3.11.75 \(\int \frac {1}{(c d^2+2 c d e x+c e^2 x^2)^{3/2}} \, dx\) [1075]

Optimal. Leaf size=41 \[ -\frac {1}{2 c e (d+e x) \sqrt {c d^2+2 c d e x+c e^2 x^2}} \]

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 41, 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 = {621} \begin {gather*} -\frac {1}{2 c e (d+e x) \sqrt {c d^2+2 c d e x+c e^2 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 621

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[2*((a + b*x + c*x^2)^(p + 1)/((2*p + 1)*(b + 2*
c*x))), x] /; FreeQ[{a, b, c, p}, x] && EqQ[b^2 - 4*a*c, 0] && LtQ[p, -1]

Rubi steps

\begin {align*} \int \frac {1}{\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}} \, dx &=-\frac {1}{2 c e (d+e x) \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 = 25, normalized size = 0.61 \begin {gather*} -\frac {d+e x}{2 e \left (c (d+e x)^2\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.54, size = 33, normalized size = 0.80

method	result	size
risch	\(-\frac {1}{2 c \left (e x +d \right ) \sqrt {\left (e x +d \right )^{2} c}\, e}\)	\(27\)
gosper	\(-\frac {e x +d}{2 e \left (x^{2} c \,e^{2}+2 c d e x +c \,d^{2}\right )^{\frac {3}{2}}}\)	\(33\)
default	\(-\frac {e x +d}{2 e \left (x^{2} c \,e^{2}+2 c d e x +c \,d^{2}\right )^{\frac {3}{2}}}\)	\(33\)
trager	\(\frac {\left (e x +2 d \right ) x \sqrt {x^{2} c \,e^{2}+2 c d e x +c \,d^{2}}}{2 c^{2} d^{2} \left (e x +d \right )^{3}}\)	\(46\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(c*e^2*x^2+2*c*d*e*x+c*d^2)^(3/2),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.28, size = 15, normalized size = 0.37 \begin {gather*} -\frac {e^{\left (-3\right )}}{2 \, {\left (d e^{\left (-1\right )} + x\right )}^{2} c^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*e^2*x^2+2*c*d*e*x+c*d^2)^(3/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 2.69, size = 67, normalized size = 1.63 \begin {gather*} -\frac {\sqrt {c x^{2} e^{2} + 2 \, c d x e + c d^{2}}}{2 \, {\left (c^{2} x^{3} e^{4} + 3 \, c^{2} d x^{2} e^{3} + 3 \, c^{2} d^{2} x e^{2} + c^{2} d^{3} e\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*e^2*x^2+2*c*d*e*x+c*d^2)^(3/2),x, algorithm="fricas")

[Out]

-1/2*sqrt(c*x^2*e^2 + 2*c*d*x*e + c*d^2)/(c^2*x^3*e^4 + 3*c^2*d*x^2*e^3 + 3*c^2*d^2*x*e^2 + c^2*d^3*e)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (c d^{2} + 2 c d e x + c e^{2} x^{2}\right )^{\frac {3}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*e**2*x**2+2*c*d*e*x+c*d**2)**(3/2),x)

[Out]

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

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Giac [A]
time = 1.61, size = 24, normalized size = 0.59 \begin {gather*} -\frac {e^{\left (-1\right )}}{2 \, {\left (x e + d\right )}^{2} 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(1/(c*e^2*x^2+2*c*d*e*x+c*d^2)^(3/2),x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 0.48, size = 37, normalized size = 0.90 \begin {gather*} -\frac {\sqrt {c\,d^2+2\,c\,d\,e\,x+c\,e^2\,x^2}}{2\,c^2\,e\,{\left (d+e\,x\right )}^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(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)/(2*c^2*e*(d + e*x)^3)

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