Optimal. Leaf size=147 \[ \frac{2 a^2 (a+b x)^{3/2}}{3 b^3 (b-c)}-\frac{2 a^2 (a+c x)^{3/2}}{3 c^3 (b-c)}+\frac{2 (a+b x)^{7/2}}{7 b^3 (b-c)}-\frac{4 a (a+b x)^{5/2}}{5 b^3 (b-c)}-\frac{2 (a+c x)^{7/2}}{7 c^3 (b-c)}+\frac{4 a (a+c x)^{5/2}}{5 c^3 (b-c)} \]
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Rubi [A] time = 0.121838, antiderivative size = 147, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08, Rules used = {2103, 43} \[ \frac{2 a^2 (a+b x)^{3/2}}{3 b^3 (b-c)}-\frac{2 a^2 (a+c x)^{3/2}}{3 c^3 (b-c)}+\frac{2 (a+b x)^{7/2}}{7 b^3 (b-c)}-\frac{4 a (a+b x)^{5/2}}{5 b^3 (b-c)}-\frac{2 (a+c x)^{7/2}}{7 c^3 (b-c)}+\frac{4 a (a+c x)^{5/2}}{5 c^3 (b-c)} \]
Antiderivative was successfully verified.
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Rule 2103
Rule 43
Rubi steps
\begin{align*} \int \frac{x^3}{\sqrt{a+b x}+\sqrt{a+c x}} \, dx &=\frac{\int x^2 \sqrt{a+b x} \, dx}{b-c}-\frac{\int x^2 \sqrt{a+c x} \, dx}{b-c}\\ &=\frac{\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}{b-c}-\frac{\int \left (\frac{a^2 \sqrt{a+c x}}{c^2}-\frac{2 a (a+c x)^{3/2}}{c^2}+\frac{(a+c x)^{5/2}}{c^2}\right ) \, dx}{b-c}\\ &=\frac{2 a^2 (a+b x)^{3/2}}{3 b^3 (b-c)}-\frac{4 a (a+b x)^{5/2}}{5 b^3 (b-c)}+\frac{2 (a+b x)^{7/2}}{7 b^3 (b-c)}-\frac{2 a^2 (a+c x)^{3/2}}{3 (b-c) c^3}+\frac{4 a (a+c x)^{5/2}}{5 (b-c) c^3}-\frac{2 (a+c x)^{7/2}}{7 (b-c) c^3}\\ \end{align*}
Mathematica [A] time = 0.218664, size = 147, normalized size = 1. \[ \frac{2 a^2 (a+b x)^{3/2}}{3 b^3 (b-c)}-\frac{2 a^2 (a+c x)^{3/2}}{3 c^3 (b-c)}+\frac{2 (a+b x)^{7/2}}{7 b^3 (b-c)}-\frac{4 a (a+b x)^{5/2}}{5 b^3 (b-c)}-\frac{2 (a+c x)^{7/2}}{7 c^3 (b-c)}+\frac{4 a (a+c x)^{5/2}}{5 c^3 (b-c)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 90, normalized size = 0.6 \begin{align*} 2\,{\frac{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}}{ \left ( b-c \right ){b}^{3}}}-2\,{\frac{1/7\, \left ( cx+a \right ) ^{7/2}-2/5\,a \left ( cx+a \right ) ^{5/2}+1/3\,{a}^{2} \left ( cx+a \right ) ^{3/2}}{ \left ( b-c \right ){c}^{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^{3}}{\sqrt{b x + a} + \sqrt{c x + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.20086, size = 250, normalized size = 1.7 \begin{align*} \frac{2 \,{\left ({\left (15 \, b^{3} c^{3} x^{3} + 3 \, a b^{2} c^{3} x^{2} - 4 \, a^{2} b c^{3} x + 8 \, a^{3} c^{3}\right )} \sqrt{b x + a} -{\left (15 \, b^{3} c^{3} x^{3} + 3 \, a b^{3} c^{2} x^{2} - 4 \, a^{2} b^{3} c x + 8 \, a^{3} b^{3}\right )} \sqrt{c x + a}\right )}}{105 \,{\left (b^{4} c^{3} - b^{3} c^{4}\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^{3}}{\sqrt{a + b x} + \sqrt{a + c x}}\, 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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