Optimal. Leaf size=155 \[ -\frac{8 a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{(b-c)^3}+\frac{8 a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a+c x}}{\sqrt{a}}\right )}{(b-c)^3}+\frac{8 a \sqrt{a+b x}}{(b-c)^3}-\frac{8 a \sqrt{a+c x}}{(b-c)^3}+\frac{2 (b+3 c) (a+b x)^{3/2}}{3 b (b-c)^3}-\frac{2 (3 b+c) (a+c x)^{3/2}}{3 c (b-c)^3} \]
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Rubi [A] time = 0.202475, antiderivative size = 155, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16, Rules used = {6690, 50, 63, 208} \[ -\frac{8 a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{(b-c)^3}+\frac{8 a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a+c x}}{\sqrt{a}}\right )}{(b-c)^3}+\frac{8 a \sqrt{a+b x}}{(b-c)^3}-\frac{8 a \sqrt{a+c x}}{(b-c)^3}+\frac{2 (b+3 c) (a+b x)^{3/2}}{3 b (b-c)^3}-\frac{2 (3 b+c) (a+c x)^{3/2}}{3 c (b-c)^3} \]
Antiderivative was successfully verified.
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Rule 6690
Rule 50
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{x^2}{\left (\sqrt{a+b x}+\sqrt{a+c x}\right )^3} \, dx &=\frac{\int \left (b \left (1+\frac{3 c}{b}\right ) \sqrt{a+b x}+\frac{4 a \sqrt{a+b x}}{x}-3 b \left (1+\frac{c}{3 b}\right ) \sqrt{a+c x}-\frac{4 a \sqrt{a+c x}}{x}\right ) \, dx}{(b-c)^3}\\ &=\frac{2 (b+3 c) (a+b x)^{3/2}}{3 b (b-c)^3}-\frac{2 (3 b+c) (a+c x)^{3/2}}{3 (b-c)^3 c}+\frac{(4 a) \int \frac{\sqrt{a+b x}}{x} \, dx}{(b-c)^3}-\frac{(4 a) \int \frac{\sqrt{a+c x}}{x} \, dx}{(b-c)^3}\\ &=\frac{8 a \sqrt{a+b x}}{(b-c)^3}+\frac{2 (b+3 c) (a+b x)^{3/2}}{3 b (b-c)^3}-\frac{8 a \sqrt{a+c x}}{(b-c)^3}-\frac{2 (3 b+c) (a+c x)^{3/2}}{3 (b-c)^3 c}+\frac{\left (4 a^2\right ) \int \frac{1}{x \sqrt{a+b x}} \, dx}{(b-c)^3}-\frac{\left (4 a^2\right ) \int \frac{1}{x \sqrt{a+c x}} \, dx}{(b-c)^3}\\ &=\frac{8 a \sqrt{a+b x}}{(b-c)^3}+\frac{2 (b+3 c) (a+b x)^{3/2}}{3 b (b-c)^3}-\frac{8 a \sqrt{a+c x}}{(b-c)^3}-\frac{2 (3 b+c) (a+c x)^{3/2}}{3 (b-c)^3 c}+\frac{\left (8 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b x}\right )}{b (b-c)^3}-\frac{\left (8 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{c}+\frac{x^2}{c}} \, dx,x,\sqrt{a+c x}\right )}{(b-c)^3 c}\\ &=\frac{8 a \sqrt{a+b x}}{(b-c)^3}+\frac{2 (b+3 c) (a+b x)^{3/2}}{3 b (b-c)^3}-\frac{8 a \sqrt{a+c x}}{(b-c)^3}-\frac{2 (3 b+c) (a+c x)^{3/2}}{3 (b-c)^3 c}-\frac{8 a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{(b-c)^3}+\frac{8 a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a+c x}}{\sqrt{a}}\right )}{(b-c)^3}\\ \end{align*}
Mathematica [A] time = 0.270552, size = 119, normalized size = 0.77 \[ \frac{2 \left (-12 a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )+12 a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a+c x}}{\sqrt{a}}\right )+\frac{(b+3 c) (a+b x)^{3/2}}{b}-\frac{(3 b+c) (a+c x)^{3/2}}{c}+12 a \sqrt{a+b x}-12 a \sqrt{a+c x}\right )}{3 (b-c)^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 148, normalized size = 1. \begin{align*}{\frac{2}{3\, \left ( b-c \right ) ^{3}} \left ( bx+a \right ) ^{{\frac{3}{2}}}}+2\,{\frac{c \left ( bx+a \right ) ^{3/2}}{ \left ( b-c \right ) ^{3}b}}-2\,{\frac{b \left ( cx+a \right ) ^{3/2}}{ \left ( b-c \right ) ^{3}c}}-{\frac{2}{3\, \left ( b-c \right ) ^{3}} \left ( cx+a \right ) ^{{\frac{3}{2}}}}+4\,{\frac{a}{ \left ( b-c \right ) ^{3}} \left ( 2\,\sqrt{bx+a}-2\,\sqrt{a}{\it Artanh} \left ({\frac{\sqrt{bx+a}}{\sqrt{a}}} \right ) \right ) }-4\,{\frac{a}{ \left ( b-c \right ) ^{3}} \left ( 2\,\sqrt{cx+a}-2\,\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{x^{2}}{{\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 = 1.31923, size = 755, normalized size = 4.87 \begin{align*} \left [-\frac{2 \,{\left (6 \, a^{\frac{3}{2}} b c \log \left (\frac{b x + 2 \, \sqrt{b x + a} \sqrt{a} + 2 \, a}{x}\right ) + 6 \, a^{\frac{3}{2}} b c \log \left (\frac{c x - 2 \, \sqrt{c x + a} \sqrt{a} + 2 \, a}{x}\right ) -{\left (13 \, a b c + 3 \, a c^{2} +{\left (b^{2} c + 3 \, b c^{2}\right )} x\right )} \sqrt{b x + a} +{\left (3 \, a b^{2} + 13 \, a b c +{\left (3 \, b^{2} c + b c^{2}\right )} x\right )} \sqrt{c x + a}\right )}}{3 \,{\left (b^{4} c - 3 \, b^{3} c^{2} + 3 \, b^{2} c^{3} - b c^{4}\right )}}, \frac{2 \,{\left (12 \, \sqrt{-a} a b c \arctan \left (\frac{\sqrt{b x + a} \sqrt{-a}}{a}\right ) - 12 \, \sqrt{-a} a b c \arctan \left (\frac{\sqrt{c x + a} \sqrt{-a}}{a}\right ) +{\left (13 \, a b c + 3 \, a c^{2} +{\left (b^{2} c + 3 \, b c^{2}\right )} x\right )} \sqrt{b x + a} -{\left (3 \, a b^{2} + 13 \, a b c +{\left (3 \, b^{2} c + b c^{2}\right )} x\right )} \sqrt{c x + a}\right )}}{3 \,{\left (b^{4} c - 3 \, b^{3} c^{2} + 3 \, b^{2} c^{3} - b c^{4}\right )}}\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{x^{2}}{\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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