Optimal. Leaf size=375 \[ \frac{24 a^2 (a+b x)^{5/2}}{5 b^3 (a-c)^3}+\frac{2 a^2 (a+3 c) (a+b x)^{3/2}}{3 b^3 (a-c)^3}-\frac{8 a^3 (a+b x)^{3/2}}{3 b^3 (a-c)^3}-\frac{24 c^2 (b x+c)^{5/2}}{5 b^3 (a-c)^3}+\frac{8 c^3 (b x+c)^{3/2}}{3 b^3 (a-c)^3}-\frac{2 c^2 (3 a+c) (b x+c)^{3/2}}{3 b^3 (a-c)^3}+\frac{8 (a+b x)^{9/2}}{9 b^3 (a-c)^3}+\frac{2 (a+3 c) (a+b x)^{7/2}}{7 b^3 (a-c)^3}-\frac{24 a (a+b x)^{7/2}}{7 b^3 (a-c)^3}-\frac{4 a (a+3 c) (a+b x)^{5/2}}{5 b^3 (a-c)^3}-\frac{8 (b x+c)^{9/2}}{9 b^3 (a-c)^3}+\frac{24 c (b x+c)^{7/2}}{7 b^3 (a-c)^3}-\frac{2 (3 a+c) (b x+c)^{7/2}}{7 b^3 (a-c)^3}+\frac{4 c (3 a+c) (b x+c)^{5/2}}{5 b^3 (a-c)^3} \]
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Rubi [A] time = 0.37164, antiderivative size = 375, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08, Rules used = {6689, 43} \[ \frac{24 a^2 (a+b x)^{5/2}}{5 b^3 (a-c)^3}+\frac{2 a^2 (a+3 c) (a+b x)^{3/2}}{3 b^3 (a-c)^3}-\frac{8 a^3 (a+b x)^{3/2}}{3 b^3 (a-c)^3}-\frac{24 c^2 (b x+c)^{5/2}}{5 b^3 (a-c)^3}+\frac{8 c^3 (b x+c)^{3/2}}{3 b^3 (a-c)^3}-\frac{2 c^2 (3 a+c) (b x+c)^{3/2}}{3 b^3 (a-c)^3}+\frac{8 (a+b x)^{9/2}}{9 b^3 (a-c)^3}+\frac{2 (a+3 c) (a+b x)^{7/2}}{7 b^3 (a-c)^3}-\frac{24 a (a+b x)^{7/2}}{7 b^3 (a-c)^3}-\frac{4 a (a+3 c) (a+b x)^{5/2}}{5 b^3 (a-c)^3}-\frac{8 (b x+c)^{9/2}}{9 b^3 (a-c)^3}+\frac{24 c (b x+c)^{7/2}}{7 b^3 (a-c)^3}-\frac{2 (3 a+c) (b x+c)^{7/2}}{7 b^3 (a-c)^3}+\frac{4 c (3 a+c) (b x+c)^{5/2}}{5 b^3 (a-c)^3} \]
Antiderivative was successfully verified.
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Rule 6689
Rule 43
Rubi steps
\begin{align*} \int \frac{x^2}{\left (\sqrt{a+b x}+\sqrt{c+b x}\right )^3} \, dx &=\frac{\int \left (a \left (1+\frac{3 c}{a}\right ) x^2 \sqrt{a+b x}+4 b x^3 \sqrt{a+b x}-3 a \left (1+\frac{c}{3 a}\right ) x^2 \sqrt{c+b x}-4 b x^3 \sqrt{c+b x}\right ) \, dx}{(a-c)^3}\\ &=\frac{(4 b) \int x^3 \sqrt{a+b x} \, dx}{(a-c)^3}-\frac{(4 b) \int x^3 \sqrt{c+b x} \, dx}{(a-c)^3}-\frac{(3 a+c) \int x^2 \sqrt{c+b x} \, dx}{(a-c)^3}+\frac{(a+3 c) \int x^2 \sqrt{a+b x} \, dx}{(a-c)^3}\\ &=\frac{(4 b) \int \left (-\frac{a^3 \sqrt{a+b x}}{b^3}+\frac{3 a^2 (a+b x)^{3/2}}{b^3}-\frac{3 a (a+b x)^{5/2}}{b^3}+\frac{(a+b x)^{7/2}}{b^3}\right ) \, dx}{(a-c)^3}-\frac{(4 b) \int \left (-\frac{c^3 \sqrt{c+b x}}{b^3}+\frac{3 c^2 (c+b x)^{3/2}}{b^3}-\frac{3 c (c+b x)^{5/2}}{b^3}+\frac{(c+b x)^{7/2}}{b^3}\right ) \, dx}{(a-c)^3}-\frac{(3 a+c) \int \left (\frac{c^2 \sqrt{c+b x}}{b^2}-\frac{2 c (c+b x)^{3/2}}{b^2}+\frac{(c+b x)^{5/2}}{b^2}\right ) \, dx}{(a-c)^3}+\frac{(a+3 c) \int \left (\frac{a^2 \sqrt{a+b x}}{b^2}-\frac{2 a (a+b x)^{3/2}}{b^2}+\frac{(a+b x)^{5/2}}{b^2}\right ) \, dx}{(a-c)^3}\\ &=-\frac{8 a^3 (a+b x)^{3/2}}{3 b^3 (a-c)^3}+\frac{2 a^2 (a+3 c) (a+b x)^{3/2}}{3 b^3 (a-c)^3}+\frac{24 a^2 (a+b x)^{5/2}}{5 b^3 (a-c)^3}-\frac{4 a (a+3 c) (a+b x)^{5/2}}{5 b^3 (a-c)^3}-\frac{24 a (a+b x)^{7/2}}{7 b^3 (a-c)^3}+\frac{2 (a+3 c) (a+b x)^{7/2}}{7 b^3 (a-c)^3}+\frac{8 (a+b x)^{9/2}}{9 b^3 (a-c)^3}+\frac{8 c^3 (c+b x)^{3/2}}{3 b^3 (a-c)^3}-\frac{2 c^2 (3 a+c) (c+b x)^{3/2}}{3 b^3 (a-c)^3}-\frac{24 c^2 (c+b x)^{5/2}}{5 b^3 (a-c)^3}+\frac{4 c (3 a+c) (c+b x)^{5/2}}{5 b^3 (a-c)^3}+\frac{24 c (c+b x)^{7/2}}{7 b^3 (a-c)^3}-\frac{2 (3 a+c) (c+b x)^{7/2}}{7 b^3 (a-c)^3}-\frac{8 (c+b x)^{9/2}}{9 b^3 (a-c)^3}\\ \end{align*}
Mathematica [A] time = 0.412515, size = 282, normalized size = 0.75 \[ \frac{2 \left (4 a^3 \sqrt{a+b x} (5 b x+18 c)-3 a^2 b x \sqrt{a+b x} (5 b x+12 c)-40 a^4 \sqrt{a+b x}+a \left (27 b^2 c x^2 \left (\sqrt{a+b x}-\sqrt{b x+c}\right )+5 b^3 x^3 \left (13 \sqrt{a+b x}-27 \sqrt{b x+c}\right )-72 c^3 \sqrt{b x+c}+36 b c^2 x \sqrt{b x+c}\right )+5 \left (b^3 c x^3 \left (27 \sqrt{a+b x}-13 \sqrt{b x+c}\right )+28 b^4 x^4 \left (\sqrt{a+b x}-\sqrt{b x+c}\right )+3 b^2 c^2 x^2 \sqrt{b x+c}+8 c^4 \sqrt{b x+c}-4 b c^3 x \sqrt{b x+c}\right )\right )}{315 b^3 (a-c)^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 294, normalized size = 0.8 \begin{align*} 2\,{\frac{a \left ( 1/7\, \left ( bx+a \right ) ^{7/2}-2/5\,a \left ( bx+a \right ) ^{5/2}+1/3\,{a}^{2} \left ( bx+a \right ) ^{3/2} \right ) }{ \left ( a-c \right ) ^{3}{b}^{3}}}+6\,{\frac{c \left ( 1/7\, \left ( bx+a \right ) ^{7/2}-2/5\,a \left ( bx+a \right ) ^{5/2}+1/3\,{a}^{2} \left ( bx+a \right ) ^{3/2} \right ) }{ \left ( a-c \right ) ^{3}{b}^{3}}}-6\,{\frac{a \left ( 1/7\, \left ( bx+c \right ) ^{7/2}-2/5\,c \left ( bx+c \right ) ^{5/2}+1/3\,{c}^{2} \left ( bx+c \right ) ^{3/2} \right ) }{ \left ( a-c \right ) ^{3}{b}^{3}}}-2\,{\frac{c \left ( 1/7\, \left ( bx+c \right ) ^{7/2}-2/5\,c \left ( bx+c \right ) ^{5/2}+1/3\,{c}^{2} \left ( bx+c \right ) ^{3/2} \right ) }{ \left ( a-c \right ) ^{3}{b}^{3}}}+8\,{\frac{1/9\, \left ( bx+a \right ) ^{9/2}-3/7\, \left ( bx+a \right ) ^{7/2}a+3/5\,{a}^{2} \left ( bx+a \right ) ^{5/2}-1/3\,{a}^{3} \left ( bx+a \right ) ^{3/2}}{ \left ( a-c \right ) ^{3}{b}^{3}}}-8\,{\frac{1/9\, \left ( bx+c \right ) ^{9/2}-3/7\,c \left ( bx+c \right ) ^{7/2}+3/5\,{c}^{2} \left ( bx+c \right ) ^{5/2}-1/3\,{c}^{3} \left ( bx+c \right ) ^{3/2}}{ \left ( a-c \right ) ^{3}{b}^{3}}} \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{b x + c}\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.945516, size = 452, normalized size = 1.21 \begin{align*} \frac{2 \,{\left ({\left (140 \, b^{4} x^{4} - 40 \, a^{4} + 72 \, a^{3} c + 5 \,{\left (13 \, a b^{3} + 27 \, b^{3} c\right )} x^{3} - 3 \,{\left (5 \, a^{2} b^{2} - 9 \, a b^{2} c\right )} x^{2} + 4 \,{\left (5 \, a^{3} b - 9 \, a^{2} b c\right )} x\right )} \sqrt{b x + a} -{\left (140 \, b^{4} x^{4} + 72 \, a c^{3} - 40 \, c^{4} + 5 \,{\left (27 \, a b^{3} + 13 \, b^{3} c\right )} x^{3} + 3 \,{\left (9 \, a b^{2} c - 5 \, b^{2} c^{2}\right )} x^{2} - 4 \,{\left (9 \, a b c^{2} - 5 \, b c^{3}\right )} x\right )} \sqrt{b x + c}\right )}}{315 \,{\left (a^{3} b^{3} - 3 \, a^{2} b^{3} c + 3 \, a b^{3} c^{2} - b^{3} c^{3}\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{b x + c}\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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