\(\int \frac {(a d e+(c d^2+a e^2) x+c d e x^2)^3}{(d+e x)^6} \, dx\) [1859]

Optimal result

Integrand size = 35, antiderivative size = 97 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3}{(d+e x)^6} \, dx=\frac {c^3 d^3 x}{e^3}+\frac {\left (c d^2-a e^2\right )^3}{2 e^4 (d+e x)^2}-\frac {3 c d \left (c d^2-a e^2\right )^2}{e^4 (d+e x)}-\frac {3 c^2 d^2 \left (c d^2-a e^2\right ) \log (d+e x)}{e^4} \]

[Out]

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

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 97, 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} \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3}{(d+e x)^6} \, dx=-\frac {3 c^2 d^2 \left (c d^2-a e^2\right ) \log (d+e x)}{e^4}-\frac {3 c d \left (c d^2-a e^2\right )^2}{e^4 (d+e x)}+\frac {\left (c d^2-a e^2\right )^3}{2 e^4 (d+e x)^2}+\frac {c^3 d^3 x}{e^3} \]

[In]

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

[Out]

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

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

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.33 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3}{(d+e x)^6} \, dx=\frac {-a^3 e^6-3 a^2 c d e^4 (d+2 e x)+3 a c^2 d^3 e^2 (3 d+4 e x)+c^3 d^3 \left (-5 d^3-4 d^2 e x+4 d e^2 x^2+2 e^3 x^3\right )-6 c^2 d^2 \left (c d^2-a e^2\right ) (d+e x)^2 \log (d+e x)}{2 e^4 (d+e x)^2} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 2.35 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.37

method	result	size
default	\(\frac {c^{3} d^{3} x}{e^{3}}-\frac {3 c d \left (a^{2} e^{4}-2 a c \,d^{2} e^{2}+c^{2} d^{4}\right )}{e^{4} \left (e x +d \right )}-\frac {e^{6} a^{3}-3 d^{2} e^{4} a^{2} c +3 d^{4} e^{2} c^{2} a -c^{3} d^{6}}{2 e^{4} \left (e x +d \right )^{2}}+\frac {3 c^{2} d^{2} \left (e^{2} a -c \,d^{2}\right ) \ln \left (e x +d \right )}{e^{4}}\)	\(133\)
risch	\(\frac {c^{3} d^{3} x}{e^{3}}+\frac {\left (-3 d \,e^{4} a^{2} c +6 d^{3} e^{2} c^{2} a -3 d^{5} c^{3}\right ) x -\frac {e^{6} a^{3}+3 d^{2} e^{4} a^{2} c -9 d^{4} e^{2} c^{2} a +5 c^{3} d^{6}}{2 e}}{e^{3} \left (e x +d \right )^{2}}+\frac {3 c^{2} d^{2} \ln \left (e x +d \right ) a}{e^{2}}-\frac {3 c^{3} d^{4} \ln \left (e x +d \right )}{e^{4}}\)	\(138\)
parallelrisch	\(\frac {6 \ln \left (e x +d \right ) x^{2} a \,c^{2} d^{2} e^{4}-6 \ln \left (e x +d \right ) x^{2} c^{3} d^{4} e^{2}+2 x^{3} c^{3} d^{3} e^{3}+12 \ln \left (e x +d \right ) x a \,c^{2} d^{3} e^{3}-12 \ln \left (e x +d \right ) x \,c^{3} d^{5} e +6 \ln \left (e x +d \right ) a \,c^{2} d^{4} e^{2}-6 \ln \left (e x +d \right ) c^{3} d^{6}-6 x \,a^{2} c d \,e^{5}+12 x a \,c^{2} d^{3} e^{3}-12 x \,c^{3} d^{5} e -e^{6} a^{3}-3 d^{2} e^{4} a^{2} c +9 d^{4} e^{2} c^{2} a -9 c^{3} d^{6}}{2 e^{4} \left (e x +d \right )^{2}}\)	\(210\)
norman	\(\frac {e^{2} c^{3} d^{3} x^{6}-\frac {d^{3} \left (a^{3} e^{7}+3 d^{2} e^{5} a^{2} c -9 a \,c^{2} d^{4} e^{3}+15 c^{3} d^{6} e \right )}{2 e^{5}}-\frac {\left (a^{3} e^{7}+21 d^{2} e^{5} a^{2} c -45 a \,c^{2} d^{4} e^{3}+103 c^{3} d^{6} e \right ) x^{3}}{2 e^{2}}-\frac {d \left (3 a^{2} c \,e^{5}-6 e^{3} a \,c^{2} d^{2}+18 d^{4} e \,c^{3}\right ) x^{4}}{e}-\frac {d \left (3 a^{3} e^{7}+27 d^{2} e^{5} a^{2} c -63 a \,c^{2} d^{4} e^{3}+123 c^{3} d^{6} e \right ) x^{2}}{2 e^{3}}-\frac {d^{2} \left (3 a^{3} e^{7}+15 d^{2} e^{5} a^{2} c -39 a \,c^{2} d^{4} e^{3}+69 c^{3} d^{6} e \right ) x}{2 e^{4}}}{\left (e x +d \right )^{5}}+\frac {3 c^{2} d^{2} \left (e^{2} a -c \,d^{2}\right ) \ln \left (e x +d \right )}{e^{4}}\)	\(293\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 209 vs. \(2 (95) = 190\).

Time = 0.27 (sec) , antiderivative size = 209, normalized size of antiderivative = 2.15 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3}{(d+e x)^6} \, dx=\frac {2 \, c^{3} d^{3} e^{3} x^{3} + 4 \, c^{3} d^{4} e^{2} x^{2} - 5 \, c^{3} d^{6} + 9 \, a c^{2} d^{4} e^{2} - 3 \, a^{2} c d^{2} e^{4} - a^{3} e^{6} - 2 \, {\left (2 \, c^{3} d^{5} e - 6 \, a c^{2} d^{3} e^{3} + 3 \, a^{2} c d e^{5}\right )} x - 6 \, {\left (c^{3} d^{6} - a c^{2} d^{4} e^{2} + {\left (c^{3} d^{4} e^{2} - a c^{2} d^{2} e^{4}\right )} x^{2} + 2 \, {\left (c^{3} d^{5} e - a c^{2} d^{3} e^{3}\right )} x\right )} \log \left (e x + d\right )}{2 \, {\left (e^{6} x^{2} + 2 \, d e^{5} x + d^{2} e^{4}\right )}} \]

[In]

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

[Out]

1/2*(2*c^3*d^3*e^3*x^3 + 4*c^3*d^4*e^2*x^2 - 5*c^3*d^6 + 9*a*c^2*d^4*e^2 - 3*a^2*c*d^2*e^4 - a^3*e^6 - 2*(2*c^
3*d^5*e - 6*a*c^2*d^3*e^3 + 3*a^2*c*d*e^5)*x - 6*(c^3*d^6 - a*c^2*d^4*e^2 + (c^3*d^4*e^2 - a*c^2*d^2*e^4)*x^2
+ 2*(c^3*d^5*e - a*c^2*d^3*e^3)*x)*log(e*x + d))/(e^6*x^2 + 2*d*e^5*x + d^2*e^4)

Sympy [A] (verification not implemented)

Time = 0.75 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.48 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3}{(d+e x)^6} \, dx=\frac {c^{3} d^{3} x}{e^{3}} + \frac {3 c^{2} d^{2} \left (a e^{2} - c d^{2}\right ) \log {\left (d + e x \right )}}{e^{4}} + \frac {- a^{3} e^{6} - 3 a^{2} c d^{2} e^{4} + 9 a c^{2} d^{4} e^{2} - 5 c^{3} d^{6} + x \left (- 6 a^{2} c d e^{5} + 12 a c^{2} d^{3} e^{3} - 6 c^{3} d^{5} e\right )}{2 d^{2} e^{4} + 4 d e^{5} x + 2 e^{6} x^{2}} \]

[In]

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

[Out]

c**3*d**3*x/e**3 + 3*c**2*d**2*(a*e**2 - c*d**2)*log(d + e*x)/e**4 + (-a**3*e**6 - 3*a**2*c*d**2*e**4 + 9*a*c*
*2*d**4*e**2 - 5*c**3*d**6 + x*(-6*a**2*c*d*e**5 + 12*a*c**2*d**3*e**3 - 6*c**3*d**5*e))/(2*d**2*e**4 + 4*d*e*
*5*x + 2*e**6*x**2)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.46 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3}{(d+e x)^6} \, dx=\frac {c^{3} d^{3} x}{e^{3}} - \frac {5 \, c^{3} d^{6} - 9 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} + a^{3} e^{6} + 6 \, {\left (c^{3} d^{5} e - 2 \, a c^{2} d^{3} e^{3} + a^{2} c d e^{5}\right )} x}{2 \, {\left (e^{6} x^{2} + 2 \, d e^{5} x + d^{2} e^{4}\right )}} - \frac {3 \, {\left (c^{3} d^{4} - a c^{2} d^{2} e^{2}\right )} \log \left (e x + d\right )}{e^{4}} \]

[In]

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

[Out]

c^3*d^3*x/e^3 - 1/2*(5*c^3*d^6 - 9*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 + a^3*e^6 + 6*(c^3*d^5*e - 2*a*c^2*d^3*e^3
+ a^2*c*d*e^5)*x)/(e^6*x^2 + 2*d*e^5*x + d^2*e^4) - 3*(c^3*d^4 - a*c^2*d^2*e^2)*log(e*x + d)/e^4

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.33 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3}{(d+e x)^6} \, dx=\frac {c^{3} d^{3} x}{e^{3}} - \frac {3 \, {\left (c^{3} d^{4} - a c^{2} d^{2} e^{2}\right )} \log \left ({\left | e x + d \right |}\right )}{e^{4}} - \frac {5 \, c^{3} d^{6} - 9 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} + a^{3} e^{6} + 6 \, {\left (c^{3} d^{5} e - 2 \, a c^{2} d^{3} e^{3} + a^{2} c d e^{5}\right )} x}{2 \, {\left (e x + d\right )}^{2} e^{4}} \]

[In]

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

[Out]

c^3*d^3*x/e^3 - 3*(c^3*d^4 - a*c^2*d^2*e^2)*log(abs(e*x + d))/e^4 - 1/2*(5*c^3*d^6 - 9*a*c^2*d^4*e^2 + 3*a^2*c
*d^2*e^4 + a^3*e^6 + 6*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)*x)/((e*x + d)^2*e^4)

Mupad [B] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.54 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3}{(d+e x)^6} \, dx=\frac {c^3\,d^3\,x}{e^3}-\frac {\ln \left (d+e\,x\right )\,\left (3\,c^3\,d^4-3\,a\,c^2\,d^2\,e^2\right )}{e^4}-\frac {\frac {a^3\,e^6+3\,a^2\,c\,d^2\,e^4-9\,a\,c^2\,d^4\,e^2+5\,c^3\,d^6}{2\,e}+x\,\left (3\,a^2\,c\,d\,e^4-6\,a\,c^2\,d^3\,e^2+3\,c^3\,d^5\right )}{d^2\,e^3+2\,d\,e^4\,x+e^5\,x^2} \]

[In]

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

[Out]

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