Optimal. Leaf size=137 \[ -\frac{5}{2} a^{3/2} B c \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )-\frac{\left (a+c x^2\right )^{5/2} (A-B x)}{3 x^3}-\frac{5 \left (a+c x^2\right )^{3/2} (a B-A c x)}{6 x^2}-\frac{5 a c \sqrt{a+c x^2} (A-B x)}{2 x}+\frac{5}{2} a A c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right ) \]
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Rubi [A] time = 0.11025, antiderivative size = 137, 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 = {813, 844, 217, 206, 266, 63, 208} \[ -\frac{5}{2} a^{3/2} B c \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )-\frac{\left (a+c x^2\right )^{5/2} (A-B x)}{3 x^3}-\frac{5 \left (a+c x^2\right )^{3/2} (a B-A c x)}{6 x^2}-\frac{5 a c \sqrt{a+c x^2} (A-B x)}{2 x}+\frac{5}{2} a A c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right ) \]
Antiderivative was successfully verified.
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Rule 813
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^4} \, dx &=-\frac{(A-B x) \left (a+c x^2\right )^{5/2}}{3 x^3}-\frac{5}{18} \int \frac{(-6 a B-6 A c x) \left (a+c x^2\right )^{3/2}}{x^3} \, dx\\ &=-\frac{5 (a B-A c x) \left (a+c x^2\right )^{3/2}}{6 x^2}-\frac{(A-B x) \left (a+c x^2\right )^{5/2}}{3 x^3}+\frac{5}{48} \int \frac{(24 a A c+24 a B c x) \sqrt{a+c x^2}}{x^2} \, dx\\ &=-\frac{5 a c (A-B x) \sqrt{a+c x^2}}{2 x}-\frac{5 (a B-A c x) \left (a+c x^2\right )^{3/2}}{6 x^2}-\frac{(A-B x) \left (a+c x^2\right )^{5/2}}{3 x^3}-\frac{5}{96} \int \frac{-48 a^2 B c-48 a A c^2 x}{x \sqrt{a+c x^2}} \, dx\\ &=-\frac{5 a c (A-B x) \sqrt{a+c x^2}}{2 x}-\frac{5 (a B-A c x) \left (a+c x^2\right )^{3/2}}{6 x^2}-\frac{(A-B x) \left (a+c x^2\right )^{5/2}}{3 x^3}+\frac{1}{2} \left (5 a^2 B c\right ) \int \frac{1}{x \sqrt{a+c x^2}} \, dx+\frac{1}{2} \left (5 a A c^2\right ) \int \frac{1}{\sqrt{a+c x^2}} \, dx\\ &=-\frac{5 a c (A-B x) \sqrt{a+c x^2}}{2 x}-\frac{5 (a B-A c x) \left (a+c x^2\right )^{3/2}}{6 x^2}-\frac{(A-B x) \left (a+c x^2\right )^{5/2}}{3 x^3}+\frac{1}{4} \left (5 a^2 B c\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+c x}} \, dx,x,x^2\right )+\frac{1}{2} \left (5 a A c^2\right ) \operatorname{Subst}\left (\int \frac{1}{1-c x^2} \, dx,x,\frac{x}{\sqrt{a+c x^2}}\right )\\ &=-\frac{5 a c (A-B x) \sqrt{a+c x^2}}{2 x}-\frac{5 (a B-A c x) \left (a+c x^2\right )^{3/2}}{6 x^2}-\frac{(A-B x) \left (a+c x^2\right )^{5/2}}{3 x^3}+\frac{5}{2} a A c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )+\frac{1}{2} \left (5 a^2 B\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{c}+\frac{x^2}{c}} \, dx,x,\sqrt{a+c x^2}\right )\\ &=-\frac{5 a c (A-B x) \sqrt{a+c x^2}}{2 x}-\frac{5 (a B-A c x) \left (a+c x^2\right )^{3/2}}{6 x^2}-\frac{(A-B x) \left (a+c x^2\right )^{5/2}}{3 x^3}+\frac{5}{2} a A c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )-\frac{5}{2} a^{3/2} B c \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )\\ \end{align*}
Mathematica [C] time = 0.0272371, size = 94, normalized size = 0.69 \[ \frac{B c \left (a+c x^2\right )^{7/2} \, _2F_1\left (2,\frac{7}{2};\frac{9}{2};\frac{c x^2}{a}+1\right )}{7 a^2}-\frac{a^2 A \sqrt{a+c x^2} \, _2F_1\left (-\frac{5}{2},-\frac{3}{2};-\frac{1}{2};-\frac{c x^2}{a}\right )}{3 x^3 \sqrt{\frac{c x^2}{a}+1}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 207, normalized size = 1.5 \begin{align*} -{\frac{A}{3\,a{x}^{3}} \left ( c{x}^{2}+a \right ) ^{{\frac{7}{2}}}}-{\frac{4\,Ac}{3\,{a}^{2}x} \left ( c{x}^{2}+a \right ) ^{{\frac{7}{2}}}}+{\frac{4\,A{c}^{2}x}{3\,{a}^{2}} \left ( c{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{5\,A{c}^{2}x}{3\,a} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{5\,A{c}^{2}x}{2}\sqrt{c{x}^{2}+a}}+{\frac{5\,aA}{2}{c}^{{\frac{3}{2}}}\ln \left ( x\sqrt{c}+\sqrt{c{x}^{2}+a} \right ) }-{\frac{B}{2\,a{x}^{2}} \left ( c{x}^{2}+a \right ) ^{{\frac{7}{2}}}}+{\frac{Bc}{2\,a} \left ( c{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{5\,Bc}{6} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{5\,Bc}{2}{a}^{{\frac{3}{2}}}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{c{x}^{2}+a} \right ) } \right ) }+{\frac{5\,aBc}{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 = 1.87295, size = 1300, normalized size = 9.49 \begin{align*} \left [\frac{15 \, A a c^{\frac{3}{2}} x^{3} \log \left (-2 \, c x^{2} - 2 \, \sqrt{c x^{2} + a} \sqrt{c} x - a\right ) + 15 \, B a^{\frac{3}{2}} c x^{3} \log \left (-\frac{c x^{2} - 2 \, \sqrt{c x^{2} + a} \sqrt{a} + 2 \, a}{x^{2}}\right ) + 2 \,{\left (2 \, B c^{2} x^{5} + 3 \, A c^{2} x^{4} + 14 \, B a c x^{3} - 14 \, A a c x^{2} - 3 \, B a^{2} x - 2 \, A a^{2}\right )} \sqrt{c x^{2} + a}}{12 \, x^{3}}, -\frac{30 \, A a \sqrt{-c} c x^{3} \arctan \left (\frac{\sqrt{-c} x}{\sqrt{c x^{2} + a}}\right ) - 15 \, B a^{\frac{3}{2}} c x^{3} \log \left (-\frac{c x^{2} - 2 \, \sqrt{c x^{2} + a} \sqrt{a} + 2 \, a}{x^{2}}\right ) - 2 \,{\left (2 \, B c^{2} x^{5} + 3 \, A c^{2} x^{4} + 14 \, B a c x^{3} - 14 \, A a c x^{2} - 3 \, B a^{2} x - 2 \, A a^{2}\right )} \sqrt{c x^{2} + a}}{12 \, x^{3}}, \frac{30 \, B \sqrt{-a} a c x^{3} \arctan \left (\frac{\sqrt{-a}}{\sqrt{c x^{2} + a}}\right ) + 15 \, A a c^{\frac{3}{2}} x^{3} \log \left (-2 \, c x^{2} - 2 \, \sqrt{c x^{2} + a} \sqrt{c} x - a\right ) + 2 \,{\left (2 \, B c^{2} x^{5} + 3 \, A c^{2} x^{4} + 14 \, B a c x^{3} - 14 \, A a c x^{2} - 3 \, B a^{2} x - 2 \, A a^{2}\right )} \sqrt{c x^{2} + a}}{12 \, x^{3}}, -\frac{15 \, A a \sqrt{-c} c x^{3} \arctan \left (\frac{\sqrt{-c} x}{\sqrt{c x^{2} + a}}\right ) - 15 \, B \sqrt{-a} a c x^{3} \arctan \left (\frac{\sqrt{-a}}{\sqrt{c x^{2} + a}}\right ) -{\left (2 \, B c^{2} x^{5} + 3 \, A c^{2} x^{4} + 14 \, B a c x^{3} - 14 \, A a c x^{2} - 3 \, B a^{2} x - 2 \, A a^{2}\right )} \sqrt{c x^{2} + a}}{6 \, x^{3}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 10.819, size = 277, normalized size = 2.02 \begin{align*} - \frac{2 A a^{\frac{3}{2}} c}{x \sqrt{1 + \frac{c x^{2}}{a}}} + \frac{A \sqrt{a} c^{2} x \sqrt{1 + \frac{c x^{2}}{a}}}{2} - \frac{2 A \sqrt{a} c^{2} x}{\sqrt{1 + \frac{c x^{2}}{a}}} - \frac{A a^{2} \sqrt{c} \sqrt{\frac{a}{c x^{2}} + 1}}{3 x^{2}} - \frac{A a c^{\frac{3}{2}} \sqrt{\frac{a}{c x^{2}} + 1}}{3} + \frac{5 A a c^{\frac{3}{2}} \operatorname{asinh}{\left (\frac{\sqrt{c} x}{\sqrt{a}} \right )}}{2} - \frac{5 B a^{\frac{3}{2}} c \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{c} x} \right )}}{2} - \frac{B a^{2} \sqrt{c} \sqrt{\frac{a}{c x^{2}} + 1}}{2 x} + \frac{2 B a^{2} \sqrt{c}}{x \sqrt{\frac{a}{c x^{2}} + 1}} + \frac{2 B a c^{\frac{3}{2}} x}{\sqrt{\frac{a}{c x^{2}} + 1}} + B c^{2} \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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.20972, size = 323, normalized size = 2.36 \begin{align*} \frac{5 \, B a^{2} c \arctan \left (-\frac{\sqrt{c} x - \sqrt{c x^{2} + a}}{\sqrt{-a}}\right )}{\sqrt{-a}} - \frac{5}{2} \, A a c^{\frac{3}{2}} \log \left ({\left | -\sqrt{c} x + \sqrt{c x^{2} + a} \right |}\right ) + \frac{1}{6} \,{\left (14 \, B a c +{\left (2 \, B c^{2} x + 3 \, A c^{2}\right )} x\right )} \sqrt{c x^{2} + a} + \frac{3 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{5} B a^{2} c + 18 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{4} A a^{2} c^{\frac{3}{2}} - 24 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{2} A a^{3} c^{\frac{3}{2}} - 3 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )} B a^{4} c + 14 \, A a^{4} c^{\frac{3}{2}}}{3 \,{\left ({\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{2} - a\right )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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