3.13.45 \(\int \frac {b d+2 c d x}{(a+b x+c x^2)^{3/2}} \, dx\) [1245]

Optimal. Leaf size=17 \[ -\frac {2 d}{\sqrt {a+b x+c x^2}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.00, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {643} \begin {gather*} -\frac {2 d}{\sqrt {a+b x+c x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*d)/Sqrt[a + b*x + c*x^2]

Rule 643

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

________________________________________________________________________________________

Mathematica [A]
time = 0.02, size = 16, normalized size = 0.94 \begin {gather*} -\frac {2 d}{\sqrt {a+x (b+c x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*d)/Sqrt[a + x*(b + c*x)]

________________________________________________________________________________________

Maple [A]
time = 0.66, size = 16, normalized size = 0.94

method	result	size
gosper	\(-\frac {2 d}{\sqrt {c \,x^{2}+b x +a}}\)	\(16\)
default	\(-\frac {2 d}{\sqrt {c \,x^{2}+b x +a}}\)	\(16\)
trager	\(-\frac {2 d}{\sqrt {c \,x^{2}+b x +a}}\)	\(16\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*c*d*x+b*d)/(c*x^2+b*x+a)^(3/2),x,method=_RETURNVERBOSE)

[Out]

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

________________________________________________________________________________________

Maxima [A]
time = 0.28, size = 15, normalized size = 0.88 \begin {gather*} -\frac {2 \, d}{\sqrt {c x^{2} + b x + a}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*d/sqrt(c*x^2 + b*x + a)

________________________________________________________________________________________

Fricas [A]
time = 3.44, size = 15, normalized size = 0.88 \begin {gather*} -\frac {2 \, d}{\sqrt {c x^{2} + b x + a}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*d/sqrt(c*x^2 + b*x + a)

________________________________________________________________________________________

Sympy [A]
time = 0.42, size = 17, normalized size = 1.00 \begin {gather*} - \frac {2 d}{\sqrt {a + b x + c x^{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*d/sqrt(a + b*x + c*x**2)

________________________________________________________________________________________

Giac [A]
time = 3.54, size = 15, normalized size = 0.88 \begin {gather*} -\frac {2 \, d}{\sqrt {c x^{2} + b x + a}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*d/sqrt(c*x^2 + b*x + a)

________________________________________________________________________________________

Mupad [B]
time = 0.53, size = 15, normalized size = 0.88 \begin {gather*} -\frac {2\,d}{\sqrt {c\,x^2+b\,x+a}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(2*d)/(a + b*x + c*x^2)^(1/2)

________________________________________________________________________________________