3.1898 \(\int \frac{(d+e x)^8}{(a d e+(c d^2+a e^2) x+c d e x^2)^4} \, dx\)

Optimal. Leaf size=146 \[ -\frac{6 e^2 \left (c d^2-a e^2\right )^2}{c^5 d^5 (a e+c d x)}-\frac{2 e \left (c d^2-a e^2\right )^3}{c^5 d^5 (a e+c d x)^2}-\frac{\left (c d^2-a e^2\right )^4}{3 c^5 d^5 (a e+c d x)^3}+\frac{4 e^3 \left (c d^2-a e^2\right ) \log (a e+c d x)}{c^5 d^5}+\frac{e^4 x}{c^4 d^4} \]

[Out]

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

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Rubi [A]  time = 0.133414, antiderivative size = 146, 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} \[ -\frac{6 e^2 \left (c d^2-a e^2\right )^2}{c^5 d^5 (a e+c d x)}-\frac{2 e \left (c d^2-a e^2\right )^3}{c^5 d^5 (a e+c d x)^2}-\frac{\left (c d^2-a e^2\right )^4}{3 c^5 d^5 (a e+c d x)^3}+\frac{4 e^3 \left (c d^2-a e^2\right ) \log (a e+c d x)}{c^5 d^5}+\frac{e^4 x}{c^4 d^4} \]

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)^4,x]

[Out]

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

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)^8}{\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)^4}{(a e+c d x)^4} \, dx\\ &=\int \left (\frac{e^4}{c^4 d^4}+\frac{\left (c d^2-a e^2\right )^4}{c^4 d^4 (a e+c d x)^4}+\frac{4 e \left (c d^2-a e^2\right )^3}{c^4 d^4 (a e+c d x)^3}+\frac{6 \left (c d^2 e-a e^3\right )^2}{c^4 d^4 (a e+c d x)^2}+\frac{4 \left (c d^2 e^3-a e^5\right )}{c^4 d^4 (a e+c d x)}\right ) \, dx\\ &=\frac{e^4 x}{c^4 d^4}-\frac{\left (c d^2-a e^2\right )^4}{3 c^5 d^5 (a e+c d x)^3}-\frac{2 e \left (c d^2-a e^2\right )^3}{c^5 d^5 (a e+c d x)^2}-\frac{6 e^2 \left (c d^2-a e^2\right )^2}{c^5 d^5 (a e+c d x)}+\frac{4 e^3 \left (c d^2-a e^2\right ) \log (a e+c d x)}{c^5 d^5}\\ \end{align*}

Mathematica [A]  time = 0.0741037, size = 194, normalized size = 1.33 \[ \frac{-3 a^2 c^2 d^2 e^4 \left (2 d^2-18 d e x+3 e^2 x^2\right )+a^3 c d e^6 (22 d-27 e x)-13 a^4 e^8+a c^3 d^3 e^2 \left (-18 d^2 e x-2 d^3+36 d e^2 x^2+9 e^3 x^3\right )-12 e^3 \left (a e^2-c d^2\right ) (a e+c d x)^3 \log (a e+c d x)-c^4 \left (18 d^6 e^2 x^2-3 d^4 e^4 x^4+6 d^7 e x+d^8\right )}{3 c^5 d^5 (a e+c d x)^3} \]

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)^4,x]

[Out]

(-13*a^4*e^8 + a^3*c*d*e^6*(22*d - 27*e*x) - 3*a^2*c^2*d^2*e^4*(2*d^2 - 18*d*e*x + 3*e^2*x^2) + a*c^3*d^3*e^2*
(-2*d^3 - 18*d^2*e*x + 36*d*e^2*x^2 + 9*e^3*x^3) - c^4*(d^8 + 6*d^7*e*x + 18*d^6*e^2*x^2 - 3*d^4*e^4*x^4) - 12
*e^3*(-(c*d^2) + a*e^2)*(a*e + c*d*x)^3*Log[a*e + c*d*x])/(3*c^5*d^5*(a*e + c*d*x)^3)

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Maple [B]  time = 0.048, size = 318, normalized size = 2.2 \begin{align*}{\frac{{e}^{4}x}{{c}^{4}{d}^{4}}}-{\frac{{a}^{4}{e}^{8}}{3\,{c}^{5}{d}^{5} \left ( cdx+ae \right ) ^{3}}}+{\frac{4\,{a}^{3}{e}^{6}}{3\,{c}^{4}{d}^{3} \left ( cdx+ae \right ) ^{3}}}-2\,{\frac{{a}^{2}{e}^{4}}{{c}^{3}d \left ( cdx+ae \right ) ^{3}}}+{\frac{4\,ad{e}^{2}}{3\,{c}^{2} \left ( cdx+ae \right ) ^{3}}}-{\frac{{d}^{3}}{3\,c \left ( cdx+ae \right ) ^{3}}}+2\,{\frac{{a}^{3}{e}^{7}}{{c}^{5}{d}^{5} \left ( cdx+ae \right ) ^{2}}}-6\,{\frac{{a}^{2}{e}^{5}}{{c}^{4}{d}^{3} \left ( cdx+ae \right ) ^{2}}}+6\,{\frac{a{e}^{3}}{{c}^{3}d \left ( cdx+ae \right ) ^{2}}}-2\,{\frac{de}{{c}^{2} \left ( cdx+ae \right ) ^{2}}}-6\,{\frac{{e}^{6}{a}^{2}}{{c}^{5}{d}^{5} \left ( cdx+ae \right ) }}+12\,{\frac{{e}^{4}a}{{c}^{4}{d}^{3} \left ( cdx+ae \right ) }}-6\,{\frac{{e}^{2}}{{c}^{3}d \left ( cdx+ae \right ) }}-4\,{\frac{{e}^{5}\ln \left ( cdx+ae \right ) a}{{c}^{5}{d}^{5}}}+4\,{\frac{{e}^{3}\ln \left ( cdx+ae \right ) }{{c}^{4}{d}^{3}}} \end{align*}

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)^4,x)

[Out]

e^4*x/c^4/d^4-1/3/c^5/d^5/(c*d*x+a*e)^3*a^4*e^8+4/3/c^4/d^3/(c*d*x+a*e)^3*a^3*e^6-2/c^3/d/(c*d*x+a*e)^3*a^2*e^
4+4/3/c^2*d/(c*d*x+a*e)^3*a*e^2-1/3/c*d^3/(c*d*x+a*e)^3+2/d^5*e^7/c^5/(c*d*x+a*e)^2*a^3-6/d^3*e^5/c^4/(c*d*x+a
*e)^2*a^2+6/d*e^3/c^3/(c*d*x+a*e)^2*a-2*d*e/c^2/(c*d*x+a*e)^2-6/c^5/d^5*e^6/(c*d*x+a*e)*a^2+12/c^4/d^3*e^4/(c*
d*x+a*e)*a-6/c^3/d*e^2/(c*d*x+a*e)-4/c^5/d^5*e^5*ln(c*d*x+a*e)*a+4/c^4/d^3*e^3*ln(c*d*x+a*e)

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Maxima [A]  time = 1.14443, size = 328, normalized size = 2.25 \begin{align*} -\frac{c^{4} d^{8} + 2 \, a c^{3} d^{6} e^{2} + 6 \, a^{2} c^{2} d^{4} e^{4} - 22 \, a^{3} c d^{2} e^{6} + 13 \, a^{4} e^{8} + 18 \,{\left (c^{4} d^{6} e^{2} - 2 \, a c^{3} d^{4} e^{4} + a^{2} c^{2} d^{2} e^{6}\right )} x^{2} + 6 \,{\left (c^{4} d^{7} e + 3 \, a c^{3} d^{5} e^{3} - 9 \, a^{2} c^{2} d^{3} e^{5} + 5 \, a^{3} c d e^{7}\right )} x}{3 \,{\left (c^{8} d^{8} x^{3} + 3 \, a c^{7} d^{7} e x^{2} + 3 \, a^{2} c^{6} d^{6} e^{2} x + a^{3} c^{5} d^{5} e^{3}\right )}} + \frac{e^{4} x}{c^{4} d^{4}} + \frac{4 \,{\left (c d^{2} e^{3} - a e^{5}\right )} \log \left (c d x + a e\right )}{c^{5} d^{5}} \end{align*}

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)^4,x, algorithm="maxima")

[Out]

-1/3*(c^4*d^8 + 2*a*c^3*d^6*e^2 + 6*a^2*c^2*d^4*e^4 - 22*a^3*c*d^2*e^6 + 13*a^4*e^8 + 18*(c^4*d^6*e^2 - 2*a*c^
3*d^4*e^4 + a^2*c^2*d^2*e^6)*x^2 + 6*(c^4*d^7*e + 3*a*c^3*d^5*e^3 - 9*a^2*c^2*d^3*e^5 + 5*a^3*c*d*e^7)*x)/(c^8
*d^8*x^3 + 3*a*c^7*d^7*e*x^2 + 3*a^2*c^6*d^6*e^2*x + a^3*c^5*d^5*e^3) + e^4*x/(c^4*d^4) + 4*(c*d^2*e^3 - a*e^5
)*log(c*d*x + a*e)/(c^5*d^5)

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Fricas [B]  time = 1.85276, size = 675, normalized size = 4.62 \begin{align*} \frac{3 \, c^{4} d^{4} e^{4} x^{4} + 9 \, a c^{3} d^{3} e^{5} x^{3} - c^{4} d^{8} - 2 \, a c^{3} d^{6} e^{2} - 6 \, a^{2} c^{2} d^{4} e^{4} + 22 \, a^{3} c d^{2} e^{6} - 13 \, a^{4} e^{8} - 9 \,{\left (2 \, c^{4} d^{6} e^{2} - 4 \, a c^{3} d^{4} e^{4} + a^{2} c^{2} d^{2} e^{6}\right )} x^{2} - 3 \,{\left (2 \, c^{4} d^{7} e + 6 \, a c^{3} d^{5} e^{3} - 18 \, a^{2} c^{2} d^{3} e^{5} + 9 \, a^{3} c d e^{7}\right )} x + 12 \,{\left (a^{3} c d^{2} e^{6} - a^{4} e^{8} +{\left (c^{4} d^{5} e^{3} - a c^{3} d^{3} e^{5}\right )} x^{3} + 3 \,{\left (a c^{3} d^{4} e^{4} - a^{2} c^{2} d^{2} e^{6}\right )} x^{2} + 3 \,{\left (a^{2} c^{2} d^{3} e^{5} - a^{3} c d e^{7}\right )} x\right )} \log \left (c d x + a e\right )}{3 \,{\left (c^{8} d^{8} x^{3} + 3 \, a c^{7} d^{7} e x^{2} + 3 \, a^{2} c^{6} d^{6} e^{2} x + a^{3} c^{5} d^{5} e^{3}\right )}} \end{align*}

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)^4,x, algorithm="fricas")

[Out]

1/3*(3*c^4*d^4*e^4*x^4 + 9*a*c^3*d^3*e^5*x^3 - c^4*d^8 - 2*a*c^3*d^6*e^2 - 6*a^2*c^2*d^4*e^4 + 22*a^3*c*d^2*e^
6 - 13*a^4*e^8 - 9*(2*c^4*d^6*e^2 - 4*a*c^3*d^4*e^4 + a^2*c^2*d^2*e^6)*x^2 - 3*(2*c^4*d^7*e + 6*a*c^3*d^5*e^3
- 18*a^2*c^2*d^3*e^5 + 9*a^3*c*d*e^7)*x + 12*(a^3*c*d^2*e^6 - a^4*e^8 + (c^4*d^5*e^3 - a*c^3*d^3*e^5)*x^3 + 3*
(a*c^3*d^4*e^4 - a^2*c^2*d^2*e^6)*x^2 + 3*(a^2*c^2*d^3*e^5 - a^3*c*d*e^7)*x)*log(c*d*x + a*e))/(c^8*d^8*x^3 +
3*a*c^7*d^7*e*x^2 + 3*a^2*c^6*d^6*e^2*x + a^3*c^5*d^5*e^3)

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Sympy [A]  time = 11.7701, size = 257, normalized size = 1.76 \begin{align*} - \frac{13 a^{4} e^{8} - 22 a^{3} c d^{2} e^{6} + 6 a^{2} c^{2} d^{4} e^{4} + 2 a c^{3} d^{6} e^{2} + c^{4} d^{8} + x^{2} \left (18 a^{2} c^{2} d^{2} e^{6} - 36 a c^{3} d^{4} e^{4} + 18 c^{4} d^{6} e^{2}\right ) + x \left (30 a^{3} c d e^{7} - 54 a^{2} c^{2} d^{3} e^{5} + 18 a c^{3} d^{5} e^{3} + 6 c^{4} d^{7} e\right )}{3 a^{3} c^{5} d^{5} e^{3} + 9 a^{2} c^{6} d^{6} e^{2} x + 9 a c^{7} d^{7} e x^{2} + 3 c^{8} d^{8} x^{3}} + \frac{e^{4} x}{c^{4} d^{4}} - \frac{4 e^{3} \left (a e^{2} - c d^{2}\right ) \log{\left (a e + c d x \right )}}{c^{5} d^{5}} \end{align*}

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)**4,x)

[Out]

-(13*a**4*e**8 - 22*a**3*c*d**2*e**6 + 6*a**2*c**2*d**4*e**4 + 2*a*c**3*d**6*e**2 + c**4*d**8 + x**2*(18*a**2*
c**2*d**2*e**6 - 36*a*c**3*d**4*e**4 + 18*c**4*d**6*e**2) + x*(30*a**3*c*d*e**7 - 54*a**2*c**2*d**3*e**5 + 18*
a*c**3*d**5*e**3 + 6*c**4*d**7*e))/(3*a**3*c**5*d**5*e**3 + 9*a**2*c**6*d**6*e**2*x + 9*a*c**7*d**7*e*x**2 + 3
*c**8*d**8*x**3) + e**4*x/(c**4*d**4) - 4*e**3*(a*e**2 - c*d**2)*log(a*e + c*d*x)/(c**5*d**5)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{sage}_{0} x \end{align*}

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)^4,x, algorithm="giac")

[Out]

sage0*x