Optimal. Leaf size=113 \[ -\frac{2 c^{5/2} (b B-A c) \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right )}{b^{9/2}}-\frac{2 c^2 (b B-A c)}{b^4 \sqrt{x}}+\frac{2 c (b B-A c)}{3 b^3 x^{3/2}}-\frac{2 (b B-A c)}{5 b^2 x^{5/2}}-\frac{2 A}{7 b x^{7/2}} \]
[Out]
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Rubi [A] time = 0.159226, antiderivative size = 113, 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 \[ -\frac{2 c^{5/2} (b B-A c) \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right )}{b^{9/2}}-\frac{2 c^2 (b B-A c)}{b^4 \sqrt{x}}+\frac{2 c (b B-A c)}{3 b^3 x^{3/2}}-\frac{2 (b B-A c)}{5 b^2 x^{5/2}}-\frac{2 A}{7 b x^{7/2}} \]
Antiderivative was successfully verified.
[In] Int[(A + B*x)/(x^(7/2)*(b*x + c*x^2)),x]
[Out]
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Rubi in Sympy [A] time = 20.1947, size = 107, normalized size = 0.95 \[ - \frac{2 A}{7 b x^{\frac{7}{2}}} + \frac{2 \left (A c - B b\right )}{5 b^{2} x^{\frac{5}{2}}} - \frac{2 c \left (A c - B b\right )}{3 b^{3} x^{\frac{3}{2}}} + \frac{2 c^{2} \left (A c - B b\right )}{b^{4} \sqrt{x}} + \frac{2 c^{\frac{5}{2}} \left (A c - B b\right ) \operatorname{atan}{\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}} \right )}}{b^{\frac{9}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)/x**(7/2)/(c*x**2+b*x),x)
[Out]
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Mathematica [A] time = 0.235231, size = 107, normalized size = 0.95 \[ \frac{2 c^{5/2} (A c-b B) \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right )}{b^{9/2}}+\frac{A \left (-30 b^3+42 b^2 c x-70 b c^2 x^2+210 c^3 x^3\right )-14 b B x \left (3 b^2-5 b c x+15 c^2 x^2\right )}{105 b^4 x^{7/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(A + B*x)/(x^(7/2)*(b*x + c*x^2)),x]
[Out]
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Maple [A] time = 0.018, size = 126, normalized size = 1.1 \[ -{\frac{2\,A}{7\,b}{x}^{-{\frac{7}{2}}}}+{\frac{2\,Ac}{5\,{b}^{2}}{x}^{-{\frac{5}{2}}}}-{\frac{2\,B}{5\,b}{x}^{-{\frac{5}{2}}}}-{\frac{2\,A{c}^{2}}{3\,{b}^{3}}{x}^{-{\frac{3}{2}}}}+{\frac{2\,Bc}{3\,{b}^{2}}{x}^{-{\frac{3}{2}}}}+2\,{\frac{A{c}^{3}}{{b}^{4}\sqrt{x}}}-2\,{\frac{B{c}^{2}}{{b}^{3}\sqrt{x}}}+2\,{\frac{A{c}^{4}}{{b}^{4}\sqrt{bc}}\arctan \left ({\frac{c\sqrt{x}}{\sqrt{bc}}} \right ) }-2\,{\frac{B{c}^{3}}{{b}^{3}\sqrt{bc}}\arctan \left ({\frac{c\sqrt{x}}{\sqrt{bc}}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)/x^(7/2)/(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^(7/2)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.300907, size = 1, normalized size = 0.01 \[ \left [-\frac{105 \,{\left (B b c^{2} - A c^{3}\right )} x^{\frac{7}{2}} \sqrt{-\frac{c}{b}} \log \left (\frac{c x + 2 \, b \sqrt{x} \sqrt{-\frac{c}{b}} - b}{c x + b}\right ) + 30 \, A b^{3} + 210 \,{\left (B b c^{2} - A c^{3}\right )} x^{3} - 70 \,{\left (B b^{2} c - A b c^{2}\right )} x^{2} + 42 \,{\left (B b^{3} - A b^{2} c\right )} x}{105 \, b^{4} x^{\frac{7}{2}}}, \frac{2 \,{\left (105 \,{\left (B b c^{2} - A c^{3}\right )} x^{\frac{7}{2}} \sqrt{\frac{c}{b}} \arctan \left (\frac{b \sqrt{\frac{c}{b}}}{c \sqrt{x}}\right ) - 15 \, A b^{3} - 105 \,{\left (B b c^{2} - A c^{3}\right )} x^{3} + 35 \,{\left (B b^{2} c - A b c^{2}\right )} x^{2} - 21 \,{\left (B b^{3} - A b^{2} c\right )} x\right )}}{105 \, b^{4} x^{\frac{7}{2}}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)/((c*x^2 + b*x)*x^(7/2)),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**(7/2)/(c*x**2+b*x),x)
[Out]
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GIAC/XCAS [A] time = 0.271709, size = 140, normalized size = 1.24 \[ -\frac{2 \,{\left (B b c^{3} - A c^{4}\right )} \arctan \left (\frac{c \sqrt{x}}{\sqrt{b c}}\right )}{\sqrt{b c} b^{4}} - \frac{2 \,{\left (105 \, B b c^{2} x^{3} - 105 \, A c^{3} x^{3} - 35 \, B b^{2} c x^{2} + 35 \, A b c^{2} x^{2} + 21 \, B b^{3} x - 21 \, A b^{2} c x + 15 \, A b^{3}\right )}}{105 \, b^{4} x^{\frac{7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)/((c*x^2 + b*x)*x^(7/2)),x, algorithm="giac")
[Out]