\(\int \frac {(a d e+(c d^2+a e^2) x+c d e x^2)^2}{(d+e x)^7} \, dx\) [1848]

Optimal result

Integrand size = 35, antiderivative size = 77 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2}{(d+e x)^7} \, dx=-\frac {\left (c d^2-a e^2\right )^2}{4 e^3 (d+e x)^4}+\frac {2 c d \left (c d^2-a e^2\right )}{3 e^3 (d+e x)^3}-\frac {c^2 d^2}{2 e^3 (d+e x)^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, Rules used = {640, 45} \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2}{(d+e x)^7} \, dx=\frac {2 c d \left (c d^2-a e^2\right )}{3 e^3 (d+e x)^3}-\frac {\left (c d^2-a e^2\right )^2}{4 e^3 (d+e x)^4}-\frac {c^2 d^2}{2 e^3 (d+e x)^2} \]

[In]

Int[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^2/(d + e*x)^7,x]

[Out]

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

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

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 {(a e+c d x)^2}{(d+e x)^5} \, dx \\ & = \int \left (\frac {\left (-c d^2+a e^2\right )^2}{e^2 (d+e x)^5}-\frac {2 c d \left (c d^2-a e^2\right )}{e^2 (d+e x)^4}+\frac {c^2 d^2}{e^2 (d+e x)^3}\right ) \, dx \\ & = -\frac {\left (c d^2-a e^2\right )^2}{4 e^3 (d+e x)^4}+\frac {2 c d \left (c d^2-a e^2\right )}{3 e^3 (d+e x)^3}-\frac {c^2 d^2}{2 e^3 (d+e x)^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.79 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2}{(d+e x)^7} \, dx=-\frac {3 a^2 e^4+2 a c d e^2 (d+4 e x)+c^2 d^2 \left (d^2+4 d e x+6 e^2 x^2\right )}{12 e^3 (d+e x)^4} \]

[In]

Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^2/(d + e*x)^7,x]

[Out]

-1/12*(3*a^2*e^4 + 2*a*c*d*e^2*(d + 4*e*x) + c^2*d^2*(d^2 + 4*d*e*x + 6*e^2*x^2))/(e^3*(d + e*x)^4)

Maple [A] (verified)

Time = 2.42 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.94

method	result	size
gosper	\(-\frac {6 x^{2} c^{2} d^{2} e^{2}+8 x a c d \,e^{3}+4 x \,c^{2} d^{3} e +3 a^{2} e^{4}+2 a c \,d^{2} e^{2}+c^{2} d^{4}}{12 e^{3} \left (e x +d \right )^{4}}\)	\(72\)
risch	\(\frac {-\frac {d^{2} c^{2} x^{2}}{2 e}-\frac {d c \left (2 e^{2} a +c \,d^{2}\right ) x}{3 e^{2}}-\frac {3 a^{2} e^{4}+2 a c \,d^{2} e^{2}+c^{2} d^{4}}{12 e^{3}}}{\left (e x +d \right )^{4}}\)	\(75\)
parallelrisch	\(\frac {-6 c^{2} d^{2} e^{3} x^{2}-8 a c d \,e^{4} x -4 c^{2} d^{3} e^{2} x -3 a^{2} e^{5}-2 a \,d^{2} e^{3} c -d^{4} e \,c^{2}}{12 e^{4} \left (e x +d \right )^{4}}\)	\(76\)
default	\(-\frac {2 c d \left (e^{2} a -c \,d^{2}\right )}{3 e^{3} \left (e x +d \right )^{3}}-\frac {a^{2} e^{4}-2 a c \,d^{2} e^{2}+c^{2} d^{4}}{4 e^{3} \left (e x +d \right )^{4}}-\frac {c^{2} d^{2}}{2 e^{3} \left (e x +d \right )^{2}}\)	\(83\)
norman	\(\frac {-\frac {d^{2} \left (3 a^{2} e^{7}+2 a \,d^{2} e^{5} c +c^{2} d^{4} e^{3}\right )}{12 e^{6}}-\frac {\left (a^{2} e^{7}+6 a \,d^{2} e^{5} c +5 c^{2} d^{4} e^{3}\right ) x^{2}}{4 e^{4}}-\frac {2 d \left (a c \,e^{5}+2 c^{2} d^{2} e^{3}\right ) x^{3}}{3 e^{3}}-\frac {d \left (a^{2} e^{7}+2 a \,d^{2} e^{5} c +c^{2} d^{4} e^{3}\right ) x}{2 e^{5}}-\frac {e \,c^{2} d^{2} x^{4}}{2}}{\left (e x +d \right )^{6}}\)	\(158\)

[In]

int((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^2/(e*x+d)^7,x,method=_RETURNVERBOSE)

[Out]

-1/12/e^3*(6*c^2*d^2*e^2*x^2+8*a*c*d*e^3*x+4*c^2*d^3*e*x+3*a^2*e^4+2*a*c*d^2*e^2+c^2*d^4)/(e*x+d)^4

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.40 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2}{(d+e x)^7} \, dx=-\frac {6 \, c^{2} d^{2} e^{2} x^{2} + c^{2} d^{4} + 2 \, a c d^{2} e^{2} + 3 \, a^{2} e^{4} + 4 \, {\left (c^{2} d^{3} e + 2 \, a c d e^{3}\right )} x}{12 \, {\left (e^{7} x^{4} + 4 \, d e^{6} x^{3} + 6 \, d^{2} e^{5} x^{2} + 4 \, d^{3} e^{4} x + d^{4} e^{3}\right )}} \]

[In]

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^2/(e*x+d)^7,x, algorithm="fricas")

[Out]

-1/12*(6*c^2*d^2*e^2*x^2 + c^2*d^4 + 2*a*c*d^2*e^2 + 3*a^2*e^4 + 4*(c^2*d^3*e + 2*a*c*d*e^3)*x)/(e^7*x^4 + 4*d
*e^6*x^3 + 6*d^2*e^5*x^2 + 4*d^3*e^4*x + d^4*e^3)

Sympy [A] (verification not implemented)

Time = 1.03 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.48 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2}{(d+e x)^7} \, dx=\frac {- 3 a^{2} e^{4} - 2 a c d^{2} e^{2} - c^{2} d^{4} - 6 c^{2} d^{2} e^{2} x^{2} + x \left (- 8 a c d e^{3} - 4 c^{2} d^{3} e\right )}{12 d^{4} e^{3} + 48 d^{3} e^{4} x + 72 d^{2} e^{5} x^{2} + 48 d e^{6} x^{3} + 12 e^{7} x^{4}} \]

[In]

integrate((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**2/(e*x+d)**7,x)

[Out]

(-3*a**2*e**4 - 2*a*c*d**2*e**2 - c**2*d**4 - 6*c**2*d**2*e**2*x**2 + x*(-8*a*c*d*e**3 - 4*c**2*d**3*e))/(12*d
**4*e**3 + 48*d**3*e**4*x + 72*d**2*e**5*x**2 + 48*d*e**6*x**3 + 12*e**7*x**4)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.40 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2}{(d+e x)^7} \, dx=-\frac {6 \, c^{2} d^{2} e^{2} x^{2} + c^{2} d^{4} + 2 \, a c d^{2} e^{2} + 3 \, a^{2} e^{4} + 4 \, {\left (c^{2} d^{3} e + 2 \, a c d e^{3}\right )} x}{12 \, {\left (e^{7} x^{4} + 4 \, d e^{6} x^{3} + 6 \, d^{2} e^{5} x^{2} + 4 \, d^{3} e^{4} x + d^{4} e^{3}\right )}} \]

[In]

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^2/(e*x+d)^7,x, algorithm="maxima")

[Out]

-1/12*(6*c^2*d^2*e^2*x^2 + c^2*d^4 + 2*a*c*d^2*e^2 + 3*a^2*e^4 + 4*(c^2*d^3*e + 2*a*c*d*e^3)*x)/(e^7*x^4 + 4*d
*e^6*x^3 + 6*d^2*e^5*x^2 + 4*d^3*e^4*x + d^4*e^3)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.92 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2}{(d+e x)^7} \, dx=-\frac {6 \, c^{2} d^{2} e^{2} x^{2} + 4 \, c^{2} d^{3} e x + 8 \, a c d e^{3} x + c^{2} d^{4} + 2 \, a c d^{2} e^{2} + 3 \, a^{2} e^{4}}{12 \, {\left (e x + d\right )}^{4} e^{3}} \]

[In]

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^2/(e*x+d)^7,x, algorithm="giac")

[Out]

-1/12*(6*c^2*d^2*e^2*x^2 + 4*c^2*d^3*e*x + 8*a*c*d*e^3*x + c^2*d^4 + 2*a*c*d^2*e^2 + 3*a^2*e^4)/((e*x + d)^4*e
^3)

Mupad [B] (verification not implemented)

Time = 9.88 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.10 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2}{(d+e x)^7} \, dx=-\frac {\frac {a^2\,e}{4}-d\,\left (\frac {c^2\,x^3}{3}-\frac {2\,a\,c\,x}{3}\right )-\frac {c^2\,e\,x^4}{12}+\frac {a\,c\,d^2}{6\,e}}{d^4+4\,d^3\,e\,x+6\,d^2\,e^2\,x^2+4\,d\,e^3\,x^3+e^4\,x^4} \]

[In]

int((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^2/(d + e*x)^7,x)

[Out]

-((a^2*e)/4 - d*((c^2*x^3)/3 - (2*a*c*x)/3) - (c^2*e*x^4)/12 + (a*c*d^2)/(6*e))/(d^4 + e^4*x^4 + 4*d*e^3*x^3 +
6*d^2*e^2*x^2 + 4*d^3*e*x)