3.803 \(\int \frac{x \sqrt{a+b x}}{\sqrt{-a-b x}} \, dx\)

Optimal. Leaf size=28 \[ \frac{x^2 \sqrt{a+b x}}{2 \sqrt{-a-b x}} \]

[Out]

(x^2*Sqrt[a + b*x])/(2*Sqrt[-a - b*x])

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Rubi [A]  time = 0.0114064, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087 \[ \frac{x^2 \sqrt{a+b x}}{2 \sqrt{-a-b x}} \]

Antiderivative was successfully verified.

[In]  Int[(x*Sqrt[a + b*x])/Sqrt[-a - b*x],x]

[Out]

(x^2*Sqrt[a + b*x])/(2*Sqrt[-a - b*x])

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{\sqrt{a + b x} \int x\, dx}{\sqrt{- a - b x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x*(b*x+a)**(1/2)/(-b*x-a)**(1/2),x)

[Out]

sqrt(a + b*x)*Integral(x, x)/sqrt(-a - b*x)

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Mathematica [A]  time = 0.00760888, size = 28, normalized size = 1. \[ \frac{x^2 \sqrt{a+b x}}{2 \sqrt{-a-b x}} \]

Antiderivative was successfully verified.

[In]  Integrate[(x*Sqrt[a + b*x])/Sqrt[-a - b*x],x]

[Out]

(x^2*Sqrt[a + b*x])/(2*Sqrt[-a - b*x])

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Maple [A]  time = 0.003, size = 23, normalized size = 0.8 \[{\frac{{x}^{2}}{2}\sqrt{bx+a}{\frac{1}{\sqrt{-bx-a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*(b*x+a)^(1/2)/(-b*x-a)^(1/2),x)

[Out]

1/2*x^2*(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/sqrt(-b*x - a),x, algorithm="maxima")

[Out]

Exception raised: RuntimeError

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Fricas [A]  time = 0.231748, size = 1, normalized size = 0.04 \[ 0 \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(b*x + a)*x/sqrt(-b*x - a),x, algorithm="fricas")

[Out]

0

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Sympy [A]  time = 7.19166, size = 27, normalized size = 0.96 \[ \frac{i a^{2}}{b^{2}} + \frac{i a x}{b} - \frac{i \left (a + b x\right )^{2}}{2 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x*(b*x+a)**(1/2)/(-b*x-a)**(1/2),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.224817, size = 28, normalized size = 1. \[ -\frac{i \,{\left ({\left (b x + a\right )}^{2} - 2 \,{\left (b x + a\right )} a\right )}}{2 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(b*x + a)*x/sqrt(-b*x - a),x, algorithm="giac")

[Out]

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