Optimal. Leaf size=164 \[ -\frac{2 a \sqrt{a+b x}}{x^2 (b-c)^3}+\frac{2 a \sqrt{a+c x}}{x^2 (b-c)^3}-\frac{(2 b+3 c) \sqrt{a+b x}}{x (b-c)^3}+\frac{(3 b+2 c) \sqrt{a+c x}}{x (b-c)^3}-\frac{3 b c \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{\sqrt{a} (b-c)^3}+\frac{3 b c \tanh ^{-1}\left (\frac{\sqrt{a+c x}}{\sqrt{a}}\right )}{\sqrt{a} (b-c)^3} \]
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Rubi [A] time = 0.1781, antiderivative size = 275, normalized size of antiderivative = 1.68, number of steps used = 16, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238, Rules used = {6690, 47, 51, 63, 208} \[ \frac{b^2 \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{\sqrt{a} (b-c)^3}-\frac{c^2 \tanh ^{-1}\left (\frac{\sqrt{a+c x}}{\sqrt{a}}\right )}{\sqrt{a} (b-c)^3}-\frac{2 a \sqrt{a+b x}}{x^2 (b-c)^3}+\frac{2 a \sqrt{a+c x}}{x^2 (b-c)^3}-\frac{b \sqrt{a+b x}}{x (b-c)^3}-\frac{(b+3 c) \sqrt{a+b x}}{x (b-c)^3}+\frac{c \sqrt{a+c x}}{x (b-c)^3}+\frac{(3 b+c) \sqrt{a+c x}}{x (b-c)^3}-\frac{b (b+3 c) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{\sqrt{a} (b-c)^3}+\frac{c (3 b+c) \tanh ^{-1}\left (\frac{\sqrt{a+c x}}{\sqrt{a}}\right )}{\sqrt{a} (b-c)^3} \]
Antiderivative was successfully verified.
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Rule 6690
Rule 47
Rule 51
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{1}{\left (\sqrt{a+b x}+\sqrt{a+c x}\right )^3} \, dx &=\frac{\int \left (\frac{4 a \sqrt{a+b x}}{x^3}+\frac{b \left (1+\frac{3 c}{b}\right ) \sqrt{a+b x}}{x^2}-\frac{4 a \sqrt{a+c x}}{x^3}-\frac{3 b \left (1+\frac{c}{3 b}\right ) \sqrt{a+c x}}{x^2}\right ) \, dx}{(b-c)^3}\\ &=\frac{(4 a) \int \frac{\sqrt{a+b x}}{x^3} \, dx}{(b-c)^3}-\frac{(4 a) \int \frac{\sqrt{a+c x}}{x^3} \, dx}{(b-c)^3}-\frac{(3 b+c) \int \frac{\sqrt{a+c x}}{x^2} \, dx}{(b-c)^3}+\frac{(b+3 c) \int \frac{\sqrt{a+b x}}{x^2} \, dx}{(b-c)^3}\\ &=-\frac{2 a \sqrt{a+b x}}{(b-c)^3 x^2}-\frac{(b+3 c) \sqrt{a+b x}}{(b-c)^3 x}+\frac{2 a \sqrt{a+c x}}{(b-c)^3 x^2}+\frac{(3 b+c) \sqrt{a+c x}}{(b-c)^3 x}+\frac{(a b) \int \frac{1}{x^2 \sqrt{a+b x}} \, dx}{(b-c)^3}-\frac{(a c) \int \frac{1}{x^2 \sqrt{a+c x}} \, dx}{(b-c)^3}-\frac{(c (3 b+c)) \int \frac{1}{x \sqrt{a+c x}} \, dx}{2 (b-c)^3}+\frac{(b (b+3 c)) \int \frac{1}{x \sqrt{a+b x}} \, dx}{2 (b-c)^3}\\ &=-\frac{2 a \sqrt{a+b x}}{(b-c)^3 x^2}-\frac{b \sqrt{a+b x}}{(b-c)^3 x}-\frac{(b+3 c) \sqrt{a+b x}}{(b-c)^3 x}+\frac{2 a \sqrt{a+c x}}{(b-c)^3 x^2}+\frac{c \sqrt{a+c x}}{(b-c)^3 x}+\frac{(3 b+c) \sqrt{a+c x}}{(b-c)^3 x}-\frac{b^2 \int \frac{1}{x \sqrt{a+b x}} \, dx}{2 (b-c)^3}+\frac{c^2 \int \frac{1}{x \sqrt{a+c x}} \, dx}{2 (b-c)^3}-\frac{(3 b+c) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{c}+\frac{x^2}{c}} \, dx,x,\sqrt{a+c x}\right )}{(b-c)^3}+\frac{(b+3 c) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b x}\right )}{(b-c)^3}\\ &=-\frac{2 a \sqrt{a+b x}}{(b-c)^3 x^2}-\frac{b \sqrt{a+b x}}{(b-c)^3 x}-\frac{(b+3 c) \sqrt{a+b x}}{(b-c)^3 x}+\frac{2 a \sqrt{a+c x}}{(b-c)^3 x^2}+\frac{c \sqrt{a+c x}}{(b-c)^3 x}+\frac{(3 b+c) \sqrt{a+c x}}{(b-c)^3 x}-\frac{b (b+3 c) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{\sqrt{a} (b-c)^3}+\frac{c (3 b+c) \tanh ^{-1}\left (\frac{\sqrt{a+c x}}{\sqrt{a}}\right )}{\sqrt{a} (b-c)^3}-\frac{b \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b x}\right )}{(b-c)^3}+\frac{c \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{c}+\frac{x^2}{c}} \, dx,x,\sqrt{a+c x}\right )}{(b-c)^3}\\ &=-\frac{2 a \sqrt{a+b x}}{(b-c)^3 x^2}-\frac{b \sqrt{a+b x}}{(b-c)^3 x}-\frac{(b+3 c) \sqrt{a+b x}}{(b-c)^3 x}+\frac{2 a \sqrt{a+c x}}{(b-c)^3 x^2}+\frac{c \sqrt{a+c x}}{(b-c)^3 x}+\frac{(3 b+c) \sqrt{a+c x}}{(b-c)^3 x}+\frac{b^2 \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{\sqrt{a} (b-c)^3}-\frac{b (b+3 c) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{\sqrt{a} (b-c)^3}-\frac{c^2 \tanh ^{-1}\left (\frac{\sqrt{a+c x}}{\sqrt{a}}\right )}{\sqrt{a} (b-c)^3}+\frac{c (3 b+c) \tanh ^{-1}\left (\frac{\sqrt{a+c x}}{\sqrt{a}}\right )}{\sqrt{a} (b-c)^3}\\ \end{align*}
Mathematica [C] time = 0.260628, size = 182, normalized size = 1.11 \[ \frac{-\frac{8 b^2 (a+b x)^{3/2} \, _2F_1\left (\frac{3}{2},3;\frac{5}{2};\frac{b x}{a}+1\right )}{a^2}+\frac{8 c^2 (a+c x)^{3/2} \, _2F_1\left (\frac{3}{2},3;\frac{5}{2};\frac{c x}{a}+1\right )}{a^2}-\frac{3 (b+3 c) \left (b x \sqrt{\frac{b x}{a}+1} \tanh ^{-1}\left (\sqrt{\frac{b x}{a}+1}\right )+a+b x\right )}{x \sqrt{a+b x}}+\frac{3 (3 b+c) \left (c x \sqrt{\frac{c x}{a}+1} \tanh ^{-1}\left (\sqrt{\frac{c x}{a}+1}\right )+a+c x\right )}{x \sqrt{a+c x}}}{3 (b-c)^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.003, size = 300, normalized size = 1.8 \begin{align*} 2\,{\frac{{b}^{2}}{ \left ( b-c \right ) ^{3}} \left ( -1/2\,{\frac{\sqrt{bx+a}}{bx}}-1/2\,{\frac{1}{\sqrt{a}}{\it Artanh} \left ({\frac{\sqrt{bx+a}}{\sqrt{a}}} \right ) } \right ) }+8\,{\frac{a{b}^{2}}{ \left ( b-c \right ) ^{3}} \left ({\frac{1}{{b}^{2}{x}^{2}} \left ( -1/8\,{\frac{ \left ( bx+a \right ) ^{3/2}}{a}}-1/8\,\sqrt{bx+a} \right ) }+1/8\,{\frac{1}{{a}^{3/2}}{\it Artanh} \left ({\frac{\sqrt{bx+a}}{\sqrt{a}}} \right ) } \right ) }-8\,{\frac{{c}^{2}a}{ \left ( b-c \right ) ^{3}} \left ({\frac{1}{{c}^{2}{x}^{2}} \left ( -1/8\,{\frac{ \left ( cx+a \right ) ^{3/2}}{a}}-1/8\,\sqrt{cx+a} \right ) }+1/8\,{\frac{1}{{a}^{3/2}}{\it Artanh} \left ({\frac{\sqrt{cx+a}}{\sqrt{a}}} \right ) } \right ) }+6\,{\frac{bc}{ \left ( b-c \right ) ^{3}} \left ( -1/2\,{\frac{\sqrt{bx+a}}{bx}}-1/2\,{\frac{1}{\sqrt{a}}{\it Artanh} \left ({\frac{\sqrt{bx+a}}{\sqrt{a}}} \right ) } \right ) }-6\,{\frac{bc}{ \left ( b-c \right ) ^{3}} \left ( -1/2\,{\frac{\sqrt{cx+a}}{cx}}-1/2\,{\frac{1}{\sqrt{a}}{\it Artanh} \left ({\frac{\sqrt{cx+a}}{\sqrt{a}}} \right ) } \right ) }-2\,{\frac{{c}^{2}}{ \left ( b-c \right ) ^{3}} \left ( -1/2\,{\frac{\sqrt{cx+a}}{cx}}-1/2\,{\frac{1}{\sqrt{a}}{\it Artanh} \left ({\frac{\sqrt{cx+a}}{\sqrt{a}}} \right ) } \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (\sqrt{b x + a} + \sqrt{c x + a}\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.32698, size = 703, normalized size = 4.29 \begin{align*} \left [-\frac{3 \, \sqrt{a} b c x^{2} \log \left (\frac{b x + 2 \, \sqrt{b x + a} \sqrt{a} + 2 \, a}{x}\right ) + 3 \, \sqrt{a} b c x^{2} \log \left (\frac{c x - 2 \, \sqrt{c x + a} \sqrt{a} + 2 \, a}{x}\right ) + 2 \,{\left (2 \, a^{2} +{\left (2 \, a b + 3 \, a c\right )} x\right )} \sqrt{b x + a} - 2 \,{\left (2 \, a^{2} +{\left (3 \, a b + 2 \, a c\right )} x\right )} \sqrt{c x + a}}{2 \,{\left (a b^{3} - 3 \, a b^{2} c + 3 \, a b c^{2} - a c^{3}\right )} x^{2}}, \frac{3 \, \sqrt{-a} b c x^{2} \arctan \left (\frac{\sqrt{b x + a} \sqrt{-a}}{a}\right ) - 3 \, \sqrt{-a} b c x^{2} \arctan \left (\frac{\sqrt{c x + a} \sqrt{-a}}{a}\right ) -{\left (2 \, a^{2} +{\left (2 \, a b + 3 \, a c\right )} x\right )} \sqrt{b x + a} +{\left (2 \, a^{2} +{\left (3 \, a b + 2 \, a c\right )} x\right )} \sqrt{c x + a}}{{\left (a b^{3} - 3 \, a b^{2} c + 3 \, a b c^{2} - a c^{3}\right )} x^{2}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (\sqrt{a + b x} + \sqrt{a + c x}\right )^{3}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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