\(\int \frac {(d+e x)^9}{(a d e+(c d^2+a e^2) x+c d e x^2)^4} \, dx\) [1897]

Optimal result

Integrand size = 35, antiderivative size = 179 \[ \int \frac {(d+e x)^9}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^4} \, dx=\frac {e^4 \left (5 c d^2-4 a e^2\right ) x}{c^5 d^5}+\frac {e^5 x^2}{2 c^4 d^4}-\frac {\left (c d^2-a e^2\right )^5}{3 c^6 d^6 (a e+c d x)^3}-\frac {5 e \left (c d^2-a e^2\right )^4}{2 c^6 d^6 (a e+c d x)^2}-\frac {10 e^2 \left (c d^2-a e^2\right )^3}{c^6 d^6 (a e+c d x)}+\frac {10 e^3 \left (c d^2-a e^2\right )^2 \log (a e+c d x)}{c^6 d^6} \]

[Out]

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

Rubi [A] (verified)

Time = 0.12 (sec) , antiderivative size = 179, 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} \[ \int \frac {(d+e x)^9}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^4} \, dx=-\frac {10 e^2 \left (c d^2-a e^2\right )^3}{c^6 d^6 (a e+c d x)}-\frac {5 e \left (c d^2-a e^2\right )^4}{2 c^6 d^6 (a e+c d x)^2}-\frac {\left (c d^2-a e^2\right )^5}{3 c^6 d^6 (a e+c d x)^3}+\frac {10 e^3 \left (c d^2-a e^2\right )^2 \log (a e+c d x)}{c^6 d^6}+\frac {e^4 x \left (5 c d^2-4 a e^2\right )}{c^5 d^5}+\frac {e^5 x^2}{2 c^4 d^4} \]

[In]

Int[(d + e*x)^9/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^4,x]

[Out]

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

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 259, normalized size of antiderivative = 1.45 \[ \int \frac {(d+e x)^9}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^4} \, dx=\frac {47 a^5 e^{10}+a^4 c d e^8 (-130 d+81 e x)+a^3 c^2 d^2 e^6 \left (110 d^2-270 d e x-9 e^2 x^2\right )-a^2 c^3 d^3 e^4 \left (20 d^3-270 d^2 e x+90 d e^2 x^2+63 e^3 x^3\right )-5 a c^4 d^4 e^2 \left (d^4+12 d^3 e x-36 d^2 e^2 x^2-18 d e^3 x^3+3 e^4 x^4\right )+c^5 d^5 \left (-2 d^5-15 d^4 e x-60 d^3 e^2 x^2+30 d e^4 x^4+3 e^5 x^5\right )+60 e^3 \left (c d^2-a e^2\right )^2 (a e+c d x)^3 \log (a e+c d x)}{6 c^6 d^6 (a e+c d x)^3} \]

[In]

Integrate[(d + e*x)^9/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^4,x]

[Out]

(47*a^5*e^10 + a^4*c*d*e^8*(-130*d + 81*e*x) + a^3*c^2*d^2*e^6*(110*d^2 - 270*d*e*x - 9*e^2*x^2) - a^2*c^3*d^3
*e^4*(20*d^3 - 270*d^2*e*x + 90*d*e^2*x^2 + 63*e^3*x^3) - 5*a*c^4*d^4*e^2*(d^4 + 12*d^3*e*x - 36*d^2*e^2*x^2 -
18*d*e^3*x^3 + 3*e^4*x^4) + c^5*d^5*(-2*d^5 - 15*d^4*e*x - 60*d^3*e^2*x^2 + 30*d*e^4*x^4 + 3*e^5*x^5) + 60*e^
3*(c*d^2 - a*e^2)^2*(a*e + c*d*x)^3*Log[a*e + c*d*x])/(6*c^6*d^6*(a*e + c*d*x)^3)

Maple [A] (verified)

Time = 2.44 (sec) , antiderivative size = 300, normalized size of antiderivative = 1.68

method	result	size
default	\(-\frac {e^{4} \left (-\frac {1}{2} c d e \,x^{2}+4 a \,e^{2} x -5 c \,d^{2} x \right )}{c^{5} d^{5}}-\frac {5 e \left (a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}\right )}{2 c^{6} d^{6} \left (c d x +a e \right )^{2}}-\frac {-a^{5} e^{10}+5 a^{4} c \,d^{2} e^{8}-10 a^{3} c^{2} d^{4} e^{6}+10 a^{2} c^{3} d^{6} e^{4}-5 a \,c^{4} d^{8} e^{2}+c^{5} d^{10}}{3 c^{6} d^{6} \left (c d x +a e \right )^{3}}+\frac {10 e^{3} \left (a^{2} e^{4}-2 a c \,d^{2} e^{2}+c^{2} d^{4}\right ) \ln \left (c d x +a e \right )}{c^{6} d^{6}}+\frac {10 e^{2} \left (e^{6} a^{3}-3 d^{2} e^{4} a^{2} c +3 d^{4} e^{2} c^{2} a -c^{3} d^{6}\right )}{c^{6} d^{6} \left (c d x +a e \right )}\)	\(300\)
risch	\(\frac {e^{5} x^{2}}{2 c^{4} d^{4}}-\frac {4 e^{6} a x}{c^{5} d^{5}}+\frac {5 e^{4} x}{c^{4} d^{3}}+\frac {\left (10 a^{3} c \,e^{8} d -30 a^{2} c^{2} e^{6} d^{3}+30 a \,c^{3} e^{4} d^{5}-10 c^{4} e^{2} d^{7}\right ) x^{2}+\frac {5 e \left (7 a^{4} e^{8}-20 a^{3} c \,d^{2} e^{6}+18 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}-c^{4} d^{8}\right ) x}{2}+\frac {47 a^{5} e^{10}-130 a^{4} c \,d^{2} e^{8}+110 a^{3} c^{2} d^{4} e^{6}-20 a^{2} c^{3} d^{6} e^{4}-5 a \,c^{4} d^{8} e^{2}-2 c^{5} d^{10}}{6 c d}}{c^{5} d^{5} \left (c d x +a e \right )^{3}}+\frac {10 e^{7} \ln \left (c d x +a e \right ) a^{2}}{c^{6} d^{6}}-\frac {20 e^{5} \ln \left (c d x +a e \right ) a}{c^{5} d^{4}}+\frac {10 e^{3} \ln \left (c d x +a e \right )}{c^{4} d^{2}}\)	\(311\)
parallelrisch	\(\frac {-540 a^{3} c^{2} d^{3} e^{7} x +270 a^{2} c^{3} d^{5} e^{5} x +270 a^{4} c d \,e^{9} x -220 a^{4} c \,d^{2} e^{8}+110 a^{3} c^{2} d^{4} e^{6}-20 a^{2} c^{3} d^{6} e^{4}-5 a \,c^{4} d^{8} e^{2}-15 c^{5} d^{9} e x +180 \ln \left (c d x +a e \right ) x^{2} a^{3} c^{2} d^{2} e^{8}-360 \ln \left (c d x +a e \right ) x^{2} a^{2} c^{3} d^{4} e^{6}+180 \ln \left (c d x +a e \right ) x \,a^{4} c d \,e^{9}-360 \ln \left (c d x +a e \right ) x \,a^{3} c^{2} d^{3} e^{7}+180 \ln \left (c d x +a e \right ) x \,a^{2} c^{3} d^{5} e^{5}+110 a^{5} e^{10}-2 c^{5} d^{10}+60 \ln \left (c d x +a e \right ) a^{5} e^{10}+30 x^{4} c^{5} d^{6} e^{4}+3 x^{5} e^{5} c^{5} d^{5}-60 x^{2} c^{5} d^{8} e^{2}+60 \ln \left (c d x +a e \right ) x^{3} a^{2} c^{3} d^{3} e^{7}-120 \ln \left (c d x +a e \right ) x^{3} a \,c^{4} d^{5} e^{5}+180 \ln \left (c d x +a e \right ) x^{2} a \,c^{4} d^{6} e^{4}-60 x a \,c^{4} d^{7} e^{3}-120 \ln \left (c d x +a e \right ) a^{4} c \,d^{2} e^{8}+60 \ln \left (c d x +a e \right ) a^{3} c^{2} d^{4} e^{6}+180 x^{2} a^{3} c^{2} d^{2} e^{8}-360 x^{2} a^{2} c^{3} d^{4} e^{6}+180 x^{2} a \,c^{4} d^{6} e^{4}-15 x^{4} a \,c^{4} d^{4} e^{6}+60 \ln \left (c d x +a e \right ) x^{3} c^{5} d^{7} e^{3}}{6 c^{6} d^{6} \left (c d x +a e \right )^{3}}\)	\(536\)
norman	\(\frac {\frac {110 a^{5} e^{10}-175 a^{4} c \,d^{2} e^{8}+11 a^{3} c^{2} d^{4} e^{6}-20 a^{2} c^{3} d^{6} e^{4}-5 a \,c^{4} d^{8} e^{2}-2 c^{5} d^{10}}{6 c^{6} d^{3}}+\frac {e^{8} x^{8}}{2 c d}+\frac {\left (110 a^{5} e^{16}+635 c \,d^{2} a^{4} e^{14}-664 a^{3} c^{2} d^{4} e^{12}-776 a^{2} c^{3} d^{6} e^{10}-491 a \,c^{4} d^{8} e^{8}-326 c^{5} d^{10} e^{6}\right ) x^{3}}{6 c^{6} d^{6} e^{3}}+\frac {\left (110 a^{5} e^{14}+95 a^{4} c \,d^{2} e^{12}-334 a^{3} c^{2} d^{4} e^{10}-122 a^{2} c^{3} d^{6} e^{8}-104 a \,c^{4} d^{8} e^{6}-37 c^{5} d^{10} e^{4}\right ) x^{2}}{2 c^{6} d^{5} e^{2}}+\frac {\left (110 a^{5} e^{12}-85 a^{4} c \,d^{2} e^{10}-124 a^{3} c^{2} d^{4} e^{8}-29 a^{2} c^{3} d^{6} e^{6}-25 a \,c^{4} d^{8} e^{4}-7 c^{5} d^{10} e^{2}\right ) x}{2 c^{6} d^{4} e}+\frac {\left (90 a^{4} e^{14}+45 a^{3} c \,d^{2} e^{12}-234 d^{4} a^{2} c^{2} e^{10}-97 a \,c^{3} d^{6} e^{8}-154 c^{4} d^{8} e^{6}\right ) x^{4}}{2 c^{5} d^{5} e^{2}}+\frac {\left (60 a^{3} e^{12}-75 c \,a^{2} d^{2} e^{10}-9 d^{4} c^{2} a \,e^{8}-88 d^{6} e^{6} c^{3}\right ) x^{5}}{2 c^{4} d^{4} e}-\frac {e^{7} \left (5 e^{2} a -13 c \,d^{2}\right ) x^{7}}{2 c^{2} d^{2}}}{\left (c d x +a e \right )^{3} \left (e x +d \right )^{3}}+\frac {10 e^{3} \left (a^{2} e^{4}-2 a c \,d^{2} e^{2}+c^{2} d^{4}\right ) \ln \left (c d x +a e \right )}{c^{6} d^{6}}\)	\(570\)

[In]

int((e*x+d)^9/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^4,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 488 vs. \(2 (173) = 346\).

Time = 0.29 (sec) , antiderivative size = 488, normalized size of antiderivative = 2.73 \[ \int \frac {(d+e x)^9}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^4} \, dx=\frac {3 \, c^{5} d^{5} e^{5} x^{5} - 2 \, c^{5} d^{10} - 5 \, a c^{4} d^{8} e^{2} - 20 \, a^{2} c^{3} d^{6} e^{4} + 110 \, a^{3} c^{2} d^{4} e^{6} - 130 \, a^{4} c d^{2} e^{8} + 47 \, a^{5} e^{10} + 15 \, {\left (2 \, c^{5} d^{6} e^{4} - a c^{4} d^{4} e^{6}\right )} x^{4} + 9 \, {\left (10 \, a c^{4} d^{5} e^{5} - 7 \, a^{2} c^{3} d^{3} e^{7}\right )} x^{3} - 3 \, {\left (20 \, c^{5} d^{8} e^{2} - 60 \, a c^{4} d^{6} e^{4} + 30 \, a^{2} c^{3} d^{4} e^{6} + 3 \, a^{3} c^{2} d^{2} e^{8}\right )} x^{2} - 3 \, {\left (5 \, c^{5} d^{9} e + 20 \, a c^{4} d^{7} e^{3} - 90 \, a^{2} c^{3} d^{5} e^{5} + 90 \, a^{3} c^{2} d^{3} e^{7} - 27 \, a^{4} c d e^{9}\right )} x + 60 \, {\left (a^{3} c^{2} d^{4} e^{6} - 2 \, a^{4} c d^{2} e^{8} + a^{5} e^{10} + {\left (c^{5} d^{7} e^{3} - 2 \, a c^{4} d^{5} e^{5} + a^{2} c^{3} d^{3} e^{7}\right )} x^{3} + 3 \, {\left (a c^{4} d^{6} e^{4} - 2 \, a^{2} c^{3} d^{4} e^{6} + a^{3} c^{2} d^{2} e^{8}\right )} x^{2} + 3 \, {\left (a^{2} c^{3} d^{5} e^{5} - 2 \, a^{3} c^{2} d^{3} e^{7} + a^{4} c d e^{9}\right )} x\right )} \log \left (c d x + a e\right )}{6 \, {\left (c^{9} d^{9} x^{3} + 3 \, a c^{8} d^{8} e x^{2} + 3 \, a^{2} c^{7} d^{7} e^{2} x + a^{3} c^{6} d^{6} e^{3}\right )}} \]

[In]

integrate((e*x+d)^9/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^4,x, algorithm="fricas")

[Out]

1/6*(3*c^5*d^5*e^5*x^5 - 2*c^5*d^10 - 5*a*c^4*d^8*e^2 - 20*a^2*c^3*d^6*e^4 + 110*a^3*c^2*d^4*e^6 - 130*a^4*c*d
^2*e^8 + 47*a^5*e^10 + 15*(2*c^5*d^6*e^4 - a*c^4*d^4*e^6)*x^4 + 9*(10*a*c^4*d^5*e^5 - 7*a^2*c^3*d^3*e^7)*x^3 -
3*(20*c^5*d^8*e^2 - 60*a*c^4*d^6*e^4 + 30*a^2*c^3*d^4*e^6 + 3*a^3*c^2*d^2*e^8)*x^2 - 3*(5*c^5*d^9*e + 20*a*c^
4*d^7*e^3 - 90*a^2*c^3*d^5*e^5 + 90*a^3*c^2*d^3*e^7 - 27*a^4*c*d*e^9)*x + 60*(a^3*c^2*d^4*e^6 - 2*a^4*c*d^2*e^
8 + a^5*e^10 + (c^5*d^7*e^3 - 2*a*c^4*d^5*e^5 + a^2*c^3*d^3*e^7)*x^3 + 3*(a*c^4*d^6*e^4 - 2*a^2*c^3*d^4*e^6 +
a^3*c^2*d^2*e^8)*x^2 + 3*(a^2*c^3*d^5*e^5 - 2*a^3*c^2*d^3*e^7 + a^4*c*d*e^9)*x)*log(c*d*x + a*e))/(c^9*d^9*x^3
+ 3*a*c^8*d^8*e*x^2 + 3*a^2*c^7*d^7*e^2*x + a^3*c^6*d^6*e^3)

Sympy [F(-1)]

Timed out. \[ \int \frac {(d+e x)^9}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^4} \, dx=\text {Timed out} \]

[In]

integrate((e*x+d)**9/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**4,x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 326, normalized size of antiderivative = 1.82 \[ \int \frac {(d+e x)^9}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^4} \, dx=-\frac {2 \, c^{5} d^{10} + 5 \, a c^{4} d^{8} e^{2} + 20 \, a^{2} c^{3} d^{6} e^{4} - 110 \, a^{3} c^{2} d^{4} e^{6} + 130 \, a^{4} c d^{2} e^{8} - 47 \, a^{5} e^{10} + 60 \, {\left (c^{5} d^{8} e^{2} - 3 \, a c^{4} d^{6} e^{4} + 3 \, a^{2} c^{3} d^{4} e^{6} - a^{3} c^{2} d^{2} e^{8}\right )} x^{2} + 15 \, {\left (c^{5} d^{9} e + 4 \, a c^{4} d^{7} e^{3} - 18 \, a^{2} c^{3} d^{5} e^{5} + 20 \, a^{3} c^{2} d^{3} e^{7} - 7 \, a^{4} c d e^{9}\right )} x}{6 \, {\left (c^{9} d^{9} x^{3} + 3 \, a c^{8} d^{8} e x^{2} + 3 \, a^{2} c^{7} d^{7} e^{2} x + a^{3} c^{6} d^{6} e^{3}\right )}} + \frac {c d e^{5} x^{2} + 2 \, {\left (5 \, c d^{2} e^{4} - 4 \, a e^{6}\right )} x}{2 \, c^{5} d^{5}} + \frac {10 \, {\left (c^{2} d^{4} e^{3} - 2 \, a c d^{2} e^{5} + a^{2} e^{7}\right )} \log \left (c d x + a e\right )}{c^{6} d^{6}} \]

[In]

integrate((e*x+d)^9/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^4,x, algorithm="maxima")

[Out]

-1/6*(2*c^5*d^10 + 5*a*c^4*d^8*e^2 + 20*a^2*c^3*d^6*e^4 - 110*a^3*c^2*d^4*e^6 + 130*a^4*c*d^2*e^8 - 47*a^5*e^1
0 + 60*(c^5*d^8*e^2 - 3*a*c^4*d^6*e^4 + 3*a^2*c^3*d^4*e^6 - a^3*c^2*d^2*e^8)*x^2 + 15*(c^5*d^9*e + 4*a*c^4*d^7
*e^3 - 18*a^2*c^3*d^5*e^5 + 20*a^3*c^2*d^3*e^7 - 7*a^4*c*d*e^9)*x)/(c^9*d^9*x^3 + 3*a*c^8*d^8*e*x^2 + 3*a^2*c^
7*d^7*e^2*x + a^3*c^6*d^6*e^3) + 1/2*(c*d*e^5*x^2 + 2*(5*c*d^2*e^4 - 4*a*e^6)*x)/(c^5*d^5) + 10*(c^2*d^4*e^3 -
2*a*c*d^2*e^5 + a^2*e^7)*log(c*d*x + a*e)/(c^6*d^6)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 299, normalized size of antiderivative = 1.67 \[ \int \frac {(d+e x)^9}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^4} \, dx=\frac {10 \, {\left (c^{2} d^{4} e^{3} - 2 \, a c d^{2} e^{5} + a^{2} e^{7}\right )} \log \left ({\left | c d x + a e \right |}\right )}{c^{6} d^{6}} - \frac {2 \, c^{5} d^{10} + 5 \, a c^{4} d^{8} e^{2} + 20 \, a^{2} c^{3} d^{6} e^{4} - 110 \, a^{3} c^{2} d^{4} e^{6} + 130 \, a^{4} c d^{2} e^{8} - 47 \, a^{5} e^{10} + 60 \, {\left (c^{5} d^{8} e^{2} - 3 \, a c^{4} d^{6} e^{4} + 3 \, a^{2} c^{3} d^{4} e^{6} - a^{3} c^{2} d^{2} e^{8}\right )} x^{2} + 15 \, {\left (c^{5} d^{9} e + 4 \, a c^{4} d^{7} e^{3} - 18 \, a^{2} c^{3} d^{5} e^{5} + 20 \, a^{3} c^{2} d^{3} e^{7} - 7 \, a^{4} c d e^{9}\right )} x}{6 \, {\left (c d x + a e\right )}^{3} c^{6} d^{6}} + \frac {c^{4} d^{4} e^{5} x^{2} + 10 \, c^{4} d^{5} e^{4} x - 8 \, a c^{3} d^{3} e^{6} x}{2 \, c^{8} d^{8}} \]

[In]

integrate((e*x+d)^9/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^4,x, algorithm="giac")

[Out]

10*(c^2*d^4*e^3 - 2*a*c*d^2*e^5 + a^2*e^7)*log(abs(c*d*x + a*e))/(c^6*d^6) - 1/6*(2*c^5*d^10 + 5*a*c^4*d^8*e^2
+ 20*a^2*c^3*d^6*e^4 - 110*a^3*c^2*d^4*e^6 + 130*a^4*c*d^2*e^8 - 47*a^5*e^10 + 60*(c^5*d^8*e^2 - 3*a*c^4*d^6*
e^4 + 3*a^2*c^3*d^4*e^6 - a^3*c^2*d^2*e^8)*x^2 + 15*(c^5*d^9*e + 4*a*c^4*d^7*e^3 - 18*a^2*c^3*d^5*e^5 + 20*a^3
*c^2*d^3*e^7 - 7*a^4*c*d*e^9)*x)/((c*d*x + a*e)^3*c^6*d^6) + 1/2*(c^4*d^4*e^5*x^2 + 10*c^4*d^5*e^4*x - 8*a*c^3
*d^3*e^6*x)/(c^8*d^8)

Mupad [B] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 331, normalized size of antiderivative = 1.85 \[ \int \frac {(d+e x)^9}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^4} \, dx=x\,\left (\frac {5\,e^4}{c^4\,d^3}-\frac {4\,a\,e^6}{c^5\,d^5}\right )-\frac {x^2\,\left (-10\,a^3\,c\,d\,e^8+30\,a^2\,c^2\,d^3\,e^6-30\,a\,c^3\,d^5\,e^4+10\,c^4\,d^7\,e^2\right )+x\,\left (-\frac {35\,a^4\,e^9}{2}+50\,a^3\,c\,d^2\,e^7-45\,a^2\,c^2\,d^4\,e^5+10\,a\,c^3\,d^6\,e^3+\frac {5\,c^4\,d^8\,e}{2}\right )+\frac {-47\,a^5\,e^{10}+130\,a^4\,c\,d^2\,e^8-110\,a^3\,c^2\,d^4\,e^6+20\,a^2\,c^3\,d^6\,e^4+5\,a\,c^4\,d^8\,e^2+2\,c^5\,d^{10}}{6\,c\,d}}{a^3\,c^5\,d^5\,e^3+3\,a^2\,c^6\,d^6\,e^2\,x+3\,a\,c^7\,d^7\,e\,x^2+c^8\,d^8\,x^3}+\frac {\ln \left (a\,e+c\,d\,x\right )\,\left (10\,a^2\,e^7-20\,a\,c\,d^2\,e^5+10\,c^2\,d^4\,e^3\right )}{c^6\,d^6}+\frac {e^5\,x^2}{2\,c^4\,d^4} \]

[In]

int((d + e*x)^9/(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^4,x)

[Out]

x*((5*e^4)/(c^4*d^3) - (4*a*e^6)/(c^5*d^5)) - (x^2*(10*c^4*d^7*e^2 - 30*a*c^3*d^5*e^4 + 30*a^2*c^2*d^3*e^6 - 1
0*a^3*c*d*e^8) + x*((5*c^4*d^8*e)/2 - (35*a^4*e^9)/2 + 10*a*c^3*d^6*e^3 + 50*a^3*c*d^2*e^7 - 45*a^2*c^2*d^4*e^
5) + (2*c^5*d^10 - 47*a^5*e^10 + 5*a*c^4*d^8*e^2 + 130*a^4*c*d^2*e^8 + 20*a^2*c^3*d^6*e^4 - 110*a^3*c^2*d^4*e^
6)/(6*c*d))/(c^8*d^8*x^3 + a^3*c^5*d^5*e^3 + 3*a*c^7*d^7*e*x^2 + 3*a^2*c^6*d^6*e^2*x) + (log(a*e + c*d*x)*(10*
a^2*e^7 + 10*c^2*d^4*e^3 - 20*a*c*d^2*e^5))/(c^6*d^6) + (e^5*x^2)/(2*c^4*d^4)