Optimal. Leaf size=92 \[ \frac{15}{4} a^2 \sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a x+b x^2}}\right )+\frac{5 b \left (a x+b x^2\right )^{3/2}}{2 x}+\frac{15}{4} a b \sqrt{a x+b x^2}-\frac{2 \left (a x+b x^2\right )^{5/2}}{x^3} \]
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Rubi [A] time = 0.124471, antiderivative size = 92, 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 \[ \frac{15}{4} a^2 \sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a x+b x^2}}\right )+\frac{5 b \left (a x+b x^2\right )^{3/2}}{2 x}+\frac{15}{4} a b \sqrt{a x+b x^2}-\frac{2 \left (a x+b x^2\right )^{5/2}}{x^3} \]
Antiderivative was successfully verified.
[In] Int[(a*x + b*x^2)^(5/2)/x^4,x]
[Out]
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Rubi in Sympy [A] time = 13.6052, size = 85, normalized size = 0.92 \[ \frac{15 a^{2} \sqrt{b} \operatorname{atanh}{\left (\frac{\sqrt{b} x}{\sqrt{a x + b x^{2}}} \right )}}{4} + \frac{15 a b \sqrt{a x + b x^{2}}}{4} + \frac{5 b \left (a x + b x^{2}\right )^{\frac{3}{2}}}{2 x} - \frac{2 \left (a x + b x^{2}\right )^{\frac{5}{2}}}{x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x**2+a*x)**(5/2)/x**4,x)
[Out]
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Mathematica [A] time = 0.0917801, size = 93, normalized size = 1.01 \[ \frac{\sqrt{a+b x} \left (\sqrt{a+b x} \left (-8 a^2+9 a b x+2 b^2 x^2\right )+15 a^2 \sqrt{b} \sqrt{x} \log \left (\sqrt{b} \sqrt{a+b x}+b \sqrt{x}\right )\right )}{4 \sqrt{x (a+b x)}} \]
Antiderivative was successfully verified.
[In] Integrate[(a*x + b*x^2)^(5/2)/x^4,x]
[Out]
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Maple [B] time = 0.008, size = 185, normalized size = 2. \[ -2\,{\frac{ \left ( b{x}^{2}+ax \right ) ^{7/2}}{a{x}^{4}}}+12\,{\frac{b \left ( b{x}^{2}+ax \right ) ^{7/2}}{{a}^{2}{x}^{3}}}-32\,{\frac{{b}^{2} \left ( b{x}^{2}+ax \right ) ^{7/2}}{{x}^{2}{a}^{3}}}+32\,{\frac{{b}^{3} \left ( b{x}^{2}+ax \right ) ^{5/2}}{{a}^{3}}}+20\,{\frac{{b}^{3} \left ( b{x}^{2}+ax \right ) ^{3/2}x}{{a}^{2}}}+10\,{\frac{{b}^{2} \left ( b{x}^{2}+ax \right ) ^{3/2}}{a}}-{\frac{15\,{b}^{2}x}{2}\sqrt{b{x}^{2}+ax}}-{\frac{15\,ab}{4}\sqrt{b{x}^{2}+ax}}+{\frac{15\,{a}^{2}}{8}\sqrt{b}\ln \left ({1 \left ({\frac{a}{2}}+bx \right ){\frac{1}{\sqrt{b}}}}+\sqrt{b{x}^{2}+ax} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x^2+a*x)^(5/2)/x^4,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((b*x^2 + a*x)^(5/2)/x^4,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.232924, size = 1, normalized size = 0.01 \[ \left [\frac{15 \, a^{2} \sqrt{b} x \log \left (2 \, b x + a + 2 \, \sqrt{b x^{2} + a x} \sqrt{b}\right ) + 2 \,{\left (2 \, b^{2} x^{2} + 9 \, a b x - 8 \, a^{2}\right )} \sqrt{b x^{2} + a x}}{8 \, x}, \frac{15 \, a^{2} \sqrt{-b} x \arctan \left (\frac{\sqrt{b x^{2} + a x}}{\sqrt{-b} x}\right ) +{\left (2 \, b^{2} x^{2} + 9 \, a b x - 8 \, a^{2}\right )} \sqrt{b x^{2} + a x}}{4 \, x}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a*x)^(5/2)/x^4,x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (x \left (a + b x\right )\right )^{\frac{5}{2}}}{x^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x**2+a*x)**(5/2)/x**4,x)
[Out]
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GIAC/XCAS [A] time = 0.226964, size = 120, normalized size = 1.3 \[ -\frac{15}{8} \, a^{2} \sqrt{b}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )} \sqrt{b} - a \right |}\right ) + \frac{2 \, a^{3}}{\sqrt{b} x - \sqrt{b x^{2} + a x}} + \frac{1}{4} \,{\left (2 \, b^{2} x + 9 \, a b\right )} \sqrt{b x^{2} + a x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a*x)^(5/2)/x^4,x, algorithm="giac")
[Out]