3.19.85 \(\int \frac {(d+e x)^9}{(a d e+(c d^2+a e^2) x+c d e x^2)^3} \, dx\) [1885]

Optimal. Leaf size=221 \[ \frac {20 e^3 \left (c d^2-a e^2\right )^3 x}{c^6 d^6}-\frac {\left (c d^2-a e^2\right )^6}{2 c^7 d^7 (a e+c d x)^2}-\frac {6 e \left (c d^2-a e^2\right )^5}{c^7 d^7 (a e+c d x)}+\frac {15 e^4 \left (c d^2-a e^2\right )^2 (a e+c d x)^2}{2 c^7 d^7}+\frac {2 e^5 \left (c d^2-a e^2\right ) (a e+c d x)^3}{c^7 d^7}+\frac {e^6 (a e+c d x)^4}{4 c^7 d^7}+\frac {15 e^2 \left (c d^2-a e^2\right )^4 \log (a e+c d x)}{c^7 d^7} \]

[Out]

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

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Rubi [A]
time = 0.19, antiderivative size = 221, 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^6 (a e+c d x)^4}{4 c^7 d^7}-\frac {6 e \left (c d^2-a e^2\right )^5}{c^7 d^7 (a e+c d x)}-\frac {\left (c d^2-a e^2\right )^6}{2 c^7 d^7 (a e+c d x)^2}+\frac {15 e^2 \left (c d^2-a e^2\right )^4 \log (a e+c d x)}{c^7 d^7}+\frac {2 e^5 \left (c d^2-a e^2\right ) (a e+c d x)^3}{c^7 d^7}+\frac {15 e^4 \left (c d^2-a e^2\right )^2 (a e+c d x)^2}{2 c^7 d^7}+\frac {20 e^3 x \left (c d^2-a e^2\right )^3}{c^6 d^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^9/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3,x]

[Out]

(20*e^3*(c*d^2 - a*e^2)^3*x)/(c^6*d^6) - (c*d^2 - a*e^2)^6/(2*c^7*d^7*(a*e + c*d*x)^2) - (6*e*(c*d^2 - a*e^2)^
5)/(c^7*d^7*(a*e + c*d*x)) + (15*e^4*(c*d^2 - a*e^2)^2*(a*e + c*d*x)^2)/(2*c^7*d^7) + (2*e^5*(c*d^2 - a*e^2)*(
a*e + c*d*x)^3)/(c^7*d^7) + (e^6*(a*e + c*d*x)^4)/(4*c^7*d^7) + (15*e^2*(c*d^2 - a*e^2)^4*Log[a*e + c*d*x])/(c
^7*d^7)

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)^9}{\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)^6}{(a e+c d x)^3} \, dx\\ &=\int \left (\frac {20 \left (c d^2 e-a e^3\right )^3}{c^6 d^6}+\frac {\left (c d^2-a e^2\right )^6}{c^6 d^6 (a e+c d x)^3}+\frac {6 e \left (c d^2-a e^2\right )^5}{c^6 d^6 (a e+c d x)^2}+\frac {15 e^2 \left (c d^2-a e^2\right )^4}{c^6 d^6 (a e+c d x)}+\frac {15 e^4 \left (c d^2-a e^2\right )^2 (a e+c d x)}{c^6 d^6}+\frac {6 \left (c d^2 e^5-a e^7\right ) (a e+c d x)^2}{c^6 d^6}+\frac {e^6 (a e+c d x)^3}{c^6 d^6}\right ) \, dx\\ &=\frac {20 e^3 \left (c d^2-a e^2\right )^3 x}{c^6 d^6}-\frac {\left (c d^2-a e^2\right )^6}{2 c^7 d^7 (a e+c d x)^2}-\frac {6 e \left (c d^2-a e^2\right )^5}{c^7 d^7 (a e+c d x)}+\frac {15 e^4 \left (c d^2-a e^2\right )^2 (a e+c d x)^2}{2 c^7 d^7}+\frac {2 e^5 \left (c d^2-a e^2\right ) (a e+c d x)^3}{c^7 d^7}+\frac {e^6 (a e+c d x)^4}{4 c^7 d^7}+\frac {15 e^2 \left (c d^2-a e^2\right )^4 \log (a e+c d x)}{c^7 d^7}\\ \end {align*}

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Mathematica [A]
time = 0.08, size = 337, normalized size = 1.52 \begin {gather*} \frac {22 a^6 e^{12}-4 a^5 c d e^{10} (27 d+4 e x)+2 a^4 c^2 d^2 e^8 \left (105 d^2+12 d e x-34 e^2 x^2\right )-4 a^3 c^3 d^3 e^6 \left (50 d^3-15 d^2 e x-63 d e^2 x^2+5 e^3 x^3\right )+5 a^2 c^4 d^4 e^4 \left (18 d^4-32 d^3 e x-66 d^2 e^2 x^2+16 d e^3 x^3+e^4 x^4\right )-2 a c^5 d^5 e^2 \left (6 d^5-60 d^4 e x-80 d^3 e^2 x^2+60 d^2 e^3 x^3+10 d e^4 x^4+e^5 x^5\right )+c^6 d^6 \left (-2 d^6-24 d^5 e x+80 d^3 e^3 x^3+30 d^2 e^4 x^4+8 d e^5 x^5+e^6 x^6\right )+60 e^2 \left (c d^2-a e^2\right )^4 (a e+c d x)^2 \log (a e+c d x)}{4 c^7 d^7 (a e+c d x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^9/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3,x]

[Out]

(22*a^6*e^12 - 4*a^5*c*d*e^10*(27*d + 4*e*x) + 2*a^4*c^2*d^2*e^8*(105*d^2 + 12*d*e*x - 34*e^2*x^2) - 4*a^3*c^3
*d^3*e^6*(50*d^3 - 15*d^2*e*x - 63*d*e^2*x^2 + 5*e^3*x^3) + 5*a^2*c^4*d^4*e^4*(18*d^4 - 32*d^3*e*x - 66*d^2*e^
2*x^2 + 16*d*e^3*x^3 + e^4*x^4) - 2*a*c^5*d^5*e^2*(6*d^5 - 60*d^4*e*x - 80*d^3*e^2*x^2 + 60*d^2*e^3*x^3 + 10*d
*e^4*x^4 + e^5*x^5) + c^6*d^6*(-2*d^6 - 24*d^5*e*x + 80*d^3*e^3*x^3 + 30*d^2*e^4*x^4 + 8*d*e^5*x^5 + e^6*x^6)
+ 60*e^2*(c*d^2 - a*e^2)^4*(a*e + c*d*x)^2*Log[a*e + c*d*x])/(4*c^7*d^7*(a*e + c*d*x)^2)

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Maple [A]
time = 0.80, size = 399, normalized size = 1.81

method	result	size
default	\(-\frac {e^{3} \left (-\frac {1}{4} c^{3} d^{3} e^{3} x^{4}+a \,c^{2} d^{2} e^{4} x^{3}-2 c^{3} d^{4} e^{2} x^{3}-3 a^{2} c d \,e^{5} x^{2}+9 a \,c^{2} d^{3} e^{3} x^{2}-\frac {15}{2} c^{3} d^{5} e \,x^{2}+10 e^{6} a^{3} x -36 e^{4} d^{2} a^{2} c x +45 d^{4} e^{2} c^{2} a x -20 d^{6} c^{3} x \right )}{c^{6} d^{6}}+\frac {6 e \left (a^{5} e^{10}-5 a^{4} c \,d^{2} e^{8}+10 a^{3} c^{2} d^{4} e^{6}-10 a^{2} c^{3} d^{6} e^{4}+5 a \,c^{4} d^{8} e^{2}-c^{5} d^{10}\right )}{c^{7} d^{7} \left (c d x +a e \right )}+\frac {15 e^{2} \left (a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}\right ) \ln \left (c d x +a e \right )}{c^{7} d^{7}}-\frac {a^{6} e^{12}-6 a^{5} c \,d^{2} e^{10}+15 a^{4} c^{2} d^{4} e^{8}-20 a^{3} c^{3} d^{6} e^{6}+15 a^{2} c^{4} d^{8} e^{4}-6 a \,c^{5} d^{10} e^{2}+c^{6} d^{12}}{2 c^{7} d^{7} \left (c d x +a e \right )^{2}}\)	\(399\)
risch	\(\frac {e^{6} x^{4}}{4 c^{3} d^{3}}-\frac {e^{7} a \,x^{3}}{c^{4} d^{4}}+\frac {2 e^{5} x^{3}}{c^{3} d^{2}}+\frac {3 e^{8} a^{2} x^{2}}{c^{5} d^{5}}-\frac {9 e^{6} a \,x^{2}}{c^{4} d^{3}}+\frac {15 e^{4} x^{2}}{2 c^{3} d}-\frac {10 e^{9} a^{3} x}{c^{6} d^{6}}+\frac {36 e^{7} a^{2} x}{c^{5} d^{4}}-\frac {45 e^{5} a x}{c^{4} d^{2}}+\frac {20 e^{3} x}{c^{3}}+\frac {\left (6 a^{5} e^{11}-30 a^{4} d^{2} e^{9} c +60 a^{3} c^{2} d^{4} e^{7}-60 a^{2} d^{6} e^{5} c^{3}+30 a \,c^{4} d^{8} e^{3}-6 d^{10} e \,c^{5}\right ) x +\frac {11 a^{6} e^{12}-54 a^{5} c \,d^{2} e^{10}+105 a^{4} c^{2} d^{4} e^{8}-100 a^{3} c^{3} d^{6} e^{6}+45 a^{2} c^{4} d^{8} e^{4}-6 a \,c^{5} d^{10} e^{2}-c^{6} d^{12}}{2 c d}}{c^{6} d^{6} \left (c d x +a e \right )^{2}}+\frac {15 e^{10} \ln \left (c d x +a e \right ) a^{4}}{c^{7} d^{7}}-\frac {60 e^{8} \ln \left (c d x +a e \right ) a^{3}}{c^{6} d^{5}}+\frac {90 e^{6} \ln \left (c d x +a e \right ) a^{2}}{c^{5} d^{3}}-\frac {60 e^{4} \ln \left (c d x +a e \right ) a}{c^{4} d}+\frac {15 d \,e^{2} \ln \left (c d x +a e \right )}{c^{3}}\)	\(432\)
norman	\(\frac {\frac {90 a^{6} e^{12}-320 a^{5} c \,d^{2} e^{10}+375 a^{4} c^{2} d^{4} e^{8}-100 a^{3} c^{3} d^{6} e^{6}-100 a^{2} c^{4} d^{8} e^{4}-12 a \,c^{5} d^{10} e^{2}-2 c^{6} d^{12}}{4 c^{7} d^{5}}+\frac {e^{8} x^{8}}{4 c d}+\frac {\left (90 a^{6} e^{16}-80 c \,d^{2} a^{5} e^{14}-425 a^{4} c^{2} d^{4} e^{12}+720 a^{3} c^{3} d^{6} e^{10}-185 a^{2} c^{4} d^{8} e^{8}-272 a \,c^{5} d^{10} e^{6}-240 c^{6} d^{12} e^{4}\right ) x^{2}}{4 c^{7} d^{7} e^{2}}+\frac {\left (60 a^{5} e^{14}-200 a^{4} c \,d^{2} e^{12}+225 a^{3} c^{2} d^{4} e^{10}-105 a^{2} c^{3} d^{6} e^{8}+70 a \,c^{4} d^{8} e^{6}-162 c^{5} d^{10} e^{4}\right ) x^{3}}{2 c^{6} d^{6} e}-\frac {e^{5} \left (10 e^{6} a^{3}-45 e^{4} d^{2} a^{2} c +81 d^{4} e^{2} c^{2} a -74 d^{6} c^{3}\right ) x^{5}}{2 c^{4} d^{4}}+\frac {e^{6} \left (5 a^{2} e^{4}-24 a c \,d^{2} e^{2}+47 c^{2} d^{4}\right ) x^{6}}{4 c^{3} d^{3}}-\frac {e^{7} \left (e^{2} a -5 c \,d^{2}\right ) x^{7}}{2 c^{2} d^{2}}+\frac {\left (90 a^{6} e^{14}-260 a^{5} c \,d^{2} e^{12}+175 a^{4} c^{2} d^{4} e^{10}+95 a^{3} c^{3} d^{6} e^{8}-80 a^{2} c^{4} d^{8} e^{6}-142 a \,c^{5} d^{10} e^{4}-14 c^{6} d^{12} e^{2}\right ) x}{2 c^{7} d^{6} e}}{\left (c d x +a e \right )^{2} \left (e x +d \right )^{2}}+\frac {15 e^{2} \left (a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}\right ) \ln \left (c d x +a e \right )}{c^{7} d^{7}}\)	\(605\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^9/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^3,x,method=_RETURNVERBOSE)

[Out]

-e^3/c^6/d^6*(-1/4*c^3*d^3*e^3*x^4+a*c^2*d^2*e^4*x^3-2*c^3*d^4*e^2*x^3-3*a^2*c*d*e^5*x^2+9*a*c^2*d^3*e^3*x^2-1
5/2*c^3*d^5*e*x^2+10*e^6*a^3*x-36*e^4*d^2*a^2*c*x+45*d^4*e^2*c^2*a*x-20*d^6*c^3*x)+6/c^7/d^7*e*(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)+15/c^7/d^7*e^2*(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)-1/2/c^7/d^7*(a^6*e^12-6*a^5*c*d^2*e^
10+15*a^4*c^2*d^4*e^8-20*a^3*c^3*d^6*e^6+15*a^2*c^4*d^8*e^4-6*a*c^5*d^10*e^2+c^6*d^12)/(c*d*x+a*e)^2

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Maxima [A]
time = 0.34, size = 384, normalized size = 1.74 \begin {gather*} -\frac {c^{6} d^{12} + 6 \, a c^{5} d^{10} e^{2} - 45 \, a^{2} c^{4} d^{8} e^{4} + 100 \, a^{3} c^{3} d^{6} e^{6} - 105 \, a^{4} c^{2} d^{4} e^{8} + 54 \, a^{5} c d^{2} e^{10} - 11 \, a^{6} e^{12} + 12 \, {\left (c^{6} d^{11} e - 5 \, a c^{5} d^{9} e^{3} + 10 \, a^{2} c^{4} d^{7} e^{5} - 10 \, a^{3} c^{3} d^{5} e^{7} + 5 \, a^{4} c^{2} d^{3} e^{9} - a^{5} c d e^{11}\right )} x}{2 \, {\left (c^{9} d^{9} x^{2} + 2 \, a c^{8} d^{8} x e + a^{2} c^{7} d^{7} e^{2}\right )}} + \frac {c^{3} d^{3} x^{4} e^{6} + 4 \, {\left (2 \, c^{3} d^{4} e^{5} - a c^{2} d^{2} e^{7}\right )} x^{3} + 6 \, {\left (5 \, c^{3} d^{5} e^{4} - 6 \, a c^{2} d^{3} e^{6} + 2 \, a^{2} c d e^{8}\right )} x^{2} + 4 \, {\left (20 \, c^{3} d^{6} e^{3} - 45 \, a c^{2} d^{4} e^{5} + 36 \, a^{2} c d^{2} e^{7} - 10 \, a^{3} e^{9}\right )} x}{4 \, c^{6} d^{6}} + \frac {15 \, {\left (c^{4} d^{8} e^{2} - 4 \, a c^{3} d^{6} e^{4} + 6 \, a^{2} c^{2} d^{4} e^{6} - 4 \, a^{3} c d^{2} e^{8} + a^{4} e^{10}\right )} \log \left (c d x + a e\right )}{c^{7} d^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^9/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^3,x, algorithm="maxima")

[Out]

-1/2*(c^6*d^12 + 6*a*c^5*d^10*e^2 - 45*a^2*c^4*d^8*e^4 + 100*a^3*c^3*d^6*e^6 - 105*a^4*c^2*d^4*e^8 + 54*a^5*c*
d^2*e^10 - 11*a^6*e^12 + 12*(c^6*d^11*e - 5*a*c^5*d^9*e^3 + 10*a^2*c^4*d^7*e^5 - 10*a^3*c^3*d^5*e^7 + 5*a^4*c^
2*d^3*e^9 - a^5*c*d*e^11)*x)/(c^9*d^9*x^2 + 2*a*c^8*d^8*x*e + a^2*c^7*d^7*e^2) + 1/4*(c^3*d^3*x^4*e^6 + 4*(2*c
^3*d^4*e^5 - a*c^2*d^2*e^7)*x^3 + 6*(5*c^3*d^5*e^4 - 6*a*c^2*d^3*e^6 + 2*a^2*c*d*e^8)*x^2 + 4*(20*c^3*d^6*e^3
- 45*a*c^2*d^4*e^5 + 36*a^2*c*d^2*e^7 - 10*a^3*e^9)*x)/(c^6*d^6) + 15*(c^4*d^8*e^2 - 4*a*c^3*d^6*e^4 + 6*a^2*c
^2*d^4*e^6 - 4*a^3*c*d^2*e^8 + a^4*e^10)*log(c*d*x + a*e)/(c^7*d^7)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 597 vs. \(2 (211) = 422\).
time = 3.33, size = 597, normalized size = 2.70 \begin {gather*} -\frac {24 \, c^{6} d^{11} x e + 2 \, c^{6} d^{12} + 12 \, a c^{5} d^{10} e^{2} + 16 \, a^{5} c d x e^{11} - 22 \, a^{6} e^{12} + 4 \, {\left (17 \, a^{4} c^{2} d^{2} x^{2} + 27 \, a^{5} c d^{2}\right )} e^{10} + 4 \, {\left (5 \, a^{3} c^{3} d^{3} x^{3} - 6 \, a^{4} c^{2} d^{3} x\right )} e^{9} - {\left (5 \, a^{2} c^{4} d^{4} x^{4} + 252 \, a^{3} c^{3} d^{4} x^{2} + 210 \, a^{4} c^{2} d^{4}\right )} e^{8} + 2 \, {\left (a c^{5} d^{5} x^{5} - 40 \, a^{2} c^{4} d^{5} x^{3} - 30 \, a^{3} c^{3} d^{5} x\right )} e^{7} - {\left (c^{6} d^{6} x^{6} - 20 \, a c^{5} d^{6} x^{4} - 330 \, a^{2} c^{4} d^{6} x^{2} - 200 \, a^{3} c^{3} d^{6}\right )} e^{6} - 8 \, {\left (c^{6} d^{7} x^{5} - 15 \, a c^{5} d^{7} x^{3} - 20 \, a^{2} c^{4} d^{7} x\right )} e^{5} - 10 \, {\left (3 \, c^{6} d^{8} x^{4} + 16 \, a c^{5} d^{8} x^{2} + 9 \, a^{2} c^{4} d^{8}\right )} e^{4} - 40 \, {\left (2 \, c^{6} d^{9} x^{3} + 3 \, a c^{5} d^{9} x\right )} e^{3} - 60 \, {\left (c^{6} d^{10} x^{2} e^{2} + 2 \, a c^{5} d^{9} x e^{3} - 8 \, a^{2} c^{4} d^{7} x e^{5} + 12 \, a^{3} c^{3} d^{5} x e^{7} - 8 \, a^{4} c^{2} d^{3} x e^{9} + 2 \, a^{5} c d x e^{11} + a^{6} e^{12} + {\left (a^{4} c^{2} d^{2} x^{2} - 4 \, a^{5} c d^{2}\right )} e^{10} - 2 \, {\left (2 \, a^{3} c^{3} d^{4} x^{2} - 3 \, a^{4} c^{2} d^{4}\right )} e^{8} + 2 \, {\left (3 \, a^{2} c^{4} d^{6} x^{2} - 2 \, a^{3} c^{3} d^{6}\right )} e^{6} - {\left (4 \, a c^{5} d^{8} x^{2} - a^{2} c^{4} d^{8}\right )} e^{4}\right )} \log \left (c d x + a e\right )}{4 \, {\left (c^{9} d^{9} x^{2} + 2 \, a c^{8} d^{8} x e + a^{2} c^{7} d^{7} e^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^9/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^3,x, algorithm="fricas")

[Out]

-1/4*(24*c^6*d^11*x*e + 2*c^6*d^12 + 12*a*c^5*d^10*e^2 + 16*a^5*c*d*x*e^11 - 22*a^6*e^12 + 4*(17*a^4*c^2*d^2*x
^2 + 27*a^5*c*d^2)*e^10 + 4*(5*a^3*c^3*d^3*x^3 - 6*a^4*c^2*d^3*x)*e^9 - (5*a^2*c^4*d^4*x^4 + 252*a^3*c^3*d^4*x
^2 + 210*a^4*c^2*d^4)*e^8 + 2*(a*c^5*d^5*x^5 - 40*a^2*c^4*d^5*x^3 - 30*a^3*c^3*d^5*x)*e^7 - (c^6*d^6*x^6 - 20*
a*c^5*d^6*x^4 - 330*a^2*c^4*d^6*x^2 - 200*a^3*c^3*d^6)*e^6 - 8*(c^6*d^7*x^5 - 15*a*c^5*d^7*x^3 - 20*a^2*c^4*d^
7*x)*e^5 - 10*(3*c^6*d^8*x^4 + 16*a*c^5*d^8*x^2 + 9*a^2*c^4*d^8)*e^4 - 40*(2*c^6*d^9*x^3 + 3*a*c^5*d^9*x)*e^3
- 60*(c^6*d^10*x^2*e^2 + 2*a*c^5*d^9*x*e^3 - 8*a^2*c^4*d^7*x*e^5 + 12*a^3*c^3*d^5*x*e^7 - 8*a^4*c^2*d^3*x*e^9
+ 2*a^5*c*d*x*e^11 + a^6*e^12 + (a^4*c^2*d^2*x^2 - 4*a^5*c*d^2)*e^10 - 2*(2*a^3*c^3*d^4*x^2 - 3*a^4*c^2*d^4)*e
^8 + 2*(3*a^2*c^4*d^6*x^2 - 2*a^3*c^3*d^6)*e^6 - (4*a*c^5*d^8*x^2 - a^2*c^4*d^8)*e^4)*log(c*d*x + a*e))/(c^9*d
^9*x^2 + 2*a*c^8*d^8*x*e + a^2*c^7*d^7*e^2)

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Sympy [A]
time = 109.73, size = 386, normalized size = 1.75 \begin {gather*} x^{3} \left (- \frac {a e^{7}}{c^{4} d^{4}} + \frac {2 e^{5}}{c^{3} d^{2}}\right ) + x^{2} \cdot \left (\frac {3 a^{2} e^{8}}{c^{5} d^{5}} - \frac {9 a e^{6}}{c^{4} d^{3}} + \frac {15 e^{4}}{2 c^{3} d}\right ) + x \left (- \frac {10 a^{3} e^{9}}{c^{6} d^{6}} + \frac {36 a^{2} e^{7}}{c^{5} d^{4}} - \frac {45 a e^{5}}{c^{4} d^{2}} + \frac {20 e^{3}}{c^{3}}\right ) + \frac {11 a^{6} e^{12} - 54 a^{5} c d^{2} e^{10} + 105 a^{4} c^{2} d^{4} e^{8} - 100 a^{3} c^{3} d^{6} e^{6} + 45 a^{2} c^{4} d^{8} e^{4} - 6 a c^{5} d^{10} e^{2} - c^{6} d^{12} + x \left (12 a^{5} c d e^{11} - 60 a^{4} c^{2} d^{3} e^{9} + 120 a^{3} c^{3} d^{5} e^{7} - 120 a^{2} c^{4} d^{7} e^{5} + 60 a c^{5} d^{9} e^{3} - 12 c^{6} d^{11} e\right )}{2 a^{2} c^{7} d^{7} e^{2} + 4 a c^{8} d^{8} e x + 2 c^{9} d^{9} x^{2}} + \frac {e^{6} x^{4}}{4 c^{3} d^{3}} + \frac {15 e^{2} \left (a e^{2} - c d^{2}\right )^{4} \log {\left (a e + c d x \right )}}{c^{7} d^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**9/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**3,x)

[Out]

x**3*(-a*e**7/(c**4*d**4) + 2*e**5/(c**3*d**2)) + x**2*(3*a**2*e**8/(c**5*d**5) - 9*a*e**6/(c**4*d**3) + 15*e*
*4/(2*c**3*d)) + x*(-10*a**3*e**9/(c**6*d**6) + 36*a**2*e**7/(c**5*d**4) - 45*a*e**5/(c**4*d**2) + 20*e**3/c**
3) + (11*a**6*e**12 - 54*a**5*c*d**2*e**10 + 105*a**4*c**2*d**4*e**8 - 100*a**3*c**3*d**6*e**6 + 45*a**2*c**4*
d**8*e**4 - 6*a*c**5*d**10*e**2 - c**6*d**12 + x*(12*a**5*c*d*e**11 - 60*a**4*c**2*d**3*e**9 + 120*a**3*c**3*d
**5*e**7 - 120*a**2*c**4*d**7*e**5 + 60*a*c**5*d**9*e**3 - 12*c**6*d**11*e))/(2*a**2*c**7*d**7*e**2 + 4*a*c**8
*d**8*e*x + 2*c**9*d**9*x**2) + e**6*x**4/(4*c**3*d**3) + 15*e**2*(a*e**2 - c*d**2)**4*log(a*e + c*d*x)/(c**7*
d**7)

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Giac [A]
time = 1.38, size = 380, normalized size = 1.72 \begin {gather*} \frac {15 \, {\left (c^{4} d^{8} e^{2} - 4 \, a c^{3} d^{6} e^{4} + 6 \, a^{2} c^{2} d^{4} e^{6} - 4 \, a^{3} c d^{2} e^{8} + a^{4} e^{10}\right )} \log \left ({\left | c d x + a e \right |}\right )}{c^{7} d^{7}} - \frac {c^{6} d^{12} + 6 \, a c^{5} d^{10} e^{2} - 45 \, a^{2} c^{4} d^{8} e^{4} + 100 \, a^{3} c^{3} d^{6} e^{6} - 105 \, a^{4} c^{2} d^{4} e^{8} + 54 \, a^{5} c d^{2} e^{10} - 11 \, a^{6} e^{12} + 12 \, {\left (c^{6} d^{11} e - 5 \, a c^{5} d^{9} e^{3} + 10 \, a^{2} c^{4} d^{7} e^{5} - 10 \, a^{3} c^{3} d^{5} e^{7} + 5 \, a^{4} c^{2} d^{3} e^{9} - a^{5} c d e^{11}\right )} x}{2 \, {\left (c d x + a e\right )}^{2} c^{7} d^{7}} + \frac {c^{9} d^{9} x^{4} e^{6} + 8 \, c^{9} d^{10} x^{3} e^{5} + 30 \, c^{9} d^{11} x^{2} e^{4} + 80 \, c^{9} d^{12} x e^{3} - 4 \, a c^{8} d^{8} x^{3} e^{7} - 36 \, a c^{8} d^{9} x^{2} e^{6} - 180 \, a c^{8} d^{10} x e^{5} + 12 \, a^{2} c^{7} d^{7} x^{2} e^{8} + 144 \, a^{2} c^{7} d^{8} x e^{7} - 40 \, a^{3} c^{6} d^{6} x e^{9}}{4 \, c^{12} d^{12}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^9/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^3,x, algorithm="giac")

[Out]

15*(c^4*d^8*e^2 - 4*a*c^3*d^6*e^4 + 6*a^2*c^2*d^4*e^6 - 4*a^3*c*d^2*e^8 + a^4*e^10)*log(abs(c*d*x + a*e))/(c^7
*d^7) - 1/2*(c^6*d^12 + 6*a*c^5*d^10*e^2 - 45*a^2*c^4*d^8*e^4 + 100*a^3*c^3*d^6*e^6 - 105*a^4*c^2*d^4*e^8 + 54
*a^5*c*d^2*e^10 - 11*a^6*e^12 + 12*(c^6*d^11*e - 5*a*c^5*d^9*e^3 + 10*a^2*c^4*d^7*e^5 - 10*a^3*c^3*d^5*e^7 + 5
*a^4*c^2*d^3*e^9 - a^5*c*d*e^11)*x)/((c*d*x + a*e)^2*c^7*d^7) + 1/4*(c^9*d^9*x^4*e^6 + 8*c^9*d^10*x^3*e^5 + 30
*c^9*d^11*x^2*e^4 + 80*c^9*d^12*x*e^3 - 4*a*c^8*d^8*x^3*e^7 - 36*a*c^8*d^9*x^2*e^6 - 180*a*c^8*d^10*x*e^5 + 12
*a^2*c^7*d^7*x^2*e^8 + 144*a^2*c^7*d^8*x*e^7 - 40*a^3*c^6*d^6*x*e^9)/(c^12*d^12)

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Mupad [B]
time = 0.13, size = 516, normalized size = 2.33 \begin {gather*} x^3\,\left (\frac {2\,e^5}{c^3\,d^2}-\frac {a\,e^7}{c^4\,d^4}\right )-x^2\,\left (\frac {3\,a^2\,e^8}{2\,c^5\,d^5}-\frac {15\,e^4}{2\,c^3\,d}+\frac {3\,a\,e\,\left (\frac {6\,e^5}{c^3\,d^2}-\frac {3\,a\,e^7}{c^4\,d^4}\right )}{2\,c\,d}\right )+\frac {x\,\left (6\,a^5\,e^{11}-30\,a^4\,c\,d^2\,e^9+60\,a^3\,c^2\,d^4\,e^7-60\,a^2\,c^3\,d^6\,e^5+30\,a\,c^4\,d^8\,e^3-6\,c^5\,d^{10}\,e\right )-\frac {-11\,a^6\,e^{12}+54\,a^5\,c\,d^2\,e^{10}-105\,a^4\,c^2\,d^4\,e^8+100\,a^3\,c^3\,d^6\,e^6-45\,a^2\,c^4\,d^8\,e^4+6\,a\,c^5\,d^{10}\,e^2+c^6\,d^{12}}{2\,c\,d}}{a^2\,c^6\,d^6\,e^2+2\,a\,c^7\,d^7\,e\,x+c^8\,d^8\,x^2}+x\,\left (\frac {20\,e^3}{c^3}-\frac {a^3\,e^9}{c^6\,d^6}-\frac {3\,a^2\,e^2\,\left (\frac {6\,e^5}{c^3\,d^2}-\frac {3\,a\,e^7}{c^4\,d^4}\right )}{c^2\,d^2}+\frac {3\,a\,e\,\left (\frac {3\,a^2\,e^8}{c^5\,d^5}-\frac {15\,e^4}{c^3\,d}+\frac {3\,a\,e\,\left (\frac {6\,e^5}{c^3\,d^2}-\frac {3\,a\,e^7}{c^4\,d^4}\right )}{c\,d}\right )}{c\,d}\right )+\frac {e^6\,x^4}{4\,c^3\,d^3}+\frac {\ln \left (a\,e+c\,d\,x\right )\,\left (15\,a^4\,e^{10}-60\,a^3\,c\,d^2\,e^8+90\,a^2\,c^2\,d^4\,e^6-60\,a\,c^3\,d^6\,e^4+15\,c^4\,d^8\,e^2\right )}{c^7\,d^7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)^9/(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^3,x)

[Out]

x^3*((2*e^5)/(c^3*d^2) - (a*e^7)/(c^4*d^4)) - x^2*((3*a^2*e^8)/(2*c^5*d^5) - (15*e^4)/(2*c^3*d) + (3*a*e*((6*e
^5)/(c^3*d^2) - (3*a*e^7)/(c^4*d^4)))/(2*c*d)) + (x*(6*a^5*e^11 - 6*c^5*d^10*e + 30*a*c^4*d^8*e^3 - 30*a^4*c*d
^2*e^9 - 60*a^2*c^3*d^6*e^5 + 60*a^3*c^2*d^4*e^7) - (c^6*d^12 - 11*a^6*e^12 + 6*a*c^5*d^10*e^2 + 54*a^5*c*d^2*
e^10 - 45*a^2*c^4*d^8*e^4 + 100*a^3*c^3*d^6*e^6 - 105*a^4*c^2*d^4*e^8)/(2*c*d))/(c^8*d^8*x^2 + a^2*c^6*d^6*e^2
+ 2*a*c^7*d^7*e*x) + x*((20*e^3)/c^3 - (a^3*e^9)/(c^6*d^6) - (3*a^2*e^2*((6*e^5)/(c^3*d^2) - (3*a*e^7)/(c^4*d
^4)))/(c^2*d^2) + (3*a*e*((3*a^2*e^8)/(c^5*d^5) - (15*e^4)/(c^3*d) + (3*a*e*((6*e^5)/(c^3*d^2) - (3*a*e^7)/(c^
4*d^4)))/(c*d)))/(c*d)) + (e^6*x^4)/(4*c^3*d^3) + (log(a*e + c*d*x)*(15*a^4*e^10 + 15*c^4*d^8*e^2 - 60*a*c^3*d
^6*e^4 - 60*a^3*c*d^2*e^8 + 90*a^2*c^2*d^4*e^6))/(c^7*d^7)

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