Optimal. Leaf size=113 \[ -\frac{2 a^{5/2} (A b-a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{b^{9/2}}+\frac{2 a^2 \sqrt{x} (A b-a B)}{b^4}-\frac{2 a x^{3/2} (A b-a B)}{3 b^3}+\frac{2 x^{5/2} (A b-a B)}{5 b^2}+\frac{2 B x^{7/2}}{7 b} \]
[Out]
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Rubi [A] time = 0.142805, 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 \[ -\frac{2 a^{5/2} (A b-a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{b^{9/2}}+\frac{2 a^2 \sqrt{x} (A b-a B)}{b^4}-\frac{2 a x^{3/2} (A b-a B)}{3 b^3}+\frac{2 x^{5/2} (A b-a B)}{5 b^2}+\frac{2 B x^{7/2}}{7 b} \]
Antiderivative was successfully verified.
[In] Int[(x^(5/2)*(A + B*x))/(a + b*x),x]
[Out]
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Rubi in Sympy [A] time = 18.8372, size = 105, normalized size = 0.93 \[ \frac{2 B x^{\frac{7}{2}}}{7 b} - \frac{2 a^{\frac{5}{2}} \left (A b - B a\right ) \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{b^{\frac{9}{2}}} + \frac{2 a^{2} \sqrt{x} \left (A b - B a\right )}{b^{4}} - \frac{2 a x^{\frac{3}{2}} \left (A b - B a\right )}{3 b^{3}} + \frac{2 x^{\frac{5}{2}} \left (A b - B a\right )}{5 b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**(5/2)*(B*x+A)/(b*x+a),x)
[Out]
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Mathematica [A] time = 0.119589, size = 101, normalized size = 0.89 \[ \frac{2 a^{5/2} (a B-A b) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{b^{9/2}}+\frac{2 \sqrt{x} \left (-105 a^3 B+35 a^2 b (3 A+B x)-7 a b^2 x (5 A+3 B x)+3 b^3 x^2 (7 A+5 B x)\right )}{105 b^4} \]
Antiderivative was successfully verified.
[In] Integrate[(x^(5/2)*(A + B*x))/(a + b*x),x]
[Out]
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Maple [A] time = 0.01, size = 126, normalized size = 1.1 \[{\frac{2\,B}{7\,b}{x}^{{\frac{7}{2}}}}+{\frac{2\,A}{5\,b}{x}^{{\frac{5}{2}}}}-{\frac{2\,Ba}{5\,{b}^{2}}{x}^{{\frac{5}{2}}}}-{\frac{2\,Aa}{3\,{b}^{2}}{x}^{{\frac{3}{2}}}}+{\frac{2\,B{a}^{2}}{3\,{b}^{3}}{x}^{{\frac{3}{2}}}}+2\,{\frac{{a}^{2}A\sqrt{x}}{{b}^{3}}}-2\,{\frac{B{a}^{3}\sqrt{x}}{{b}^{4}}}-2\,{\frac{{a}^{3}A}{{b}^{3}\sqrt{ab}}\arctan \left ({\frac{\sqrt{x}b}{\sqrt{ab}}} \right ) }+2\,{\frac{B{a}^{4}}{{b}^{4}\sqrt{ab}}\arctan \left ({\frac{\sqrt{x}b}{\sqrt{ab}}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^(5/2)*(B*x+A)/(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)*x^(5/2)/(b*x + a),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.220915, size = 1, normalized size = 0.01 \[ \left [-\frac{105 \,{\left (B a^{3} - A a^{2} b\right )} \sqrt{-\frac{a}{b}} \log \left (\frac{b x - 2 \, b \sqrt{x} \sqrt{-\frac{a}{b}} - a}{b x + a}\right ) - 2 \,{\left (15 \, B b^{3} x^{3} - 105 \, B a^{3} + 105 \, A a^{2} b - 21 \,{\left (B a b^{2} - A b^{3}\right )} x^{2} + 35 \,{\left (B a^{2} b - A a b^{2}\right )} x\right )} \sqrt{x}}{105 \, b^{4}}, \frac{2 \,{\left (105 \,{\left (B a^{3} - A a^{2} b\right )} \sqrt{\frac{a}{b}} \arctan \left (\frac{\sqrt{x}}{\sqrt{\frac{a}{b}}}\right ) +{\left (15 \, B b^{3} x^{3} - 105 \, B a^{3} + 105 \, A a^{2} b - 21 \,{\left (B a b^{2} - A b^{3}\right )} x^{2} + 35 \,{\left (B a^{2} b - A a b^{2}\right )} x\right )} \sqrt{x}\right )}}{105 \, b^{4}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*x^(5/2)/(b*x + a),x, algorithm="fricas")
[Out]
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Sympy [A] time = 52.3262, size = 162, normalized size = 1.43 \[ - \frac{2 A a^{\frac{5}{2}} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{b^{\frac{7}{2}}} + \frac{2 A a^{2} \sqrt{x}}{b^{3}} - \frac{2 A a x^{\frac{3}{2}}}{3 b^{2}} + \frac{2 A x^{\frac{5}{2}}}{5 b} + \frac{2 B a^{\frac{7}{2}} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{b^{\frac{9}{2}}} - \frac{2 B a^{3} \sqrt{x}}{b^{4}} + \frac{2 B a^{2} x^{\frac{3}{2}}}{3 b^{3}} - \frac{2 B a x^{\frac{5}{2}}}{5 b^{2}} + \frac{2 B x^{\frac{7}{2}}}{7 b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**(5/2)*(B*x+A)/(b*x+a),x)
[Out]
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GIAC/XCAS [A] time = 0.256701, size = 155, normalized size = 1.37 \[ \frac{2 \,{\left (B a^{4} - A a^{3} b\right )} \arctan \left (\frac{b \sqrt{x}}{\sqrt{a b}}\right )}{\sqrt{a b} b^{4}} + \frac{2 \,{\left (15 \, B b^{6} x^{\frac{7}{2}} - 21 \, B a b^{5} x^{\frac{5}{2}} + 21 \, A b^{6} x^{\frac{5}{2}} + 35 \, B a^{2} b^{4} x^{\frac{3}{2}} - 35 \, A a b^{5} x^{\frac{3}{2}} - 105 \, B a^{3} b^{3} \sqrt{x} + 105 \, A a^{2} b^{4} \sqrt{x}\right )}}{105 \, b^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*x^(5/2)/(b*x + a),x, algorithm="giac")
[Out]