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

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 = 0.0032819, 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, Rules used = {23, 30} \[ \frac{x^2 \sqrt{a+b x}}{2 \sqrt{-a-b x}} \]

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])

Rule 23

Int[(u_.)*((a_) + (b_.)*(v_))^(m_)*((c_) + (d_.)*(v_))^(n_), x_Symbol] :> Dist[(a + b*v)^m/(c + d*v)^m, Int[u*
(c + d*v)^(m + n), x], x] /; FreeQ[{a, b, c, d, m, n}, x] && EqQ[b*c - a*d, 0] &&  !(IntegerQ[m] || IntegerQ[n
] || GtQ[b/d, 0])

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

\begin{align*} \int \frac{x \sqrt{a+b x}}{\sqrt{-a-b x}} \, dx &=\frac{\sqrt{a+b x} \int x \, dx}{\sqrt{-a-b x}}\\ &=\frac{x^2 \sqrt{a+b x}}{2 \sqrt{-a-b x}}\\ \end{align*}

Mathematica [A]  time = 0.004716, 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 = 0.001, size = 23, normalized size = 0.8 \begin{align*}{\frac{{x}^{2}}{2}\sqrt{bx+a}{\frac{1}{\sqrt{-bx-a}}}} \end{align*}

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(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: RuntimeError

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Fricas [A]  time = 2.12023, size = 4, normalized size = 0.14 \begin{align*} 0 \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(b*x+a)^(1/2)/(-b*x-a)^(1/2),x, algorithm="fricas")

[Out]

0

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Sympy [C]  time = 5.14335, size = 27, normalized size = 0.96 \begin{align*} \frac{i a^{2}}{b^{2}} + \frac{i a x}{b} - \frac{i \left (a + b x\right )^{2}}{2 b^{2}} \end{align*}

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 [C]  time = 2.55571, size = 28, normalized size = 1. \begin{align*} -\frac{i \,{\left ({\left (b x + a\right )}^{2} - 2 \,{\left (b x + a\right )} a\right )}}{2 \, b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(b*x+a)^(1/2)/(-b*x-a)^(1/2),x, algorithm="giac")

[Out]

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