\(\int \frac {(d+e x)^7}{(a d e+(c d^2+a e^2) x+c d e x^2)^4} \, dx\) [1899]

Optimal result

Integrand size = 35, antiderivative size = 122 \[ \int \frac {(d+e x)^7}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^4} \, dx=-\frac {\left (c d^2-a e^2\right )^3}{3 c^4 d^4 (a e+c d x)^3}-\frac {3 e \left (c d^2-a e^2\right )^2}{2 c^4 d^4 (a e+c d x)^2}-\frac {3 e^2 \left (c d^2-a e^2\right )}{c^4 d^4 (a e+c d x)}+\frac {e^3 \log (a e+c d x)}{c^4 d^4} \]

[Out]

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

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 122, 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} \[ \int \frac {(d+e x)^7}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^4} \, dx=\frac {e^3 \log (a e+c d x)}{c^4 d^4}-\frac {3 e^2 \left (c d^2-a e^2\right )}{c^4 d^4 (a e+c d x)}-\frac {3 e \left (c d^2-a e^2\right )^2}{2 c^4 d^4 (a e+c d x)^2}-\frac {\left (c d^2-a e^2\right )^3}{3 c^4 d^4 (a e+c d x)^3} \]

[In]

Int[(d + e*x)^7/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^4,x]

[Out]

-1/3*(c*d^2 - a*e^2)^3/(c^4*d^4*(a*e + c*d*x)^3) - (3*e*(c*d^2 - a*e^2)^2)/(2*c^4*d^4*(a*e + c*d*x)^2) - (3*e^
2*(c*d^2 - a*e^2))/(c^4*d^4*(a*e + c*d*x)) + (e^3*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)^4} \, dx \\ & = \int \left (\frac {\left (c d^2-a e^2\right )^3}{c^3 d^3 (a e+c d x)^4}+\frac {3 e \left (c d^2-a e^2\right )^2}{c^3 d^3 (a e+c d x)^3}+\frac {3 \left (c d^2 e^2-a e^4\right )}{c^3 d^3 (a e+c d x)^2}+\frac {e^3}{c^3 d^3 (a e+c d x)}\right ) \, dx \\ & = -\frac {\left (c d^2-a e^2\right )^3}{3 c^4 d^4 (a e+c d x)^3}-\frac {3 e \left (c d^2-a e^2\right )^2}{2 c^4 d^4 (a e+c d x)^2}-\frac {3 e^2 \left (c d^2-a e^2\right )}{c^4 d^4 (a e+c d x)}+\frac {e^3 \log (a e+c d x)}{c^4 d^4} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.81 \[ \int \frac {(d+e x)^7}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^4} \, dx=\frac {-\frac {\left (c d^2-a e^2\right ) \left (11 a^2 e^4+a c d e^2 (5 d+27 e x)+c^2 d^2 \left (2 d^2+9 d e x+18 e^2 x^2\right )\right )}{(a e+c d x)^3}+6 e^3 \log (a e+c d x)}{6 c^4 d^4} \]

[In]

Integrate[(d + e*x)^7/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^4,x]

[Out]

(-(((c*d^2 - a*e^2)*(11*a^2*e^4 + a*c*d*e^2*(5*d + 27*e*x) + c^2*d^2*(2*d^2 + 9*d*e*x + 18*e^2*x^2)))/(a*e + c
*d*x)^3) + 6*e^3*Log[a*e + c*d*x])/(6*c^4*d^4)

Maple [A] (verified)

Time = 2.90 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.19

method	result	size
risch	\(\frac {\frac {3 e^{2} \left (e^{2} a -c \,d^{2}\right ) x^{2}}{c^{2} d^{2}}+\frac {3 e \left (3 a^{2} e^{4}-2 a c \,d^{2} e^{2}-c^{2} d^{4}\right ) x}{2 c^{3} d^{3}}+\frac {11 e^{6} a^{3}-6 d^{2} e^{4} a^{2} c -3 d^{4} e^{2} c^{2} a -2 c^{3} d^{6}}{6 c^{4} d^{4}}}{\left (c d x +a e \right )^{3}}+\frac {e^{3} \ln \left (c d x +a e \right )}{c^{4} d^{4}}\)	\(145\)
default	\(-\frac {3 e \left (a^{2} e^{4}-2 a c \,d^{2} e^{2}+c^{2} d^{4}\right )}{2 c^{4} d^{4} \left (c d x +a e \right )^{2}}-\frac {-e^{6} a^{3}+3 d^{2} e^{4} a^{2} c -3 d^{4} e^{2} c^{2} a +c^{3} d^{6}}{3 c^{4} d^{4} \left (c d x +a e \right )^{3}}+\frac {e^{3} \ln \left (c d x +a e \right )}{c^{4} d^{4}}+\frac {3 e^{2} \left (e^{2} a -c \,d^{2}\right )}{c^{4} d^{4} \left (c d x +a e \right )}\)	\(156\)
parallelrisch	\(\frac {6 \ln \left (c d x +a e \right ) x^{3} c^{3} d^{3} e^{3}+18 \ln \left (c d x +a e \right ) x^{2} a \,c^{2} d^{2} e^{4}+18 \ln \left (c d x +a e \right ) x \,a^{2} c d \,e^{5}+18 x^{2} a \,c^{2} d^{2} e^{4}-18 x^{2} c^{3} d^{4} e^{2}+6 \ln \left (c d x +a e \right ) a^{3} e^{6}+27 x \,a^{2} c d \,e^{5}-18 x a \,c^{2} d^{3} e^{3}-9 x \,c^{3} d^{5} e +11 e^{6} a^{3}-6 d^{2} e^{4} a^{2} c -3 d^{4} e^{2} c^{2} a -2 c^{3} d^{6}}{6 c^{4} d^{4} \left (c d x +a e \right )^{3}}\)	\(207\)
norman	\(\frac {\frac {11 e^{6} a^{3}-6 d^{2} e^{4} a^{2} c -3 d^{4} e^{2} c^{2} a -2 c^{3} d^{6}}{6 c^{4} d}+\frac {\left (11 a^{3} e^{12}+75 c \,a^{2} d^{2} e^{10}-3 d^{4} c^{2} a \,e^{8}-83 d^{6} e^{6} c^{3}\right ) x^{3}}{6 c^{4} d^{4} e^{3}}+\frac {\left (11 a^{3} e^{10}+21 a^{2} c \,d^{2} e^{8}-15 a \,d^{4} e^{6} c^{2}-17 c^{3} d^{6} e^{4}\right ) x^{2}}{2 c^{4} d^{3} e^{2}}+\frac {\left (11 a^{3} e^{8}+3 a^{2} c \,d^{2} e^{6}-9 a \,c^{2} d^{4} e^{4}-5 c^{3} d^{6} e^{2}\right ) x}{2 c^{4} d^{2} e}+\frac {3 \left (3 a^{2} e^{10}+4 a c \,d^{2} e^{8}-7 c^{2} d^{4} e^{6}\right ) x^{4}}{2 c^{3} d^{3} e^{2}}+\frac {3 \left (a \,e^{8}-c \,e^{6} d^{2}\right ) x^{5}}{c^{2} d^{2} e}}{\left (c d x +a e \right )^{3} \left (e x +d \right )^{3}}+\frac {e^{3} \ln \left (c d x +a e \right )}{c^{4} d^{4}}\)	\(334\)

[In]

int((e*x+d)^7/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^4,x,method=_RETURNVERBOSE)

[Out]

(3*e^2*(a*e^2-c*d^2)/c^2/d^2*x^2+3/2*e*(3*a^2*e^4-2*a*c*d^2*e^2-c^2*d^4)/c^3/d^3*x+1/6*(11*a^3*e^6-6*a^2*c*d^2
*e^4-3*a*c^2*d^4*e^2-2*c^3*d^6)/c^4/d^4)/(c*d*x+a*e)^3+e^3*ln(c*d*x+a*e)/c^4/d^4

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.78 \[ \int \frac {(d+e x)^7}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^4} \, dx=-\frac {2 \, c^{3} d^{6} + 3 \, a c^{2} d^{4} e^{2} + 6 \, a^{2} c d^{2} e^{4} - 11 \, a^{3} e^{6} + 18 \, {\left (c^{3} d^{4} e^{2} - a c^{2} d^{2} e^{4}\right )} x^{2} + 9 \, {\left (c^{3} d^{5} e + 2 \, a c^{2} d^{3} e^{3} - 3 \, a^{2} c d e^{5}\right )} x - 6 \, {\left (c^{3} d^{3} e^{3} x^{3} + 3 \, a c^{2} d^{2} e^{4} x^{2} + 3 \, a^{2} c d e^{5} x + a^{3} e^{6}\right )} \log \left (c d x + a e\right )}{6 \, {\left (c^{7} d^{7} x^{3} + 3 \, a c^{6} d^{6} e x^{2} + 3 \, a^{2} c^{5} d^{5} e^{2} x + a^{3} c^{4} d^{4} e^{3}\right )}} \]

[In]

integrate((e*x+d)^7/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^4,x, algorithm="fricas")

[Out]

-1/6*(2*c^3*d^6 + 3*a*c^2*d^4*e^2 + 6*a^2*c*d^2*e^4 - 11*a^3*e^6 + 18*(c^3*d^4*e^2 - a*c^2*d^2*e^4)*x^2 + 9*(c
^3*d^5*e + 2*a*c^2*d^3*e^3 - 3*a^2*c*d*e^5)*x - 6*(c^3*d^3*e^3*x^3 + 3*a*c^2*d^2*e^4*x^2 + 3*a^2*c*d*e^5*x + a
^3*e^6)*log(c*d*x + a*e))/(c^7*d^7*x^3 + 3*a*c^6*d^6*e*x^2 + 3*a^2*c^5*d^5*e^2*x + a^3*c^4*d^4*e^3)

Sympy [A] (verification not implemented)

Time = 10.57 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.55 \[ \int \frac {(d+e x)^7}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^4} \, dx=\frac {11 a^{3} e^{6} - 6 a^{2} c d^{2} e^{4} - 3 a c^{2} d^{4} e^{2} - 2 c^{3} d^{6} + x^{2} \cdot \left (18 a c^{2} d^{2} e^{4} - 18 c^{3} d^{4} e^{2}\right ) + x \left (27 a^{2} c d e^{5} - 18 a c^{2} d^{3} e^{3} - 9 c^{3} d^{5} e\right )}{6 a^{3} c^{4} d^{4} e^{3} + 18 a^{2} c^{5} d^{5} e^{2} x + 18 a c^{6} d^{6} e x^{2} + 6 c^{7} d^{7} x^{3}} + \frac {e^{3} \log {\left (a e + c d x \right )}}{c^{4} d^{4}} \]

[In]

integrate((e*x+d)**7/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**4,x)

[Out]

(11*a**3*e**6 - 6*a**2*c*d**2*e**4 - 3*a*c**2*d**4*e**2 - 2*c**3*d**6 + x**2*(18*a*c**2*d**2*e**4 - 18*c**3*d*
*4*e**2) + x*(27*a**2*c*d*e**5 - 18*a*c**2*d**3*e**3 - 9*c**3*d**5*e))/(6*a**3*c**4*d**4*e**3 + 18*a**2*c**5*d
**5*e**2*x + 18*a*c**6*d**6*e*x**2 + 6*c**7*d**7*x**3) + e**3*log(a*e + c*d*x)/(c**4*d**4)

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.47 \[ \int \frac {(d+e x)^7}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^4} \, dx=-\frac {2 \, c^{3} d^{6} + 3 \, a c^{2} d^{4} e^{2} + 6 \, a^{2} c d^{2} e^{4} - 11 \, a^{3} e^{6} + 18 \, {\left (c^{3} d^{4} e^{2} - a c^{2} d^{2} e^{4}\right )} x^{2} + 9 \, {\left (c^{3} d^{5} e + 2 \, a c^{2} d^{3} e^{3} - 3 \, a^{2} c d e^{5}\right )} x}{6 \, {\left (c^{7} d^{7} x^{3} + 3 \, a c^{6} d^{6} e x^{2} + 3 \, a^{2} c^{5} d^{5} e^{2} x + a^{3} c^{4} d^{4} e^{3}\right )}} + \frac {e^{3} \log \left (c d x + a e\right )}{c^{4} d^{4}} \]

[In]

integrate((e*x+d)^7/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^4,x, algorithm="maxima")

[Out]

-1/6*(2*c^3*d^6 + 3*a*c^2*d^4*e^2 + 6*a^2*c*d^2*e^4 - 11*a^3*e^6 + 18*(c^3*d^4*e^2 - a*c^2*d^2*e^4)*x^2 + 9*(c
^3*d^5*e + 2*a*c^2*d^3*e^3 - 3*a^2*c*d*e^5)*x)/(c^7*d^7*x^3 + 3*a*c^6*d^6*e*x^2 + 3*a^2*c^5*d^5*e^2*x + a^3*c^
4*d^4*e^3) + e^3*log(c*d*x + a*e)/(c^4*d^4)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.16 \[ \int \frac {(d+e x)^7}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^4} \, dx=\frac {e^{3} \log \left ({\left | c d x + a e \right |}\right )}{c^{4} d^{4}} - \frac {18 \, {\left (c^{2} d^{3} e^{2} - a c d e^{4}\right )} x^{2} + 9 \, {\left (c^{2} d^{4} e + 2 \, a c d^{2} e^{3} - 3 \, a^{2} e^{5}\right )} x + \frac {2 \, c^{3} d^{6} + 3 \, a c^{2} d^{4} e^{2} + 6 \, a^{2} c d^{2} e^{4} - 11 \, a^{3} e^{6}}{c d}}{6 \, {\left (c d x + a e\right )}^{3} c^{3} d^{3}} \]

[In]

integrate((e*x+d)^7/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^4,x, algorithm="giac")

[Out]

e^3*log(abs(c*d*x + a*e))/(c^4*d^4) - 1/6*(18*(c^2*d^3*e^2 - a*c*d*e^4)*x^2 + 9*(c^2*d^4*e + 2*a*c*d^2*e^3 - 3
*a^2*e^5)*x + (2*c^3*d^6 + 3*a*c^2*d^4*e^2 + 6*a^2*c*d^2*e^4 - 11*a^3*e^6)/(c*d))/((c*d*x + a*e)^3*c^3*d^3)

Mupad [B] (verification not implemented)

Time = 9.85 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.46 \[ \int \frac {(d+e x)^7}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^4} \, dx=\frac {e^3\,\ln \left (a\,e+c\,d\,x\right )}{c^4\,d^4}-\frac {\frac {-11\,a^3\,e^6+6\,a^2\,c\,d^2\,e^4+3\,a\,c^2\,d^4\,e^2+2\,c^3\,d^6}{6\,c^4\,d^4}+\frac {3\,x\,\left (-3\,a^2\,e^5+2\,a\,c\,d^2\,e^3+c^2\,d^4\,e\right )}{2\,c^3\,d^3}-\frac {3\,e^2\,x^2\,\left (a\,e^2-c\,d^2\right )}{c^2\,d^2}}{a^3\,e^3+3\,a^2\,c\,d\,e^2\,x+3\,a\,c^2\,d^2\,e\,x^2+c^3\,d^3\,x^3} \]

[In]

int((d + e*x)^7/(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^4,x)

[Out]

(e^3*log(a*e + c*d*x))/(c^4*d^4) - ((2*c^3*d^6 - 11*a^3*e^6 + 3*a*c^2*d^4*e^2 + 6*a^2*c*d^2*e^4)/(6*c^4*d^4) +
(3*x*(c^2*d^4*e - 3*a^2*e^5 + 2*a*c*d^2*e^3))/(2*c^3*d^3) - (3*e^2*x^2*(a*e^2 - c*d^2))/(c^2*d^2))/(a^3*e^3 +
c^3*d^3*x^3 + 3*a^2*c*d*e^2*x + 3*a*c^2*d^2*e*x^2)