Optimal. Leaf size=129 \[ \frac{5 A b \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{2 a^{7/2}}-\frac{5 A \sqrt{a+b x^2}}{2 a^3 x^2}-\frac{8 B \sqrt{a+b x^2}}{3 a^3 x}+\frac{5 A+4 B x}{3 a^2 x^2 \sqrt{a+b x^2}}+\frac{A+B x}{3 a x^2 \left (a+b x^2\right )^{3/2}} \]
[Out]
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Rubi [A] time = 0.405455, antiderivative size = 129, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3 \[ \frac{5 A b \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{2 a^{7/2}}-\frac{5 A \sqrt{a+b x^2}}{2 a^3 x^2}-\frac{8 B \sqrt{a+b x^2}}{3 a^3 x}+\frac{5 A+4 B x}{3 a^2 x^2 \sqrt{a+b x^2}}+\frac{A+B x}{3 a x^2 \left (a+b x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
[In] Int[(A + B*x)/(x^3*(a + b*x^2)^(5/2)),x]
[Out]
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Rubi in Sympy [A] time = 48.7299, size = 119, normalized size = 0.92 \[ - \frac{5 A \sqrt{a + b x^{2}}}{2 a^{3} x^{2}} + \frac{5 A b \operatorname{atanh}{\left (\frac{\sqrt{a + b x^{2}}}{\sqrt{a}} \right )}}{2 a^{\frac{7}{2}}} - \frac{8 B \sqrt{a + b x^{2}}}{3 a^{3} x} + \frac{A + B x}{3 a x^{2} \left (a + b x^{2}\right )^{\frac{3}{2}}} + \frac{5 A + 4 B x}{3 a^{2} x^{2} \sqrt{a + b x^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)/x**3/(b*x**2+a)**(5/2),x)
[Out]
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Mathematica [A] time = 0.354827, size = 103, normalized size = 0.8 \[ \frac{-\frac{\sqrt{a} \left (3 a^2 (A+2 B x)+4 a b x^2 (5 A+6 B x)+b^2 x^4 (15 A+16 B x)\right )}{x^2 \left (a+b x^2\right )^{3/2}}+15 A b \log \left (\sqrt{a} \sqrt{a+b x^2}+a\right )-15 A b \log (x)}{6 a^{7/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(A + B*x)/(x^3*(a + b*x^2)^(5/2)),x]
[Out]
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Maple [A] time = 0.012, size = 134, normalized size = 1. \[ -{\frac{A}{2\,a{x}^{2}} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}}-{\frac{5\,Ab}{6\,{a}^{2}} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}}-{\frac{5\,Ab}{2\,{a}^{3}}{\frac{1}{\sqrt{b{x}^{2}+a}}}}+{\frac{5\,Ab}{2}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{2}+a} \right ) } \right ){a}^{-{\frac{7}{2}}}}-{\frac{B}{ax} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}}-{\frac{4\,bBx}{3\,{a}^{2}} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}}-{\frac{8\,bBx}{3\,{a}^{3}}{\frac{1}{\sqrt{b{x}^{2}+a}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)/x^3/(b*x^2+a)^(5/2),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)/((b*x^2 + a)^(5/2)*x^3),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.266771, size = 1, normalized size = 0.01 \[ \left [-\frac{2 \,{\left (16 \, B b^{2} x^{5} + 15 \, A b^{2} x^{4} + 24 \, B a b x^{3} + 20 \, A a b x^{2} + 6 \, B a^{2} x + 3 \, A a^{2}\right )} \sqrt{b x^{2} + a} \sqrt{a} - 15 \,{\left (A b^{3} x^{6} + 2 \, A a b^{2} x^{4} + A a^{2} b x^{2}\right )} \log \left (-\frac{{\left (b x^{2} + 2 \, a\right )} \sqrt{a} + 2 \, \sqrt{b x^{2} + a} a}{x^{2}}\right )}{12 \,{\left (a^{3} b^{2} x^{6} + 2 \, a^{4} b x^{4} + a^{5} x^{2}\right )} \sqrt{a}}, -\frac{{\left (16 \, B b^{2} x^{5} + 15 \, A b^{2} x^{4} + 24 \, B a b x^{3} + 20 \, A a b x^{2} + 6 \, B a^{2} x + 3 \, A a^{2}\right )} \sqrt{b x^{2} + a} \sqrt{-a} - 15 \,{\left (A b^{3} x^{6} + 2 \, A a b^{2} x^{4} + A a^{2} b x^{2}\right )} \arctan \left (\frac{\sqrt{-a}}{\sqrt{b x^{2} + a}}\right )}{6 \,{\left (a^{3} b^{2} x^{6} + 2 \, a^{4} b x^{4} + a^{5} x^{2}\right )} \sqrt{-a}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)/((b*x^2 + a)^(5/2)*x^3),x, algorithm="fricas")
[Out]
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Sympy [A] time = 55.7557, size = 1034, normalized size = 8.02 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x+A)/x**3/(b*x**2+a)**(5/2),x)
[Out]
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GIAC/XCAS [A] time = 0.221876, size = 266, normalized size = 2.06 \[ -\frac{{\left ({\left (\frac{5 \, B b^{2} x}{a^{3}} + \frac{6 \, A b^{2}}{a^{3}}\right )} x + \frac{6 \, B b}{a^{2}}\right )} x + \frac{7 \, A b}{a^{2}}}{3 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}}} - \frac{5 \, A b \arctan \left (-\frac{\sqrt{b} x - \sqrt{b x^{2} + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{3}} + \frac{{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{3} A b + 2 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} B a \sqrt{b} +{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )} A a b - 2 \, B a^{2} \sqrt{b}}{{\left ({\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} - a\right )}^{2} a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)/((b*x^2 + a)^(5/2)*x^3),x, algorithm="giac")
[Out]