\(\int (d+e x) \sqrt {c d^2+2 c d e x+c e^2 x^2} \, dx\) [1031]

Optimal result

Integrand size = 30, antiderivative size = 34 \[ \int (d+e x) \sqrt {c d^2+2 c d e x+c e^2 x^2} \, dx=\frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}}{3 c e} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 34, 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} \[ \int (d+e x) \sqrt {c d^2+2 c d e x+c e^2 x^2} \, dx=\frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}}{3 c e} \]

[In]

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

[Out]

(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(3/2)/(3*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 {\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}}{3 c e} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.68 \[ \int (d+e x) \sqrt {c d^2+2 c d e x+c e^2 x^2} \, dx=\frac {\left (c (d+e x)^2\right )^{3/2}}{3 c e} \]

[In]

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

[Out]

(c*(d + e*x)^2)^(3/2)/(3*c*e)

Maple [A] (verified)

Time = 2.30 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.71

method	result	size
risch	\(\frac {\left (e x +d \right )^{2} \sqrt {c \left (e x +d \right )^{2}}}{3 e}\)	\(24\)
pseudoelliptic	\(\frac {\left (e x +d \right )^{2} \sqrt {c \left (e x +d \right )^{2}}}{3 e}\)	\(24\)
default	\(\frac {\left (e x +d \right )^{2} \sqrt {c \,x^{2} e^{2}+2 x c d e +c \,d^{2}}}{3 e}\)	\(35\)
gosper	\(\frac {x \left (x^{2} e^{2}+3 d e x +3 d^{2}\right ) \sqrt {c \,x^{2} e^{2}+2 x c d e +c \,d^{2}}}{3 e x +3 d}\)	\(51\)
trager	\(\frac {x \left (x^{2} e^{2}+3 d e x +3 d^{2}\right ) \sqrt {c \,x^{2} e^{2}+2 x c d e +c \,d^{2}}}{3 e x +3 d}\)	\(51\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.37 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.53 \[ \int (d+e x) \sqrt {c d^2+2 c d e x+c e^2 x^2} \, dx=\frac {\sqrt {c e^{2} x^{2} + 2 \, c d e x + c d^{2}} {\left (e^{2} x^{3} + 3 \, d e x^{2} + 3 \, d^{2} x\right )}}{3 \, {\left (e x + d\right )}} \]

[In]

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

[Out]

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 107 vs. \(2 (29) = 58\).

Time = 0.09 (sec) , antiderivative size = 107, normalized size of antiderivative = 3.15 \[ \int (d+e x) \sqrt {c d^2+2 c d e x+c e^2 x^2} \, dx=\begin {cases} \frac {d^{2} \sqrt {c d^{2} + 2 c d e x + c e^{2} x^{2}}}{3 e} + \frac {2 d x \sqrt {c d^{2} + 2 c d e x + c e^{2} x^{2}}}{3} + \frac {e x^{2} \sqrt {c d^{2} + 2 c d e x + c e^{2} x^{2}}}{3} & \text {for}\: e \neq 0 \\d x \sqrt {c d^{2}} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((d**2*sqrt(c*d**2 + 2*c*d*e*x + c*e**2*x**2)/(3*e) + 2*d*x*sqrt(c*d**2 + 2*c*d*e*x + c*e**2*x**2)/3
+ e*x**2*sqrt(c*d**2 + 2*c*d*e*x + c*e**2*x**2)/3, Ne(e, 0)), (d*x*sqrt(c*d**2), True))

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.88 \[ \int (d+e x) \sqrt {c d^2+2 c d e x+c e^2 x^2} \, dx=\frac {{\left (c e^{2} x^{2} + 2 \, c d e x + c d^{2}\right )}^{\frac {3}{2}}}{3 \, c e} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.62 \[ \int (d+e x) \sqrt {c d^2+2 c d e x+c e^2 x^2} \, dx=\frac {{\left (e x + d\right )}^{3} \sqrt {c} \mathrm {sgn}\left (e x + d\right )}{3 \, e} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 10.03 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.00 \[ \int (d+e x) \sqrt {c d^2+2 c d e x+c e^2 x^2} \, dx=\frac {{\left (d+e\,x\right )}^2\,\sqrt {c\,d^2+2\,c\,d\,e\,x+c\,e^2\,x^2}}{3\,e} \]

[In]

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

[Out]

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