Optimal. Leaf size=107 \[ \frac{\sqrt{b} (5 A b-3 a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{a^{7/2}}+\frac{5 A b-3 a B}{a^3 \sqrt{x}}-\frac{5 A b-3 a B}{3 a^2 b x^{3/2}}+\frac{A b-a B}{a b x^{3/2} (a+b x)} \]
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Rubi [A] time = 0.138668, antiderivative size = 107, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ \frac{\sqrt{b} (5 A b-3 a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{a^{7/2}}+\frac{5 A b-3 a B}{a^3 \sqrt{x}}-\frac{5 A b-3 a B}{3 a^2 b x^{3/2}}+\frac{A b-a B}{a b x^{3/2} (a+b x)} \]
Antiderivative was successfully verified.
[In] Int[(A + B*x)/(x^(5/2)*(a + b*x)^2),x]
[Out]
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Rubi in Sympy [A] time = 16.8235, size = 97, normalized size = 0.91 \[ \frac{A b - B a}{a b x^{\frac{3}{2}} \left (a + b x\right )} - \frac{5 A b - 3 B a}{3 a^{2} b x^{\frac{3}{2}}} + \frac{5 A b - 3 B a}{a^{3} \sqrt{x}} + \frac{\sqrt{b} \left (5 A b - 3 B a\right ) \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{a^{\frac{7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)/x**(5/2)/(b*x+a)**2,x)
[Out]
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Mathematica [A] time = 0.15924, size = 90, normalized size = 0.84 \[ \frac{\sqrt{b} (5 A b-3 a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{a^{7/2}}+\frac{-2 a^2 (A+3 B x)+a b x (10 A-9 B x)+15 A b^2 x^2}{3 a^3 x^{3/2} (a+b x)} \]
Antiderivative was successfully verified.
[In] Integrate[(A + B*x)/(x^(5/2)*(a + b*x)^2),x]
[Out]
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Maple [A] time = 0.024, size = 113, normalized size = 1.1 \[ -{\frac{2\,A}{3\,{a}^{2}}{x}^{-{\frac{3}{2}}}}+4\,{\frac{Ab}{\sqrt{x}{a}^{3}}}-2\,{\frac{B}{\sqrt{x}{a}^{2}}}+{\frac{{b}^{2}A}{{a}^{3} \left ( bx+a \right ) }\sqrt{x}}-{\frac{Bb}{{a}^{2} \left ( bx+a \right ) }\sqrt{x}}+5\,{\frac{{b}^{2}A}{{a}^{3}\sqrt{ab}}\arctan \left ({\frac{\sqrt{x}b}{\sqrt{ab}}} \right ) }-3\,{\frac{Bb}{{a}^{2}\sqrt{ab}}\arctan \left ({\frac{\sqrt{x}b}{\sqrt{ab}}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)/x^(5/2)/(b*x+a)^2,x)
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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)^2*x^(5/2)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.235961, size = 1, normalized size = 0.01 \[ \left [-\frac{4 \, A a^{2} + 6 \,{\left (3 \, B a b - 5 \, A b^{2}\right )} x^{2} + 3 \,{\left ({\left (3 \, B a b - 5 \, A b^{2}\right )} x^{2} +{\left (3 \, B a^{2} - 5 \, A a b\right )} x\right )} \sqrt{x} \sqrt{-\frac{b}{a}} \log \left (\frac{b x + 2 \, a \sqrt{x} \sqrt{-\frac{b}{a}} - a}{b x + a}\right ) + 4 \,{\left (3 \, B a^{2} - 5 \, A a b\right )} x}{6 \,{\left (a^{3} b x^{2} + a^{4} x\right )} \sqrt{x}}, -\frac{2 \, A a^{2} + 3 \,{\left (3 \, B a b - 5 \, A b^{2}\right )} x^{2} - 3 \,{\left ({\left (3 \, B a b - 5 \, A b^{2}\right )} x^{2} +{\left (3 \, B a^{2} - 5 \, A a b\right )} x\right )} \sqrt{x} \sqrt{\frac{b}{a}} \arctan \left (\frac{a \sqrt{\frac{b}{a}}}{b \sqrt{x}}\right ) + 2 \,{\left (3 \, B a^{2} - 5 \, A a b\right )} x}{3 \,{\left (a^{3} b x^{2} + a^{4} x\right )} \sqrt{x}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)/((b*x + a)^2*x^(5/2)),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**(5/2)/(b*x+a)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.23504, size = 115, normalized size = 1.07 \[ -\frac{{\left (3 \, B a b - 5 \, A b^{2}\right )} \arctan \left (\frac{b \sqrt{x}}{\sqrt{a b}}\right )}{\sqrt{a b} a^{3}} - \frac{B a b \sqrt{x} - A b^{2} \sqrt{x}}{{\left (b x + a\right )} a^{3}} - \frac{2 \,{\left (3 \, B a x - 6 \, A b x + A a\right )}}{3 \, a^{3} x^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)/((b*x + a)^2*x^(5/2)),x, algorithm="giac")
[Out]