Optimal. Leaf size=69 \[ \frac{2 \sqrt{b} (A b-a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{a^{5/2}}+\frac{2 (A b-a B)}{a^2 \sqrt{x}}-\frac{2 A}{3 a x^{3/2}} \]
[Out]
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Rubi [A] time = 0.102135, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ \frac{2 \sqrt{b} (A b-a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{a^{5/2}}+\frac{2 (A b-a B)}{a^2 \sqrt{x}}-\frac{2 A}{3 a x^{3/2}} \]
Antiderivative was successfully verified.
[In] Int[(A + B*x)/(x^(5/2)*(a + b*x)),x]
[Out]
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Rubi in Sympy [A] time = 11.5715, size = 65, normalized size = 0.94 \[ - \frac{2 A}{3 a x^{\frac{3}{2}}} + \frac{2 \left (A b - B a\right )}{a^{2} \sqrt{x}} + \frac{2 \sqrt{b} \left (A b - B a\right ) \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{a^{\frac{5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)/x**(5/2)/(b*x+a),x)
[Out]
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Mathematica [A] time = 0.0808479, size = 64, normalized size = 0.93 \[ \frac{2 \sqrt{b} (A b-a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{a^{5/2}}-\frac{2 (a (A+3 B x)-3 A b x)}{3 a^2 x^{3/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(A + B*x)/(x^(5/2)*(a + b*x)),x]
[Out]
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Maple [A] time = 0.016, size = 78, normalized size = 1.1 \[ -{\frac{2\,A}{3\,a}{x}^{-{\frac{3}{2}}}}+2\,{\frac{Ab}{\sqrt{x}{a}^{2}}}-2\,{\frac{B}{\sqrt{x}a}}+2\,{\frac{A{b}^{2}}{{a}^{2}\sqrt{ab}}\arctan \left ({\frac{\sqrt{x}b}{\sqrt{ab}}} \right ) }-2\,{\frac{Bb}{a\sqrt{ab}}\arctan \left ({\frac{\sqrt{x}b}{\sqrt{ab}}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)/x^(5/2)/(b*x+a),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^(5/2)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.221126, size = 1, normalized size = 0.01 \[ \left [-\frac{3 \,{\left (B a - A b\right )} x^{\frac{3}{2}} \sqrt{-\frac{b}{a}} \log \left (\frac{b x + 2 \, a \sqrt{x} \sqrt{-\frac{b}{a}} - a}{b x + a}\right ) + 2 \, A a + 6 \,{\left (B a - A b\right )} x}{3 \, a^{2} x^{\frac{3}{2}}}, \frac{2 \,{\left (3 \,{\left (B a - A b\right )} x^{\frac{3}{2}} \sqrt{\frac{b}{a}} \arctan \left (\frac{a \sqrt{\frac{b}{a}}}{b \sqrt{x}}\right ) - A a - 3 \,{\left (B a - A b\right )} x\right )}}{3 \, a^{2} x^{\frac{3}{2}}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)/((b*x + a)*x^(5/2)),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{A + B x}{x^{\frac{5}{2}} \left (a + b x\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x+A)/x**(5/2)/(b*x+a),x)
[Out]
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GIAC/XCAS [A] time = 0.214453, size = 74, normalized size = 1.07 \[ -\frac{2 \,{\left (B a b - A b^{2}\right )} \arctan \left (\frac{b \sqrt{x}}{\sqrt{a b}}\right )}{\sqrt{a b} a^{2}} - \frac{2 \,{\left (3 \, B a x - 3 \, A b x + A a\right )}}{3 \, a^{2} x^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)/((b*x + a)*x^(5/2)),x, algorithm="giac")
[Out]