3.14.93 \(\int \frac {b+2 c x}{(a+b x+c x^2)^{3/2}} \, dx\)

Optimal. Leaf size=16 \[ -\frac {2}{\sqrt {a+b x+c x^2}} \]

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Rubi [A]  time = 0.01, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {629} \begin {gather*} -\frac {2}{\sqrt {a+b x+c x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(b + 2*c*x)/(a + b*x + c*x^2)^(3/2),x]

[Out]

-2/Sqrt[a + b*x + c*x^2]

Rule 629

Int[((d_) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d*(a + b*x + c*x^2)^(p +
1))/(b*(p + 1)), x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[2*c*d - b*e, 0] && NeQ[p, -1]

Rubi steps

\begin {align*} \int \frac {b+2 c x}{\left (a+b x+c x^2\right )^{3/2}} \, dx &=-\frac {2}{\sqrt {a+b x+c x^2}}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 15, normalized size = 0.94 \begin {gather*} -\frac {2}{\sqrt {a+x (b+c x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(b + 2*c*x)/(a + b*x + c*x^2)^(3/2),x]

[Out]

-2/Sqrt[a + x*(b + c*x)]

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IntegrateAlgebraic [A]  time = 0.02, size = 16, normalized size = 1.00 \begin {gather*} -\frac {2}{\sqrt {a+b x+c x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(b + 2*c*x)/(a + b*x + c*x^2)^(3/2),x]

[Out]

-2/Sqrt[a + b*x + c*x^2]

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fricas [A]  time = 0.43, size = 14, normalized size = 0.88 \begin {gather*} -\frac {2}{\sqrt {c x^{2} + b x + a}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*x+b)/(c*x^2+b*x+a)^(3/2),x, algorithm="fricas")

[Out]

-2/sqrt(c*x^2 + b*x + a)

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giac [A]  time = 0.21, size = 14, normalized size = 0.88 \begin {gather*} -\frac {2}{\sqrt {c x^{2} + b x + a}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*x+b)/(c*x^2+b*x+a)^(3/2),x, algorithm="giac")

[Out]

-2/sqrt(c*x^2 + b*x + a)

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maple [A]  time = 0.05, size = 15, normalized size = 0.94 \begin {gather*} -\frac {2}{\sqrt {c \,x^{2}+b x +a}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*c*x+b)/(c*x^2+b*x+a)^(3/2),x)

[Out]

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

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maxima [A]  time = 0.51, size = 14, normalized size = 0.88 \begin {gather*} -\frac {2}{\sqrt {c x^{2} + b x + a}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*x+b)/(c*x^2+b*x+a)^(3/2),x, algorithm="maxima")

[Out]

-2/sqrt(c*x^2 + b*x + a)

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mupad [B]  time = 1.89, size = 14, normalized size = 0.88 \begin {gather*} -\frac {2}{\sqrt {c\,x^2+b\,x+a}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b + 2*c*x)/(a + b*x + c*x^2)^(3/2),x)

[Out]

-2/(a + b*x + c*x^2)^(1/2)

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sympy [A]  time = 0.92, size = 15, normalized size = 0.94 \begin {gather*} - \frac {2}{\sqrt {a + b x + c x^{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*x+b)/(c*x**2+b*x+a)**(3/2),x)

[Out]

-2/sqrt(a + b*x + c*x**2)

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