\(\int \frac {(d+e x)^6}{(c d^2+2 c d e x+c e^2 x^2)^3} \, dx\) [1020]

Optimal result

Integrand size = 30, antiderivative size = 5 \[ \int \frac {(d+e x)^6}{\left (c d^2+2 c d e x+c e^2 x^2\right )^3} \, dx=\frac {x}{c^3} \]

[Out]

x/c^3

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 5, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {27, 8} \[ \int \frac {(d+e x)^6}{\left (c d^2+2 c d e x+c e^2 x^2\right )^3} \, dx=\frac {x}{c^3} \]

[In]

Int[(d + e*x)^6/(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^3,x]

[Out]

x/c^3

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{c^3} \, dx \\ & = \frac {x}{c^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 5, normalized size of antiderivative = 1.00 \[ \int \frac {(d+e x)^6}{\left (c d^2+2 c d e x+c e^2 x^2\right )^3} \, dx=\frac {x}{c^3} \]

[In]

Integrate[(d + e*x)^6/(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^3,x]

[Out]

x/c^3

Maple [A] (verified)

Time = 2.61 (sec) , antiderivative size = 6, normalized size of antiderivative = 1.20

method	result	size
default	\(\frac {x}{c^{3}}\)	\(6\)
risch	\(\frac {x}{c^{3}}\)	\(6\)
norman	\(\frac {\frac {e^{5} x^{6}}{c}-\frac {5 d^{6}}{e c}-\frac {24 d^{5} x}{c}-\frac {15 e^{3} d^{2} x^{4}}{c}-\frac {40 d^{3} e^{2} x^{3}}{c}-\frac {45 d^{4} e \,x^{2}}{c}}{c^{2} \left (e x +d \right )^{5}}\)	\(83\)

[In]

int((e*x+d)^6/(c*e^2*x^2+2*c*d*e*x+c*d^2)^3,x,method=_RETURNVERBOSE)

[Out]

x/c^3

Fricas [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 5, normalized size of antiderivative = 1.00 \[ \int \frac {(d+e x)^6}{\left (c d^2+2 c d e x+c e^2 x^2\right )^3} \, dx=\frac {x}{c^{3}} \]

[In]

integrate((e*x+d)^6/(c*e^2*x^2+2*c*d*e*x+c*d^2)^3,x, algorithm="fricas")

[Out]

x/c^3

Sympy [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.60 \[ \int \frac {(d+e x)^6}{\left (c d^2+2 c d e x+c e^2 x^2\right )^3} \, dx=\frac {x}{c^{3}} \]

[In]

integrate((e*x+d)**6/(c*e**2*x**2+2*c*d*e*x+c*d**2)**3,x)

[Out]

x/c**3

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 5, normalized size of antiderivative = 1.00 \[ \int \frac {(d+e x)^6}{\left (c d^2+2 c d e x+c e^2 x^2\right )^3} \, dx=\frac {x}{c^{3}} \]

[In]

integrate((e*x+d)^6/(c*e^2*x^2+2*c*d*e*x+c*d^2)^3,x, algorithm="maxima")

[Out]

x/c^3

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 5, normalized size of antiderivative = 1.00 \[ \int \frac {(d+e x)^6}{\left (c d^2+2 c d e x+c e^2 x^2\right )^3} \, dx=\frac {x}{c^{3}} \]

[In]

integrate((e*x+d)^6/(c*e^2*x^2+2*c*d*e*x+c*d^2)^3,x, algorithm="giac")

[Out]

x/c^3

Mupad [B] (verification not implemented)

Time = 0.01 (sec) , antiderivative size = 5, normalized size of antiderivative = 1.00 \[ \int \frac {(d+e x)^6}{\left (c d^2+2 c d e x+c e^2 x^2\right )^3} \, dx=\frac {x}{c^3} \]

[In]

int((d + e*x)^6/(c*d^2 + c*e^2*x^2 + 2*c*d*e*x)^3,x)

[Out]

x/c^3