Optimal. Leaf size=94 \[ \frac{5 a^3 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a x+b x^2}}\right )}{8 \sqrt{b}}-\frac{5}{8} a (a+2 b x) \sqrt{a x+b x^2}+\frac{2 \left (a x+b x^2\right )^{5/2}}{x^2}-\frac{5}{3} b \left (a x+b x^2\right )^{3/2} \]
[Out]
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Rubi [A] time = 0.114336, antiderivative size = 94, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.294 \[ \frac{5 a^3 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a x+b x^2}}\right )}{8 \sqrt{b}}-\frac{5}{8} a (a+2 b x) \sqrt{a x+b x^2}+\frac{2 \left (a x+b x^2\right )^{5/2}}{x^2}-\frac{5}{3} b \left (a x+b x^2\right )^{3/2} \]
Antiderivative was successfully verified.
[In] Int[(a*x + b*x^2)^(5/2)/x^3,x]
[Out]
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Rubi in Sympy [A] time = 12.3092, size = 88, normalized size = 0.94 \[ \frac{5 a^{3} \operatorname{atanh}{\left (\frac{\sqrt{b} x}{\sqrt{a x + b x^{2}}} \right )}}{8 \sqrt{b}} - \frac{5 a \left (a + 2 b x\right ) \sqrt{a x + b x^{2}}}{8} - \frac{5 b \left (a x + b x^{2}\right )^{\frac{3}{2}}}{3} + \frac{2 \left (a x + b x^{2}\right )^{\frac{5}{2}}}{x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x**2+a*x)**(5/2)/x**3,x)
[Out]
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Mathematica [A] time = 0.0940536, size = 82, normalized size = 0.87 \[ \frac{1}{24} \sqrt{x (a+b x)} \left (\frac{15 a^3 \log \left (\sqrt{b} \sqrt{a+b x}+b \sqrt{x}\right )}{\sqrt{b} \sqrt{x} \sqrt{a+b x}}+33 a^2+26 a b x+8 b^2 x^2\right ) \]
Antiderivative was successfully verified.
[In] Integrate[(a*x + b*x^2)^(5/2)/x^3,x]
[Out]
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Maple [B] time = 0.007, size = 158, normalized size = 1.7 \[ 2\,{\frac{ \left ( b{x}^{2}+ax \right ) ^{7/2}}{a{x}^{3}}}-{\frac{16\,b}{3\,{a}^{2}{x}^{2}} \left ( b{x}^{2}+ax \right ) ^{{\frac{7}{2}}}}+{\frac{16\,{b}^{2}}{3\,{a}^{2}} \left ( b{x}^{2}+ax \right ) ^{{\frac{5}{2}}}}+{\frac{10\,{b}^{2}x}{3\,a} \left ( b{x}^{2}+ax \right ) ^{{\frac{3}{2}}}}+{\frac{5\,b}{3} \left ( b{x}^{2}+ax \right ) ^{{\frac{3}{2}}}}-{\frac{5\,abx}{4}\sqrt{b{x}^{2}+ax}}-{\frac{5\,{a}^{2}}{8}\sqrt{b{x}^{2}+ax}}+{\frac{5\,{a}^{3}}{16}\ln \left ({1 \left ({\frac{a}{2}}+bx \right ){\frac{1}{\sqrt{b}}}}+\sqrt{b{x}^{2}+ax} \right ){\frac{1}{\sqrt{b}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x^2+a*x)^(5/2)/x^3,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^2 + a*x)^(5/2)/x^3,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.237574, size = 1, normalized size = 0.01 \[ \left [\frac{15 \, a^{3} \log \left ({\left (2 \, b x + a\right )} \sqrt{b} + 2 \, \sqrt{b x^{2} + a x} b\right ) + 2 \,{\left (8 \, b^{2} x^{2} + 26 \, a b x + 33 \, a^{2}\right )} \sqrt{b x^{2} + a x} \sqrt{b}}{48 \, \sqrt{b}}, \frac{15 \, a^{3} \arctan \left (\frac{\sqrt{b x^{2} + a x} \sqrt{-b}}{b x}\right ) +{\left (8 \, b^{2} x^{2} + 26 \, a b x + 33 \, a^{2}\right )} \sqrt{b x^{2} + a x} \sqrt{-b}}{24 \, \sqrt{-b}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a*x)^(5/2)/x^3,x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (x \left (a + b x\right )\right )^{\frac{5}{2}}}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x**2+a*x)**(5/2)/x**3,x)
[Out]
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GIAC/XCAS [A] time = 0.222701, size = 97, normalized size = 1.03 \[ -\frac{5 \, a^{3}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )} \sqrt{b} - a \right |}\right )}{16 \, \sqrt{b}} + \frac{1}{24} \, \sqrt{b x^{2} + a x}{\left (33 \, a^{2} + 2 \,{\left (4 \, b^{2} x + 13 \, a b\right )} x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a*x)^(5/2)/x^3,x, algorithm="giac")
[Out]