Optimal. Leaf size=24 \[ -\frac{2 (b+2 c x)}{b^2 \sqrt{b x+c x^2}} \]
[Out]
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Rubi [A] time = 0.0145355, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ -\frac{2 (b+2 c x)}{b^2 \sqrt{b x+c x^2}} \]
Antiderivative was successfully verified.
[In] Int[(b*x + c*x^2)^(-3/2),x]
[Out]
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Rubi in Sympy [A] time = 1.54375, size = 24, normalized size = 1. \[ - \frac{2 b + 4 c x}{b^{2} \sqrt{b x + c x^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(c*x**2+b*x)**(3/2),x)
[Out]
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Mathematica [A] time = 0.0218273, size = 22, normalized size = 0.92 \[ -\frac{2 (b+2 c x)}{b^2 \sqrt{x (b+c x)}} \]
Antiderivative was successfully verified.
[In] Integrate[(b*x + c*x^2)^(-3/2),x]
[Out]
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Maple [A] time = 0.005, size = 29, normalized size = 1.2 \[ -2\,{\frac{ \left ( 2\,cx+b \right ) x \left ( cx+b \right ) }{{b}^{2} \left ( c{x}^{2}+bx \right ) ^{3/2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(c*x^2+b*x)^(3/2),x)
[Out]
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Maxima [A] time = 0.713439, size = 47, normalized size = 1.96 \[ -\frac{4 \, c x}{\sqrt{c x^{2} + b x} b^{2}} - \frac{2}{\sqrt{c x^{2} + b x} b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x)^(-3/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.216463, size = 30, normalized size = 1.25 \[ -\frac{2 \,{\left (2 \, c x + b\right )}}{\sqrt{c x^{2} + b x} b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x)^(-3/2),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (b x + c x^{2}\right )^{\frac{3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(c*x**2+b*x)**(3/2),x)
[Out]
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GIAC/XCAS [A] time = 0.221073, size = 32, normalized size = 1.33 \[ -\frac{2 \,{\left (\frac{2 \, c x}{b^{2}} + \frac{1}{b}\right )}}{\sqrt{c x^{2} + b x}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x)^(-3/2),x, algorithm="giac")
[Out]