\(\int \frac {(d+e x)^6}{(a d e+(c d^2+a e^2) x+c d e x^2)^2} \, dx\) [1876]

Optimal result

Integrand size = 35, antiderivative size = 145 \[ \int \frac {(d+e x)^6}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx=\frac {e^2 \left (6 c^2 d^4-8 a c d^2 e^2+3 a^2 e^4\right ) x}{c^4 d^4}+\frac {e^3 \left (2 c d^2-a e^2\right ) x^2}{c^3 d^3}+\frac {e^4 x^3}{3 c^2 d^2}-\frac {\left (c d^2-a e^2\right )^4}{c^5 d^5 (a e+c d x)}+\frac {4 e \left (c d^2-a e^2\right )^3 \log (a e+c d x)}{c^5 d^5} \]

[Out]

e^2*(3*a^2*e^4-8*a*c*d^2*e^2+6*c^2*d^4)*x/c^4/d^4+e^3*(-a*e^2+2*c*d^2)*x^2/c^3/d^3+1/3*e^4*x^3/c^2/d^2-(-a*e^2
+c*d^2)^4/c^5/d^5/(c*d*x+a*e)+4*e*(-a*e^2+c*d^2)^3*ln(c*d*x+a*e)/c^5/d^5

Rubi [A] (verified)

Time = 0.11 (sec) , antiderivative size = 145, 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} \[ \int \frac {(d+e x)^6}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx=\frac {e^2 x \left (3 a^2 e^4-8 a c d^2 e^2+6 c^2 d^4\right )}{c^4 d^4}-\frac {\left (c d^2-a e^2\right )^4}{c^5 d^5 (a e+c d x)}+\frac {4 e \left (c d^2-a e^2\right )^3 \log (a e+c d x)}{c^5 d^5}+\frac {e^3 x^2 \left (2 c d^2-a e^2\right )}{c^3 d^3}+\frac {e^4 x^3}{3 c^2 d^2} \]

[In]

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

[Out]

(e^2*(6*c^2*d^4 - 8*a*c*d^2*e^2 + 3*a^2*e^4)*x)/(c^4*d^4) + (e^3*(2*c*d^2 - a*e^2)*x^2)/(c^3*d^3) + (e^4*x^3)/
(3*c^2*d^2) - (c*d^2 - a*e^2)^4/(c^5*d^5*(a*e + c*d*x)) + (4*e*(c*d^2 - a*e^2)^3*Log[a*e + c*d*x])/(c^5*d^5)

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)^4}{(a e+c d x)^2} \, dx \\ & = \int \left (\frac {6 c^2 d^4 e^2-8 a c d^2 e^4+3 a^2 e^6}{c^4 d^4}+\frac {2 e^3 \left (2 c d^2-a e^2\right ) x}{c^3 d^3}+\frac {e^4 x^2}{c^2 d^2}+\frac {\left (c d^2-a e^2\right )^4}{c^4 d^4 (a e+c d x)^2}+\frac {4 e \left (c d^2-a e^2\right )^3}{c^4 d^4 (a e+c d x)}\right ) \, dx \\ & = \frac {e^2 \left (6 c^2 d^4-8 a c d^2 e^2+3 a^2 e^4\right ) x}{c^4 d^4}+\frac {e^3 \left (2 c d^2-a e^2\right ) x^2}{c^3 d^3}+\frac {e^4 x^3}{3 c^2 d^2}-\frac {\left (c d^2-a e^2\right )^4}{c^5 d^5 (a e+c d x)}+\frac {4 e \left (c d^2-a e^2\right )^3 \log (a e+c d x)}{c^5 d^5} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.35 \[ \int \frac {(d+e x)^6}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx=\frac {-3 a^4 e^8+3 a^3 c d e^6 (4 d+3 e x)-6 a^2 c^2 d^2 e^4 \left (3 d^2+4 d e x-e^2 x^2\right )+2 a c^3 d^3 e^2 \left (6 d^3+9 d^2 e x-9 d e^2 x^2-e^3 x^3\right )+c^4 d^4 \left (-3 d^4+18 d^2 e^2 x^2+6 d e^3 x^3+e^4 x^4\right )-12 e \left (-c d^2+a e^2\right )^3 (a e+c d x) \log (a e+c d x)}{3 c^5 d^5 (a e+c d x)} \]

[In]

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

[Out]

(-3*a^4*e^8 + 3*a^3*c*d*e^6*(4*d + 3*e*x) - 6*a^2*c^2*d^2*e^4*(3*d^2 + 4*d*e*x - e^2*x^2) + 2*a*c^3*d^3*e^2*(6
*d^3 + 9*d^2*e*x - 9*d*e^2*x^2 - e^3*x^3) + c^4*d^4*(-3*d^4 + 18*d^2*e^2*x^2 + 6*d*e^3*x^3 + e^4*x^4) - 12*e*(
-(c*d^2) + a*e^2)^3*(a*e + c*d*x)*Log[a*e + c*d*x])/(3*c^5*d^5*(a*e + c*d*x))

Maple [A] (verified)

Time = 2.40 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.43

method	result	size
default	\(\frac {e^{2} \left (\frac {1}{3} x^{3} c^{2} d^{2} e^{2}-x^{2} a c d \,e^{3}+2 x^{2} c^{2} d^{3} e +3 a^{2} e^{4} x -8 a c \,d^{2} e^{2} x +6 c^{2} d^{4} x \right )}{c^{4} d^{4}}-\frac {4 e \left (e^{6} a^{3}-3 d^{2} e^{4} a^{2} c +3 d^{4} e^{2} c^{2} a -c^{3} d^{6}\right ) \ln \left (c d x +a e \right )}{c^{5} d^{5}}-\frac {a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}}{c^{5} d^{5} \left (c d x +a e \right )}\)	\(208\)
risch	\(\frac {e^{4} x^{3}}{3 c^{2} d^{2}}-\frac {e^{5} x^{2} a}{c^{3} d^{3}}+\frac {2 e^{3} x^{2}}{c^{2} d}+\frac {3 e^{6} a^{2} x}{c^{4} d^{4}}-\frac {8 e^{4} a x}{c^{3} d^{2}}+\frac {6 e^{2} x}{c^{2}}-\frac {4 e^{7} \ln \left (c d x +a e \right ) a^{3}}{c^{5} d^{5}}+\frac {12 e^{5} \ln \left (c d x +a e \right ) a^{2}}{c^{4} d^{3}}-\frac {12 e^{3} \ln \left (c d x +a e \right ) a}{c^{3} d}+\frac {4 d e \ln \left (c d x +a e \right )}{c^{2}}-\frac {a^{4} e^{8}}{c^{5} d^{5} \left (c d x +a e \right )}+\frac {4 a^{3} e^{6}}{c^{4} d^{3} \left (c d x +a e \right )}-\frac {6 a^{2} e^{4}}{c^{3} d \left (c d x +a e \right )}+\frac {4 d a \,e^{2}}{c^{2} \left (c d x +a e \right )}-\frac {d^{3}}{c \left (c d x +a e \right )}\)	\(275\)
norman	\(\frac {-\frac {4 a^{4} e^{8}-10 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}+2 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}}{c^{5} d^{4}}+\frac {e^{5} x^{5}}{3 c d}-\frac {\left (4 a^{4} e^{10}-10 a^{3} c \,d^{2} e^{8}+8 a^{2} c^{2} d^{4} e^{6}-4 a \,c^{3} d^{6} e^{4}+7 c^{4} d^{8} e^{2}\right ) x}{c^{5} d^{5} e}+\frac {2 e^{3} \left (3 a^{2} e^{4}-10 a c \,d^{2} e^{2}+12 c^{2} d^{4}\right ) x^{3}}{3 c^{3} d^{3}}-\frac {e^{4} \left (2 e^{2} a -7 c \,d^{2}\right ) x^{4}}{3 c^{2} d^{2}}}{\left (c d x +a e \right ) \left (e x +d \right )}-\frac {4 e \left (e^{6} a^{3}-3 d^{2} e^{4} a^{2} c +3 d^{4} e^{2} c^{2} a -c^{3} d^{6}\right ) \ln \left (c d x +a e \right )}{c^{5} d^{5}}\)	\(293\)
parallelrisch	\(-\frac {-c^{4} d^{4} e^{4} x^{4}+2 a \,c^{3} d^{3} e^{5} x^{3}-6 c^{4} d^{5} e^{3} x^{3}+12 \ln \left (c d x +a e \right ) x \,a^{3} c d \,e^{7}-36 \ln \left (c d x +a e \right ) x \,a^{2} c^{2} d^{3} e^{5}+36 \ln \left (c d x +a e \right ) x a \,c^{3} d^{5} e^{3}-12 \ln \left (c d x +a e \right ) x \,c^{4} d^{7} e -6 a^{2} c^{2} d^{2} e^{6} x^{2}+18 a \,c^{3} d^{4} e^{4} x^{2}-18 c^{4} d^{6} e^{2} x^{2}+12 \ln \left (c d x +a e \right ) a^{4} e^{8}-36 \ln \left (c d x +a e \right ) a^{3} c \,d^{2} e^{6}+36 \ln \left (c d x +a e \right ) a^{2} c^{2} d^{4} e^{4}-12 \ln \left (c d x +a e \right ) a \,c^{3} d^{6} e^{2}+12 a^{4} e^{8}-36 a^{3} c \,d^{2} e^{6}+36 a^{2} c^{2} d^{4} e^{4}-12 a \,c^{3} d^{6} e^{2}+3 c^{4} d^{8}}{3 c^{5} d^{5} \left (c d x +a e \right )}\)	\(330\)

[In]

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

[Out]

e^2/c^4/d^4*(1/3*x^3*c^2*d^2*e^2-x^2*a*c*d*e^3+2*x^2*c^2*d^3*e+3*a^2*e^4*x-8*a*c*d^2*e^2*x+6*c^2*d^4*x)-4/c^5/
d^5*e*(a^3*e^6-3*a^2*c*d^2*e^4+3*a*c^2*d^4*e^2-c^3*d^6)*ln(c*d*x+a*e)-(a^4*e^8-4*a^3*c*d^2*e^6+6*a^2*c^2*d^4*e
^4-4*a*c^3*d^6*e^2+c^4*d^8)/c^5/d^5/(c*d*x+a*e)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 305 vs. \(2 (143) = 286\).

Time = 0.29 (sec) , antiderivative size = 305, normalized size of antiderivative = 2.10 \[ \int \frac {(d+e x)^6}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx=\frac {c^{4} d^{4} e^{4} x^{4} - 3 \, c^{4} d^{8} + 12 \, a c^{3} d^{6} e^{2} - 18 \, a^{2} c^{2} d^{4} e^{4} + 12 \, a^{3} c d^{2} e^{6} - 3 \, a^{4} e^{8} + 2 \, {\left (3 \, c^{4} d^{5} e^{3} - a c^{3} d^{3} e^{5}\right )} x^{3} + 6 \, {\left (3 \, c^{4} d^{6} e^{2} - 3 \, a c^{3} d^{4} e^{4} + a^{2} c^{2} d^{2} e^{6}\right )} x^{2} + 3 \, {\left (6 \, a c^{3} d^{5} e^{3} - 8 \, a^{2} c^{2} d^{3} e^{5} + 3 \, a^{3} c d e^{7}\right )} x + 12 \, {\left (a c^{3} d^{6} e^{2} - 3 \, a^{2} c^{2} d^{4} e^{4} + 3 \, a^{3} c d^{2} e^{6} - a^{4} e^{8} + {\left (c^{4} d^{7} e - 3 \, a c^{3} d^{5} e^{3} + 3 \, a^{2} c^{2} d^{3} e^{5} - a^{3} c d e^{7}\right )} x\right )} \log \left (c d x + a e\right )}{3 \, {\left (c^{6} d^{6} x + a c^{5} d^{5} e\right )}} \]

[In]

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

[Out]

1/3*(c^4*d^4*e^4*x^4 - 3*c^4*d^8 + 12*a*c^3*d^6*e^2 - 18*a^2*c^2*d^4*e^4 + 12*a^3*c*d^2*e^6 - 3*a^4*e^8 + 2*(3
*c^4*d^5*e^3 - a*c^3*d^3*e^5)*x^3 + 6*(3*c^4*d^6*e^2 - 3*a*c^3*d^4*e^4 + a^2*c^2*d^2*e^6)*x^2 + 3*(6*a*c^3*d^5
*e^3 - 8*a^2*c^2*d^3*e^5 + 3*a^3*c*d*e^7)*x + 12*(a*c^3*d^6*e^2 - 3*a^2*c^2*d^4*e^4 + 3*a^3*c*d^2*e^6 - a^4*e^
8 + (c^4*d^7*e - 3*a*c^3*d^5*e^3 + 3*a^2*c^2*d^3*e^5 - a^3*c*d*e^7)*x)*log(c*d*x + a*e))/(c^6*d^6*x + a*c^5*d^
5*e)

Sympy [A] (verification not implemented)

Time = 0.51 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.28 \[ \int \frac {(d+e x)^6}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx=x^{2} \left (- \frac {a e^{5}}{c^{3} d^{3}} + \frac {2 e^{3}}{c^{2} d}\right ) + x \left (\frac {3 a^{2} e^{6}}{c^{4} d^{4}} - \frac {8 a e^{4}}{c^{3} d^{2}} + \frac {6 e^{2}}{c^{2}}\right ) + \frac {- a^{4} e^{8} + 4 a^{3} c d^{2} e^{6} - 6 a^{2} c^{2} d^{4} e^{4} + 4 a c^{3} d^{6} e^{2} - c^{4} d^{8}}{a c^{5} d^{5} e + c^{6} d^{6} x} + \frac {e^{4} x^{3}}{3 c^{2} d^{2}} - \frac {4 e \left (a e^{2} - c d^{2}\right )^{3} \log {\left (a e + c d x \right )}}{c^{5} d^{5}} \]

[In]

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

[Out]

x**2*(-a*e**5/(c**3*d**3) + 2*e**3/(c**2*d)) + x*(3*a**2*e**6/(c**4*d**4) - 8*a*e**4/(c**3*d**2) + 6*e**2/c**2
) + (-a**4*e**8 + 4*a**3*c*d**2*e**6 - 6*a**2*c**2*d**4*e**4 + 4*a*c**3*d**6*e**2 - c**4*d**8)/(a*c**5*d**5*e
+ c**6*d**6*x) + e**4*x**3/(3*c**2*d**2) - 4*e*(a*e**2 - c*d**2)**3*log(a*e + c*d*x)/(c**5*d**5)

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.48 \[ \int \frac {(d+e x)^6}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx=-\frac {c^{4} d^{8} - 4 \, a c^{3} d^{6} e^{2} + 6 \, a^{2} c^{2} d^{4} e^{4} - 4 \, a^{3} c d^{2} e^{6} + a^{4} e^{8}}{c^{6} d^{6} x + a c^{5} d^{5} e} + \frac {c^{2} d^{2} e^{4} x^{3} + 3 \, {\left (2 \, c^{2} d^{3} e^{3} - a c d e^{5}\right )} x^{2} + 3 \, {\left (6 \, c^{2} d^{4} e^{2} - 8 \, a c d^{2} e^{4} + 3 \, a^{2} e^{6}\right )} x}{3 \, c^{4} d^{4}} + \frac {4 \, {\left (c^{3} d^{6} e - 3 \, a c^{2} d^{4} e^{3} + 3 \, a^{2} c d^{2} e^{5} - a^{3} e^{7}\right )} \log \left (c d x + a e\right )}{c^{5} d^{5}} \]

[In]

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

[Out]

-(c^4*d^8 - 4*a*c^3*d^6*e^2 + 6*a^2*c^2*d^4*e^4 - 4*a^3*c*d^2*e^6 + a^4*e^8)/(c^6*d^6*x + a*c^5*d^5*e) + 1/3*(
c^2*d^2*e^4*x^3 + 3*(2*c^2*d^3*e^3 - a*c*d*e^5)*x^2 + 3*(6*c^2*d^4*e^2 - 8*a*c*d^2*e^4 + 3*a^2*e^6)*x)/(c^4*d^
4) + 4*(c^3*d^6*e - 3*a*c^2*d^4*e^3 + 3*a^2*c*d^2*e^5 - a^3*e^7)*log(c*d*x + a*e)/(c^5*d^5)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 222, normalized size of antiderivative = 1.53 \[ \int \frac {(d+e x)^6}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx=\frac {4 \, {\left (c^{3} d^{6} e - 3 \, a c^{2} d^{4} e^{3} + 3 \, a^{2} c d^{2} e^{5} - a^{3} e^{7}\right )} \log \left ({\left | c d x + a e \right |}\right )}{c^{5} d^{5}} - \frac {c^{4} d^{8} - 4 \, a c^{3} d^{6} e^{2} + 6 \, a^{2} c^{2} d^{4} e^{4} - 4 \, a^{3} c d^{2} e^{6} + a^{4} e^{8}}{{\left (c d x + a e\right )} c^{5} d^{5}} + \frac {c^{4} d^{4} e^{4} x^{3} + 6 \, c^{4} d^{5} e^{3} x^{2} - 3 \, a c^{3} d^{3} e^{5} x^{2} + 18 \, c^{4} d^{6} e^{2} x - 24 \, a c^{3} d^{4} e^{4} x + 9 \, a^{2} c^{2} d^{2} e^{6} x}{3 \, c^{6} d^{6}} \]

[In]

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

[Out]

4*(c^3*d^6*e - 3*a*c^2*d^4*e^3 + 3*a^2*c*d^2*e^5 - a^3*e^7)*log(abs(c*d*x + a*e))/(c^5*d^5) - (c^4*d^8 - 4*a*c
^3*d^6*e^2 + 6*a^2*c^2*d^4*e^4 - 4*a^3*c*d^2*e^6 + a^4*e^8)/((c*d*x + a*e)*c^5*d^5) + 1/3*(c^4*d^4*e^4*x^3 + 6
*c^4*d^5*e^3*x^2 - 3*a*c^3*d^3*e^5*x^2 + 18*c^4*d^6*e^2*x - 24*a*c^3*d^4*e^4*x + 9*a^2*c^2*d^2*e^6*x)/(c^6*d^6
)

Mupad [B] (verification not implemented)

Time = 9.76 (sec) , antiderivative size = 242, normalized size of antiderivative = 1.67 \[ \int \frac {(d+e x)^6}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx=x^2\,\left (\frac {2\,e^3}{c^2\,d}-\frac {a\,e^5}{c^3\,d^3}\right )-x\,\left (\frac {a^2\,e^6}{c^4\,d^4}-\frac {6\,e^2}{c^2}+\frac {2\,a\,e\,\left (\frac {4\,e^3}{c^2\,d}-\frac {2\,a\,e^5}{c^3\,d^3}\right )}{c\,d}\right )-\frac {\ln \left (a\,e+c\,d\,x\right )\,\left (4\,a^3\,e^7-12\,a^2\,c\,d^2\,e^5+12\,a\,c^2\,d^4\,e^3-4\,c^3\,d^6\,e\right )}{c^5\,d^5}-\frac {a^4\,e^8-4\,a^3\,c\,d^2\,e^6+6\,a^2\,c^2\,d^4\,e^4-4\,a\,c^3\,d^6\,e^2+c^4\,d^8}{c\,d\,\left (x\,c^5\,d^5+a\,e\,c^4\,d^4\right )}+\frac {e^4\,x^3}{3\,c^2\,d^2} \]

[In]

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

[Out]

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