Optimal. Leaf size=69 \[ \frac{5 b^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{a^{7/2}}+\frac{5 b}{a^3 \sqrt{x}}-\frac{5}{3 a^2 x^{3/2}}+\frac{1}{a x^{3/2} (a+b x)} \]
[Out]
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Rubi [A] time = 0.0580196, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308 \[ \frac{5 b^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{a^{7/2}}+\frac{5 b}{a^3 \sqrt{x}}-\frac{5}{3 a^2 x^{3/2}}+\frac{1}{a x^{3/2} (a+b x)} \]
Antiderivative was successfully verified.
[In] Int[1/(x^(5/2)*(a + b*x)^2),x]
[Out]
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Rubi in Sympy [A] time = 11.9465, size = 65, normalized size = 0.94 \[ \frac{1}{a x^{\frac{3}{2}} \left (a + b x\right )} - \frac{5}{3 a^{2} x^{\frac{3}{2}}} + \frac{5 b}{a^{3} \sqrt{x}} + \frac{5 b^{\frac{3}{2}} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{a^{\frac{7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x**(5/2)/(b*x+a)**2,x)
[Out]
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Mathematica [A] time = 0.0654992, size = 68, normalized size = 0.99 \[ \frac{5 b^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{a^{7/2}}+\frac{-2 a^2+10 a b x+15 b^2 x^2}{3 a^3 x^{3/2} (a+b x)} \]
Antiderivative was successfully verified.
[In] Integrate[1/(x^(5/2)*(a + b*x)^2),x]
[Out]
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Maple [A] time = 0.022, size = 60, normalized size = 0.9 \[ -{\frac{2}{3\,{a}^{2}}{x}^{-{\frac{3}{2}}}}+4\,{\frac{b}{{a}^{3}\sqrt{x}}}+{\frac{{b}^{2}}{{a}^{3} \left ( bx+a \right ) }\sqrt{x}}+5\,{\frac{{b}^{2}}{{a}^{3}\sqrt{ab}}\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x^(5/2)/(b*x+a)^2,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)^2*x^(5/2)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.25695, size = 1, normalized size = 0.01 \[ \left [\frac{30 \, b^{2} x^{2} + 20 \, a b x + 15 \,{\left (b^{2} x^{2} + a b x\right )} \sqrt{x} \sqrt{-\frac{b}{a}} \log \left (\frac{b x + 2 \, a \sqrt{x} \sqrt{-\frac{b}{a}} - a}{b x + a}\right ) - 4 \, a^{2}}{6 \,{\left (a^{3} b x^{2} + a^{4} x\right )} \sqrt{x}}, \frac{15 \, b^{2} x^{2} + 10 \, a b x - 15 \,{\left (b^{2} x^{2} + a b x\right )} \sqrt{x} \sqrt{\frac{b}{a}} \arctan \left (\frac{a \sqrt{\frac{b}{a}}}{b \sqrt{x}}\right ) - 2 \, a^{2}}{3 \,{\left (a^{3} b x^{2} + a^{4} x\right )} \sqrt{x}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x + a)^2*x^(5/2)),x, algorithm="fricas")
[Out]
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Sympy [A] time = 11.6904, size = 991, normalized size = 14.36 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x**(5/2)/(b*x+a)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.202231, size = 78, normalized size = 1.13 \[ \frac{5 \, b^{2} \arctan \left (\frac{b \sqrt{x}}{\sqrt{a b}}\right )}{\sqrt{a b} a^{3}} + \frac{b^{2} \sqrt{x}}{{\left (b x + a\right )} a^{3}} + \frac{2 \,{\left (6 \, b x - a\right )}}{3 \, a^{3} x^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x + a)^2*x^(5/2)),x, algorithm="giac")
[Out]