Optimal. Leaf size=135 \[ -\frac{1}{2} a^{3/2} (2 a B+5 A b) \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )+\frac{\left (a+b x^2\right )^{5/2} (2 a B+5 A b)}{10 a}+\frac{1}{6} \left (a+b x^2\right )^{3/2} (2 a B+5 A b)+\frac{1}{2} a \sqrt{a+b x^2} (2 a B+5 A b)-\frac{A \left (a+b x^2\right )^{7/2}}{2 a x^2} \]
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Rubi [A] time = 0.267193, antiderivative size = 135, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ -\frac{1}{2} a^{3/2} (2 a B+5 A b) \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )+\frac{\left (a+b x^2\right )^{5/2} (2 a B+5 A b)}{10 a}+\frac{1}{6} \left (a+b x^2\right )^{3/2} (2 a B+5 A b)+\frac{1}{2} a \sqrt{a+b x^2} (2 a B+5 A b)-\frac{A \left (a+b x^2\right )^{7/2}}{2 a x^2} \]
Antiderivative was successfully verified.
[In] Int[((a + b*x^2)^(5/2)*(A + B*x^2))/x^3,x]
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Rubi in Sympy [A] time = 22.277, size = 119, normalized size = 0.88 \[ - \frac{A \left (a + b x^{2}\right )^{\frac{7}{2}}}{2 a x^{2}} - a^{\frac{3}{2}} \left (\frac{5 A b}{2} + B a\right ) \operatorname{atanh}{\left (\frac{\sqrt{a + b x^{2}}}{\sqrt{a}} \right )} + \frac{a \sqrt{a + b x^{2}} \left (5 A b + 2 B a\right )}{2} + \left (a + b x^{2}\right )^{\frac{3}{2}} \left (\frac{5 A b}{6} + \frac{B a}{3}\right ) + \frac{\left (a + b x^{2}\right )^{\frac{5}{2}} \left (\frac{5 A b}{2} + B a\right )}{5 a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x**2+a)**(5/2)*(B*x**2+A)/x**3,x)
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Mathematica [A] time = 0.272987, size = 117, normalized size = 0.87 \[ \frac{1}{30} \left (-15 a^{3/2} (2 a B+5 A b) \log \left (\sqrt{a} \sqrt{a+b x^2}+a\right )+15 a^{3/2} \log (x) (2 a B+5 A b)+\sqrt{a+b x^2} \left (-\frac{15 a^2 A}{x^2}+2 b x^2 (11 a B+5 A b)+2 a (23 a B+35 A b)+6 b^2 B x^4\right )\right ) \]
Antiderivative was successfully verified.
[In] Integrate[((a + b*x^2)^(5/2)*(A + B*x^2))/x^3,x]
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Maple [A] time = 0.012, size = 161, normalized size = 1.2 \[ -{\frac{A}{2\,a{x}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{2}}}}+{\frac{Ab}{2\,a} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{5\,Ab}{6} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{5\,Ab}{2}{a}^{{\frac{3}{2}}}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{2}+a} \right ) } \right ) }+{\frac{5\,abA}{2}\sqrt{b{x}^{2}+a}}+{\frac{B}{5} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{Ba}{3} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-B{a}^{{\frac{5}{2}}}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{2}+a} \right ) } \right ) +B\sqrt{b{x}^{2}+a}{a}^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x^2+a)^(5/2)*(B*x^2+A)/x^3,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((B*x^2 + A)*(b*x^2 + a)^(5/2)/x^3,x, algorithm="maxima")
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Fricas [A] time = 0.257479, size = 1, normalized size = 0.01 \[ \left [\frac{15 \,{\left (2 \, B a^{2} + 5 \, A a b\right )} \sqrt{a} x^{2} \log \left (-\frac{b x^{2} - 2 \, \sqrt{b x^{2} + a} \sqrt{a} + 2 \, a}{x^{2}}\right ) + 2 \,{\left (6 \, B b^{2} x^{6} + 2 \,{\left (11 \, B a b + 5 \, A b^{2}\right )} x^{4} - 15 \, A a^{2} + 2 \,{\left (23 \, B a^{2} + 35 \, A a b\right )} x^{2}\right )} \sqrt{b x^{2} + a}}{60 \, x^{2}}, -\frac{15 \,{\left (2 \, B a^{2} + 5 \, A a b\right )} \sqrt{-a} x^{2} \arctan \left (\frac{a}{\sqrt{b x^{2} + a} \sqrt{-a}}\right ) -{\left (6 \, B b^{2} x^{6} + 2 \,{\left (11 \, B a b + 5 \, A b^{2}\right )} x^{4} - 15 \, A a^{2} + 2 \,{\left (23 \, B a^{2} + 35 \, A a b\right )} x^{2}\right )} \sqrt{b x^{2} + a}}{30 \, x^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)*(b*x^2 + a)^(5/2)/x^3,x, algorithm="fricas")
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Sympy [A] time = 57.2278, size = 296, normalized size = 2.19 \[ - \frac{5 A a^{\frac{3}{2}} b \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x} \right )}}{2} - \frac{A a^{2} \sqrt{b} \sqrt{\frac{a}{b x^{2}} + 1}}{2 x} + \frac{2 A a^{2} \sqrt{b}}{x \sqrt{\frac{a}{b x^{2}} + 1}} + \frac{2 A a b^{\frac{3}{2}} x}{\sqrt{\frac{a}{b x^{2}} + 1}} + A b^{2} \left (\begin{cases} \frac{\sqrt{a} x^{2}}{2} & \text{for}\: b = 0 \\\frac{\left (a + b x^{2}\right )^{\frac{3}{2}}}{3 b} & \text{otherwise} \end{cases}\right ) - B a^{\frac{5}{2}} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x} \right )} + \frac{B a^{3}}{\sqrt{b} x \sqrt{\frac{a}{b x^{2}} + 1}} + \frac{B a^{2} \sqrt{b} x}{\sqrt{\frac{a}{b x^{2}} + 1}} + 2 B a b \left (\begin{cases} \frac{\sqrt{a} x^{2}}{2} & \text{for}\: b = 0 \\\frac{\left (a + b x^{2}\right )^{\frac{3}{2}}}{3 b} & \text{otherwise} \end{cases}\right ) + B b^{2} \left (\begin{cases} - \frac{2 a^{2} \sqrt{a + b x^{2}}}{15 b^{2}} + \frac{a x^{2} \sqrt{a + b x^{2}}}{15 b} + \frac{x^{4} \sqrt{a + b x^{2}}}{5} & \text{for}\: b \neq 0 \\\frac{\sqrt{a} x^{4}}{4} & \text{otherwise} \end{cases}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x**2+a)**(5/2)*(B*x**2+A)/x**3,x)
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GIAC/XCAS [A] time = 0.243336, size = 188, normalized size = 1.39 \[ \frac{6 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} B b + 10 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} B a b + 30 \, \sqrt{b x^{2} + a} B a^{2} b + 10 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} A b^{2} + 60 \, \sqrt{b x^{2} + a} A a b^{2} - \frac{15 \, \sqrt{b x^{2} + a} A a^{2} b}{x^{2}} + \frac{15 \,{\left (2 \, B a^{3} b + 5 \, A a^{2} b^{2}\right )} \arctan \left (\frac{\sqrt{b x^{2} + a}}{\sqrt{-a}}\right )}{\sqrt{-a}}}{30 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)*(b*x^2 + a)^(5/2)/x^3,x, algorithm="giac")
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