Optimal. Leaf size=153 \[ \frac{5 b^2 (7 A b-6 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{16 a^{9/2}}-\frac{5 b \sqrt{a+b x^2} (7 A b-6 a B)}{16 a^4 x^2}+\frac{5 \sqrt{a+b x^2} (7 A b-6 a B)}{24 a^3 x^4}-\frac{7 A b-6 a B}{6 a^2 x^4 \sqrt{a+b x^2}}-\frac{A}{6 a x^6 \sqrt{a+b x^2}} \]
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Rubi [A] time = 0.306121, antiderivative size = 153, 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 \[ \frac{5 b^2 (7 A b-6 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{16 a^{9/2}}-\frac{5 b \sqrt{a+b x^2} (7 A b-6 a B)}{16 a^4 x^2}+\frac{5 \sqrt{a+b x^2} (7 A b-6 a B)}{24 a^3 x^4}-\frac{7 A b-6 a B}{6 a^2 x^4 \sqrt{a+b x^2}}-\frac{A}{6 a x^6 \sqrt{a+b x^2}} \]
Antiderivative was successfully verified.
[In] Int[(A + B*x^2)/(x^7*(a + b*x^2)^(3/2)),x]
[Out]
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Rubi in Sympy [A] time = 25.6741, size = 148, normalized size = 0.97 \[ - \frac{A}{6 a x^{6} \sqrt{a + b x^{2}}} - \frac{7 A b - 6 B a}{6 a^{2} x^{4} \sqrt{a + b x^{2}}} + \frac{5 \sqrt{a + b x^{2}} \left (7 A b - 6 B a\right )}{24 a^{3} x^{4}} - \frac{5 b \sqrt{a + b x^{2}} \left (7 A b - 6 B a\right )}{16 a^{4} x^{2}} + \frac{5 b^{2} \left (7 A b - 6 B a\right ) \operatorname{atanh}{\left (\frac{\sqrt{a + b x^{2}}}{\sqrt{a}} \right )}}{16 a^{\frac{9}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x**2+A)/x**7/(b*x**2+a)**(3/2),x)
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Mathematica [A] time = 0.336931, size = 143, normalized size = 0.93 \[ \frac{\frac{\sqrt{a} \left (-4 a^3 \left (2 A+3 B x^2\right )+2 a^2 b x^2 \left (7 A+15 B x^2\right )+5 a b^2 x^4 \left (18 B x^2-7 A\right )-105 A b^3 x^6\right )}{x^6 \sqrt{a+b x^2}}+15 b^2 (7 A b-6 a B) \log \left (\sqrt{a} \sqrt{a+b x^2}+a\right )+15 b^2 \log (x) (6 a B-7 A b)}{48 a^{9/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(A + B*x^2)/(x^7*(a + b*x^2)^(3/2)),x]
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Maple [A] time = 0.018, size = 197, normalized size = 1.3 \[ -{\frac{A}{6\,a{x}^{6}}{\frac{1}{\sqrt{b{x}^{2}+a}}}}+{\frac{7\,Ab}{24\,{a}^{2}{x}^{4}}{\frac{1}{\sqrt{b{x}^{2}+a}}}}-{\frac{35\,{b}^{2}A}{48\,{a}^{3}{x}^{2}}{\frac{1}{\sqrt{b{x}^{2}+a}}}}-{\frac{35\,A{b}^{3}}{16\,{a}^{4}}{\frac{1}{\sqrt{b{x}^{2}+a}}}}+{\frac{35\,A{b}^{3}}{16}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{2}+a} \right ) } \right ){a}^{-{\frac{9}{2}}}}-{\frac{B}{4\,a{x}^{4}}{\frac{1}{\sqrt{b{x}^{2}+a}}}}+{\frac{5\,Bb}{8\,{a}^{2}{x}^{2}}{\frac{1}{\sqrt{b{x}^{2}+a}}}}+{\frac{15\,B{b}^{2}}{8\,{a}^{3}}{\frac{1}{\sqrt{b{x}^{2}+a}}}}-{\frac{15\,B{b}^{2}}{8}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{2}+a} \right ) } \right ){a}^{-{\frac{7}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x^2+A)/x^7/(b*x^2+a)^(3/2),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((B*x^2 + A)/((b*x^2 + a)^(3/2)*x^7),x, algorithm="maxima")
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Fricas [A] time = 0.234418, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left (15 \,{\left (6 \, B a b^{2} - 7 \, A b^{3}\right )} x^{6} + 5 \,{\left (6 \, B a^{2} b - 7 \, A a b^{2}\right )} x^{4} - 8 \, A a^{3} - 2 \,{\left (6 \, B a^{3} - 7 \, A a^{2} b\right )} x^{2}\right )} \sqrt{b x^{2} + a} \sqrt{a} - 15 \,{\left ({\left (6 \, B a b^{3} - 7 \, A b^{4}\right )} x^{8} +{\left (6 \, B a^{2} b^{2} - 7 \, A a b^{3}\right )} x^{6}\right )} \log \left (-\frac{{\left (b x^{2} + 2 \, a\right )} \sqrt{a} + 2 \, \sqrt{b x^{2} + a} a}{x^{2}}\right )}{96 \,{\left (a^{4} b x^{8} + a^{5} x^{6}\right )} \sqrt{a}}, \frac{{\left (15 \,{\left (6 \, B a b^{2} - 7 \, A b^{3}\right )} x^{6} + 5 \,{\left (6 \, B a^{2} b - 7 \, A a b^{2}\right )} x^{4} - 8 \, A a^{3} - 2 \,{\left (6 \, B a^{3} - 7 \, A a^{2} b\right )} x^{2}\right )} \sqrt{b x^{2} + a} \sqrt{-a} - 15 \,{\left ({\left (6 \, B a b^{3} - 7 \, A b^{4}\right )} x^{8} +{\left (6 \, B a^{2} b^{2} - 7 \, A a b^{3}\right )} x^{6}\right )} \arctan \left (\frac{\sqrt{-a}}{\sqrt{b x^{2} + a}}\right )}{48 \,{\left (a^{4} b x^{8} + a^{5} x^{6}\right )} \sqrt{-a}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)/((b*x^2 + a)^(3/2)*x^7),x, algorithm="fricas")
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Sympy [A] time = 91.6751, size = 236, normalized size = 1.54 \[ A \left (- \frac{1}{6 a \sqrt{b} x^{7} \sqrt{\frac{a}{b x^{2}} + 1}} + \frac{7 \sqrt{b}}{24 a^{2} x^{5} \sqrt{\frac{a}{b x^{2}} + 1}} - \frac{35 b^{\frac{3}{2}}}{48 a^{3} x^{3} \sqrt{\frac{a}{b x^{2}} + 1}} - \frac{35 b^{\frac{5}{2}}}{16 a^{4} x \sqrt{\frac{a}{b x^{2}} + 1}} + \frac{35 b^{3} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x} \right )}}{16 a^{\frac{9}{2}}}\right ) + B \left (- \frac{1}{4 a \sqrt{b} x^{5} \sqrt{\frac{a}{b x^{2}} + 1}} + \frac{5 \sqrt{b}}{8 a^{2} x^{3} \sqrt{\frac{a}{b x^{2}} + 1}} + \frac{15 b^{\frac{3}{2}}}{8 a^{3} x \sqrt{\frac{a}{b x^{2}} + 1}} - \frac{15 b^{2} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x} \right )}}{8 a^{\frac{7}{2}}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x**2+A)/x**7/(b*x**2+a)**(3/2),x)
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GIAC/XCAS [A] time = 0.231239, size = 243, normalized size = 1.59 \[ \frac{5 \,{\left (6 \, B a b^{2} - 7 \, A b^{3}\right )} \arctan \left (\frac{\sqrt{b x^{2} + a}}{\sqrt{-a}}\right )}{16 \, \sqrt{-a} a^{4}} + \frac{B a b^{2} - A b^{3}}{\sqrt{b x^{2} + a} a^{4}} + \frac{42 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} B a b^{2} - 96 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} B a^{2} b^{2} + 54 \, \sqrt{b x^{2} + a} B a^{3} b^{2} - 57 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} A b^{3} + 136 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} A a b^{3} - 87 \, \sqrt{b x^{2} + a} A a^{2} b^{3}}{48 \, a^{4} b^{3} x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)/((b*x^2 + a)^(3/2)*x^7),x, algorithm="giac")
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