Integrand size = 25, antiderivative size = 174 \[ \int \frac {1}{x^2 \left (\sqrt {a+b x}+\sqrt {a+c x}\right )^2} \, dx=-\frac {2 a}{3 (b-c)^2 x^3}-\frac {b+c}{2 (b-c)^2 x^2}-\frac {(b+c) \sqrt {a+b x} \sqrt {a+c x}}{4 a^2 (b-c) x}-\frac {(b+c) \sqrt {a+b x} (a+c x)^{3/2}}{2 a^2 (b-c)^2 x^2}+\frac {2 (a+b x)^{3/2} (a+c x)^{3/2}}{3 a^2 (b-c)^2 x^3}+\frac {(b+c) \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a+c x}}\right )}{4 a^2} \] Output:
-2/3*a/(b-c)^2/x^3-1/2*(b+c)/(b-c)^2/x^2-1/4*(b+c)*(b*x+a)^(1/2)*(c*x+a)^( 1/2)/a^2/(b-c)/x-1/2*(b+c)*(b*x+a)^(1/2)*(c*x+a)^(3/2)/a^2/(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
Time = 10.29 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.88 \[ \int \frac {1}{x^2 \left (\sqrt {a+b x}+\sqrt {a+c x}\right )^2} \, dx=\frac {-8 a^3+2 a (b+c) x \sqrt {a+b x} \sqrt {a+c x}+\left (-3 b^2+2 b c-3 c^2\right ) x^2 \sqrt {a+b x} \sqrt {a+c x}+a^2 \left (-6 b x-6 c x+8 \sqrt {a+b x} \sqrt {a+c x}\right )+3 (b-c)^2 (b+c) x^3 \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a+c x}}\right )}{12 a^2 (b-c)^2 x^3} \] Input:
Integrate[1/(x^2*(Sqrt[a + b*x] + Sqrt[a + c*x])^2),x]
Output:
(-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]) + 3*(b - c)^2*(b + c)*x^3*ArcTanh[Sqrt[a + b*x]/Sqrt[a + c*x]])/(12*a^2*(b - c)^2*x^3)
Time = 0.72 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.92, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {7241, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {1}{x^2 \left (\sqrt {a+b x}+\sqrt {a+c x}\right )^2} \, dx\) |
| \(\Big \downarrow \) 7241 |
| \(\displaystyle \frac {\int \left (\frac {2 a}{x^4}-\frac {2 \sqrt {a+b x} \sqrt {a+c x}}{x^4}+\frac {b+c}{x^3}\right )dx}{(b-c)^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {(b+c) (b-c)^2 \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a+c x}}\right )}{4 a^2}-\frac {\left (b^2-c^2\right ) \sqrt {a+b x} \sqrt {a+c x}}{4 a^2 x}+\frac {2 (a+b x)^{3/2} (a+c x)^{3/2}}{3 a^2 x^3}-\frac {(b+c) \sqrt {a+b x} (a+c x)^{3/2}}{2 a^2 x^2}-\frac {2 a}{3 x^3}-\frac {b+c}{2 x^2}}{(b-c)^2}\) |
Input:
Int[1/(x^2*(Sqrt[a + b*x] + Sqrt[a + c*x])^2),x]
Output:
((-2*a)/(3*x^3) - (b + c)/(2*x^2) - ((b^2 - c^2)*Sqrt[a + b*x]*Sqrt[a + c* x])/(4*a^2*x) - ((b + c)*Sqrt[a + b*x]*(a + c*x)^(3/2))/(2*a^2*x^2) + (2*( a + b*x)^(3/2)*(a + c*x)^(3/2))/(3*a^2*x^3) + ((b - c)^2*(b + c)*ArcTanh[S qrt[a + b*x]/Sqrt[a + c*x]])/(4*a^2))/(b - c)^2
Int[(u_.)*((e_.)*Sqrt[(a_.) + (b_.)*(x_)^(n_.)] + (f_.)*Sqrt[(c_.) + (d_.)* (x_)^(n_.)])^(m_), x_Symbol] :> Simp[(b*e^2 - d*f^2)^m Int[ExpandIntegran d[(u*x^(m*n))/(e*Sqrt[a + b*x^n] - f*Sqrt[c + d*x^n])^m, x], x], x] /; Free Q[{a, b, c, d, e, f, n}, x] && ILtQ[m, 0] && EqQ[a*e^2 - c*f^2, 0]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.02 (sec) , antiderivative size = 457, normalized size of antiderivative = 2.63
| method | result | size |
| default | \(-\frac {b}{2 x^{2} \left (b -c \right )^{2}}-\frac {c}{2 x^{2} \left (b -c \right )^{2}}-\frac {2 a}{3 \left (b -c \right )^{2} x^{3}}-\frac {\sqrt {b x +a}\, \sqrt {c x +a}\, \left (-3 \ln \left (\frac {a \left (2 \sqrt {b c \,x^{2}+a b x +a c x +a^{2}}\, \operatorname {csgn}\left (a \right )+b x +c x +2 a \right )}{x}\right ) x^{3} b^{3}+3 \ln \left (\frac {a \left (2 \sqrt {b c \,x^{2}+a b x +a c x +a^{2}}\, \operatorname {csgn}\left (a \right )+b x +c x +2 a \right )}{x}\right ) x^{3} b^{2} c +3 \ln \left (\frac {a \left (2 \sqrt {b c \,x^{2}+a b x +a c x +a^{2}}\, \operatorname {csgn}\left (a \right )+b x +c x +2 a \right )}{x}\right ) x^{3} b \,c^{2}-3 \ln \left (\frac {a \left (2 \sqrt {b c \,x^{2}+a b x +a c x +a^{2}}\, \operatorname {csgn}\left (a \right )+b x +c x +2 a \right )}{x}\right ) x^{3} c^{3}+6 \sqrt {b c \,x^{2}+a b x +a c x +a^{2}}\, \operatorname {csgn}\left (a \right ) x^{2} b^{2}-4 \sqrt {b c \,x^{2}+a b x +a c x +a^{2}}\, \operatorname {csgn}\left (a \right ) x^{2} b c +6 \sqrt {b c \,x^{2}+a b x +a c x +a^{2}}\, \operatorname {csgn}\left (a \right ) x^{2} c^{2}-4 \,\operatorname {csgn}\left (a \right ) a \sqrt {b c \,x^{2}+a b x +a c x +a^{2}}\, x b -4 \,\operatorname {csgn}\left (a \right ) a \sqrt {b c \,x^{2}+a b x +a c x +a^{2}}\, x c -16 \sqrt {b c \,x^{2}+a b x +a c x +a^{2}}\, a^{2} \operatorname {csgn}\left (a \right )\right ) \operatorname {csgn}\left (a \right )}{24 \left (b -c \right )^{2} a^{2} \sqrt {b c \,x^{2}+a b x +a c x +a^{2}}\, x^{3}}\) | \(457\) |
Input:
int(1/x^2/((b*x+a)^(1/2)+(c*x+a)^(1/2))^2,x,method=_RETURNVERBOSE)
Output:
-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)*csgn (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)*cs gn(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*(b*c*x^2+a*b*x+a*c*x+a^2)^(1/2)*csgn(a)* x^2*b^2-4*(b*c*x^2+a*b*x+a*c*x+a^2)^(1/2)*csgn(a)*x^2*b*c+6*(b*c*x^2+a*b*x +a*c*x+a^2)^(1/2)*csgn(a)*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
Time = 0.08 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.05 \[ \int \frac {1}{x^2 \left (\sqrt {a+b x}+\sqrt {a+c x}\right )^2} \, dx=-\frac {12 \, {\left (b^{3} - b^{2} c - b c^{2} + c^{3}\right )} x^{3} \log \left (-\frac {{\left (b + c\right )} x - 2 \, \sqrt {b x + a} \sqrt {c x + a} + 2 \, a}{x}\right ) + {\left (5 \, b^{3} + 3 \, b^{2} c + 3 \, b c^{2} + 5 \, c^{3}\right )} x^{3} + 64 \, a^{3} + 8 \, {\left ({\left (3 \, b^{2} - 2 \, b c + 3 \, c^{2}\right )} x^{2} - 8 \, a^{2} - 2 \, {\left (a b + a c\right )} x\right )} \sqrt {b x + a} \sqrt {c x + a} + 48 \, {\left (a^{2} b + a^{2} c\right )} x}{96 \, {\left (a^{2} b^{2} - 2 \, a^{2} b c + a^{2} c^{2}\right )} x^{3}} \] Input:
integrate(1/x^2/((b*x+a)^(1/2)+(c*x+a)^(1/2))^2,x, algorithm="fricas")
Output:
-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)
\[ \int \frac {1}{x^2 \left (\sqrt {a+b x}+\sqrt {a+c x}\right )^2} \, dx=\int \frac {1}{x^{2} \left (\sqrt {a + b x} + \sqrt {a + c x}\right )^{2}}\, dx \] Input:
integrate(1/x**2/((b*x+a)**(1/2)+(c*x+a)**(1/2))**2,x)
Output:
Integral(1/(x**2*(sqrt(a + b*x) + sqrt(a + c*x))**2), x)
\[ \int \frac {1}{x^2 \left (\sqrt {a+b x}+\sqrt {a+c x}\right )^2} \, dx=\int { \frac {1}{x^{2} {\left (\sqrt {b x + a} + \sqrt {c x + a}\right )}^{2}} \,d x } \] Input:
integrate(1/x^2/((b*x+a)^(1/2)+(c*x+a)^(1/2))^2,x, algorithm="maxima")
Output:
integrate(1/(x^2*(sqrt(b*x + a) + sqrt(c*x + a))^2), x)
Leaf count of result is larger than twice the leaf count of optimal. 802 vs. \(2 (146) = 292\).
Time = 2.29 (sec) , antiderivative size = 802, normalized size of antiderivative = 4.61 \[ \int \frac {1}{x^2 \left (\sqrt {a+b x}+\sqrt {a+c x}\right )^2} \, dx =\text {Too large to display} \] Input:
integrate(1/x^2/((b*x+a)^(1/2)+(c*x+a)^(1/2))^2,x, algorithm="giac")
Output:
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)*sq rt(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)*(sqrt(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)*sq rt(b*x + a) - sqrt(a*b^2 + (b*x + a)*b*c - a*b*c))^4*a^3*abs(b) + 3*(5*b^1 1 - 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^1 0*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)*s qrt(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)
Time = 53.52 (sec) , antiderivative size = 1290, normalized size of antiderivative = 7.41 \[ \int \frac {1}{x^2 \left (\sqrt {a+b x}+\sqrt {a+c x}\right )^2} \, dx=\text {Too large to display} \] Input:
int(1/(x^2*((a + b*x)^(1/2) + (a + c*x)^(1/2))^2),x)
Output:
(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...
Time = 0.40 (sec) , antiderivative size = 363, normalized size of antiderivative = 2.09 \[ \int \frac {1}{x^2 \left (\sqrt {a+b x}+\sqrt {a+c x}\right )^2} \, dx=\frac {16 \sqrt {c x +a}\, \sqrt {b x +a}\, a^{2}+4 \sqrt {c x +a}\, \sqrt {b x +a}\, a b x +4 \sqrt {c x +a}\, \sqrt {b x +a}\, a c x -6 \sqrt {c x +a}\, \sqrt {b x +a}\, b^{2} x^{2}+4 \sqrt {c x +a}\, \sqrt {b x +a}\, b c \,x^{2}-6 \sqrt {c x +a}\, \sqrt {b x +a}\, c^{2} x^{2}+3 \,\mathrm {log}\left (-2 \sqrt {c x +a}\, \sqrt {b x +a}\, b -2 a b -b^{2} x -b c x \right ) b^{3} x^{3}-3 \,\mathrm {log}\left (-2 \sqrt {c x +a}\, \sqrt {b x +a}\, b -2 a b -b^{2} x -b c x \right ) b^{2} c \,x^{3}-3 \,\mathrm {log}\left (-2 \sqrt {c x +a}\, \sqrt {b x +a}\, b -2 a b -b^{2} x -b c x \right ) b \,c^{2} x^{3}+3 \,\mathrm {log}\left (-2 \sqrt {c x +a}\, \sqrt {b x +a}\, b -2 a b -b^{2} x -b c x \right ) c^{3} x^{3}-3 \,\mathrm {log}\left (b x \right ) b^{3} x^{3}+3 \,\mathrm {log}\left (b x \right ) b^{2} c \,x^{3}+3 \,\mathrm {log}\left (b x \right ) b \,c^{2} x^{3}-3 \,\mathrm {log}\left (b x \right ) c^{3} x^{3}-16 a^{3}-12 a^{2} b x -12 a^{2} c x}{24 a^{2} x^{3} \left (b^{2}-2 b c +c^{2}\right )} \] Input:
int(1/x^2/((b*x+a)^(1/2)+(c*x+a)^(1/2))^2,x)
Output:
(16*sqrt(a + c*x)*sqrt(a + b*x)*a**2 + 4*sqrt(a + c*x)*sqrt(a + b*x)*a*b*x + 4*sqrt(a + c*x)*sqrt(a + b*x)*a*c*x - 6*sqrt(a + c*x)*sqrt(a + b*x)*b** 2*x**2 + 4*sqrt(a + c*x)*sqrt(a + b*x)*b*c*x**2 - 6*sqrt(a + c*x)*sqrt(a + b*x)*c**2*x**2 + 3*log( - 2*sqrt(a + c*x)*sqrt(a + b*x)*b - 2*a*b - b**2* x - b*c*x)*b**3*x**3 - 3*log( - 2*sqrt(a + c*x)*sqrt(a + b*x)*b - 2*a*b - b**2*x - b*c*x)*b**2*c*x**3 - 3*log( - 2*sqrt(a + c*x)*sqrt(a + b*x)*b - 2 *a*b - b**2*x - b*c*x)*b*c**2*x**3 + 3*log( - 2*sqrt(a + c*x)*sqrt(a + b*x )*b - 2*a*b - b**2*x - b*c*x)*c**3*x**3 - 3*log(b*x)*b**3*x**3 + 3*log(b*x )*b**2*c*x**3 + 3*log(b*x)*b*c**2*x**3 - 3*log(b*x)*c**3*x**3 - 16*a**3 - 12*a**2*b*x - 12*a**2*c*x)/(24*a**2*x**3*(b**2 - 2*b*c + c**2))