Optimal. Leaf size=68 \[ -\frac{2 b^{5/2} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{a^{7/2}}-\frac{2 b^2}{a^3 \sqrt{x}}+\frac{2 b}{3 a^2 x^{3/2}}-\frac{2}{5 a x^{5/2}} \]
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Rubi [A] time = 0.0589313, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ -\frac{2 b^{5/2} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{a^{7/2}}-\frac{2 b^2}{a^3 \sqrt{x}}+\frac{2 b}{3 a^2 x^{3/2}}-\frac{2}{5 a x^{5/2}} \]
Antiderivative was successfully verified.
[In] Int[1/(x^(7/2)*(a + b*x)),x]
[Out]
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Rubi in Sympy [A] time = 11.8811, size = 65, normalized size = 0.96 \[ - \frac{2}{5 a x^{\frac{5}{2}}} + \frac{2 b}{3 a^{2} x^{\frac{3}{2}}} - \frac{2 b^{2}}{a^{3} \sqrt{x}} - \frac{2 b^{\frac{5}{2}} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{a^{\frac{7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x**(7/2)/(b*x+a),x)
[Out]
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Mathematica [A] time = 0.0449019, size = 61, normalized size = 0.9 \[ -\frac{2 b^{5/2} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{a^{7/2}}-\frac{2 \left (3 a^2-5 a b x+15 b^2 x^2\right )}{15 a^3 x^{5/2}} \]
Antiderivative was successfully verified.
[In] Integrate[1/(x^(7/2)*(a + b*x)),x]
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Maple [A] time = 0.014, size = 54, normalized size = 0.8 \[ -{\frac{2}{5\,a}{x}^{-{\frac{5}{2}}}}-2\,{\frac{{b}^{2}}{{a}^{3}\sqrt{x}}}+{\frac{2\,b}{3\,{a}^{2}}{x}^{-{\frac{3}{2}}}}-2\,{\frac{{b}^{3}}{{a}^{3}\sqrt{ab}}\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x^(7/2)/(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(1/((b*x + a)*x^(7/2)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.227736, size = 1, normalized size = 0.01 \[ \left [\frac{15 \, b^{2} x^{\frac{5}{2}} \sqrt{-\frac{b}{a}} \log \left (\frac{b x - 2 \, a \sqrt{x} \sqrt{-\frac{b}{a}} - a}{b x + a}\right ) - 30 \, b^{2} x^{2} + 10 \, a b x - 6 \, a^{2}}{15 \, a^{3} x^{\frac{5}{2}}}, \frac{2 \,{\left (15 \, b^{2} x^{\frac{5}{2}} \sqrt{\frac{b}{a}} \arctan \left (\frac{a \sqrt{\frac{b}{a}}}{b \sqrt{x}}\right ) - 15 \, b^{2} x^{2} + 5 \, a b x - 3 \, a^{2}\right )}}{15 \, a^{3} x^{\frac{5}{2}}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x + a)*x^(7/2)),x, algorithm="fricas")
[Out]
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Sympy [A] time = 12.2853, size = 65, normalized size = 0.96 \[ - \frac{2}{5 a x^{\frac{5}{2}}} + \frac{2 b}{3 a^{2} x^{\frac{3}{2}}} - \frac{2 b^{2}}{a^{3} \sqrt{x}} - \frac{2 b^{\frac{5}{2}} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{a^{\frac{7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x**(7/2)/(b*x+a),x)
[Out]
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GIAC/XCAS [A] time = 0.203083, size = 70, normalized size = 1.03 \[ -\frac{2 \, b^{3} \arctan \left (\frac{b \sqrt{x}}{\sqrt{a b}}\right )}{\sqrt{a b} a^{3}} - \frac{2 \,{\left (15 \, b^{2} x^{2} - 5 \, a b x + 3 \, a^{2}\right )}}{15 \, a^{3} x^{\frac{5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x + a)*x^(7/2)),x, algorithm="giac")
[Out]