Optimal. Leaf size=28 \[ \frac{x^3 \sqrt{a+b x}}{3 \sqrt{-a-b x}} \]
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Rubi [A] time = 0.0114941, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08 \[ \frac{x^3 \sqrt{a+b x}}{3 \sqrt{-a-b x}} \]
Antiderivative was successfully verified.
[In] Int[(x^2*Sqrt[a + b*x])/Sqrt[-a - b*x],x]
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Rubi in Sympy [A] time = 3.54842, size = 22, normalized size = 0.79 \[ \frac{x^{3} \sqrt{a + b x}}{3 \sqrt{- a - b x}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**2*(b*x+a)**(1/2)/(-b*x-a)**(1/2),x)
[Out]
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Mathematica [A] time = 0.013278, size = 28, normalized size = 1. \[ \frac{x^3 \sqrt{a+b x}}{3 \sqrt{-a-b x}} \]
Antiderivative was successfully verified.
[In] Integrate[(x^2*Sqrt[a + b*x])/Sqrt[-a - b*x],x]
[Out]
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Maple [A] time = 0.003, size = 23, normalized size = 0.8 \[{\frac{{x}^{3}}{3}\sqrt{bx+a}{\frac{1}{\sqrt{-bx-a}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^2*(b*x+a)^(1/2)/(-b*x-a)^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(b*x + a)*x^2/sqrt(-b*x - a),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.232351, size = 1, normalized size = 0.04 \[ 0 \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(b*x + a)*x^2/sqrt(-b*x - a),x, algorithm="fricas")
[Out]
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Sympy [A] time = 9.65608, size = 44, normalized size = 1.57 \[ - \frac{i a^{3}}{b^{3}} - \frac{i a^{2} x}{b^{2}} + \frac{i a \left (a + b x\right )^{2}}{b^{3}} - \frac{i \left (a + b x\right )^{3}}{3 b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**2*(b*x+a)**(1/2)/(-b*x-a)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.228834, size = 58, normalized size = 2.07 \[ -\frac{i \,{\left ({\left (b x + a\right )}^{3} b^{4} - 3 \,{\left (b x + a\right )}^{2} a b^{4} + 3 \,{\left (b x + a\right )} a^{2} b^{4}\right )}}{3 \, b^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(b*x + a)*x^2/sqrt(-b*x - a),x, algorithm="giac")
[Out]