\(\int \frac {(d+e x)^6}{(a d e+(c d^2+a e^2) x+c d e x^2)^3} \, dx\) [1888]

Optimal result

Integrand size = 35, antiderivative size = 111 \[ \int \frac {(d+e x)^6}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3} \, dx=\frac {e^3 x}{c^3 d^3}-\frac {\left (c d^2-a e^2\right )^3}{2 c^4 d^4 (a e+c d x)^2}-\frac {3 e \left (c d^2-a e^2\right )^2}{c^4 d^4 (a e+c d x)}+\frac {3 e^2 \left (c d^2-a e^2\right ) \log (a e+c d x)}{c^4 d^4} \]

[Out]

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

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 111, 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} \[ \int \frac {(d+e x)^6}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3} \, dx=-\frac {3 e \left (c d^2-a e^2\right )^2}{c^4 d^4 (a e+c d x)}-\frac {\left (c d^2-a e^2\right )^3}{2 c^4 d^4 (a e+c d x)^2}+\frac {3 e^2 \left (c d^2-a e^2\right ) \log (a e+c d x)}{c^4 d^4}+\frac {e^3 x}{c^3 d^3} \]

[In]

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

[Out]

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

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.25 \[ \int \frac {(d+e x)^6}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3} \, dx=\frac {-5 a^3 e^6+a^2 c d e^4 (9 d-4 e x)+a c^2 d^2 e^2 \left (-3 d^2+12 d e x+4 e^2 x^2\right )-c^3 \left (d^6+6 d^5 e x-2 d^3 e^3 x^3\right )-6 e^2 \left (-c d^2+a e^2\right ) (a e+c d x)^2 \log (a e+c d x)}{2 c^4 d^4 (a e+c d x)^2} \]

[In]

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

[Out]

(-5*a^3*e^6 + a^2*c*d*e^4*(9*d - 4*e*x) + a*c^2*d^2*e^2*(-3*d^2 + 12*d*e*x + 4*e^2*x^2) - c^3*(d^6 + 6*d^5*e*x
- 2*d^3*e^3*x^3) - 6*e^2*(-(c*d^2) + a*e^2)*(a*e + c*d*x)^2*Log[a*e + c*d*x])/(2*c^4*d^4*(a*e + c*d*x)^2)

Maple [A] (verified)

Time = 2.87 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.32

method	result	size
default	\(\frac {e^{3} x}{c^{3} d^{3}}-\frac {-e^{6} a^{3}+3 d^{2} e^{4} a^{2} c -3 d^{4} e^{2} c^{2} a +c^{3} d^{6}}{2 c^{4} d^{4} \left (c d x +a e \right )^{2}}-\frac {3 e^{2} \left (e^{2} a -c \,d^{2}\right ) \ln \left (c d x +a e \right )}{c^{4} d^{4}}-\frac {3 e \left (a^{2} e^{4}-2 a c \,d^{2} e^{2}+c^{2} d^{4}\right )}{c^{4} d^{4} \left (c d x +a e \right )}\)	\(147\)
risch	\(\frac {e^{3} x}{c^{3} d^{3}}+\frac {\left (-3 a^{2} e^{5}+6 a \,d^{2} e^{3} c -3 d^{4} e \,c^{2}\right ) x -\frac {5 e^{6} a^{3}-9 d^{2} e^{4} a^{2} c +3 d^{4} e^{2} c^{2} a +c^{3} d^{6}}{2 c d}}{c^{3} d^{3} \left (c d x +a e \right )^{2}}-\frac {3 e^{4} \ln \left (c d x +a e \right ) a}{c^{4} d^{4}}+\frac {3 e^{2} \ln \left (c d x +a e \right )}{c^{3} d^{2}}\)	\(150\)
parallelrisch	\(-\frac {6 \ln \left (c d x +a e \right ) x^{2} a \,c^{2} d^{2} e^{4}-6 \ln \left (c d x +a e \right ) x^{2} c^{3} d^{4} e^{2}-2 x^{3} c^{3} d^{3} e^{3}+12 \ln \left (c d x +a e \right ) x \,a^{2} c d \,e^{5}-12 \ln \left (c d x +a e \right ) x a \,c^{2} d^{3} e^{3}+6 \ln \left (c d x +a e \right ) a^{3} e^{6}-6 \ln \left (c d x +a e \right ) a^{2} c \,d^{2} e^{4}+12 x \,a^{2} c d \,e^{5}-12 x a \,c^{2} d^{3} e^{3}+6 x \,c^{3} d^{5} e +9 e^{6} a^{3}-9 d^{2} e^{4} a^{2} c +3 d^{4} e^{2} c^{2} a +c^{3} d^{6}}{2 c^{4} d^{4} \left (c d x +a e \right )^{2}}\)	\(234\)
norman	\(\frac {\frac {e^{5} x^{5}}{c d}-\frac {9 e^{6} a^{3}-5 d^{2} e^{4} a^{2} c +3 d^{4} e^{2} c^{2} a +c^{3} d^{6}}{2 c^{4} d^{2}}-\frac {\left (9 a^{3} e^{10}+19 a^{2} c \,d^{2} e^{8}-5 d^{4} c^{2} a \,e^{6}+17 c^{3} d^{6} e^{4}\right ) x^{2}}{2 c^{4} d^{4} e^{2}}-\frac {\left (9 a^{3} e^{8}+a^{2} c \,d^{2} e^{6}+a \,c^{2} d^{4} e^{4}+4 c^{3} d^{6} e^{2}\right ) x}{c^{4} d^{3} e}-\frac {2 \left (3 a^{2} e^{8}-a c \,d^{2} e^{6}+3 d^{4} e^{4} c^{2}\right ) x^{3}}{c^{3} d^{3} e}}{\left (c d x +a e \right )^{2} \left (e x +d \right )^{2}}-\frac {3 e^{2} \left (e^{2} a -c \,d^{2}\right ) \ln \left (c d x +a e \right )}{c^{4} d^{4}}\)	\(270\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 227 vs. \(2 (109) = 218\).

Time = 0.31 (sec) , antiderivative size = 227, normalized size of antiderivative = 2.05 \[ \int \frac {(d+e x)^6}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3} \, dx=\frac {2 \, c^{3} d^{3} e^{3} x^{3} + 4 \, a c^{2} d^{2} e^{4} x^{2} - c^{3} d^{6} - 3 \, a c^{2} d^{4} e^{2} + 9 \, a^{2} c d^{2} e^{4} - 5 \, a^{3} e^{6} - 2 \, {\left (3 \, c^{3} d^{5} e - 6 \, a c^{2} d^{3} e^{3} + 2 \, a^{2} c d e^{5}\right )} x + 6 \, {\left (a^{2} c d^{2} e^{4} - a^{3} e^{6} + {\left (c^{3} d^{4} e^{2} - a c^{2} d^{2} e^{4}\right )} x^{2} + 2 \, {\left (a c^{2} d^{3} e^{3} - a^{2} c d e^{5}\right )} x\right )} \log \left (c d x + a e\right )}{2 \, {\left (c^{6} d^{6} x^{2} + 2 \, a c^{5} d^{5} e x + a^{2} c^{4} d^{4} e^{2}\right )}} \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.77 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.47 \[ \int \frac {(d+e x)^6}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3} \, dx=\frac {- 5 a^{3} e^{6} + 9 a^{2} c d^{2} e^{4} - 3 a c^{2} d^{4} e^{2} - c^{3} d^{6} + x \left (- 6 a^{2} c d e^{5} + 12 a c^{2} d^{3} e^{3} - 6 c^{3} d^{5} e\right )}{2 a^{2} c^{4} d^{4} e^{2} + 4 a c^{5} d^{5} e x + 2 c^{6} d^{6} x^{2}} + \frac {e^{3} x}{c^{3} d^{3}} - \frac {3 e^{2} \left (a e^{2} - c d^{2}\right ) \log {\left (a e + c d x \right )}}{c^{4} d^{4}} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.41 \[ \int \frac {(d+e x)^6}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3} \, dx=-\frac {c^{3} d^{6} + 3 \, a c^{2} d^{4} e^{2} - 9 \, a^{2} c d^{2} e^{4} + 5 \, a^{3} e^{6} + 6 \, {\left (c^{3} d^{5} e - 2 \, a c^{2} d^{3} e^{3} + a^{2} c d e^{5}\right )} x}{2 \, {\left (c^{6} d^{6} x^{2} + 2 \, a c^{5} d^{5} e x + a^{2} c^{4} d^{4} e^{2}\right )}} + \frac {e^{3} x}{c^{3} d^{3}} + \frac {3 \, {\left (c d^{2} e^{2} - a e^{4}\right )} \log \left (c d x + a e\right )}{c^{4} d^{4}} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.23 \[ \int \frac {(d+e x)^6}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3} \, dx=\frac {e^{3} x}{c^{3} d^{3}} + \frac {3 \, {\left (c d^{2} e^{2} - a e^{4}\right )} \log \left ({\left | c d x + a e \right |}\right )}{c^{4} d^{4}} - \frac {c^{3} d^{6} + 3 \, a c^{2} d^{4} e^{2} - 9 \, a^{2} c d^{2} e^{4} + 5 \, a^{3} e^{6} + 6 \, {\left (c^{3} d^{5} e - 2 \, a c^{2} d^{3} e^{3} + a^{2} c d e^{5}\right )} x}{2 \, {\left (c d x + a e\right )}^{2} c^{4} d^{4}} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 9.95 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.47 \[ \int \frac {(d+e x)^6}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3} \, dx=\frac {e^3\,x}{c^3\,d^3}-\frac {x\,\left (3\,a^2\,e^5-6\,a\,c\,d^2\,e^3+3\,c^2\,d^4\,e\right )+\frac {5\,a^3\,e^6-9\,a^2\,c\,d^2\,e^4+3\,a\,c^2\,d^4\,e^2+c^3\,d^6}{2\,c\,d}}{a^2\,c^3\,d^3\,e^2+2\,a\,c^4\,d^4\,e\,x+c^5\,d^5\,x^2}-\frac {\ln \left (a\,e+c\,d\,x\right )\,\left (3\,a\,e^4-3\,c\,d^2\,e^2\right )}{c^4\,d^4} \]

[In]

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

[Out]

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