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

Optimal result

Integrand size = 24, antiderivative size = 41 \[ \int \frac {1}{\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}} \, dx=-\frac {1}{2 c e (d+e x) \sqrt {c d^2+2 c d e x+c e^2 x^2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {621} \[ \int \frac {1}{\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}} \, dx=-\frac {1}{2 c e (d+e x) \sqrt {c d^2+2 c d e x+c e^2 x^2}} \]

[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*} \text {integral}& = -\frac {1}{2 c e (d+e x) \sqrt {c d^2+2 c d e x+c e^2 x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.61 \[ \int \frac {1}{\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}} \, dx=-\frac {d+e x}{2 e \left (c (d+e x)^2\right )^{3/2}} \]

[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))

Maple [A] (verified)

Time = 2.46 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.66

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

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.68 \[ \int \frac {1}{\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}} \, dx=-\frac {\sqrt {c e^{2} x^{2} + 2 \, c d e x + c d^{2}}}{2 \, {\left (c^{2} e^{4} x^{3} + 3 \, c^{2} d e^{3} x^{2} + 3 \, c^{2} d^{2} e^{2} x + c^{2} d^{3} e\right )}} \]

[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*e^2*x^2 + 2*c*d*e*x + c*d^2)/(c^2*e^4*x^3 + 3*c^2*d*e^3*x^2 + 3*c^2*d^2*e^2*x + c^2*d^3*e)

Sympy [F]

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

[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)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.44 \[ \int \frac {1}{\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}} \, dx=-\frac {1}{2 \, \left (c e^{2}\right )^{\frac {3}{2}} {\left (x + \frac {d}{e}\right )}^{2}} \]

[In]

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

[Out]

-1/2/((c*e^2)^(3/2)*(x + d/e)^2)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.56 \[ \int \frac {1}{\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}} \, dx=-\frac {1}{2 \, {\left (e x + d\right )}^{2} c^{\frac {3}{2}} e \mathrm {sgn}\left (e x + d\right )} \]

[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*x + d)^2*c^(3/2)*e*sgn(e*x + d))

Mupad [B] (verification not implemented)

Time = 9.93 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.90 \[ \int \frac {1}{\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}} \, dx=-\frac {\sqrt {c\,d^2+2\,c\,d\,e\,x+c\,e^2\,x^2}}{2\,c^2\,e\,{\left (d+e\,x\right )}^3} \]

[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)