Optimal. Leaf size=149 \[ \frac{5 A c^4 \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )}{128 a^{3/2}}+\frac{5 A c^3 \sqrt{a+c x^2}}{128 a x^2}+\frac{5 A c^2 \left (a+c x^2\right )^{3/2}}{192 a x^4}+\frac{A c \left (a+c x^2\right )^{5/2}}{48 a x^6}-\frac{A \left (a+c x^2\right )^{7/2}}{8 a x^8}-\frac{B \left (a+c x^2\right )^{7/2}}{7 a x^7} \]
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Rubi [A] time = 0.0989955, antiderivative size = 149, normalized size of antiderivative = 1., number of steps used = 8, 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} \[ \frac{5 A c^4 \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )}{128 a^{3/2}}+\frac{5 A c^3 \sqrt{a+c x^2}}{128 a x^2}+\frac{5 A c^2 \left (a+c x^2\right )^{3/2}}{192 a x^4}+\frac{A c \left (a+c x^2\right )^{5/2}}{48 a x^6}-\frac{A \left (a+c x^2\right )^{7/2}}{8 a x^8}-\frac{B \left (a+c x^2\right )^{7/2}}{7 a x^7} \]
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) \left (a+c x^2\right )^{5/2}}{x^9} \, dx &=-\frac{A \left (a+c x^2\right )^{7/2}}{8 a x^8}-\frac{\int \frac{(-8 a B+A c x) \left (a+c x^2\right )^{5/2}}{x^8} \, dx}{8 a}\\ &=-\frac{A \left (a+c x^2\right )^{7/2}}{8 a x^8}-\frac{B \left (a+c x^2\right )^{7/2}}{7 a x^7}-\frac{(A c) \int \frac{\left (a+c x^2\right )^{5/2}}{x^7} \, dx}{8 a}\\ &=-\frac{A \left (a+c x^2\right )^{7/2}}{8 a x^8}-\frac{B \left (a+c x^2\right )^{7/2}}{7 a x^7}-\frac{(A c) \operatorname{Subst}\left (\int \frac{(a+c x)^{5/2}}{x^4} \, dx,x,x^2\right )}{16 a}\\ &=\frac{A c \left (a+c x^2\right )^{5/2}}{48 a x^6}-\frac{A \left (a+c x^2\right )^{7/2}}{8 a x^8}-\frac{B \left (a+c x^2\right )^{7/2}}{7 a x^7}-\frac{\left (5 A c^2\right ) \operatorname{Subst}\left (\int \frac{(a+c x)^{3/2}}{x^3} \, dx,x,x^2\right )}{96 a}\\ &=\frac{5 A c^2 \left (a+c x^2\right )^{3/2}}{192 a x^4}+\frac{A c \left (a+c x^2\right )^{5/2}}{48 a x^6}-\frac{A \left (a+c x^2\right )^{7/2}}{8 a x^8}-\frac{B \left (a+c x^2\right )^{7/2}}{7 a x^7}-\frac{\left (5 A c^3\right ) \operatorname{Subst}\left (\int \frac{\sqrt{a+c x}}{x^2} \, dx,x,x^2\right )}{128 a}\\ &=\frac{5 A c^3 \sqrt{a+c x^2}}{128 a x^2}+\frac{5 A c^2 \left (a+c x^2\right )^{3/2}}{192 a x^4}+\frac{A c \left (a+c x^2\right )^{5/2}}{48 a x^6}-\frac{A \left (a+c x^2\right )^{7/2}}{8 a x^8}-\frac{B \left (a+c x^2\right )^{7/2}}{7 a x^7}-\frac{\left (5 A c^4\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+c x}} \, dx,x,x^2\right )}{256 a}\\ &=\frac{5 A c^3 \sqrt{a+c x^2}}{128 a x^2}+\frac{5 A c^2 \left (a+c x^2\right )^{3/2}}{192 a x^4}+\frac{A c \left (a+c x^2\right )^{5/2}}{48 a x^6}-\frac{A \left (a+c x^2\right )^{7/2}}{8 a x^8}-\frac{B \left (a+c x^2\right )^{7/2}}{7 a x^7}-\frac{\left (5 A c^3\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{c}+\frac{x^2}{c}} \, dx,x,\sqrt{a+c x^2}\right )}{128 a}\\ &=\frac{5 A c^3 \sqrt{a+c x^2}}{128 a x^2}+\frac{5 A c^2 \left (a+c x^2\right )^{3/2}}{192 a x^4}+\frac{A c \left (a+c x^2\right )^{5/2}}{48 a x^6}-\frac{A \left (a+c x^2\right )^{7/2}}{8 a x^8}-\frac{B \left (a+c x^2\right )^{7/2}}{7 a x^7}+\frac{5 A c^4 \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )}{128 a^{3/2}}\\ \end{align*}
Mathematica [C] time = 0.0189238, size = 53, normalized size = 0.36 \[ -\frac{\left (a+c x^2\right )^{7/2} \left (a^4 B+A c^4 x^7 \, _2F_1\left (\frac{7}{2},5;\frac{9}{2};\frac{c x^2}{a}+1\right )\right )}{7 a^5 x^7} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.019, size = 185, normalized size = 1.2 \begin{align*} -{\frac{B}{7\,a{x}^{7}} \left ( c{x}^{2}+a \right ) ^{{\frac{7}{2}}}}-{\frac{A}{8\,a{x}^{8}} \left ( c{x}^{2}+a \right ) ^{{\frac{7}{2}}}}+{\frac{Ac}{48\,{a}^{2}{x}^{6}} \left ( c{x}^{2}+a \right ) ^{{\frac{7}{2}}}}+{\frac{A{c}^{2}}{192\,{a}^{3}{x}^{4}} \left ( c{x}^{2}+a \right ) ^{{\frac{7}{2}}}}+{\frac{A{c}^{3}}{128\,{a}^{4}{x}^{2}} \left ( c{x}^{2}+a \right ) ^{{\frac{7}{2}}}}-{\frac{A{c}^{4}}{128\,{a}^{4}} \left ( c{x}^{2}+a \right ) ^{{\frac{5}{2}}}}-{\frac{5\,A{c}^{4}}{384\,{a}^{3}} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{5\,A{c}^{4}}{128}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{c{x}^{2}+a} \right ) } \right ){a}^{-{\frac{3}{2}}}}-{\frac{5\,A{c}^{4}}{128\,{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 = 2.0236, size = 666, normalized size = 4.47 \begin{align*} \left [\frac{105 \, A \sqrt{a} c^{4} x^{8} \log \left (-\frac{c x^{2} + 2 \, \sqrt{c x^{2} + a} \sqrt{a} + 2 \, a}{x^{2}}\right ) - 2 \,{\left (384 \, B a c^{3} x^{7} + 105 \, A a c^{3} x^{6} + 1152 \, B a^{2} c^{2} x^{5} + 826 \, A a^{2} c^{2} x^{4} + 1152 \, B a^{3} c x^{3} + 952 \, A a^{3} c x^{2} + 384 \, B a^{4} x + 336 \, A a^{4}\right )} \sqrt{c x^{2} + a}}{5376 \, a^{2} x^{8}}, -\frac{105 \, A \sqrt{-a} c^{4} x^{8} \arctan \left (\frac{\sqrt{-a}}{\sqrt{c x^{2} + a}}\right ) +{\left (384 \, B a c^{3} x^{7} + 105 \, A a c^{3} x^{6} + 1152 \, B a^{2} c^{2} x^{5} + 826 \, A a^{2} c^{2} x^{4} + 1152 \, B a^{3} c x^{3} + 952 \, A a^{3} c x^{2} + 384 \, B a^{4} x + 336 \, A a^{4}\right )} \sqrt{c x^{2} + a}}{2688 \, a^{2} x^{8}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 27.5989, size = 609, normalized size = 4.09 \begin{align*} - \frac{A a^{3}}{8 \sqrt{c} x^{9} \sqrt{\frac{a}{c x^{2}} + 1}} - \frac{23 A a^{2} \sqrt{c}}{48 x^{7} \sqrt{\frac{a}{c x^{2}} + 1}} - \frac{127 A a c^{\frac{3}{2}}}{192 x^{5} \sqrt{\frac{a}{c x^{2}} + 1}} - \frac{133 A c^{\frac{5}{2}}}{384 x^{3} \sqrt{\frac{a}{c x^{2}} + 1}} - \frac{5 A c^{\frac{7}{2}}}{128 a x \sqrt{\frac{a}{c x^{2}} + 1}} + \frac{5 A c^{4} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{c} x} \right )}}{128 a^{\frac{3}{2}}} - \frac{15 B 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 B 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 B 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 B 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 B 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 B 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 B a c^{\frac{3}{2}} \sqrt{\frac{a}{c x^{2}} + 1}}{5 x^{4}} - \frac{7 B c^{\frac{5}{2}} \sqrt{\frac{a}{c x^{2}} + 1}}{15 x^{2}} - \frac{B c^{\frac{7}{2}} \sqrt{\frac{a}{c x^{2}} + 1}}{15 a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.25284, size = 663, normalized size = 4.45 \begin{align*} -\frac{5 \, A c^{4} \arctan \left (-\frac{\sqrt{c} x - \sqrt{c x^{2} + a}}{\sqrt{-a}}\right )}{64 \, \sqrt{-a} a} + \frac{105 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{15} A c^{4} + 2688 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{14} B a c^{\frac{7}{2}} + 2779 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{13} A a c^{4} - 2688 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{12} B a^{2} c^{\frac{7}{2}} + 6265 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{11} A a^{2} c^{4} + 13440 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{10} B a^{3} c^{\frac{7}{2}} + 12355 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{9} A a^{3} c^{4} - 13440 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{8} B a^{4} c^{\frac{7}{2}} + 12355 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{7} A a^{4} c^{4} + 8064 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{6} B a^{5} c^{\frac{7}{2}} + 6265 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{5} A a^{5} c^{4} - 8064 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{4} B a^{6} c^{\frac{7}{2}} + 2779 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{3} A a^{6} c^{4} + 384 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{2} B a^{7} c^{\frac{7}{2}} + 105 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )} A a^{7} c^{4} - 384 \, B a^{8} c^{\frac{7}{2}}}{1344 \,{\left ({\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{2} - a\right )}^{8} a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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