Optimal. Leaf size=95 \[ \frac{5 a \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a-b x}}\right )}{b^{7/2}}-\frac{5 \sqrt{x} \sqrt{a-b x}}{b^3}-\frac{10 x^{3/2}}{3 b^2 \sqrt{a-b x}}+\frac{2 x^{5/2}}{3 b (a-b x)^{3/2}} \]
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Rubi [A] time = 0.0731161, antiderivative size = 95, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{5 a \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a-b x}}\right )}{b^{7/2}}-\frac{5 \sqrt{x} \sqrt{a-b x}}{b^3}-\frac{10 x^{3/2}}{3 b^2 \sqrt{a-b x}}+\frac{2 x^{5/2}}{3 b (a-b x)^{3/2}} \]
Antiderivative was successfully verified.
[In] Int[x^(5/2)/(a - b*x)^(5/2),x]
[Out]
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Rubi in Sympy [A] time = 11.5081, size = 85, normalized size = 0.89 \[ - \frac{5 a \operatorname{atan}{\left (\frac{\sqrt{a - b x}}{\sqrt{b} \sqrt{x}} \right )}}{b^{\frac{7}{2}}} + \frac{2 x^{\frac{5}{2}}}{3 b \left (a - b x\right )^{\frac{3}{2}}} - \frac{10 x^{\frac{3}{2}}}{3 b^{2} \sqrt{a - b x}} - \frac{5 \sqrt{x} \sqrt{a - b x}}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**(5/2)/(-b*x+a)**(5/2),x)
[Out]
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Mathematica [A] time = 0.143678, size = 72, normalized size = 0.76 \[ \frac{5 a \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a-b x}}\right )}{b^{7/2}}-\frac{\sqrt{x} \left (15 a^2-20 a b x+3 b^2 x^2\right )}{3 b^3 (a-b x)^{3/2}} \]
Antiderivative was successfully verified.
[In] Integrate[x^(5/2)/(a - b*x)^(5/2),x]
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Maple [B] time = 0.052, size = 160, normalized size = 1.7 \[ -{\frac{1}{{b}^{3}}\sqrt{x}\sqrt{-bx+a}}+{1 \left ({\frac{5\,a}{2}\arctan \left ({1\sqrt{b} \left ( x-{\frac{a}{2\,b}} \right ){\frac{1}{\sqrt{-b{x}^{2}+ax}}}} \right ){b}^{-{\frac{7}{2}}}}+{\frac{14\,a}{3\,{b}^{4}}\sqrt{-b \left ( x-{\frac{a}{b}} \right ) ^{2}- \left ( x-{\frac{a}{b}} \right ) a} \left ( x-{\frac{a}{b}} \right ) ^{-1}}+{\frac{2\,{a}^{2}}{3\,{b}^{5}}\sqrt{-b \left ( x-{\frac{a}{b}} \right ) ^{2}- \left ( x-{\frac{a}{b}} \right ) a} \left ( x-{\frac{a}{b}} \right ) ^{-2}} \right ) \sqrt{x \left ( -bx+a \right ) }{\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{-bx+a}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^(5/2)/(-b*x+a)^(5/2),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(x^(5/2)/(-b*x + a)^(5/2),x, algorithm="maxima")
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Fricas [A] time = 0.22544, size = 1, normalized size = 0.01 \[ \left [\frac{15 \,{\left (a b x - a^{2}\right )} \sqrt{-b x + a} \sqrt{x} \log \left (-2 \, \sqrt{-b x + a} b \sqrt{x} -{\left (2 \, b x - a\right )} \sqrt{-b}\right ) + 2 \,{\left (3 \, b^{2} x^{3} - 20 \, a b x^{2} + 15 \, a^{2} x\right )} \sqrt{-b}}{6 \,{\left (b^{4} x - a b^{3}\right )} \sqrt{-b x + a} \sqrt{-b} \sqrt{x}}, -\frac{15 \,{\left (a b x - a^{2}\right )} \sqrt{-b x + a} \sqrt{x} \arctan \left (\frac{\sqrt{-b x + a}}{\sqrt{b} \sqrt{x}}\right ) -{\left (3 \, b^{2} x^{3} - 20 \, a b x^{2} + 15 \, a^{2} x\right )} \sqrt{b}}{3 \,{\left (b^{4} x - a b^{3}\right )} \sqrt{-b x + a} \sqrt{b} \sqrt{x}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^(5/2)/(-b*x + a)^(5/2),x, algorithm="fricas")
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Sympy [A] time = 84.9273, size = 971, normalized size = 10.22 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**(5/2)/(-b*x+a)**(5/2),x)
[Out]
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GIAC/XCAS [A] time = 0.235694, size = 298, normalized size = 3.14 \[ \frac{{\left (\frac{15 \, a{\rm ln}\left ({\left (\sqrt{-b x + a} \sqrt{-b} - \sqrt{{\left (b x - a\right )} b + a b}\right )}^{2}\right )}{\sqrt{-b} b^{2}} - \frac{6 \, \sqrt{{\left (b x - a\right )} b + a b} \sqrt{-b x + a}}{b^{3}} - \frac{8 \,{\left (9 \, a^{2}{\left (\sqrt{-b x + a} \sqrt{-b} - \sqrt{{\left (b x - a\right )} b + a b}\right )}^{4} - 12 \, a^{3}{\left (\sqrt{-b x + a} \sqrt{-b} - \sqrt{{\left (b x - a\right )} b + a b}\right )}^{2} b + 7 \, a^{4} b^{2}\right )}}{{\left ({\left (\sqrt{-b x + a} \sqrt{-b} - \sqrt{{\left (b x - a\right )} b + a b}\right )}^{2} - a b\right )}^{3} \sqrt{-b} b}\right )}{\left | b \right |}}{6 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^(5/2)/(-b*x + a)^(5/2),x, algorithm="giac")
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