Optimal. Leaf size=48 \[ \frac{4 c \left (b x+c x^2\right )^{3/2}}{15 b^2 x^3}-\frac{2 \left (b x+c x^2\right )^{3/2}}{5 b x^4} \]
[Out]
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Rubi [A] time = 0.0602688, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ \frac{4 c \left (b x+c x^2\right )^{3/2}}{15 b^2 x^3}-\frac{2 \left (b x+c x^2\right )^{3/2}}{5 b x^4} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[b*x + c*x^2]/x^4,x]
[Out]
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Rubi in Sympy [A] time = 6.23142, size = 42, normalized size = 0.88 \[ - \frac{2 \left (b x + c x^{2}\right )^{\frac{3}{2}}}{5 b x^{4}} + \frac{4 c \left (b x + c x^{2}\right )^{\frac{3}{2}}}{15 b^{2} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c*x**2+b*x)**(1/2)/x**4,x)
[Out]
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Mathematica [A] time = 0.0198937, size = 40, normalized size = 0.83 \[ \frac{2 \sqrt{x (b+c x)} \left (-3 b^2-b c x+2 c^2 x^2\right )}{15 b^2 x^3} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[b*x + c*x^2]/x^4,x]
[Out]
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Maple [A] time = 0.006, size = 33, normalized size = 0.7 \[ -{\frac{ \left ( 2\,cx+2\,b \right ) \left ( -2\,cx+3\,b \right ) }{15\,{b}^{2}{x}^{3}}\sqrt{c{x}^{2}+bx}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c*x^2+b*x)^(1/2)/x^4,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(sqrt(c*x^2 + b*x)/x^4,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.217756, size = 51, normalized size = 1.06 \[ \frac{2 \,{\left (2 \, c^{2} x^{2} - b c x - 3 \, b^{2}\right )} \sqrt{c x^{2} + b x}}{15 \, b^{2} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(c*x^2 + b*x)/x^4,x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{x \left (b + c x\right )}}{x^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x**2+b*x)**(1/2)/x**4,x)
[Out]
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GIAC/XCAS [A] time = 0.212456, size = 144, normalized size = 3. \[ \frac{2 \,{\left (15 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{3} c^{\frac{3}{2}} + 25 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{2} b c + 15 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} b^{2} \sqrt{c} + 3 \, b^{3}\right )}}{15 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(c*x^2 + b*x)/x^4,x, algorithm="giac")
[Out]