\(\int \frac {(d+e x)^6}{(c d^2+2 c d e x+c e^2 x^2)^2} \, dx\) [1008]

Optimal result

Integrand size = 30, antiderivative size = 17 \[ \int \frac {(d+e x)^6}{\left (c d^2+2 c d e x+c e^2 x^2\right )^2} \, dx=\frac {(d+e x)^3}{3 c^2 e} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {27, 12, 32} \[ \int \frac {(d+e x)^6}{\left (c d^2+2 c d e x+c e^2 x^2\right )^2} \, dx=\frac {(d+e x)^3}{3 c^2 e} \]

[In]

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

[Out]

(d + e*x)^3/(3*c^2*e)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, 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]

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]

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

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00 \[ \int \frac {(d+e x)^6}{\left (c d^2+2 c d e x+c e^2 x^2\right )^2} \, dx=\frac {(d+e x)^3}{3 c^2 e} \]

[In]

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

[Out]

(d + e*x)^3/(3*c^2*e)

Maple [A] (verified)

Time = 2.41 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.94

method	result	size
default	\(\frac {\left (e x +d \right )^{3}}{3 c^{2} e}\)	\(16\)
gosper	\(\frac {x \left (x^{2} e^{2}+3 d e x +3 d^{2}\right )}{3 c^{2}}\)	\(25\)
parallelrisch	\(\frac {e^{2} x^{3}+3 d e \,x^{2}+3 x \,d^{2}}{3 c^{2}}\)	\(27\)
risch	\(\frac {e^{2} x^{3}}{3 c^{2}}+\frac {e d \,x^{2}}{c^{2}}+\frac {x \,d^{2}}{c^{2}}+\frac {d^{3}}{3 c^{2} e}\)	\(41\)
norman	\(\frac {\frac {e^{5} x^{6}}{3 c}+\frac {2 d \,e^{4} x^{5}}{c}-\frac {19 d^{6}}{3 e c}+\frac {5 e^{3} d^{2} x^{4}}{c}-\frac {15 d^{4} e \,x^{2}}{c}-\frac {18 d^{5} x}{c}}{c \left (e x +d \right )^{3}}\)	\(82\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.53 \[ \int \frac {(d+e x)^6}{\left (c d^2+2 c d e x+c e^2 x^2\right )^2} \, dx=\frac {e^{2} x^{3} + 3 \, d e x^{2} + 3 \, d^{2} x}{3 \, c^{2}} \]

[In]

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

[Out]

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 29 vs. \(2 (12) = 24\).

Time = 0.04 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.71 \[ \int \frac {(d+e x)^6}{\left (c d^2+2 c d e x+c e^2 x^2\right )^2} \, dx=\frac {d^{2} x}{c^{2}} + \frac {d e x^{2}}{c^{2}} + \frac {e^{2} x^{3}}{3 c^{2}} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.53 \[ \int \frac {(d+e x)^6}{\left (c d^2+2 c d e x+c e^2 x^2\right )^2} \, dx=\frac {e^{2} x^{3} + 3 \, d e x^{2} + 3 \, d^{2} x}{3 \, c^{2}} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.53 \[ \int \frac {(d+e x)^6}{\left (c d^2+2 c d e x+c e^2 x^2\right )^2} \, dx=\frac {e^{2} x^{3} + 3 \, d e x^{2} + 3 \, d^{2} x}{3 \, c^{2}} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.41 \[ \int \frac {(d+e x)^6}{\left (c d^2+2 c d e x+c e^2 x^2\right )^2} \, dx=\frac {x\,\left (3\,d^2+3\,d\,e\,x+e^2\,x^2\right )}{3\,c^2} \]

[In]

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

[Out]

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