Optimal. Leaf size=118 \[ -\frac{5 a^6 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a x+b x^2}}\right )}{512 b^{7/2}}+\frac{5 a^4 (a+2 b x) \sqrt{a x+b x^2}}{512 b^3}-\frac{5 a^2 (a+2 b x) \left (a x+b x^2\right )^{3/2}}{192 b^2}+\frac{(a+2 b x) \left (a x+b x^2\right )^{5/2}}{12 b} \]
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Rubi [A] time = 0.0954816, antiderivative size = 118, 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 \[ -\frac{5 a^6 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a x+b x^2}}\right )}{512 b^{7/2}}+\frac{5 a^4 (a+2 b x) \sqrt{a x+b x^2}}{512 b^3}-\frac{5 a^2 (a+2 b x) \left (a x+b x^2\right )^{3/2}}{192 b^2}+\frac{(a+2 b x) \left (a x+b x^2\right )^{5/2}}{12 b} \]
Antiderivative was successfully verified.
[In] Int[(a*x + b*x^2)^(5/2),x]
[Out]
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Rubi in Sympy [A] time = 11.2079, size = 110, normalized size = 0.93 \[ - \frac{5 a^{6} \operatorname{atanh}{\left (\frac{\sqrt{b} x}{\sqrt{a x + b x^{2}}} \right )}}{512 b^{\frac{7}{2}}} + \frac{5 a^{4} \left (a + 2 b x\right ) \sqrt{a x + b x^{2}}}{512 b^{3}} - \frac{5 a^{2} \left (a + 2 b x\right ) \left (a x + b x^{2}\right )^{\frac{3}{2}}}{192 b^{2}} + \frac{\left (a + 2 b x\right ) \left (a x + b x^{2}\right )^{\frac{5}{2}}}{12 b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x**2+a*x)**(5/2),x)
[Out]
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Mathematica [A] time = 0.138311, size = 122, normalized size = 1.03 \[ \frac{\sqrt{x (a+b x)} \left (\sqrt{b} \left (15 a^5-10 a^4 b x+8 a^3 b^2 x^2+432 a^2 b^3 x^3+640 a b^4 x^4+256 b^5 x^5\right )-\frac{15 a^6 \log \left (\sqrt{b} \sqrt{a+b x}+b \sqrt{x}\right )}{\sqrt{x} \sqrt{a+b x}}\right )}{1536 b^{7/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(a*x + b*x^2)^(5/2),x]
[Out]
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Maple [A] time = 0.005, size = 134, normalized size = 1.1 \[{\frac{2\,bx+a}{12\,b} \left ( b{x}^{2}+ax \right ) ^{{\frac{5}{2}}}}-{\frac{5\,{a}^{2}x}{96\,b} \left ( b{x}^{2}+ax \right ) ^{{\frac{3}{2}}}}-{\frac{5\,{a}^{3}}{192\,{b}^{2}} \left ( b{x}^{2}+ax \right ) ^{{\frac{3}{2}}}}+{\frac{5\,{a}^{4}x}{256\,{b}^{2}}\sqrt{b{x}^{2}+ax}}+{\frac{5\,{a}^{5}}{512\,{b}^{3}}\sqrt{b{x}^{2}+ax}}-{\frac{5\,{a}^{6}}{1024}\ln \left ({1 \left ({\frac{a}{2}}+bx \right ){\frac{1}{\sqrt{b}}}}+\sqrt{b{x}^{2}+ax} \right ){b}^{-{\frac{7}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x^2+a*x)^(5/2),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, algorithm="maxima")
[Out]
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Fricas [A] time = 0.231287, size = 1, normalized size = 0.01 \[ \left [\frac{15 \, a^{6} \log \left ({\left (2 \, b x + a\right )} \sqrt{b} - 2 \, \sqrt{b x^{2} + a x} b\right ) + 2 \,{\left (256 \, b^{5} x^{5} + 640 \, a b^{4} x^{4} + 432 \, a^{2} b^{3} x^{3} + 8 \, a^{3} b^{2} x^{2} - 10 \, a^{4} b x + 15 \, a^{5}\right )} \sqrt{b x^{2} + a x} \sqrt{b}}{3072 \, b^{\frac{7}{2}}}, -\frac{15 \, a^{6} \arctan \left (\frac{\sqrt{b x^{2} + a x} \sqrt{-b}}{b x}\right ) -{\left (256 \, b^{5} x^{5} + 640 \, a b^{4} x^{4} + 432 \, a^{2} b^{3} x^{3} + 8 \, a^{3} b^{2} x^{2} - 10 \, a^{4} b x + 15 \, a^{5}\right )} \sqrt{b x^{2} + a x} \sqrt{-b}}{1536 \, \sqrt{-b} b^{3}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a*x)^(5/2),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \left (a x + b x^{2}\right )^{\frac{5}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x**2+a*x)**(5/2),x)
[Out]
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GIAC/XCAS [A] time = 0.223985, size = 144, normalized size = 1.22 \[ \frac{5 \, a^{6}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )} \sqrt{b} - a \right |}\right )}{1024 \, b^{\frac{7}{2}}} + \frac{1}{1536} \, \sqrt{b x^{2} + a x}{\left (\frac{15 \, a^{5}}{b^{3}} - 2 \,{\left (\frac{5 \, a^{4}}{b^{2}} - 4 \,{\left (\frac{a^{3}}{b} + 2 \,{\left (27 \, a^{2} + 8 \,{\left (2 \, b^{2} x + 5 \, a b\right )} x\right )} x\right )} x\right )} x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a*x)^(5/2),x, algorithm="giac")
[Out]