\(\int \frac {(d+e x)^2}{(a d e+(c d^2+a e^2) x+c d e x^2)^2} \, dx\) [1880]

Optimal result

Integrand size = 35, antiderivative size = 18 \[ \int \frac {(d+e x)^2}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx=-\frac {1}{c d (a e+c d x)} \]

[Out]

-1/c/d/(c*d*x+a*e)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, Rules used = {640, 32} \[ \int \frac {(d+e x)^2}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx=-\frac {1}{c d (a e+c d x)} \]

[In]

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

[Out]

-(1/(c*d*(a*e + c*d*x)))

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rule 640

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

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

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {(d+e x)^2}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx=-\frac {1}{c d (a e+c d x)} \]

[In]

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

[Out]

-(1/(c*d*(a*e + c*d*x)))

Maple [A] (verified)

Time = 2.76 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.06

method	result	size
gosper	\(-\frac {1}{c d \left (c d x +a e \right )}\)	\(19\)
default	\(-\frac {1}{c d \left (c d x +a e \right )}\)	\(19\)
risch	\(-\frac {1}{c d \left (c d x +a e \right )}\)	\(19\)
parallelrisch	\(-\frac {1}{c d \left (c d x +a e \right )}\)	\(19\)
norman	\(\frac {-\frac {1}{c}-\frac {e x}{c d}}{\left (c d x +a e \right ) \left (e x +d \right )}\)	\(35\)

[In]

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

[Out]

-1/c/d/(c*d*x+a*e)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {(d+e x)^2}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx=-\frac {1}{c^{2} d^{2} x + a c d e} \]

[In]

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

[Out]

-1/(c^2*d^2*x + a*c*d*e)

Sympy [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.94 \[ \int \frac {(d+e x)^2}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx=- \frac {1}{a c d e + c^{2} d^{2} x} \]

[In]

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

[Out]

-1/(a*c*d*e + c**2*d**2*x)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {(d+e x)^2}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx=-\frac {1}{c^{2} d^{2} x + a c d e} \]

[In]

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

[Out]

-1/(c^2*d^2*x + a*c*d*e)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {(d+e x)^2}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx=-\frac {1}{{\left (c d x + a e\right )} c d} \]

[In]

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

[Out]

-1/((c*d*x + a*e)*c*d)

Mupad [B] (verification not implemented)

Time = 9.70 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {(d+e x)^2}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx=-\frac {1}{c\,d\,\left (a\,e+c\,d\,x\right )} \]

[In]

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

[Out]

-1/(c*d*(a*e + c*d*x))