Optimal. Leaf size=189 \[ -\frac{b^4 (7 A b-10 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{256 a^{9/2}}+\frac{b^3 \sqrt{a+b x^2} (7 A b-10 a B)}{256 a^4 x^2}-\frac{b^2 \sqrt{a+b x^2} (7 A b-10 a B)}{384 a^3 x^4}+\frac{b \sqrt{a+b x^2} (7 A b-10 a B)}{480 a^2 x^6}+\frac{\sqrt{a+b x^2} (7 A b-10 a B)}{80 a x^8}-\frac{A \left (a+b x^2\right )^{3/2}}{10 a x^{10}} \]
[Out]
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Rubi [A] time = 0.371806, antiderivative size = 189, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273 \[ -\frac{b^4 (7 A b-10 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{256 a^{9/2}}+\frac{b^3 \sqrt{a+b x^2} (7 A b-10 a B)}{256 a^4 x^2}-\frac{b^2 \sqrt{a+b x^2} (7 A b-10 a B)}{384 a^3 x^4}+\frac{b \sqrt{a+b x^2} (7 A b-10 a B)}{480 a^2 x^6}+\frac{\sqrt{a+b x^2} (7 A b-10 a B)}{80 a x^8}-\frac{A \left (a+b x^2\right )^{3/2}}{10 a x^{10}} \]
Antiderivative was successfully verified.
[In] Int[(Sqrt[a + b*x^2]*(A + B*x^2))/x^11,x]
[Out]
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Rubi in Sympy [A] time = 31.8797, size = 177, normalized size = 0.94 \[ - \frac{A \left (a + b x^{2}\right )^{\frac{3}{2}}}{10 a x^{10}} + \frac{\sqrt{a + b x^{2}} \left (7 A b - 10 B a\right )}{80 a x^{8}} + \frac{b \sqrt{a + b x^{2}} \left (7 A b - 10 B a\right )}{480 a^{2} x^{6}} - \frac{b^{2} \sqrt{a + b x^{2}} \left (7 A b - 10 B a\right )}{384 a^{3} x^{4}} + \frac{b^{3} \sqrt{a + b x^{2}} \left (7 A b - 10 B a\right )}{256 a^{4} x^{2}} - \frac{b^{4} \left (7 A b - 10 B a\right ) \operatorname{atanh}{\left (\frac{\sqrt{a + b x^{2}}}{\sqrt{a}} \right )}}{256 a^{\frac{9}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x**2+A)*(b*x**2+a)**(1/2)/x**11,x)
[Out]
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Mathematica [A] time = 0.251981, size = 169, normalized size = 0.89 \[ -\frac{b^4 (7 A b-10 a B) \log \left (\sqrt{a} \sqrt{a+b x^2}+a\right )}{256 a^{9/2}}+\frac{b^4 \log (x) (7 A b-10 a B)}{256 a^{9/2}}+\sqrt{a+b x^2} \left (-\frac{b^3 (10 a B-7 A b)}{256 a^4 x^2}+\frac{b^2 (10 a B-7 A b)}{384 a^3 x^4}-\frac{b (10 a B-7 A b)}{480 a^2 x^6}+\frac{-10 a B-A b}{80 a x^8}-\frac{A}{10 x^{10}}\right ) \]
Antiderivative was successfully verified.
[In] Integrate[(Sqrt[a + b*x^2]*(A + B*x^2))/x^11,x]
[Out]
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Maple [A] time = 0.038, size = 281, normalized size = 1.5 \[ -{\frac{A}{10\,a{x}^{10}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{7\,Ab}{80\,{a}^{2}{x}^{8}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{7\,{b}^{2}A}{96\,{a}^{3}{x}^{6}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{7\,A{b}^{3}}{128\,{a}^{4}{x}^{4}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{7\,A{b}^{4}}{256\,{a}^{5}{x}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{7\,A{b}^{5}}{256}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{2}+a} \right ) } \right ){a}^{-{\frac{9}{2}}}}+{\frac{7\,A{b}^{5}}{256\,{a}^{5}}\sqrt{b{x}^{2}+a}}-{\frac{B}{8\,a{x}^{8}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{5\,Bb}{48\,{a}^{2}{x}^{6}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{5\,B{b}^{2}}{64\,{a}^{3}{x}^{4}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{5\,B{b}^{3}}{128\,{a}^{4}{x}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{5\,B{b}^{4}}{128}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{2}+a} \right ) } \right ){a}^{-{\frac{7}{2}}}}-{\frac{5\,B{b}^{4}}{128\,{a}^{4}}\sqrt{b{x}^{2}+a}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x^2+A)*(b*x^2+a)^(1/2)/x^11,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^2 + A)*sqrt(b*x^2 + a)/x^11,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.410732, size = 1, normalized size = 0.01 \[ \left [-\frac{15 \,{\left (10 \, B a b^{4} - 7 \, A b^{5}\right )} x^{10} \log \left (-\frac{{\left (b x^{2} + 2 \, a\right )} \sqrt{a} - 2 \, \sqrt{b x^{2} + a} a}{x^{2}}\right ) + 2 \,{\left (15 \,{\left (10 \, B a b^{3} - 7 \, A b^{4}\right )} x^{8} - 10 \,{\left (10 \, B a^{2} b^{2} - 7 \, A a b^{3}\right )} x^{6} + 384 \, A a^{4} + 8 \,{\left (10 \, B a^{3} b - 7 \, A a^{2} b^{2}\right )} x^{4} + 48 \,{\left (10 \, B a^{4} + A a^{3} b\right )} x^{2}\right )} \sqrt{b x^{2} + a} \sqrt{a}}{7680 \, a^{\frac{9}{2}} x^{10}}, \frac{15 \,{\left (10 \, B a b^{4} - 7 \, A b^{5}\right )} x^{10} \arctan \left (\frac{\sqrt{-a}}{\sqrt{b x^{2} + a}}\right ) -{\left (15 \,{\left (10 \, B a b^{3} - 7 \, A b^{4}\right )} x^{8} - 10 \,{\left (10 \, B a^{2} b^{2} - 7 \, A a b^{3}\right )} x^{6} + 384 \, A a^{4} + 8 \,{\left (10 \, B a^{3} b - 7 \, A a^{2} b^{2}\right )} x^{4} + 48 \,{\left (10 \, B a^{4} + A a^{3} b\right )} x^{2}\right )} \sqrt{b x^{2} + a} \sqrt{-a}}{3840 \, \sqrt{-a} a^{4} x^{10}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)*sqrt(b*x^2 + a)/x^11,x, algorithm="fricas")
[Out]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x**2+A)*(b*x**2+a)**(1/2)/x**11,x)
[Out]
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GIAC/XCAS [A] time = 0.239782, size = 311, normalized size = 1.65 \[ -\frac{\frac{15 \,{\left (10 \, B a b^{5} - 7 \, A b^{6}\right )} \arctan \left (\frac{\sqrt{b x^{2} + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{4}} + \frac{150 \,{\left (b x^{2} + a\right )}^{\frac{9}{2}} B a b^{5} - 700 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} B a^{2} b^{5} + 1280 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} B a^{3} b^{5} - 580 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} B a^{4} b^{5} - 150 \, \sqrt{b x^{2} + a} B a^{5} b^{5} - 105 \,{\left (b x^{2} + a\right )}^{\frac{9}{2}} A b^{6} + 490 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} A a b^{6} - 896 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} A a^{2} b^{6} + 790 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} A a^{3} b^{6} + 105 \, \sqrt{b x^{2} + a} A a^{4} b^{6}}{a^{4} b^{5} x^{10}}}{3840 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)*sqrt(b*x^2 + a)/x^11,x, algorithm="giac")
[Out]