3.13.37 \(\int \frac {b d+2 c d x}{\sqrt {a+b x+c x^2}} \, dx\) [1237]

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

[Out]

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

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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*} 2 d \sqrt {a+b x+c x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(b*d + 2*c*d*x)/Sqrt[a + b*x + c*x^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}{\sqrt {a+b x+c x^2}} \, dx &=2 d \sqrt {a+b x+c x^2}\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 16, normalized size = 0.94 \begin {gather*} 2 d \sqrt {a+x (b+c x)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

2*d*Sqrt[a + x*(b + c*x)]

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Maple [A]
time = 0.65, size = 16, normalized size = 0.94

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*c*d*x+b*d)/(c*x^2+b*x+a)^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.27, size = 15, normalized size = 0.88 \begin {gather*} 2 \, \sqrt {c x^{2} + b x + a} d \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)^(1/2),x, algorithm="maxima")

[Out]

2*sqrt(c*x^2 + b*x + a)*d

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Fricas [A]
time = 3.30, size = 15, normalized size = 0.88 \begin {gather*} 2 \, \sqrt {c x^{2} + b x + a} d \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)^(1/2),x, algorithm="fricas")

[Out]

2*sqrt(c*x^2 + b*x + a)*d

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Sympy [A]
time = 0.05, size = 15, normalized size = 0.88 \begin {gather*} 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)**(1/2),x)

[Out]

2*d*sqrt(a + b*x + c*x**2)

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Giac [A]
time = 3.76, size = 15, normalized size = 0.88 \begin {gather*} 2 \, \sqrt {c x^{2} + b x + a} d \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)^(1/2),x, algorithm="giac")

[Out]

2*sqrt(c*x^2 + b*x + a)*d

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Mupad [B]
time = 0.53, size = 15, normalized size = 0.88 \begin {gather*} 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)^(1/2),x)

[Out]

2*d*(a + b*x + c*x^2)^(1/2)

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