2*b*c^4 + 96*a^2*b^4*c - 384*a^2*b^2*c^3 + 96*a^2*b^3*c^2))/((a
+ c*x)^(1/2) - a^(1/2))^7 + (((a + b*x)^(1/2) - a^(1/2))^3*(32*a^2*b^5 - 64*a^2*b^4*c + 32*a^2*b^3*c^2))/((a
+ c*x)^(1/2) - a^(1/2))^3 + (((a + b*x)^(1/2) - a^(1/2))^9*(32*a^2*c^5 - 64*a^2*b*c^4 + 32*a^2*b^2*c^3))/((a +
c*x)^(1/2) - a^(1/2))^9) - (((c*(8*b*c + 3*b^2 + 3*c^2))/(16*a^2*(b - c)^2) - (c*(17*b*c + 4*b^2 + 4*c^2))/(3
2*a^2*(b - c)^2))*((a + b*x)^(1/2) - a^(1/2)))/((a + c*x)^(1/2) - a^(1/2)) - (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))/(8*a^2) - ((2*a)/3 + x*(b/2 + c/2))/(x^3*(b^2 - 2*b*c + c^2)) + (c^3*((a + b*x)^(1/2) - a^(1/2))^3)/(96*a^
2*(b - c)^2*((a + c*x)^(1/2) - a^(1/2))^3)

________________________________________________________________________________________