3.1897 \(\int \frac{(d+e x)^9}{(a d e+(c d^2+a e^2) x+c d e x^2)^4} \, dx\)

Optimal. Leaf size=179 \[ \frac{e^4 x \left (5 c d^2-4 a e^2\right )}{c^5 d^5}-\frac{10 e^2 \left (c d^2-a e^2\right )^3}{c^6 d^6 (a e+c d x)}-\frac{5 e \left (c d^2-a e^2\right )^4}{2 c^6 d^6 (a e+c d x)^2}-\frac{\left (c d^2-a e^2\right )^5}{3 c^6 d^6 (a e+c d x)^3}+\frac{10 e^3 \left (c d^2-a e^2\right )^2 \log (a e+c d x)}{c^6 d^6}+\frac{e^5 x^2}{2 c^4 d^4} \]

[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)

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Rubi [A]  time = 0.173534, antiderivative size = 179, 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} \[ \frac{e^4 x \left (5 c d^2-4 a e^2\right )}{c^5 d^5}-\frac{10 e^2 \left (c d^2-a e^2\right )^3}{c^6 d^6 (a e+c d x)}-\frac{5 e \left (c d^2-a e^2\right )^4}{2 c^6 d^6 (a e+c d x)^2}-\frac{\left (c d^2-a e^2\right )^5}{3 c^6 d^6 (a e+c d x)^3}+\frac{10 e^3 \left (c d^2-a e^2\right )^2 \log (a e+c d x)}{c^6 d^6}+\frac{e^5 x^2}{2 c^4 d^4} \]

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)^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 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 )^4} \, dx &=\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]  time = 0.0941166, size = 259, normalized size = 1.45 \[ \frac{a^3 c^2 d^2 e^6 \left (110 d^2-270 d e x-9 e^2 x^2\right )-a^2 c^3 d^3 e^4 \left (-270 d^2 e x+20 d^3+90 d e^2 x^2+63 e^3 x^3\right )+a^4 c d e^8 (81 e x-130 d)+47 a^5 e^{10}-5 a c^4 d^4 e^2 \left (-36 d^2 e^2 x^2+12 d^3 e x+d^4-18 d e^3 x^3+3 e^4 x^4\right )+60 e^3 \left (c d^2-a e^2\right )^2 (a e+c d x)^3 \log (a e+c d x)+c^5 d^5 \left (-60 d^3 e^2 x^2-15 d^4 e x-2 d^5+30 d e^4 x^4+3 e^5 x^5\right )}{6 c^6 d^6 (a e+c d x)^3} \]

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)^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)

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Maple [B]  time = 0.049, size = 436, normalized size = 2.4 \begin{align*}{\frac{{e}^{5}{x}^{2}}{2\,{c}^{4}{d}^{4}}}-4\,{\frac{a{e}^{6}x}{{c}^{5}{d}^{5}}}+5\,{\frac{{e}^{4}x}{{c}^{4}{d}^{3}}}+{\frac{{a}^{5}{e}^{10}}{3\,{c}^{6}{d}^{6} \left ( cdx+ae \right ) ^{3}}}-{\frac{5\,{a}^{4}{e}^{8}}{3\,{c}^{5}{d}^{4} \left ( cdx+ae \right ) ^{3}}}+{\frac{10\,{a}^{3}{e}^{6}}{3\,{c}^{4}{d}^{2} \left ( cdx+ae \right ) ^{3}}}-{\frac{10\,{a}^{2}{e}^{4}}{3\,{c}^{3} \left ( cdx+ae \right ) ^{3}}}+{\frac{5\,a{d}^{2}{e}^{2}}{3\,{c}^{2} \left ( cdx+ae \right ) ^{3}}}-{\frac{{d}^{4}}{3\,c \left ( cdx+ae \right ) ^{3}}}-{\frac{5\,{a}^{4}{e}^{9}}{2\,{c}^{6}{d}^{6} \left ( cdx+ae \right ) ^{2}}}+10\,{\frac{{e}^{7}{a}^{3}}{{c}^{5}{d}^{4} \left ( cdx+ae \right ) ^{2}}}-15\,{\frac{{a}^{2}{e}^{5}}{{c}^{4}{d}^{2} \left ( cdx+ae \right ) ^{2}}}+10\,{\frac{a{e}^{3}}{{c}^{3} \left ( cdx+ae \right ) ^{2}}}-{\frac{5\,{d}^{2}e}{2\,{c}^{2} \left ( cdx+ae \right ) ^{2}}}+10\,{\frac{{e}^{8}{a}^{3}}{{c}^{6}{d}^{6} \left ( cdx+ae \right ) }}-30\,{\frac{{e}^{6}{a}^{2}}{{c}^{5}{d}^{4} \left ( cdx+ae \right ) }}+30\,{\frac{{e}^{4}a}{{c}^{4}{d}^{2} \left ( cdx+ae \right ) }}-10\,{\frac{{e}^{2}}{{c}^{3} \left ( cdx+ae \right ) }}+10\,{\frac{{e}^{7}\ln \left ( cdx+ae \right ){a}^{2}}{{c}^{6}{d}^{6}}}-20\,{\frac{{e}^{5}\ln \left ( cdx+ae \right ) a}{{c}^{5}{d}^{4}}}+10\,{\frac{{e}^{3}\ln \left ( cdx+ae \right ) }{{c}^{4}{d}^{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)^4,x)

[Out]

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

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Maxima [A]  time = 1.09504, size = 440, normalized size = 2.46 \begin{align*} -\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}} \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)^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)

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Fricas [B]  time = 1.93881, size = 976, normalized size = 5.45 \begin{align*} \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 )}} \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)^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)

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Sympy [A]  time = 42.7509, size = 335, normalized size = 1.87 \begin{align*} \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} + x^{2} \left (60 a^{3} c^{2} d^{2} e^{8} - 180 a^{2} c^{3} d^{4} e^{6} + 180 a c^{4} d^{6} e^{4} - 60 c^{5} d^{8} e^{2}\right ) + x \left (105 a^{4} c d e^{9} - 300 a^{3} c^{2} d^{3} e^{7} + 270 a^{2} c^{3} d^{5} e^{5} - 60 a c^{4} d^{7} e^{3} - 15 c^{5} d^{9} e\right )}{6 a^{3} c^{6} d^{6} e^{3} + 18 a^{2} c^{7} d^{7} e^{2} x + 18 a c^{8} d^{8} e x^{2} + 6 c^{9} d^{9} x^{3}} + \frac{e^{5} x^{2}}{2 c^{4} d^{4}} - \frac{x \left (4 a e^{6} - 5 c d^{2} e^{4}\right )}{c^{5} d^{5}} + \frac{10 e^{3} \left (a e^{2} - c d^{2}\right )^{2} \log{\left (a e + c d x \right )}}{c^{6} d^{6}} \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)**4,x)

[Out]

(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 + x**2*(60*a**3*c**2*d**2*e**8 - 180*a**2*c**3*d**4*e**6 + 180*a*c**4*d**6*e**4 - 60*c**5*d**8*
e**2) + x*(105*a**4*c*d*e**9 - 300*a**3*c**2*d**3*e**7 + 270*a**2*c**3*d**5*e**5 - 60*a*c**4*d**7*e**3 - 15*c*
*5*d**9*e))/(6*a**3*c**6*d**6*e**3 + 18*a**2*c**7*d**7*e**2*x + 18*a*c**8*d**8*e*x**2 + 6*c**9*d**9*x**3) + e*
*5*x**2/(2*c**4*d**4) - x*(4*a*e**6 - 5*c*d**2*e**4)/(c**5*d**5) + 10*e**3*(a*e**2 - c*d**2)**2*log(a*e + c*d*
x)/(c**6*d**6)

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Giac [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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)^4,x, algorithm="giac")

[Out]

Timed out