Optimal. Leaf size=101 \[ -\frac{5 a^4 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a x+b x^2}}\right )}{64 b^{3/2}}+\frac{5 a^2 (a+2 b x) \sqrt{a x+b x^2}}{64 b}+\frac{5}{24} a \left (a x+b x^2\right )^{3/2}+\frac{\left (a x+b x^2\right )^{5/2}}{4 x} \]
[Out]
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Rubi [A] time = 0.116534, antiderivative size = 101, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235 \[ -\frac{5 a^4 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a x+b x^2}}\right )}{64 b^{3/2}}+\frac{5 a^2 (a+2 b x) \sqrt{a x+b x^2}}{64 b}+\frac{5}{24} a \left (a x+b x^2\right )^{3/2}+\frac{\left (a x+b x^2\right )^{5/2}}{4 x} \]
Antiderivative was successfully verified.
[In] Int[(a*x + b*x^2)^(5/2)/x^2,x]
[Out]
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Rubi in Sympy [A] time = 12.8237, size = 90, normalized size = 0.89 \[ - \frac{5 a^{4} \operatorname{atanh}{\left (\frac{\sqrt{b} x}{\sqrt{a x + b x^{2}}} \right )}}{64 b^{\frac{3}{2}}} + \frac{5 a^{2} \left (a + 2 b x\right ) \sqrt{a x + b x^{2}}}{64 b} + \frac{5 a \left (a x + b x^{2}\right )^{\frac{3}{2}}}{24} + \frac{\left (a x + b x^{2}\right )^{\frac{5}{2}}}{4 x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x**2+a*x)**(5/2)/x**2,x)
[Out]
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Mathematica [A] time = 0.116319, size = 100, normalized size = 0.99 \[ \frac{\sqrt{x (a+b x)} \left (\sqrt{b} \left (15 a^3+118 a^2 b x+136 a b^2 x^2+48 b^3 x^3\right )-\frac{15 a^4 \log \left (\sqrt{b} \sqrt{a+b x}+b \sqrt{x}\right )}{\sqrt{x} \sqrt{a+b x}}\right )}{192 b^{3/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(a*x + b*x^2)^(5/2)/x^2,x]
[Out]
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Maple [A] time = 0.007, size = 135, normalized size = 1.3 \[{\frac{2}{3\,a{x}^{2}} \left ( b{x}^{2}+ax \right ) ^{{\frac{7}{2}}}}-{\frac{2\,b}{3\,a} \left ( b{x}^{2}+ax \right ) ^{{\frac{5}{2}}}}-{\frac{5\,bx}{12} \left ( b{x}^{2}+ax \right ) ^{{\frac{3}{2}}}}-{\frac{5\,a}{24} \left ( b{x}^{2}+ax \right ) ^{{\frac{3}{2}}}}+{\frac{5\,{a}^{2}x}{32}\sqrt{b{x}^{2}+ax}}+{\frac{5\,{a}^{3}}{64\,b}\sqrt{b{x}^{2}+ax}}-{\frac{5\,{a}^{4}}{128}\ln \left ({1 \left ({\frac{a}{2}}+bx \right ){\frac{1}{\sqrt{b}}}}+\sqrt{b{x}^{2}+ax} \right ){b}^{-{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x^2+a*x)^(5/2)/x^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((b*x^2 + a*x)^(5/2)/x^2,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.239614, size = 1, normalized size = 0.01 \[ \left [\frac{15 \, a^{4} \log \left ({\left (2 \, b x + a\right )} \sqrt{b} - 2 \, \sqrt{b x^{2} + a x} b\right ) + 2 \,{\left (48 \, b^{3} x^{3} + 136 \, a b^{2} x^{2} + 118 \, a^{2} b x + 15 \, a^{3}\right )} \sqrt{b x^{2} + a x} \sqrt{b}}{384 \, b^{\frac{3}{2}}}, -\frac{15 \, a^{4} \arctan \left (\frac{\sqrt{b x^{2} + a x} \sqrt{-b}}{b x}\right ) -{\left (48 \, b^{3} x^{3} + 136 \, a b^{2} x^{2} + 118 \, a^{2} b x + 15 \, a^{3}\right )} \sqrt{b x^{2} + a x} \sqrt{-b}}{192 \, \sqrt{-b} b}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a*x)^(5/2)/x^2,x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (x \left (a + b x\right )\right )^{\frac{5}{2}}}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x**2+a*x)**(5/2)/x**2,x)
[Out]
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GIAC/XCAS [A] time = 0.223946, size = 113, normalized size = 1.12 \[ \frac{5 \, a^{4}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )} \sqrt{b} - a \right |}\right )}{128 \, b^{\frac{3}{2}}} + \frac{1}{192} \, \sqrt{b x^{2} + a x}{\left (\frac{15 \, a^{3}}{b} + 2 \,{\left (59 \, a^{2} + 4 \,{\left (6 \, b^{2} x + 17 \, a b\right )} x\right )} x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a*x)^(5/2)/x^2,x, algorithm="giac")
[Out]