Optimal. Leaf size=123 \[ \frac{\sqrt{a+b x} (a+c x)^{3/2}}{a x^2 (b-c)^2}-\frac{a}{x^2 (b-c)^2}+\frac{\sqrt{a+b x} \sqrt{a+c x}}{2 a x (b-c)}-\frac{\tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a+c x}}\right )}{2 a}-\frac{b+c}{x (b-c)^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.201447, antiderivative size = 123, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16, Rules used = {6690, 94, 93, 208} \[ \frac{\sqrt{a+b x} (a+c x)^{3/2}}{a x^2 (b-c)^2}-\frac{a}{x^2 (b-c)^2}+\frac{\sqrt{a+b x} \sqrt{a+c x}}{2 a x (b-c)}-\frac{\tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a+c x}}\right )}{2 a}-\frac{b+c}{x (b-c)^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6690
Rule 94
Rule 93
Rule 208
Rubi steps
\begin{align*} \int \frac{1}{x \left (\sqrt{a+b x}+\sqrt{a+c x}\right )^2} \, dx &=\frac{\int \left (\frac{2 a}{x^3}+\frac{b \left (1+\frac{c}{b}\right )}{x^2}-\frac{2 \sqrt{a+b x} \sqrt{a+c x}}{x^3}\right ) \, dx}{(b-c)^2}\\ &=-\frac{a}{(b-c)^2 x^2}-\frac{b+c}{(b-c)^2 x}-\frac{2 \int \frac{\sqrt{a+b x} \sqrt{a+c x}}{x^3} \, dx}{(b-c)^2}\\ &=-\frac{a}{(b-c)^2 x^2}-\frac{b+c}{(b-c)^2 x}+\frac{\sqrt{a+b x} (a+c x)^{3/2}}{a (b-c)^2 x^2}-\frac{\int \frac{\sqrt{a+c x}}{x^2 \sqrt{a+b x}} \, dx}{2 (b-c)}\\ &=-\frac{a}{(b-c)^2 x^2}-\frac{b+c}{(b-c)^2 x}+\frac{\sqrt{a+b x} \sqrt{a+c x}}{2 a (b-c) x}+\frac{\sqrt{a+b x} (a+c x)^{3/2}}{a (b-c)^2 x^2}+\frac{1}{4} \int \frac{1}{x \sqrt{a+b x} \sqrt{a+c x}} \, dx\\ &=-\frac{a}{(b-c)^2 x^2}-\frac{b+c}{(b-c)^2 x}+\frac{\sqrt{a+b x} \sqrt{a+c x}}{2 a (b-c) x}+\frac{\sqrt{a+b x} (a+c x)^{3/2}}{a (b-c)^2 x^2}+\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{-a+a x^2} \, dx,x,\frac{\sqrt{a+b x}}{\sqrt{a+c x}}\right )\\ &=-\frac{a}{(b-c)^2 x^2}-\frac{b+c}{(b-c)^2 x}+\frac{\sqrt{a+b x} \sqrt{a+c x}}{2 a (b-c) x}+\frac{\sqrt{a+b x} (a+c x)^{3/2}}{a (b-c)^2 x^2}-\frac{\tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a+c x}}\right )}{2 a}\\ \end{align*}
Mathematica [A] time = 0.147289, size = 109, normalized size = 0.89 \[ \frac{-2 a^2-x^2 (b-c)^2 \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a+c x}}\right )+2 a \left (\sqrt{a+b x} \sqrt{a+c x}-b x-c x\right )+x (b+c) \sqrt{a+b x} \sqrt{a+c x}}{2 a x^2 (b-c)^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] time = 0.015, size = 313, normalized size = 2.5 \begin{align*} -{\frac{b}{x \left ( b-c \right ) ^{2}}}-{\frac{c}{x \left ( b-c \right ) ^{2}}}-{\frac{a}{ \left ( b-c \right ) ^{2}{x}^{2}}}+{\frac{{\it csgn} \left ( a \right ) }{4\, \left ( b-c \right ) ^{2}a{x}^{2}}\sqrt{bx+a}\sqrt{cx+a} \left ( -\ln \left ({\frac{a}{x} \left ( 2\,{\it csgn} \left ( a \right ) \sqrt{bc{x}^{2}+abx+acx+{a}^{2}}+bx+cx+2\,a \right ) } \right ){x}^{2}{b}^{2}+2\,\ln \left ({\frac{a \left ( 2\,{\it csgn} \left ( a \right ) \sqrt{bc{x}^{2}+abx+acx+{a}^{2}}+bx+cx+2\,a \right ) }{x}} \right ){x}^{2}bc-\ln \left ({\frac{a}{x} \left ( 2\,{\it csgn} \left ( a \right ) \sqrt{bc{x}^{2}+abx+acx+{a}^{2}}+bx+cx+2\,a \right ) } \right ){x}^{2}{c}^{2}+2\,{\it csgn} \left ( a \right ) \sqrt{bc{x}^{2}+abx+acx+{a}^{2}}xb+2\,{\it csgn} \left ( a \right ) \sqrt{bc{x}^{2}+abx+acx+{a}^{2}}xc+4\,{\it csgn} \left ( a \right ) a\sqrt{bc{x}^{2}+abx+acx+{a}^{2}} \right ){\frac{1}{\sqrt{bc{x}^{2}+abx+acx+{a}^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x{\left (\sqrt{b x + a} + \sqrt{c x + a}\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.30788, size = 308, normalized size = 2.5 \begin{align*} \frac{4 \,{\left (b^{2} - 2 \, b c + c^{2}\right )} x^{2} \log \left (-\frac{{\left (b + c\right )} x - 2 \, \sqrt{b x + a} \sqrt{c x + a} + 2 \, a}{x}\right ) +{\left (b^{2} + 2 \, b c + c^{2}\right )} x^{2} + 8 \,{\left ({\left (b + c\right )} x + 2 \, a\right )} \sqrt{b x + a} \sqrt{c x + a} - 16 \, a^{2} - 16 \,{\left (a b + a c\right )} x}{16 \,{\left (a b^{2} - 2 \, a b c + a c^{2}\right )} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x \left (\sqrt{a + b x} + \sqrt{a + c x}\right )^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]