Optimal. Leaf size=136 \[ -\frac{2 b^{7/2} (A b-a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{a^{11/2}}-\frac{2 b^3 (A b-a B)}{a^5 \sqrt{x}}+\frac{2 b^2 (A b-a B)}{3 a^4 x^{3/2}}-\frac{2 b (A b-a B)}{5 a^3 x^{5/2}}+\frac{2 (A b-a B)}{7 a^2 x^{7/2}}-\frac{2 A}{9 a x^{9/2}} \]
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Rubi [A] time = 0.191084, antiderivative size = 136, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ -\frac{2 b^{7/2} (A b-a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{a^{11/2}}-\frac{2 b^3 (A b-a B)}{a^5 \sqrt{x}}+\frac{2 b^2 (A b-a B)}{3 a^4 x^{3/2}}-\frac{2 b (A b-a B)}{5 a^3 x^{5/2}}+\frac{2 (A b-a B)}{7 a^2 x^{7/2}}-\frac{2 A}{9 a x^{9/2}} \]
Antiderivative was successfully verified.
[In] Int[(A + B*x)/(x^(11/2)*(a + b*x)),x]
[Out]
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Rubi in Sympy [A] time = 24.4444, size = 129, normalized size = 0.95 \[ - \frac{2 A}{9 a x^{\frac{9}{2}}} + \frac{2 \left (A b - B a\right )}{7 a^{2} x^{\frac{7}{2}}} - \frac{2 b \left (A b - B a\right )}{5 a^{3} x^{\frac{5}{2}}} + \frac{2 b^{2} \left (A b - B a\right )}{3 a^{4} x^{\frac{3}{2}}} - \frac{2 b^{3} \left (A b - B a\right )}{a^{5} \sqrt{x}} - \frac{2 b^{\frac{7}{2}} \left (A b - B a\right ) \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{a^{\frac{11}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)/x**(11/2)/(b*x+a),x)
[Out]
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Mathematica [A] time = 0.175334, size = 122, normalized size = 0.9 \[ \frac{2 b^{7/2} (a B-A b) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{a^{11/2}}-\frac{2 \left (5 a^4 (7 A+9 B x)-9 a^3 b x (5 A+7 B x)+21 a^2 b^2 x^2 (3 A+5 B x)-105 a b^3 x^3 (A+3 B x)+315 A b^4 x^4\right )}{315 a^5 x^{9/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(A + B*x)/(x^(11/2)*(a + b*x)),x]
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Maple [A] time = 0.017, size = 150, normalized size = 1.1 \[ -{\frac{2\,A}{9\,a}{x}^{-{\frac{9}{2}}}}+{\frac{2\,Ab}{7\,{a}^{2}}{x}^{-{\frac{7}{2}}}}-{\frac{2\,B}{7\,a}{x}^{-{\frac{7}{2}}}}-2\,{\frac{{b}^{4}A}{{a}^{5}\sqrt{x}}}+2\,{\frac{{b}^{3}B}{{a}^{4}\sqrt{x}}}-{\frac{2\,{b}^{2}A}{5\,{a}^{3}}{x}^{-{\frac{5}{2}}}}+{\frac{2\,Bb}{5\,{a}^{2}}{x}^{-{\frac{5}{2}}}}+{\frac{2\,{b}^{3}A}{3\,{a}^{4}}{x}^{-{\frac{3}{2}}}}-{\frac{2\,{b}^{2}B}{3\,{a}^{3}}{x}^{-{\frac{3}{2}}}}-2\,{\frac{A{b}^{5}}{{a}^{5}\sqrt{ab}}\arctan \left ({\frac{\sqrt{x}b}{\sqrt{ab}}} \right ) }+2\,{\frac{{b}^{4}B}{{a}^{4}\sqrt{ab}}\arctan \left ({\frac{\sqrt{x}b}{\sqrt{ab}}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)/x^(11/2)/(b*x+a),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 + a)*x^(11/2)),x, algorithm="maxima")
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Fricas [A] time = 0.224919, size = 1, normalized size = 0.01 \[ \left [-\frac{315 \,{\left (B a b^{3} - A b^{4}\right )} x^{\frac{9}{2}} \sqrt{-\frac{b}{a}} \log \left (\frac{b x - 2 \, a \sqrt{x} \sqrt{-\frac{b}{a}} - a}{b x + a}\right ) + 70 \, A a^{4} - 630 \,{\left (B a b^{3} - A b^{4}\right )} x^{4} + 210 \,{\left (B a^{2} b^{2} - A a b^{3}\right )} x^{3} - 126 \,{\left (B a^{3} b - A a^{2} b^{2}\right )} x^{2} + 90 \,{\left (B a^{4} - A a^{3} b\right )} x}{315 \, a^{5} x^{\frac{9}{2}}}, -\frac{2 \,{\left (315 \,{\left (B a b^{3} - A b^{4}\right )} x^{\frac{9}{2}} \sqrt{\frac{b}{a}} \arctan \left (\frac{a \sqrt{\frac{b}{a}}}{b \sqrt{x}}\right ) + 35 \, A a^{4} - 315 \,{\left (B a b^{3} - A b^{4}\right )} x^{4} + 105 \,{\left (B a^{2} b^{2} - A a b^{3}\right )} x^{3} - 63 \,{\left (B a^{3} b - A a^{2} b^{2}\right )} x^{2} + 45 \,{\left (B a^{4} - A a^{3} b\right )} x\right )}}{315 \, a^{5} x^{\frac{9}{2}}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)/((b*x + a)*x^(11/2)),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+A)/x**(11/2)/(b*x+a),x)
[Out]
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GIAC/XCAS [A] time = 0.219479, size = 173, normalized size = 1.27 \[ \frac{2 \,{\left (B a b^{4} - A b^{5}\right )} \arctan \left (\frac{b \sqrt{x}}{\sqrt{a b}}\right )}{\sqrt{a b} a^{5}} + \frac{2 \,{\left (315 \, B a b^{3} x^{4} - 315 \, A b^{4} x^{4} - 105 \, B a^{2} b^{2} x^{3} + 105 \, A a b^{3} x^{3} + 63 \, B a^{3} b x^{2} - 63 \, A a^{2} b^{2} x^{2} - 45 \, B a^{4} x + 45 \, A a^{3} b x - 35 \, A a^{4}\right )}}{315 \, a^{5} x^{\frac{9}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)/((b*x + a)*x^(11/2)),x, algorithm="giac")
[Out]