Optimal. Leaf size=113 \[ \frac{2 b^{5/2} (A b-a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{a^{9/2}}+\frac{2 b^2 (A b-a B)}{a^4 \sqrt{x}}-\frac{2 b (A b-a B)}{3 a^3 x^{3/2}}+\frac{2 (A b-a B)}{5 a^2 x^{5/2}}-\frac{2 A}{7 a x^{7/2}} \]
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Rubi [A] time = 0.158595, antiderivative size = 113, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ \frac{2 b^{5/2} (A b-a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{a^{9/2}}+\frac{2 b^2 (A b-a B)}{a^4 \sqrt{x}}-\frac{2 b (A b-a B)}{3 a^3 x^{3/2}}+\frac{2 (A b-a B)}{5 a^2 x^{5/2}}-\frac{2 A}{7 a x^{7/2}} \]
Antiderivative was successfully verified.
[In] Int[(A + B*x)/(x^(9/2)*(a + b*x)),x]
[Out]
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Rubi in Sympy [A] time = 19.5921, size = 107, normalized size = 0.95 \[ - \frac{2 A}{7 a x^{\frac{7}{2}}} + \frac{2 \left (A b - B a\right )}{5 a^{2} x^{\frac{5}{2}}} - \frac{2 b \left (A b - B a\right )}{3 a^{3} x^{\frac{3}{2}}} + \frac{2 b^{2} \left (A b - B a\right )}{a^{4} \sqrt{x}} + \frac{2 b^{\frac{5}{2}} \left (A b - B a\right ) \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{a^{\frac{9}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)/x**(9/2)/(b*x+a),x)
[Out]
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Mathematica [A] time = 0.141294, size = 103, normalized size = 0.91 \[ \frac{2 b^{5/2} (A b-a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{a^{9/2}}+\frac{-6 a^3 (5 A+7 B x)+14 a^2 b x (3 A+5 B x)-70 a b^2 x^2 (A+3 B x)+210 A b^3 x^3}{105 a^4 x^{7/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(A + B*x)/(x^(9/2)*(a + b*x)),x]
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Maple [A] time = 0.017, size = 126, normalized size = 1.1 \[ -{\frac{2\,A}{7\,a}{x}^{-{\frac{7}{2}}}}+{\frac{2\,Ab}{5\,{a}^{2}}{x}^{-{\frac{5}{2}}}}-{\frac{2\,B}{5\,a}{x}^{-{\frac{5}{2}}}}-{\frac{2\,{b}^{2}A}{3\,{a}^{3}}{x}^{-{\frac{3}{2}}}}+{\frac{2\,Bb}{3\,{a}^{2}}{x}^{-{\frac{3}{2}}}}+2\,{\frac{{b}^{3}A}{{a}^{4}\sqrt{x}}}-2\,{\frac{{b}^{2}B}{{a}^{3}\sqrt{x}}}+2\,{\frac{A{b}^{4}}{{a}^{4}\sqrt{ab}}\arctan \left ({\frac{\sqrt{x}b}{\sqrt{ab}}} \right ) }-2\,{\frac{{b}^{3}B}{{a}^{3}\sqrt{ab}}\arctan \left ({\frac{\sqrt{x}b}{\sqrt{ab}}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)/x^(9/2)/(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^(9/2)),x, algorithm="maxima")
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Fricas [A] time = 0.223307, size = 1, normalized size = 0.01 \[ \left [-\frac{105 \,{\left (B a b^{2} - A b^{3}\right )} x^{\frac{7}{2}} \sqrt{-\frac{b}{a}} \log \left (\frac{b x + 2 \, a \sqrt{x} \sqrt{-\frac{b}{a}} - a}{b x + a}\right ) + 30 \, A a^{3} + 210 \,{\left (B a b^{2} - A b^{3}\right )} x^{3} - 70 \,{\left (B a^{2} b - A a b^{2}\right )} x^{2} + 42 \,{\left (B a^{3} - A a^{2} b\right )} x}{105 \, a^{4} x^{\frac{7}{2}}}, \frac{2 \,{\left (105 \,{\left (B a b^{2} - A b^{3}\right )} x^{\frac{7}{2}} \sqrt{\frac{b}{a}} \arctan \left (\frac{a \sqrt{\frac{b}{a}}}{b \sqrt{x}}\right ) - 15 \, A a^{3} - 105 \,{\left (B a b^{2} - A b^{3}\right )} x^{3} + 35 \,{\left (B a^{2} b - A a b^{2}\right )} x^{2} - 21 \,{\left (B a^{3} - A a^{2} b\right )} x\right )}}{105 \, a^{4} x^{\frac{7}{2}}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)/((b*x + a)*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)/(b*x+a),x)
[Out]
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GIAC/XCAS [A] time = 0.213955, size = 140, normalized size = 1.24 \[ -\frac{2 \,{\left (B a b^{3} - A b^{4}\right )} \arctan \left (\frac{b \sqrt{x}}{\sqrt{a b}}\right )}{\sqrt{a b} a^{4}} - \frac{2 \,{\left (105 \, B a b^{2} x^{3} - 105 \, A b^{3} x^{3} - 35 \, B a^{2} b x^{2} + 35 \, A a b^{2} x^{2} + 21 \, B a^{3} x - 21 \, A a^{2} b x + 15 \, A a^{3}\right )}}{105 \, a^{4} x^{\frac{7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)/((b*x + a)*x^(9/2)),x, algorithm="giac")
[Out]