Optimal. Leaf size=132 \[ \frac{1}{16} a^2 \sqrt{a+c x^2} (16 A+5 B x)-a^{5/2} A \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )+\frac{5 a^3 B \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{16 \sqrt{c}}+\frac{1}{24} a \left (a+c x^2\right )^{3/2} (8 A+5 B x)+\frac{1}{30} \left (a+c x^2\right )^{5/2} (6 A+5 B x) \]
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Rubi [A] time = 0.121926, antiderivative size = 132, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.35, Rules used = {815, 844, 217, 206, 266, 63, 208} \[ \frac{1}{16} a^2 \sqrt{a+c x^2} (16 A+5 B x)-a^{5/2} A \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )+\frac{5 a^3 B \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{16 \sqrt{c}}+\frac{1}{24} a \left (a+c x^2\right )^{3/2} (8 A+5 B x)+\frac{1}{30} \left (a+c x^2\right )^{5/2} (6 A+5 B x) \]
Antiderivative was successfully verified.
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Rule 815
Rule 844
Rule 217
Rule 206
Rule 266
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{(A+B x) \left (a+c x^2\right )^{5/2}}{x} \, dx &=\frac{1}{30} (6 A+5 B x) \left (a+c x^2\right )^{5/2}+\frac{\int \frac{(6 a A c+5 a B c x) \left (a+c x^2\right )^{3/2}}{x} \, dx}{6 c}\\ &=\frac{1}{24} a (8 A+5 B x) \left (a+c x^2\right )^{3/2}+\frac{1}{30} (6 A+5 B x) \left (a+c x^2\right )^{5/2}+\frac{\int \frac{\left (24 a^2 A c^2+15 a^2 B c^2 x\right ) \sqrt{a+c x^2}}{x} \, dx}{24 c^2}\\ &=\frac{1}{16} a^2 (16 A+5 B x) \sqrt{a+c x^2}+\frac{1}{24} a (8 A+5 B x) \left (a+c x^2\right )^{3/2}+\frac{1}{30} (6 A+5 B x) \left (a+c x^2\right )^{5/2}+\frac{\int \frac{48 a^3 A c^3+15 a^3 B c^3 x}{x \sqrt{a+c x^2}} \, dx}{48 c^3}\\ &=\frac{1}{16} a^2 (16 A+5 B x) \sqrt{a+c x^2}+\frac{1}{24} a (8 A+5 B x) \left (a+c x^2\right )^{3/2}+\frac{1}{30} (6 A+5 B x) \left (a+c x^2\right )^{5/2}+\left (a^3 A\right ) \int \frac{1}{x \sqrt{a+c x^2}} \, dx+\frac{1}{16} \left (5 a^3 B\right ) \int \frac{1}{\sqrt{a+c x^2}} \, dx\\ &=\frac{1}{16} a^2 (16 A+5 B x) \sqrt{a+c x^2}+\frac{1}{24} a (8 A+5 B x) \left (a+c x^2\right )^{3/2}+\frac{1}{30} (6 A+5 B x) \left (a+c x^2\right )^{5/2}+\frac{1}{2} \left (a^3 A\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+c x}} \, dx,x,x^2\right )+\frac{1}{16} \left (5 a^3 B\right ) \operatorname{Subst}\left (\int \frac{1}{1-c x^2} \, dx,x,\frac{x}{\sqrt{a+c x^2}}\right )\\ &=\frac{1}{16} a^2 (16 A+5 B x) \sqrt{a+c x^2}+\frac{1}{24} a (8 A+5 B x) \left (a+c x^2\right )^{3/2}+\frac{1}{30} (6 A+5 B x) \left (a+c x^2\right )^{5/2}+\frac{5 a^3 B \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{16 \sqrt{c}}+\frac{\left (a^3 A\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{c}+\frac{x^2}{c}} \, dx,x,\sqrt{a+c x^2}\right )}{c}\\ &=\frac{1}{16} a^2 (16 A+5 B x) \sqrt{a+c x^2}+\frac{1}{24} a (8 A+5 B x) \left (a+c x^2\right )^{3/2}+\frac{1}{30} (6 A+5 B x) \left (a+c x^2\right )^{5/2}+\frac{5 a^3 B \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{16 \sqrt{c}}-a^{5/2} A \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )\\ \end{align*}
Mathematica [A] time = 0.334138, size = 139, normalized size = 1.05 \[ \frac{1}{240} \sqrt{a+c x^2} \left (a^2 (368 A+165 B x)+2 a c x^2 (88 A+65 B x)+8 c^2 x^4 (6 A+5 B x)\right )-a^{5/2} A \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )+\frac{5 a^{7/2} B \sqrt{\frac{c x^2}{a}+1} \sinh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{16 \sqrt{c} \sqrt{a+c x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 138, normalized size = 1.1 \begin{align*}{\frac{Bx}{6} \left ( c{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{5\,aBx}{24} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{5\,{a}^{2}Bx}{16}\sqrt{c{x}^{2}+a}}+{\frac{5\,B{a}^{3}}{16}\ln \left ( x\sqrt{c}+\sqrt{c{x}^{2}+a} \right ){\frac{1}{\sqrt{c}}}}+{\frac{A}{5} \left ( c{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{aA}{3} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-A{a}^{{\frac{5}{2}}}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{c{x}^{2}+a} \right ) } \right ) +A\sqrt{c{x}^{2}+a}{a}^{2} \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 = 1.80889, size = 1353, normalized size = 10.25 \begin{align*} \left [\frac{75 \, B a^{3} \sqrt{c} \log \left (-2 \, c x^{2} - 2 \, \sqrt{c x^{2} + a} \sqrt{c} x - a\right ) + 240 \, A a^{\frac{5}{2}} c \log \left (-\frac{c x^{2} - 2 \, \sqrt{c x^{2} + a} \sqrt{a} + 2 \, a}{x^{2}}\right ) + 2 \,{\left (40 \, B c^{3} x^{5} + 48 \, A c^{3} x^{4} + 130 \, B a c^{2} x^{3} + 176 \, A a c^{2} x^{2} + 165 \, B a^{2} c x + 368 \, A a^{2} c\right )} \sqrt{c x^{2} + a}}{480 \, c}, -\frac{75 \, B a^{3} \sqrt{-c} \arctan \left (\frac{\sqrt{-c} x}{\sqrt{c x^{2} + a}}\right ) - 120 \, A a^{\frac{5}{2}} c \log \left (-\frac{c x^{2} - 2 \, \sqrt{c x^{2} + a} \sqrt{a} + 2 \, a}{x^{2}}\right ) -{\left (40 \, B c^{3} x^{5} + 48 \, A c^{3} x^{4} + 130 \, B a c^{2} x^{3} + 176 \, A a c^{2} x^{2} + 165 \, B a^{2} c x + 368 \, A a^{2} c\right )} \sqrt{c x^{2} + a}}{240 \, c}, \frac{480 \, A \sqrt{-a} a^{2} c \arctan \left (\frac{\sqrt{-a}}{\sqrt{c x^{2} + a}}\right ) + 75 \, B a^{3} \sqrt{c} \log \left (-2 \, c x^{2} - 2 \, \sqrt{c x^{2} + a} \sqrt{c} x - a\right ) + 2 \,{\left (40 \, B c^{3} x^{5} + 48 \, A c^{3} x^{4} + 130 \, B a c^{2} x^{3} + 176 \, A a c^{2} x^{2} + 165 \, B a^{2} c x + 368 \, A a^{2} c\right )} \sqrt{c x^{2} + a}}{480 \, c}, -\frac{75 \, B a^{3} \sqrt{-c} \arctan \left (\frac{\sqrt{-c} x}{\sqrt{c x^{2} + a}}\right ) - 240 \, A \sqrt{-a} a^{2} c \arctan \left (\frac{\sqrt{-a}}{\sqrt{c x^{2} + a}}\right ) -{\left (40 \, B c^{3} x^{5} + 48 \, A c^{3} x^{4} + 130 \, B a c^{2} x^{3} + 176 \, A a c^{2} x^{2} + 165 \, B a^{2} c x + 368 \, A a^{2} c\right )} \sqrt{c x^{2} + a}}{240 \, c}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 33.2409, size = 323, normalized size = 2.45 \begin{align*} - A a^{\frac{5}{2}} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{c} x} \right )} + \frac{A a^{3}}{\sqrt{c} x \sqrt{\frac{a}{c x^{2}} + 1}} + \frac{A a^{2} \sqrt{c} x}{\sqrt{\frac{a}{c x^{2}} + 1}} + 2 A a c \left (\begin{cases} \frac{\sqrt{a} x^{2}}{2} & \text{for}\: c = 0 \\\frac{\left (a + c x^{2}\right )^{\frac{3}{2}}}{3 c} & \text{otherwise} \end{cases}\right ) + A c^{2} \left (\begin{cases} - \frac{2 a^{2} \sqrt{a + c x^{2}}}{15 c^{2}} + \frac{a x^{2} \sqrt{a + c x^{2}}}{15 c} + \frac{x^{4} \sqrt{a + c x^{2}}}{5} & \text{for}\: c \neq 0 \\\frac{\sqrt{a} x^{4}}{4} & \text{otherwise} \end{cases}\right ) + \frac{B a^{\frac{5}{2}} x \sqrt{1 + \frac{c x^{2}}{a}}}{2} + \frac{3 B a^{\frac{5}{2}} x}{16 \sqrt{1 + \frac{c x^{2}}{a}}} + \frac{35 B a^{\frac{3}{2}} c x^{3}}{48 \sqrt{1 + \frac{c x^{2}}{a}}} + \frac{17 B \sqrt{a} c^{2} x^{5}}{24 \sqrt{1 + \frac{c x^{2}}{a}}} + \frac{5 B a^{3} \operatorname{asinh}{\left (\frac{\sqrt{c} x}{\sqrt{a}} \right )}}{16 \sqrt{c}} + \frac{B c^{3} x^{7}}{6 \sqrt{a} \sqrt{1 + \frac{c x^{2}}{a}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1564, size = 169, normalized size = 1.28 \begin{align*} \frac{2 \, A a^{3} \arctan \left (-\frac{\sqrt{c} x - \sqrt{c x^{2} + a}}{\sqrt{-a}}\right )}{\sqrt{-a}} - \frac{5 \, B a^{3} \log \left ({\left | -\sqrt{c} x + \sqrt{c x^{2} + a} \right |}\right )}{16 \, \sqrt{c}} + \frac{1}{240} \,{\left (368 \, A a^{2} +{\left (165 \, B a^{2} + 2 \,{\left (88 \, A a c +{\left (65 \, B a c + 4 \,{\left (5 \, B c^{2} x + 6 \, A c^{2}\right )} x\right )} x\right )} x\right )} x\right )} \sqrt{c x^{2} + a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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