\(\int \frac {\sqrt {c d^2+2 c d e x+c e^2 x^2}}{d+e x} \, dx\) [1033]

Optimal result

Integrand size = 32, antiderivative size = 28 \[ \int \frac {\sqrt {c d^2+2 c d e x+c e^2 x^2}}{d+e x} \, dx=\frac {\sqrt {c d^2+2 c d e x+c e^2 x^2}}{e} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 28, 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} \[ \int \frac {\sqrt {c d^2+2 c d e x+c e^2 x^2}}{d+e x} \, dx=\frac {\sqrt {c d^2+2 c d e x+c e^2 x^2}}{e} \]

[In]

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

[Out]

Sqrt[c*d^2 + 2*c*d*e*x + c*e^2*x^2]/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]

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}{\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}}{e} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.61 \[ \int \frac {\sqrt {c d^2+2 c d e x+c e^2 x^2}}{d+e x} \, dx=\frac {\sqrt {c (d+e x)^2}}{e} \]

[In]

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

[Out]

Sqrt[c*(d + e*x)^2]/e

Maple [A] (verified)

Time = 2.87 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.57

method	result	size
pseudoelliptic	\(\frac {\sqrt {c \left (e x +d \right )^{2}}}{e}\)	\(16\)
risch	\(\frac {\sqrt {c \left (e x +d \right )^{2}}\, x}{e x +d}\)	\(21\)
default	\(\frac {x \sqrt {c \,x^{2} e^{2}+2 x c d e +c \,d^{2}}}{e x +d}\)	\(32\)
trager	\(\frac {x \sqrt {c \,x^{2} e^{2}+2 x c d e +c \,d^{2}}}{e x +d}\)	\(32\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.11 \[ \int \frac {\sqrt {c d^2+2 c d e x+c e^2 x^2}}{d+e x} \, dx=\frac {\sqrt {c e^{2} x^{2} + 2 \, c d e x + c d^{2}} x}{e x + d} \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.74 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.32 \[ \int \frac {\sqrt {c d^2+2 c d e x+c e^2 x^2}}{d+e x} \, dx=\begin {cases} \frac {\sqrt {c d^{2} + 2 c d e x + c e^{2} x^{2}}}{e} & \text {for}\: e \neq 0 \\\frac {x \sqrt {c d^{2}}}{d} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

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

Maxima [F(-2)]

Exception generated. \[ \int \frac {\sqrt {c d^2+2 c d e x+c e^2 x^2}}{d+e x} \, dx=\text {Exception raised: RuntimeError} \]

[In]

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

[Out]

Exception raised: RuntimeError >> ECL says: expt: undefined: 0 to a negative exponent.

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.86 \[ \int \frac {\sqrt {c d^2+2 c d e x+c e^2 x^2}}{d+e x} \, dx={\left (x \mathrm {sgn}\left (e x + d\right ) + \frac {d \mathrm {sgn}\left (e x + d\right )}{e}\right )} \sqrt {c} \]

[In]

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

[Out]

(x*sgn(e*x + d) + d*sgn(e*x + d)/e)*sqrt(c)

Mupad [B] (verification not implemented)

Time = 10.04 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.54 \[ \int \frac {\sqrt {c d^2+2 c d e x+c e^2 x^2}}{d+e x} \, dx=\frac {\sqrt {c\,{\left (d+e\,x\right )}^2}}{e} \]

[In]

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

[Out]

(c*(d + e*x)^2)^(1/2)/e