Optimal. Leaf size=147 \[ -\frac{8 A c^2 \sqrt{a+c x^2}}{15 a^3 x}+\frac{4 A c \sqrt{a+c x^2}}{15 a^2 x^3}-\frac{3 B c^2 \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )}{8 a^{5/2}}+\frac{3 B c \sqrt{a+c x^2}}{8 a^2 x^2}-\frac{A \sqrt{a+c x^2}}{5 a x^5}-\frac{B \sqrt{a+c x^2}}{4 a x^4} \]
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Rubi [A] time = 0.12864, antiderivative size = 147, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {835, 807, 266, 63, 208} \[ -\frac{8 A c^2 \sqrt{a+c x^2}}{15 a^3 x}+\frac{4 A c \sqrt{a+c x^2}}{15 a^2 x^3}-\frac{3 B c^2 \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )}{8 a^{5/2}}+\frac{3 B c \sqrt{a+c x^2}}{8 a^2 x^2}-\frac{A \sqrt{a+c x^2}}{5 a x^5}-\frac{B \sqrt{a+c x^2}}{4 a x^4} \]
Antiderivative was successfully verified.
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Rule 835
Rule 807
Rule 266
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{A+B x}{x^6 \sqrt{a+c x^2}} \, dx &=-\frac{A \sqrt{a+c x^2}}{5 a x^5}-\frac{\int \frac{-5 a B+4 A c x}{x^5 \sqrt{a+c x^2}} \, dx}{5 a}\\ &=-\frac{A \sqrt{a+c x^2}}{5 a x^5}-\frac{B \sqrt{a+c x^2}}{4 a x^4}+\frac{\int \frac{-16 a A c-15 a B c x}{x^4 \sqrt{a+c x^2}} \, dx}{20 a^2}\\ &=-\frac{A \sqrt{a+c x^2}}{5 a x^5}-\frac{B \sqrt{a+c x^2}}{4 a x^4}+\frac{4 A c \sqrt{a+c x^2}}{15 a^2 x^3}-\frac{\int \frac{45 a^2 B c-32 a A c^2 x}{x^3 \sqrt{a+c x^2}} \, dx}{60 a^3}\\ &=-\frac{A \sqrt{a+c x^2}}{5 a x^5}-\frac{B \sqrt{a+c x^2}}{4 a x^4}+\frac{4 A c \sqrt{a+c x^2}}{15 a^2 x^3}+\frac{3 B c \sqrt{a+c x^2}}{8 a^2 x^2}+\frac{\int \frac{64 a^2 A c^2+45 a^2 B c^2 x}{x^2 \sqrt{a+c x^2}} \, dx}{120 a^4}\\ &=-\frac{A \sqrt{a+c x^2}}{5 a x^5}-\frac{B \sqrt{a+c x^2}}{4 a x^4}+\frac{4 A c \sqrt{a+c x^2}}{15 a^2 x^3}+\frac{3 B c \sqrt{a+c x^2}}{8 a^2 x^2}-\frac{8 A c^2 \sqrt{a+c x^2}}{15 a^3 x}+\frac{\left (3 B c^2\right ) \int \frac{1}{x \sqrt{a+c x^2}} \, dx}{8 a^2}\\ &=-\frac{A \sqrt{a+c x^2}}{5 a x^5}-\frac{B \sqrt{a+c x^2}}{4 a x^4}+\frac{4 A c \sqrt{a+c x^2}}{15 a^2 x^3}+\frac{3 B c \sqrt{a+c x^2}}{8 a^2 x^2}-\frac{8 A c^2 \sqrt{a+c x^2}}{15 a^3 x}+\frac{\left (3 B c^2\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+c x}} \, dx,x,x^2\right )}{16 a^2}\\ &=-\frac{A \sqrt{a+c x^2}}{5 a x^5}-\frac{B \sqrt{a+c x^2}}{4 a x^4}+\frac{4 A c \sqrt{a+c x^2}}{15 a^2 x^3}+\frac{3 B c \sqrt{a+c x^2}}{8 a^2 x^2}-\frac{8 A c^2 \sqrt{a+c x^2}}{15 a^3 x}+\frac{(3 B c) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{c}+\frac{x^2}{c}} \, dx,x,\sqrt{a+c x^2}\right )}{8 a^2}\\ &=-\frac{A \sqrt{a+c x^2}}{5 a x^5}-\frac{B \sqrt{a+c x^2}}{4 a x^4}+\frac{4 A c \sqrt{a+c x^2}}{15 a^2 x^3}+\frac{3 B c \sqrt{a+c x^2}}{8 a^2 x^2}-\frac{8 A c^2 \sqrt{a+c x^2}}{15 a^3 x}-\frac{3 B c^2 \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )}{8 a^{5/2}}\\ \end{align*}
Mathematica [C] time = 0.0201174, size = 72, normalized size = 0.49 \[ -\frac{\sqrt{a+c x^2} \left (A \left (3 a^2-4 a c x^2+8 c^2 x^4\right )+15 B c^2 x^5 \, _2F_1\left (\frac{1}{2},3;\frac{3}{2};\frac{c x^2}{a}+1\right )\right )}{15 a^3 x^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.011, size = 129, normalized size = 0.9 \begin{align*} -{\frac{B}{4\,a{x}^{4}}\sqrt{c{x}^{2}+a}}+{\frac{3\,Bc}{8\,{a}^{2}{x}^{2}}\sqrt{c{x}^{2}+a}}-{\frac{3\,B{c}^{2}}{8}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{c{x}^{2}+a} \right ) } \right ){a}^{-{\frac{5}{2}}}}-{\frac{A}{5\,a{x}^{5}}\sqrt{c{x}^{2}+a}}+{\frac{4\,Ac}{15\,{a}^{2}{x}^{3}}\sqrt{c{x}^{2}+a}}-{\frac{8\,A{c}^{2}}{15\,{a}^{3}x}\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 = 1.56022, size = 467, normalized size = 3.18 \begin{align*} \left [\frac{45 \, B \sqrt{a} c^{2} x^{5} \log \left (-\frac{c x^{2} - 2 \, \sqrt{c x^{2} + a} \sqrt{a} + 2 \, a}{x^{2}}\right ) - 2 \,{\left (64 \, A c^{2} x^{4} - 45 \, B a c x^{3} - 32 \, A a c x^{2} + 30 \, B a^{2} x + 24 \, A a^{2}\right )} \sqrt{c x^{2} + a}}{240 \, a^{3} x^{5}}, \frac{45 \, B \sqrt{-a} c^{2} x^{5} \arctan \left (\frac{\sqrt{-a}}{\sqrt{c x^{2} + a}}\right ) -{\left (64 \, A c^{2} x^{4} - 45 \, B a c x^{3} - 32 \, A a c x^{2} + 30 \, B a^{2} x + 24 \, A a^{2}\right )} \sqrt{c x^{2} + a}}{120 \, a^{3} x^{5}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 7.10817, size = 408, normalized size = 2.78 \begin{align*} - \frac{3 A a^{4} c^{\frac{9}{2}} \sqrt{\frac{a}{c x^{2}} + 1}}{15 a^{5} c^{4} x^{4} + 30 a^{4} c^{5} x^{6} + 15 a^{3} c^{6} x^{8}} - \frac{2 A a^{3} c^{\frac{11}{2}} x^{2} \sqrt{\frac{a}{c x^{2}} + 1}}{15 a^{5} c^{4} x^{4} + 30 a^{4} c^{5} x^{6} + 15 a^{3} c^{6} x^{8}} - \frac{3 A a^{2} c^{\frac{13}{2}} x^{4} \sqrt{\frac{a}{c x^{2}} + 1}}{15 a^{5} c^{4} x^{4} + 30 a^{4} c^{5} x^{6} + 15 a^{3} c^{6} x^{8}} - \frac{12 A a c^{\frac{15}{2}} x^{6} \sqrt{\frac{a}{c x^{2}} + 1}}{15 a^{5} c^{4} x^{4} + 30 a^{4} c^{5} x^{6} + 15 a^{3} c^{6} x^{8}} - \frac{8 A c^{\frac{17}{2}} x^{8} \sqrt{\frac{a}{c x^{2}} + 1}}{15 a^{5} c^{4} x^{4} + 30 a^{4} c^{5} x^{6} + 15 a^{3} c^{6} x^{8}} - \frac{B}{4 \sqrt{c} x^{5} \sqrt{\frac{a}{c x^{2}} + 1}} + \frac{B \sqrt{c}}{8 a x^{3} \sqrt{\frac{a}{c x^{2}} + 1}} + \frac{3 B c^{\frac{3}{2}}}{8 a^{2} x \sqrt{\frac{a}{c x^{2}} + 1}} - \frac{3 B c^{2} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{c} x} \right )}}{8 a^{\frac{5}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.20971, size = 325, normalized size = 2.21 \begin{align*} \frac{3 \, B c^{2} \arctan \left (-\frac{\sqrt{c} x - \sqrt{c x^{2} + a}}{\sqrt{-a}}\right )}{4 \, \sqrt{-a} a^{2}} - \frac{45 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{9} B c^{2} - 210 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{7} B a c^{2} - 640 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{4} A a^{2} c^{\frac{5}{2}} + 210 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{3} B a^{3} c^{2} + 320 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{2} A a^{3} c^{\frac{5}{2}} - 45 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )} B a^{4} c^{2} - 64 \, A a^{4} c^{\frac{5}{2}}}{60 \,{\left ({\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{2} - a\right )}^{5} a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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