Optimal. Leaf size=51 \[ \frac{8 c (b+2 c x)}{3 b^3 \sqrt{b x+c x^2}}-\frac{2}{3 b x \sqrt{b x+c x^2}} \]
[Out]
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Rubi [A] time = 0.0498431, antiderivative size = 51, 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 \[ \frac{8 c (b+2 c x)}{3 b^3 \sqrt{b x+c x^2}}-\frac{2}{3 b x \sqrt{b x+c x^2}} \]
Antiderivative was successfully verified.
[In] Int[1/(x*(b*x + c*x^2)^(3/2)),x]
[Out]
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Rubi in Sympy [A] time = 5.32063, size = 46, normalized size = 0.9 \[ - \frac{2}{3 b x \sqrt{b x + c x^{2}}} + \frac{4 c \left (2 b + 4 c x\right )}{3 b^{3} \sqrt{b x + c x^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x/(c*x**2+b*x)**(3/2),x)
[Out]
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Mathematica [A] time = 0.0321132, size = 40, normalized size = 0.78 \[ \frac{2 \left (-b^2+4 b c x+8 c^2 x^2\right )}{3 b^3 x \sqrt{x (b+c x)}} \]
Antiderivative was successfully verified.
[In] Integrate[1/(x*(b*x + c*x^2)^(3/2)),x]
[Out]
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Maple [A] time = 0.007, size = 39, normalized size = 0.8 \[ -{\frac{ \left ( 2\,cx+2\,b \right ) \left ( -8\,{c}^{2}{x}^{2}-4\,bcx+{b}^{2} \right ) }{3\,{b}^{3}} \left ( c{x}^{2}+bx \right ) ^{-{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x/(c*x^2+b*x)^(3/2),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(1/((c*x^2 + b*x)^(3/2)*x),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.21999, size = 51, normalized size = 1. \[ \frac{2 \,{\left (8 \, c^{2} x^{2} + 4 \, b c x - b^{2}\right )}}{3 \, \sqrt{c x^{2} + b x} b^{3} x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((c*x^2 + b*x)^(3/2)*x),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x \left (x \left (b + c x\right )\right )^{\frac{3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x/(c*x**2+b*x)**(3/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (c x^{2} + b x\right )}^{\frac{3}{2}} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((c*x^2 + b*x)^(3/2)*x),x, algorithm="giac")
[Out]