3.1885 \(\int \frac{(d+e x)^9}{(a d e+(c d^2+a e^2) x+c d e x^2)^3} \, dx\)

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

[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)

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Rubi [A]  time = 0.261405, antiderivative size = 221, normalized size of antiderivative = 1., 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} \[ \frac{e^6 (a e+c d x)^4}{4 c^7 d^7}+\frac{2 e^5 \left (c d^2-a e^2\right ) (a e+c d x)^3}{c^7 d^7}+\frac{15 e^4 \left (c d^2-a e^2\right )^2 (a e+c d x)^2}{2 c^7 d^7}+\frac{20 e^3 x \left (c d^2-a e^2\right )^3}{c^6 d^6}-\frac{6 e \left (c d^2-a e^2\right )^5}{c^7 d^7 (a e+c d x)}-\frac{\left (c d^2-a e^2\right )^6}{2 c^7 d^7 (a e+c d x)^2}+\frac{15 e^2 \left (c d^2-a e^2\right )^4 \log (a e+c d x)}{c^7 d^7} \]

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 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]

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])

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*}

Mathematica [A]  time = 0.129065, size = 337, normalized size = 1.52 \[ \frac{2 a^4 c^2 d^2 e^8 \left (105 d^2+12 d e x-34 e^2 x^2\right )-4 a^3 c^3 d^3 e^6 \left (-15 d^2 e x+50 d^3-63 d e^2 x^2+5 e^3 x^3\right )+5 a^2 c^4 d^4 e^4 \left (-66 d^2 e^2 x^2-32 d^3 e x+18 d^4+16 d e^3 x^3+e^4 x^4\right )-4 a^5 c d e^{10} (27 d+4 e x)+22 a^6 e^{12}-2 a c^5 d^5 e^2 \left (-80 d^3 e^2 x^2+60 d^2 e^3 x^3-60 d^4 e x+6 d^5+10 d e^4 x^4+e^5 x^5\right )+60 e^2 \left (c d^2-a e^2\right )^4 (a e+c d x)^2 \log (a e+c d x)+c^6 d^6 \left (80 d^3 e^3 x^3+30 d^2 e^4 x^4-24 d^5 e x-2 d^6+8 d e^5 x^5+e^6 x^6\right )}{4 c^7 d^7 (a e+c d x)^2} \]

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 [B]  time = 0.051, size = 544, normalized size = 2.5 \begin{align*} -{\frac{15\,{a}^{4}{e}^{8}}{2\,{c}^{5}{d}^{3} \left ( cdx+ae \right ) ^{2}}}+15\,{\frac{{e}^{10}\ln \left ( cdx+ae \right ){a}^{4}}{{c}^{7}{d}^{7}}}-60\,{\frac{{e}^{8}\ln \left ( cdx+ae \right ){a}^{3}}{{c}^{6}{d}^{5}}}+3\,{\frac{{e}^{8}{x}^{2}{a}^{2}}{{c}^{5}{d}^{5}}}+30\,{\frac{ad{e}^{3}}{{c}^{3} \left ( cdx+ae \right ) }}-{\frac{{a}^{6}{e}^{12}}{2\,{c}^{7}{d}^{7} \left ( cdx+ae \right ) ^{2}}}+3\,{\frac{{a}^{5}{e}^{10}}{{c}^{6}{d}^{5} \left ( cdx+ae \right ) ^{2}}}+{\frac{{e}^{6}{x}^{4}}{4\,{c}^{3}{d}^{3}}}+2\,{\frac{{e}^{5}{x}^{3}}{{c}^{3}{d}^{2}}}+{\frac{15\,{e}^{4}{x}^{2}}{2\,{c}^{3}d}}-6\,{\frac{{d}^{3}e}{{c}^{2} \left ( cdx+ae \right ) }}+15\,{\frac{d{e}^{2}\ln \left ( cdx+ae \right ) }{{c}^{3}}}+90\,{\frac{{e}^{6}\ln \left ( cdx+ae \right ){a}^{2}}{{c}^{5}{d}^{3}}}-60\,{\frac{{e}^{4}\ln \left ( cdx+ae \right ) a}{{c}^{4}d}}-{\frac{{e}^{7}{x}^{3}a}{{c}^{4}{d}^{4}}}-9\,{\frac{{e}^{6}{x}^{2}a}{{c}^{4}{d}^{3}}}-10\,{\frac{{a}^{3}{e}^{9}x}{{c}^{6}{d}^{6}}}+36\,{\frac{{a}^{2}{e}^{7}x}{{c}^{5}{d}^{4}}}-45\,{\frac{a{e}^{5}x}{{c}^{4}{d}^{2}}}+20\,{\frac{{e}^{3}x}{{c}^{3}}}-{\frac{{d}^{5}}{2\,c \left ( cdx+ae \right ) ^{2}}}-60\,{\frac{{a}^{2}{e}^{5}}{{c}^{4}d \left ( cdx+ae \right ) }}+60\,{\frac{{e}^{7}{a}^{3}}{{c}^{5}{d}^{3} \left ( cdx+ae \right ) }}+3\,{\frac{{d}^{3}{e}^{2}a}{{c}^{2} \left ( cdx+ae \right ) ^{2}}}+6\,{\frac{{a}^{5}{e}^{11}}{{c}^{7}{d}^{7} \left ( cdx+ae \right ) }}-30\,{\frac{{e}^{9}{a}^{4}}{{c}^{6}{d}^{5} \left ( cdx+ae \right ) }}-{\frac{15\,{a}^{2}d{e}^{4}}{2\,{c}^{3} \left ( cdx+ae \right ) ^{2}}}+10\,{\frac{{a}^{3}{e}^{6}}{{c}^{4}d \left ( cdx+ae \right ) ^{2}}} \end{align*}

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)

[Out]

-15/2/c^5/d^3/(c*d*x+a*e)^2*a^4*e^8+15/c^7/d^7*e^10*ln(c*d*x+a*e)*a^4-60/c^6/d^5*e^8*ln(c*d*x+a*e)*a^3+3*e^8/c
^5/d^5*x^2*a^2+30*d*e^3/c^3/(c*d*x+a*e)*a-1/2/c^7/d^7/(c*d*x+a*e)^2*a^6*e^12+3/c^6/d^5/(c*d*x+a*e)^2*a^5*e^10+
1/4*e^6/c^3/d^3*x^4+2*e^5/c^3/d^2*x^3+15/2*e^4/c^3/d*x^2-6*d^3*e/c^2/(c*d*x+a*e)+15/c^3*d*e^2*ln(c*d*x+a*e)+90
/c^5/d^3*e^6*ln(c*d*x+a*e)*a^2-60/c^4/d*e^4*ln(c*d*x+a*e)*a-e^7/c^4/d^4*x^3*a-9*e^6/c^4/d^3*x^2*a-10*e^9/c^6/d
^6*a^3*x+36*e^7/c^5/d^4*a^2*x-45*e^5/c^4/d^2*a*x+20*e^3/c^3*x-1/2/c*d^5/(c*d*x+a*e)^2-60/d*e^5/c^4/(c*d*x+a*e)
*a^2+60/d^3*e^7/c^5/(c*d*x+a*e)*a^3+3/c^2*d^3/(c*d*x+a*e)^2*a*e^2+6/d^7*e^11/c^7/(c*d*x+a*e)*a^5-30/d^5*e^9/c^
6/(c*d*x+a*e)*a^4-15/2/c^3*d/(c*d*x+a*e)^2*a^2*e^4+10/c^4/d/(c*d*x+a*e)^2*a^3*e^6

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Maxima [A]  time = 1.07531, size = 551, normalized size = 2.49 \begin{align*} -\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} e x + a^{2} c^{7} d^{7} e^{2}\right )}} + \frac{c^{3} d^{3} e^{6} x^{4} + 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{align*}

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*e*x + a^2*c^7*d^7*e^2) + 1/4*(c^3*d^3*e^6*x^4 + 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]  time = 1.8488, size = 1220, normalized size = 5.52 \begin{align*} \frac{c^{6} d^{6} e^{6} x^{6} - 2 \, c^{6} d^{12} - 12 \, a c^{5} d^{10} e^{2} + 90 \, a^{2} c^{4} d^{8} e^{4} - 200 \, a^{3} c^{3} d^{6} e^{6} + 210 \, a^{4} c^{2} d^{4} e^{8} - 108 \, a^{5} c d^{2} e^{10} + 22 \, a^{6} e^{12} + 2 \,{\left (4 \, c^{6} d^{7} e^{5} - a c^{5} d^{5} e^{7}\right )} x^{5} + 5 \,{\left (6 \, c^{6} d^{8} e^{4} - 4 \, a c^{5} d^{6} e^{6} + a^{2} c^{4} d^{4} e^{8}\right )} x^{4} + 20 \,{\left (4 \, c^{6} d^{9} e^{3} - 6 \, a c^{5} d^{7} e^{5} + 4 \, a^{2} c^{4} d^{5} e^{7} - a^{3} c^{3} d^{3} e^{9}\right )} x^{3} + 2 \,{\left (80 \, a c^{5} d^{8} e^{4} - 165 \, a^{2} c^{4} d^{6} e^{6} + 126 \, a^{3} c^{3} d^{4} e^{8} - 34 \, a^{4} c^{2} d^{2} e^{10}\right )} x^{2} - 4 \,{\left (6 \, c^{6} d^{11} e - 30 \, a c^{5} d^{9} e^{3} + 40 \, a^{2} c^{4} d^{7} e^{5} - 15 \, a^{3} c^{3} d^{5} e^{7} - 6 \, a^{4} c^{2} d^{3} e^{9} + 4 \, a^{5} c d e^{11}\right )} x + 60 \,{\left (a^{2} c^{4} d^{8} e^{4} - 4 \, a^{3} c^{3} d^{6} e^{6} + 6 \, a^{4} c^{2} d^{4} e^{8} - 4 \, a^{5} c d^{2} e^{10} + a^{6} e^{12} +{\left (c^{6} d^{10} e^{2} - 4 \, a c^{5} d^{8} e^{4} + 6 \, a^{2} c^{4} d^{6} e^{6} - 4 \, a^{3} c^{3} d^{4} e^{8} + a^{4} c^{2} d^{2} e^{10}\right )} x^{2} + 2 \,{\left (a c^{5} d^{9} e^{3} - 4 \, a^{2} c^{4} d^{7} e^{5} + 6 \, a^{3} c^{3} d^{5} e^{7} - 4 \, a^{4} c^{2} d^{3} e^{9} + a^{5} c d e^{11}\right )} x\right )} \log \left (c d x + a e\right )}{4 \,{\left (c^{9} d^{9} x^{2} + 2 \, a c^{8} d^{8} e x + a^{2} c^{7} d^{7} e^{2}\right )}} \end{align*}

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*(c^6*d^6*e^6*x^6 - 2*c^6*d^12 - 12*a*c^5*d^10*e^2 + 90*a^2*c^4*d^8*e^4 - 200*a^3*c^3*d^6*e^6 + 210*a^4*c^2
*d^4*e^8 - 108*a^5*c*d^2*e^10 + 22*a^6*e^12 + 2*(4*c^6*d^7*e^5 - a*c^5*d^5*e^7)*x^5 + 5*(6*c^6*d^8*e^4 - 4*a*c
^5*d^6*e^6 + a^2*c^4*d^4*e^8)*x^4 + 20*(4*c^6*d^9*e^3 - 6*a*c^5*d^7*e^5 + 4*a^2*c^4*d^5*e^7 - a^3*c^3*d^3*e^9)
*x^3 + 2*(80*a*c^5*d^8*e^4 - 165*a^2*c^4*d^6*e^6 + 126*a^3*c^3*d^4*e^8 - 34*a^4*c^2*d^2*e^10)*x^2 - 4*(6*c^6*d
^11*e - 30*a*c^5*d^9*e^3 + 40*a^2*c^4*d^7*e^5 - 15*a^3*c^3*d^5*e^7 - 6*a^4*c^2*d^3*e^9 + 4*a^5*c*d*e^11)*x + 6
0*(a^2*c^4*d^8*e^4 - 4*a^3*c^3*d^6*e^6 + 6*a^4*c^2*d^4*e^8 - 4*a^5*c*d^2*e^10 + a^6*e^12 + (c^6*d^10*e^2 - 4*a
*c^5*d^8*e^4 + 6*a^2*c^4*d^6*e^6 - 4*a^3*c^3*d^4*e^8 + a^4*c^2*d^2*e^10)*x^2 + 2*(a*c^5*d^9*e^3 - 4*a^2*c^4*d^
7*e^5 + 6*a^3*c^3*d^5*e^7 - 4*a^4*c^2*d^3*e^9 + a^5*c*d*e^11)*x)*log(c*d*x + a*e))/(c^9*d^9*x^2 + 2*a*c^8*d^8*
e*x + a^2*c^7*d^7*e^2)

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Sympy [A]  time = 16.1664, size = 386, normalized size = 1.75 \begin{align*} \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{x^{3} \left (a e^{7} - 2 c d^{2} e^{5}\right )}{c^{4} d^{4}} + \frac{x^{2} \left (6 a^{2} e^{8} - 18 a c d^{2} e^{6} + 15 c^{2} d^{4} e^{4}\right )}{2 c^{5} d^{5}} - \frac{x \left (10 a^{3} e^{9} - 36 a^{2} c d^{2} e^{7} + 45 a c^{2} d^{4} e^{5} - 20 c^{3} d^{6} e^{3}\right )}{c^{6} d^{6}} + \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{align*}

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]

(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) - x**3*(a*e**7 - 2*c*d**2*e**5)/(c**4*d**4) + x**2*(6*a**2*
e**8 - 18*a*c*d**2*e**6 + 15*c**2*d**4*e**4)/(2*c**5*d**5) - x*(10*a**3*e**9 - 36*a**2*c*d**2*e**7 + 45*a*c**2
*d**4*e**5 - 20*c**3*d**6*e**3)/(c**6*d**6) + 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 [B]  time = 18.4971, size = 1442, normalized size = 6.52 \begin{align*} \text{result too large to display} \end{align*}

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^9*d^18*e^2 - 9*a*c^8*d^16*e^4 + 36*a^2*c^7*d^14*e^6 - 84*a^3*c^6*d^12*e^8 + 126*a^4*c^5*d^10*e^10 - 126*
a^5*c^4*d^8*e^12 + 84*a^6*c^3*d^6*e^14 - 36*a^7*c^2*d^4*e^16 + 9*a^8*c*d^2*e^18 - a^9*e^20)*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^11*d^15 - 4*a*c^10*d^13*e^2 + 6*a^2*c^9*d^11*e^
4 - 4*a^3*c^8*d^9*e^6 + a^4*c^7*d^7*e^8)*sqrt(-c^2*d^4 + 2*a*c*d^2*e^2 - a^2*e^4)) + 15/2*(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^2*e + c*d^2*x + a*x*e^2 + a*d*e)/(c^7*d
^7) - 1/2*(c^10*d^22 + 2*a*c^9*d^20*e^2 - 63*a^2*c^8*d^18*e^4 + 312*a^3*c^7*d^16*e^6 - 798*a^4*c^6*d^14*e^8 +
1260*a^5*c^5*d^12*e^10 - 1302*a^6*c^4*d^10*e^12 + 888*a^7*c^3*d^8*e^14 - 387*a^8*c^2*d^6*e^16 + 98*a^9*c*d^4*e
^18 - 11*a^10*d^2*e^20 + 12*(c^10*d^19*e^3 - 9*a*c^9*d^17*e^5 + 36*a^2*c^8*d^15*e^7 - 84*a^3*c^7*d^13*e^9 + 12
6*a^4*c^6*d^11*e^11 - 126*a^5*c^5*d^9*e^13 + 84*a^6*c^4*d^7*e^15 - 36*a^7*c^3*d^5*e^17 + 9*a^8*c^2*d^3*e^19 -
a^9*c*d*e^21)*x^3 + (25*c^10*d^20*e^2 - 214*a*c^9*d^18*e^4 + 801*a^2*c^8*d^16*e^6 - 1704*a^3*c^7*d^14*e^8 + 22
26*a^4*c^6*d^12*e^10 - 1764*a^5*c^5*d^10*e^12 + 714*a^6*c^4*d^8*e^14 + 24*a^7*c^3*d^6*e^16 - 171*a^8*c^2*d^4*e
^18 + 74*a^9*c*d^2*e^20 - 11*a^10*e^22)*x^2 + 2*(7*c^10*d^21*e - 52*a*c^9*d^19*e^3 + 153*a^2*c^8*d^17*e^5 - 19
2*a^3*c^7*d^15*e^7 - 42*a^4*c^6*d^13*e^9 + 504*a^5*c^5*d^11*e^11 - 798*a^6*c^4*d^9*e^13 + 672*a^7*c^3*d^7*e^15
- 333*a^8*c^2*d^5*e^17 + 92*a^9*c*d^3*e^19 - 11*a^10*d*e^21)*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^7*d^7) + 1/4*(c^9*d^9*x^4*e^18 + 8*c^9*d^10*x^3*e^17 + 30*c^9*d^11*x^2*e
^16 + 80*c^9*d^12*x*e^15 - 4*a*c^8*d^8*x^3*e^19 - 36*a*c^8*d^9*x^2*e^18 - 180*a*c^8*d^10*x*e^17 + 12*a^2*c^7*d
^7*x^2*e^20 + 144*a^2*c^7*d^8*x*e^19 - 40*a^3*c^6*d^6*x*e^21)*e^(-12)/(c^12*d^12)