Optimal. Leaf size=111 \[ -\frac{c \sqrt{a+c x^2} (3 A+8 B x)}{8 x^2}-\frac{\left (a+c x^2\right )^{3/2} (3 A+4 B x)}{12 x^4}-\frac{3 A c^2 \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )}{8 \sqrt{a}}+B c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right ) \]
[Out]
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Rubi [A] time = 0.264941, antiderivative size = 111, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.35 \[ -\frac{c \sqrt{a+c x^2} (3 A+8 B x)}{8 x^2}-\frac{\left (a+c x^2\right )^{3/2} (3 A+4 B x)}{12 x^4}-\frac{3 A c^2 \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )}{8 \sqrt{a}}+B c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right ) \]
Antiderivative was successfully verified.
[In] Int[((A + B*x)*(a + c*x^2)^(3/2))/x^5,x]
[Out]
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Rubi in Sympy [A] time = 33.6342, size = 104, normalized size = 0.94 \[ - \frac{3 A c^{2} \operatorname{atanh}{\left (\frac{\sqrt{a + c x^{2}}}{\sqrt{a}} \right )}}{8 \sqrt{a}} + B c^{\frac{3}{2}} \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{a + c x^{2}}} \right )} - \frac{c \left (6 A + 16 B x\right ) \sqrt{a + c x^{2}}}{16 x^{2}} - \frac{\left (3 A + 4 B x\right ) \left (a + c x^{2}\right )^{\frac{3}{2}}}{12 x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)*(c*x**2+a)**(3/2)/x**5,x)
[Out]
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Mathematica [A] time = 0.350929, size = 120, normalized size = 1.08 \[ -\frac{\sqrt{a+c x^2} \left (a (6 A+8 B x)+c x^2 (15 A+32 B x)\right )}{24 x^4}-\frac{3 A c^2 \log \left (\sqrt{a} \sqrt{a+c x^2}+a\right )}{8 \sqrt{a}}+\frac{3 A c^2 \log (x)}{8 \sqrt{a}}+B c^{3/2} \log \left (\sqrt{c} \sqrt{a+c x^2}+c x\right ) \]
Antiderivative was successfully verified.
[In] Integrate[((A + B*x)*(a + c*x^2)^(3/2))/x^5,x]
[Out]
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Maple [B] time = 0.015, size = 202, normalized size = 1.8 \[ -{\frac{A}{4\,a{x}^{4}} \left ( c{x}^{2}+a \right ) ^{{\frac{5}{2}}}}-{\frac{Ac}{8\,{a}^{2}{x}^{2}} \left ( c{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{A{c}^{2}}{8\,{a}^{2}} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{3\,A{c}^{2}}{8}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{c{x}^{2}+a} \right ) } \right ){\frac{1}{\sqrt{a}}}}+{\frac{3\,A{c}^{2}}{8\,a}\sqrt{c{x}^{2}+a}}-{\frac{B}{3\,a{x}^{3}} \left ( c{x}^{2}+a \right ) ^{{\frac{5}{2}}}}-{\frac{2\,Bc}{3\,{a}^{2}x} \left ( c{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{2\,B{c}^{2}x}{3\,{a}^{2}} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{B{c}^{2}x}{a}\sqrt{c{x}^{2}+a}}+B{c}^{{\frac{3}{2}}}\ln \left ( \sqrt{c}x+\sqrt{c{x}^{2}+a} \right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)*(c*x^2+a)^(3/2)/x^5,x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)^(3/2)*(B*x + A)/x^5,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.309644, size = 1, normalized size = 0.01 \[ \left [\frac{24 \, B \sqrt{a} c^{\frac{3}{2}} x^{4} \log \left (-2 \, c x^{2} - 2 \, \sqrt{c x^{2} + a} \sqrt{c} x - a\right ) + 9 \, A c^{2} x^{4} \log \left (-\frac{{\left (c x^{2} + 2 \, a\right )} \sqrt{a} - 2 \, \sqrt{c x^{2} + a} a}{x^{2}}\right ) - 2 \,{\left (32 \, B c x^{3} + 15 \, A c x^{2} + 8 \, B a x + 6 \, A a\right )} \sqrt{c x^{2} + a} \sqrt{a}}{48 \, \sqrt{a} x^{4}}, \frac{48 \, B \sqrt{a} \sqrt{-c} c x^{4} \arctan \left (\frac{c x}{\sqrt{c x^{2} + a} \sqrt{-c}}\right ) + 9 \, A c^{2} x^{4} \log \left (-\frac{{\left (c x^{2} + 2 \, a\right )} \sqrt{a} - 2 \, \sqrt{c x^{2} + a} a}{x^{2}}\right ) - 2 \,{\left (32 \, B c x^{3} + 15 \, A c x^{2} + 8 \, B a x + 6 \, A a\right )} \sqrt{c x^{2} + a} \sqrt{a}}{48 \, \sqrt{a} x^{4}}, -\frac{9 \, A c^{2} x^{4} \arctan \left (\frac{\sqrt{-a}}{\sqrt{c x^{2} + a}}\right ) - 12 \, B \sqrt{-a} c^{\frac{3}{2}} x^{4} \log \left (-2 \, c x^{2} - 2 \, \sqrt{c x^{2} + a} \sqrt{c} x - a\right ) +{\left (32 \, B c x^{3} + 15 \, A c x^{2} + 8 \, B a x + 6 \, A a\right )} \sqrt{c x^{2} + a} \sqrt{-a}}{24 \, \sqrt{-a} x^{4}}, \frac{24 \, B \sqrt{-a} \sqrt{-c} c x^{4} \arctan \left (\frac{c x}{\sqrt{c x^{2} + a} \sqrt{-c}}\right ) - 9 \, A c^{2} x^{4} \arctan \left (\frac{\sqrt{-a}}{\sqrt{c x^{2} + a}}\right ) -{\left (32 \, B c x^{3} + 15 \, A c x^{2} + 8 \, B a x + 6 \, A a\right )} \sqrt{c x^{2} + a} \sqrt{-a}}{24 \, \sqrt{-a} x^{4}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)^(3/2)*(B*x + A)/x^5,x, algorithm="fricas")
[Out]
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Sympy [A] time = 24.3675, size = 236, normalized size = 2.13 \[ - \frac{A a^{2}}{4 \sqrt{c} x^{5} \sqrt{\frac{a}{c x^{2}} + 1}} - \frac{3 A a \sqrt{c}}{8 x^{3} \sqrt{\frac{a}{c x^{2}} + 1}} - \frac{A c^{\frac{3}{2}} \sqrt{\frac{a}{c x^{2}} + 1}}{2 x} - \frac{A c^{\frac{3}{2}}}{8 x \sqrt{\frac{a}{c x^{2}} + 1}} - \frac{3 A c^{2} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{c} x} \right )}}{8 \sqrt{a}} - \frac{B \sqrt{a} c}{x \sqrt{1 + \frac{c x^{2}}{a}}} - \frac{B a \sqrt{c} \sqrt{\frac{a}{c x^{2}} + 1}}{3 x^{2}} - \frac{B c^{\frac{3}{2}} \sqrt{\frac{a}{c x^{2}} + 1}}{3} + B c^{\frac{3}{2}} \operatorname{asinh}{\left (\frac{\sqrt{c} x}{\sqrt{a}} \right )} - \frac{B c^{2} x}{\sqrt{a} \sqrt{1 + \frac{c x^{2}}{a}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x+A)*(c*x**2+a)**(3/2)/x**5,x)
[Out]
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GIAC/XCAS [A] time = 0.288447, size = 385, normalized size = 3.47 \[ \frac{3 \, A c^{2} \arctan \left (-\frac{\sqrt{c} x - \sqrt{c x^{2} + a}}{\sqrt{-a}}\right )}{4 \, \sqrt{-a}} - B c^{\frac{3}{2}}{\rm ln}\left ({\left | -\sqrt{c} x + \sqrt{c x^{2} + a} \right |}\right ) + \frac{15 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{7} A c^{2} + 48 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{6} B a c^{\frac{3}{2}} + 9 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{5} A a c^{2} - 96 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{4} B a^{2} c^{\frac{3}{2}} + 9 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{3} A a^{2} c^{2} + 80 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{2} B a^{3} c^{\frac{3}{2}} + 15 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )} A a^{3} c^{2} - 32 \, B a^{4} c^{\frac{3}{2}}}{12 \,{\left ({\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{2} - a\right )}^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)^(3/2)*(B*x + A)/x^5,x, algorithm="giac")
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