Optimal. Leaf size=136 \[ a^{5/2} (-B) \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )+\frac{15}{8} a^2 A \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )+\frac{1}{8} a \sqrt{a+c x^2} (8 a B+15 A c x)-\frac{\left (a+c x^2\right )^{5/2} (5 A-B x)}{5 x}+\frac{1}{12} \left (a+c x^2\right )^{3/2} (4 a B+15 A c x) \]
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Rubi [A] time = 0.348791, antiderivative size = 136, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4 \[ a^{5/2} (-B) \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )+\frac{15}{8} a^2 A \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )+\frac{1}{8} a \sqrt{a+c x^2} (8 a B+15 A c x)-\frac{\left (a+c x^2\right )^{5/2} (5 A-B x)}{5 x}+\frac{1}{12} \left (a+c x^2\right )^{3/2} (4 a B+15 A c x) \]
Antiderivative was successfully verified.
[In] Int[((A + B*x)*(a + c*x^2)^(5/2))/x^2,x]
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Rubi in Sympy [A] time = 44.241, size = 124, normalized size = 0.91 \[ \frac{15 A a^{2} \sqrt{c} \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{a + c x^{2}}} \right )}}{8} - B a^{\frac{5}{2}} \operatorname{atanh}{\left (\frac{\sqrt{a + c x^{2}}}{\sqrt{a}} \right )} + \frac{a \sqrt{a + c x^{2}} \left (30 A c x + 16 B a\right )}{16} + \frac{\left (a + c x^{2}\right )^{\frac{3}{2}} \left (30 A c x + 8 B a\right )}{24} - \frac{\left (5 A - B x\right ) \left (a + c x^{2}\right )^{\frac{5}{2}}}{5 x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)*(c*x**2+a)**(5/2)/x**2,x)
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Mathematica [A] time = 0.222108, size = 135, normalized size = 0.99 \[ -a^{5/2} B \log \left (\sqrt{a} \sqrt{a+c x^2}+a\right )+a^{5/2} B \log (x)+\frac{\sqrt{a+c x^2} \left (-8 a^2 (15 A-23 B x)+a c x^2 (135 A+88 B x)+6 c^2 x^4 (5 A+4 B x)\right )}{120 x}+\frac{15}{8} a^2 A \sqrt{c} \log \left (\sqrt{c} \sqrt{a+c x^2}+c x\right ) \]
Antiderivative was successfully verified.
[In] Integrate[((A + B*x)*(a + c*x^2)^(5/2))/x^2,x]
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Maple [A] time = 0.013, size = 158, normalized size = 1.2 \[ -{\frac{A}{ax} \left ( c{x}^{2}+a \right ) ^{{\frac{7}{2}}}}+{\frac{Acx}{a} \left ( c{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{5\,Acx}{4} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{15\,aAcx}{8}\sqrt{c{x}^{2}+a}}+{\frac{15\,A{a}^{2}}{8}\sqrt{c}\ln \left ( \sqrt{c}x+\sqrt{c{x}^{2}+a} \right ) }+{\frac{B}{5} \left ( c{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{Ba}{3} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-B{a}^{{\frac{5}{2}}}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{c{x}^{2}+a} \right ) } \right ) +B\sqrt{c{x}^{2}+a}{a}^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)*(c*x^2+a)^(5/2)/x^2,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)^(5/2)*(B*x + A)/x^2,x, algorithm="maxima")
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Fricas [A] time = 0.314231, size = 1, normalized size = 0.01 \[ \left [\frac{225 \, A a^{2} \sqrt{c} x \log \left (-2 \, c x^{2} - 2 \, \sqrt{c x^{2} + a} \sqrt{c} x - a\right ) + 120 \, B a^{\frac{5}{2}} x \log \left (-\frac{c x^{2} - 2 \, \sqrt{c x^{2} + a} \sqrt{a} + 2 \, a}{x^{2}}\right ) + 2 \,{\left (24 \, B c^{2} x^{5} + 30 \, A c^{2} x^{4} + 88 \, B a c x^{3} + 135 \, A a c x^{2} + 184 \, B a^{2} x - 120 \, A a^{2}\right )} \sqrt{c x^{2} + a}}{240 \, x}, \frac{225 \, A a^{2} \sqrt{-c} x \arctan \left (\frac{c x}{\sqrt{c x^{2} + a} \sqrt{-c}}\right ) + 60 \, B a^{\frac{5}{2}} x \log \left (-\frac{c x^{2} - 2 \, \sqrt{c x^{2} + a} \sqrt{a} + 2 \, a}{x^{2}}\right ) +{\left (24 \, B c^{2} x^{5} + 30 \, A c^{2} x^{4} + 88 \, B a c x^{3} + 135 \, A a c x^{2} + 184 \, B a^{2} x - 120 \, A a^{2}\right )} \sqrt{c x^{2} + a}}{120 \, x}, -\frac{240 \, B \sqrt{-a} a^{2} x \arctan \left (\frac{a}{\sqrt{c x^{2} + a} \sqrt{-a}}\right ) - 225 \, A a^{2} \sqrt{c} x \log \left (-2 \, c x^{2} - 2 \, \sqrt{c x^{2} + a} \sqrt{c} x - a\right ) - 2 \,{\left (24 \, B c^{2} x^{5} + 30 \, A c^{2} x^{4} + 88 \, B a c x^{3} + 135 \, A a c x^{2} + 184 \, B a^{2} x - 120 \, A a^{2}\right )} \sqrt{c x^{2} + a}}{240 \, x}, \frac{225 \, A a^{2} \sqrt{-c} x \arctan \left (\frac{c x}{\sqrt{c x^{2} + a} \sqrt{-c}}\right ) - 120 \, B \sqrt{-a} a^{2} x \arctan \left (\frac{a}{\sqrt{c x^{2} + a} \sqrt{-a}}\right ) +{\left (24 \, B c^{2} x^{5} + 30 \, A c^{2} x^{4} + 88 \, B a c x^{3} + 135 \, A a c x^{2} + 184 \, B a^{2} x - 120 \, A a^{2}\right )} \sqrt{c x^{2} + a}}{120 \, x}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)^(5/2)*(B*x + A)/x^2,x, algorithm="fricas")
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Sympy [A] time = 29.7623, size = 318, normalized size = 2.34 \[ - \frac{A a^{\frac{5}{2}}}{x \sqrt{1 + \frac{c x^{2}}{a}}} + A a^{\frac{3}{2}} c x \sqrt{1 + \frac{c x^{2}}{a}} - \frac{7 A a^{\frac{3}{2}} c x}{8 \sqrt{1 + \frac{c x^{2}}{a}}} + \frac{3 A \sqrt{a} c^{2} x^{3}}{8 \sqrt{1 + \frac{c x^{2}}{a}}} + \frac{15 A a^{2} \sqrt{c} \operatorname{asinh}{\left (\frac{\sqrt{c} x}{\sqrt{a}} \right )}}{8} + \frac{A c^{3} x^{5}}{4 \sqrt{a} \sqrt{1 + \frac{c x^{2}}{a}}} - B a^{\frac{5}{2}} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{c} x} \right )} + \frac{B a^{3}}{\sqrt{c} x \sqrt{\frac{a}{c x^{2}} + 1}} + \frac{B a^{2} \sqrt{c} x}{\sqrt{\frac{a}{c x^{2}} + 1}} + 2 B a c \left (\begin{cases} \frac{\sqrt{a} x^{2}}{2} & \text{for}\: c = 0 \\\frac{\left (a + c x^{2}\right )^{\frac{3}{2}}}{3 c} & \text{otherwise} \end{cases}\right ) + B c^{2} \left (\begin{cases} - \frac{2 a^{2} \sqrt{a + c x^{2}}}{15 c^{2}} + \frac{a x^{2} \sqrt{a + c x^{2}}}{15 c} + \frac{x^{4} \sqrt{a + c x^{2}}}{5} & \text{for}\: c \neq 0 \\\frac{\sqrt{a} x^{4}}{4} & \text{otherwise} \end{cases}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x+A)*(c*x**2+a)**(5/2)/x**2,x)
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GIAC/XCAS [A] time = 0.280555, size = 203, normalized size = 1.49 \[ \frac{2 \, B a^{3} \arctan \left (-\frac{\sqrt{c} x - \sqrt{c x^{2} + a}}{\sqrt{-a}}\right )}{\sqrt{-a}} - \frac{15}{8} \, A a^{2} \sqrt{c}{\rm ln}\left ({\left | -\sqrt{c} x + \sqrt{c x^{2} + a} \right |}\right ) + \frac{2 \, A a^{3} \sqrt{c}}{{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{2} - a} + \frac{1}{120} \,{\left (184 \, B a^{2} +{\left (135 \, A a c + 2 \,{\left (44 \, B a c + 3 \,{\left (4 \, B c^{2} x + 5 \, A c^{2}\right )} x\right )} x\right )} x\right )} \sqrt{c x^{2} + a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)^(5/2)*(B*x + A)/x^2,x, algorithm="giac")
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