Optimal. Leaf size=115 \[ -\frac{A \left (a+c x^2\right )^{7/2}}{7 a x^7}-\frac{5 B c^2 \sqrt{a+c x^2}}{16 x^2}-\frac{5 B c^3 \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )}{16 \sqrt{a}}-\frac{B \left (a+c x^2\right )^{5/2}}{6 x^6}-\frac{5 B c \left (a+c x^2\right )^{3/2}}{24 x^4} \]
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Rubi [A] time = 0.0725691, antiderivative size = 115, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {807, 266, 47, 63, 208} \[ -\frac{A \left (a+c x^2\right )^{7/2}}{7 a x^7}-\frac{5 B c^2 \sqrt{a+c x^2}}{16 x^2}-\frac{5 B c^3 \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )}{16 \sqrt{a}}-\frac{B \left (a+c x^2\right )^{5/2}}{6 x^6}-\frac{5 B c \left (a+c x^2\right )^{3/2}}{24 x^4} \]
Antiderivative was successfully verified.
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Rule 807
Rule 266
Rule 47
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{(A+B x) \left (a+c x^2\right )^{5/2}}{x^8} \, dx &=-\frac{A \left (a+c x^2\right )^{7/2}}{7 a x^7}+B \int \frac{\left (a+c x^2\right )^{5/2}}{x^7} \, dx\\ &=-\frac{A \left (a+c x^2\right )^{7/2}}{7 a x^7}+\frac{1}{2} B \operatorname{Subst}\left (\int \frac{(a+c x)^{5/2}}{x^4} \, dx,x,x^2\right )\\ &=-\frac{B \left (a+c x^2\right )^{5/2}}{6 x^6}-\frac{A \left (a+c x^2\right )^{7/2}}{7 a x^7}+\frac{1}{12} (5 B c) \operatorname{Subst}\left (\int \frac{(a+c x)^{3/2}}{x^3} \, dx,x,x^2\right )\\ &=-\frac{5 B c \left (a+c x^2\right )^{3/2}}{24 x^4}-\frac{B \left (a+c x^2\right )^{5/2}}{6 x^6}-\frac{A \left (a+c x^2\right )^{7/2}}{7 a x^7}+\frac{1}{16} \left (5 B c^2\right ) \operatorname{Subst}\left (\int \frac{\sqrt{a+c x}}{x^2} \, dx,x,x^2\right )\\ &=-\frac{5 B c^2 \sqrt{a+c x^2}}{16 x^2}-\frac{5 B c \left (a+c x^2\right )^{3/2}}{24 x^4}-\frac{B \left (a+c x^2\right )^{5/2}}{6 x^6}-\frac{A \left (a+c x^2\right )^{7/2}}{7 a x^7}+\frac{1}{32} \left (5 B c^3\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+c x}} \, dx,x,x^2\right )\\ &=-\frac{5 B c^2 \sqrt{a+c x^2}}{16 x^2}-\frac{5 B c \left (a+c x^2\right )^{3/2}}{24 x^4}-\frac{B \left (a+c x^2\right )^{5/2}}{6 x^6}-\frac{A \left (a+c x^2\right )^{7/2}}{7 a x^7}+\frac{1}{16} \left (5 B c^2\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{c}+\frac{x^2}{c}} \, dx,x,\sqrt{a+c x^2}\right )\\ &=-\frac{5 B c^2 \sqrt{a+c x^2}}{16 x^2}-\frac{5 B c \left (a+c x^2\right )^{3/2}}{24 x^4}-\frac{B \left (a+c x^2\right )^{5/2}}{6 x^6}-\frac{A \left (a+c x^2\right )^{7/2}}{7 a x^7}-\frac{5 B c^3 \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )}{16 \sqrt{a}}\\ \end{align*}
Mathematica [A] time = 0.0993478, size = 123, normalized size = 1.07 \[ \frac{-\frac{\left (a+c x^2\right ) \left (2 a^2 c x^2 (72 A+91 B x)+8 a^3 (6 A+7 B x)+3 a c^2 x^4 (48 A+77 B x)+48 A c^3 x^6\right )}{a x^7}-105 B c^3 \sqrt{\frac{c x^2}{a}+1} \tanh ^{-1}\left (\sqrt{\frac{c x^2}{a}+1}\right )}{336 \sqrt{a+c x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.016, size = 164, normalized size = 1.4 \begin{align*} -{\frac{A}{7\,a{x}^{7}} \left ( c{x}^{2}+a \right ) ^{{\frac{7}{2}}}}-{\frac{B}{6\,a{x}^{6}} \left ( c{x}^{2}+a \right ) ^{{\frac{7}{2}}}}-{\frac{Bc}{24\,{a}^{2}{x}^{4}} \left ( c{x}^{2}+a \right ) ^{{\frac{7}{2}}}}-{\frac{B{c}^{2}}{16\,{a}^{3}{x}^{2}} \left ( c{x}^{2}+a \right ) ^{{\frac{7}{2}}}}+{\frac{B{c}^{3}}{16\,{a}^{3}} \left ( c{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{5\,B{c}^{3}}{48\,{a}^{2}} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{5\,B{c}^{3}}{16}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{c{x}^{2}+a} \right ) } \right ){\frac{1}{\sqrt{a}}}}+{\frac{5\,B{c}^{3}}{16\,a}\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 = 1.81048, size = 578, normalized size = 5.03 \begin{align*} \left [\frac{105 \, B \sqrt{a} c^{3} x^{7} \log \left (-\frac{c x^{2} - 2 \, \sqrt{c x^{2} + a} \sqrt{a} + 2 \, a}{x^{2}}\right ) - 2 \,{\left (48 \, A c^{3} x^{6} + 231 \, B a c^{2} x^{5} + 144 \, A a c^{2} x^{4} + 182 \, B a^{2} c x^{3} + 144 \, A a^{2} c x^{2} + 56 \, B a^{3} x + 48 \, A a^{3}\right )} \sqrt{c x^{2} + a}}{672 \, a x^{7}}, \frac{105 \, B \sqrt{-a} c^{3} x^{7} \arctan \left (\frac{\sqrt{-a}}{\sqrt{c x^{2} + a}}\right ) -{\left (48 \, A c^{3} x^{6} + 231 \, B a c^{2} x^{5} + 144 \, A a c^{2} x^{4} + 182 \, B a^{2} c x^{3} + 144 \, A a^{2} c x^{2} + 56 \, B a^{3} x + 48 \, A a^{3}\right )} \sqrt{c x^{2} + a}}{336 \, a x^{7}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 19.4185, size = 605, normalized size = 5.26 \begin{align*} - \frac{15 A a^{7} c^{\frac{9}{2}} \sqrt{\frac{a}{c x^{2}} + 1}}{105 a^{5} c^{4} x^{6} + 210 a^{4} c^{5} x^{8} + 105 a^{3} c^{6} x^{10}} - \frac{33 A a^{6} c^{\frac{11}{2}} x^{2} \sqrt{\frac{a}{c x^{2}} + 1}}{105 a^{5} c^{4} x^{6} + 210 a^{4} c^{5} x^{8} + 105 a^{3} c^{6} x^{10}} - \frac{17 A a^{5} c^{\frac{13}{2}} x^{4} \sqrt{\frac{a}{c x^{2}} + 1}}{105 a^{5} c^{4} x^{6} + 210 a^{4} c^{5} x^{8} + 105 a^{3} c^{6} x^{10}} - \frac{3 A a^{4} c^{\frac{15}{2}} x^{6} \sqrt{\frac{a}{c x^{2}} + 1}}{105 a^{5} c^{4} x^{6} + 210 a^{4} c^{5} x^{8} + 105 a^{3} c^{6} x^{10}} - \frac{12 A a^{3} c^{\frac{17}{2}} x^{8} \sqrt{\frac{a}{c x^{2}} + 1}}{105 a^{5} c^{4} x^{6} + 210 a^{4} c^{5} x^{8} + 105 a^{3} c^{6} x^{10}} - \frac{8 A a^{2} c^{\frac{19}{2}} x^{10} \sqrt{\frac{a}{c x^{2}} + 1}}{105 a^{5} c^{4} x^{6} + 210 a^{4} c^{5} x^{8} + 105 a^{3} c^{6} x^{10}} - \frac{2 A a c^{\frac{3}{2}} \sqrt{\frac{a}{c x^{2}} + 1}}{5 x^{4}} - \frac{7 A c^{\frac{5}{2}} \sqrt{\frac{a}{c x^{2}} + 1}}{15 x^{2}} - \frac{A c^{\frac{7}{2}} \sqrt{\frac{a}{c x^{2}} + 1}}{15 a} - \frac{B a^{3}}{6 \sqrt{c} x^{7} \sqrt{\frac{a}{c x^{2}} + 1}} - \frac{17 B a^{2} \sqrt{c}}{24 x^{5} \sqrt{\frac{a}{c x^{2}} + 1}} - \frac{35 B a c^{\frac{3}{2}}}{48 x^{3} \sqrt{\frac{a}{c x^{2}} + 1}} - \frac{B c^{\frac{5}{2}} \sqrt{\frac{a}{c x^{2}} + 1}}{2 x} - \frac{3 B c^{\frac{5}{2}}}{16 x \sqrt{\frac{a}{c x^{2}} + 1}} - \frac{5 B c^{3} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{c} x} \right )}}{16 \sqrt{a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.20925, size = 427, normalized size = 3.71 \begin{align*} \frac{5 \, B c^{3} \arctan \left (-\frac{\sqrt{c} x - \sqrt{c x^{2} + a}}{\sqrt{-a}}\right )}{8 \, \sqrt{-a}} + \frac{231 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{13} B c^{3} + 336 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{12} A c^{\frac{7}{2}} - 196 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{11} B a c^{3} + 595 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{9} B a^{2} c^{3} + 1680 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{8} A a^{2} c^{\frac{7}{2}} - 595 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{5} B a^{4} c^{3} + 1008 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{4} A a^{4} c^{\frac{7}{2}} + 196 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{3} B a^{5} c^{3} - 231 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )} B a^{6} c^{3} + 48 \, A a^{6} c^{\frac{7}{2}}}{168 \,{\left ({\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{2} - a\right )}^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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