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

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

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Rubi [A]  time = 0.00, antiderivative size = 28, normalized size of antiderivative = 1.00, 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} \begin {gather*} \frac {x^2 \sqrt {a+b x}}{2 \sqrt {-a-b x}} \end {gather*}

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*}

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Mathematica [A]  time = 0.01, size = 28, normalized size = 1.00 \begin {gather*} \frac {x^2 \sqrt {a+b x}}{2 \sqrt {-a-b x}} \end {gather*}

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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IntegrateAlgebraic [F]  time = 0.09, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x \sqrt {a+b x}}{\sqrt {-a-b x}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

Defer[IntegrateAlgebraic][(x*Sqrt[a + b*x])/Sqrt[-a - b*x], x]

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fricas [A]  time = 1.98, size = 1, normalized size = 0.04 \begin {gather*} 0 \end {gather*}

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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giac [C]  time = 1.55, size = 21, normalized size = 0.75 \begin {gather*} -\frac {i \, {\left ({\left (b x + a\right )}^{2} - 2 \, {\left (b x + a\right )} a\right )}}{2 \, b^{2}} \end {gather*}

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

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maple [A]  time = 0.00, size = 23, normalized size = 0.82 \begin {gather*} \frac {\sqrt {b x +a}\, x^{2}}{2 \sqrt {-b x -a}} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}

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 >> ECL says: Error executing code in Maxima: expt: undefined: 0 to a negative e
xponent.

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {x\,\sqrt {a+b\,x}}{\sqrt {-a-b\,x}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [C]  time = 2.79, size = 27, normalized size = 0.96 \begin {gather*} \frac {i a^{2}}{b^{2}} + \frac {i a x}{b} - \frac {i \left (a + b x\right )^{2}}{2 b^{2}} \end {gather*}

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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