\(\int \frac {1}{(d+e x) \sqrt {c d^2+2 c d e x+c e^2 x^2}} \, dx\) [1067]

Optimal result

Integrand size = 32, antiderivative size = 29 \[ \int \frac {1}{(d+e x) \sqrt {c d^2+2 c d e x+c e^2 x^2}} \, dx=-\frac {1}{e \sqrt {c d^2+2 c d e x+c e^2 x^2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {657, 643} \[ \int \frac {1}{(d+e x) \sqrt {c d^2+2 c d e x+c e^2 x^2}} \, dx=-\frac {1}{e \sqrt {c d^2+2 c d e x+c e^2 x^2}} \]

[In]

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

[Out]

-(1/(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]

Rule 657

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[e^(m - 1)/c^((m - 1)/2
), Int[(d + e*x)*(a + b*x + c*x^2)^(p + (m - 1)/2), x], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[b^2 - 4*a*c,
0] &&  !IntegerQ[p] && EqQ[2*c*d - b*e, 0] && IntegerQ[(m - 1)/2]

Rubi steps \begin{align*} \text {integral}& = c \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}{e \sqrt {c d^2+2 c d e x+c e^2 x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.62 \[ \int \frac {1}{(d+e x) \sqrt {c d^2+2 c d e x+c e^2 x^2}} \, dx=-\frac {1}{e \sqrt {c (d+e x)^2}} \]

[In]

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

[Out]

-(1/(e*Sqrt[c*(d + e*x)^2]))

Maple [A] (verified)

Time = 2.43 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.59

method	result	size
risch	\(-\frac {1}{\sqrt {c \left (e x +d \right )^{2}}\, e}\)	\(17\)
pseudoelliptic	\(-\frac {1}{\sqrt {c \left (e x +d \right )^{2}}\, e}\)	\(17\)
gosper	\(-\frac {1}{e \sqrt {c \,x^{2} e^{2}+2 x c d e +c \,d^{2}}}\)	\(28\)
default	\(-\frac {1}{e \sqrt {c \,x^{2} e^{2}+2 x c d e +c \,d^{2}}}\)	\(28\)
trager	\(\frac {x \sqrt {c \,x^{2} e^{2}+2 x c d e +c \,d^{2}}}{d c \left (e x +d \right )^{2}}\)	\(38\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.69 \[ \int \frac {1}{(d+e x) \sqrt {c d^2+2 c d e x+c e^2 x^2}} \, dx=-\frac {\sqrt {c e^{2} x^{2} + 2 \, c d e x + c d^{2}}}{c e^{3} x^{2} + 2 \, c d e^{2} x + c d^{2} e} \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 1.21 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.41 \[ \int \frac {1}{(d+e x) \sqrt {c d^2+2 c d e x+c e^2 x^2}} \, dx=\begin {cases} - \frac {1}{e \sqrt {c d^{2} + 2 c d e x + c e^{2} x^{2}}} & \text {for}\: e \neq 0 \\\frac {x}{d \sqrt {c d^{2}}} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.66 \[ \int \frac {1}{(d+e x) \sqrt {c d^2+2 c d e x+c e^2 x^2}} \, dx=-\frac {1}{\sqrt {c} e^{2} x + \sqrt {c} d e} \]

[In]

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

[Out]

-1/(sqrt(c)*e^2*x + sqrt(c)*d*e)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.79 \[ \int \frac {1}{(d+e x) \sqrt {c d^2+2 c d e x+c e^2 x^2}} \, dx=-\frac {1}{{\left (e x + d\right )} \sqrt {c} e \mathrm {sgn}\left (e x + d\right )} \]

[In]

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

[Out]

-1/((e*x + d)*sqrt(c)*e*sgn(e*x + d))

Mupad [B] (verification not implemented)

Time = 9.71 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.93 \[ \int \frac {1}{(d+e x) \sqrt {c d^2+2 c d e x+c e^2 x^2}} \, dx=-\frac {1}{e\,\sqrt {c\,d^2+2\,c\,d\,e\,x+c\,e^2\,x^2}} \]

[In]

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

[Out]

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