\(\int \frac {c d^2+2 c d e x+c e^2 x^2}{d+e x} \, dx\) [981]

Optimal result

Integrand size = 28, antiderivative size = 14 \[ \int \frac {c d^2+2 c d e x+c e^2 x^2}{d+e x} \, dx=c d x+\frac {1}{2} c e x^2 \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.036, Rules used = {24} \[ \int \frac {c d^2+2 c d e x+c e^2 x^2}{d+e x} \, dx=c d x+\frac {1}{2} c e x^2 \]

[In]

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

[Out]

c*d*x + (c*e*x^2)/2

Rule 24

Int[(u_.)*((a_) + (b_.)*(v_))^(m_)*((A_.) + (B_.)*(v_) + (C_.)*(v_)^2), x_Symbol] :> Dist[1/b^2, Int[u*(a + b*
v)^(m + 1)*Simp[b*B - a*C + b*C*v, x], x], x] /; FreeQ[{a, b, A, B, C}, x] && EqQ[A*b^2 - a*b*B + a^2*C, 0] &&
LeQ[m, -1]

Rubi steps \begin{align*} \text {integral}& = \frac {\int \left (c d e^2+c e^3 x\right ) \, dx}{e^2} \\ & = c d x+\frac {1}{2} c e x^2 \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int \frac {c d^2+2 c d e x+c e^2 x^2}{d+e x} \, dx=c \left (d x+\frac {e x^2}{2}\right ) \]

[In]

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

[Out]

c*(d*x + (e*x^2)/2)

Maple [A] (verified)

Time = 2.25 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86

method	result	size
gosper	\(\frac {x \left (e x +2 d \right ) c}{2}\)	\(12\)
default	\(c \left (\frac {1}{2} e \,x^{2}+d x \right )\)	\(13\)
norman	\(c d x +\frac {1}{2} c e \,x^{2}\)	\(13\)
risch	\(c d x +\frac {1}{2} c e \,x^{2}\)	\(13\)
parallelrisch	\(c d x +\frac {1}{2} c e \,x^{2}\)	\(13\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.34 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86 \[ \int \frac {c d^2+2 c d e x+c e^2 x^2}{d+e x} \, dx=\frac {1}{2} \, c e x^{2} + c d x \]

[In]

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

[Out]

1/2*c*e*x^2 + c*d*x

Sympy [A] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86 \[ \int \frac {c d^2+2 c d e x+c e^2 x^2}{d+e x} \, dx=c d x + \frac {c e x^{2}}{2} \]

[In]

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

[Out]

c*d*x + c*e*x**2/2

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86 \[ \int \frac {c d^2+2 c d e x+c e^2 x^2}{d+e x} \, dx=\frac {1}{2} \, c e x^{2} + c d x \]

[In]

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

[Out]

1/2*c*e*x^2 + c*d*x

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86 \[ \int \frac {c d^2+2 c d e x+c e^2 x^2}{d+e x} \, dx=\frac {1}{2} \, c e x^{2} + c d x \]

[In]

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

[Out]

1/2*c*e*x^2 + c*d*x

Mupad [B] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.79 \[ \int \frac {c d^2+2 c d e x+c e^2 x^2}{d+e x} \, dx=\frac {c\,x\,\left (2\,d+e\,x\right )}{2} \]

[In]

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

[Out]

(c*x*(2*d + e*x))/2