3.19.86 \(\int \frac {(d+e x)^8}{(a d e+(c d^2+a e^2) x+c d e x^2)^3} \, dx\) [1886]

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

[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 = 0.14, 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 = {640, 45} \begin {gather*} \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} \end {gather*}

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 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*} \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 = 0.06, size = 262, normalized size = 1.42 \begin {gather*} \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 )+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 )-60 e^2 \left (-c d^2+a e^2\right )^3 (a e+c d x)^2 \log (a e+c d x)}{6 c^6 d^6 (a e+c d x)^2} \end {gather*}

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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Maple [A]
time = 0.71, size = 297, normalized size = 1.61

method	result	size
default	\(\frac {e^{3} \left (\frac {1}{3} c^{2} d^{2} e^{2} x^{3}-\frac {3}{2} a c d \,e^{3} x^{2}+\frac {5}{2} c^{2} d^{3} e \,x^{2}+6 a^{2} e^{4} x -15 a c \,d^{2} e^{2} x +10 c^{2} d^{4} x \right )}{c^{5} d^{5}}-\frac {5 e \left (a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}\right )}{c^{6} d^{6} \left (c d x +a e \right )}-\frac {10 e^{2} \left (e^{6} a^{3}-3 e^{4} d^{2} a^{2} c +3 d^{4} e^{2} c^{2} a -d^{6} c^{3}\right ) \ln \left (c d x +a e \right )}{c^{6} d^{6}}-\frac {-a^{5} e^{10}+5 a^{4} c \,d^{2} e^{8}-10 a^{3} c^{2} d^{4} e^{6}+10 a^{2} c^{3} d^{6} e^{4}-5 a \,c^{4} d^{8} e^{2}+c^{5} d^{10}}{2 c^{6} d^{6} \left (c d x +a e \right )^{2}}\)	\(297\)
risch	\(\frac {e^{5} x^{3}}{3 c^{3} d^{3}}-\frac {3 e^{6} a \,x^{2}}{2 c^{4} d^{4}}+\frac {5 e^{4} x^{2}}{2 c^{3} d^{2}}+\frac {6 e^{7} a^{2} x}{c^{5} d^{5}}-\frac {15 e^{5} a x}{c^{4} d^{3}}+\frac {10 e^{3} x}{c^{3} d}+\frac {\left (-5 a^{4} e^{9}+20 e^{7} a^{3} d^{2} c -30 a^{2} e^{5} d^{4} c^{2}+20 a \,e^{3} d^{6} c^{3}-5 e \,d^{8} c^{4}\right ) x -\frac {9 a^{5} e^{10}-35 a^{4} c \,d^{2} e^{8}+50 a^{3} c^{2} d^{4} e^{6}-30 a^{2} c^{3} d^{6} e^{4}+5 a \,c^{4} d^{8} e^{2}+c^{5} d^{10}}{2 c d}}{c^{5} d^{5} \left (c d x +a e \right )^{2}}-\frac {10 e^{8} \ln \left (c d x +a e \right ) a^{3}}{c^{6} d^{6}}+\frac {30 e^{6} \ln \left (c d x +a e \right ) a^{2}}{c^{5} d^{4}}-\frac {30 e^{4} \ln \left (c d x +a e \right ) a}{c^{4} d^{2}}+\frac {10 e^{2} \ln \left (c d x +a e \right )}{c^{3}}\)	\(321\)
norman	\(\frac {-\frac {90 a^{5} e^{10}-230 a^{4} c \,d^{2} e^{8}+145 a^{3} c^{2} d^{4} e^{6}+45 a^{2} c^{3} d^{6} e^{4}+15 a \,c^{4} d^{8} e^{2}+3 c^{5} d^{10}}{6 d^{4} c^{6}}+\frac {e^{7} x^{7}}{3 c d}-\frac {\left (90 a^{5} e^{14}+10 a^{4} c \,d^{2} e^{12}-415 a^{3} c^{2} d^{4} e^{10}+305 a^{2} c^{3} d^{6} e^{8}+190 a \,c^{4} d^{8} e^{6}+198 c^{5} d^{10} e^{4}\right ) x^{2}}{6 c^{6} d^{6} e^{2}}-\frac {\left (90 a^{5} e^{12}-170 a^{4} c \,d^{2} e^{10}+5 a^{3} c^{2} d^{4} e^{8}+100 a^{2} d^{6} e^{6} c^{3}+90 a \,c^{4} d^{8} e^{4}+18 c^{5} d^{10} e^{2}\right ) x}{3 c^{6} d^{5} e}-\frac {\left (60 a^{4} e^{12}-140 a^{3} c \,d^{2} e^{10}+85 d^{4} a^{2} c^{2} e^{8}-20 a \,c^{3} d^{6} e^{6}+120 c^{4} d^{8} e^{4}\right ) x^{3}}{3 c^{5} d^{5} e}+\frac {e^{5} \left (10 a^{2} e^{4}-35 a c \,d^{2} e^{2}+46 c^{2} d^{4}\right ) x^{5}}{3 c^{3} d^{3}}-\frac {e^{6} \left (5 e^{2} a -19 c \,d^{2}\right ) x^{6}}{6 c^{2} d^{2}}}{\left (c d x +a e \right )^{2} \left (e x +d \right )^{2}}-\frac {10 e^{2} \left (e^{6} a^{3}-3 e^{4} d^{2} a^{2} c +3 d^{4} e^{2} c^{2} a -d^{6} c^{3}\right ) \ln \left (c d x +a e \right )}{c^{6} d^{6}}\)	\(482\)

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,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.31, size = 293, normalized size = 1.58 \begin {gather*} -\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} x e + a^{2} c^{6} d^{6} e^{2}\right )}} + \frac {2 \, c^{2} d^{2} x^{3} e^{5} + 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}} \end {gather*}

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*x*e + a^2*c^6*d^6*e^2) + 1/6*(2*c^2*d^2*x^3*e^5 + 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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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 455 vs. \(2 (173) = 346\).
time = 2.53, size = 455, normalized size = 2.46 \begin {gather*} -\frac {30 \, c^{5} d^{9} x e + 3 \, c^{5} d^{10} + 15 \, a c^{4} d^{8} e^{2} - 6 \, a^{4} c d x e^{9} + 27 \, a^{5} e^{10} - 21 \, {\left (3 \, a^{3} c^{2} d^{2} x^{2} + 5 \, a^{4} c d^{2}\right )} e^{8} - 10 \, {\left (2 \, a^{2} c^{3} d^{3} x^{3} + 3 \, a^{3} c^{2} d^{3} x\right )} e^{7} + 5 \, {\left (a c^{4} d^{4} x^{4} + 33 \, a^{2} c^{3} d^{4} x^{2} + 30 \, a^{3} c^{2} d^{4}\right )} e^{6} - 2 \, {\left (c^{5} d^{5} x^{5} - 30 \, a c^{4} d^{5} x^{3} - 60 \, a^{2} c^{3} d^{5} x\right )} e^{5} - 15 \, {\left (c^{5} d^{6} x^{4} + 8 \, a c^{4} d^{6} x^{2} + 6 \, a^{2} c^{3} d^{6}\right )} e^{4} - 60 \, {\left (c^{5} d^{7} x^{3} + 2 \, a c^{4} d^{7} x\right )} e^{3} - 60 \, {\left (c^{5} d^{8} x^{2} e^{2} + 2 \, a c^{4} d^{7} x e^{3} - 6 \, a^{2} c^{3} d^{5} x e^{5} + 6 \, a^{3} c^{2} d^{3} x e^{7} - 2 \, a^{4} c d x e^{9} - a^{5} e^{10} - {\left (a^{3} c^{2} d^{2} x^{2} - 3 \, a^{4} c d^{2}\right )} e^{8} + 3 \, {\left (a^{2} c^{3} d^{4} x^{2} - a^{3} c^{2} d^{4}\right )} e^{6} - {\left (3 \, a c^{4} d^{6} x^{2} - a^{2} c^{3} d^{6}\right )} e^{4}\right )} \log \left (c d x + a e\right )}{6 \, {\left (c^{8} d^{8} x^{2} + 2 \, a c^{7} d^{7} x e + a^{2} c^{6} d^{6} e^{2}\right )}} \end {gather*}

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*(30*c^5*d^9*x*e + 3*c^5*d^10 + 15*a*c^4*d^8*e^2 - 6*a^4*c*d*x*e^9 + 27*a^5*e^10 - 21*(3*a^3*c^2*d^2*x^2 +
5*a^4*c*d^2)*e^8 - 10*(2*a^2*c^3*d^3*x^3 + 3*a^3*c^2*d^3*x)*e^7 + 5*(a*c^4*d^4*x^4 + 33*a^2*c^3*d^4*x^2 + 30*
a^3*c^2*d^4)*e^6 - 2*(c^5*d^5*x^5 - 30*a*c^4*d^5*x^3 - 60*a^2*c^3*d^5*x)*e^5 - 15*(c^5*d^6*x^4 + 8*a*c^4*d^6*x
^2 + 6*a^2*c^3*d^6)*e^4 - 60*(c^5*d^7*x^3 + 2*a*c^4*d^7*x)*e^3 - 60*(c^5*d^8*x^2*e^2 + 2*a*c^4*d^7*x*e^3 - 6*a
^2*c^3*d^5*x*e^5 + 6*a^3*c^2*d^3*x*e^7 - 2*a^4*c*d*x*e^9 - a^5*e^10 - (a^3*c^2*d^2*x^2 - 3*a^4*c*d^2)*e^8 + 3*
(a^2*c^3*d^4*x^2 - a^3*c^2*d^4)*e^6 - (3*a*c^4*d^6*x^2 - a^2*c^3*d^6)*e^4)*log(c*d*x + a*e))/(c^8*d^8*x^2 + 2*
a*c^7*d^7*x*e + a^2*c^6*d^6*e^2)

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Sympy [A]
time = 46.27, size = 303, normalized size = 1.64 \begin {gather*} 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}} \end {gather*}

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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Giac [A]
time = 0.80, size = 285, normalized size = 1.54 \begin {gather*} \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 ({\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} - 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 d x + a e\right )}^{2} c^{6} d^{6}} + \frac {2 \, c^{6} d^{6} x^{3} e^{5} + 15 \, c^{6} d^{7} x^{2} e^{4} + 60 \, c^{6} d^{8} x e^{3} - 9 \, a c^{5} d^{5} x^{2} e^{6} - 90 \, a c^{5} d^{6} x e^{5} + 36 \, a^{2} c^{4} d^{4} x e^{7}}{6 \, c^{9} d^{9}} \end {gather*}

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^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(abs(c*d*x + a*e))/(c^6*d^6) - 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 + 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)/((c*d*x + a*e)^2*c^6*d^6) + 1/6*(2*
c^6*d^6*x^3*e^5 + 15*c^6*d^7*x^2*e^4 + 60*c^6*d^8*x*e^3 - 9*a*c^5*d^5*x^2*e^6 - 90*a*c^5*d^6*x*e^5 + 36*a^2*c^
4*d^4*x*e^7)/(c^9*d^9)

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Mupad [B]
time = 0.62, size = 341, normalized size = 1.84 \begin {gather*} 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} \end {gather*}

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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