3.802 \(\int \frac{x^2 \sqrt{a+b x}}{\sqrt{-a-b x}} \, dx\)

Optimal. Leaf size=28 \[ \frac{x^3 \sqrt{a+b x}}{3 \sqrt{-a-b x}} \]

[Out]

(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]

[Out]

(x^3*Sqrt[a + b*x])/(3*Sqrt[-a - b*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]

x**3*sqrt(a + b*x)/(3*sqrt(-a - b*x))

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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]

(x^3*Sqrt[a + b*x])/(3*Sqrt[-a - b*x])

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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]

1/3*x^3*(b*x+a)^(1/2)/(-b*x-a)^(1/2)

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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]

Exception raised: RuntimeError

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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]

0

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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]

-I*a**3/b**3 - I*a**2*x/b**2 + I*a*(a + b*x)**2/b**3 - I*(a + b*x)**3/(3*b**3)

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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]

-1/3*I*((b*x + a)^3*b^4 - 3*(b*x + a)^2*a*b^4 + 3*(b*x + a)*a^2*b^4)/b^7