\(\int \frac {d+e x}{\sqrt {c d^2+2 c d e x+c e^2 x^2}} \, dx\) [1065]

Optimal result

Integrand size = 30, antiderivative size = 31 \[ \int \frac {d+e x}{\sqrt {c d^2+2 c d e x+c e^2 x^2}} \, dx=\frac {\sqrt {c d^2+2 c d e x+c e^2 x^2}}{c e} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.033, Rules used = {643} \[ \int \frac {d+e x}{\sqrt {c d^2+2 c d e x+c e^2 x^2}} \, dx=\frac {\sqrt {c d^2+2 c d e x+c e^2 x^2}}{c e} \]

[In]

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

[Out]

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

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

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.65 \[ \int \frac {d+e x}{\sqrt {c d^2+2 c d e x+c e^2 x^2}} \, dx=\frac {\sqrt {c (d+e x)^2}}{c e} \]

[In]

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

[Out]

Sqrt[c*(d + e*x)^2]/(c*e)

Maple [A] (verified)

Time = 2.46 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.61

method	result	size
risch	\(\frac {\left (e x +d \right ) x}{\sqrt {c \left (e x +d \right )^{2}}}\)	\(19\)
pseudoelliptic	\(\frac {\sqrt {c \left (e x +d \right )^{2}}}{c e}\)	\(19\)
default	\(\frac {x \left (e x +d \right )}{\sqrt {c \,x^{2} e^{2}+2 x c d e +c \,d^{2}}}\)	\(30\)
trager	\(\frac {x \sqrt {c \,x^{2} e^{2}+2 x c d e +c \,d^{2}}}{c \left (e x +d \right )}\)	\(35\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.10 \[ \int \frac {d+e x}{\sqrt {c d^2+2 c d e x+c e^2 x^2}} \, dx=\frac {\sqrt {c e^{2} x^{2} + 2 \, c d e x + c d^{2}} x}{c e x + c d} \]

[In]

integrate((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)*x/(c*e*x + c*d)

Sympy [A] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.26 \[ \int \frac {d+e x}{\sqrt {c d^2+2 c d e x+c e^2 x^2}} \, dx=\begin {cases} \frac {\sqrt {c d^{2} + 2 c d e x + c e^{2} x^{2}}}{c e} & \text {for}\: e \neq 0 \\\frac {d x}{\sqrt {c d^{2}}} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.94 \[ \int \frac {d+e x}{\sqrt {c d^2+2 c d e x+c e^2 x^2}} \, dx=\frac {\sqrt {c e^{2} x^{2} + 2 \, c d e x + c d^{2}}}{c e} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.42 \[ \int \frac {d+e x}{\sqrt {c d^2+2 c d e x+c e^2 x^2}} \, dx=\frac {x}{\sqrt {c} \mathrm {sgn}\left (e x + d\right )} \]

[In]

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

[Out]

x/(sqrt(c)*sgn(e*x + d))

Mupad [B] (verification not implemented)

Time = 10.12 (sec) , antiderivative size = 109, normalized size of antiderivative = 3.52 \[ \int \frac {d+e x}{\sqrt {c d^2+2 c d e x+c e^2 x^2}} \, dx=\frac {\sqrt {c\,d^2+2\,c\,d\,e\,x+c\,e^2\,x^2}}{c\,e}-\frac {c\,d\,e^2\,\ln \left (\sqrt {c\,{\left (d+e\,x\right )}^2}\,\sqrt {c\,e^2}+c\,d\,e+c\,e^2\,x\right )}{{\left (c\,e^2\right )}^{3/2}}+\frac {c\,d\,e^2\,\ln \left (c\,x\,e^2+c\,d\,e\right )\,\mathrm {sign}\left (c\,e\,\left (d+e\,x\right )\right )}{{\left (c\,e^2\right )}^{3/2}} \]

[In]

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

[Out]

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