\(\int \frac {(d+e x)^7}{(a d e+(c d^2+a e^2) x+c d e x^2)^2} \, dx\) [1875]

Optimal result

Integrand size = 35, antiderivative size = 184 \[ \int \frac {(d+e x)^7}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx=\frac {10 e^2 \left (c d^2-a e^2\right )^3 x}{c^5 d^5}-\frac {\left (c d^2-a e^2\right )^5}{c^6 d^6 (a e+c d x)}+\frac {5 e^3 \left (c d^2-a e^2\right )^2 (a e+c d x)^2}{c^6 d^6}+\frac {5 e^4 \left (c d^2-a e^2\right ) (a e+c d x)^3}{3 c^6 d^6}+\frac {e^5 (a e+c d x)^4}{4 c^6 d^6}+\frac {5 e \left (c d^2-a e^2\right )^4 \log (a e+c d x)}{c^6 d^6} \]

[Out]

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

Rubi [A] (verified)

Time = 0.14 (sec) , antiderivative size = 184, 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} \[ \int \frac {(d+e x)^7}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx=\frac {e^5 (a e+c d x)^4}{4 c^6 d^6}-\frac {\left (c d^2-a e^2\right )^5}{c^6 d^6 (a e+c d x)}+\frac {5 e \left (c d^2-a e^2\right )^4 \log (a e+c d x)}{c^6 d^6}+\frac {5 e^4 \left (c d^2-a e^2\right ) (a e+c d x)^3}{3 c^6 d^6}+\frac {5 e^3 \left (c d^2-a e^2\right )^2 (a e+c d x)^2}{c^6 d^6}+\frac {10 e^2 x \left (c d^2-a e^2\right )^3}{c^5 d^5} \]

[In]

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

[Out]

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

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)^5}{(a e+c d x)^2} \, dx \\ & = \int \left (\frac {10 e^2 \left (c d^2-a e^2\right )^3}{c^5 d^5}+\frac {\left (c d^2-a e^2\right )^5}{c^5 d^5 (a e+c d x)^2}+\frac {5 e \left (c d^2-a e^2\right )^4}{c^5 d^5 (a e+c d x)}+\frac {10 e^3 \left (c d^2-a e^2\right )^2 (a e+c d x)}{c^5 d^5}+\frac {5 \left (c d^2 e^4-a e^6\right ) (a e+c d x)^2}{c^5 d^5}+\frac {e^5 (a e+c d x)^3}{c^5 d^5}\right ) \, dx \\ & = \frac {10 e^2 \left (c d^2-a e^2\right )^3 x}{c^5 d^5}-\frac {\left (c d^2-a e^2\right )^5}{c^6 d^6 (a e+c d x)}+\frac {5 e^3 \left (c d^2-a e^2\right )^2 (a e+c d x)^2}{c^6 d^6}+\frac {5 e^4 \left (c d^2-a e^2\right ) (a e+c d x)^3}{3 c^6 d^6}+\frac {e^5 (a e+c d x)^4}{4 c^6 d^6}+\frac {5 e \left (c d^2-a e^2\right )^4 \log (a e+c d x)}{c^6 d^6} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 263, normalized size of antiderivative = 1.43 \[ \int \frac {(d+e x)^7}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx=\frac {12 a^5 e^{10}-12 a^4 c d e^8 (5 d+4 e x)+30 a^3 c^2 d^2 e^6 \left (4 d^2+6 d e x-e^2 x^2\right )-10 a^2 c^3 d^3 e^4 \left (12 d^3+24 d^2 e x-12 d e^2 x^2-e^3 x^3\right )+5 a c^4 d^4 e^2 \left (12 d^4+24 d^3 e x-36 d^2 e^2 x^2-8 d e^3 x^3-e^4 x^4\right )+c^5 d^5 \left (-12 d^5+120 d^3 e^2 x^2+60 d^2 e^3 x^3+20 d e^4 x^4+3 e^5 x^5\right )+60 e \left (c d^2-a e^2\right )^4 (a e+c d x) \log (a e+c d x)}{12 c^6 d^6 (a e+c d x)} \]

[In]

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

[Out]

(12*a^5*e^10 - 12*a^4*c*d*e^8*(5*d + 4*e*x) + 30*a^3*c^2*d^2*e^6*(4*d^2 + 6*d*e*x - e^2*x^2) - 10*a^2*c^3*d^3*
e^4*(12*d^3 + 24*d^2*e*x - 12*d*e^2*x^2 - e^3*x^3) + 5*a*c^4*d^4*e^2*(12*d^4 + 24*d^3*e*x - 36*d^2*e^2*x^2 - 8
*d*e^3*x^3 - e^4*x^4) + c^5*d^5*(-12*d^5 + 120*d^3*e^2*x^2 + 60*d^2*e^3*x^3 + 20*d*e^4*x^4 + 3*e^5*x^5) + 60*e
*(c*d^2 - a*e^2)^4*(a*e + c*d*x)*Log[a*e + c*d*x])/(12*c^6*d^6*(a*e + c*d*x))

Maple [A] (verified)

Time = 2.35 (sec) , antiderivative size = 298, normalized size of antiderivative = 1.62

method	result	size
default	\(-\frac {e^{2} \left (-\frac {1}{4} x^{4} c^{3} d^{3} e^{3}+\frac {2}{3} x^{3} a \,c^{2} d^{2} e^{4}-\frac {5}{3} x^{3} c^{3} d^{4} e^{2}-\frac {3}{2} x^{2} a^{2} c d \,e^{5}+5 x^{2} a \,c^{2} d^{3} e^{3}-5 x^{2} c^{3} d^{5} e +4 e^{6} a^{3} x -15 d^{2} e^{4} a^{2} c x +20 d^{4} e^{2} c^{2} a x -10 c^{3} d^{6} x \right )}{c^{5} d^{5}}+\frac {5 e \left (a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}\right ) \ln \left (c d x +a e \right )}{c^{6} d^{6}}-\frac {-a^{5} e^{10}+5 a^{4} c \,d^{2} e^{8}-10 a^{3} c^{2} d^{4} e^{6}+10 a^{2} c^{3} d^{6} e^{4}-5 a \,c^{4} d^{8} e^{2}+c^{5} d^{10}}{c^{6} d^{6} \left (c d x +a e \right )}\)	\(298\)
risch	\(\frac {e^{5} x^{4}}{4 c^{2} d^{2}}-\frac {2 e^{6} x^{3} a}{3 c^{3} d^{3}}+\frac {5 e^{4} x^{3}}{3 c^{2} d}+\frac {3 e^{7} x^{2} a^{2}}{2 c^{4} d^{4}}-\frac {5 e^{5} x^{2} a}{c^{3} d^{2}}+\frac {5 e^{3} x^{2}}{c^{2}}-\frac {4 e^{8} a^{3} x}{c^{5} d^{5}}+\frac {15 e^{6} a^{2} x}{c^{4} d^{3}}-\frac {20 e^{4} a x}{c^{3} d}+\frac {10 e^{2} d x}{c^{2}}+\frac {a^{5} e^{10}}{c^{6} d^{6} \left (c d x +a e \right )}-\frac {5 a^{4} e^{8}}{c^{5} d^{4} \left (c d x +a e \right )}+\frac {10 a^{3} e^{6}}{c^{4} d^{2} \left (c d x +a e \right )}-\frac {10 a^{2} e^{4}}{c^{3} \left (c d x +a e \right )}+\frac {5 d^{2} a \,e^{2}}{c^{2} \left (c d x +a e \right )}-\frac {d^{4}}{c \left (c d x +a e \right )}+\frac {5 e^{9} \ln \left (c d x +a e \right ) a^{4}}{c^{6} d^{6}}-\frac {20 e^{7} \ln \left (c d x +a e \right ) a^{3}}{c^{5} d^{4}}+\frac {30 e^{5} \ln \left (c d x +a e \right ) a^{2}}{c^{4} d^{2}}-\frac {20 e^{3} \ln \left (c d x +a e \right ) a}{c^{3}}+\frac {5 d^{2} e \ln \left (c d x +a e \right )}{c^{2}}\)	\(378\)
norman	\(\frac {\frac {10 a^{5} e^{10}-35 a^{4} c \,d^{2} e^{8}+40 a^{3} c^{2} d^{4} e^{6}-10 a^{2} c^{3} d^{6} e^{4}-10 a \,c^{4} d^{8} e^{2}-2 c^{5} d^{10}}{2 c^{6} d^{5}}+\frac {e^{6} x^{6}}{4 c d}+\frac {\left (10 a^{5} e^{12}-35 a^{4} c \,d^{2} e^{10}+45 a^{3} c^{2} d^{4} e^{8}-30 a^{2} c^{3} d^{6} e^{6}+20 a \,c^{4} d^{8} e^{4}-22 c^{5} d^{10} e^{2}\right ) x}{2 c^{6} d^{6} e}-\frac {5 e^{3} \left (3 e^{6} a^{3}-13 d^{2} e^{4} a^{2} c +22 d^{4} e^{2} c^{2} a -18 c^{3} d^{6}\right ) x^{3}}{6 c^{4} d^{4}}+\frac {5 e^{4} \left (2 a^{2} e^{4}-9 a c \,d^{2} e^{2}+16 c^{2} d^{4}\right ) x^{4}}{12 c^{3} d^{3}}-\frac {e^{5} \left (5 e^{2} a -23 c \,d^{2}\right ) x^{5}}{12 c^{2} d^{2}}}{\left (c d x +a e \right ) \left (e x +d \right )}+\frac {5 e \left (a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}\right ) \ln \left (c d x +a e \right )}{c^{6} d^{6}}\)	\(390\)
parallelrisch	\(\frac {-240 a^{4} c \,d^{2} e^{8}+360 a^{3} c^{2} d^{4} e^{6}-240 a^{2} c^{3} d^{6} e^{4}+60 a \,c^{4} d^{8} e^{2}+60 \ln \left (c d x +a e \right ) x \,c^{5} d^{9} e +60 \ln \left (c d x +a e \right ) x \,a^{4} c d \,e^{9}-240 \ln \left (c d x +a e \right ) x \,a^{3} c^{2} d^{3} e^{7}+360 \ln \left (c d x +a e \right ) x \,a^{2} c^{3} d^{5} e^{5}-240 \ln \left (c d x +a e \right ) x a \,c^{4} d^{7} e^{3}+60 a^{5} e^{10}-12 c^{5} d^{10}+60 \ln \left (c d x +a e \right ) a^{5} e^{10}+20 x^{4} c^{5} d^{6} e^{4}+3 x^{5} e^{5} c^{5} d^{5}+60 x^{3} c^{5} d^{7} e^{3}+120 x^{2} c^{5} d^{8} e^{2}-240 \ln \left (c d x +a e \right ) a^{4} c \,d^{2} e^{8}+360 \ln \left (c d x +a e \right ) a^{3} c^{2} d^{4} e^{6}-240 \ln \left (c d x +a e \right ) a^{2} c^{3} d^{6} e^{4}+60 \ln \left (c d x +a e \right ) a \,c^{4} d^{8} e^{2}+10 x^{3} a^{2} c^{3} d^{3} e^{7}-40 x^{3} a \,c^{4} d^{5} e^{5}-30 x^{2} a^{3} c^{2} d^{2} e^{8}+120 x^{2} a^{2} c^{3} d^{4} e^{6}-180 x^{2} a \,c^{4} d^{6} e^{4}-5 x^{4} a \,c^{4} d^{4} e^{6}}{12 c^{6} d^{6} \left (c d x +a e \right )}\)	\(454\)

[In]

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

[Out]

-e^2/c^5/d^5*(-1/4*x^4*c^3*d^3*e^3+2/3*x^3*a*c^2*d^2*e^4-5/3*x^3*c^3*d^4*e^2-3/2*x^2*a^2*c*d*e^5+5*x^2*a*c^2*d
^3*e^3-5*x^2*c^3*d^5*e+4*e^6*a^3*x-15*d^2*e^4*a^2*c*x+20*d^4*e^2*c^2*a*x-10*c^3*d^6*x)+5/c^6/d^6*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)*ln(c*d*x+a*e)-1/c^6/d^6*(-a^5*e^10+5*a^4*c*d^2*e^8-10
*a^3*c^2*d^4*e^6+10*a^2*c^3*d^6*e^4-5*a*c^4*d^8*e^2+c^5*d^10)/(c*d*x+a*e)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 417 vs. \(2 (180) = 360\).

Time = 0.58 (sec) , antiderivative size = 417, normalized size of antiderivative = 2.27 \[ \int \frac {(d+e x)^7}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx=\frac {3 \, c^{5} d^{5} e^{5} x^{5} - 12 \, c^{5} d^{10} + 60 \, a c^{4} d^{8} e^{2} - 120 \, a^{2} c^{3} d^{6} e^{4} + 120 \, a^{3} c^{2} d^{4} e^{6} - 60 \, a^{4} c d^{2} e^{8} + 12 \, a^{5} e^{10} + 5 \, {\left (4 \, c^{5} d^{6} e^{4} - a c^{4} d^{4} e^{6}\right )} x^{4} + 10 \, {\left (6 \, c^{5} d^{7} e^{3} - 4 \, a c^{4} d^{5} e^{5} + a^{2} c^{3} d^{3} e^{7}\right )} x^{3} + 30 \, {\left (4 \, c^{5} d^{8} e^{2} - 6 \, a c^{4} d^{6} e^{4} + 4 \, a^{2} c^{3} d^{4} e^{6} - a^{3} c^{2} d^{2} e^{8}\right )} x^{2} + 12 \, {\left (10 \, a c^{4} d^{7} e^{3} - 20 \, a^{2} c^{3} d^{5} e^{5} + 15 \, a^{3} c^{2} d^{3} e^{7} - 4 \, a^{4} c d e^{9}\right )} x + 60 \, {\left (a c^{4} d^{8} e^{2} - 4 \, a^{2} c^{3} d^{6} e^{4} + 6 \, a^{3} c^{2} d^{4} e^{6} - 4 \, a^{4} c d^{2} e^{8} + a^{5} e^{10} + {\left (c^{5} d^{9} e - 4 \, a c^{4} d^{7} e^{3} + 6 \, a^{2} c^{3} d^{5} e^{5} - 4 \, a^{3} c^{2} d^{3} e^{7} + a^{4} c d e^{9}\right )} x\right )} \log \left (c d x + a e\right )}{12 \, {\left (c^{7} d^{7} x + a c^{6} d^{6} e\right )}} \]

[In]

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

[Out]

1/12*(3*c^5*d^5*e^5*x^5 - 12*c^5*d^10 + 60*a*c^4*d^8*e^2 - 120*a^2*c^3*d^6*e^4 + 120*a^3*c^2*d^4*e^6 - 60*a^4*
c*d^2*e^8 + 12*a^5*e^10 + 5*(4*c^5*d^6*e^4 - a*c^4*d^4*e^6)*x^4 + 10*(6*c^5*d^7*e^3 - 4*a*c^4*d^5*e^5 + a^2*c^
3*d^3*e^7)*x^3 + 30*(4*c^5*d^8*e^2 - 6*a*c^4*d^6*e^4 + 4*a^2*c^3*d^4*e^6 - a^3*c^2*d^2*e^8)*x^2 + 12*(10*a*c^4
*d^7*e^3 - 20*a^2*c^3*d^5*e^5 + 15*a^3*c^2*d^3*e^7 - 4*a^4*c*d*e^9)*x + 60*(a*c^4*d^8*e^2 - 4*a^2*c^3*d^6*e^4
+ 6*a^3*c^2*d^4*e^6 - 4*a^4*c*d^2*e^8 + a^5*e^10 + (c^5*d^9*e - 4*a*c^4*d^7*e^3 + 6*a^2*c^3*d^5*e^5 - 4*a^3*c^
2*d^3*e^7 + a^4*c*d*e^9)*x)*log(c*d*x + a*e))/(c^7*d^7*x + a*c^6*d^6*e)

Sympy [A] (verification not implemented)

Time = 1.14 (sec) , antiderivative size = 264, normalized size of antiderivative = 1.43 \[ \int \frac {(d+e x)^7}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx=x^{3} \left (- \frac {2 a e^{6}}{3 c^{3} d^{3}} + \frac {5 e^{4}}{3 c^{2} d}\right ) + x^{2} \cdot \left (\frac {3 a^{2} e^{7}}{2 c^{4} d^{4}} - \frac {5 a e^{5}}{c^{3} d^{2}} + \frac {5 e^{3}}{c^{2}}\right ) + x \left (- \frac {4 a^{3} e^{8}}{c^{5} d^{5}} + \frac {15 a^{2} e^{6}}{c^{4} d^{3}} - \frac {20 a e^{4}}{c^{3} d} + \frac {10 d e^{2}}{c^{2}}\right ) + \frac {a^{5} e^{10} - 5 a^{4} c d^{2} e^{8} + 10 a^{3} c^{2} d^{4} e^{6} - 10 a^{2} c^{3} d^{6} e^{4} + 5 a c^{4} d^{8} e^{2} - c^{5} d^{10}}{a c^{6} d^{6} e + c^{7} d^{7} x} + \frac {e^{5} x^{4}}{4 c^{2} d^{2}} + \frac {5 e \left (a e^{2} - c d^{2}\right )^{4} \log {\left (a e + c d x \right )}}{c^{6} d^{6}} \]

[In]

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

[Out]

x**3*(-2*a*e**6/(3*c**3*d**3) + 5*e**4/(3*c**2*d)) + x**2*(3*a**2*e**7/(2*c**4*d**4) - 5*a*e**5/(c**3*d**2) +
5*e**3/c**2) + x*(-4*a**3*e**8/(c**5*d**5) + 15*a**2*e**6/(c**4*d**3) - 20*a*e**4/(c**3*d) + 10*d*e**2/c**2) +
(a**5*e**10 - 5*a**4*c*d**2*e**8 + 10*a**3*c**2*d**4*e**6 - 10*a**2*c**3*d**6*e**4 + 5*a*c**4*d**8*e**2 - c**
5*d**10)/(a*c**6*d**6*e + c**7*d**7*x) + e**5*x**4/(4*c**2*d**2) + 5*e*(a*e**2 - c*d**2)**4*log(a*e + c*d*x)/(
c**6*d**6)

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 300, normalized size of antiderivative = 1.63 \[ \int \frac {(d+e x)^7}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx=-\frac {c^{5} d^{10} - 5 \, a c^{4} d^{8} e^{2} + 10 \, a^{2} c^{3} d^{6} e^{4} - 10 \, a^{3} c^{2} d^{4} e^{6} + 5 \, a^{4} c d^{2} e^{8} - a^{5} e^{10}}{c^{7} d^{7} x + a c^{6} d^{6} e} + \frac {3 \, c^{3} d^{3} e^{5} x^{4} + 4 \, {\left (5 \, c^{3} d^{4} e^{4} - 2 \, a c^{2} d^{2} e^{6}\right )} x^{3} + 6 \, {\left (10 \, c^{3} d^{5} e^{3} - 10 \, a c^{2} d^{3} e^{5} + 3 \, a^{2} c d e^{7}\right )} x^{2} + 12 \, {\left (10 \, c^{3} d^{6} e^{2} - 20 \, a c^{2} d^{4} e^{4} + 15 \, a^{2} c d^{2} e^{6} - 4 \, a^{3} e^{8}\right )} x}{12 \, c^{5} d^{5}} + \frac {5 \, {\left (c^{4} d^{8} e - 4 \, a c^{3} d^{6} e^{3} + 6 \, a^{2} c^{2} d^{4} e^{5} - 4 \, a^{3} c d^{2} e^{7} + a^{4} e^{9}\right )} \log \left (c d x + a e\right )}{c^{6} d^{6}} \]

[In]

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

[Out]

-(c^5*d^10 - 5*a*c^4*d^8*e^2 + 10*a^2*c^3*d^6*e^4 - 10*a^3*c^2*d^4*e^6 + 5*a^4*c*d^2*e^8 - a^5*e^10)/(c^7*d^7*
x + a*c^6*d^6*e) + 1/12*(3*c^3*d^3*e^5*x^4 + 4*(5*c^3*d^4*e^4 - 2*a*c^2*d^2*e^6)*x^3 + 6*(10*c^3*d^5*e^3 - 10*
a*c^2*d^3*e^5 + 3*a^2*c*d*e^7)*x^2 + 12*(10*c^3*d^6*e^2 - 20*a*c^2*d^4*e^4 + 15*a^2*c*d^2*e^6 - 4*a^3*e^8)*x)/
(c^5*d^5) + 5*(c^4*d^8*e - 4*a*c^3*d^6*e^3 + 6*a^2*c^2*d^4*e^5 - 4*a^3*c*d^2*e^7 + a^4*e^9)*log(c*d*x + a*e)/(
c^6*d^6)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 312, normalized size of antiderivative = 1.70 \[ \int \frac {(d+e x)^7}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx=\frac {5 \, {\left (c^{4} d^{8} e - 4 \, a c^{3} d^{6} e^{3} + 6 \, a^{2} c^{2} d^{4} e^{5} - 4 \, a^{3} c d^{2} e^{7} + a^{4} e^{9}\right )} \log \left ({\left | c d x + a e \right |}\right )}{c^{6} d^{6}} - \frac {c^{5} d^{10} - 5 \, a c^{4} d^{8} e^{2} + 10 \, a^{2} c^{3} d^{6} e^{4} - 10 \, a^{3} c^{2} d^{4} e^{6} + 5 \, a^{4} c d^{2} e^{8} - a^{5} e^{10}}{{\left (c d x + a e\right )} c^{6} d^{6}} + \frac {3 \, c^{6} d^{6} e^{5} x^{4} + 20 \, c^{6} d^{7} e^{4} x^{3} - 8 \, a c^{5} d^{5} e^{6} x^{3} + 60 \, c^{6} d^{8} e^{3} x^{2} - 60 \, a c^{5} d^{6} e^{5} x^{2} + 18 \, a^{2} c^{4} d^{4} e^{7} x^{2} + 120 \, c^{6} d^{9} e^{2} x - 240 \, a c^{5} d^{7} e^{4} x + 180 \, a^{2} c^{4} d^{5} e^{6} x - 48 \, a^{3} c^{3} d^{3} e^{8} x}{12 \, c^{8} d^{8}} \]

[In]

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

[Out]

5*(c^4*d^8*e - 4*a*c^3*d^6*e^3 + 6*a^2*c^2*d^4*e^5 - 4*a^3*c*d^2*e^7 + a^4*e^9)*log(abs(c*d*x + a*e))/(c^6*d^6
) - (c^5*d^10 - 5*a*c^4*d^8*e^2 + 10*a^2*c^3*d^6*e^4 - 10*a^3*c^2*d^4*e^6 + 5*a^4*c*d^2*e^8 - a^5*e^10)/((c*d*
x + a*e)*c^6*d^6) + 1/12*(3*c^6*d^6*e^5*x^4 + 20*c^6*d^7*e^4*x^3 - 8*a*c^5*d^5*e^6*x^3 + 60*c^6*d^8*e^3*x^2 -
60*a*c^5*d^6*e^5*x^2 + 18*a^2*c^4*d^4*e^7*x^2 + 120*c^6*d^9*e^2*x - 240*a*c^5*d^7*e^4*x + 180*a^2*c^4*d^5*e^6*
x - 48*a^3*c^3*d^3*e^8*x)/(c^8*d^8)

Mupad [B] (verification not implemented)

Time = 9.65 (sec) , antiderivative size = 387, normalized size of antiderivative = 2.10 \[ \int \frac {(d+e x)^7}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx=x\,\left (\frac {10\,d\,e^2}{c^2}+\frac {2\,a\,e\,\left (\frac {a^2\,e^7}{c^4\,d^4}-\frac {10\,e^3}{c^2}+\frac {2\,a\,e\,\left (\frac {5\,e^4}{c^2\,d}-\frac {2\,a\,e^6}{c^3\,d^3}\right )}{c\,d}\right )}{c\,d}-\frac {a^2\,e^2\,\left (\frac {5\,e^4}{c^2\,d}-\frac {2\,a\,e^6}{c^3\,d^3}\right )}{c^2\,d^2}\right )+x^3\,\left (\frac {5\,e^4}{3\,c^2\,d}-\frac {2\,a\,e^6}{3\,c^3\,d^3}\right )-x^2\,\left (\frac {a^2\,e^7}{2\,c^4\,d^4}-\frac {5\,e^3}{c^2}+\frac {a\,e\,\left (\frac {5\,e^4}{c^2\,d}-\frac {2\,a\,e^6}{c^3\,d^3}\right )}{c\,d}\right )+\frac {e^5\,x^4}{4\,c^2\,d^2}+\frac {\ln \left (a\,e+c\,d\,x\right )\,\left (5\,a^4\,e^9-20\,a^3\,c\,d^2\,e^7+30\,a^2\,c^2\,d^4\,e^5-20\,a\,c^3\,d^6\,e^3+5\,c^4\,d^8\,e\right )}{c^6\,d^6}+\frac {a^5\,e^{10}-5\,a^4\,c\,d^2\,e^8+10\,a^3\,c^2\,d^4\,e^6-10\,a^2\,c^3\,d^6\,e^4+5\,a\,c^4\,d^8\,e^2-c^5\,d^{10}}{c\,d\,\left (x\,c^6\,d^6+a\,e\,c^5\,d^5\right )} \]

[In]

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

[Out]

x*((10*d*e^2)/c^2 + (2*a*e*((a^2*e^7)/(c^4*d^4) - (10*e^3)/c^2 + (2*a*e*((5*e^4)/(c^2*d) - (2*a*e^6)/(c^3*d^3)
))/(c*d)))/(c*d) - (a^2*e^2*((5*e^4)/(c^2*d) - (2*a*e^6)/(c^3*d^3)))/(c^2*d^2)) + x^3*((5*e^4)/(3*c^2*d) - (2*
a*e^6)/(3*c^3*d^3)) - x^2*((a^2*e^7)/(2*c^4*d^4) - (5*e^3)/c^2 + (a*e*((5*e^4)/(c^2*d) - (2*a*e^6)/(c^3*d^3)))
/(c*d)) + (e^5*x^4)/(4*c^2*d^2) + (log(a*e + c*d*x)*(5*a^4*e^9 + 5*c^4*d^8*e - 20*a*c^3*d^6*e^3 - 20*a^3*c*d^2
*e^7 + 30*a^2*c^2*d^4*e^5))/(c^6*d^6) + (a^5*e^10 - c^5*d^10 + 5*a*c^4*d^8*e^2 - 5*a^4*c*d^2*e^8 - 10*a^2*c^3*
d^6*e^4 + 10*a^3*c^2*d^4*e^6)/(c*d*(c^6*d^6*x + a*c^5*d^5*e))