3.1886 \(\int \frac {(d+e x)^8}{(a d e+(c d^2+a e^2) x+c d e x^2)^3} \, dx\)

Optimal. Leaf size=185 \[ \frac {e^3 x \left (6 a^2 e^4-15 a c d^2 e^2+10 c^2 d^4\right )}{c^5 d^5}-\frac {5 e \left (c d^2-a e^2\right )^4}{c^6 d^6 (a e+c d x)}-\frac {\left (c d^2-a e^2\right )^5}{2 c^6 d^6 (a e+c d x)^2}+\frac {10 e^2 \left (c d^2-a e^2\right )^3 \log (a e+c d x)}{c^6 d^6}+\frac {e^4 x^2 \left (5 c d^2-3 a e^2\right )}{2 c^4 d^4}+\frac {e^5 x^3}{3 c^3 d^3} \]

[Out]

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

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Rubi [A]  time = 0.18, antiderivative size = 185, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, Rules used = {626, 43} \[ \frac {e^3 x \left (6 a^2 e^4-15 a c d^2 e^2+10 c^2 d^4\right )}{c^5 d^5}+\frac {e^4 x^2 \left (5 c d^2-3 a e^2\right )}{2 c^4 d^4}-\frac {5 e \left (c d^2-a e^2\right )^4}{c^6 d^6 (a e+c d x)}-\frac {\left (c d^2-a e^2\right )^5}{2 c^6 d^6 (a e+c d x)^2}+\frac {10 e^2 \left (c d^2-a e^2\right )^3 \log (a e+c d x)}{c^6 d^6}+\frac {e^5 x^3}{3 c^3 d^3} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^8/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3,x]

[Out]

(e^3*(10*c^2*d^4 - 15*a*c*d^2*e^2 + 6*a^2*e^4)*x)/(c^5*d^5) + (e^4*(5*c*d^2 - 3*a*e^2)*x^2)/(2*c^4*d^4) + (e^5
*x^3)/(3*c^3*d^3) - (c*d^2 - a*e^2)^5/(2*c^6*d^6*(a*e + c*d*x)^2) - (5*e*(c*d^2 - a*e^2)^4)/(c^6*d^6*(a*e + c*
d*x)) + (10*e^2*(c*d^2 - a*e^2)^3*Log[a*e + c*d*x])/(c^6*d^6)

Rule 43

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 626

Int[((d_) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[(d + e*x)^(m + p)*(a
/d + (c*x)/e)^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*} \int \frac {(d+e x)^8}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3} \, dx &=\int \frac {(d+e x)^5}{(a e+c d x)^3} \, dx\\ &=\int \left (\frac {10 c^2 d^4 e^3-15 a c d^2 e^5+6 a^2 e^7}{c^5 d^5}+\frac {e^4 \left (5 c d^2-3 a e^2\right ) x}{c^4 d^4}+\frac {e^5 x^2}{c^3 d^3}+\frac {\left (c d^2-a e^2\right )^5}{c^5 d^5 (a e+c d x)^3}+\frac {5 e \left (c d^2-a e^2\right )^4}{c^5 d^5 (a e+c d x)^2}+\frac {10 e^2 \left (c d^2-a e^2\right )^3}{c^5 d^5 (a e+c d x)}\right ) \, dx\\ &=\frac {e^3 \left (10 c^2 d^4-15 a c d^2 e^2+6 a^2 e^4\right ) x}{c^5 d^5}+\frac {e^4 \left (5 c d^2-3 a e^2\right ) x^2}{2 c^4 d^4}+\frac {e^5 x^3}{3 c^3 d^3}-\frac {\left (c d^2-a e^2\right )^5}{2 c^6 d^6 (a e+c d x)^2}-\frac {5 e \left (c d^2-a e^2\right )^4}{c^6 d^6 (a e+c d x)}+\frac {10 e^2 \left (c d^2-a e^2\right )^3 \log (a e+c d x)}{c^6 d^6}\\ \end {align*}

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Mathematica [A]  time = 0.09, size = 262, normalized size = 1.42 \[ \frac {-27 a^5 e^{10}+3 a^4 c d e^8 (35 d+2 e x)+3 a^3 c^2 d^2 e^6 \left (-50 d^2+10 d e x+21 e^2 x^2\right )+5 a^2 c^3 d^3 e^4 \left (18 d^3-24 d^2 e x-33 d e^2 x^2+4 e^3 x^3\right )-5 a c^4 d^4 e^2 \left (3 d^4-24 d^3 e x-24 d^2 e^2 x^2+12 d e^3 x^3+e^4 x^4\right )-60 e^2 \left (a e^2-c d^2\right )^3 (a e+c d x)^2 \log (a e+c d x)+c^5 d^5 \left (-3 d^5-30 d^4 e x+60 d^2 e^3 x^3+15 d e^4 x^4+2 e^5 x^5\right )}{6 c^6 d^6 (a e+c d x)^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^8/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3,x]

[Out]

(-27*a^5*e^10 + 3*a^4*c*d*e^8*(35*d + 2*e*x) + 3*a^3*c^2*d^2*e^6*(-50*d^2 + 10*d*e*x + 21*e^2*x^2) + 5*a^2*c^3
*d^3*e^4*(18*d^3 - 24*d^2*e*x - 33*d*e^2*x^2 + 4*e^3*x^3) - 5*a*c^4*d^4*e^2*(3*d^4 - 24*d^3*e*x - 24*d^2*e^2*x
^2 + 12*d*e^3*x^3 + e^4*x^4) + c^5*d^5*(-3*d^5 - 30*d^4*e*x + 60*d^2*e^3*x^3 + 15*d*e^4*x^4 + 2*e^5*x^5) - 60*
e^2*(-(c*d^2) + a*e^2)^3*(a*e + c*d*x)^2*Log[a*e + c*d*x])/(6*c^6*d^6*(a*e + c*d*x)^2)

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fricas [B]  time = 1.18, size = 469, normalized size = 2.54 \[ \frac {2 \, c^{5} d^{5} e^{5} x^{5} - 3 \, c^{5} d^{10} - 15 \, a c^{4} d^{8} e^{2} + 90 \, a^{2} c^{3} d^{6} e^{4} - 150 \, a^{3} c^{2} d^{4} e^{6} + 105 \, a^{4} c d^{2} e^{8} - 27 \, a^{5} e^{10} + 5 \, {\left (3 \, c^{5} d^{6} e^{4} - a c^{4} d^{4} e^{6}\right )} x^{4} + 20 \, {\left (3 \, c^{5} d^{7} e^{3} - 3 \, a c^{4} d^{5} e^{5} + a^{2} c^{3} d^{3} e^{7}\right )} x^{3} + 3 \, {\left (40 \, a c^{4} d^{6} e^{4} - 55 \, a^{2} c^{3} d^{4} e^{6} + 21 \, a^{3} c^{2} d^{2} e^{8}\right )} x^{2} - 6 \, {\left (5 \, c^{5} d^{9} e - 20 \, a c^{4} d^{7} e^{3} + 20 \, a^{2} c^{3} d^{5} e^{5} - 5 \, a^{3} c^{2} d^{3} e^{7} - a^{4} c d e^{9}\right )} x + 60 \, {\left (a^{2} c^{3} d^{6} e^{4} - 3 \, a^{3} c^{2} d^{4} e^{6} + 3 \, a^{4} c d^{2} e^{8} - a^{5} e^{10} + {\left (c^{5} d^{8} e^{2} - 3 \, a c^{4} d^{6} e^{4} + 3 \, a^{2} c^{3} d^{4} e^{6} - a^{3} c^{2} d^{2} e^{8}\right )} x^{2} + 2 \, {\left (a c^{4} d^{7} e^{3} - 3 \, a^{2} c^{3} d^{5} e^{5} + 3 \, a^{3} c^{2} d^{3} e^{7} - a^{4} c d e^{9}\right )} x\right )} \log \left (c d x + a e\right )}{6 \, {\left (c^{8} d^{8} x^{2} + 2 \, a c^{7} d^{7} e x + a^{2} c^{6} d^{6} e^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^8/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^3,x, algorithm="fricas")

[Out]

1/6*(2*c^5*d^5*e^5*x^5 - 3*c^5*d^10 - 15*a*c^4*d^8*e^2 + 90*a^2*c^3*d^6*e^4 - 150*a^3*c^2*d^4*e^6 + 105*a^4*c*
d^2*e^8 - 27*a^5*e^10 + 5*(3*c^5*d^6*e^4 - a*c^4*d^4*e^6)*x^4 + 20*(3*c^5*d^7*e^3 - 3*a*c^4*d^5*e^5 + a^2*c^3*
d^3*e^7)*x^3 + 3*(40*a*c^4*d^6*e^4 - 55*a^2*c^3*d^4*e^6 + 21*a^3*c^2*d^2*e^8)*x^2 - 6*(5*c^5*d^9*e - 20*a*c^4*
d^7*e^3 + 20*a^2*c^3*d^5*e^5 - 5*a^3*c^2*d^3*e^7 - a^4*c*d*e^9)*x + 60*(a^2*c^3*d^6*e^4 - 3*a^3*c^2*d^4*e^6 +
3*a^4*c*d^2*e^8 - a^5*e^10 + (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 + 2*(a*
c^4*d^7*e^3 - 3*a^2*c^3*d^5*e^5 + 3*a^3*c^2*d^3*e^7 - a^4*c*d*e^9)*x)*log(c*d*x + a*e))/(c^8*d^8*x^2 + 2*a*c^7
*d^7*e*x + a^2*c^6*d^6*e^2)

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giac [B]  time = 12.11, size = 933, normalized size = 5.04 \[ \frac {10 \, {\left (c^{8} d^{16} e^{2} - 8 \, a c^{7} d^{14} e^{4} + 28 \, a^{2} c^{6} d^{12} e^{6} - 56 \, a^{3} c^{5} d^{10} e^{8} + 70 \, a^{4} c^{4} d^{8} e^{10} - 56 \, a^{5} c^{3} d^{6} e^{12} + 28 \, a^{6} c^{2} d^{4} e^{14} - 8 \, a^{7} c d^{2} e^{16} + a^{8} e^{18}\right )} \arctan \left (\frac {2 \, c d x e + c d^{2} + a e^{2}}{\sqrt {-c^{2} d^{4} + 2 \, a c d^{2} e^{2} - a^{2} e^{4}}}\right )}{{\left (c^{10} d^{14} - 4 \, a c^{9} d^{12} e^{2} + 6 \, a^{2} c^{8} d^{10} e^{4} - 4 \, a^{3} c^{7} d^{8} e^{6} + a^{4} c^{6} d^{6} e^{8}\right )} \sqrt {-c^{2} d^{4} + 2 \, a c d^{2} e^{2} - a^{2} e^{4}}} + \frac {5 \, {\left (c^{3} d^{6} e^{2} - 3 \, a c^{2} d^{4} e^{4} + 3 \, a^{2} c d^{2} e^{6} - a^{3} e^{8}\right )} \log \left (c d x^{2} e + c d^{2} x + a x e^{2} + a d e\right )}{c^{6} d^{6}} - \frac {c^{9} d^{20} + a c^{8} d^{18} e^{2} - 44 \, a^{2} c^{7} d^{16} e^{4} + 196 \, a^{3} c^{6} d^{14} e^{6} - 434 \, a^{4} c^{5} d^{12} e^{8} + 574 \, a^{5} c^{4} d^{10} e^{10} - 476 \, a^{6} c^{3} d^{8} e^{12} + 244 \, a^{7} c^{2} d^{6} e^{14} - 71 \, a^{8} c d^{4} e^{16} + 9 \, a^{9} d^{2} e^{18} + 10 \, {\left (c^{9} d^{17} e^{3} - 8 \, a c^{8} d^{15} e^{5} + 28 \, a^{2} c^{7} d^{13} e^{7} - 56 \, a^{3} c^{6} d^{11} e^{9} + 70 \, a^{4} c^{5} d^{9} e^{11} - 56 \, a^{5} c^{4} d^{7} e^{13} + 28 \, a^{6} c^{3} d^{5} e^{15} - 8 \, a^{7} c^{2} d^{3} e^{17} + a^{8} c d e^{19}\right )} x^{3} + 3 \, {\left (7 \, c^{9} d^{18} e^{2} - 53 \, a c^{8} d^{16} e^{4} + 172 \, a^{2} c^{7} d^{14} e^{6} - 308 \, a^{3} c^{6} d^{12} e^{8} + 322 \, a^{4} c^{5} d^{10} e^{10} - 182 \, a^{5} c^{4} d^{8} e^{12} + 28 \, a^{6} c^{3} d^{6} e^{14} + 28 \, a^{7} c^{2} d^{4} e^{16} - 17 \, a^{8} c d^{2} e^{18} + 3 \, a^{9} e^{20}\right )} x^{2} + 6 \, {\left (2 \, c^{9} d^{19} e - 13 \, a c^{8} d^{17} e^{3} + 32 \, a^{2} c^{7} d^{15} e^{5} - 28 \, a^{3} c^{6} d^{13} e^{7} - 28 \, a^{4} c^{5} d^{11} e^{9} + 98 \, a^{5} c^{4} d^{9} e^{11} - 112 \, a^{6} c^{3} d^{7} e^{13} + 68 \, a^{7} c^{2} d^{5} e^{15} - 22 \, a^{8} c d^{3} e^{17} + 3 \, a^{9} d e^{19}\right )} x}{2 \, {\left (c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )}^{2} {\left (c d x^{2} e + c d^{2} x + a x e^{2} + a d e\right )}^{2} c^{6} d^{6}} + \frac {{\left (2 \, c^{6} d^{6} x^{3} e^{14} + 15 \, c^{6} d^{7} x^{2} e^{13} + 60 \, c^{6} d^{8} x e^{12} - 9 \, a c^{5} d^{5} x^{2} e^{15} - 90 \, a c^{5} d^{6} x e^{14} + 36 \, a^{2} c^{4} d^{4} x e^{16}\right )} e^{\left (-9\right )}}{6 \, c^{9} d^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^8/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^3,x, algorithm="giac")

[Out]

10*(c^8*d^16*e^2 - 8*a*c^7*d^14*e^4 + 28*a^2*c^6*d^12*e^6 - 56*a^3*c^5*d^10*e^8 + 70*a^4*c^4*d^8*e^10 - 56*a^5
*c^3*d^6*e^12 + 28*a^6*c^2*d^4*e^14 - 8*a^7*c*d^2*e^16 + a^8*e^18)*arctan((2*c*d*x*e + c*d^2 + a*e^2)/sqrt(-c^
2*d^4 + 2*a*c*d^2*e^2 - a^2*e^4))/((c^10*d^14 - 4*a*c^9*d^12*e^2 + 6*a^2*c^8*d^10*e^4 - 4*a^3*c^7*d^8*e^6 + a^
4*c^6*d^6*e^8)*sqrt(-c^2*d^4 + 2*a*c*d^2*e^2 - a^2*e^4)) + 5*(c^3*d^6*e^2 - 3*a*c^2*d^4*e^4 + 3*a^2*c*d^2*e^6
- a^3*e^8)*log(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e)/(c^6*d^6) - 1/2*(c^9*d^20 + a*c^8*d^18*e^2 - 44*a^2*c^7*
d^16*e^4 + 196*a^3*c^6*d^14*e^6 - 434*a^4*c^5*d^12*e^8 + 574*a^5*c^4*d^10*e^10 - 476*a^6*c^3*d^8*e^12 + 244*a^
7*c^2*d^6*e^14 - 71*a^8*c*d^4*e^16 + 9*a^9*d^2*e^18 + 10*(c^9*d^17*e^3 - 8*a*c^8*d^15*e^5 + 28*a^2*c^7*d^13*e^
7 - 56*a^3*c^6*d^11*e^9 + 70*a^4*c^5*d^9*e^11 - 56*a^5*c^4*d^7*e^13 + 28*a^6*c^3*d^5*e^15 - 8*a^7*c^2*d^3*e^17
+ a^8*c*d*e^19)*x^3 + 3*(7*c^9*d^18*e^2 - 53*a*c^8*d^16*e^4 + 172*a^2*c^7*d^14*e^6 - 308*a^3*c^6*d^12*e^8 + 3
22*a^4*c^5*d^10*e^10 - 182*a^5*c^4*d^8*e^12 + 28*a^6*c^3*d^6*e^14 + 28*a^7*c^2*d^4*e^16 - 17*a^8*c*d^2*e^18 +
3*a^9*e^20)*x^2 + 6*(2*c^9*d^19*e - 13*a*c^8*d^17*e^3 + 32*a^2*c^7*d^15*e^5 - 28*a^3*c^6*d^13*e^7 - 28*a^4*c^5
*d^11*e^9 + 98*a^5*c^4*d^9*e^11 - 112*a^6*c^3*d^7*e^13 + 68*a^7*c^2*d^5*e^15 - 22*a^8*c*d^3*e^17 + 3*a^9*d*e^1
9)*x)/((c^2*d^4 - 2*a*c*d^2*e^2 + a^2*e^4)^2*(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e)^2*c^6*d^6) + 1/6*(2*c^6*d
^6*x^3*e^14 + 15*c^6*d^7*x^2*e^13 + 60*c^6*d^8*x*e^12 - 9*a*c^5*d^5*x^2*e^15 - 90*a*c^5*d^6*x*e^14 + 36*a^2*c^
4*d^4*x*e^16)*e^(-9)/(c^9*d^9)

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maple [B]  time = 0.05, size = 412, normalized size = 2.23 \[ \frac {a^{5} e^{10}}{2 \left (c d x +a e \right )^{2} c^{6} d^{6}}-\frac {5 a^{4} e^{8}}{2 \left (c d x +a e \right )^{2} c^{5} d^{4}}+\frac {5 a^{3} e^{6}}{\left (c d x +a e \right )^{2} c^{4} d^{2}}-\frac {5 a^{2} e^{4}}{\left (c d x +a e \right )^{2} c^{3}}+\frac {5 a \,d^{2} e^{2}}{2 \left (c d x +a e \right )^{2} c^{2}}-\frac {d^{4}}{2 \left (c d x +a e \right )^{2} c}+\frac {e^{5} x^{3}}{3 c^{3} d^{3}}-\frac {5 a^{4} e^{9}}{\left (c d x +a e \right ) c^{6} d^{6}}+\frac {20 a^{3} e^{7}}{\left (c d x +a e \right ) c^{5} d^{4}}-\frac {30 a^{2} e^{5}}{\left (c d x +a e \right ) c^{4} d^{2}}+\frac {20 a \,e^{3}}{\left (c d x +a e \right ) c^{3}}-\frac {3 a \,e^{6} x^{2}}{2 c^{4} d^{4}}-\frac {5 d^{2} e}{\left (c d x +a e \right ) c^{2}}+\frac {5 e^{4} x^{2}}{2 c^{3} d^{2}}-\frac {10 a^{3} e^{8} \ln \left (c d x +a e \right )}{c^{6} d^{6}}+\frac {30 a^{2} e^{6} \ln \left (c d x +a e \right )}{c^{5} d^{4}}+\frac {6 a^{2} e^{7} x}{c^{5} d^{5}}-\frac {30 a \,e^{4} \ln \left (c d x +a e \right )}{c^{4} d^{2}}-\frac {15 a \,e^{5} x}{c^{4} d^{3}}+\frac {10 e^{3} x}{c^{3} d}+\frac {10 e^{2} \ln \left (c d x +a e \right )}{c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^8/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^3,x)

[Out]

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

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maxima [A]  time = 1.28, size = 310, normalized size = 1.68 \[ -\frac {c^{5} d^{10} + 5 \, a c^{4} d^{8} e^{2} - 30 \, a^{2} c^{3} d^{6} e^{4} + 50 \, a^{3} c^{2} d^{4} e^{6} - 35 \, a^{4} c d^{2} e^{8} + 9 \, a^{5} e^{10} + 10 \, {\left (c^{5} d^{9} e - 4 \, a c^{4} d^{7} e^{3} + 6 \, a^{2} c^{3} d^{5} e^{5} - 4 \, a^{3} c^{2} d^{3} e^{7} + a^{4} c d e^{9}\right )} x}{2 \, {\left (c^{8} d^{8} x^{2} + 2 \, a c^{7} d^{7} e x + a^{2} c^{6} d^{6} e^{2}\right )}} + \frac {2 \, c^{2} d^{2} e^{5} x^{3} + 3 \, {\left (5 \, c^{2} d^{3} e^{4} - 3 \, a c d e^{6}\right )} x^{2} + 6 \, {\left (10 \, c^{2} d^{4} e^{3} - 15 \, a c d^{2} e^{5} + 6 \, a^{2} e^{7}\right )} x}{6 \, c^{5} d^{5}} + \frac {10 \, {\left (c^{3} d^{6} e^{2} - 3 \, a c^{2} d^{4} e^{4} + 3 \, a^{2} c d^{2} e^{6} - a^{3} e^{8}\right )} \log \left (c d x + a e\right )}{c^{6} d^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^8/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^3,x, algorithm="maxima")

[Out]

-1/2*(c^5*d^10 + 5*a*c^4*d^8*e^2 - 30*a^2*c^3*d^6*e^4 + 50*a^3*c^2*d^4*e^6 - 35*a^4*c*d^2*e^8 + 9*a^5*e^10 + 1
0*(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)/(c^8*d^8*x^2 + 2*a*c^
7*d^7*e*x + a^2*c^6*d^6*e^2) + 1/6*(2*c^2*d^2*e^5*x^3 + 3*(5*c^2*d^3*e^4 - 3*a*c*d*e^6)*x^2 + 6*(10*c^2*d^4*e^
3 - 15*a*c*d^2*e^5 + 6*a^2*e^7)*x)/(c^5*d^5) + 10*(c^3*d^6*e^2 - 3*a*c^2*d^4*e^4 + 3*a^2*c*d^2*e^6 - a^3*e^8)*
log(c*d*x + a*e)/(c^6*d^6)

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mupad [B]  time = 0.62, size = 341, normalized size = 1.84 \[ x^2\,\left (\frac {5\,e^4}{2\,c^3\,d^2}-\frac {3\,a\,e^6}{2\,c^4\,d^4}\right )-x\,\left (\frac {3\,a^2\,e^7}{c^5\,d^5}-\frac {10\,e^3}{c^3\,d}+\frac {3\,a\,e\,\left (\frac {5\,e^4}{c^3\,d^2}-\frac {3\,a\,e^6}{c^4\,d^4}\right )}{c\,d}\right )-\frac {x\,\left (5\,a^4\,e^9-20\,a^3\,c\,d^2\,e^7+30\,a^2\,c^2\,d^4\,e^5-20\,a\,c^3\,d^6\,e^3+5\,c^4\,d^8\,e\right )+\frac {9\,a^5\,e^{10}-35\,a^4\,c\,d^2\,e^8+50\,a^3\,c^2\,d^4\,e^6-30\,a^2\,c^3\,d^6\,e^4+5\,a\,c^4\,d^8\,e^2+c^5\,d^{10}}{2\,c\,d}}{a^2\,c^5\,d^5\,e^2+2\,a\,c^6\,d^6\,e\,x+c^7\,d^7\,x^2}-\frac {\ln \left (a\,e+c\,d\,x\right )\,\left (10\,a^3\,e^8-30\,a^2\,c\,d^2\,e^6+30\,a\,c^2\,d^4\,e^4-10\,c^3\,d^6\,e^2\right )}{c^6\,d^6}+\frac {e^5\,x^3}{3\,c^3\,d^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)^8/(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^3,x)

[Out]

x^2*((5*e^4)/(2*c^3*d^2) - (3*a*e^6)/(2*c^4*d^4)) - x*((3*a^2*e^7)/(c^5*d^5) - (10*e^3)/(c^3*d) + (3*a*e*((5*e
^4)/(c^3*d^2) - (3*a*e^6)/(c^4*d^4)))/(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) + (9*a^5*e^10 + c^5*d^10 + 5*a*c^4*d^8*e^2 - 35*a^4*c*d^2*e^8 - 30*a^2*c^3*d^6*e^4 +
50*a^3*c^2*d^4*e^6)/(2*c*d))/(c^7*d^7*x^2 + a^2*c^5*d^5*e^2 + 2*a*c^6*d^6*e*x) - (log(a*e + c*d*x)*(10*a^3*e^8
- 10*c^3*d^6*e^2 + 30*a*c^2*d^4*e^4 - 30*a^2*c*d^2*e^6))/(c^6*d^6) + (e^5*x^3)/(3*c^3*d^3)

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sympy [A]  time = 4.85, size = 303, normalized size = 1.64 \[ x^{2} \left (- \frac {3 a e^{6}}{2 c^{4} d^{4}} + \frac {5 e^{4}}{2 c^{3} d^{2}}\right ) + x \left (\frac {6 a^{2} e^{7}}{c^{5} d^{5}} - \frac {15 a e^{5}}{c^{4} d^{3}} + \frac {10 e^{3}}{c^{3} d}\right ) + \frac {- 9 a^{5} e^{10} + 35 a^{4} c d^{2} e^{8} - 50 a^{3} c^{2} d^{4} e^{6} + 30 a^{2} c^{3} d^{6} e^{4} - 5 a c^{4} d^{8} e^{2} - c^{5} d^{10} + x \left (- 10 a^{4} c d e^{9} + 40 a^{3} c^{2} d^{3} e^{7} - 60 a^{2} c^{3} d^{5} e^{5} + 40 a c^{4} d^{7} e^{3} - 10 c^{5} d^{9} e\right )}{2 a^{2} c^{6} d^{6} e^{2} + 4 a c^{7} d^{7} e x + 2 c^{8} d^{8} x^{2}} + \frac {e^{5} x^{3}}{3 c^{3} d^{3}} - \frac {10 e^{2} \left (a e^{2} - c d^{2}\right )^{3} \log {\left (a e + c d x \right )}}{c^{6} d^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**8/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**3,x)

[Out]

x**2*(-3*a*e**6/(2*c**4*d**4) + 5*e**4/(2*c**3*d**2)) + x*(6*a**2*e**7/(c**5*d**5) - 15*a*e**5/(c**4*d**3) + 1
0*e**3/(c**3*d)) + (-9*a**5*e**10 + 35*a**4*c*d**2*e**8 - 50*a**3*c**2*d**4*e**6 + 30*a**2*c**3*d**6*e**4 - 5*
a*c**4*d**8*e**2 - c**5*d**10 + x*(-10*a**4*c*d*e**9 + 40*a**3*c**2*d**3*e**7 - 60*a**2*c**3*d**5*e**5 + 40*a*
c**4*d**7*e**3 - 10*c**5*d**9*e))/(2*a**2*c**6*d**6*e**2 + 4*a*c**7*d**7*e*x + 2*c**8*d**8*x**2) + e**5*x**3/(
3*c**3*d**3) - 10*e**2*(a*e**2 - c*d**2)**3*log(a*e + c*d*x)/(c**6*d**6)

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