\(\int \frac {(d+e x)^6}{(a d e+(c d^2+a e^2) x+c d e x^2)^4} \, dx\) [1900]

Optimal result

Integrand size = 35, antiderivative size = 35 \[ \int \frac {(d+e x)^6}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^4} \, dx=-\frac {(d+e x)^3}{3 \left (c d^2-a e^2\right ) (a e+c d x)^3} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 35, 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, 37} \[ \int \frac {(d+e x)^6}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^4} \, dx=-\frac {(d+e x)^3}{3 \left (c d^2-a e^2\right ) (a e+c d x)^3} \]

[In]

Int[(d + e*x)^6/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^4,x]

[Out]

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

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n +
1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[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 {(d+e x)^2}{(a e+c d x)^4} \, dx \\ & = -\frac {(d+e x)^3}{3 \left (c d^2-a e^2\right ) (a e+c d x)^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.86 \[ \int \frac {(d+e x)^6}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^4} \, dx=-\frac {a^2 e^4+a c d e^2 (d+3 e x)+c^2 d^2 \left (d^2+3 d e x+3 e^2 x^2\right )}{3 c^3 d^3 (a e+c d x)^3} \]

[In]

Integrate[(d + e*x)^6/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^4,x]

[Out]

-1/3*(a^2*e^4 + a*c*d*e^2*(d + 3*e*x) + c^2*d^2*(d^2 + 3*d*e*x + 3*e^2*x^2))/(c^3*d^3*(a*e + c*d*x)^3)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(75\) vs. \(2(33)=66\).

Time = 2.42 (sec) , antiderivative size = 76, normalized size of antiderivative = 2.17

method	result	size
gosper	\(-\frac {3 x^{2} c^{2} d^{2} e^{2}+3 x a c d \,e^{3}+3 x \,c^{2} d^{3} e +a^{2} e^{4}+a c \,d^{2} e^{2}+c^{2} d^{4}}{3 c^{3} d^{3} \left (c d x +a e \right )^{3}}\)	\(76\)
parallelrisch	\(\frac {-3 x^{2} c^{2} d^{2} e^{2}-3 x a c d \,e^{3}-3 x \,c^{2} d^{3} e -a^{2} e^{4}-a c \,d^{2} e^{2}-c^{2} d^{4}}{3 c^{3} d^{3} \left (c d x +a e \right )^{3}}\)	\(79\)
risch	\(\frac {-\frac {e^{2} x^{2}}{d c}-\frac {e \left (e^{2} a +c \,d^{2}\right ) x}{c^{2} d^{2}}-\frac {a^{2} e^{4}+a c \,d^{2} e^{2}+c^{2} d^{4}}{3 c^{3} d^{3}}}{\left (c d x +a e \right )^{3}}\)	\(80\)
default	\(\frac {e \left (e^{2} a -c \,d^{2}\right )}{c^{3} d^{3} \left (c d x +a e \right )^{2}}-\frac {a^{2} e^{4}-2 a c \,d^{2} e^{2}+c^{2} d^{4}}{3 c^{3} d^{3} \left (c d x +a e \right )^{3}}-\frac {e^{2}}{c^{3} d^{3} \left (c d x +a e \right )}\)	\(96\)
norman	\(\frac {-\frac {e^{5} x^{5}}{c d}+\frac {\left (-a^{2} e^{6}-2 a c \,d^{2} e^{4}-2 c^{2} d^{4} e^{2}\right ) x}{c^{3} d e}+\frac {\left (-a \,e^{8}-4 c \,e^{6} d^{2}\right ) x^{4}}{c^{2} d^{2} e^{2}}+\frac {\left (-a^{2} e^{8}-4 a c \,d^{2} e^{6}-5 c^{2} e^{4} d^{4}\right ) x^{2}}{c^{3} d^{2} e^{2}}+\frac {-a^{2} e^{4}-a c \,d^{2} e^{2}-c^{2} d^{4}}{3 c^{3}}+\frac {\left (-a^{2} e^{10}-10 a c \,d^{2} e^{8}-19 c^{2} d^{4} e^{6}\right ) x^{3}}{3 c^{3} d^{3} e^{3}}}{\left (c d x +a e \right )^{3} \left (e x +d \right )^{3}}\)	\(223\)

[In]

int((e*x+d)^6/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^4,x,method=_RETURNVERBOSE)

[Out]

-1/3*(3*c^2*d^2*e^2*x^2+3*a*c*d*e^3*x+3*c^2*d^3*e*x+a^2*e^4+a*c*d^2*e^2+c^2*d^4)/c^3/d^3/(c*d*x+a*e)^3

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 113 vs. \(2 (33) = 66\).

Time = 0.27 (sec) , antiderivative size = 113, normalized size of antiderivative = 3.23 \[ \int \frac {(d+e x)^6}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^4} \, dx=-\frac {3 \, c^{2} d^{2} e^{2} x^{2} + c^{2} d^{4} + a c d^{2} e^{2} + a^{2} e^{4} + 3 \, {\left (c^{2} d^{3} e + a c d e^{3}\right )} x}{3 \, {\left (c^{6} d^{6} x^{3} + 3 \, a c^{5} d^{5} e x^{2} + 3 \, a^{2} c^{4} d^{4} e^{2} x + a^{3} c^{3} d^{3} e^{3}\right )}} \]

[In]

integrate((e*x+d)^6/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^4,x, algorithm="fricas")

[Out]

-1/3*(3*c^2*d^2*e^2*x^2 + c^2*d^4 + a*c*d^2*e^2 + a^2*e^4 + 3*(c^2*d^3*e + a*c*d*e^3)*x)/(c^6*d^6*x^3 + 3*a*c^
5*d^5*e*x^2 + 3*a^2*c^4*d^4*e^2*x + a^3*c^3*d^3*e^3)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 121 vs. \(2 (29) = 58\).

Time = 0.71 (sec) , antiderivative size = 121, normalized size of antiderivative = 3.46 \[ \int \frac {(d+e x)^6}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^4} \, dx=\frac {- a^{2} e^{4} - a c d^{2} e^{2} - c^{2} d^{4} - 3 c^{2} d^{2} e^{2} x^{2} + x \left (- 3 a c d e^{3} - 3 c^{2} d^{3} e\right )}{3 a^{3} c^{3} d^{3} e^{3} + 9 a^{2} c^{4} d^{4} e^{2} x + 9 a c^{5} d^{5} e x^{2} + 3 c^{6} d^{6} x^{3}} \]

[In]

integrate((e*x+d)**6/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**4,x)

[Out]

(-a**2*e**4 - a*c*d**2*e**2 - c**2*d**4 - 3*c**2*d**2*e**2*x**2 + x*(-3*a*c*d*e**3 - 3*c**2*d**3*e))/(3*a**3*c
**3*d**3*e**3 + 9*a**2*c**4*d**4*e**2*x + 9*a*c**5*d**5*e*x**2 + 3*c**6*d**6*x**3)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 113 vs. \(2 (33) = 66\).

Time = 0.19 (sec) , antiderivative size = 113, normalized size of antiderivative = 3.23 \[ \int \frac {(d+e x)^6}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^4} \, dx=-\frac {3 \, c^{2} d^{2} e^{2} x^{2} + c^{2} d^{4} + a c d^{2} e^{2} + a^{2} e^{4} + 3 \, {\left (c^{2} d^{3} e + a c d e^{3}\right )} x}{3 \, {\left (c^{6} d^{6} x^{3} + 3 \, a c^{5} d^{5} e x^{2} + 3 \, a^{2} c^{4} d^{4} e^{2} x + a^{3} c^{3} d^{3} e^{3}\right )}} \]

[In]

integrate((e*x+d)^6/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^4,x, algorithm="maxima")

[Out]

-1/3*(3*c^2*d^2*e^2*x^2 + c^2*d^4 + a*c*d^2*e^2 + a^2*e^4 + 3*(c^2*d^3*e + a*c*d*e^3)*x)/(c^6*d^6*x^3 + 3*a*c^
5*d^5*e*x^2 + 3*a^2*c^4*d^4*e^2*x + a^3*c^3*d^3*e^3)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 75 vs. \(2 (33) = 66\).

Time = 0.29 (sec) , antiderivative size = 75, normalized size of antiderivative = 2.14 \[ \int \frac {(d+e x)^6}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^4} \, dx=-\frac {3 \, c^{2} d^{2} e^{2} x^{2} + 3 \, c^{2} d^{3} e x + 3 \, a c d e^{3} x + c^{2} d^{4} + a c d^{2} e^{2} + a^{2} e^{4}}{3 \, {\left (c d x + a e\right )}^{3} c^{3} d^{3}} \]

[In]

integrate((e*x+d)^6/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^4,x, algorithm="giac")

[Out]

-1/3*(3*c^2*d^2*e^2*x^2 + 3*c^2*d^3*e*x + 3*a*c*d*e^3*x + c^2*d^4 + a*c*d^2*e^2 + a^2*e^4)/((c*d*x + a*e)^3*c^
3*d^3)

Mupad [B] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 81, normalized size of antiderivative = 2.31 \[ \int \frac {(d+e x)^6}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^4} \, dx=-\frac {\frac {d}{3\,c}+e\,\left (\frac {x}{c}-\frac {x^3}{3\,a}\right )+\frac {a\,e^2}{3\,c^2\,d}}{a^3\,e^3+3\,a^2\,c\,d\,e^2\,x+3\,a\,c^2\,d^2\,e\,x^2+c^3\,d^3\,x^3} \]

[In]

int((d + e*x)^6/(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^4,x)

[Out]

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