\(\int \frac {(d+e x)^4}{(a d e+(c d^2+a e^2) x+c d e x^2)^3} \, dx\) [1890]

Optimal result

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

[Out]

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

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

[In]

Int[(d + e*x)^4/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3,x]

[Out]

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

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}{(a e+c d x)^3} \, dx \\ & = -\frac {(d+e x)^2}{2 \left (c d^2-a e^2\right ) (a e+c d x)^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00 \[ \int \frac {(d+e x)^4}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3} \, dx=-\frac {a e^2+c d (d+2 e x)}{2 c^2 d^2 (a e+c d x)^2} \]

[In]

Integrate[(d + e*x)^4/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3,x]

[Out]

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

Maple [A] (verified)

Time = 2.40 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.03

method	result	size
gosper	\(-\frac {2 x c d e +e^{2} a +c \,d^{2}}{2 \left (c d x +a e \right )^{2} c^{2} d^{2}}\)	\(36\)
parallelrisch	\(\frac {-2 x c d e -e^{2} a -c \,d^{2}}{2 c^{2} d^{2} \left (c d x +a e \right )^{2}}\)	\(38\)
risch	\(\frac {-\frac {e x}{c d}-\frac {e^{2} a +c \,d^{2}}{2 c^{2} d^{2}}}{\left (c d x +a e \right )^{2}}\)	\(42\)
default	\(-\frac {-e^{2} a +c \,d^{2}}{2 c^{2} d^{2} \left (c d x +a e \right )^{2}}-\frac {e}{c^{2} d^{2} \left (c d x +a e \right )}\)	\(51\)
norman	\(\frac {-\frac {e^{3} x^{3}}{c d}+\frac {\left (-e^{4} a -2 d^{2} e^{2} c \right ) x}{c^{2} d e}+\frac {-e^{2} a -c \,d^{2}}{2 c^{2}}+\frac {\left (-a \,e^{6}-5 c \,d^{2} e^{4}\right ) x^{2}}{2 c^{2} d^{2} e^{2}}}{\left (c d x +a e \right )^{2} \left (e x +d \right )^{2}}\)	\(109\)

[In]

int((e*x+d)^4/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^3,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.60 \[ \int \frac {(d+e x)^4}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3} \, dx=-\frac {2 \, c d e x + c d^{2} + a e^{2}}{2 \, {\left (c^{4} d^{4} x^{2} + 2 \, a c^{3} d^{3} e x + a^{2} c^{2} d^{2} e^{2}\right )}} \]

[In]

integrate((e*x+d)^4/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^3,x, algorithm="fricas")

[Out]

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 60 vs. \(2 (29) = 58\).

Time = 0.18 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.71 \[ \int \frac {(d+e x)^4}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3} \, dx=\frac {- a e^{2} - c d^{2} - 2 c d e x}{2 a^{2} c^{2} d^{2} e^{2} + 4 a c^{3} d^{3} e x + 2 c^{4} d^{4} x^{2}} \]

[In]

integrate((e*x+d)**4/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**3,x)

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.60 \[ \int \frac {(d+e x)^4}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3} \, dx=-\frac {2 \, c d e x + c d^{2} + a e^{2}}{2 \, {\left (c^{4} d^{4} x^{2} + 2 \, a c^{3} d^{3} e x + a^{2} c^{2} d^{2} e^{2}\right )}} \]

[In]

integrate((e*x+d)^4/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^3,x, algorithm="maxima")

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00 \[ \int \frac {(d+e x)^4}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3} \, dx=-\frac {2 \, c d e x + c d^{2} + a e^{2}}{2 \, {\left (c d x + a e\right )}^{2} c^{2} d^{2}} \]

[In]

integrate((e*x+d)^4/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^3,x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.23 \[ \int \frac {(d+e x)^4}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3} \, dx=-\frac {\frac {1}{2\,c}-\frac {x^2}{2\,a}}{a^2\,e^2+2\,a\,c\,d\,e\,x+c^2\,d^2\,x^2} \]

[In]

int((d + e*x)^4/(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^3,x)

[Out]

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