Optimal. Leaf size=95 \[ \frac{3 A c \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )}{2 a^{5/2}}-\frac{3 A \sqrt{a+c x^2}}{2 a^2 x^2}-\frac{2 B \sqrt{a+c x^2}}{a^2 x}+\frac{A+B x}{a x^2 \sqrt{a+c x^2}} \]
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Rubi [A] time = 0.258422, antiderivative size = 95, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3 \[ \frac{3 A c \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )}{2 a^{5/2}}-\frac{3 A \sqrt{a+c x^2}}{2 a^2 x^2}-\frac{2 B \sqrt{a+c x^2}}{a^2 x}+\frac{A+B x}{a x^2 \sqrt{a+c x^2}} \]
Antiderivative was successfully verified.
[In] Int[(A + B*x)/(x^3*(a + c*x^2)^(3/2)),x]
[Out]
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Rubi in Sympy [A] time = 30.1005, size = 88, normalized size = 0.93 \[ - \frac{3 A \sqrt{a + c x^{2}}}{2 a^{2} x^{2}} + \frac{3 A c \operatorname{atanh}{\left (\frac{\sqrt{a + c x^{2}}}{\sqrt{a}} \right )}}{2 a^{\frac{5}{2}}} - \frac{2 B \sqrt{a + c x^{2}}}{a^{2} x} + \frac{A + B x}{a x^{2} \sqrt{a + c x^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)/x**3/(c*x**2+a)**(3/2),x)
[Out]
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Mathematica [A] time = 0.2502, size = 83, normalized size = 0.87 \[ \frac{-\frac{\sqrt{a} \left (a (A+2 B x)+c x^2 (3 A+4 B x)\right )}{x^2 \sqrt{a+c x^2}}+3 A c \log \left (\sqrt{a} \sqrt{a+c x^2}+a\right )-3 A c \log (x)}{2 a^{5/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(A + B*x)/(x^3*(a + c*x^2)^(3/2)),x]
[Out]
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Maple [A] time = 0.013, size = 101, normalized size = 1.1 \[ -{\frac{A}{2\,a{x}^{2}}{\frac{1}{\sqrt{c{x}^{2}+a}}}}-{\frac{3\,Ac}{2\,{a}^{2}}{\frac{1}{\sqrt{c{x}^{2}+a}}}}+{\frac{3\,Ac}{2}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{c{x}^{2}+a} \right ) } \right ){a}^{-{\frac{5}{2}}}}-{\frac{B}{ax}{\frac{1}{\sqrt{c{x}^{2}+a}}}}-2\,{\frac{Bcx}{{a}^{2}\sqrt{c{x}^{2}+a}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)/x^3/(c*x^2+a)^(3/2),x)
[Out]
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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((B*x + A)/((c*x^2 + a)^(3/2)*x^3),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.294541, size = 1, normalized size = 0.01 \[ \left [-\frac{2 \,{\left (4 \, B c x^{3} + 3 \, A c x^{2} + 2 \, B a x + A a\right )} \sqrt{c x^{2} + a} \sqrt{a} - 3 \,{\left (A c^{2} x^{4} + A a c x^{2}\right )} \log \left (-\frac{{\left (c x^{2} + 2 \, a\right )} \sqrt{a} + 2 \, \sqrt{c x^{2} + a} a}{x^{2}}\right )}{4 \,{\left (a^{2} c x^{4} + a^{3} x^{2}\right )} \sqrt{a}}, -\frac{{\left (4 \, B c x^{3} + 3 \, A c x^{2} + 2 \, B a x + A a\right )} \sqrt{c x^{2} + a} \sqrt{-a} - 3 \,{\left (A c^{2} x^{4} + A a c x^{2}\right )} \arctan \left (\frac{\sqrt{-a}}{\sqrt{c x^{2} + a}}\right )}{2 \,{\left (a^{2} c x^{4} + a^{3} x^{2}\right )} \sqrt{-a}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)/((c*x^2 + a)^(3/2)*x^3),x, algorithm="fricas")
[Out]
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Sympy [A] time = 26.6863, size = 124, normalized size = 1.31 \[ A \left (- \frac{1}{2 a \sqrt{c} x^{3} \sqrt{\frac{a}{c x^{2}} + 1}} - \frac{3 \sqrt{c}}{2 a^{2} x \sqrt{\frac{a}{c x^{2}} + 1}} + \frac{3 c \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{c} x} \right )}}{2 a^{\frac{5}{2}}}\right ) + B \left (- \frac{1}{a \sqrt{c} x^{2} \sqrt{\frac{a}{c x^{2}} + 1}} - \frac{2 \sqrt{c}}{a^{2} \sqrt{\frac{a}{c x^{2}} + 1}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x+A)/x**3/(c*x**2+a)**(3/2),x)
[Out]
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GIAC/XCAS [A] time = 0.281255, size = 231, normalized size = 2.43 \[ -\frac{\frac{B c x}{a^{2}} + \frac{A c}{a^{2}}}{\sqrt{c x^{2} + a}} - \frac{3 \, A c \arctan \left (-\frac{\sqrt{c} x - \sqrt{c x^{2} + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{2}} + \frac{{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{3} A c + 2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{2} B a \sqrt{c} +{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )} A a c - 2 \, B a^{2} \sqrt{c}}{{\left ({\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{2} - a\right )}^{2} a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)/((c*x^2 + a)^(3/2)*x^3),x, algorithm="giac")
[Out]