Optimal. Leaf size=136 \[ \frac{2 c^{7/2} (b B-A c) \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right )}{b^{11/2}}+\frac{2 c^3 (b B-A c)}{b^5 \sqrt{x}}-\frac{2 c^2 (b B-A c)}{3 b^4 x^{3/2}}+\frac{2 c (b B-A c)}{5 b^3 x^{5/2}}-\frac{2 (b B-A c)}{7 b^2 x^{7/2}}-\frac{2 A}{9 b x^{9/2}} \]
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Rubi [A] time = 0.191533, antiderivative size = 136, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ \frac{2 c^{7/2} (b B-A c) \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right )}{b^{11/2}}+\frac{2 c^3 (b B-A c)}{b^5 \sqrt{x}}-\frac{2 c^2 (b B-A c)}{3 b^4 x^{3/2}}+\frac{2 c (b B-A c)}{5 b^3 x^{5/2}}-\frac{2 (b B-A c)}{7 b^2 x^{7/2}}-\frac{2 A}{9 b x^{9/2}} \]
Antiderivative was successfully verified.
[In] Int[(A + B*x)/(x^(9/2)*(b*x + c*x^2)),x]
[Out]
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Rubi in Sympy [A] time = 25.005, size = 129, normalized size = 0.95 \[ - \frac{2 A}{9 b x^{\frac{9}{2}}} + \frac{2 \left (A c - B b\right )}{7 b^{2} x^{\frac{7}{2}}} - \frac{2 c \left (A c - B b\right )}{5 b^{3} x^{\frac{5}{2}}} + \frac{2 c^{2} \left (A c - B b\right )}{3 b^{4} x^{\frac{3}{2}}} - \frac{2 c^{3} \left (A c - B b\right )}{b^{5} \sqrt{x}} - \frac{2 c^{\frac{7}{2}} \left (A c - B b\right ) \operatorname{atan}{\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}} \right )}}{b^{\frac{11}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)/x**(9/2)/(c*x**2+b*x),x)
[Out]
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Mathematica [A] time = 0.304365, size = 129, normalized size = 0.95 \[ \frac{2 c^{7/2} (b B-A c) \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right )}{b^{11/2}}-\frac{2 \left (A \left (35 b^4-45 b^3 c x+63 b^2 c^2 x^2-105 b c^3 x^3+315 c^4 x^4\right )+3 b B x \left (15 b^3-21 b^2 c x+35 b c^2 x^2-105 c^3 x^3\right )\right )}{315 b^5 x^{9/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(A + B*x)/(x^(9/2)*(b*x + c*x^2)),x]
[Out]
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Maple [A] time = 0.02, size = 150, normalized size = 1.1 \[ -{\frac{2\,A}{9\,b}{x}^{-{\frac{9}{2}}}}+{\frac{2\,Ac}{7\,{b}^{2}}{x}^{-{\frac{7}{2}}}}-{\frac{2\,B}{7\,b}{x}^{-{\frac{7}{2}}}}-2\,{\frac{{c}^{4}A}{{b}^{5}\sqrt{x}}}+2\,{\frac{B{c}^{3}}{{b}^{4}\sqrt{x}}}-{\frac{2\,A{c}^{2}}{5\,{b}^{3}}{x}^{-{\frac{5}{2}}}}+{\frac{2\,Bc}{5\,{b}^{2}}{x}^{-{\frac{5}{2}}}}+{\frac{2\,A{c}^{3}}{3\,{b}^{4}}{x}^{-{\frac{3}{2}}}}-{\frac{2\,B{c}^{2}}{3\,{b}^{3}}{x}^{-{\frac{3}{2}}}}-2\,{\frac{A{c}^{5}}{{b}^{5}\sqrt{bc}}\arctan \left ({\frac{c\sqrt{x}}{\sqrt{bc}}} \right ) }+2\,{\frac{{c}^{4}B}{{b}^{4}\sqrt{bc}}\arctan \left ({\frac{c\sqrt{x}}{\sqrt{bc}}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)/x^(9/2)/(c*x^2+b*x),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)/((c*x^2 + b*x)*x^(9/2)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.302253, size = 1, normalized size = 0.01 \[ \left [-\frac{315 \,{\left (B b c^{3} - A c^{4}\right )} x^{\frac{9}{2}} \sqrt{-\frac{c}{b}} \log \left (\frac{c x - 2 \, b \sqrt{x} \sqrt{-\frac{c}{b}} - b}{c x + b}\right ) + 70 \, A b^{4} - 630 \,{\left (B b c^{3} - A c^{4}\right )} x^{4} + 210 \,{\left (B b^{2} c^{2} - A b c^{3}\right )} x^{3} - 126 \,{\left (B b^{3} c - A b^{2} c^{2}\right )} x^{2} + 90 \,{\left (B b^{4} - A b^{3} c\right )} x}{315 \, b^{5} x^{\frac{9}{2}}}, -\frac{2 \,{\left (315 \,{\left (B b c^{3} - A c^{4}\right )} x^{\frac{9}{2}} \sqrt{\frac{c}{b}} \arctan \left (\frac{b \sqrt{\frac{c}{b}}}{c \sqrt{x}}\right ) + 35 \, A b^{4} - 315 \,{\left (B b c^{3} - A c^{4}\right )} x^{4} + 105 \,{\left (B b^{2} c^{2} - A b c^{3}\right )} x^{3} - 63 \,{\left (B b^{3} c - A b^{2} c^{2}\right )} x^{2} + 45 \,{\left (B b^{4} - A b^{3} c\right )} x\right )}}{315 \, b^{5} x^{\frac{9}{2}}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)/((c*x^2 + b*x)*x^(9/2)),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**(9/2)/(c*x**2+b*x),x)
[Out]
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GIAC/XCAS [A] time = 0.270923, size = 173, normalized size = 1.27 \[ \frac{2 \,{\left (B b c^{4} - A c^{5}\right )} \arctan \left (\frac{c \sqrt{x}}{\sqrt{b c}}\right )}{\sqrt{b c} b^{5}} + \frac{2 \,{\left (315 \, B b c^{3} x^{4} - 315 \, A c^{4} x^{4} - 105 \, B b^{2} c^{2} x^{3} + 105 \, A b c^{3} x^{3} + 63 \, B b^{3} c x^{2} - 63 \, A b^{2} c^{2} x^{2} - 45 \, B b^{4} x + 45 \, A b^{3} c x - 35 \, A b^{4}\right )}}{315 \, b^{5} x^{\frac{9}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)/((c*x^2 + b*x)*x^(9/2)),x, algorithm="giac")
[Out]