3.11.74 \(\int \frac {d+e x}{(c d^2+2 c d e x+c e^2 x^2)^{3/2}} \, dx\) [1074]

Optimal. Leaf size=32 \[ -\frac {1}{c e \sqrt {c d^2+2 c d e x+c e^2 x^2}} \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.033, Rules used = {643} \begin {gather*} -\frac {1}{c e \sqrt {c d^2+2 c d e x+c e^2 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-(1/(c*e*Sqrt[c*d^2 + 2*c*d*e*x + c*e^2*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 {d+e x}{\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}} \, dx &=-\frac {1}{c e \sqrt {c d^2+2 c d e x+c e^2 x^2}}\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 21, normalized size = 0.66 \begin {gather*} -\frac {1}{c e \sqrt {c (d+e x)^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-(1/(c*e*Sqrt[c*(d + e*x)^2]))

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Maple [A]
time = 0.54, size = 35, normalized size = 1.09

method	result	size
risch	\(-\frac {1}{c \sqrt {\left (e x +d \right )^{2} c}\, e}\)	\(20\)
gosper	\(-\frac {\left (e x +d \right )^{2}}{e \left (x^{2} c \,e^{2}+2 c d e x +c \,d^{2}\right )^{\frac {3}{2}}}\)	\(35\)
default	\(-\frac {\left (e x +d \right )^{2}}{e \left (x^{2} c \,e^{2}+2 c d e x +c \,d^{2}\right )^{\frac {3}{2}}}\)	\(35\)
trager	\(\frac {x \sqrt {x^{2} c \,e^{2}+2 c d e x +c \,d^{2}}}{c^{2} d \left (e x +d \right )^{2}}\)	\(38\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(e*x+d)^2/e/(c*e^2*x^2+2*c*d*e*x+c*d^2)^(3/2)

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Maxima [A]
time = 0.27, size = 29, normalized size = 0.91 \begin {gather*} -\frac {e^{\left (-1\right )}}{\sqrt {c x^{2} e^{2} + 2 \, c d x e + c d^{2}} c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-e^(-1)/(sqrt(c*x^2*e^2 + 2*c*d*x*e + c*d^2)*c)

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Fricas [A]
time = 3.47, size = 54, normalized size = 1.69 \begin {gather*} -\frac {\sqrt {c x^{2} e^{2} + 2 \, c d x e + c d^{2}}}{c^{2} x^{2} e^{3} + 2 \, c^{2} d x e^{2} + c^{2} d^{2} e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(c*x^2*e^2 + 2*c*d*x*e + c*d^2)/(c^2*x^2*e^3 + 2*c^2*d*x*e^2 + c^2*d^2*e)

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Sympy [A]
time = 0.52, size = 42, normalized size = 1.31 \begin {gather*} \begin {cases} - \frac {1}{c e \sqrt {c d^{2} + 2 c d e x + c e^{2} x^{2}}} & \text {for}\: e \neq 0 \\\frac {d x}{\left (c d^{2}\right )^{\frac {3}{2}}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-1/(c*e*sqrt(c*d**2 + 2*c*d*e*x + c*e**2*x**2)), Ne(e, 0)), (d*x/(c*d**2)**(3/2), True))

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Giac [A]
time = 1.66, size = 24, normalized size = 0.75 \begin {gather*} -\frac {e^{\left (-1\right )}}{{\left (x e + d\right )} c^{\frac {3}{2}} \mathrm {sgn}\left (x e + d\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-e^(-1)/((x*e + d)*c^(3/2)*sgn(x*e + d))

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Mupad [B]
time = 0.46, size = 37, normalized size = 1.16 \begin {gather*} -\frac {\sqrt {c\,d^2+2\,c\,d\,e\,x+c\,e^2\,x^2}}{c^2\,e\,{\left (d+e\,x\right )}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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