Optimal. Leaf size=99 \[ \frac{A c^2 \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )}{8 a^{3/2}}+\frac{A c \sqrt{a+c x^2}}{8 a x^2}-\frac{A \left (a+c x^2\right )^{3/2}}{4 a x^4}-\frac{B \left (a+c x^2\right )^{3/2}}{3 a x^3} \]
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Rubi [A] time = 0.0617995, antiderivative size = 99, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {835, 807, 266, 47, 63, 208} \[ \frac{A c^2 \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )}{8 a^{3/2}}+\frac{A c \sqrt{a+c x^2}}{8 a x^2}-\frac{A \left (a+c x^2\right )^{3/2}}{4 a x^4}-\frac{B \left (a+c x^2\right )^{3/2}}{3 a x^3} \]
Antiderivative was successfully verified.
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Rule 835
Rule 807
Rule 266
Rule 47
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{(A+B x) \sqrt{a+c x^2}}{x^5} \, dx &=-\frac{A \left (a+c x^2\right )^{3/2}}{4 a x^4}-\frac{\int \frac{(-4 a B+A c x) \sqrt{a+c x^2}}{x^4} \, dx}{4 a}\\ &=-\frac{A \left (a+c x^2\right )^{3/2}}{4 a x^4}-\frac{B \left (a+c x^2\right )^{3/2}}{3 a x^3}-\frac{(A c) \int \frac{\sqrt{a+c x^2}}{x^3} \, dx}{4 a}\\ &=-\frac{A \left (a+c x^2\right )^{3/2}}{4 a x^4}-\frac{B \left (a+c x^2\right )^{3/2}}{3 a x^3}-\frac{(A c) \operatorname{Subst}\left (\int \frac{\sqrt{a+c x}}{x^2} \, dx,x,x^2\right )}{8 a}\\ &=\frac{A c \sqrt{a+c x^2}}{8 a x^2}-\frac{A \left (a+c x^2\right )^{3/2}}{4 a x^4}-\frac{B \left (a+c x^2\right )^{3/2}}{3 a x^3}-\frac{\left (A c^2\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+c x}} \, dx,x,x^2\right )}{16 a}\\ &=\frac{A c \sqrt{a+c x^2}}{8 a x^2}-\frac{A \left (a+c x^2\right )^{3/2}}{4 a x^4}-\frac{B \left (a+c x^2\right )^{3/2}}{3 a x^3}-\frac{(A c) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{c}+\frac{x^2}{c}} \, dx,x,\sqrt{a+c x^2}\right )}{8 a}\\ &=\frac{A c \sqrt{a+c x^2}}{8 a x^2}-\frac{A \left (a+c x^2\right )^{3/2}}{4 a x^4}-\frac{B \left (a+c x^2\right )^{3/2}}{3 a x^3}+\frac{A c^2 \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )}{8 a^{3/2}}\\ \end{align*}
Mathematica [C] time = 0.0151004, size = 53, normalized size = 0.54 \[ -\frac{\left (a+c x^2\right )^{3/2} \left (a^2 B+A c^2 x^3 \, _2F_1\left (\frac{3}{2},3;\frac{5}{2};\frac{c x^2}{a}+1\right )\right )}{3 a^3 x^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 107, normalized size = 1.1 \begin{align*} -{\frac{B}{3\,a{x}^{3}} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{A}{4\,a{x}^{4}} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{Ac}{8\,{a}^{2}{x}^{2}} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{A{c}^{2}}{8}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{c{x}^{2}+a} \right ) } \right ){a}^{-{\frac{3}{2}}}}-{\frac{A{c}^{2}}{8\,{a}^{2}}\sqrt{c{x}^{2}+a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.73329, size = 412, normalized size = 4.16 \begin{align*} \left [\frac{3 \, A \sqrt{a} c^{2} x^{4} \log \left (-\frac{c x^{2} + 2 \, \sqrt{c x^{2} + a} \sqrt{a} + 2 \, a}{x^{2}}\right ) - 2 \,{\left (8 \, B a c x^{3} + 3 \, A a c x^{2} + 8 \, B a^{2} x + 6 \, A a^{2}\right )} \sqrt{c x^{2} + a}}{48 \, a^{2} x^{4}}, -\frac{3 \, A \sqrt{-a} c^{2} x^{4} \arctan \left (\frac{\sqrt{-a}}{\sqrt{c x^{2} + a}}\right ) +{\left (8 \, B a c x^{3} + 3 \, A a c x^{2} + 8 \, B a^{2} x + 6 \, A a^{2}\right )} \sqrt{c x^{2} + a}}{24 \, a^{2} x^{4}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 5.90189, size = 144, normalized size = 1.45 \begin{align*} - \frac{A a}{4 \sqrt{c} x^{5} \sqrt{\frac{a}{c x^{2}} + 1}} - \frac{3 A \sqrt{c}}{8 x^{3} \sqrt{\frac{a}{c x^{2}} + 1}} - \frac{A c^{\frac{3}{2}}}{8 a x \sqrt{\frac{a}{c x^{2}} + 1}} + \frac{A c^{2} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{c} x} \right )}}{8 a^{\frac{3}{2}}} - \frac{B \sqrt{c} \sqrt{\frac{a}{c x^{2}} + 1}}{3 x^{2}} - \frac{B c^{\frac{3}{2}} \sqrt{\frac{a}{c x^{2}} + 1}}{3 a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.15874, size = 360, normalized size = 3.64 \begin{align*} -\frac{A c^{2} \arctan \left (-\frac{\sqrt{c} x - \sqrt{c x^{2} + a}}{\sqrt{-a}}\right )}{4 \, \sqrt{-a} a} + \frac{3 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{7} A c^{2} + 24 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{6} B a c^{\frac{3}{2}} + 21 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{5} A a c^{2} - 24 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{4} B a^{2} c^{\frac{3}{2}} + 21 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{3} A a^{2} c^{2} + 8 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{2} B a^{3} c^{\frac{3}{2}} + 3 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )} A a^{3} c^{2} - 8 \, B a^{4} c^{\frac{3}{2}}}{12 \,{\left ({\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{2} - a\right )}^{4} a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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